LMFDB - Embedded newform 3640.2.a.q.1.3

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:	$3$
Coefficient field:	3.3.1573.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 2 $ x^3 - x^2 - 7*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			1.3
Root			$3.06871$ 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)

$f(q)$	$=$	$q+3.06871 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.41697 q^{9} +O(q^{10})$	$q+3.06871 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.41697 q^{9} +4.00000 q^{11} -1.00000 q^{13} +3.06871 q^{15} +0.720448 q^{17} +1.27955 q^{19} -3.06871 q^{21} -2.00000 q^{23} +1.00000 q^{25} +10.4857 q^{27} -3.06871 q^{29} -5.76523 q^{31} +12.2748 q^{33} -1.00000 q^{35} +9.20612 q^{37} -3.06871 q^{39} +3.27955 q^{41} +10.1374 q^{43} +6.41697 q^{45} +0.696520 q^{47} +1.00000 q^{49} +2.21084 q^{51} -4.69652 q^{53} +4.00000 q^{55} +3.92657 q^{57} +11.0687 q^{59} -2.00000 q^{61} -6.41697 q^{63} -1.00000 q^{65} +1.41697 q^{67} -6.13742 q^{69} -0.559104 q^{71} -14.9714 q^{73} +3.06871 q^{75} -4.00000 q^{77} -5.41697 q^{79} +12.9266 q^{81} -8.97135 q^{83} +0.720448 q^{85} -9.41697 q^{87} +14.0401 q^{89} +1.00000 q^{91} -17.6918 q^{93} +1.27955 q^{95} -0.137416 q^{97} +25.6679 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9	$ 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9} + 12 q^{11} - 3 q^{13} + q^{15} - q^{17} + 7 q^{19} - q^{21} - 6 q^{23} + 3 q^{25} + 10 q^{27} - q^{29} + q^{31} + 4 q^{33} - 3 q^{35} + 3 q^{37} - q^{39} + 13 q^{41} + 14 q^{43} + 6 q^{45} - 8 q^{47} + 3 q^{49} + 18 q^{51} - 4 q^{53} + 12 q^{55} - 16 q^{57} + 25 q^{59} - 6 q^{61} - 6 q^{63} - 3 q^{65} - 9 q^{67} - 2 q^{69} - 8 q^{71} - 2 q^{73} + q^{75} - 12 q^{77} - 3 q^{79} + 11 q^{81} + 16 q^{83} - q^{85} - 15 q^{87} - 9 q^{89} + 3 q^{91} - 7 q^{93} + 7 q^{95} + 16 q^{97} + 24 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9 + 12 * q^11 - 3 * q^13 + q^15 - q^17 + 7 * q^19 - q^21 - 6 * q^23 + 3 * q^25 + 10 * q^27 - q^29 + q^31 + 4 * q^33 - 3 * q^35 + 3 * q^37 - q^39 + 13 * q^41 + 14 * q^43 + 6 * q^45 - 8 * q^47 + 3 * q^49 + 18 * q^51 - 4 * q^53 + 12 * q^55 - 16 * q^57 + 25 * q^59 - 6 * q^61 - 6 * q^63 - 3 * q^65 - 9 * q^67 - 2 * q^69 - 8 * q^71 - 2 * q^73 + q^75 - 12 * q^77 - 3 * q^79 + 11 * q^81 + 16 * q^83 - q^85 - 15 * q^87 - 9 * q^89 + 3 * q^91 - 7 * q^93 + 7 * q^95 + 16 * q^97 + 24 * 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
$3$	3.06871	1.77172	0.885860	−	0.463953i	$-0.153569\pi$
$3$	3.06871	1.77172	0.885860	+	0.463953i	$0.153569\pi$
$4$	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
$9$	6.41697	2.13899
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.06871	0.792337
$16$	0	0
$17$	0.720448	0.174734	0.0873671	−	0.996176i	$-0.472155\pi$
$17$	0.720448	0.174734	0.0873671	+	0.996176i	$0.472155\pi$
$18$	0	0
$19$	1.27955	0.293549	0.146775	−	0.989170i	$-0.453111\pi$
$19$	1.27955	0.293549	0.146775	+	0.989170i	$0.453111\pi$
$20$	0	0
$21$	−3.06871	−0.669647
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	10.4857	2.01797
$28$	0	0
$29$	−3.06871	−0.569845	−0.284922	−	0.958551i	$-0.591968\pi$
$29$	−3.06871	−0.569845	−0.284922	+	0.958551i	$0.591968\pi$
$30$	0	0
$31$	−5.76523	−1.03547	−0.517733	−	0.855542i	$-0.673224\pi$
$31$	−5.76523	−1.03547	−0.517733	+	0.855542i	$0.673224\pi$
$32$	0	0
$33$	12.2748	2.13677
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	9.20612	1.51348	0.756739	−	0.653717i	$-0.226792\pi$
$37$	9.20612	1.51348	0.756739	+	0.653717i	$0.226792\pi$
$38$	0	0
$39$	−3.06871	−0.491387
$40$	0	0
$41$	3.27955	0.512180	0.256090	−	0.966653i	$-0.417566\pi$
$41$	3.27955	0.512180	0.256090	+	0.966653i	$0.417566\pi$
$42$	0	0
$43$	10.1374	1.54594	0.772971	−	0.634442i	$-0.218770\pi$
$43$	10.1374	1.54594	0.772971	+	0.634442i	$0.218770\pi$
$44$	0	0
$45$	6.41697	0.956585
$46$	0	0
$47$	0.696520	0.101598	0.0507989	−	0.998709i	$-0.483823\pi$
$47$	0.696520	0.101598	0.0507989	+	0.998709i	$0.483823\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.21084	0.309580
$52$	0	0
$53$	−4.69652	−0.645117	−0.322558	−	0.946550i	$-0.604543\pi$
$53$	−4.69652	−0.645117	−0.322558	+	0.946550i	$0.604543\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	3.92657	0.520087
$58$	0	0
$59$	11.0687	1.44102	0.720512	−	0.693443i	$-0.243907\pi$
