LMFDB - Embedded newform 3640.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.27460$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27460 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.17380 q^{9} +O(q^{10})$	$q-2.27460 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.17380 q^{9} -3.56912 q^{11} +1.00000 q^{13} +2.27460 q^{15} -1.60468 q^{17} +4.94452 q^{19} -2.27460 q^{21} -1.01992 q^{23} +1.00000 q^{25} +1.87928 q^{27} +1.84372 q^{29} -5.83155 q^{31} +8.11832 q^{33} -1.00000 q^{35} +9.41284 q^{37} -2.27460 q^{39} -1.41525 q^{41} -3.75855 q^{43} -2.17380 q^{45} -2.01217 q^{47} +1.00000 q^{49} +3.65000 q^{51} -2.58905 q^{53} +3.56912 q^{55} -11.2468 q^{57} +4.81163 q^{59} +8.53703 q^{61} +2.17380 q^{63} -1.00000 q^{65} +1.41525 q^{67} +2.31992 q^{69} -8.75080 q^{71} +0.229279 q^{73} -2.27460 q^{75} -3.56912 q^{77} +5.96444 q^{79} -10.7960 q^{81} -8.54920 q^{83} +1.60468 q^{85} -4.19372 q^{87} +10.1913 q^{89} +1.00000 q^{91} +13.2644 q^{93} -4.94452 q^{95} +7.67527 q^{97} -7.75855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 + 4 * q^7	$ 4 q - 4 q^{5} + 4 q^{7} - q^{11} + 4 q^{13} - 11 q^{17} - 3 q^{19} - 9 q^{23} + 4 q^{25} + 3 q^{27} - 15 q^{29} - 6 q^{31} + q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} - 3 q^{47} + 4 q^{49} - 4 q^{51} - 2 q^{53} + q^{55} - 28 q^{57} - 3 q^{59} + 21 q^{61} - 4 q^{65} + 6 q^{67} - 23 q^{69} - 16 q^{71} + 15 q^{73} - q^{77} + 6 q^{79} - 8 q^{81} - 16 q^{83} + 11 q^{85} - 13 q^{87} + q^{89} + 4 q^{91} - 2 q^{93} + 3 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^7 - q^11 + 4 * q^13 - 11 * q^17 - 3 * q^19 - 9 * q^23 + 4 * q^25 + 3 * q^27 - 15 * q^29 - 6 * q^31 + q^33 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 - 3 * q^47 + 4 * q^49 - 4 * q^51 - 2 * q^53 + q^55 - 28 * q^57 - 3 * q^59 + 21 * q^61 - 4 * q^65 + 6 * q^67 - 23 * q^69 - 16 * q^71 + 15 * q^73 - q^77 + 6 * q^79 - 8 * q^81 - 16 * q^83 + 11 * q^85 - 13 * q^87 + q^89 + 4 * q^91 - 2 * q^93 + 3 * q^95 - 9 * q^97 - 22 * 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
$3$	−2.27460	−1.31324	−0.656620	−	0.754222i	$-0.728014\pi$
$3$	−2.27460	−1.31324	−0.656620	+	0.754222i	$0.728014\pi$
$4$	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
$9$	2.17380	0.724600
$10$	0	0
$11$	−3.56912	−1.07613	−0.538065	−	0.842903i	$-0.680845\pi$
$11$	−3.56912	−1.07613	−0.538065	+	0.842903i	$0.680845\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.27460	0.587299
$16$	0	0
$17$	−1.60468	−0.389191	−0.194596	−	0.980884i	$-0.562339\pi$
$17$	−1.60468	−0.389191	−0.194596	+	0.980884i	$0.562339\pi$
$18$	0	0
$19$	4.94452	1.13435	0.567175	−	0.823597i	$-0.308036\pi$
$19$	4.94452	1.13435	0.567175	+	0.823597i	$0.308036\pi$
$20$	0	0
$21$	−2.27460	−0.496358
$22$	0	0
$23$	−1.01992	−0.212669	−0.106334	−	0.994330i	$-0.533911\pi$
$23$	−1.01992	−0.212669	−0.106334	+	0.994330i	$0.533911\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.87928	0.361667
$28$	0	0
$29$	1.84372	0.342370	0.171185	−	0.985239i	$-0.445240\pi$
$29$	1.84372	0.342370	0.171185	+	0.985239i	$0.445240\pi$
$30$	0	0
$31$	−5.83155	−1.04738	−0.523689	−	0.851910i	$-0.675445\pi$
$31$	−5.83155	−1.04738	−0.523689	+	0.851910i	$0.675445\pi$
$32$	0	0
$33$	8.11832	1.41322
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	9.41284	1.54746	0.773731	−	0.633515i	$-0.218388\pi$
$37$	9.41284	1.54746	0.773731	+	0.633515i	$0.218388\pi$
$38$	0	0
$39$	−2.27460	−0.364227
$40$	0	0
$41$	−1.41525	−0.221024	−0.110512	−	0.993875i	$-0.535249\pi$
$41$	−1.41525	−0.221024	−0.110512	+	0.993875i	$0.535249\pi$
$42$	0	0
$43$	−3.75855	−0.573174	−0.286587	−	0.958054i	$-0.592521\pi$
$43$	−3.75855	−0.573174	−0.286587	+	0.958054i	$0.592521\pi$
$44$	0	0
$45$	−2.17380	−0.324051
$46$	0	0
$47$	−2.01217	−0.293505	−0.146753	−	0.989173i	$-0.546882\pi$
$47$	−2.01217	−0.293505	−0.146753	+	0.989173i	$0.546882\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.65000	0.511102
$52$	0	0
$53$	−2.58905	−0.355633	−0.177816	−	0.984064i	$-0.556903\pi$
$53$	−2.58905	−0.355633	−0.177816	+	0.984064i	$0.556903\pi$
$54$	0	0
$55$	3.56912	0.481260
$56$	0	0
$57$	−11.2468	−1.48967
$58$	0	0
$59$	4.81163	0.626420	0.313210	−	0.949684i	$-0.398596\pi$
$59$	4.81163	0.626420	0.313210	+	0.949684i	$0.398596\pi$
$60$	0	0
$61$	8.53703	1.09305	0.546527	−	0.837441i	$-0.315949\pi$
$61$	8.53703	1.09305	0.546527	+	0.837441i	$0.315949\pi$
$62$	0	0
$63$	2.17380	0.273873
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	1.41525	0.172900	0.0864500	−	0.996256i	$-0.472448\pi$
$67$	1.41525	0.172900	0.0864500	+	0.996256i	$0.472448\pi$
$68$	0	0
$69$	2.31992	0.279285
$70$	0	0
$71$	−8.75080	−1.03853	−0.519264	−	0.854614i	$-0.673794\pi$
