LMFDB - Embedded newform 3640.2.a.ba.1.5

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 33x^3 - 41x^2 - 31x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.436282$ 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)

$f(q)$	$=$	$q+1.43628 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.937093 q^{9} +O(q^{10})$	$q+1.43628 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.937093 q^{9} +2.45660 q^{11} +1.00000 q^{13} +1.43628 q^{15} +4.33865 q^{17} -6.88620 q^{19} +1.43628 q^{21} +6.07654 q^{23} +1.00000 q^{25} -5.65478 q^{27} -9.48180 q^{29} +5.26109 q^{31} +3.52837 q^{33} +1.00000 q^{35} +7.80863 q^{37} +1.43628 q^{39} +9.50703 q^{41} +9.43910 q^{43} -0.937093 q^{45} +10.0599 q^{47} +1.00000 q^{49} +6.23153 q^{51} +0.911613 q^{53} +2.45660 q^{55} -9.89053 q^{57} -13.1177 q^{59} -1.31318 q^{61} -0.937093 q^{63} +1.00000 q^{65} +0.840682 q^{67} +8.72763 q^{69} -1.73072 q^{71} -0.421623 q^{73} +1.43628 q^{75} +2.45660 q^{77} -0.101325 q^{79} -5.31058 q^{81} -17.7414 q^{83} +4.33865 q^{85} -13.6185 q^{87} +13.3066 q^{89} +1.00000 q^{91} +7.55641 q^{93} -6.88620 q^{95} +11.6408 q^{97} -2.30206 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * 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
$3$	1.43628	0.829238	0.414619	−	0.909995i	$-0.363915\pi$
$3$	1.43628	0.829238	0.414619	+	0.909995i	$0.363915\pi$
$4$	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
$9$	−0.937093	−0.312364
$10$	0	0
$11$	2.45660	0.740692	0.370346	−	0.928894i	$-0.379239\pi$
$11$	2.45660	0.740692	0.370346	+	0.928894i	$0.379239\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.43628	0.370847
$16$	0	0
$17$	4.33865	1.05228	0.526139	−	0.850399i	$-0.323639\pi$
$17$	4.33865	1.05228	0.526139	+	0.850399i	$0.323639\pi$
$18$	0	0
$19$	−6.88620	−1.57980	−0.789901	−	0.613234i	$-0.789868\pi$
$19$	−6.88620	−1.57980	−0.789901	+	0.613234i	$0.789868\pi$
$20$	0	0
$21$	1.43628	0.313423
$22$	0	0
$23$	6.07654	1.26705	0.633523	−	0.773724i	$-0.281608\pi$
$23$	6.07654	1.26705	0.633523	+	0.773724i	$0.281608\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65478	−1.08826
$28$	0	0
$29$	−9.48180	−1.76073	−0.880363	−	0.474300i	$-0.842701\pi$
$29$	−9.48180	−1.76073	−0.880363	+	0.474300i	$0.842701\pi$
$30$	0	0
$31$	5.26109	0.944919	0.472460	−	0.881352i	$-0.343366\pi$
$31$	5.26109	0.944919	0.472460	+	0.881352i	$0.343366\pi$
$32$	0	0
$33$	3.52837	0.614210
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	7.80863	1.28373	0.641866	−	0.766817i	$-0.278161\pi$
$37$	7.80863	1.28373	0.641866	+	0.766817i	$0.278161\pi$
$38$	0	0
$39$	1.43628	0.229989
$40$	0	0
$41$	9.50703	1.48475	0.742374	−	0.669986i	$-0.233700\pi$
$41$	9.50703	1.48475	0.742374	+	0.669986i	$0.233700\pi$
$42$	0	0
$43$	9.43910	1.43945	0.719724	−	0.694260i	$-0.244268\pi$
$43$	9.43910	1.43945	0.719724	+	0.694260i	$0.244268\pi$
$44$	0	0
$45$	−0.937093	−0.139694
$46$	0	0
$47$	10.0599	1.46739	0.733695	−	0.679478i	$-0.237794\pi$
$47$	10.0599	1.46739	0.733695	+	0.679478i	$0.237794\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.23153	0.872589
$52$	0	0
$53$	0.911613	0.125220	0.0626099	−	0.998038i	$-0.480058\pi$
$53$	0.911613	0.125220	0.0626099	+	0.998038i	$0.480058\pi$
$54$	0	0
$55$	2.45660	0.331247
$56$	0	0
$57$	−9.89053	−1.31003
$58$	0	0
$59$	−13.1177	−1.70778	−0.853892	−	0.520451i	$-0.825764\pi$
$59$	−13.1177	−1.70778	−0.853892	+	0.520451i	$0.825764\pi$
$60$	0	0
$61$	−1.31318	−0.168135	−0.0840674	−	0.996460i	$-0.526791\pi$
$61$	−1.31318	−0.168135	−0.0840674	+	0.996460i	$0.526791\pi$
$62$	0	0
$63$	−0.937093	−0.118063
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	0.840682	0.102706	0.0513528	−	0.998681i	$-0.483647\pi$
$67$	0.840682	0.102706	0.0513528	+	0.998681i	$0.483647\pi$
$68$	0	0
$69$	8.72763	1.05068
$70$	0	0
$71$	−1.73072	−0.205399	−0.102699	−	0.994712i	$-0.532748\pi$
$71$	−1.73072	−0.205399	−0.102699	+	0.994712i	$0.532748\pi$
$72$	0	0
$73$	−0.421623	−0.0493473	−0.0246736	−	0.999696i	$-0.507855\pi$
$73$	−0.421623	−0.0493473	−0.0246736	+	0.999696i	$0.507855\pi$
$74$	0	0
$75$	1.43628	0.165848
$76$	0	0
$77$	2.45660	0.279955
$78$	0	0
$79$	−0.101325	−0.0113999	−0.00569996	−	0.999984i	$-0.501814\pi$
$79$	−0.101325	−0.0113999	−0.00569996	+	0.999984i	$0.501814\pi$
$80$	0	0
$81$	−5.31058	−0.590064
$82$	0	0
$83$	−17.7414	−1.94737	−0.973685	−	0.227900i	$-0.926814\pi$
$83$	−17.7414	−1.94737	−0.973685	+	0.227900i	$0.926814\pi$
$84$	0	0
$85$	4.33865	0.470593
$86$	0	0
$87$	−13.6185	−1.46006
