LMFDB - Embedded newform 6040.2.a.n.1.8

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 $ x^13 - 4x^12 - 14x^11 + 70x^10 + 41x^9 - 403x^8 + 109x^7 + 870x^6 - 444x^5 - 708x^4 + 373x^3 + 108x^2 - 82x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.278392$ 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)

$f(q)$	$=$	$q-0.278392 q^{3} -1.00000 q^{5} -2.48850 q^{7} -2.92250 q^{9} +O(q^{10})$	$q-0.278392 q^{3} -1.00000 q^{5} -2.48850 q^{7} -2.92250 q^{9} -4.96782 q^{11} +3.11362 q^{13} +0.278392 q^{15} +5.66870 q^{17} +6.08593 q^{19} +0.692779 q^{21} +1.52724 q^{23} +1.00000 q^{25} +1.64878 q^{27} -3.59331 q^{29} +2.97603 q^{31} +1.38300 q^{33} +2.48850 q^{35} -2.11268 q^{37} -0.866808 q^{39} -7.11601 q^{41} +5.15280 q^{43} +2.92250 q^{45} +11.0453 q^{47} -0.807370 q^{49} -1.57812 q^{51} -4.27272 q^{53} +4.96782 q^{55} -1.69428 q^{57} +3.02727 q^{59} -2.41978 q^{61} +7.27263 q^{63} -3.11362 q^{65} +5.86623 q^{67} -0.425170 q^{69} +5.61335 q^{71} -8.02047 q^{73} -0.278392 q^{75} +12.3624 q^{77} -9.74831 q^{79} +8.30849 q^{81} +8.75861 q^{83} -5.66870 q^{85} +1.00035 q^{87} -2.72185 q^{89} -7.74824 q^{91} -0.828505 q^{93} -6.08593 q^{95} -4.79633 q^{97} +14.5184 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * 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
$3$	−0.278392	−0.160730	−0.0803649	−	0.996766i	$-0.525609\pi$
$3$	−0.278392	−0.160730	−0.0803649	+	0.996766i	$0.525609\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.48850	−0.940564	−0.470282	−	0.882516i	$-0.655848\pi$
$7$	−2.48850	−0.940564	−0.470282	+	0.882516i	$0.655848\pi$
$8$	0	0
$9$	−2.92250	−0.974166
$10$	0	0
$11$	−4.96782	−1.49785	−0.748927	−	0.662653i	$-0.769431\pi$
$11$	−4.96782	−1.49785	−0.748927	+	0.662653i	$0.769431\pi$
$12$	0	0
$13$	3.11362	0.863563	0.431782	−	0.901978i	$-0.357885\pi$
$13$	3.11362	0.863563	0.431782	+	0.901978i	$0.357885\pi$
$14$	0	0
$15$	0.278392	0.0718805
$16$	0	0
$17$	5.66870	1.37486	0.687430	−	0.726250i	$-0.258739\pi$
$17$	5.66870	1.37486	0.687430	+	0.726250i	$0.258739\pi$
$18$	0	0
$19$	6.08593	1.39621	0.698104	−	0.715996i	$-0.254027\pi$
$19$	6.08593	1.39621	0.698104	+	0.715996i	$0.254027\pi$
$20$	0	0
$21$	0.692779	0.151177
$22$	0	0
$23$	1.52724	0.318451	0.159225	−	0.987242i	$-0.449100\pi$
$23$	1.52724	0.318451	0.159225	+	0.987242i	$0.449100\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.64878	0.317307
$28$	0	0
$29$	−3.59331	−0.667260	−0.333630	−	0.942704i	$-0.608274\pi$
$29$	−3.59331	−0.667260	−0.333630	+	0.942704i	$0.608274\pi$
$30$	0	0
$31$	2.97603	0.534512	0.267256	−	0.963626i	$-0.413883\pi$
$31$	2.97603	0.534512	0.267256	+	0.963626i	$0.413883\pi$
$32$	0	0
$33$	1.38300	0.240750
$34$	0	0
$35$	2.48850	0.420633
$36$	0	0
$37$	−2.11268	−0.347322	−0.173661	−	0.984805i	$-0.555560\pi$
$37$	−2.11268	−0.347322	−0.173661	+	0.984805i	$0.555560\pi$
$38$	0	0
$39$	−0.866808	−0.138800
$40$	0	0
$41$	−7.11601	−1.11133	−0.555667	−	0.831405i	$-0.687537\pi$
$41$	−7.11601	−1.11133	−0.555667	+	0.831405i	$0.687537\pi$
$42$	0	0
$43$	5.15280	0.785794	0.392897	−	0.919582i	$-0.371473\pi$
$43$	5.15280	0.785794	0.392897	+	0.919582i	$0.371473\pi$
$44$	0	0
$45$	2.92250	0.435660
$46$	0	0
$47$	11.0453	1.61113	0.805563	−	0.592510i	$-0.201863\pi$
$47$	11.0453	1.61113	0.805563	+	0.592510i	$0.201863\pi$
$48$	0	0
$49$	−0.807370	−0.115339
$50$	0	0
$51$	−1.57812	−0.220981
$52$	0	0
$53$	−4.27272	−0.586904	−0.293452	−	0.955974i	$-0.594804\pi$
$53$	−4.27272	−0.586904	−0.293452	+	0.955974i	$0.594804\pi$
$54$	0	0
$55$	4.96782	0.669860
$56$	0	0
$57$	−1.69428	−0.224412
$58$	0	0
$59$	3.02727	0.394117	0.197058	−	0.980392i	$-0.436861\pi$
$59$	3.02727	0.394117	0.197058	+	0.980392i	$0.436861\pi$
$60$	0	0
$61$	−2.41978	−0.309821	−0.154911	−	0.987928i	$-0.549509\pi$
$61$	−2.41978	−0.309821	−0.154911	+	0.987928i	$0.549509\pi$
$62$	0	0
$63$	7.27263	0.916266
$64$	0	0
$65$	−3.11362	−0.386197
$66$	0	0
$67$	5.86623	0.716674	0.358337	−	0.933592i	$-0.383344\pi$
$67$	5.86623	0.716674	0.358337	+	0.933592i	$0.383344\pi$
$68$	0	0
$69$	−0.425170	−0.0511845
$70$	0	0
$71$	5.61335	0.666182	0.333091	−	0.942895i	$-0.391908\pi$
$71$	5.61335	0.666182	0.333091	+	0.942895i	$0.391908\pi$
$72$	0	0
$73$	−8.02047	−0.938725	−0.469362	−	0.883006i	$-0.655516\pi$
$73$	−8.02047	−0.938725	−0.469362	+	0.883006i	$0.655516\pi$
$74$	0	0
$75$	−0.278392	−0.0321460
$76$	0	0
$77$	12.3624	1.40883
$78$	0	0
$79$	−9.74831	−1.09677	−0.548385	−	0.836226i	$-0.684757\pi$