$59$	11.0687	1.44102	0.720512	+	0.693443i	$0.243907\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−6.41697	−0.808462
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	1.41697	0.173110	0.0865551	−	0.996247i	$-0.472414\pi$
$67$	1.41697	0.173110	0.0865551	+	0.996247i	$0.472414\pi$
$68$	0	0
$69$	−6.13742	−0.738858
$70$	0	0
$71$	−0.559104	−0.0663535	−0.0331767	−	0.999450i	$-0.510562\pi$
$71$	−0.559104	−0.0663535	−0.0331767	+	0.999450i	$0.510562\pi$
$72$	0	0
$73$	−14.9714	−1.75226	−0.876132	−	0.482071i	$-0.839885\pi$
$73$	−14.9714	−1.75226	−0.876132	+	0.482071i	$0.839885\pi$
$74$	0	0
$75$	3.06871	0.354344
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−5.41697	−0.609456	−0.304728	−	0.952439i	$-0.598566\pi$
$79$	−5.41697	−0.609456	−0.304728	+	0.952439i	$0.598566\pi$
$80$	0	0
$81$	12.9266	1.43629
$82$	0	0
$83$	−8.97135	−0.984734	−0.492367	−	0.870388i	$-0.663868\pi$
$83$	−8.97135	−0.984734	−0.492367	+	0.870388i	$0.663868\pi$
$84$	0	0
$85$	0.720448	0.0781436
$86$	0	0
$87$	−9.41697	−1.00960
$88$	0	0
$89$	14.0401	1.48824	0.744122	−	0.668044i	$-0.232868\pi$
$89$	14.0401	1.48824	0.744122	+	0.668044i	$0.232868\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−17.6918	−1.83455
$94$	0	0
$95$	1.27955	0.131279
$96$	0	0
$97$	−0.137416	−0.0139524	−0.00697622	−	0.999976i	$-0.502221\pi$
$97$	−0.137416	−0.0139524	−0.00697622	+	0.999976i	$0.502221\pi$
$98$	0	0
$99$	25.6679	2.57972
$100$	0	0
$101$	18.9714	1.88772	0.943860	−	0.330346i	$-0.107165\pi$
$101$	18.9714	1.88772	0.943860	+	0.330346i	$0.107165\pi$
$102$	0	0
$103$	1.14214	0.112538	0.0562690	−	0.998416i	$-0.482080\pi$
$103$	1.14214	0.112538	0.0562690	+	0.998416i	$0.482080\pi$
$104$	0	0
$105$	−3.06871	−0.299475
$106$	0	0
$107$	−15.1088	−1.46062	−0.730310	−	0.683116i	$-0.760624\pi$
$107$	−15.1088	−1.46062	−0.730310	+	0.683116i	$0.760624\pi$
$108$	0	0
$109$	−12.6965	−1.21611	−0.608053	−	0.793896i	$-0.708049\pi$
$109$	−12.6965	−1.21611	−0.608053	+	0.793896i	$0.708049\pi$
$110$	0	0
$111$	28.2509	2.68146
$112$	0	0
$113$	−16.1374	−1.51808	−0.759040	−	0.651044i	$-0.774331\pi$
$113$	−16.1374	−1.51808	−0.759040	+	0.651044i	$0.774331\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−6.41697	−0.593249
$118$	0	0
$119$	−0.720448	−0.0660434
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	10.0640	0.907439
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	0	0
$129$	31.1088	2.73897
$130$	0	0
$131$	−8.97135	−0.783831	−0.391915	−	0.920001i	$-0.628187\pi$
$131$	−8.97135	−0.783831	−0.391915	+	0.920001i	$0.628187\pi$
$132$	0	0
$133$	−1.27955	−0.110951
$134$	0	0
$135$	10.4857	0.902463
$136$	0	0
$137$	−5.20612	−0.444789	−0.222395	−	0.974957i	$-0.571387\pi$
$137$	−5.20612	−0.444789	−0.222395	+	0.974957i	$0.571387\pi$
$138$	0	0
$139$	−6.83394	−0.579647	−0.289823	−	0.957080i	$-0.593597\pi$
$139$	−6.83394	−0.579647	−0.289823	+	0.957080i	$0.593597\pi$
$140$	0	0
$141$	2.13742	0.180003
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−3.06871	−0.254842
$146$	0	0
$147$	3.06871	0.253103
$148$	0	0
$149$	−4.69652	−0.384754	−0.192377	−	0.981321i	$-0.561620\pi$
$149$	−4.69652	−0.384754	−0.192377	+	0.981321i	$0.561620\pi$
$150$	0	0
$151$	−4.83394	−0.393380	−0.196690	−	0.980466i	$-0.563019\pi$
$151$	−4.83394	−0.393380	−0.196690	+	0.980466i	$0.563019\pi$
$152$	0	0
$153$	4.62309	0.373755
$154$	0	0
$155$	−5.76523	−0.463074
$156$	0	0
$157$	15.3435	1.22455	0.612274	−	0.790646i	$-0.290255\pi$
$157$	15.3435	1.22455	0.612274	+	0.790646i	$0.290255\pi$
$158$	0	0
$159$	−14.4122	−1.14297
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	13.7652	1.07818	0.539088	−	0.842249i	$-0.318769\pi$
$163$	13.7652	1.07818	0.539088	+	0.842249i	$0.318769\pi$
$164$	0	0
$165$	12.2748	0.955594
$166$	0	0
$167$	2.13742	0.165398	0.0826991	−	0.996575i	$-0.473646\pi$
$167$	2.13742	0.165398	0.0826991	+	0.996575i	$0.473646\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.21084	0.627899
$172$	0	0
$173$	−9.95050	−0.756522	−0.378261	−	0.925699i	$-0.623478\pi$
$173$	−9.95050	−0.756522	−0.378261	+	0.925699i	$0.623478\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	33.9666	2.55309
$178$	0	0
$179$	−15.1327	−1.13107	−0.565535	−	0.824724i	$-0.691330\pi$
$179$	−15.1327	−1.13107	−0.565535	+	0.824724i	$0.691330\pi$
$180$	0	0
$181$	7.86258	0.584421	0.292211	−	0.956354i	$-0.405609\pi$