$71$	−8.75080	−1.03853	−0.519264	+	0.854614i	$0.673794\pi$
$72$	0	0
$73$	0.229279	0.0268351	0.0134175	−	0.999910i	$-0.495729\pi$
$73$	0.229279	0.0268351	0.0134175	+	0.999910i	$0.495729\pi$
$74$	0	0
$75$	−2.27460	−0.262648
$76$	0	0
$77$	−3.56912	−0.406739
$78$	0	0
$79$	5.96444	0.671052	0.335526	−	0.942031i	$-0.391086\pi$
$79$	5.96444	0.671052	0.335526	+	0.942031i	$0.391086\pi$
$80$	0	0
$81$	−10.7960	−1.19956
$82$	0	0
$83$	−8.54920	−0.938396	−0.469198	−	0.883093i	$-0.655457\pi$
$83$	−8.54920	−0.938396	−0.469198	+	0.883093i	$0.655457\pi$
$84$	0	0
$85$	1.60468	0.174052
$86$	0	0
$87$	−4.19372	−0.449614
$88$	0	0
$89$	10.1913	1.08028	0.540139	−	0.841576i	$-0.318372\pi$
$89$	10.1913	1.08028	0.540139	+	0.841576i	$0.318372\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	13.2644	1.37546
$94$	0	0
$95$	−4.94452	−0.507297
$96$	0	0
$97$	7.67527	0.779306	0.389653	−	0.920962i	$-0.372595\pi$
$97$	7.67527	0.779306	0.389653	+	0.920962i	$0.372595\pi$
$98$	0	0
$99$	−7.75855	−0.779764
$100$	0	0
$101$	19.0550	1.89604	0.948020	−	0.318211i	$-0.103082\pi$
$101$	19.0550	1.89604	0.948020	+	0.318211i	$0.103082\pi$
$102$	0	0
$103$	−14.0828	−1.38762	−0.693808	−	0.720160i	$-0.744068\pi$
$103$	−14.0828	−1.38762	−0.693808	+	0.720160i	$0.744068\pi$
$104$	0	0
$105$	2.27460	0.221978
$106$	0	0
$107$	−12.8569	−1.24293	−0.621464	−	0.783443i	$-0.713462\pi$
$107$	−12.8569	−1.24293	−0.621464	+	0.783443i	$0.713462\pi$
$108$	0	0
$109$	1.55695	0.149129	0.0745645	−	0.997216i	$-0.476243\pi$
$109$	1.55695	0.149129	0.0745645	+	0.997216i	$0.476243\pi$
$110$	0	0
$111$	−21.4104	−2.03219
$112$	0	0
$113$	−3.98783	−0.375144	−0.187572	−	0.982251i	$-0.560062\pi$
$113$	−3.98783	−0.375144	−0.187572	+	0.982251i	$0.560062\pi$
$114$	0	0
$115$	1.01992	0.0951084
$116$	0	0
$117$	2.17380	0.200968
$118$	0	0
$119$	−1.60468	−0.147101
$120$	0	0
$121$	1.73863	0.158057
$122$	0	0
$123$	3.21912	0.290258
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.63248	−0.411066	−0.205533	−	0.978650i	$-0.565893\pi$
$127$	−4.63248	−0.411066	−0.205533	+	0.978650i	$0.565893\pi$
$128$	0	0
$129$	8.54920	0.752715
$130$	0	0
$131$	9.09839	0.794930	0.397465	−	0.917617i	$-0.369890\pi$
$131$	9.09839	0.794930	0.397465	+	0.917617i	$0.369890\pi$
$132$	0	0
$133$	4.94452	0.428744
$134$	0	0
$135$	−1.87928	−0.161742
$136$	0	0
$137$	−18.1515	−1.55078	−0.775392	−	0.631480i	$-0.782448\pi$
$137$	−18.1515	−1.55078	−0.775392	+	0.631480i	$0.782448\pi$
$138$	0	0
$139$	−2.28130	−0.193497	−0.0967485	−	0.995309i	$-0.530844\pi$
$139$	−2.28130	−0.193497	−0.0967485	+	0.995309i	$0.530844\pi$
$140$	0	0
$141$	4.57688	0.385443
$142$	0	0
$143$	−3.56912	−0.298465
$144$	0	0
$145$	−1.84372	−0.153113
$146$	0	0
$147$	−2.27460	−0.187606
$148$	0	0
$149$	−13.9167	−1.14010	−0.570051	−	0.821609i	$-0.693077\pi$
$149$	−13.9167	−1.14010	−0.570051	+	0.821609i	$0.693077\pi$
$150$	0	0
$151$	2.69520	0.219332	0.109666	−	0.993969i	$-0.465022\pi$
$151$	2.69520	0.219332	0.109666	+	0.993969i	$0.465022\pi$
$152$	0	0
$153$	−3.48825	−0.282008
$154$	0	0
$155$	5.83155	0.468401
$156$	0	0
$157$	−21.6495	−1.72782	−0.863908	−	0.503649i	$-0.831990\pi$
$157$	−21.6495	−1.72782	−0.863908	+	0.503649i	$0.831990\pi$
$158$	0	0
$159$	5.88904	0.467031
$160$	0	0
$161$	−1.01992	−0.0803813
$162$	0	0
$163$	−1.73757	−0.136097	−0.0680485	−	0.997682i	$-0.521677\pi$
$163$	−1.73757	−0.136097	−0.0680485	+	0.997682i	$0.521677\pi$
$164$	0	0
$165$	−8.11832	−0.632010
$166$	0	0
$167$	−18.4139	−1.42491	−0.712455	−	0.701718i	$-0.752417\pi$
$167$	−18.4139	−1.42491	−0.712455	+	0.701718i	$0.752417\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.7484	0.821950
$172$	0	0
$173$	−0.882739	−0.0671134	−0.0335567	−	0.999437i	$-0.510683\pi$
$173$	−0.882739	−0.0671134	−0.0335567	+	0.999437i	$0.510683\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.9445	−0.822640
$178$	0	0
$179$	10.6320	0.794670	0.397335	−	0.917674i	$-0.369935\pi$
$179$	10.6320	0.794670	0.397335	+	0.917674i	$0.369935\pi$
$180$	0	0
$181$	13.3520	0.992447	0.496224	−	0.868195i	$-0.334720\pi$
$181$	13.3520	0.992447	0.496224	+	0.868195i	$0.334720\pi$
$182$	0	0
$183$	−19.4183	−1.43544
$184$	0	0
$185$	−9.41284	−0.692046
$186$	0	0
$187$	5.72729	0.418821
$188$	0	0
$189$	1.87928	0.136697
$190$	0	0
$191$	−10.5702	−0.764831	−0.382416	−	0.923990i	$-0.624908\pi$
$191$	−10.5702	−0.764831	−0.382416	+	0.923990i	$0.624908\pi$
$192$	0	0
$193$	−12.0584	−0.867984	−0.433992	−	0.900917i	$-0.642895\pi$
$193$	−12.0584	−0.867984	−0.433992	+	0.900917i	$0.642895\pi$
$194$	0	0
$195$	2.27460	0.162887
$196$	0	0
$197$	−10.6778	−0.760762	−0.380381	−	0.924830i	$-0.624207\pi$
$197$	−10.6778	−0.760762	−0.380381	+	0.924830i	$0.624207\pi$
$198$	0	0