$79$	−9.74831	−1.09677	−0.548385	+	0.836226i	$0.684757\pi$
$80$	0	0
$81$	8.30849	0.923165
$82$	0	0
$83$	8.75861	0.961382	0.480691	−	0.876890i	$-0.340386\pi$
$83$	8.75861	0.961382	0.480691	+	0.876890i	$0.340386\pi$
$84$	0	0
$85$	−5.66870	−0.614856
$86$	0	0
$87$	1.00035	0.107249
$88$	0	0
$89$	−2.72185	−0.288515	−0.144258	−	0.989540i	$-0.546079\pi$
$89$	−2.72185	−0.288515	−0.144258	+	0.989540i	$0.546079\pi$
$90$	0	0
$91$	−7.74824	−0.812237
$92$	0	0
$93$	−0.828505	−0.0859119
$94$	0	0
$95$	−6.08593	−0.624404
$96$	0	0
$97$	−4.79633	−0.486993	−0.243497	−	0.969902i	$-0.578294\pi$
$97$	−4.79633	−0.486993	−0.243497	+	0.969902i	$0.578294\pi$
$98$	0	0
$99$	14.5184	1.45916
$100$	0	0
$101$	−4.40128	−0.437944	−0.218972	−	0.975731i	$-0.570270\pi$
$101$	−4.40128	−0.437944	−0.218972	+	0.975731i	$0.570270\pi$
$102$	0	0
$103$	−6.87919	−0.677826	−0.338913	−	0.940818i	$-0.610059\pi$
$103$	−6.87919	−0.677826	−0.338913	+	0.940818i	$0.610059\pi$
$104$	0	0
$105$	−0.692779	−0.0676083
$106$	0	0
$107$	−16.3773	−1.58325	−0.791627	−	0.611004i	$-0.790766\pi$
$107$	−16.3773	−1.58325	−0.791627	+	0.611004i	$0.790766\pi$
$108$	0	0
$109$	9.80073	0.938740	0.469370	−	0.883002i	$-0.344481\pi$
$109$	9.80073	0.938740	0.469370	+	0.883002i	$0.344481\pi$
$110$	0	0
$111$	0.588153	0.0558250
$112$	0	0
$113$	−9.92602	−0.933762	−0.466881	−	0.884320i	$-0.654622\pi$
$113$	−9.92602	−0.933762	−0.466881	+	0.884320i	$0.654622\pi$
$114$	0	0
$115$	−1.52724	−0.142415
$116$	0	0
$117$	−9.09955	−0.841254
$118$	0	0
$119$	−14.1065	−1.29315
$120$	0	0
$121$	13.6792	1.24357
$122$	0	0
$123$	1.98104	0.178625
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.26555	0.733449	0.366725	−	0.930330i	$-0.380479\pi$
$127$	8.26555	0.733449	0.366725	+	0.930330i	$0.380479\pi$
$128$	0	0
$129$	−1.43450	−0.126301
$130$	0	0
$131$	−15.9121	−1.39025	−0.695125	−	0.718889i	$-0.744651\pi$
$131$	−15.9121	−1.39025	−0.695125	+	0.718889i	$0.744651\pi$
$132$	0	0
$133$	−15.1448	−1.31322
$134$	0	0
$135$	−1.64878	−0.141904
$136$	0	0
$137$	−4.56095	−0.389668	−0.194834	−	0.980836i	$-0.562417\pi$
$137$	−4.56095	−0.389668	−0.194834	+	0.980836i	$0.562417\pi$
$138$	0	0
$139$	−13.5077	−1.14571	−0.572855	−	0.819656i	$-0.694164\pi$
$139$	−13.5077	−1.14571	−0.572855	+	0.819656i	$0.694164\pi$
$140$	0	0
$141$	−3.07493	−0.258956
$142$	0	0
$143$	−15.4679	−1.29349
$144$	0	0
$145$	3.59331	0.298408
$146$	0	0
$147$	0.224765	0.0185383
$148$	0	0
$149$	−16.0360	−1.31372	−0.656862	−	0.754011i	$-0.728116\pi$
$149$	−16.0360	−1.31372	−0.656862	+	0.754011i	$0.728116\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−16.5668	−1.33934
$154$	0	0
$155$	−2.97603	−0.239041
$156$	0	0
$157$	4.55909	0.363855	0.181928	−	0.983312i	$-0.441766\pi$
$157$	4.55909	0.363855	0.181928	+	0.983312i	$0.441766\pi$
$158$	0	0
$159$	1.18949	0.0943329
$160$	0	0
$161$	−3.80052	−0.299523
$162$	0	0
$163$	15.3540	1.20262	0.601309	−	0.799017i	$-0.294646\pi$
$163$	15.3540	1.20262	0.601309	+	0.799017i	$0.294646\pi$
$164$	0	0
$165$	−1.38300	−0.107667
$166$	0	0
$167$	−11.4726	−0.887780	−0.443890	−	0.896081i	$-0.646402\pi$
$167$	−11.4726	−0.887780	−0.443890	+	0.896081i	$0.646402\pi$
$168$	0	0
$169$	−3.30537	−0.254259
$170$	0	0
$171$	−17.7861	−1.36014
$172$	0	0
$173$	−7.23849	−0.550332	−0.275166	−	0.961397i	$-0.588733\pi$
$173$	−7.23849	−0.550332	−0.275166	+	0.961397i	$0.588733\pi$
$174$	0	0
$175$	−2.48850	−0.188113
$176$	0	0
$177$	−0.842767	−0.0633463
$178$	0	0
$179$	−8.48771	−0.634401	−0.317201	−	0.948358i	$-0.602743\pi$
$179$	−8.48771	−0.634401	−0.317201	+	0.948358i	$0.602743\pi$
$180$	0	0
$181$	8.68167	0.645303	0.322652	−	0.946518i	$-0.395426\pi$
$181$	8.68167	0.645303	0.322652	+	0.946518i	$0.395426\pi$
$182$	0	0
$183$	0.673648	0.0497975
$184$	0	0
$185$	2.11268	0.155327
$186$	0	0
$187$	−28.1611	−2.05934
$188$	0	0
$189$	−4.10298	−0.298448
$190$	0	0
$191$	−13.8033	−0.998774	−0.499387	−	0.866379i	$-0.666441\pi$
$191$	−13.8033	−0.998774	−0.499387	+	0.866379i	$0.666441\pi$
$192$	0	0
$193$	−14.0034	−1.00799	−0.503995	−	0.863707i	$-0.668137\pi$
$193$	−14.0034	−1.00799	−0.503995	+	0.863707i	$0.668137\pi$
$194$	0	0
$195$	0.866808	0.0620734
$196$	0	0
$197$	−6.80323	−0.484710	−0.242355	−	0.970188i	$-0.577920\pi$
$197$	−6.80323	−0.484710	−0.242355	+	0.970188i	$0.577920\pi$
$198$	0	0
$199$	19.3521	1.37183	0.685916	−	0.727681i	$-0.259402\pi$
$199$	19.3521	1.37183	0.685916	+	0.727681i	$0.259402\pi$
$200$	0	0
$201$	−1.63311	−0.115191
$202$	0	0
$203$	8.94194	0.627601