$88$	0	0
$89$	13.3066	1.41050	0.705249	−	0.708960i	$-0.250835\pi$
$89$	13.3066	1.41050	0.705249	+	0.708960i	$0.250835\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	7.55641	0.783563
$94$	0	0
$95$	−6.88620	−0.706509
$96$	0	0
$97$	11.6408	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$97$	11.6408	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$98$	0	0
$99$	−2.30206	−0.231366
$100$	0	0
$101$	−15.4162	−1.53397	−0.766987	−	0.641663i	$-0.778245\pi$
$101$	−15.4162	−1.53397	−0.766987	+	0.641663i	$0.778245\pi$
$102$	0	0
$103$	11.8488	1.16749	0.583747	−	0.811936i	$-0.301586\pi$
$103$	11.8488	1.16749	0.583747	+	0.811936i	$0.301586\pi$
$104$	0	0
$105$	1.43628	0.140167
$106$	0	0
$107$	18.2306	1.76242	0.881211	−	0.472724i	$-0.156729\pi$
$107$	18.2306	1.76242	0.881211	+	0.472724i	$0.156729\pi$
$108$	0	0
$109$	−5.52516	−0.529214	−0.264607	−	0.964356i	$-0.585242\pi$
$109$	−5.52516	−0.529214	−0.264607	+	0.964356i	$0.585242\pi$
$110$	0	0
$111$	11.2154	1.06452
$112$	0	0
$113$	2.44061	0.229593	0.114797	−	0.993389i	$-0.463378\pi$
$113$	2.44061	0.229593	0.114797	+	0.993389i	$0.463378\pi$
$114$	0	0
$115$	6.07654	0.566641
$116$	0	0
$117$	−0.937093	−0.0866343
$118$	0	0
$119$	4.33865	0.397724
$120$	0	0
$121$	−4.96514	−0.451376
$122$	0	0
$123$	13.6548	1.23121
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.18661	0.194030	0.0970152	−	0.995283i	$-0.469070\pi$
$127$	2.18661	0.194030	0.0970152	+	0.995283i	$0.469070\pi$
$128$	0	0
$129$	13.5572	1.19365
$130$	0	0
$131$	−3.44475	−0.300969	−0.150485	−	0.988612i	$-0.548083\pi$
$131$	−3.44475	−0.300969	−0.150485	+	0.988612i	$0.548083\pi$
$132$	0	0
$133$	−6.88620	−0.597109
$134$	0	0
$135$	−5.65478	−0.486686
$136$	0	0
$137$	−0.197129	−0.0168419	−0.00842093	−	0.999965i	$-0.502680\pi$
$137$	−0.197129	−0.0168419	−0.00842093	+	0.999965i	$0.502680\pi$
$138$	0	0
$139$	22.8688	1.93971	0.969854	−	0.243688i	$-0.0783573\pi$
$139$	22.8688	1.93971	0.969854	+	0.243688i	$0.0783573\pi$
$140$	0	0
$141$	14.4489	1.21682
$142$	0	0
$143$	2.45660	0.205431
$144$	0	0
$145$	−9.48180	−0.787421
$146$	0	0
$147$	1.43628	0.118463
$148$	0	0
$149$	−5.13798	−0.420920	−0.210460	−	0.977602i	$-0.567496\pi$
$149$	−5.13798	−0.420920	−0.210460	+	0.977602i	$0.567496\pi$
$150$	0	0
$151$	20.5371	1.67128	0.835642	−	0.549274i	$-0.185096\pi$
$151$	20.5371	1.67128	0.835642	+	0.549274i	$0.185096\pi$
$152$	0	0
$153$	−4.06572	−0.328694
$154$	0	0
$155$	5.26109	0.422581
$156$	0	0
$157$	−7.32709	−0.584765	−0.292383	−	0.956301i	$-0.594448\pi$
$157$	−7.32709	−0.584765	−0.292383	+	0.956301i	$0.594448\pi$
$158$	0	0
$159$	1.30933	0.103837
$160$	0	0
$161$	6.07654	0.478899
$162$	0	0
$163$	−15.2769	−1.19658	−0.598291	−	0.801279i	$-0.704153\pi$
$163$	−15.2769	−1.19658	−0.598291	+	0.801279i	$0.704153\pi$
$164$	0	0
$165$	3.52837	0.274683
$166$	0	0
$167$	−20.4204	−1.58018	−0.790089	−	0.612993i	$-0.789966\pi$
$167$	−20.4204	−1.58018	−0.790089	+	0.612993i	$0.789966\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.45301	0.493474
$172$	0	0
$173$	2.75033	0.209104	0.104552	−	0.994519i	$-0.466659\pi$
$173$	2.75033	0.209104	0.104552	+	0.994519i	$0.466659\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−18.8408	−1.41616
$178$	0	0
$179$	8.02804	0.600044	0.300022	−	0.953932i	$-0.403006\pi$
$179$	8.02804	0.600044	0.300022	+	0.953932i	$0.403006\pi$
$180$	0	0
$181$	−2.14096	−0.159136	−0.0795682	−	0.996829i	$-0.525354\pi$
$181$	−2.14096	−0.159136	−0.0795682	+	0.996829i	$0.525354\pi$
$182$	0	0
$183$	−1.88609	−0.139424
$184$	0	0
$185$	7.80863	0.574102
$186$	0	0
$187$	10.6583	0.779414
$188$	0	0
$189$	−5.65478	−0.411325
$190$	0	0
$191$	−12.7818	−0.924855	−0.462428	−	0.886657i	$-0.653021\pi$
$191$	−12.7818	−0.924855	−0.462428	+	0.886657i	$0.653021\pi$
$192$	0	0
$193$	−24.0095	−1.72824	−0.864119	−	0.503287i	$-0.832124\pi$
$193$	−24.0095	−1.72824	−0.864119	+	0.503287i	$0.832124\pi$
$194$	0	0
$195$	1.43628	0.102854
$196$	0	0
$197$	7.18599	0.511981	0.255990	−	0.966679i	$-0.417598\pi$
$197$	7.18599	0.511981	0.255990	+	0.966679i	$0.417598\pi$
$198$	0	0
$199$	−25.6507	−1.81833	−0.909166	−	0.416433i	$-0.863280\pi$
$199$	−25.6507	−1.81833	−0.909166	+	0.416433i	$0.863280\pi$
$200$	0	0
$201$	1.20746	0.0851674
$202$	0	0
$203$	−9.48180	−0.665492
$204$	0	0
$205$	9.50703	0.664000
$206$	0	0
$207$	−5.69428	−0.395780
$208$	0	0
$209$	−16.9166	−1.17015
$210$	0	0
$211$	−6.98601	−0.480937	−0.240469	−	0.970657i	$-0.577301\pi$
$211$	−6.98601	−0.480937	−0.240469	+	0.970657i	$0.577301\pi$