$199$	−25.8179	−1.83018	−0.915092	−	0.403245i	$-0.867882\pi$
$199$	−25.8179	−1.83018	−0.915092	+	0.403245i	$0.867882\pi$
$200$	0	0
$201$	−3.21912	−0.227059
$202$	0	0
$203$	1.84372	0.129404
$204$	0	0
$205$	1.41525	0.0988451
$206$	0	0
$207$	−2.21711	−0.154100
$208$	0	0
$209$	−17.6476	−1.22071
$210$	0	0
$211$	−2.87822	−0.198145	−0.0990724	−	0.995080i	$-0.531588\pi$
$211$	−2.87822	−0.198145	−0.0990724	+	0.995080i	$0.531588\pi$
$212$	0	0
$213$	19.9046	1.36384
$214$	0	0
$215$	3.75855	0.256331
$216$	0	0
$217$	−5.83155	−0.395871
$218$	0	0
$219$	−0.521518	−0.0352409
$220$	0	0
$221$	−1.60468	−0.107942
$222$	0	0
$223$	15.7103	1.05204	0.526020	−	0.850472i	$-0.323684\pi$
$223$	15.7103	1.05204	0.526020	+	0.850472i	$0.323684\pi$
$224$	0	0
$225$	2.17380	0.144920
$226$	0	0
$227$	−16.0663	−1.06636	−0.533179	−	0.846002i	$-0.679003\pi$
$227$	−16.0663	−1.06636	−0.533179	+	0.846002i	$0.679003\pi$
$228$	0	0
$229$	−5.46938	−0.361427	−0.180713	−	0.983536i	$-0.557841\pi$
$229$	−5.46938	−0.361427	−0.180713	+	0.983536i	$0.557841\pi$
$230$	0	0
$231$	8.11832	0.534146
$232$	0	0
$233$	3.28783	0.215393	0.107696	−	0.994184i	$-0.465653\pi$
$233$	3.28783	0.215393	0.107696	+	0.994184i	$0.465653\pi$
$234$	0	0
$235$	2.01217	0.131259
$236$	0	0
$237$	−13.5667	−0.881253
$238$	0	0
$239$	−24.2765	−1.57032	−0.785158	−	0.619296i	$-0.787418\pi$
$239$	−24.2765	−1.57032	−0.785158	+	0.619296i	$0.787418\pi$
$240$	0	0
$241$	23.2766	1.49938	0.749689	−	0.661790i	$-0.230203\pi$
$241$	23.2766	1.49938	0.749689	+	0.661790i	$0.230203\pi$
$242$	0	0
$243$	18.9187	1.21364
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.94452	0.314612
$248$	0	0
$249$	19.4460	1.23234
$250$	0	0
$251$	−9.22152	−0.582057	−0.291029	−	0.956714i	$-0.593998\pi$
$251$	−9.22152	−0.582057	−0.291029	+	0.956714i	$0.593998\pi$
$252$	0	0
$253$	3.64023	0.228860
$254$	0	0
$255$	−3.65000	−0.228572
$256$	0	0
$257$	3.99519	0.249213	0.124607	−	0.992206i	$-0.460233\pi$
$257$	3.99519	0.249213	0.124607	+	0.992206i	$0.460233\pi$
$258$	0	0
$259$	9.41284	0.584886
$260$	0	0
$261$	4.00788	0.248081
$262$	0	0
$263$	−11.4195	−0.704159	−0.352079	−	0.935970i	$-0.614525\pi$
$263$	−11.4195	−0.704159	−0.352079	+	0.935970i	$0.614525\pi$
$264$	0	0
$265$	2.58905	0.159044
$266$	0	0
$267$	−23.1812	−1.41866
$268$	0	0
$269$	−22.6627	−1.38177	−0.690885	−	0.722965i	$-0.742779\pi$
$269$	−22.6627	−1.38177	−0.690885	+	0.722965i	$0.742779\pi$
$270$	0	0
$271$	−19.7103	−1.19732	−0.598658	−	0.801005i	$-0.704299\pi$
$271$	−19.7103	−1.19732	−0.598658	+	0.801005i	$0.704299\pi$
$272$	0	0
$273$	−2.27460	−0.137665
$274$	0	0
$275$	−3.56912	−0.215226
$276$	0	0
$277$	−29.3594	−1.76403	−0.882017	−	0.471218i	$-0.843815\pi$
$277$	−29.3594	−1.76403	−0.882017	+	0.471218i	$0.843815\pi$
$278$	0	0
$279$	−12.6766	−0.758929
$280$	0	0
$281$	−21.5123	−1.28332	−0.641658	−	0.766991i	$-0.721753\pi$
$281$	−21.5123	−1.28332	−0.641658	+	0.766991i	$0.721753\pi$
$282$	0	0
$283$	15.1027	0.897762	0.448881	−	0.893592i	$-0.351823\pi$
$283$	15.1027	0.897762	0.448881	+	0.893592i	$0.351823\pi$
$284$	0	0
$285$	11.2468	0.666203
$286$	0	0
$287$	−1.41525	−0.0835394
$288$	0	0
$289$	−14.4250	−0.848530
$290$	0	0
$291$	−17.4582	−1.02342
$292$	0	0
$293$	22.8569	1.33532	0.667659	−	0.744468i	$-0.267297\pi$
$293$	22.8569	1.33532	0.667659	+	0.744468i	$0.267297\pi$
$294$	0	0
$295$	−4.81163	−0.280144
$296$	0	0
$297$	−6.70736	−0.389201
$298$	0	0
$299$	−1.01992	−0.0589837
$300$	0	0
$301$	−3.75855	−0.216639
$302$	0	0
$303$	−43.3424	−2.48996
$304$	0	0
$305$	−8.53703	−0.488829
$306$	0	0
$307$	−26.9874	−1.54025	−0.770127	−	0.637890i	$-0.779807\pi$
$307$	−26.9874	−1.54025	−0.770127	+	0.637890i	$0.779807\pi$
$308$	0	0
$309$	32.0326	1.82227
$310$	0	0
$311$	−12.9232	−0.732810	−0.366405	−	0.930455i	$-0.619412\pi$
$311$	−12.9232	−0.732810	−0.366405	+	0.930455i	$0.619412\pi$
$312$	0	0
$313$	−5.35871	−0.302892	−0.151446	−	0.988466i	$-0.548393\pi$
$313$	−5.35871	−0.302892	−0.151446	+	0.988466i	$0.548393\pi$
$314$	0	0
$315$	−2.17380	−0.122480
$316$	0	0
$317$	16.5566	0.929909	0.464954	−	0.885335i	$-0.346071\pi$
$317$	16.5566	0.929909	0.464954	+	0.885335i	$0.346071\pi$
$318$	0	0
$319$	−6.58046	−0.368435
$320$	0	0
$321$	29.2444	1.63226
$322$	0	0
$323$	−7.93436	−0.441480
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−3.54144	−0.195842
$328$	0	0
$329$	−2.01217	−0.110934
$330$	0	0
$331$	−4.98008	−0.273730	−0.136865	−	0.990590i	$-0.543703\pi$
$331$	−4.98008	−0.273730	−0.136865	+	0.990590i	$0.543703\pi$
$332$	0	0
$333$	20.4616	1.12129
$334$	0	0
$335$	−1.41525	−0.0773232
$336$	0	0
$337$	−18.2801	−0.995779	−0.497889	−	0.867241i	$-0.665891\pi$
$337$	−18.2801	−0.995779	−0.497889	+	0.867241i	$0.665891\pi$
$338$	0	0