$204$	0	0
$205$	7.11601	0.497004
$206$	0	0
$207$	−4.46334	−0.310224
$208$	0	0
$209$	−30.2338	−2.09132
$210$	0	0
$211$	12.5105	0.861261	0.430630	−	0.902528i	$-0.358291\pi$
$211$	12.5105	0.861261	0.430630	+	0.902528i	$0.358291\pi$
$212$	0	0
$213$	−1.56271	−0.107075
$214$	0	0
$215$	−5.15280	−0.351418
$216$	0	0
$217$	−7.40586	−0.502743
$218$	0	0
$219$	2.23284	0.150881
$220$	0	0
$221$	17.6502	1.18728
$222$	0	0
$223$	0.941235	0.0630298	0.0315149	−	0.999503i	$-0.489967\pi$
$223$	0.941235	0.0630298	0.0315149	+	0.999503i	$0.489967\pi$
$224$	0	0
$225$	−2.92250	−0.194833
$226$	0	0
$227$	−1.56790	−0.104065	−0.0520327	−	0.998645i	$-0.516570\pi$
$227$	−1.56790	−0.104065	−0.0520327	+	0.998645i	$0.516570\pi$
$228$	0	0
$229$	9.12747	0.603160	0.301580	−	0.953441i	$-0.402486\pi$
$229$	9.12747	0.603160	0.301580	+	0.953441i	$0.402486\pi$
$230$	0	0
$231$	−3.44160	−0.226441
$232$	0	0
$233$	−9.22291	−0.604213	−0.302107	−	0.953274i	$-0.597690\pi$
$233$	−9.22291	−0.604213	−0.302107	+	0.953274i	$0.597690\pi$
$234$	0	0
$235$	−11.0453	−0.720517
$236$	0	0
$237$	2.71385	0.176284
$238$	0	0
$239$	−17.4375	−1.12794	−0.563968	−	0.825797i	$-0.690726\pi$
$239$	−17.4375	−1.12794	−0.563968	+	0.825797i	$0.690726\pi$
$240$	0	0
$241$	−18.1637	−1.17002	−0.585012	−	0.811025i	$-0.698910\pi$
$241$	−18.1637	−1.17002	−0.585012	+	0.811025i	$0.698910\pi$
$242$	0	0
$243$	−7.25935	−0.465687
$244$	0	0
$245$	0.807370	0.0515809
$246$	0	0
$247$	18.9493	1.20571
$248$	0	0
$249$	−2.43833	−0.154523
$250$	0	0
$251$	6.26556	0.395479	0.197739	−	0.980255i	$-0.436640\pi$
$251$	6.26556	0.395479	0.197739	+	0.980255i	$0.436640\pi$
$252$	0	0
$253$	−7.58703	−0.476992
$254$	0	0
$255$	1.57812	0.0988258
$256$	0	0
$257$	19.0525	1.18846	0.594231	−	0.804294i	$-0.297456\pi$
$257$	19.0525	1.18846	0.594231	+	0.804294i	$0.297456\pi$
$258$	0	0
$259$	5.25740	0.326679
$260$	0	0
$261$	10.5014	0.650022
$262$	0	0
$263$	6.20948	0.382893	0.191446	−	0.981503i	$-0.438682\pi$
$263$	6.20948	0.382893	0.191446	+	0.981503i	$0.438682\pi$
$264$	0	0
$265$	4.27272	0.262471
$266$	0	0
$267$	0.757741	0.0463730
$268$	0	0
$269$	−14.7631	−0.900123	−0.450061	−	0.892998i	$-0.648598\pi$
$269$	−14.7631	−0.900123	−0.450061	+	0.892998i	$0.648598\pi$
$270$	0	0
$271$	−16.1014	−0.978090	−0.489045	−	0.872258i	$-0.662655\pi$
$271$	−16.1014	−0.978090	−0.489045	+	0.872258i	$0.662655\pi$
$272$	0	0
$273$	2.15705	0.130551
$274$	0	0
$275$	−4.96782	−0.299571
$276$	0	0
$277$	0.832720	0.0500333	0.0250167	−	0.999687i	$-0.492036\pi$
$277$	0.832720	0.0500333	0.0250167	+	0.999687i	$0.492036\pi$
$278$	0	0
$279$	−8.69745	−0.520703
$280$	0	0
$281$	29.9274	1.78532	0.892659	−	0.450733i	$-0.148837\pi$
$281$	29.9274	1.78532	0.892659	+	0.450733i	$0.148837\pi$
$282$	0	0
$283$	7.32358	0.435341	0.217671	−	0.976022i	$-0.430154\pi$
$283$	7.32358	0.435341	0.217671	+	0.976022i	$0.430154\pi$
$284$	0	0
$285$	1.69428	0.100360
$286$	0	0
$287$	17.7082	1.04528
$288$	0	0
$289$	15.1341	0.890242
$290$	0	0
$291$	1.33526	0.0782743
$292$	0	0
$293$	−13.7169	−0.801352	−0.400676	−	0.916220i	$-0.631225\pi$
$293$	−13.7169	−0.801352	−0.400676	+	0.916220i	$0.631225\pi$
$294$	0	0
$295$	−3.02727	−0.176254
$296$	0	0
$297$	−8.19082	−0.475280
$298$	0	0
$299$	4.75523	0.275002
$300$	0	0
$301$	−12.8227	−0.739090
$302$	0	0
$303$	1.22528	0.0703906
$304$	0	0
$305$	2.41978	0.138556
$306$	0	0
$307$	−26.2158	−1.49621	−0.748107	−	0.663578i	$-0.769037\pi$
$307$	−26.2158	−1.49621	−0.748107	+	0.663578i	$0.769037\pi$
$308$	0	0
$309$	1.91511	0.108947
$310$	0	0
$311$	31.8800	1.80775	0.903873	−	0.427800i	$-0.140711\pi$
$311$	31.8800	1.80775	0.903873	+	0.427800i	$0.140711\pi$
$312$	0	0
$313$	25.0436	1.41555	0.707774	−	0.706439i	$-0.249700\pi$
$313$	25.0436	1.41555	0.707774	+	0.706439i	$0.249700\pi$
$314$	0	0
$315$	−7.27263	−0.409767
$316$	0	0
$317$	−10.5657	−0.593431	−0.296715	−	0.954966i	$-0.595891\pi$
$317$	−10.5657	−0.593431	−0.296715	+	0.954966i	$0.595891\pi$
$318$	0	0
$319$	17.8509	0.999458
$320$	0	0
$321$	4.55932	0.254476
$322$	0	0
$323$	34.4993	1.91959
$324$	0	0
$325$	3.11362	0.172713
$326$	0	0
$327$	−2.72845	−0.150883
$328$	0	0
$329$	−27.4863	−1.51537
$330$	0	0
$331$	−2.75092	−0.151204	−0.0756020	−	0.997138i	$-0.524088\pi$
$331$	−2.75092	−0.151204	−0.0756020	+	0.997138i	$0.524088\pi$
$332$	0	0
$333$	6.17429	0.338349
$334$	0	0
$335$	−5.86623	−0.320506
$336$	0	0
$337$	2.62484	0.142984	0.0714920	−	0.997441i	$-0.477224\pi$
$337$	2.62484	0.142984	0.0714920	+	0.997441i	$0.477224\pi$