$212$	0	0
$213$	−2.48581	−0.170325
$214$	0	0
$215$	9.43910	0.643741
$216$	0	0
$217$	5.26109	0.357146
$218$	0	0
$219$	−0.605570	−0.0409206
$220$	0	0
$221$	4.33865	0.291849
$222$	0	0
$223$	24.4720	1.63877	0.819384	−	0.573245i	$-0.194316\pi$
$223$	24.4720	1.63877	0.819384	+	0.573245i	$0.194316\pi$
$224$	0	0
$225$	−0.937093	−0.0624729
$226$	0	0
$227$	8.70833	0.577992	0.288996	−	0.957330i	$-0.406679\pi$
$227$	8.70833	0.577992	0.288996	+	0.957330i	$0.406679\pi$
$228$	0	0
$229$	3.21509	0.212459	0.106230	−	0.994342i	$-0.466122\pi$
$229$	3.21509	0.212459	0.106230	+	0.994342i	$0.466122\pi$
$230$	0	0
$231$	3.52837	0.232149
$232$	0	0
$233$	−14.8283	−0.971435	−0.485718	−	0.874116i	$-0.661442\pi$
$233$	−14.8283	−0.971435	−0.485718	+	0.874116i	$0.661442\pi$
$234$	0	0
$235$	10.0599	0.656237
$236$	0	0
$237$	−0.145531	−0.00945325
$238$	0	0
$239$	9.35454	0.605095	0.302548	−	0.953134i	$-0.402163\pi$
$239$	9.35454	0.605095	0.302548	+	0.953134i	$0.402163\pi$
$240$	0	0
$241$	−11.8289	−0.761967	−0.380984	−	0.924582i	$-0.624415\pi$
$241$	−11.8289	−0.761967	−0.380984	+	0.924582i	$0.624415\pi$
$242$	0	0
$243$	9.33684	0.598959
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−6.88620	−0.438158
$248$	0	0
$249$	−25.4816	−1.61483
$250$	0	0
$251$	12.0058	0.757797	0.378899	−	0.925438i	$-0.376303\pi$
$251$	12.0058	0.757797	0.378899	+	0.925438i	$0.376303\pi$
$252$	0	0
$253$	14.9276	0.938491
$254$	0	0
$255$	6.23153	0.390234
$256$	0	0
$257$	19.0950	1.19111	0.595556	−	0.803314i	$-0.296932\pi$
$257$	19.0950	1.19111	0.595556	+	0.803314i	$0.296932\pi$
$258$	0	0
$259$	7.80863	0.485205
$260$	0	0
$261$	8.88533	0.549988
$262$	0	0
$263$	17.4885	1.07839	0.539193	−	0.842182i	$-0.318729\pi$
$263$	17.4885	1.07839	0.539193	+	0.842182i	$0.318729\pi$
$264$	0	0
$265$	0.911613	0.0560000
$266$	0	0
$267$	19.1120	1.16964
$268$	0	0
$269$	−6.96351	−0.424573	−0.212286	−	0.977207i	$-0.568091\pi$
$269$	−6.96351	−0.424573	−0.212286	+	0.977207i	$0.568091\pi$
$270$	0	0
$271$	−16.3196	−0.991347	−0.495674	−	0.868509i	$-0.665079\pi$
$271$	−16.3196	−0.991347	−0.495674	+	0.868509i	$0.665079\pi$
$272$	0	0
$273$	1.43628	0.0869278
$274$	0	0
$275$	2.45660	0.148138
$276$	0	0
$277$	−15.5483	−0.934206	−0.467103	−	0.884203i	$-0.654702\pi$
$277$	−15.5483	−0.934206	−0.467103	+	0.884203i	$0.654702\pi$
$278$	0	0
$279$	−4.93013	−0.295159
$280$	0	0
$281$	5.07106	0.302514	0.151257	−	0.988494i	$-0.451668\pi$
$281$	5.07106	0.302514	0.151257	+	0.988494i	$0.451668\pi$
$282$	0	0
$283$	9.69095	0.576067	0.288034	−	0.957620i	$-0.406998\pi$
$283$	9.69095	0.576067	0.288034	+	0.957620i	$0.406998\pi$
$284$	0	0
$285$	−9.89053	−0.585864
$286$	0	0
$287$	9.50703	0.561182
$288$	0	0
$289$	1.82392	0.107290
$290$	0	0
$291$	16.7195	0.980115
$292$	0	0
$293$	−18.4320	−1.07681	−0.538406	−	0.842686i	$-0.680973\pi$
$293$	−18.4320	−1.07681	−0.538406	+	0.842686i	$0.680973\pi$
$294$	0	0
$295$	−13.1177	−0.763744
$296$	0	0
$297$	−13.8915	−0.806067
$298$	0	0
$299$	6.07654	0.351416
$300$	0	0
$301$	9.43910	0.544061
$302$	0	0
$303$	−22.1421	−1.27203
$304$	0	0
$305$	−1.31318	−0.0751922
$306$	0	0
$307$	17.1899	0.981078	0.490539	−	0.871419i	$-0.336800\pi$
$307$	17.1899	0.981078	0.490539	+	0.871419i	$0.336800\pi$
$308$	0	0
$309$	17.0182	0.968131
$310$	0	0
$311$	−17.8537	−1.01239	−0.506194	−	0.862420i	$-0.668948\pi$
$311$	−17.8537	−1.01239	−0.506194	+	0.862420i	$0.668948\pi$
$312$	0	0
$313$	−13.7383	−0.776532	−0.388266	−	0.921547i	$-0.626926\pi$
$313$	−13.7383	−0.776532	−0.388266	+	0.921547i	$0.626926\pi$
$314$	0	0
$315$	−0.937093	−0.0527992
$316$	0	0
$317$	−17.6751	−0.992732	−0.496366	−	0.868113i	$-0.665333\pi$
$317$	−17.6751	−0.992732	−0.496366	+	0.868113i	$0.665333\pi$
$318$	0	0
$319$	−23.2930	−1.30416
$320$	0	0
$321$	26.1843	1.46147
$322$	0	0
$323$	−29.8768	−1.66239
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−7.93568	−0.438844
$328$	0	0
$329$	10.0599	0.554622
$330$	0	0
$331$	−10.4996	−0.577111	−0.288555	−	0.957463i	$-0.593175\pi$
$331$	−10.4996	−0.577111	−0.288555	+	0.957463i	$0.593175\pi$
$332$	0	0
$333$	−7.31742	−0.400992
$334$	0	0
$335$	0.840682	0.0459313
$336$	0	0
$337$	−33.7282	−1.83729	−0.918645	−	0.395085i	$-0.870715\pi$
$337$	−33.7282	−1.83729	−0.918645	+	0.395085i	$0.870715\pi$
$338$	0	0
$339$	3.50541	0.190387
$340$	0	0
$341$	12.9244	0.699894
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	8.72763	0.469880
$346$	0	0
$347$	21.6449	1.16196	0.580980	−	0.813918i	$-0.302670\pi$