$339$	9.07071	0.492654
$340$	0	0
$341$	20.8135	1.12711
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.31992	−0.124900
$346$	0	0
$347$	−2.85483	−0.153255	−0.0766277	−	0.997060i	$-0.524415\pi$
$347$	−2.85483	−0.153255	−0.0766277	+	0.997060i	$0.524415\pi$
$348$	0	0
$349$	12.8360	0.687093	0.343547	−	0.939136i	$-0.388372\pi$
$349$	12.8360	0.687093	0.343547	+	0.939136i	$0.388372\pi$
$350$	0	0
$351$	1.87928	0.100308
$352$	0	0
$353$	2.81832	0.150004	0.0750021	−	0.997183i	$-0.476104\pi$
$353$	2.81832	0.150004	0.0750021	+	0.997183i	$0.476104\pi$
$354$	0	0
$355$	8.75080	0.464444
$356$	0	0
$357$	3.65000	0.193178
$358$	0	0
$359$	−2.87034	−0.151491	−0.0757454	−	0.997127i	$-0.524134\pi$
$359$	−2.87034	−0.151491	−0.0757454	+	0.997127i	$0.524134\pi$
$360$	0	0
$361$	5.44828	0.286751
$362$	0	0
$363$	−3.95468	−0.207567
$364$	0	0
$365$	−0.229279	−0.0120010
$366$	0	0
$367$	22.1968	1.15866	0.579331	−	0.815092i	$-0.303314\pi$
$367$	22.1968	1.15866	0.579331	+	0.815092i	$0.303314\pi$
$368$	0	0
$369$	−3.07646	−0.160154
$370$	0	0
$371$	−2.58905	−0.134417
$372$	0	0
$373$	3.83825	0.198737	0.0993685	−	0.995051i	$-0.468318\pi$
$373$	3.83825	0.198737	0.0993685	+	0.995051i	$0.468318\pi$
$374$	0	0
$375$	2.27460	0.117460
$376$	0	0
$377$	1.84372	0.0949564
$378$	0	0
$379$	−6.49360	−0.333554	−0.166777	−	0.985995i	$-0.553336\pi$
$379$	−6.49360	−0.333554	−0.166777	+	0.985995i	$0.553336\pi$
$380$	0	0
$381$	10.5370	0.539828
$382$	0	0
$383$	−7.08623	−0.362089	−0.181045	−	0.983475i	$-0.557948\pi$
$383$	−7.08623	−0.362089	−0.181045	+	0.983475i	$0.557948\pi$
$384$	0	0
$385$	3.56912	0.181899
$386$	0	0
$387$	−8.17034	−0.415322
$388$	0	0
$389$	−22.4255	−1.13702	−0.568510	−	0.822676i	$-0.692480\pi$
$389$	−22.4255	−1.13702	−0.568510	+	0.822676i	$0.692480\pi$
$390$	0	0
$391$	1.63665	0.0827689
$392$	0	0
$393$	−20.6952	−1.04393
$394$	0	0
$395$	−5.96444	−0.300104
$396$	0	0
$397$	2.31256	0.116064	0.0580320	−	0.998315i	$-0.481517\pi$
$397$	2.31256	0.116064	0.0580320	+	0.998315i	$0.481517\pi$
$398$	0	0
$399$	−11.2468	−0.563044
$400$	0	0
$401$	23.6318	1.18012	0.590059	−	0.807360i	$-0.299105\pi$
$401$	23.6318	1.18012	0.590059	+	0.807360i	$0.299105\pi$
$402$	0	0
$403$	−5.83155	−0.290490
$404$	0	0
$405$	10.7960	0.536457
$406$	0	0
$407$	−33.5956	−1.66527
$408$	0	0
$409$	24.0992	1.19163	0.595815	−	0.803122i	$-0.296829\pi$
$409$	24.0992	1.19163	0.595815	+	0.803122i	$0.296829\pi$
$410$	0	0
$411$	41.2873	2.03655
$412$	0	0
$413$	4.81163	0.236765
$414$	0	0
$415$	8.54920	0.419664
$416$	0	0
$417$	5.18903	0.254108
$418$	0	0
$419$	6.55656	0.320309	0.160154	−	0.987092i	$-0.448801\pi$
$419$	6.55656	0.320309	0.160154	+	0.987092i	$0.448801\pi$
$420$	0	0
$421$	−15.4689	−0.753906	−0.376953	−	0.926232i	$-0.623028\pi$
$421$	−15.4689	−0.753906	−0.376953	+	0.926232i	$0.623028\pi$
$422$	0	0
$423$	−4.37405	−0.212674
$424$	0	0
$425$	−1.60468	−0.0778383
$426$	0	0
$427$	8.53703	0.413136
$428$	0	0
$429$	8.11832	0.391956
$430$	0	0
$431$	9.13824	0.440174	0.220087	−	0.975480i	$-0.429366\pi$
$431$	9.13824	0.440174	0.220087	+	0.975480i	$0.429366\pi$
$432$	0	0
$433$	−7.54049	−0.362373	−0.181186	−	0.983449i	$-0.557994\pi$
$433$	−7.54049	−0.362373	−0.181186	+	0.983449i	$0.557994\pi$
$434$	0	0
$435$	4.19372	0.201074
$436$	0	0
$437$	−5.04304	−0.241241
$438$	0	0
$439$	27.7489	1.32438	0.662192	−	0.749334i	$-0.269626\pi$
$439$	27.7489	1.32438	0.662192	+	0.749334i	$0.269626\pi$
$440$	0	0
$441$	2.17380	0.103514
$442$	0	0
$443$	34.6038	1.64407	0.822037	−	0.569434i	$-0.192838\pi$
$443$	34.6038	1.64407	0.822037	+	0.569434i	$0.192838\pi$
$444$	0	0
$445$	−10.1913	−0.483115
$446$	0	0
$447$	31.6549	1.49723
$448$	0	0
$449$	3.16951	0.149578	0.0747891	−	0.997199i	$-0.476172\pi$
$449$	3.16951	0.149578	0.0747891	+	0.997199i	$0.476172\pi$
$450$	0	0
$451$	5.05119	0.237851
$452$	0	0
$453$	−6.13049	−0.288035
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−8.16698	−0.382035	−0.191018	−	0.981587i	$-0.561179\pi$
$457$	−8.16698	−0.382035	−0.191018	+	0.981587i	$0.561179\pi$
$458$	0	0
$459$	−3.01563	−0.140758
$460$	0	0
$461$	9.60708	0.447446	0.223723	−	0.974653i	$-0.428179\pi$
$461$	9.60708	0.447446	0.223723	+	0.974653i	$0.428179\pi$
$462$	0	0
$463$	0.430360	0.0200005	0.0100003	−	0.999950i	$-0.496817\pi$
$463$	0.430360	0.0200005	0.0100003	+	0.999950i	$0.496817\pi$
$464$	0	0
$465$	−13.2644	−0.615124
$466$	0	0
$467$	−12.6873	−0.587099	−0.293550	−	0.955944i	$-0.594837\pi$
$467$	−12.6873	−0.587099	−0.293550	+	0.955944i	$0.594837\pi$
$468$	0	0
$469$	1.41525	0.0653500
$470$	0	0
$471$	49.2439	2.26904
$472$	0	0
$473$	13.4147	0.616810
$474$	0	0
$475$	4.94452	0.226870
$476$	0	0
$477$	−5.62806	−0.257691
$478$	0	0
$479$	30.7405	1.40457	0.702285	−	0.711896i	$-0.252163\pi$