$338$	0	0
$339$	2.76333	0.150083
$340$	0	0
$341$	−14.7844	−0.800620
$342$	0	0
$343$	19.4286	1.04905
$344$	0	0
$345$	0.425170	0.0228904
$346$	0	0
$347$	16.4464	0.882891	0.441445	−	0.897288i	$-0.354466\pi$
$347$	16.4464	0.882891	0.441445	+	0.897288i	$0.354466\pi$
$348$	0	0
$349$	−7.41807	−0.397080	−0.198540	−	0.980093i	$-0.563620\pi$
$349$	−7.41807	−0.397080	−0.198540	+	0.980093i	$0.563620\pi$
$350$	0	0
$351$	5.13367	0.274015
$352$	0	0
$353$	2.28446	0.121590	0.0607948	−	0.998150i	$-0.480636\pi$
$353$	2.28446	0.121590	0.0607948	+	0.998150i	$0.480636\pi$
$354$	0	0
$355$	−5.61335	−0.297926
$356$	0	0
$357$	3.92715	0.207847
$358$	0	0
$359$	16.6318	0.877792	0.438896	−	0.898538i	$-0.355370\pi$
$359$	16.6318	0.877792	0.438896	+	0.898538i	$0.355370\pi$
$360$	0	0
$361$	18.0386	0.949399
$362$	0	0
$363$	−3.80819	−0.199878
$364$	0	0
$365$	8.02047	0.419810
$366$	0	0
$367$	−28.0993	−1.46677	−0.733385	−	0.679813i	$-0.762061\pi$
$367$	−28.0993	−1.46677	−0.733385	+	0.679813i	$0.762061\pi$
$368$	0	0
$369$	20.7965	1.08262
$370$	0	0
$371$	10.6327	0.552021
$372$	0	0
$373$	−32.4023	−1.67773	−0.838864	−	0.544340i	$-0.816780\pi$
$373$	−32.4023	−1.67773	−0.838864	+	0.544340i	$0.816780\pi$
$374$	0	0
$375$	0.278392	0.0143761
$376$	0	0
$377$	−11.1882	−0.576221
$378$	0	0
$379$	−12.3367	−0.633694	−0.316847	−	0.948477i	$-0.602624\pi$
$379$	−12.3367	−0.633694	−0.316847	+	0.948477i	$0.602624\pi$
$380$	0	0
$381$	−2.30106	−0.117887
$382$	0	0
$383$	−26.5979	−1.35909	−0.679545	−	0.733634i	$-0.737823\pi$
$383$	−26.5979	−1.35909	−0.679545	+	0.733634i	$0.737823\pi$
$384$	0	0
$385$	−12.3624	−0.630047
$386$	0	0
$387$	−15.0590	−0.765494
$388$	0	0
$389$	−3.80265	−0.192802	−0.0964010	−	0.995343i	$-0.530733\pi$
$389$	−3.80265	−0.192802	−0.0964010	+	0.995343i	$0.530733\pi$
$390$	0	0
$391$	8.65743	0.437825
$392$	0	0
$393$	4.42982	0.223455
$394$	0	0
$395$	9.74831	0.490491
$396$	0	0
$397$	−18.3199	−0.919449	−0.459725	−	0.888062i	$-0.652052\pi$
$397$	−18.3199	−0.919449	−0.459725	+	0.888062i	$0.652052\pi$
$398$	0	0
$399$	4.21621	0.211074
$400$	0	0
$401$	15.1945	0.758777	0.379389	−	0.925237i	$-0.376134\pi$
$401$	15.1945	0.758777	0.379389	+	0.925237i	$0.376134\pi$
$402$	0	0
$403$	9.26624	0.461584
$404$	0	0
$405$	−8.30849	−0.412852
$406$	0	0
$407$	10.4954	0.520237
$408$	0	0
$409$	−23.2852	−1.15138	−0.575691	−	0.817668i	$-0.695267\pi$
$409$	−23.2852	−1.15138	−0.575691	+	0.817668i	$0.695267\pi$
$410$	0	0
$411$	1.26973	0.0626312
$412$	0	0
$413$	−7.53335	−0.370692
$414$	0	0
$415$	−8.75861	−0.429943
$416$	0	0
$417$	3.76045	0.184150
$418$	0	0
$419$	28.3123	1.38315	0.691574	−	0.722306i	$-0.256918\pi$
$419$	28.3123	1.38315	0.691574	+	0.722306i	$0.256918\pi$
$420$	0	0
$421$	25.8489	1.25980	0.629900	−	0.776677i	$-0.283096\pi$
$421$	25.8489	1.25980	0.629900	+	0.776677i	$0.283096\pi$
$422$	0	0
$423$	−32.2799	−1.56950
$424$	0	0
$425$	5.66870	0.274972
$426$	0	0
$427$	6.02162	0.291407
$428$	0	0
$429$	4.30614	0.207903
$430$	0	0
$431$	−13.8729	−0.668232	−0.334116	−	0.942532i	$-0.608438\pi$
$431$	−13.8729	−0.668232	−0.334116	+	0.942532i	$0.608438\pi$
$432$	0	0
$433$	−38.9078	−1.86979	−0.934896	−	0.354923i	$-0.884507\pi$
$433$	−38.9078	−1.86979	−0.934896	+	0.354923i	$0.884507\pi$
$434$	0	0
$435$	−1.00035	−0.0479630
$436$	0	0
$437$	9.29465	0.444623
$438$	0	0
$439$	21.1680	1.01029	0.505147	−	0.863033i	$-0.331438\pi$
$439$	21.1680	1.01029	0.505147	+	0.863033i	$0.331438\pi$
$440$	0	0
$441$	2.35954	0.112359
$442$	0	0
$443$	1.54163	0.0732452	0.0366226	−	0.999329i	$-0.488340\pi$
$443$	1.54163	0.0732452	0.0366226	+	0.999329i	$0.488340\pi$
$444$	0	0
$445$	2.72185	0.129028
$446$	0	0
$447$	4.46431	0.211154
$448$	0	0
$449$	−32.2609	−1.52248	−0.761242	−	0.648468i	$-0.775410\pi$
$449$	−32.2609	−1.52248	−0.761242	+	0.648468i	$0.775410\pi$
$450$	0	0
$451$	35.3511	1.66462
$452$	0	0
$453$	−0.278392	−0.0130800
$454$	0	0
$455$	7.74824	0.363243
$456$	0	0
$457$	−15.4099	−0.720845	−0.360423	−	0.932789i	$-0.617368\pi$
$457$	−15.4099	−0.720845	−0.360423	+	0.932789i	$0.617368\pi$
$458$	0	0
$459$	9.34642	0.436253
$460$	0	0
$461$	16.5534	0.770966	0.385483	−	0.922715i	$-0.374035\pi$
$461$	16.5534	0.770966	0.385483	+	0.922715i	$0.374035\pi$
$462$	0	0
$463$	28.0601	1.30406	0.652032	−	0.758191i	$-0.273917\pi$
$463$	28.0601	1.30406	0.652032	+	0.758191i	$0.273917\pi$
$464$	0	0
$465$	0.828505	0.0384210
$466$	0	0
$467$	4.32053	0.199931	0.0999653	−	0.994991i	$-0.468127\pi$
$467$	4.32053	0.199931	0.0999653	+	0.994991i	$0.468127\pi$