$347$	21.6449	1.16196	0.580980	+	0.813918i	$0.302670\pi$
$348$	0	0
$349$	−14.7257	−0.788247	−0.394123	−	0.919058i	$-0.628952\pi$
$349$	−14.7257	−0.788247	−0.394123	+	0.919058i	$0.628952\pi$
$350$	0	0
$351$	−5.65478	−0.301830
$352$	0	0
$353$	32.3257	1.72052	0.860261	−	0.509853i	$-0.170300\pi$
$353$	32.3257	1.72052	0.860261	+	0.509853i	$0.170300\pi$
$354$	0	0
$355$	−1.73072	−0.0918572
$356$	0	0
$357$	6.23153	0.329808
$358$	0	0
$359$	−0.247553	−0.0130653	−0.00653267	−	0.999979i	$-0.502079\pi$
$359$	−0.247553	−0.0130653	−0.00653267	+	0.999979i	$0.502079\pi$
$360$	0	0
$361$	28.4198	1.49578
$362$	0	0
$363$	−7.13134	−0.374298
$364$	0	0
$365$	−0.421623	−0.0220688
$366$	0	0
$367$	22.8348	1.19197	0.595983	−	0.802997i	$-0.296763\pi$
$367$	22.8348	1.19197	0.595983	+	0.802997i	$0.296763\pi$
$368$	0	0
$369$	−8.90897	−0.463782
$370$	0	0
$371$	0.911613	0.0473286
$372$	0	0
$373$	−19.3237	−1.00054	−0.500272	−	0.865868i	$-0.666767\pi$
$373$	−19.3237	−1.00054	−0.500272	+	0.865868i	$0.666767\pi$
$374$	0	0
$375$	1.43628	0.0741693
$376$	0	0
$377$	−9.48180	−0.488338
$378$	0	0
$379$	−4.77820	−0.245440	−0.122720	−	0.992441i	$-0.539162\pi$
$379$	−4.77820	−0.245440	−0.122720	+	0.992441i	$0.539162\pi$
$380$	0	0
$381$	3.14059	0.160897
$382$	0	0
$383$	−1.84568	−0.0943100	−0.0471550	−	0.998888i	$-0.515015\pi$
$383$	−1.84568	−0.0943100	−0.0471550	+	0.998888i	$0.515015\pi$
$384$	0	0
$385$	2.45660	0.125200
$386$	0	0
$387$	−8.84531	−0.449632
$388$	0	0
$389$	14.1654	0.718213	0.359107	−	0.933297i	$-0.383081\pi$
$389$	14.1654	0.718213	0.359107	+	0.933297i	$0.383081\pi$
$390$	0	0
$391$	26.3640	1.33329
$392$	0	0
$393$	−4.94764	−0.249575
$394$	0	0
$395$	−0.101325	−0.00509820
$396$	0	0
$397$	8.61070	0.432159	0.216079	−	0.976376i	$-0.430673\pi$
$397$	8.61070	0.432159	0.216079	+	0.976376i	$0.430673\pi$
$398$	0	0
$399$	−9.89053	−0.495146
$400$	0	0
$401$	7.71934	0.385485	0.192743	−	0.981249i	$-0.438262\pi$
$401$	7.71934	0.385485	0.192743	+	0.981249i	$0.438262\pi$
$402$	0	0
$403$	5.26109	0.262073
$404$	0	0
$405$	−5.31058	−0.263885
$406$	0	0
$407$	19.1827	0.950849
$408$	0	0
$409$	−1.45969	−0.0721769	−0.0360885	−	0.999349i	$-0.511490\pi$
$409$	−1.45969	−0.0721769	−0.0360885	+	0.999349i	$0.511490\pi$
$410$	0	0
$411$	−0.283133	−0.0139659
$412$	0	0
$413$	−13.1177	−0.645481
$414$	0	0
$415$	−17.7414	−0.870890
$416$	0	0
$417$	32.8461	1.60848
$418$	0	0
$419$	0.0638988	0.00312166	0.00156083	−	0.999999i	$-0.499503\pi$
$419$	0.0638988	0.00312166	0.00156083	+	0.999999i	$0.499503\pi$
$420$	0	0
$421$	−32.8293	−1.60000	−0.800002	−	0.599997i	$-0.795168\pi$
$421$	−32.8293	−1.60000	−0.800002	+	0.599997i	$0.795168\pi$
$422$	0	0
$423$	−9.42708	−0.458361
$424$	0	0
$425$	4.33865	0.210456
$426$	0	0
$427$	−1.31318	−0.0635490
$428$	0	0
$429$	3.52837	0.170351
$430$	0	0
$431$	21.9343	1.05654	0.528268	−	0.849078i	$-0.322842\pi$
$431$	21.9343	1.05654	0.528268	+	0.849078i	$0.322842\pi$
$432$	0	0
$433$	−1.15485	−0.0554984	−0.0277492	−	0.999615i	$-0.508834\pi$
$433$	−1.15485	−0.0554984	−0.0277492	+	0.999615i	$0.508834\pi$
$434$	0	0
$435$	−13.6185	−0.652959
$436$	0	0
$437$	−41.8443	−2.00168
$438$	0	0
$439$	−16.6135	−0.792921	−0.396461	−	0.918052i	$-0.629762\pi$
$439$	−16.6135	−0.792921	−0.396461	+	0.918052i	$0.629762\pi$
$440$	0	0
$441$	−0.937093	−0.0446235
$442$	0	0
$443$	38.0229	1.80652	0.903261	−	0.429091i	$-0.141166\pi$
$443$	38.0229	1.80652	0.903261	+	0.429091i	$0.141166\pi$
$444$	0	0
$445$	13.3066	0.630794
$446$	0	0
$447$	−7.37959	−0.349043
$448$	0	0
$449$	33.5771	1.58460	0.792302	−	0.610129i	$-0.208883\pi$
$449$	33.5771	1.58460	0.792302	+	0.610129i	$0.208883\pi$
$450$	0	0
$451$	23.3549	1.09974
$452$	0	0
$453$	29.4971	1.38589
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−17.8034	−0.832807	−0.416403	−	0.909180i	$-0.636710\pi$
$457$	−17.8034	−0.832807	−0.416403	+	0.909180i	$0.636710\pi$
$458$	0	0
$459$	−24.5341	−1.14515
$460$	0	0
$461$	0.446548	0.0207978	0.0103989	−	0.999946i	$-0.496690\pi$
$461$	0.446548	0.0207978	0.0103989	+	0.999946i	$0.496690\pi$
$462$	0	0
$463$	−32.6706	−1.51833	−0.759165	−	0.650898i	$-0.774393\pi$
$463$	−32.6706	−1.51833	−0.759165	+	0.650898i	$0.774393\pi$
$464$	0	0
$465$	7.55641	0.350420
$466$	0	0
$467$	26.0324	1.20464	0.602319	−	0.798256i	$-0.294244\pi$
$467$	26.0324	1.20464	0.602319	+	0.798256i	$0.294244\pi$
$468$	0	0
$469$	0.840682	0.0388191
$470$	0	0
$471$	−10.5238	−0.484909
$472$	0	0
$473$	23.1881	1.06619
$474$	0	0
$475$	−6.88620	−0.315961
$476$	0	0
$477$	−0.854267	−0.0391142