$479$	30.7405	1.40457	0.702285	+	0.711896i	$0.252163\pi$
$480$	0	0
$481$	9.41284	0.429189
$482$	0	0
$483$	2.31992	0.105560
$484$	0	0
$485$	−7.67527	−0.348516
$486$	0	0
$487$	−31.9344	−1.44708	−0.723542	−	0.690280i	$-0.757487\pi$
$487$	−31.9344	−1.44708	−0.723542	+	0.690280i	$0.757487\pi$
$488$	0	0
$489$	3.95228	0.178728
$490$	0	0
$491$	0.886095	0.0399889	0.0199945	−	0.999800i	$-0.493635\pi$
$491$	0.886095	0.0399889	0.0199945	+	0.999800i	$0.493635\pi$
$492$	0	0
$493$	−2.95858	−0.133248
$494$	0	0
$495$	7.75855	0.348721
$496$	0	0
$497$	−8.75080	−0.392527
$498$	0	0
$499$	−6.13383	−0.274588	−0.137294	−	0.990530i	$-0.543840\pi$
$499$	−6.13383	−0.274588	−0.137294	+	0.990530i	$0.543840\pi$
$500$	0	0
$501$	41.8842	1.87125
$502$	0	0
$503$	8.29200	0.369722	0.184861	−	0.982765i	$-0.440817\pi$
$503$	8.29200	0.369722	0.184861	+	0.982765i	$0.440817\pi$
$504$	0	0
$505$	−19.0550	−0.847935
$506$	0	0
$507$	−2.27460	−0.101018
$508$	0	0
$509$	24.2202	1.07354	0.536770	−	0.843728i	$-0.319644\pi$
$509$	24.2202	1.07354	0.536770	+	0.843728i	$0.319644\pi$
$510$	0	0
$511$	0.229279	0.0101427
$512$	0	0
$513$	9.29212	0.410257
$514$	0	0
$515$	14.0828	0.620561
$516$	0	0
$517$	7.18168	0.315850
$518$	0	0
$519$	2.00788	0.0881360
$520$	0	0
$521$	21.2045	0.928988	0.464494	−	0.885576i	$-0.346236\pi$
$521$	21.2045	0.928988	0.464494	+	0.885576i	$0.346236\pi$
$522$	0	0
$523$	−29.9761	−1.31076	−0.655382	−	0.755298i	$-0.727492\pi$
$523$	−29.9761	−1.31076	−0.655382	+	0.755298i	$0.727492\pi$
$524$	0	0
$525$	−2.27460	−0.0992716
$526$	0	0
$527$	9.35776	0.407630
$528$	0	0
$529$	−21.9598	−0.954772
$530$	0	0
$531$	10.4595	0.453904
$532$	0	0
$533$	−1.41525	−0.0613011
$534$	0	0
$535$	12.8569	0.555854
$536$	0	0
$537$	−24.1834	−1.04359
$538$	0	0
$539$	−3.56912	−0.153733
$540$	0	0
$541$	22.5757	0.970603	0.485302	−	0.874347i	$-0.338710\pi$
$541$	22.5757	0.970603	0.485302	+	0.874347i	$0.338710\pi$
$542$	0	0
$543$	−30.3705	−1.30332
$544$	0	0
$545$	−1.55695	−0.0666925
$546$	0	0
$547$	−21.1733	−0.905304	−0.452652	−	0.891687i	$-0.649522\pi$
$547$	−21.1733	−0.905304	−0.452652	+	0.891687i	$0.649522\pi$
$548$	0	0
$549$	18.5578	0.792027
$550$	0	0
$551$	9.11631	0.388368
$552$	0	0
$553$	5.96444	0.253634
$554$	0	0
$555$	21.4104	0.908822
$556$	0	0
$557$	−12.1019	−0.512772	−0.256386	−	0.966574i	$-0.582532\pi$
$557$	−12.1019	−0.512772	−0.256386	+	0.966574i	$0.582532\pi$
$558$	0	0
$559$	−3.75855	−0.158970
$560$	0	0
$561$	−13.0273	−0.550012
$562$	0	0
$563$	1.82915	0.0770893	0.0385447	−	0.999257i	$-0.487728\pi$
$563$	1.82915	0.0770893	0.0385447	+	0.999257i	$0.487728\pi$
$564$	0	0
$565$	3.98783	0.167769
$566$	0	0
$567$	−10.7960	−0.453389
$568$	0	0
$569$	−15.8401	−0.664053	−0.332027	−	0.943270i	$-0.607732\pi$
$569$	−15.8401	−0.664053	−0.332027	+	0.943270i	$0.607732\pi$
$570$	0	0
$571$	35.6686	1.49268	0.746342	−	0.665563i	$-0.231809\pi$
$571$	35.6686	1.49268	0.746342	+	0.665563i	$0.231809\pi$
$572$	0	0
$573$	24.0429	1.00441
$574$	0	0
$575$	−1.01992	−0.0425338
$576$	0	0
$577$	−7.67193	−0.319387	−0.159693	−	0.987167i	$-0.551051\pi$
$577$	−7.67193	−0.319387	−0.159693	+	0.987167i	$0.551051\pi$
$578$	0	0
$579$	27.4281	1.13987
$580$	0	0
$581$	−8.54920	−0.354680
$582$	0	0
$583$	9.24062	0.382707
$584$	0	0
$585$	−2.17380	−0.0898755
$586$	0	0
$587$	3.22888	0.133270	0.0666351	−	0.997777i	$-0.478774\pi$
$587$	3.22888	0.133270	0.0666351	+	0.997777i	$0.478774\pi$
$588$	0	0
$589$	−28.8342	−1.18809
$590$	0	0
$591$	24.2877	0.999063
$592$	0	0
$593$	−35.4061	−1.45396	−0.726978	−	0.686661i	$-0.759076\pi$
$593$	−35.4061	−1.45396	−0.726978	+	0.686661i	$0.759076\pi$
$594$	0	0
$595$	1.60468	0.0657854
$596$	0	0
$597$	58.7254	2.40347
$598$	0	0
$599$	26.3900	1.07827	0.539133	−	0.842221i	$-0.318752\pi$
$599$	26.3900	1.07827	0.539133	+	0.842221i	$0.318752\pi$
$600$	0	0
$601$	37.4569	1.52790	0.763950	−	0.645275i	$-0.223257\pi$
$601$	37.4569	1.52790	0.763950	+	0.645275i	$0.223257\pi$
$602$	0	0
$603$	3.07646	0.125283
$604$	0	0
$605$	−1.73863	−0.0706853
$606$	0	0
$607$	−1.14347	−0.0464121	−0.0232060	−	0.999731i	$-0.507387\pi$
$607$	−1.14347	−0.0464121	−0.0232060	+	0.999731i	$0.507387\pi$
$608$	0	0
$609$	−4.19372	−0.169938
$610$	0	0
$611$	−2.01217	−0.0814036
$612$	0	0
$613$	34.9996	1.41362	0.706810	−	0.707403i	$-0.250133\pi$
$613$	34.9996	1.41362	0.706810	+	0.707403i	$0.250133\pi$
$614$	0	0
$615$	−3.21912	−0.129807
$616$	0	0
$617$	−24.2579	−0.976587	−0.488293	−	0.872680i	$-0.662380\pi$
$617$	−24.2579	−0.976587	−0.488293	+	0.872680i	$0.662380\pi$
$618$	0	0
$619$	23.1368	0.929946	0.464973	−	0.885325i	$-0.346064\pi$
$619$	23.1368	0.929946	0.464973	+	0.885325i	$0.346064\pi$
$620$	0	0
$621$	−1.91672	−0.0769153
$622$	0	0