$181$	7.86258	0.584421	0.292211	+	0.956354i	$0.405609\pi$
$182$	0	0
$183$	−6.13742	−0.453691
$184$	0	0
$185$	9.20612	0.676848
$186$	0	0
$187$	2.88179	0.210737
$188$	0	0
$189$	−10.4857	−0.762721
$190$	0	0
$191$	−26.0401	−1.88419	−0.942096	−	0.335343i	$-0.891148\pi$
$191$	−26.0401	−1.88419	−0.942096	+	0.335343i	$0.891148\pi$
$192$	0	0
$193$	−0.857864	−0.0617504	−0.0308752	−	0.999523i	$-0.509829\pi$
$193$	−0.857864	−0.0617504	−0.0308752	+	0.999523i	$0.509829\pi$
$194$	0	0
$195$	−3.06871	−0.219755
$196$	0	0
$197$	20.7366	1.47742	0.738710	−	0.674023i	$-0.235435\pi$
$197$	20.7366	1.47742	0.738710	+	0.674023i	$0.235435\pi$
$198$	0	0
$199$	0.744376	0.0527674	0.0263837	−	0.999652i	$-0.491601\pi$
$199$	0.744376	0.0527674	0.0263837	+	0.999652i	$0.491601\pi$
$200$	0	0
$201$	4.34826	0.306703
$202$	0	0
$203$	3.06871	0.215381
$204$	0	0
$205$	3.27955	0.229054
$206$	0	0
$207$	−12.8339	−0.892020
$208$	0	0
$209$	5.11821	0.354034
$210$	0	0
$211$	28.2509	1.94487	0.972436	−	0.233169i	$-0.0749096\pi$
$211$	28.2509	1.94487	0.972436	+	0.233169i	$0.0749096\pi$
$212$	0	0
$213$	−1.71573	−0.117560
$214$	0	0
$215$	10.1374	0.691366
$216$	0	0
$217$	5.76523	0.391369
$218$	0	0
$219$	−45.9427	−3.10452
$220$	0	0
$221$	−0.720448	−0.0484626
$222$	0	0
$223$	−1.44090	−0.0964895	−0.0482448	−	0.998836i	$-0.515363\pi$
$223$	−1.44090	−0.0964895	−0.0482448	+	0.998836i	$0.515363\pi$
$224$	0	0
$225$	6.41697	0.427798
$226$	0	0
$227$	19.8531	1.31770	0.658850	−	0.752275i	$-0.271043\pi$
$227$	19.8531	1.31770	0.658850	+	0.752275i	$0.271043\pi$
$228$	0	0
$229$	18.2509	1.20605	0.603027	−	0.797721i	$-0.293961\pi$
$229$	18.2509	1.20605	0.603027	+	0.797721i	$0.293961\pi$
$230$	0	0
$231$	−12.2748	−0.807625
$232$	0	0
$233$	−27.2462	−1.78496	−0.892478	−	0.451090i	$-0.851035\pi$
$233$	−27.2462	−1.78496	−0.892478	+	0.451090i	$0.851035\pi$
$234$	0	0
$235$	0.696520	0.0454359
$236$	0	0
$237$	−16.6231	−1.07979
$238$	0	0
$239$	−4.88179	−0.315777	−0.157888	−	0.987457i	$-0.550469\pi$
$239$	−4.88179	−0.315777	−0.157888	+	0.987457i	$0.550469\pi$
$240$	0	0
$241$	−2.16134	−0.139224	−0.0696122	−	0.997574i	$-0.522176\pi$
$241$	−2.16134	−0.139224	−0.0696122	+	0.997574i	$0.522176\pi$
$242$	0	0
$243$	8.21084	0.526726
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−1.27955	−0.0814159
$248$	0	0
$249$	−27.5305	−1.74467
$250$	0	0
$251$	10.1374	0.639868	0.319934	−	0.947440i	$-0.396339\pi$
$251$	10.1374	0.639868	0.319934	+	0.947440i	$0.396339\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	2.21084	0.138448
$256$	0	0
$257$	−15.5783	−0.971748	−0.485874	−	0.874029i	$-0.661499\pi$
$257$	−15.5783	−0.971748	−0.485874	+	0.874029i	$0.661499\pi$
$258$	0	0
$259$	−9.20612	−0.572041
$260$	0	0
$261$	−19.6918	−1.21889
$262$	0	0
$263$	−4.88179	−0.301024	−0.150512	−	0.988608i	$-0.548092\pi$
$263$	−4.88179	−0.301024	−0.150512	+	0.988608i	$0.548092\pi$
$264$	0	0
$265$	−4.69652	−0.288505
$266$	0	0
$267$	43.0848	2.63675
$268$	0	0
$269$	−6.69652	−0.408294	−0.204147	−	0.978940i	$-0.565442\pi$
$269$	−6.69652	−0.408294	−0.204147	+	0.978940i	$0.565442\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	3.06871	0.185727
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	−36.9953	−2.21485
$280$	0	0
$281$	16.8339	1.00423	0.502114	−	0.864801i	$-0.332556\pi$
$281$	16.8339	1.00423	0.502114	+	0.864801i	$0.332556\pi$
$282$	0	0
$283$	−6.99528	−0.415826	−0.207913	−	0.978147i	$-0.566667\pi$
$283$	−6.99528	−0.415826	−0.207913	+	0.978147i	$0.566667\pi$
$284$	0	0
$285$	3.92657	0.232590
$286$	0	0
$287$	−3.27955	−0.193586
$288$	0	0
$289$	−16.4810	−0.969468
$290$	0	0
$291$	−0.421688	−0.0247198
$292$	0	0
$293$	−26.5018	−1.54825	−0.774126	−	0.633032i	$-0.781810\pi$
$293$	−26.5018	−1.54825	−0.774126	+	0.633032i	$0.781810\pi$
$294$	0	0
$295$	11.0687	0.644445
$296$	0	0
$297$	41.9427	2.43376
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	−10.1374	−0.584311
$302$	0	0
$303$	58.2175	3.34451
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−8.69652	−0.496337	−0.248168	−	0.968717i	$-0.579829\pi$
$307$	−8.69652	−0.496337	−0.248168	+	0.968717i	$0.579829\pi$
$308$	0	0
$309$	3.50488	0.199386
$310$	0	0
$311$	1.44090	0.0817057	0.0408529	−	0.999165i	$-0.486993\pi$
$311$	1.44090	0.0817057	0.0408529	+	0.999165i	$0.486993\pi$
$312$	0	0
$313$	−20.1775	−1.14050	−0.570249	−	0.821472i	$-0.693153\pi$