$468$	0	0
$469$	−14.5981	−0.674078
$470$	0	0
$471$	−1.26922	−0.0584824
$472$	0	0
$473$	−25.5982	−1.17700
$474$	0	0
$475$	6.08593	0.279242
$476$	0	0
$477$	12.4870	0.571742
$478$	0	0
$479$	−25.5107	−1.16561	−0.582807	−	0.812610i	$-0.698046\pi$
$479$	−25.5107	−1.16561	−0.582807	+	0.812610i	$0.698046\pi$
$480$	0	0
$481$	−6.57808	−0.299934
$482$	0	0
$483$	1.05804	0.0481423
$484$	0	0
$485$	4.79633	0.217790
$486$	0	0
$487$	−7.90383	−0.358157	−0.179078	−	0.983835i	$-0.557312\pi$
$487$	−7.90383	−0.358157	−0.179078	+	0.983835i	$0.557312\pi$
$488$	0	0
$489$	−4.27443	−0.193296
$490$	0	0
$491$	14.7640	0.666292	0.333146	−	0.942875i	$-0.391890\pi$
$491$	14.7640	0.666292	0.333146	+	0.942875i	$0.391890\pi$
$492$	0	0
$493$	−20.3694	−0.917390
$494$	0	0
$495$	−14.5184	−0.652555
$496$	0	0
$497$	−13.9688	−0.626587
$498$	0	0
$499$	−13.1287	−0.587720	−0.293860	−	0.955849i	$-0.594940\pi$
$499$	−13.1287	−0.587720	−0.293860	+	0.955849i	$0.594940\pi$
$500$	0	0
$501$	3.19389	0.142693
$502$	0	0
$503$	−25.4685	−1.13559	−0.567793	−	0.823172i	$-0.692202\pi$
$503$	−25.4685	−1.13559	−0.567793	+	0.823172i	$0.692202\pi$
$504$	0	0
$505$	4.40128	0.195854
$506$	0	0
$507$	0.920188	0.0408670
$508$	0	0
$509$	2.14902	0.0952537	0.0476269	−	0.998865i	$-0.484834\pi$
$509$	2.14902	0.0952537	0.0476269	+	0.998865i	$0.484834\pi$
$510$	0	0
$511$	19.9589	0.882931
$512$	0	0
$513$	10.0343	0.443027
$514$	0	0
$515$	6.87919	0.303133
$516$	0	0
$517$	−54.8712	−2.41323
$518$	0	0
$519$	2.01514	0.0884548
$520$	0	0
$521$	−25.0848	−1.09899	−0.549493	−	0.835498i	$-0.685179\pi$
$521$	−25.0848	−1.09899	−0.549493	+	0.835498i	$0.685179\pi$
$522$	0	0
$523$	2.02794	0.0886757	0.0443378	−	0.999017i	$-0.485882\pi$
$523$	2.02794	0.0886757	0.0443378	+	0.999017i	$0.485882\pi$
$524$	0	0
$525$	0.692779	0.0302353
$526$	0	0
$527$	16.8702	0.734879
$528$	0	0
$529$	−20.6676	−0.898589
$530$	0	0
$531$	−8.84718	−0.383935
$532$	0	0
$533$	−22.1566	−0.959707
$534$	0	0
$535$	16.3773	0.708053
$536$	0	0
$537$	2.36291	0.101967
$538$	0	0
$539$	4.01087	0.172760
$540$	0	0
$541$	24.7123	1.06247	0.531233	−	0.847226i	$-0.321729\pi$
$541$	24.7123	1.06247	0.531233	+	0.847226i	$0.321729\pi$
$542$	0	0
$543$	−2.41691	−0.103719
$544$	0	0
$545$	−9.80073	−0.419817
$546$	0	0
$547$	−27.2769	−1.16628	−0.583138	−	0.812373i	$-0.698175\pi$
$547$	−27.2769	−1.16628	−0.583138	+	0.812373i	$0.698175\pi$
$548$	0	0
$549$	7.07180	0.301817
$550$	0	0
$551$	−21.8686	−0.931635
$552$	0	0
$553$	24.2587	1.03158
$554$	0	0
$555$	−0.588153	−0.0249657
$556$	0	0
$557$	4.56129	0.193268	0.0966339	−	0.995320i	$-0.469192\pi$
$557$	4.56129	0.193268	0.0966339	+	0.995320i	$0.469192\pi$
$558$	0	0
$559$	16.0439	0.678583
$560$	0	0
$561$	7.83982	0.330997
$562$	0	0
$563$	−4.31152	−0.181709	−0.0908545	−	0.995864i	$-0.528960\pi$
$563$	−4.31152	−0.181709	−0.0908545	+	0.995864i	$0.528960\pi$
$564$	0	0
$565$	9.92602	0.417591
$566$	0	0
$567$	−20.6757	−0.868296
$568$	0	0
$569$	−30.6598	−1.28532	−0.642662	−	0.766150i	$-0.722170\pi$
$569$	−30.6598	−1.28532	−0.642662	+	0.766150i	$0.722170\pi$
$570$	0	0
$571$	17.0766	0.714635	0.357318	−	0.933983i	$-0.383691\pi$
$571$	17.0766	0.714635	0.357318	+	0.933983i	$0.383691\pi$
$572$	0	0
$573$	3.84274	0.160533
$574$	0	0
$575$	1.52724	0.0636901
$576$	0	0
$577$	−29.6945	−1.23620	−0.618100	−	0.786100i	$-0.712097\pi$
$577$	−29.6945	−1.23620	−0.618100	+	0.786100i	$0.712097\pi$
$578$	0	0
$579$	3.89845	0.162014
$580$	0	0
$581$	−21.7958	−0.904242
$582$	0	0
$583$	21.2261	0.879096
$584$	0	0
$585$	9.09955	0.376220
$586$	0	0
$587$	−31.4428	−1.29778	−0.648892	−	0.760881i	$-0.724767\pi$
$587$	−31.4428	−1.29778	−0.648892	+	0.760881i	$0.724767\pi$
$588$	0	0
$589$	18.1119	0.746290
$590$	0	0
$591$	1.89397	0.0779074
$592$	0	0
$593$	44.4386	1.82488	0.912438	−	0.409216i	$-0.134198\pi$
$593$	44.4386	1.82488	0.912438	+	0.409216i	$0.134198\pi$
$594$	0	0
$595$	14.1065	0.578312
$596$	0	0
$597$	−5.38747	−0.220494
$598$	0	0
$599$	14.4620	0.590902	0.295451	−	0.955358i	$-0.404530\pi$
$599$	14.4620	0.590902	0.295451	+	0.955358i	$0.404530\pi$
$600$	0	0
$601$	−12.0583	−0.491868	−0.245934	−	0.969287i	$-0.579095\pi$
$601$	−12.0583	−0.491868	−0.245934	+	0.969287i	$0.579095\pi$
$602$	0	0
$603$	−17.1440	−0.698159
$604$	0	0
$605$	−13.6792	−0.556139
$606$	0	0
$607$	−37.8412	−1.53593	−0.767963	−	0.640495i	$-0.778729\pi$
$607$	−37.8412	−1.53593	−0.767963	+	0.640495i	$0.778729\pi$
$608$	0	0
$609$	−2.48937	−0.100874
$610$	0	0
$611$	34.3909	1.39131