$623$	10.1913	0.408307
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	40.1412	1.60308
$628$	0	0
$629$	−15.1046	−0.602259
$630$	0	0
$631$	29.7623	1.18482	0.592410	−	0.805637i	$-0.298177\pi$
$631$	29.7623	1.18482	0.592410	+	0.805637i	$0.298177\pi$
$632$	0	0
$633$	6.54679	0.260212
$634$	0	0
$635$	4.63248	0.183834
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−19.0225	−0.752517
$640$	0	0
$641$	−9.86281	−0.389558	−0.194779	−	0.980847i	$-0.562399\pi$
$641$	−9.86281	−0.389558	−0.194779	+	0.980847i	$0.562399\pi$
$642$	0	0
$643$	−23.3554	−0.921045	−0.460523	−	0.887648i	$-0.652338\pi$
$643$	−23.3554	−0.921045	−0.460523	+	0.887648i	$0.652338\pi$
$644$	0	0
$645$	−8.54920	−0.336624
$646$	0	0
$647$	6.68997	0.263010	0.131505	−	0.991316i	$-0.458019\pi$
$647$	6.68997	0.263010	0.131505	+	0.991316i	$0.458019\pi$
$648$	0	0
$649$	−17.1733	−0.674110
$650$	0	0
$651$	13.2644	0.519874
$652$	0	0
$653$	−1.15375	−0.0451499	−0.0225749	−	0.999745i	$-0.507186\pi$
$653$	−1.15375	−0.0451499	−0.0225749	+	0.999745i	$0.507186\pi$
$654$	0	0
$655$	−9.09839	−0.355504
$656$	0	0
$657$	0.498406	0.0194447
$658$	0	0
$659$	12.2600	0.477583	0.238791	−	0.971071i	$-0.423249\pi$
$659$	12.2600	0.477583	0.238791	+	0.971071i	$0.423249\pi$
$660$	0	0
$661$	−50.5667	−1.96682	−0.983409	−	0.181402i	$-0.941937\pi$
$661$	−50.5667	−1.96682	−0.983409	+	0.181402i	$0.941937\pi$
$662$	0	0
$663$	3.65000	0.141754
$664$	0	0
$665$	−4.94452	−0.191740
$666$	0	0
$667$	−1.88045	−0.0728115
$668$	0	0
$669$	−35.7346	−1.38158
$670$	0	0
$671$	−30.4697	−1.17627
$672$	0	0
$673$	25.7766	0.993615	0.496808	−	0.867861i	$-0.334505\pi$
$673$	25.7766	0.993615	0.496808	+	0.867861i	$0.334505\pi$
$674$	0	0
$675$	1.87928	0.0723334
$676$	0	0
$677$	−32.5079	−1.24938	−0.624690	−	0.780873i	$-0.714775\pi$
$677$	−32.5079	−1.24938	−0.624690	+	0.780873i	$0.714775\pi$
$678$	0	0
$679$	7.67527	0.294550
$680$	0	0
$681$	36.5444	1.40038
$682$	0	0
$683$	26.8056	1.02569	0.512845	−	0.858481i	$-0.328592\pi$
$683$	26.8056	1.02569	0.512845	+	0.858481i	$0.328592\pi$
$684$	0	0
$685$	18.1515	0.693532
$686$	0	0
$687$	12.4406	0.474640
$688$	0	0
$689$	−2.58905	−0.0986348
$690$	0	0
$691$	−26.8662	−1.02204	−0.511019	−	0.859569i	$-0.670732\pi$
$691$	−26.8662	−1.02204	−0.511019	+	0.859569i	$0.670732\pi$
$692$	0	0
$693$	−7.75855	−0.294723
$694$	0	0
$695$	2.28130	0.0865345
$696$	0	0
$697$	2.27101	0.0860208
$698$	0	0
$699$	−7.47848	−0.282862
$700$	0	0
$701$	−37.2560	−1.40714	−0.703571	−	0.710625i	$-0.748412\pi$
$701$	−37.2560	−1.40714	−0.703571	+	0.710625i	$0.748412\pi$
$702$	0	0
$703$	46.5420	1.75536
$704$	0	0
$705$	−4.57688	−0.172375
$706$	0	0
$707$	19.0550	0.716636
$708$	0	0
$709$	24.3044	0.912771	0.456386	−	0.889782i	$-0.349144\pi$
$709$	24.3044	0.912771	0.456386	+	0.889782i	$0.349144\pi$
$710$	0	0
$711$	12.9655	0.486244
$712$	0	0
$713$	5.94774	0.222745
$714$	0	0
$715$	3.56912	0.133478
$716$	0	0
$717$	55.2193	2.06220
$718$	0	0
$719$	35.5872	1.32718	0.663589	−	0.748097i	$-0.269032\pi$
$719$	35.5872	1.32718	0.663589	+	0.748097i	$0.269032\pi$
$720$	0	0
$721$	−14.0828	−0.524469
$722$	0	0
$723$	−52.9449	−1.96904
$724$	0	0
$725$	1.84372	0.0684740
$726$	0	0
$727$	28.6052	1.06091	0.530454	−	0.847713i	$-0.322021\pi$
$727$	28.6052	1.06091	0.530454	+	0.847713i	$0.322021\pi$
$728$	0	0
$729$	−10.6445	−0.394242
$730$	0	0
$731$	6.03126	0.223074
$732$	0	0
$733$	0.447858	0.0165420	0.00827101	−	0.999966i	$-0.497367\pi$
$733$	0.447858	0.0165420	0.00827101	+	0.999966i	$0.497367\pi$
$734$	0	0
$735$	2.27460	0.0838998
$736$	0	0
$737$	−5.05119	−0.186063
$738$	0	0
$739$	−25.3594	−0.932859	−0.466430	−	0.884558i	$-0.654460\pi$
$739$	−25.3594	−0.932859	−0.466430	+	0.884558i	$0.654460\pi$
$740$	0	0
$741$	−11.2468	−0.413161
$742$	0	0
$743$	−9.74209	−0.357403	−0.178701	−	0.983903i	$-0.557190\pi$
$743$	−9.74209	−0.357403	−0.178701	+	0.983903i	$0.557190\pi$
$744$	0	0
$745$	13.9167	0.509869
$746$	0	0
$747$	−18.5842	−0.679962
$748$	0	0
$749$	−12.8569	−0.469783
$750$	0	0
$751$	−19.7711	−0.721459	−0.360730	−	0.932670i	$-0.617472\pi$
$751$	−19.7711	−0.721459	−0.360730	+	0.932670i	$0.617472\pi$
$752$	0	0
$753$	20.9753	0.764381
$754$	0	0
$755$	−2.69520	−0.0980882
$756$	0	0
$757$	−32.2281	−1.17135	−0.585674	−	0.810547i	$-0.699170\pi$
$757$	−32.2281	−1.17135	−0.585674	+	0.810547i	$0.699170\pi$
$758$	0	0
$759$	−8.28007	−0.300547
$760$	0	0
$761$	3.98260	0.144369	0.0721846	−	0.997391i	$-0.477003\pi$
$761$	3.98260	0.144369	0.0721846	+	0.997391i	$0.477003\pi$
$762$	0	0
$763$	1.55695	0.0563655
$764$	0	0
$765$	3.48825	0.126118
$766$	0	0
$767$	4.81163	0.173738
$768$	0	0
$769$	28.2509	1.01875	0.509377	−	0.860543i	$-0.329876\pi$
$769$	28.2509	1.01875	0.509377	+	0.860543i	$0.329876\pi$