$478$	0	0
$479$	−37.9523	−1.73408	−0.867042	−	0.498235i	$-0.833982\pi$
$479$	−37.9523	−1.73408	−0.867042	+	0.498235i	$0.833982\pi$
$480$	0	0
$481$	7.80863	0.356043
$482$	0	0
$483$	8.72763	0.397121
$484$	0	0
$485$	11.6408	0.528583
$486$	0	0
$487$	13.7893	0.624854	0.312427	−	0.949942i	$-0.398858\pi$
$487$	13.7893	0.624854	0.312427	+	0.949942i	$0.398858\pi$
$488$	0	0
$489$	−21.9420	−0.992252
$490$	0	0
$491$	3.85196	0.173836	0.0869182	−	0.996215i	$-0.472298\pi$
$491$	3.85196	0.173836	0.0869182	+	0.996215i	$0.472298\pi$
$492$	0	0
$493$	−41.1383	−1.85277
$494$	0	0
$495$	−2.30206	−0.103470
$496$	0	0
$497$	−1.73072	−0.0776335
$498$	0	0
$499$	−6.23504	−0.279119	−0.139559	−	0.990214i	$-0.544569\pi$
$499$	−6.23504	−0.279119	−0.139559	+	0.990214i	$0.544569\pi$
$500$	0	0
$501$	−29.3294	−1.31034
$502$	0	0
$503$	26.7247	1.19160	0.595798	−	0.803134i	$-0.296836\pi$
$503$	26.7247	1.19160	0.595798	+	0.803134i	$0.296836\pi$
$504$	0	0
$505$	−15.4162	−0.686014
$506$	0	0
$507$	1.43628	0.0637875
$508$	0	0
$509$	−36.6308	−1.62363	−0.811817	−	0.583912i	$-0.801521\pi$
$509$	−36.6308	−1.62363	−0.811817	+	0.583912i	$0.801521\pi$
$510$	0	0
$511$	−0.421623	−0.0186515
$512$	0	0
$513$	38.9399	1.71924
$514$	0	0
$515$	11.8488	0.522119
$516$	0	0
$517$	24.7132	1.08688
$518$	0	0
$519$	3.95025	0.173397
$520$	0	0
$521$	−29.6797	−1.30029	−0.650146	−	0.759809i	$-0.725292\pi$
$521$	−29.6797	−1.30029	−0.650146	+	0.759809i	$0.725292\pi$
$522$	0	0
$523$	−17.4861	−0.764615	−0.382307	−	0.924035i	$-0.624870\pi$
$523$	−17.4861	−0.764615	−0.382307	+	0.924035i	$0.624870\pi$
$524$	0	0
$525$	1.43628	0.0626845
$526$	0	0
$527$	22.8260	0.994318
$528$	0	0
$529$	13.9244	0.605407
$530$	0	0
$531$	12.2925	0.533451
$532$	0	0
$533$	9.50703	0.411795
$534$	0	0
$535$	18.2306	0.788179
$536$	0	0
$537$	11.5305	0.497579
$538$	0	0
$539$	2.45660	0.105813
$540$	0	0
$541$	24.9919	1.07449	0.537244	−	0.843427i	$-0.319465\pi$
$541$	24.9919	1.07449	0.537244	+	0.843427i	$0.319465\pi$
$542$	0	0
$543$	−3.07502	−0.131962
$544$	0	0
$545$	−5.52516	−0.236672
$546$	0	0
$547$	−21.2845	−0.910058	−0.455029	−	0.890477i	$-0.650371\pi$
$547$	−21.2845	−0.910058	−0.455029	+	0.890477i	$0.650371\pi$
$548$	0	0
$549$	1.23057	0.0525193
$550$	0	0
$551$	65.2936	2.78160
$552$	0	0
$553$	−0.101325	−0.00430877
$554$	0	0
$555$	11.2154	0.476067
$556$	0	0
$557$	−39.6410	−1.67964	−0.839822	−	0.542863i	$-0.817340\pi$
$557$	−39.6410	−1.67964	−0.839822	+	0.542863i	$0.817340\pi$
$558$	0	0
$559$	9.43910	0.399231
$560$	0	0
$561$	15.3084	0.646319
$562$	0	0
$563$	0.763279	0.0321684	0.0160842	−	0.999871i	$-0.494880\pi$
$563$	0.763279	0.0321684	0.0160842	+	0.999871i	$0.494880\pi$
$564$	0	0
$565$	2.44061	0.102677
$566$	0	0
$567$	−5.31058	−0.223023
$568$	0	0
$569$	2.26717	0.0950447	0.0475223	−	0.998870i	$-0.484867\pi$
$569$	2.26717	0.0950447	0.0475223	+	0.998870i	$0.484867\pi$
$570$	0	0
$571$	−22.6503	−0.947884	−0.473942	−	0.880556i	$-0.657169\pi$
$571$	−22.6503	−0.947884	−0.473942	+	0.880556i	$0.657169\pi$
$572$	0	0
$573$	−18.3582	−0.766925
$574$	0	0
$575$	6.07654	0.253409
$576$	0	0
$577$	−45.5916	−1.89800	−0.949002	−	0.315270i	$-0.897905\pi$
$577$	−45.5916	−1.89800	−0.949002	+	0.315270i	$0.897905\pi$
$578$	0	0
$579$	−34.4844	−1.43312
$580$	0	0
$581$	−17.7414	−0.736036
$582$	0	0
$583$	2.23947	0.0927492
$584$	0	0
$585$	−0.937093	−0.0387440
$586$	0	0
$587$	22.1360	0.913652	0.456826	−	0.889556i	$-0.348986\pi$
$587$	22.1360	0.913652	0.456826	+	0.889556i	$0.348986\pi$
$588$	0	0
$589$	−36.2289	−1.49279
$590$	0	0
$591$	10.3211	0.424554
$592$	0	0
$593$	−43.9264	−1.80384	−0.901921	−	0.431901i	$-0.857843\pi$
$593$	−43.9264	−1.80384	−0.901921	+	0.431901i	$0.857843\pi$
$594$	0	0
$595$	4.33865	0.177867
$596$	0	0
$597$	−36.8417	−1.50783
$598$	0	0
$599$	27.6258	1.12876	0.564379	−	0.825516i	$-0.309116\pi$
$599$	27.6258	1.12876	0.564379	+	0.825516i	$0.309116\pi$
$600$	0	0
$601$	−22.6616	−0.924387	−0.462193	−	0.886779i	$-0.652937\pi$
$601$	−22.6616	−0.924387	−0.462193	+	0.886779i	$0.652937\pi$
$602$	0	0
$603$	−0.787797	−0.0320816
$604$	0	0
$605$	−4.96514	−0.201861
$606$	0	0
$607$	3.89819	0.158223	0.0791113	−	0.996866i	$-0.474792\pi$
$607$	3.89819	0.158223	0.0791113	+	0.996866i	$0.474792\pi$
$608$	0	0
$609$	−13.6185	−0.551851
$610$	0	0
$611$	10.0599	0.406981
$612$	0	0
$613$	−34.6665	−1.40017	−0.700083	−	0.714062i	$-0.746854\pi$
$613$	−34.6665	−1.40017	−0.700083	+	0.714062i	$0.746854\pi$
$614$	0	0
$615$	13.6548	0.550614
$616$	0	0