$612$	0	0
$613$	28.8870	1.16673	0.583366	−	0.812209i	$-0.301735\pi$
$613$	28.8870	1.16673	0.583366	+	0.812209i	$0.301735\pi$
$614$	0	0
$615$	−1.98104	−0.0798833
$616$	0	0
$617$	−13.4146	−0.540053	−0.270026	−	0.962853i	$-0.587032\pi$
$617$	−13.4146	−0.540053	−0.270026	+	0.962853i	$0.587032\pi$
$618$	0	0
$619$	−24.0547	−0.966842	−0.483421	−	0.875388i	$-0.660606\pi$
$619$	−24.0547	−0.966842	−0.483421	+	0.875388i	$0.660606\pi$
$620$	0	0
$621$	2.51807	0.101047
$622$	0	0
$623$	6.77332	0.271367
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.41686	0.336137
$628$	0	0
$629$	−11.9761	−0.477519
$630$	0	0
$631$	23.7872	0.946952	0.473476	−	0.880807i	$-0.342999\pi$
$631$	23.7872	0.946952	0.473476	+	0.880807i	$0.342999\pi$
$632$	0	0
$633$	−3.48283	−0.138430
$634$	0	0
$635$	−8.26555	−0.328008
$636$	0	0
$637$	−2.51384	−0.0996021
$638$	0	0
$639$	−16.4050	−0.648972
$640$	0	0
$641$	−25.6708	−1.01394	−0.506969	−	0.861965i	$-0.669234\pi$
$641$	−25.6708	−1.01394	−0.506969	+	0.861965i	$0.669234\pi$
$642$	0	0
$643$	3.38323	0.133422	0.0667108	−	0.997772i	$-0.478750\pi$
$643$	3.38323	0.133422	0.0667108	+	0.997772i	$0.478750\pi$
$644$	0	0
$645$	1.43450	0.0564833
$646$	0	0
$647$	−38.2639	−1.50431	−0.752154	−	0.658987i	$-0.770985\pi$
$647$	−38.2639	−1.50431	−0.752154	+	0.658987i	$0.770985\pi$
$648$	0	0
$649$	−15.0389	−0.590329
$650$	0	0
$651$	2.06173	0.0808057
$652$	0	0
$653$	14.8181	0.579876	0.289938	−	0.957045i	$-0.406365\pi$
$653$	14.8181	0.579876	0.289938	+	0.957045i	$0.406365\pi$
$654$	0	0
$655$	15.9121	0.621739
$656$	0	0
$657$	23.4398	0.914474
$658$	0	0
$659$	−32.4301	−1.26330	−0.631649	−	0.775255i	$-0.717621\pi$
$659$	−32.4301	−1.26330	−0.631649	+	0.775255i	$0.717621\pi$
$660$	0	0
$661$	−11.0475	−0.429699	−0.214849	−	0.976647i	$-0.568926\pi$
$661$	−11.0475	−0.429699	−0.214849	+	0.976647i	$0.568926\pi$
$662$	0	0
$663$	−4.91367	−0.190831
$664$	0	0
$665$	15.1448	0.587292
$666$	0	0
$667$	−5.48782	−0.212489
$668$	0	0
$669$	−0.262033	−0.0101308
$670$	0	0
$671$	12.0210	0.464067
$672$	0	0
$673$	10.2716	0.395941	0.197970	−	0.980208i	$-0.436565\pi$
$673$	10.2716	0.395941	0.197970	+	0.980208i	$0.436565\pi$
$674$	0	0
$675$	1.64878	0.0634615
$676$	0	0
$677$	−6.04775	−0.232434	−0.116217	−	0.993224i	$-0.537077\pi$
$677$	−6.04775	−0.232434	−0.116217	+	0.993224i	$0.537077\pi$
$678$	0	0
$679$	11.9357	0.458049
$680$	0	0
$681$	0.436492	0.0167264
$682$	0	0
$683$	16.8170	0.643486	0.321743	−	0.946827i	$-0.395731\pi$
$683$	16.8170	0.643486	0.321743	+	0.946827i	$0.395731\pi$
$684$	0	0
$685$	4.56095	0.174265
$686$	0	0
$687$	−2.54102	−0.0969458
$688$	0	0
$689$	−13.3036	−0.506828
$690$	0	0
$691$	16.3412	0.621649	0.310824	−	0.950467i	$-0.399395\pi$
$691$	16.3412	0.621649	0.310824	+	0.950467i	$0.399395\pi$
$692$	0	0
$693$	−36.1291	−1.37243
$694$	0	0
$695$	13.5077	0.512377
$696$	0	0
$697$	−40.3385	−1.52793
$698$	0	0
$699$	2.56759	0.0971150
$700$	0	0
$701$	−12.9149	−0.487790	−0.243895	−	0.969802i	$-0.578425\pi$
$701$	−12.9149	−0.487790	−0.243895	+	0.969802i	$0.578425\pi$
$702$	0	0
$703$	−12.8576	−0.484934
$704$	0	0
$705$	3.07493	0.115809
$706$	0	0
$707$	10.9526	0.411914
$708$	0	0
$709$	41.3468	1.55281	0.776406	−	0.630233i	$-0.217041\pi$
$709$	41.3468	1.55281	0.776406	+	0.630233i	$0.217041\pi$
$710$	0	0
$711$	28.4894	1.06844
$712$	0	0
$713$	4.54510	0.170216
$714$	0	0
$715$	15.4679	0.578467
$716$	0	0
$717$	4.85446	0.181293
$718$	0	0
$719$	22.4264	0.836364	0.418182	−	0.908363i	$-0.362667\pi$
$719$	22.4264	0.836364	0.418182	+	0.908363i	$0.362667\pi$
$720$	0	0
$721$	17.1189	0.637539
$722$	0	0
$723$	5.05662	0.188058
$724$	0	0
$725$	−3.59331	−0.133452
$726$	0	0
$727$	11.9682	0.443875	0.221937	−	0.975061i	$-0.428762\pi$
$727$	11.9682	0.443875	0.221937	+	0.975061i	$0.428762\pi$
$728$	0	0
$729$	−22.9045	−0.848315
$730$	0	0
$731$	29.2096	1.08036
$732$	0	0
$733$	16.6710	0.615757	0.307879	−	0.951426i	$-0.400381\pi$
$733$	16.6710	0.615757	0.307879	+	0.951426i	$0.400381\pi$
$734$	0	0
$735$	−0.224765	−0.00829059
$736$	0	0
$737$	−29.1424	−1.07347
$738$	0	0
$739$	33.9328	1.24824	0.624119	−	0.781329i	$-0.285458\pi$
$739$	33.9328	1.24824	0.624119	+	0.781329i	$0.285458\pi$
$740$	0	0
$741$	−5.27533	−0.193794
$742$	0	0
$743$	−20.4178	−0.749058	−0.374529	−	0.927215i	$-0.622196\pi$
$743$	−20.4178	−0.749058	−0.374529	+	0.927215i	$0.622196\pi$
$744$	0	0
$745$	16.0360	0.587515
$746$	0	0
$747$	−25.5970	−0.936546
$748$	0	0
$749$	40.7549	1.48915
$750$	0	0