$770$	0	0
$771$	−9.08745	−0.327277
$772$	0	0
$773$	−35.4762	−1.27599	−0.637995	−	0.770040i	$-0.720236\pi$
$773$	−35.4762	−1.27599	−0.637995	+	0.770040i	$0.720236\pi$
$774$	0	0
$775$	−5.83155	−0.209475
$776$	0	0
$777$	−21.4104	−0.768095
$778$	0	0
$779$	−6.99772	−0.250719
$780$	0	0
$781$	31.2327	1.11759
$782$	0	0
$783$	3.46486	0.123824
$784$	0	0
$785$	21.6495	0.772703
$786$	0	0
$787$	−6.30082	−0.224600	−0.112300	−	0.993674i	$-0.535822\pi$
$787$	−6.30082	−0.224600	−0.112300	+	0.993674i	$0.535822\pi$
$788$	0	0
$789$	25.9749	0.924730
$790$	0	0
$791$	−3.98783	−0.141791
$792$	0	0
$793$	8.53703	0.303159
$794$	0	0
$795$	−5.88904	−0.208863
$796$	0	0
$797$	−2.11766	−0.0750113	−0.0375057	−	0.999296i	$-0.511941\pi$
$797$	−2.11766	−0.0750113	−0.0375057	+	0.999296i	$0.511941\pi$
$798$	0	0
$799$	3.22888	0.114230
$800$	0	0
$801$	22.1539	0.782769
$802$	0	0
$803$	−0.818325	−0.0288781
$804$	0	0
$805$	1.01992	0.0359476
$806$	0	0
$807$	51.5486	1.81460
$808$	0	0
$809$	−37.9584	−1.33455	−0.667273	−	0.744813i	$-0.732539\pi$
$809$	−37.9584	−1.33455	−0.667273	+	0.744813i	$0.732539\pi$
$810$	0	0
$811$	−35.1179	−1.23316	−0.616579	−	0.787293i	$-0.711482\pi$
$811$	−35.1179	−1.23316	−0.616579	+	0.787293i	$0.711482\pi$
$812$	0	0
$813$	44.8330	1.57236
$814$	0	0
$815$	1.73757	0.0608644
$816$	0	0
$817$	−18.5842	−0.650180
$818$	0	0
$819$	2.17380	0.0759587
$820$	0	0
$821$	26.3896	0.921002	0.460501	−	0.887659i	$-0.347670\pi$
$821$	26.3896	0.921002	0.460501	+	0.887659i	$0.347670\pi$
$822$	0	0
$823$	−46.7181	−1.62849	−0.814245	−	0.580522i	$-0.802849\pi$
$823$	−46.7181	−1.62849	−0.814245	+	0.580522i	$0.802849\pi$
$824$	0	0
$825$	8.11832	0.282644
$826$	0	0
$827$	11.3749	0.395543	0.197772	−	0.980248i	$-0.436630\pi$
$827$	11.3749	0.395543	0.197772	+	0.980248i	$0.436630\pi$
$828$	0	0
$829$	21.7829	0.756551	0.378276	−	0.925693i	$-0.376517\pi$
$829$	21.7829	0.756551	0.378276	+	0.925693i	$0.376517\pi$
$830$	0	0
$831$	66.7808	2.31660
$832$	0	0
$833$	−1.60468	−0.0555988
$834$	0	0
$835$	18.4139	0.637239
$836$	0	0
$837$	−10.9591	−0.378802
$838$	0	0
$839$	−34.2712	−1.18317	−0.591587	−	0.806241i	$-0.701499\pi$
$839$	−34.2712	−1.18317	−0.591587	+	0.806241i	$0.701499\pi$
$840$	0	0
$841$	−25.6007	−0.882783
$842$	0	0
$843$	48.9318	1.68530
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	1.73863	0.0597400
$848$	0	0
$849$	−34.3526	−1.17898
$850$	0	0
$851$	−9.60038	−0.329097
$852$	0	0
$853$	−29.9193	−1.02442	−0.512208	−	0.858861i	$-0.671172\pi$
$853$	−29.9193	−1.02442	−0.512208	+	0.858861i	$0.671172\pi$
$854$	0	0
$855$	−10.7484	−0.367587
$856$	0	0
$857$	28.5079	0.973811	0.486906	−	0.873455i	$-0.338126\pi$
$857$	28.5079	0.973811	0.486906	+	0.873455i	$0.338126\pi$
$858$	0	0
$859$	24.5477	0.837558	0.418779	−	0.908088i	$-0.362458\pi$
$859$	24.5477	0.837558	0.418779	+	0.908088i	$0.362458\pi$
$860$	0	0
$861$	3.21912	0.109707
$862$	0	0
$863$	9.09557	0.309617	0.154808	−	0.987945i	$-0.450524\pi$
$863$	9.09557	0.309617	0.154808	+	0.987945i	$0.450524\pi$
$864$	0	0
$865$	0.882739	0.0300140
$866$	0	0
$867$	32.8111	1.11432
$868$	0	0
$869$	−21.2878	−0.722140
$870$	0	0
$871$	1.41525	0.0479538
$872$	0	0
$873$	16.6845	0.564685
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	53.8858	1.81960	0.909798	−	0.415052i	$-0.136237\pi$
$877$	53.8858	1.81960	0.909798	+	0.415052i	$0.136237\pi$
$878$	0	0
$879$	−51.9904	−1.75359
$880$	0	0
$881$	−12.7216	−0.428603	−0.214302	−	0.976768i	$-0.568748\pi$
$881$	−12.7216	−0.428603	−0.214302	+	0.976768i	$0.568748\pi$
$882$	0	0
$883$	−12.4647	−0.419470	−0.209735	−	0.977758i	$-0.567260\pi$
$883$	−12.4647	−0.419470	−0.209735	+	0.977758i	$0.567260\pi$
$884$	0	0
$885$	10.9445	0.367896
$886$	0	0
$887$	46.6488	1.56631	0.783157	−	0.621824i	$-0.213608\pi$
$887$	46.6488	1.56631	0.783157	+	0.621824i	$0.213608\pi$
$888$	0	0
$889$	−4.63248	−0.155368
$890$	0	0
$891$	38.5322	1.29088
$892$	0	0
$893$	−9.94921	−0.332938
$894$	0	0
$895$	−10.6320	−0.355387
$896$	0	0
$897$	2.31992	0.0774598
$898$	0	0
$899$	−10.7517	−0.358591
$900$	0	0
$901$	4.15458	0.138409
$902$	0	0
$903$	8.54920	0.284499
$904$	0	0
$905$	−13.3520	−0.443836
$906$	0	0
$907$	0.848362	0.0281694	0.0140847	−	0.999901i	$-0.495517\pi$
$907$	0.848362	0.0281694	0.0140847	+	0.999901i	$0.495517\pi$
$908$	0	0
$909$	41.4217	1.37387
$910$	0	0
$911$	17.2726	0.572266	0.286133	−	0.958190i	$-0.407630\pi$
$911$	17.2726	0.572266	0.286133	+	0.958190i	$0.407630\pi$
$912$	0	0
$913$	30.5131	1.00984
$914$	0	0
$915$	19.4183	0.641950
$916$	0	0
$917$	9.09839	0.300455
$918$	0	0
$919$	0.879155	0.0290006	0.0145003	−	0.999895i	$-0.495384\pi$
$919$	0.879155	0.0290006	0.0145003	+	0.999895i	$0.495384\pi$
$920$	0	0
$921$	61.3856	2.02272