$617$	−33.4003	−1.34465	−0.672323	−	0.740258i	$-0.734704\pi$
$617$	−33.4003	−1.34465	−0.672323	+	0.740258i	$0.734704\pi$
$618$	0	0
$619$	34.3682	1.38137	0.690687	−	0.723154i	$-0.257308\pi$
$619$	34.3682	1.38137	0.690687	+	0.723154i	$0.257308\pi$
$620$	0	0
$621$	−34.3615	−1.37888
$622$	0	0
$623$	13.3066	0.533118
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.2970	−0.970330
$628$	0	0
$629$	33.8790	1.35084
$630$	0	0
$631$	2.15203	0.0856711	0.0428356	−	0.999082i	$-0.486361\pi$
$631$	2.15203	0.0856711	0.0428356	+	0.999082i	$0.486361\pi$
$632$	0	0
$633$	−10.0339	−0.398811
$634$	0	0
$635$	2.18661	0.0867730
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	1.62185	0.0641593
$640$	0	0
$641$	−22.6622	−0.895102	−0.447551	−	0.894258i	$-0.647704\pi$
$641$	−22.6622	−0.895102	−0.447551	+	0.894258i	$0.647704\pi$
$642$	0	0
$643$	−19.7761	−0.779894	−0.389947	−	0.920837i	$-0.627507\pi$
$643$	−19.7761	−0.779894	−0.389947	+	0.920837i	$0.627507\pi$
$644$	0	0
$645$	13.5572	0.533815
$646$	0	0
$647$	−43.4315	−1.70747	−0.853734	−	0.520710i	$-0.825667\pi$
$647$	−43.4315	−1.70747	−0.853734	+	0.520710i	$0.825667\pi$
$648$	0	0
$649$	−32.2250	−1.26494
$650$	0	0
$651$	7.55641	0.296159
$652$	0	0
$653$	35.1687	1.37626	0.688129	−	0.725588i	$-0.258432\pi$
$653$	35.1687	1.37626	0.688129	+	0.725588i	$0.258432\pi$
$654$	0	0
$655$	−3.44475	−0.134598
$656$	0	0
$657$	0.395100	0.0154143
$658$	0	0
$659$	−19.0958	−0.743867	−0.371934	−	0.928259i	$-0.621305\pi$
$659$	−19.0958	−0.743867	−0.371934	+	0.928259i	$0.621305\pi$
$660$	0	0
$661$	−34.8904	−1.35708	−0.678540	−	0.734564i	$-0.737387\pi$
$661$	−34.8904	−1.35708	−0.678540	+	0.734564i	$0.737387\pi$
$662$	0	0
$663$	6.23153	0.242013
$664$	0	0
$665$	−6.88620	−0.267035
$666$	0	0
$667$	−57.6166	−2.23092
$668$	0	0
$669$	35.1487	1.35893
$670$	0	0
$671$	−3.22594	−0.124536
$672$	0	0
$673$	32.0765	1.23646	0.618229	−	0.785998i	$-0.287850\pi$
$673$	32.0765	1.23646	0.618229	+	0.785998i	$0.287850\pi$
$674$	0	0
$675$	−5.65478	−0.217652
$676$	0	0
$677$	−38.7511	−1.48933	−0.744664	−	0.667440i	$-0.767390\pi$
$677$	−38.7511	−1.48933	−0.744664	+	0.667440i	$0.767390\pi$
$678$	0	0
$679$	11.6408	0.446734
$680$	0	0
$681$	12.5076	0.479293
$682$	0	0
$683$	14.6019	0.558725	0.279362	−	0.960186i	$-0.409877\pi$
$683$	14.6019	0.558725	0.279362	+	0.960186i	$0.409877\pi$
$684$	0	0
$685$	−0.197129	−0.00753191
$686$	0	0
$687$	4.61778	0.176179
$688$	0	0
$689$	0.911613	0.0347297
$690$	0	0
$691$	2.09750	0.0797925	0.0398963	−	0.999204i	$-0.487297\pi$
$691$	2.09750	0.0797925	0.0398963	+	0.999204i	$0.487297\pi$
$692$	0	0
$693$	−2.30206	−0.0874480
$694$	0	0
$695$	22.8688	0.867463
$696$	0	0
$697$	41.2477	1.56237
$698$	0	0
$699$	−21.2977	−0.805551
$700$	0	0
$701$	14.8460	0.560724	0.280362	−	0.959894i	$-0.409546\pi$
$701$	14.8460	0.560724	0.280362	+	0.959894i	$0.409546\pi$
$702$	0	0
$703$	−53.7718	−2.02804
$704$	0	0
$705$	14.4489	0.544177
$706$	0	0
$707$	−15.4162	−0.579788
$708$	0	0
$709$	19.8215	0.744410	0.372205	−	0.928150i	$-0.378602\pi$
$709$	19.8215	0.744410	0.372205	+	0.928150i	$0.378602\pi$
$710$	0	0
$711$	0.0949507	0.00356093
$712$	0	0
$713$	31.9692	1.19726
$714$	0	0
$715$	2.45660	0.0918715
$716$	0	0
$717$	13.4358	0.501768
$718$	0	0
$719$	−37.9720	−1.41612	−0.708059	−	0.706153i	$-0.750429\pi$
$719$	−37.9720	−1.41612	−0.708059	+	0.706153i	$0.750429\pi$
$720$	0	0
$721$	11.8488	0.441271
$722$	0	0
$723$	−16.9897	−0.631852
$724$	0	0
$725$	−9.48180	−0.352145
$726$	0	0
$727$	21.6603	0.803334	0.401667	−	0.915786i	$-0.368431\pi$
$727$	21.6603	0.803334	0.401667	+	0.915786i	$0.368431\pi$
$728$	0	0
$729$	29.3421	1.08674
$730$	0	0
$731$	40.9530	1.51470
$732$	0	0
$733$	11.4745	0.423819	0.211910	−	0.977289i	$-0.432032\pi$
$733$	11.4745	0.423819	0.211910	+	0.977289i	$0.432032\pi$
$734$	0	0
$735$	1.43628	0.0529781
$736$	0	0
$737$	2.06522	0.0760732
$738$	0	0
$739$	11.0005	0.404658	0.202329	−	0.979318i	$-0.435149\pi$
$739$	11.0005	0.404658	0.202329	+	0.979318i	$0.435149\pi$
$740$	0	0
$741$	−9.89053	−0.363338
$742$	0	0
$743$	−24.9940	−0.916942	−0.458471	−	0.888709i	$-0.651603\pi$
$743$	−24.9940	−0.916942	−0.458471	+	0.888709i	$0.651603\pi$
$744$	0	0
$745$	−5.13798	−0.188241
$746$	0	0
$747$	16.6253	0.608289
$748$	0	0
$749$	18.2306	0.666133
$750$	0	0
$751$	−2.21449	−0.0808077	−0.0404039	−	0.999183i	$-0.512864\pi$
$751$	−2.21449	−0.0808077	−0.0404039	+	0.999183i	$0.512864\pi$
$752$	0	0
$753$	17.2437	0.628394
$754$	0	0
$755$	20.5371	0.747421
$756$	0	0