$751$	33.8956	1.23687	0.618435	−	0.785836i	$-0.287767\pi$
$751$	33.8956	1.23687	0.618435	+	0.785836i	$0.287767\pi$
$752$	0	0
$753$	−1.74428	−0.0635652
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	1.83679	0.0667592	0.0333796	−	0.999443i	$-0.489373\pi$
$757$	1.83679	0.0667592	0.0333796	+	0.999443i	$0.489373\pi$
$758$	0	0
$759$	2.11217	0.0766669
$760$	0	0
$761$	13.3087	0.482441	0.241220	−	0.970470i	$-0.422452\pi$
$761$	13.3087	0.482441	0.241220	+	0.970470i	$0.422452\pi$
$762$	0	0
$763$	−24.3891	−0.882946
$764$	0	0
$765$	16.5668	0.598972
$766$	0	0
$767$	9.42576	0.340344
$768$	0	0
$769$	38.6414	1.39344	0.696722	−	0.717341i	$-0.254641\pi$
$769$	38.6414	1.39344	0.696722	+	0.717341i	$0.254641\pi$
$770$	0	0
$771$	−5.30407	−0.191021
$772$	0	0
$773$	−37.1953	−1.33782	−0.668912	−	0.743342i	$-0.733240\pi$
$773$	−37.1953	−1.33782	−0.668912	+	0.743342i	$0.733240\pi$
$774$	0	0
$775$	2.97603	0.106902
$776$	0	0
$777$	−1.46362	−0.0525070
$778$	0	0
$779$	−43.3076	−1.55165
$780$	0	0
$781$	−27.8861	−0.997844
$782$	0	0
$783$	−5.92456	−0.211727
$784$	0	0
$785$	−4.55909	−0.162721
$786$	0	0
$787$	−27.3750	−0.975813	−0.487907	−	0.872896i	$-0.662239\pi$
$787$	−27.3750	−0.975813	−0.487907	+	0.872896i	$0.662239\pi$
$788$	0	0
$789$	−1.72867	−0.0615423
$790$	0	0
$791$	24.7009	0.878263
$792$	0	0
$793$	−7.53428	−0.267550
$794$	0	0
$795$	−1.18949	−0.0421870
$796$	0	0
$797$	36.4365	1.29065	0.645323	−	0.763910i	$-0.276723\pi$
$797$	36.4365	1.29065	0.645323	+	0.763910i	$0.276723\pi$
$798$	0	0
$799$	62.6126	2.21507
$800$	0	0
$801$	7.95459	0.281062
$802$	0	0
$803$	39.8442	1.40607
$804$	0	0
$805$	3.80052	0.133951
$806$	0	0
$807$	4.10993	0.144677
$808$	0	0
$809$	11.1202	0.390966	0.195483	−	0.980707i	$-0.437373\pi$
$809$	11.1202	0.390966	0.195483	+	0.980707i	$0.437373\pi$
$810$	0	0
$811$	−28.0150	−0.983741	−0.491871	−	0.870668i	$-0.663687\pi$
$811$	−28.0150	−0.983741	−0.491871	+	0.870668i	$0.663687\pi$
$812$	0	0
$813$	4.48250	0.157208
$814$	0	0
$815$	−15.3540	−0.537827
$816$	0	0
$817$	31.3596	1.09713
$818$	0	0
$819$	22.6442	0.791253
$820$	0	0
$821$	22.5277	0.786222	0.393111	−	0.919491i	$-0.371399\pi$
$821$	22.5277	0.786222	0.393111	+	0.919491i	$0.371399\pi$
$822$	0	0
$823$	−4.54193	−0.158322	−0.0791609	−	0.996862i	$-0.525224\pi$
$823$	−4.54193	−0.158322	−0.0791609	+	0.996862i	$0.525224\pi$
$824$	0	0
$825$	1.38300	0.0481499
$826$	0	0
$827$	10.7531	0.373922	0.186961	−	0.982367i	$-0.440136\pi$
$827$	10.7531	0.373922	0.186961	+	0.982367i	$0.440136\pi$
$828$	0	0
$829$	37.5116	1.30283	0.651415	−	0.758721i	$-0.274176\pi$
$829$	37.5116	1.30283	0.651415	+	0.758721i	$0.274176\pi$
$830$	0	0
$831$	−0.231823	−0.00804184
$832$	0	0
$833$	−4.57673	−0.158574
$834$	0	0
$835$	11.4726	0.397027
$836$	0	0
$837$	4.90682	0.169604
$838$	0	0
$839$	−49.9270	−1.72367	−0.861836	−	0.507188i	$-0.830685\pi$
$839$	−49.9270	−1.72367	−0.861836	+	0.507188i	$0.830685\pi$
$840$	0	0
$841$	−16.0881	−0.554764
$842$	0	0
$843$	−8.33155	−0.286954
$844$	0	0
$845$	3.30537	0.113708
$846$	0	0
$847$	−34.0407	−1.16965
$848$	0	0
$849$	−2.03883	−0.0699723
$850$	0	0
$851$	−3.22655	−0.110605
$852$	0	0
$853$	16.2139	0.555152	0.277576	−	0.960704i	$-0.410469\pi$
$853$	16.2139	0.555152	0.277576	+	0.960704i	$0.410469\pi$
$854$	0	0
$855$	17.7861	0.608273
$856$	0	0
$857$	−45.0880	−1.54018	−0.770089	−	0.637937i	$-0.779788\pi$
$857$	−45.0880	−1.54018	−0.770089	+	0.637937i	$0.779788\pi$
$858$	0	0
$859$	11.5412	0.393781	0.196891	−	0.980425i	$-0.436916\pi$
$859$	11.5412	0.393781	0.196891	+	0.980425i	$0.436916\pi$
$860$	0	0
$861$	−4.92982	−0.168008
$862$	0	0
$863$	−3.50984	−0.119476	−0.0597381	−	0.998214i	$-0.519027\pi$
$863$	−3.50984	−0.119476	−0.0597381	+	0.998214i	$0.519027\pi$
$864$	0	0
$865$	7.23849	0.246116
$866$	0	0
$867$	−4.21322	−0.143088
$868$	0	0
$869$	48.4278	1.64280
$870$	0	0
$871$	18.2652	0.618893
$872$	0	0
$873$	14.0173	0.474412
$874$	0	0
$875$	2.48850	0.0841266
$876$	0	0
$877$	−19.6242	−0.662664	−0.331332	−	0.943514i	$-0.607498\pi$
$877$	−19.6242	−0.662664	−0.331332	+	0.943514i	$0.607498\pi$
$878$	0	0
$879$	3.81869	0.128801
$880$	0	0
$881$	−20.0289	−0.674791	−0.337396	−	0.941363i	$-0.609546\pi$
$881$	−20.0289	−0.674791	−0.337396	+	0.941363i	$0.609546\pi$
$882$	0	0
$883$	−40.0942	−1.34928	−0.674639	−	0.738148i	$-0.735701\pi$
$883$	−40.0942	−1.34928	−0.674639	+	0.738148i	$0.735701\pi$
$884$	0	0
$885$	0.842767	0.0283293
$886$	0	0
$887$	27.5528	0.925133	0.462566	−	0.886585i	$-0.346929\pi$