$313$	−20.1775	−1.14050	−0.570249	+	0.821472i	$0.693153\pi$
$314$	0	0
$315$	−6.41697	−0.361555
$316$	0	0
$317$	−2.27483	−0.127767	−0.0638836	−	0.997957i	$-0.520349\pi$
$317$	−2.27483	−0.127767	−0.0638836	+	0.997957i	$0.520349\pi$
$318$	0	0
$319$	−12.2748	−0.687259
$320$	0	0
$321$	−46.3644	−2.58781
$322$	0	0
$323$	0.921851	0.0512931
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−38.9619	−2.15460
$328$	0	0
$329$	−0.696520	−0.0384004
$330$	0	0
$331$	19.1566	1.05294	0.526472	−	0.850193i	$-0.323515\pi$
$331$	19.1566	1.05294	0.526472	+	0.850193i	$0.323515\pi$
$332$	0	0
$333$	59.0754	3.23731
$334$	0	0
$335$	1.41697	0.0774172
$336$	0	0
$337$	−11.3930	−0.620618	−0.310309	−	0.950636i	$-0.600433\pi$
$337$	−11.3930	−0.620618	−0.310309	+	0.950636i	$0.600433\pi$
$338$	0	0
$339$	−49.5210	−2.68961
$340$	0	0
$341$	−23.0609	−1.24882
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.13742	−0.330427
$346$	0	0
$347$	11.8531	0.636310	0.318155	−	0.948039i	$-0.396937\pi$
$347$	11.8531	0.636310	0.318155	+	0.948039i	$0.396937\pi$
$348$	0	0
$349$	−26.5992	−1.42382	−0.711910	−	0.702270i	$-0.752170\pi$
$349$	−26.5992	−1.42382	−0.711910	+	0.702270i	$0.752170\pi$
$350$	0	0
$351$	−10.4857	−0.559684
$352$	0	0
$353$	−2.27483	−0.121077	−0.0605385	−	0.998166i	$-0.519282\pi$
$353$	−2.27483	−0.121077	−0.0605385	+	0.998166i	$0.519282\pi$
$354$	0	0
$355$	−0.559104	−0.0296742
$356$	0	0
$357$	−2.21084	−0.117010
$358$	0	0
$359$	16.1374	0.851700	0.425850	−	0.904794i	$-0.359975\pi$
$359$	16.1374	0.851700	0.425850	+	0.904794i	$0.359975\pi$
$360$	0	0
$361$	−17.3627	−0.913829
$362$	0	0
$363$	15.3435	0.805327
$364$	0	0
$365$	−14.9714	−0.783636
$366$	0	0
$367$	−19.1566	−0.999968	−0.499984	−	0.866035i	$-0.666661\pi$
$367$	−19.1566	−0.999968	−0.499984	+	0.866035i	$0.666661\pi$
$368$	0	0
$369$	21.0448	1.09555
$370$	0	0
$371$	4.69652	0.243831
$372$	0	0
$373$	−17.7157	−0.917286	−0.458643	−	0.888621i	$-0.651664\pi$
$373$	−17.7157	−0.917286	−0.458643	+	0.888621i	$0.651664\pi$
$374$	0	0
$375$	3.06871	0.158467
$376$	0	0
$377$	3.06871	0.158046
$378$	0	0
$379$	−0.696520	−0.0357778	−0.0178889	−	0.999840i	$-0.505695\pi$
$379$	−0.696520	−0.0357778	−0.0178889	+	0.999840i	$0.505695\pi$
$380$	0	0
$381$	30.6871	1.57215
$382$	0	0
$383$	15.9521	0.815116	0.407558	−	0.913179i	$-0.366380\pi$
$383$	15.9521	0.815116	0.407558	+	0.913179i	$0.366380\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	65.0515	3.30675
$388$	0	0
$389$	−15.9666	−0.809540	−0.404770	−	0.914418i	$-0.632648\pi$
$389$	−15.9666	−0.809540	−0.404770	+	0.914418i	$0.632648\pi$
$390$	0	0
$391$	−1.44090	−0.0728692
$392$	0	0
$393$	−27.5305	−1.38873
$394$	0	0
$395$	−5.41697	−0.272557
$396$	0	0
$397$	−11.4409	−0.574202	−0.287101	−	0.957900i	$-0.592692\pi$
$397$	−11.4409	−0.574202	−0.287101	+	0.957900i	$0.592692\pi$
$398$	0	0
$399$	−3.92657	−0.196574
$400$	0	0
$401$	−0.833935	−0.0416447	−0.0208224	−	0.999783i	$-0.506628\pi$
$401$	−0.833935	−0.0416447	−0.0208224	+	0.999783i	$0.506628\pi$
$402$	0	0
$403$	5.76523	0.287186
$404$	0	0
$405$	12.9266	0.642326
$406$	0	0
$407$	36.8245	1.82532
$408$	0	0
$409$	18.0896	0.894471	0.447236	−	0.894416i	$-0.352409\pi$
$409$	18.0896	0.894471	0.447236	+	0.894416i	$0.352409\pi$
$410$	0	0
$411$	−15.9761	−0.788042
$412$	0	0
$413$	−11.0687	−0.544656
$414$	0	0
$415$	−8.97135	−0.440386
$416$	0	0
$417$	−20.9714	−1.02697
$418$	0	0
$419$	3.10877	0.151873	0.0759366	−	0.997113i	$-0.475805\pi$
$419$	3.10877	0.151873	0.0759366	+	0.997113i	$0.475805\pi$
$420$	0	0
$421$	0.971351	0.0473408	0.0236704	−	0.999720i	$-0.492465\pi$
$421$	0.971351	0.0473408	0.0236704	+	0.999720i	$0.492465\pi$
$422$	0	0
$423$	4.46954	0.217317
$424$	0	0
$425$	0.720448	0.0349469
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−12.2748	−0.592634
$430$	0	0
$431$	−11.8626	−0.571401	−0.285700	−	0.958319i	$-0.592226\pi$
$431$	−11.8626	−0.571401	−0.285700	+	0.958319i	$0.592226\pi$
$432$	0	0
$433$	−5.88651	−0.282888	−0.141444	−	0.989946i	$-0.545174\pi$
$433$	−5.88651	−0.282888	−0.141444	+	0.989946i	$0.545174\pi$
$434$	0	0
$435$	−9.41697	−0.451509
$436$	0	0
$437$	−2.55910	−0.122419
$438$	0	0
$439$	−25.9427	−1.23818	−0.619089	−	0.785321i	$-0.712498\pi$
$439$	−25.9427	−1.23818	−0.619089	+	0.785321i	$0.712498\pi$
$440$	0	0
$441$	6.41697	0.305570
$442$	0	0