$757$	−14.5484	−0.528771	−0.264386	−	0.964417i	$-0.585169\pi$
$757$	−14.5484	−0.528771	−0.264386	+	0.964417i	$0.585169\pi$
$758$	0	0
$759$	21.4403	0.778232
$760$	0	0
$761$	12.7138	0.460875	0.230437	−	0.973087i	$-0.425984\pi$
$761$	12.7138	0.460875	0.230437	+	0.973087i	$0.425984\pi$
$762$	0	0
$763$	−5.52516	−0.200024
$764$	0	0
$765$	−4.06572	−0.146996
$766$	0	0
$767$	−13.1177	−0.473654
$768$	0	0
$769$	4.00360	0.144373	0.0721867	−	0.997391i	$-0.477002\pi$
$769$	4.00360	0.144373	0.0721867	+	0.997391i	$0.477002\pi$
$770$	0	0
$771$	27.4258	0.987715
$772$	0	0
$773$	−48.4344	−1.74206	−0.871031	−	0.491227i	$-0.836548\pi$
$773$	−48.4344	−1.74206	−0.871031	+	0.491227i	$0.836548\pi$
$774$	0	0
$775$	5.26109	0.188984
$776$	0	0
$777$	11.2154	0.402350
$778$	0	0
$779$	−65.4673	−2.34561
$780$	0	0
$781$	−4.25169	−0.152137
$782$	0	0
$783$	53.6175	1.91613
$784$	0	0
$785$	−7.32709	−0.261515
$786$	0	0
$787$	−0.994065	−0.0354346	−0.0177173	−	0.999843i	$-0.505640\pi$
$787$	−0.994065	−0.0354346	−0.0177173	+	0.999843i	$0.505640\pi$
$788$	0	0
$789$	25.1184	0.894238
$790$	0	0
$791$	2.44061	0.0867781
$792$	0	0
$793$	−1.31318	−0.0466322
$794$	0	0
$795$	1.30933	0.0464373
$796$	0	0
$797$	−9.80183	−0.347199	−0.173599	−	0.984816i	$-0.555540\pi$
$797$	−9.80183	−0.347199	−0.173599	+	0.984816i	$0.555540\pi$
$798$	0	0
$799$	43.6465	1.54410
$800$	0	0
$801$	−12.4695	−0.440589
$802$	0	0
$803$	−1.03576	−0.0365511
$804$	0	0
$805$	6.07654	0.214170
$806$	0	0
$807$	−10.0016	−0.352072
$808$	0	0
$809$	7.42998	0.261224	0.130612	−	0.991434i	$-0.458306\pi$
$809$	7.42998	0.261224	0.130612	+	0.991434i	$0.458306\pi$
$810$	0	0
$811$	−38.3140	−1.34539	−0.672693	−	0.739922i	$-0.734862\pi$
$811$	−38.3140	−1.34539	−0.672693	+	0.739922i	$0.734862\pi$
$812$	0	0
$813$	−23.4396	−0.822063
$814$	0	0
$815$	−15.2769	−0.535128
$816$	0	0
$817$	−64.9995	−2.27405
$818$	0	0
$819$	−0.937093	−0.0327447
$820$	0	0
$821$	25.4455	0.888055	0.444027	−	0.896013i	$-0.353549\pi$
$821$	25.4455	0.888055	0.444027	+	0.896013i	$0.353549\pi$
$822$	0	0
$823$	32.2100	1.12277	0.561384	−	0.827555i	$-0.310269\pi$
$823$	32.2100	1.12277	0.561384	+	0.827555i	$0.310269\pi$
$824$	0	0
$825$	3.52837	0.122842
$826$	0	0
$827$	26.3829	0.917424	0.458712	−	0.888585i	$-0.348311\pi$
$827$	26.3829	0.917424	0.458712	+	0.888585i	$0.348311\pi$
$828$	0	0
$829$	−29.4626	−1.02328	−0.511640	−	0.859200i	$-0.670962\pi$
$829$	−29.4626	−1.02328	−0.511640	+	0.859200i	$0.670962\pi$
$830$	0	0
$831$	−22.3317	−0.774680
$832$	0	0
$833$	4.33865	0.150325
$834$	0	0
$835$	−20.4204	−0.706677
$836$	0	0
$837$	−29.7503	−1.02832
$838$	0	0
$839$	−43.2061	−1.49164	−0.745821	−	0.666147i	$-0.767942\pi$
$839$	−43.2061	−1.49164	−0.745821	+	0.666147i	$0.767942\pi$
$840$	0	0
$841$	60.9045	2.10016
$842$	0	0
$843$	7.28348	0.250856
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−4.96514	−0.170604
$848$	0	0
$849$	13.9189	0.477697
$850$	0	0
$851$	47.4495	1.62655
$852$	0	0
$853$	13.8241	0.473329	0.236665	−	0.971591i	$-0.423946\pi$
$853$	13.8241	0.473329	0.236665	+	0.971591i	$0.423946\pi$
$854$	0	0
$855$	6.45301	0.220688
$856$	0	0
$857$	−13.7140	−0.468461	−0.234231	−	0.972181i	$-0.575257\pi$
$857$	−13.7140	−0.468461	−0.234231	+	0.972181i	$0.575257\pi$
$858$	0	0
$859$	−35.7695	−1.22044	−0.610220	−	0.792232i	$-0.708919\pi$
$859$	−35.7695	−1.22044	−0.610220	+	0.792232i	$0.708919\pi$
$860$	0	0
$861$	13.6548	0.465354
$862$	0	0
$863$	−40.6181	−1.38266	−0.691328	−	0.722541i	$-0.742974\pi$
$863$	−40.6181	−1.38266	−0.691328	+	0.722541i	$0.742974\pi$
$864$	0	0
$865$	2.75033	0.0935140
$866$	0	0
$867$	2.61967	0.0889686
$868$	0	0
$869$	−0.248914	−0.00844383
$870$	0	0
$871$	0.840682	0.0284854
$872$	0	0
$873$	−10.9085	−0.369198
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−10.4916	−0.354277	−0.177138	−	0.984186i	$-0.556684\pi$
$877$	−10.4916	−0.354277	−0.177138	+	0.984186i	$0.556684\pi$
$878$	0	0
$879$	−26.4736	−0.892933
$880$	0	0
$881$	52.4686	1.76771	0.883856	−	0.467759i	$-0.154939\pi$
$881$	52.4686	1.76771	0.883856	+	0.467759i	$0.154939\pi$
$882$	0	0
$883$	38.1425	1.28360	0.641798	−	0.766874i	$-0.278189\pi$
$883$	38.1425	1.28360	0.641798	+	0.766874i	$0.278189\pi$
$884$	0	0
$885$	−18.8408	−0.633326
$886$	0	0
$887$	−18.1079	−0.608005	−0.304003	−	0.952671i	$-0.598323\pi$
$887$	−18.1079	−0.608005	−0.304003	+	0.952671i	$0.598323\pi$
$888$	0	0
$889$	2.18661	0.0733366
$890$	0	0
$891$	−13.0459	−0.437056
$892$	0	0
$893$	−69.2747	−2.31819
$894$	0	0
$895$	8.02804	0.268348
$896$	0	0