$887$	27.5528	0.925133	0.462566	+	0.886585i	$0.346929\pi$
$888$	0	0
$889$	−20.5688	−0.689856
$890$	0	0
$891$	−41.2751	−1.38277
$892$	0	0
$893$	67.2211	2.24947
$894$	0	0
$895$	8.48771	0.283713
$896$	0	0
$897$	−1.32382	−0.0442010
$898$	0	0
$899$	−10.6938	−0.356658
$900$	0	0
$901$	−24.2208	−0.806911
$902$	0	0
$903$	3.56975	0.118794
$904$	0	0
$905$	−8.68167	−0.288588
$906$	0	0
$907$	8.23878	0.273564	0.136782	−	0.990601i	$-0.456324\pi$
$907$	8.23878	0.273564	0.136782	+	0.990601i	$0.456324\pi$
$908$	0	0
$909$	12.8627	0.426630
$910$	0	0
$911$	−26.0752	−0.863910	−0.431955	−	0.901895i	$-0.642176\pi$
$911$	−26.0752	−0.863910	−0.431955	+	0.901895i	$0.642176\pi$
$912$	0	0
$913$	−43.5112	−1.44001
$914$	0	0
$915$	−0.673648	−0.0222701
$916$	0	0
$917$	39.5974	1.30762
$918$	0	0
$919$	34.7765	1.14717	0.573586	−	0.819145i	$-0.305552\pi$
$919$	34.7765	1.14717	0.573586	+	0.819145i	$0.305552\pi$
$920$	0	0
$921$	7.29827	0.240486
$922$	0	0
$923$	17.4779	0.575290
$924$	0	0
$925$	−2.11268	−0.0694644
$926$	0	0
$927$	20.1044	0.660315
$928$	0	0
$929$	−10.9586	−0.359540	−0.179770	−	0.983709i	$-0.557535\pi$
$929$	−10.9586	−0.359540	−0.179770	+	0.983709i	$0.557535\pi$
$930$	0	0
$931$	−4.91360	−0.161037
$932$	0	0
$933$	−8.87513	−0.290559
$934$	0	0
$935$	28.1611	0.920965
$936$	0	0
$937$	3.74215	0.122251	0.0611254	−	0.998130i	$-0.480531\pi$
$937$	3.74215	0.122251	0.0611254	+	0.998130i	$0.480531\pi$
$938$	0	0
$939$	−6.97194	−0.227521
$940$	0	0
$941$	43.6599	1.42327	0.711635	−	0.702549i	$-0.247955\pi$
$941$	43.6599	1.42327	0.711635	+	0.702549i	$0.247955\pi$
$942$	0	0
$943$	−10.8678	−0.353905
$944$	0	0
$945$	4.10298	0.133470
$946$	0	0
$947$	−2.95533	−0.0960352	−0.0480176	−	0.998846i	$-0.515290\pi$
$947$	−2.95533	−0.0960352	−0.0480176	+	0.998846i	$0.515290\pi$
$948$	0	0
$949$	−24.9727	−0.810648
$950$	0	0
$951$	2.94142	0.0953820
$952$	0	0
$953$	−41.3557	−1.33964	−0.669822	−	0.742522i	$-0.733630\pi$
$953$	−41.3557	−1.33964	−0.669822	+	0.742522i	$0.733630\pi$
$954$	0	0
$955$	13.8033	0.446666
$956$	0	0
$957$	−4.96955	−0.160643
$958$	0	0
$959$	11.3499	0.366508
$960$	0	0
$961$	−22.1432	−0.714297
$962$	0	0
$963$	47.8627	1.54235
$964$	0	0
$965$	14.0034	0.450787
$966$	0	0
$967$	7.72625	0.248459	0.124230	−	0.992253i	$-0.460354\pi$
$967$	7.72625	0.248459	0.124230	+	0.992253i	$0.460354\pi$
$968$	0	0
$969$	−9.60434	−0.308536
$970$	0	0
$971$	−49.4549	−1.58708	−0.793542	−	0.608516i	$-0.791765\pi$
$971$	−49.4549	−1.58708	−0.793542	+	0.608516i	$0.791765\pi$
$972$	0	0
$973$	33.6140	1.07761
$974$	0	0
$975$	−0.866808	−0.0277601
$976$	0	0
$977$	26.9376	0.861811	0.430905	−	0.902397i	$-0.358194\pi$
$977$	26.9376	0.861811	0.430905	+	0.902397i	$0.358194\pi$
$978$	0	0
$979$	13.5216	0.432154
$980$	0	0
$981$	−28.6426	−0.914489
$982$	0	0
$983$	−22.0603	−0.703614	−0.351807	−	0.936073i	$-0.614433\pi$
$983$	−22.0603	−0.703614	−0.351807	+	0.936073i	$0.614433\pi$
$984$	0	0
$985$	6.80323	0.216769
$986$	0	0
$987$	7.65197	0.243565
$988$	0	0
$989$	7.86953	0.250237
$990$	0	0
$991$	17.5918	0.558821	0.279410	−	0.960172i	$-0.409861\pi$
$991$	17.5918	0.558821	0.279410	+	0.960172i	$0.409861\pi$
$992$	0	0
$993$	0.765834	0.0243030
$994$	0	0
$995$	−19.3521	−0.613502
$996$	0	0
$997$	−44.4629	−1.40815	−0.704077	−	0.710124i	$-0.748639\pi$
$997$	−44.4629	−1.40815	−0.704077	+	0.710124i	$0.748639\pi$
$998$	0	0
$999$	−3.48333	−0.110208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.8	✓	13	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)

\(f(q)\)	\(=\)	\(q-0.278392 q^{3} -1.00000 q^{5} -2.48850 q^{7} -2.92250 q^{9} +O(q^{10})\)	\(q-0.278392 q^{3} -1.00000 q^{5} -2.48850 q^{7} -2.92250 q^{9} -4.96782 q^{11} +3.11362 q^{13} +0.278392 q^{15} +5.66870 q^{17} +6.08593 q^{19} +0.692779 q^{21} +1.52724 q^{23} +1.00000 q^{25} +1.64878 q^{27} -3.59331 q^{29} +2.97603 q^{31} +1.38300 q^{33} +2.48850 q^{35} -2.11268 q^{37} -0.866808 q^{39} -7.11601 q^{41} +5.15280 q^{43} +2.92250 q^{45} +11.0453 q^{47} -0.807370 q^{49} -1.57812 q^{51} -4.27272 q^{53} +4.96782 q^{55} -1.69428 q^{57} +3.02727 q^{59} -2.41978 q^{61} +7.27263 q^{63} -3.11362 q^{65} +5.86623 q^{67} -0.425170 q^{69} +5.61335 q^{71} -8.02047 q^{73} -0.278392 q^{75} +12.3624 q^{77} -9.74831 q^{79} +8.30849 q^{81} +8.75861 q^{83} -5.66870 q^{85} +1.00035 q^{87} -2.72185 q^{89} -7.74824 q^{91} -0.828505 q^{93} -6.08593 q^{95} -4.79633 q^{97} +14.5184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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