$443$	33.2462	1.57957	0.789787	−	0.613381i	$-0.210191\pi$
$443$	33.2462	1.57957	0.789787	+	0.613381i	$0.210191\pi$
$444$	0	0
$445$	14.0401	0.665563
$446$	0	0
$447$	−14.4122	−0.681676
$448$	0	0
$449$	−1.25562	−0.0592566	−0.0296283	−	0.999561i	$-0.509432\pi$
$449$	−1.25562	−0.0592566	−0.0296283	+	0.999561i	$0.509432\pi$
$450$	0	0
$451$	13.1182	0.617712
$452$	0	0
$453$	−14.8339	−0.696959
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−34.5992	−1.61848	−0.809240	−	0.587478i	$-0.800121\pi$
$457$	−34.5992	−1.61848	−0.809240	+	0.587478i	$0.800121\pi$
$458$	0	0
$459$	7.55438	0.352608
$460$	0	0
$461$	−1.95050	−0.0908438	−0.0454219	−	0.998968i	$-0.514463\pi$
$461$	−1.95050	−0.0908438	−0.0454219	+	0.998968i	$0.514463\pi$
$462$	0	0
$463$	25.6918	1.19400	0.597000	−	0.802242i	$-0.296359\pi$
$463$	25.6918	1.19400	0.597000	+	0.802242i	$0.296359\pi$
$464$	0	0
$465$	−17.6918	−0.820438
$466$	0	0
$467$	20.4362	0.945673	0.472837	−	0.881150i	$-0.343230\pi$
$467$	20.4362	0.945673	0.472837	+	0.881150i	$0.343230\pi$
$468$	0	0
$469$	−1.41697	−0.0654295
$470$	0	0
$471$	47.0848	2.16955
$472$	0	0
$473$	40.5497	1.86448
$474$	0	0
$475$	1.27955	0.0587099
$476$	0	0
$477$	−30.1374	−1.37990
$478$	0	0
$479$	−10.3149	−0.471299	−0.235650	−	0.971838i	$-0.575722\pi$
$479$	−10.3149	−0.471299	−0.235650	+	0.971838i	$0.575722\pi$
$480$	0	0
$481$	−9.20612	−0.419763
$482$	0	0
$483$	6.13742	0.279262
$484$	0	0
$485$	−0.137416	−0.00623972
$486$	0	0
$487$	−15.6278	−0.708164	−0.354082	−	0.935214i	$-0.615207\pi$
$487$	−15.6278	−0.708164	−0.354082	+	0.935214i	$0.615207\pi$
$488$	0	0
$489$	42.2415	1.91022
$490$	0	0
$491$	−9.66787	−0.436305	−0.218152	−	0.975915i	$-0.570003\pi$
$491$	−9.66787	−0.436305	−0.218152	+	0.975915i	$0.570003\pi$
$492$	0	0
$493$	−2.21084	−0.0995714
$494$	0	0
$495$	25.6679	1.15368
$496$	0	0
$497$	0.559104	0.0250792
$498$	0	0
$499$	25.6679	1.14905	0.574526	−	0.818486i	$-0.305186\pi$
$499$	25.6679	1.14905	0.574526	+	0.818486i	$0.305186\pi$
$500$	0	0
$501$	6.55910	0.293039
$502$	0	0
$503$	−43.3357	−1.93225	−0.966123	−	0.258084i	$-0.916909\pi$
$503$	−43.3357	−1.93225	−0.966123	+	0.258084i	$0.916909\pi$
$504$	0	0
$505$	18.9714	0.844214
$506$	0	0
$507$	3.06871	0.136286
$508$	0	0
$509$	8.85786	0.392618	0.196309	−	0.980542i	$-0.437104\pi$
$509$	8.85786	0.392618	0.196309	+	0.980542i	$0.437104\pi$
$510$	0	0
$511$	14.9714	0.662294
$512$	0	0
$513$	13.4170	0.592374
$514$	0	0
$515$	1.14214	0.0503285
$516$	0	0
$517$	2.78608	0.122532
$518$	0	0
$519$	−30.5352	−1.34035
$520$	0	0
$521$	17.5783	0.770120	0.385060	−	0.922892i	$-0.374181\pi$
$521$	17.5783	0.770120	0.385060	+	0.922892i	$0.374181\pi$
$522$	0	0
$523$	−25.5783	−1.11846	−0.559231	−	0.829012i	$-0.688903\pi$
$523$	−25.5783	−1.11846	−0.559231	+	0.829012i	$0.688903\pi$
$524$	0	0
$525$	−3.06871	−0.133929
$526$	0	0
$527$	−4.15355	−0.180931
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	71.0275	3.08233
$532$	0	0
$533$	−3.27955	−0.142053
$534$	0	0
$535$	−15.1088	−0.653209
$536$	0	0
$537$	−46.4378	−2.00394
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	−34.4122	−1.47950	−0.739749	−	0.672883i	$-0.765056\pi$
$541$	−34.4122	−1.47950	−0.739749	+	0.672883i	$0.765056\pi$
$542$	0	0
$543$	24.1280	1.03543
$544$	0	0
$545$	−12.6965	−0.543859
$546$	0	0
$547$	12.6965	0.542864	0.271432	−	0.962458i	$-0.412503\pi$
$547$	12.6965	0.542864	0.271432	+	0.962458i	$0.412503\pi$
$548$	0	0
$549$	−12.8339	−0.547739
$550$	0	0
$551$	−3.92657	−0.167278
$552$	0	0
$553$	5.41697	0.230353
$554$	0	0
$555$	28.2509	1.19918
$556$	0	0
$557$	33.6345	1.42514	0.712570	−	0.701601i	$-0.247531\pi$
$557$	33.6345	1.42514	0.712570	+	0.701601i	$0.247531\pi$
$558$	0	0
$559$	−10.1374	−0.428767
$560$	0	0
$561$	8.84338	0.373368
$562$	0	0
$563$	32.6632	1.37659	0.688294	−	0.725432i	$-0.258360\pi$
$563$	32.6632	1.37659	0.688294	+	0.725432i	$0.258360\pi$
$564$	0	0
$565$	−16.1374	−0.678906
$566$	0	0
$567$	−12.9266	−0.542865
$568$	0	0
$569$	0.186916	0.00783593	0.00391796	−	0.999992i	$-0.498753\pi$
$569$	0.186916	0.00783593	0.00391796	+	0.999992i	$0.498753\pi$
$570$	0	0
$571$	10.8834	0.455458	0.227729	−	0.973725i	$-0.426870\pi$
$571$	10.8834	0.455458	0.227729	+	0.973725i	$0.426870\pi$
$572$	0	0
$573$	−79.9093	−3.33826
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	−14.4217	−0.600383	−0.300191	−	0.953879i	$-0.597051\pi$