$897$	8.72763	0.291407
$898$	0	0
$899$	−49.8846	−1.66374
$900$	0	0
$901$	3.95518	0.131766
$902$	0	0
$903$	13.5572	0.451156
$904$	0	0
$905$	−2.14096	−0.0711679
$906$	0	0
$907$	2.89500	0.0961268	0.0480634	−	0.998844i	$-0.484695\pi$
$907$	2.89500	0.0961268	0.0480634	+	0.998844i	$0.484695\pi$
$908$	0	0
$909$	14.4465	0.479159
$910$	0	0
$911$	−23.4547	−0.777089	−0.388545	−	0.921430i	$-0.627022\pi$
$911$	−23.4547	−0.777089	−0.388545	+	0.921430i	$0.627022\pi$
$912$	0	0
$913$	−43.5834	−1.44240
$914$	0	0
$915$	−1.88609	−0.0623522
$916$	0	0
$917$	−3.44475	−0.113756
$918$	0	0
$919$	−8.08152	−0.266585	−0.133292	−	0.991077i	$-0.542555\pi$
$919$	−8.08152	−0.266585	−0.133292	+	0.991077i	$0.542555\pi$
$920$	0	0
$921$	24.6895	0.813547
$922$	0	0
$923$	−1.73072	−0.0569674
$924$	0	0
$925$	7.80863	0.256746
$926$	0	0
$927$	−11.1034	−0.364684
$928$	0	0
$929$	−8.96664	−0.294186	−0.147093	−	0.989123i	$-0.546992\pi$
$929$	−8.96664	−0.294186	−0.147093	+	0.989123i	$0.546992\pi$
$930$	0	0
$931$	−6.88620	−0.225686
$932$	0	0
$933$	−25.6429	−0.839511
$934$	0	0
$935$	10.6583	0.348564
$936$	0	0
$937$	42.8667	1.40039	0.700197	−	0.713950i	$-0.253096\pi$
$937$	42.8667	1.40039	0.700197	+	0.713950i	$0.253096\pi$
$938$	0	0
$939$	−19.7320	−0.643930
$940$	0	0
$941$	0.609833	0.0198800	0.00994000	−	0.999951i	$-0.496836\pi$
$941$	0.609833	0.0198800	0.00994000	+	0.999951i	$0.496836\pi$
$942$	0	0
$943$	57.7699	1.88125
$944$	0	0
$945$	−5.65478	−0.183950
$946$	0	0
$947$	−8.14107	−0.264549	−0.132275	−	0.991213i	$-0.542228\pi$
$947$	−8.14107	−0.264549	−0.132275	+	0.991213i	$0.542228\pi$
$948$	0	0
$949$	−0.421623	−0.0136865
$950$	0	0
$951$	−25.3864	−0.823211
$952$	0	0
$953$	9.76910	0.316452	0.158226	−	0.987403i	$-0.449423\pi$
$953$	9.76910	0.316452	0.158226	+	0.987403i	$0.449423\pi$
$954$	0	0
$955$	−12.7818	−0.413608
$956$	0	0
$957$	−33.4553	−1.08146
$958$	0	0
$959$	−0.197129	−0.00636563
$960$	0	0
$961$	−3.32095	−0.107127
$962$	0	0
$963$	−17.0838	−0.550517
$964$	0	0
$965$	−24.0095	−0.772892
$966$	0	0
$967$	−28.0509	−0.902055	−0.451028	−	0.892510i	$-0.648942\pi$
$967$	−28.0509	−0.902055	−0.451028	+	0.892510i	$0.648942\pi$
$968$	0	0
$969$	−42.9116	−1.37852
$970$	0	0
$971$	19.9713	0.640910	0.320455	−	0.947264i	$-0.396164\pi$
$971$	19.9713	0.640910	0.320455	+	0.947264i	$0.396164\pi$
$972$	0	0
$973$	22.8688	0.733140
$974$	0	0
$975$	1.43628	0.0459978
$976$	0	0
$977$	−16.5813	−0.530483	−0.265241	−	0.964182i	$-0.585452\pi$
$977$	−16.5813	−0.530483	−0.265241	+	0.964182i	$0.585452\pi$
$978$	0	0
$979$	32.6890	1.04474
$980$	0	0
$981$	5.17758	0.165308
$982$	0	0
$983$	−38.7232	−1.23508	−0.617539	−	0.786540i	$-0.711870\pi$
$983$	−38.7232	−1.23508	−0.617539	+	0.786540i	$0.711870\pi$
$984$	0	0
$985$	7.18599	0.228965
$986$	0	0
$987$	14.4489	0.459913
$988$	0	0
$989$	57.3571	1.82385
$990$	0	0
$991$	38.2484	1.21500	0.607500	−	0.794320i	$-0.292173\pi$
$991$	38.2484	1.21500	0.607500	+	0.794320i	$0.292173\pi$
$992$	0	0
$993$	−15.0804	−0.478562
$994$	0	0
$995$	−25.6507	−0.813183
$996$	0	0
$997$	−49.0833	−1.55448	−0.777242	−	0.629202i	$-0.783382\pi$
$997$	−49.0833	−1.55448	−0.777242	+	0.629202i	$0.783382\pi$
$998$	0	0
$999$	−44.1561	−1.39704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.ba.1.5	✓	7
4.3	odd	2		7280.2.a.cf.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.ba.1.5	✓	7	1.1	even	1	trivial
7280.2.a.cf.1.3		7	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)

\(f(q)\)	\(=\)	\(q+1.43628 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.937093 q^{9} +O(q^{10})\)	\(q+1.43628 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.937093 q^{9} +2.45660 q^{11} +1.00000 q^{13} +1.43628 q^{15} +4.33865 q^{17} -6.88620 q^{19} +1.43628 q^{21} +6.07654 q^{23} +1.00000 q^{25} -5.65478 q^{27} -9.48180 q^{29} +5.26109 q^{31} +3.52837 q^{33} +1.00000 q^{35} +7.80863 q^{37} +1.43628 q^{39} +9.50703 q^{41} +9.43910 q^{43} -0.937093 q^{45} +10.0599 q^{47} +1.00000 q^{49} +6.23153 q^{51} +0.911613 q^{53} +2.45660 q^{55} -9.89053 q^{57} -13.1177 q^{59} -1.31318 q^{61} -0.937093 q^{63} +1.00000 q^{65} +0.840682 q^{67} +8.72763 q^{69} -1.73072 q^{71} -0.421623 q^{73} +1.43628 q^{75} +2.45660 q^{77} -0.101325 q^{79} -5.31058 q^{81} -17.7414 q^{83} +4.33865 q^{85} -13.6185 q^{87} +13.3066 q^{89} +1.00000 q^{91} +7.55641 q^{93} -6.88620 q^{95} +11.6408 q^{97} -2.30206 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * q^99

Introduction

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