$922$	0	0
$923$	−8.75080	−0.288036
$924$	0	0
$925$	9.41284	0.309492
$926$	0	0
$927$	−30.6131	−1.00547
$928$	0	0
$929$	−28.3896	−0.931432	−0.465716	−	0.884934i	$-0.654203\pi$
$929$	−28.3896	−0.931432	−0.465716	+	0.884934i	$0.654203\pi$
$930$	0	0
$931$	4.94452	0.162050
$932$	0	0
$933$	29.3952	0.962356
$934$	0	0
$935$	−5.72729	−0.187302
$936$	0	0
$937$	−17.7129	−0.578657	−0.289328	−	0.957230i	$-0.593432\pi$
$937$	−17.7129	−0.578657	−0.289328	+	0.957230i	$0.593432\pi$
$938$	0	0
$939$	12.1889	0.397770
$940$	0	0
$941$	−17.9427	−0.584914	−0.292457	−	0.956279i	$-0.594473\pi$
$941$	−17.9427	−0.584914	−0.292457	+	0.956279i	$0.594473\pi$
$942$	0	0
$943$	1.44344	0.0470050
$944$	0	0
$945$	−1.87928	−0.0611329
$946$	0	0
$947$	22.2493	0.723006	0.361503	−	0.932371i	$-0.382264\pi$
$947$	22.2493	0.723006	0.361503	+	0.932371i	$0.382264\pi$
$948$	0	0
$949$	0.229279	0.00744271
$950$	0	0
$951$	−37.6595	−1.22119
$952$	0	0
$953$	−6.17515	−0.200033	−0.100016	−	0.994986i	$-0.531889\pi$
$953$	−6.17515	−0.200033	−0.100016	+	0.994986i	$0.531889\pi$
$954$	0	0
$955$	10.5702	0.342043
$956$	0	0
$957$	14.9679	0.483844
$958$	0	0
$959$	−18.1515	−0.586142
$960$	0	0
$961$	3.00699	0.0969996
$962$	0	0
$963$	−27.9484	−0.900625
$964$	0	0
$965$	12.0584	0.388174
$966$	0	0
$967$	18.2734	0.587632	0.293816	−	0.955862i	$-0.405075\pi$
$967$	18.2734	0.587632	0.293816	+	0.955862i	$0.405075\pi$
$968$	0	0
$969$	18.0475	0.579769
$970$	0	0
$971$	11.9967	0.384991	0.192496	−	0.981298i	$-0.438342\pi$
$971$	11.9967	0.384991	0.192496	+	0.981298i	$0.438342\pi$
$972$	0	0
$973$	−2.28130	−0.0731350
$974$	0	0
$975$	−2.27460	−0.0728455
$976$	0	0
$977$	−46.0601	−1.47359	−0.736796	−	0.676115i	$-0.763662\pi$
$977$	−46.0601	−1.47359	−0.736796	+	0.676115i	$0.763662\pi$
$978$	0	0
$979$	−36.3741	−1.16252
$980$	0	0
$981$	3.38450	0.108059
$982$	0	0
$983$	−4.25003	−0.135555	−0.0677775	−	0.997700i	$-0.521591\pi$
$983$	−4.25003	−0.135555	−0.0677775	+	0.997700i	$0.521591\pi$
$984$	0	0
$985$	10.6778	0.340223
$986$	0	0
$987$	4.57688	0.145684
$988$	0	0
$989$	3.83344	0.121896
$990$	0	0
$991$	8.83503	0.280654	0.140327	−	0.990105i	$-0.455185\pi$
$991$	8.83503	0.280654	0.140327	+	0.990105i	$0.455185\pi$
$992$	0	0
$993$	11.3277	0.359473
$994$	0	0
$995$	25.8179	0.818483
$996$	0	0
$997$	−51.7377	−1.63855	−0.819275	−	0.573401i	$-0.805624\pi$
$997$	−51.7377	−1.63855	−0.819275	+	0.573401i	$0.805624\pi$
$998$	0	0
$999$	17.6893	0.559666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.t.1.1	✓	4
4.3	odd	2		7280.2.a.bv.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.t.1.1	✓	4	1.1	even	1	trivial
7280.2.a.bv.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27460 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.17380 q^{9} +O(q^{10})\)	\(q-2.27460 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.17380 q^{9} -3.56912 q^{11} +1.00000 q^{13} +2.27460 q^{15} -1.60468 q^{17} +4.94452 q^{19} -2.27460 q^{21} -1.01992 q^{23} +1.00000 q^{25} +1.87928 q^{27} +1.84372 q^{29} -5.83155 q^{31} +8.11832 q^{33} -1.00000 q^{35} +9.41284 q^{37} -2.27460 q^{39} -1.41525 q^{41} -3.75855 q^{43} -2.17380 q^{45} -2.01217 q^{47} +1.00000 q^{49} +3.65000 q^{51} -2.58905 q^{53} +3.56912 q^{55} -11.2468 q^{57} +4.81163 q^{59} +8.53703 q^{61} +2.17380 q^{63} -1.00000 q^{65} +1.41525 q^{67} +2.31992 q^{69} -8.75080 q^{71} +0.229279 q^{73} -2.27460 q^{75} -3.56912 q^{77} +5.96444 q^{79} -10.7960 q^{81} -8.54920 q^{83} +1.60468 q^{85} -4.19372 q^{87} +10.1913 q^{89} +1.00000 q^{91} +13.2644 q^{93} -4.94452 q^{95} +7.67527 q^{97} -7.75855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 + 4 * q^7	\( 4 q - 4 q^{5} + 4 q^{7} - q^{11} + 4 q^{13} - 11 q^{17} - 3 q^{19} - 9 q^{23} + 4 q^{25} + 3 q^{27} - 15 q^{29} - 6 q^{31} + q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} - 3 q^{47} + 4 q^{49} - 4 q^{51} - 2 q^{53} + q^{55} - 28 q^{57} - 3 q^{59} + 21 q^{61} - 4 q^{65} + 6 q^{67} - 23 q^{69} - 16 q^{71} + 15 q^{73} - q^{77} + 6 q^{79} - 8 q^{81} - 16 q^{83} + 11 q^{85} - 13 q^{87} + q^{89} + 4 q^{91} - 2 q^{93} + 3 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^7 - q^11 + 4 * q^13 - 11 * q^17 - 3 * q^19 - 9 * q^23 + 4 * q^25 + 3 * q^27 - 15 * q^29 - 6 * q^31 + q^33 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 - 3 * q^47 + 4 * q^49 - 4 * q^51 - 2 * q^53 + q^55 - 28 * q^57 - 3 * q^59 + 21 * q^61 - 4 * q^65 + 6 * q^67 - 23 * q^69 - 16 * q^71 + 15 * q^73 - q^77 + 6 * q^79 - 8 * q^81 - 16 * q^83 + 11 * q^85 - 13 * q^87 + q^89 + 4 * q^91 - 2 * q^93 + 3 * q^95 - 9 * q^97 - 22 * q^99

Introduction

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Database

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