$577$	−14.4217	−0.600383	−0.300191	+	0.953879i	$0.597051\pi$
$578$	0	0
$579$	−2.63253	−0.109404
$580$	0	0
$581$	8.97135	0.372194
$582$	0	0
$583$	−18.7861	−0.778040
$584$	0	0
$585$	−6.41697	−0.265309
$586$	0	0
$587$	29.9906	1.23784	0.618921	−	0.785453i	$-0.287570\pi$
$587$	29.9906	1.23784	0.618921	+	0.785453i	$0.287570\pi$
$588$	0	0
$589$	−7.37691	−0.303960
$590$	0	0
$591$	63.6345	2.61757
$592$	0	0
$593$	−26.1280	−1.07295	−0.536474	−	0.843917i	$-0.680244\pi$
$593$	−26.1280	−1.07295	−0.536474	+	0.843917i	$0.680244\pi$
$594$	0	0
$595$	−0.720448	−0.0295355
$596$	0	0
$597$	2.28427	0.0934891
$598$	0	0
$599$	9.39304	0.383789	0.191895	−	0.981416i	$-0.438537\pi$
$599$	9.39304	0.383789	0.191895	+	0.981416i	$0.438537\pi$
$600$	0	0
$601$	0.412247	0.0168159	0.00840795	−	0.999965i	$-0.497324\pi$
$601$	0.412247	0.0168159	0.00840795	+	0.999965i	$0.497324\pi$
$602$	0	0
$603$	9.09264	0.370281
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	31.4331	1.27583	0.637915	−	0.770107i	$-0.279797\pi$
$607$	31.4331	1.27583	0.637915	+	0.770107i	$0.279797\pi$
$608$	0	0
$609$	9.41697	0.381595
$610$	0	0
$611$	−0.696520	−0.0281782
$612$	0	0
$613$	41.0609	1.65843	0.829217	−	0.558926i	$-0.188787\pi$
$613$	41.0609	1.65843	0.829217	+	0.558926i	$0.188787\pi$
$614$	0	0
$615$	10.0640	0.405819
$616$	0	0
$617$	−41.6345	−1.67614	−0.838071	−	0.545561i	$-0.816317\pi$
$617$	−41.6345	−1.67614	−0.838071	+	0.545561i	$0.816317\pi$
$618$	0	0
$619$	7.61837	0.306208	0.153104	−	0.988210i	$-0.451073\pi$
$619$	7.61837	0.306208	0.153104	+	0.988210i	$0.451073\pi$
$620$	0	0
$621$	−20.9714	−0.841551
$622$	0	0
$623$	−14.0401	−0.562503
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.7063	0.627249
$628$	0	0
$629$	6.63253	0.264456
$630$	0	0
$631$	43.5689	1.73445	0.867225	−	0.497917i	$-0.165902\pi$
$631$	43.5689	1.73445	0.867225	+	0.497917i	$0.165902\pi$
$632$	0	0
$633$	86.6938	3.44577
$634$	0	0
$635$	10.0000	0.396838
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−3.58775	−0.141929
$640$	0	0
$641$	−11.7174	−0.462808	−0.231404	−	0.972858i	$-0.574332\pi$
$641$	−11.7174	−0.462808	−0.231404	+	0.972858i	$0.574332\pi$
$642$	0	0
$643$	−40.9714	−1.61575	−0.807876	−	0.589352i	$-0.799383\pi$
$643$	−40.9714	−1.61575	−0.807876	+	0.589352i	$0.799383\pi$
$644$	0	0
$645$	31.1088	1.22491
$646$	0	0
$647$	25.4904	1.00213	0.501065	−	0.865409i	$-0.332942\pi$
$647$	25.4904	1.00213	0.501065	+	0.865409i	$0.332942\pi$
$648$	0	0
$649$	44.2748	1.73794
$650$	0	0
$651$	17.6918	0.693396
$652$	0	0
$653$	25.1983	0.986087	0.493043	−	0.870005i	$-0.335884\pi$
$653$	25.1983	0.986087	0.493043	+	0.870005i	$0.335884\pi$
$654$	0	0
$655$	−8.97135	−0.350540
$656$	0	0
$657$	−96.0707	−3.74807
$658$	0	0
$659$	12.8006	0.498639	0.249320	−	0.968421i	$-0.419793\pi$
$659$	12.8006	0.498639	0.249320	+	0.968421i	$0.419793\pi$
$660$	0	0
$661$	2.02393	0.0787217	0.0393608	−	0.999225i	$-0.487468\pi$
$661$	2.02393	0.0787217	0.0393608	+	0.999225i	$0.487468\pi$
$662$	0	0
$663$	−2.21084	−0.0858621
$664$	0	0
$665$	−1.27955	−0.0496189
$666$	0	0
$667$	6.13742	0.237642
$668$	0	0
$669$	−4.42169	−0.170952
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−51.5210	−1.98599	−0.992995	−	0.118160i	$-0.962300\pi$
$673$	−51.5210	−1.98599	−0.992995	+	0.118160i	$0.962300\pi$
$674$	0	0
$675$	10.4857	0.403594
$676$	0	0
$677$	−8.33213	−0.320230	−0.160115	−	0.987098i	$-0.551186\pi$
$677$	−8.33213	−0.320230	−0.160115	+	0.987098i	$0.551186\pi$
$678$	0	0
$679$	0.137416	0.00527353
$680$	0	0
$681$	60.9235	2.33459
$682$	0	0
$683$	24.4779	0.936620	0.468310	−	0.883564i	$-0.344863\pi$
$683$	24.4779	0.936620	0.468310	+	0.883564i	$0.344863\pi$
$684$	0	0
$685$	−5.20612	−0.198916
$686$	0	0
$687$	56.0067	2.13679
$688$	0	0
$689$	4.69652	0.178923
$690$	0	0
$691$	−33.0114	−1.25581	−0.627907	−	0.778289i	$-0.716088\pi$
$691$	−33.0114	−1.25581	−0.627907	+	0.778289i	$0.716088\pi$
$692$	0	0
$693$	−25.6679	−0.975042
$694$	0	0
$695$	−6.83394	−0.259226
$696$	0	0
$697$	2.36275	0.0894954
$698$	0	0
$699$	−83.6106	−3.16244
$700$	0	0
$701$	38.4779	1.45329	0.726645	−	0.687013i	$-0.241079\pi$
$701$	38.4779	1.45329	0.726645	+	0.687013i	$0.241079\pi$
$702$	0	0
$703$	11.7797	0.444280
$704$	0	0
$705$	2.13742	0.0804997
$706$	0	0
$707$	−18.9714	−0.713491
$708$	0	0
$709$	−43.1566	−1.62078	−0.810390	−	0.585890i	$-0.800745\pi$
$709$	−43.1566	−1.62078	−0.810390	+	0.585890i	$0.800745\pi$
$710$	0	0
$711$	−34.7605	−1.30362
$712$	0	0
$713$	11.5305	0.431819
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−14.9808	−0.559468
$718$	0	0
$719$	−12.2270	−0.455989	−0.227995	−	0.973662i	$-0.573217\pi$
$719$	−12.2270	−0.455989	−0.227995	+	0.973662i	$0.573217\pi$
$720$	0	0
$721$	−1.14214	−0.0425354
$722$	0	0
$723$	−6.63253	−0.246667
$724$	0	0
$725$	−3.06871	−0.113969
$726$	0	0
$727$	12.1775	0.451638	0.225819	−	0.974169i	$-0.427494\pi$
$727$	12.1775	0.451638	0.225819	+	0.974169i	$0.427494\pi$
$728$	0	0
$729$	−13.5830	−0.503075
$730$	0	0
$731$	7.30348	0.270129
$732$	0	0
$733$	16.8818	0.623543	0.311772	−	0.950157i	$-0.399078\pi$
$733$	16.8818	0.623543	0.311772	+	0.950157i	$0.399078\pi$
$734$	0	0
$735$	3.06871	0.113191
$736$	0	0
$737$	5.66787	0.208779
$738$	0	0
$739$	13.9906	0.514651	0.257326	−	0.966325i	$-0.417159\pi$
$739$	13.9906	0.514651	0.257326	+	0.966325i	$0.417159\pi$
$740$	0	0
$741$	−3.92657	−0.144246
$742$	0	0
$743$	−44.8006	−1.64357	−0.821787	−	0.569795i	$-0.807022\pi$
$743$	−44.8006	−1.64357	−0.821787	+	0.569795i	$0.807022\pi$
$744$	0	0
$745$	−4.69652	−0.172067
$746$	0	0
$747$	−57.5689	−2.10633
$748$	0	0
$749$	15.1088	0.552062
$750$	0	0
$751$	−19.9026	−0.726258	−0.363129	−	0.931739i	$-0.618292\pi$
$751$	−19.9026	−0.726258	−0.363129	+	0.931739i	$0.618292\pi$
$752$	0	0
$753$	31.1088	1.13367
$754$	0	0
$755$	−4.83394	−0.175925
$756$	0	0
$757$	−43.5783	−1.58388	−0.791940	−	0.610598i	$-0.790929\pi$
$757$	−43.5783	−1.58388	−0.791940	+	0.610598i	$0.790929\pi$
$758$	0	0
$759$	−24.5497	−0.891096
$760$	0	0
$761$	31.8036	1.15288	0.576441	−	0.817139i	$-0.304441\pi$
$761$	31.8036	1.15288	0.576441	+	0.817139i	$0.304441\pi$
$762$	0	0
$763$	12.6965	0.459645
$764$	0	0
$765$	4.62309	0.167148
$766$	0	0
$767$	−11.0687	−0.399668
$768$	0	0
$769$	−42.6392	−1.53761	−0.768805	−	0.639483i	$-0.779148\pi$
$769$	−42.6392	−1.53761	−0.768805	+	0.639483i	$0.779148\pi$
$770$	0	0
$771$	−47.8053	−1.72167
$772$	0	0
$773$	3.39304	0.122039	0.0610196	−	0.998137i	$-0.480565\pi$
$773$	3.39304	0.122039	0.0610196	+	0.998137i	$0.480565\pi$
$774$	0	0
$775$	−5.76523	−0.207093
$776$	0	0
$777$	−28.2509	−1.01350
$778$	0	0
$779$	4.19636	0.150350
$780$	0	0
$781$	−2.23642	−0.0800253
$782$	0	0
$783$	−32.1775	−1.14993
$784$	0	0
$785$	15.3435	0.547634
$786$	0	0
$787$	35.3357	1.25958	0.629792	−	0.776764i	$-0.283140\pi$
$787$	35.3357	1.25958	0.629792	+	0.776764i	$0.283140\pi$
$788$	0	0
$789$	−14.9808	−0.533330
$790$	0	0
$791$	16.1374	0.573780
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−14.4122	−0.511150
$796$	0	0
$797$	46.4044	1.64373	0.821865	−	0.569682i	$-0.192934\pi$
$797$	46.4044	1.64373	0.821865	+	0.569682i	$0.192934\pi$
$798$	0	0
$799$	0.501806	0.0177526
$800$	0	0
$801$	90.0946	3.18334
$802$	0	0
$803$	−59.8854	−2.11331
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−20.5497	−0.723382
$808$	0	0
$809$	0.810007	0.0284783	0.0142392	−	0.999899i	$-0.495467\pi$
$809$	0.810007	0.0284783	0.0142392	+	0.999899i	$0.495467\pi$
$810$	0	0
$811$	−5.57831	−0.195881	−0.0979405	−	0.995192i	$-0.531225\pi$
$811$	−5.57831	−0.195881	−0.0979405	+	0.995192i	$0.531225\pi$
$812$	0	0
$813$	36.8245	1.29149
$814$	0	0
$815$	13.7652	0.482175
$816$	0	0
$817$	12.9714	0.453810
$818$	0	0
$819$	6.41697	0.224227
$820$	0	0
$821$	−19.9010	−0.694550	−0.347275	−	0.937763i	$-0.612893\pi$
$821$	−19.9010	−0.694550	−0.347275	+	0.937763i	$0.612893\pi$
$822$	0	0
$823$	−30.4217	−1.06043	−0.530217	−	0.847862i	$-0.677889\pi$
$823$	−30.4217	−1.06043	−0.530217	+	0.847862i	$0.677889\pi$
$824$	0	0
$825$	12.2748	0.427355
$826$	0	0
$827$	−54.4924	−1.89489	−0.947443	−	0.319926i	$-0.896342\pi$
$827$	−54.4924	−1.89489	−0.947443	+	0.319926i	$0.896342\pi$
$828$	0	0
$829$	33.8531	1.17577	0.587884	−	0.808945i	$-0.299961\pi$
$829$	33.8531	1.17577	0.587884	+	0.808945i	$0.299961\pi$
$830$	0	0
$831$	−24.5497	−0.851618
$832$	0	0
$833$	0.720448	0.0249620
$834$	0	0
$835$	2.13742	0.0739683
$836$	0	0
$837$	−60.4523	−2.08954
$838$	0	0
$839$	14.7861	0.510472	0.255236	−	0.966879i	$-0.417847\pi$
$839$	14.7861	0.510472	0.255236	+	0.966879i	$0.417847\pi$
$840$	0	0
$841$	−19.5830	−0.675277
$842$	0	0
$843$	51.6584	1.77921
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−21.4665	−0.736727
$850$	0	0
$851$	−18.4122	−0.631164
$852$	0	0
$853$	−36.3644	−1.24509	−0.622547	−	0.782583i	$-0.713902\pi$
$853$	−36.3644	−1.24509	−0.622547	+	0.782583i	$0.713902\pi$
$854$	0	0
$855$	8.21084	0.280805
$856$	0	0
$857$	−43.2384	−1.47700	−0.738498	−	0.674256i	$-0.764464\pi$
$857$	−43.2384	−1.47700	−0.738498	+	0.674256i	$0.764464\pi$
$858$	0	0
$859$	−4.22697	−0.144223	−0.0721113	−	0.997397i	$-0.522974\pi$
$859$	−4.22697	−0.144223	−0.0721113	+	0.997397i	$0.522974\pi$
$860$	0	0
$861$	−10.0640	−0.342980
$862$	0	0
$863$	32.8006	1.11654	0.558272	−	0.829658i	$-0.311465\pi$
$863$	32.8006	1.11654	0.558272	+	0.829658i	$0.311465\pi$
$864$	0	0
$865$	−9.95050	−0.338327
$866$	0	0
$867$	−50.5752	−1.71763
$868$	0	0
$869$	−21.6679	−0.735032
$870$	0	0
$871$	−1.41697	−0.0480121
$872$	0	0
$873$	−0.881792	−0.0298441
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−47.4170	−1.60116	−0.800579	−	0.599228i	$-0.795474\pi$
$877$	−47.4170	−1.60116	−0.800579	+	0.599228i	$0.795474\pi$
$878$	0	0
$879$	−81.3263	−2.74307
$880$	0	0
$881$	−44.0228	−1.48317	−0.741583	−	0.670861i	$-0.765925\pi$
$881$	−44.0228	−1.48317	−0.741583	+	0.670861i	$0.765925\pi$
$882$	0	0
$883$	−43.1566	−1.45234	−0.726168	−	0.687517i	$-0.758701\pi$
$883$	−43.1566	−1.45234	−0.726168	+	0.687517i	$0.758701\pi$
$884$	0	0
$885$	33.9666	1.14178
$886$	0	0
$887$	−37.6662	−1.26471	−0.632354	−	0.774680i	$-0.717911\pi$
$887$	−37.6662	−1.26471	−0.632354	+	0.774680i	$0.717911\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	51.7063	1.73223
$892$	0	0
$893$	0.891233	0.0298240
$894$	0	0
$895$	−15.1327	−0.505830
$896$	0	0
$897$	6.13742	0.204922
$898$	0	0
$899$	17.6918	0.590055
$900$	0	0
$901$	−3.38360	−0.112724
$902$	0	0
$903$	−31.1088	−1.03523
$904$	0	0
$905$	7.86258	0.261361
$906$	0	0
$907$	−35.6106	−1.18243	−0.591215	−	0.806514i	$-0.701351\pi$
$907$	−35.6106	−1.18243	−0.591215	+	0.806514i	$0.701351\pi$
$908$	0	0
$909$	121.739	4.03781
$910$	0	0
$911$	39.9093	1.32226	0.661128	−	0.750273i	$-0.270078\pi$
$911$	39.9093	1.32226	0.661128	+	0.750273i	$0.270078\pi$
$912$	0	0
$913$	−35.8854	−1.18763
$914$	0	0
$915$	−6.13742	−0.202897
$916$	0	0
$917$	8.97135	0.296260
$918$	0	0
$919$	−33.1967	−1.09506	−0.547529	−	0.836787i	$-0.684431\pi$
$919$	−33.1967	−1.09506	−0.547529	+	0.836787i	$0.684431\pi$
$920$	0	0
$921$	−26.6871	−0.879369
$922$	0	0
$923$	0.559104	0.0184031
$924$	0	0
$925$	9.20612	0.302695
$926$	0	0
$927$	7.32905	0.240718
$928$	0	0
$929$	39.8036	1.30592	0.652958	−	0.757394i	$-0.273528\pi$
$929$	39.8036	1.30592	0.652958	+	0.757394i	$0.273528\pi$
$930$	0	0
$931$	1.27955	0.0419356
$932$	0	0
$933$	4.42169	0.144760
$934$	0	0
$935$	2.88179	0.0942447
$936$	0	0
$937$	11.2317	0.366924	0.183462	−	0.983027i	$-0.441270\pi$
$937$	11.2317	0.366924	0.183462	+	0.983027i	$0.441270\pi$
$938$	0	0
$939$	−61.9188	−2.02064
$940$	0	0
$941$	8.03337	0.261880	0.130940	−	0.991390i	$-0.458200\pi$
$941$	8.03337	0.261880	0.130940	+	0.991390i	$0.458200\pi$
$942$	0	0
$943$	−6.55910	−0.213594
$944$	0	0
$945$	−10.4857	−0.341099
$946$	0	0
$947$	56.5257	1.83684	0.918420	−	0.395607i	$-0.129466\pi$
$947$	56.5257	1.83684	0.918420	+	0.395607i	$0.129466\pi$
$948$	0	0
$949$	14.9714	0.485991
$950$	0	0
$951$	−6.98079	−0.226368
$952$	0	0
$953$	−29.5783	−0.958135	−0.479068	−	0.877778i	$-0.659025\pi$
$953$	−29.5783	−0.958135	−0.479068	+	0.877778i	$0.659025\pi$
$954$	0	0
$955$	−26.0401	−0.842637
$956$	0	0
$957$	−37.6679	−1.21763
$958$	0	0
$959$	5.20612	0.168114
$960$	0	0
$961$	2.23785	0.0721887
$962$	0	0
$963$	−96.9525	−3.12425
$964$	0	0
$965$	−0.857864	−0.0276156
$966$	0	0
$967$	11.9026	0.382763	0.191382	−	0.981516i	$-0.438703\pi$
$967$	11.9026	0.382763	0.191382	+	0.981516i	$0.438703\pi$
$968$	0	0
$969$	2.82889	0.0908770
$970$	0	0
$971$	−15.1566	−0.486399	−0.243200	−	0.969976i	$-0.578197\pi$
$971$	−15.1566	−0.486399	−0.243200	+	0.969976i	$0.578197\pi$
$972$	0	0
$973$	6.83394	0.219086
$974$	0	0
$975$	−3.06871	−0.0982773
$976$	0	0
$977$	−21.4075	−0.684887	−0.342444	−	0.939538i	$-0.611255\pi$
$977$	−21.4075	−0.684887	−0.342444	+	0.939538i	$0.611255\pi$
$978$	0	0
$979$	56.1602	1.79489
$980$	0	0
$981$	−81.4732	−2.60124
$982$	0	0
$983$	23.3035	0.743266	0.371633	−	0.928380i	$-0.378798\pi$
$983$	23.3035	0.743266	0.371633	+	0.928380i	$0.378798\pi$
$984$	0	0
$985$	20.7366	0.660722
$986$	0	0
$987$	−2.13742	−0.0680347
$988$	0	0
$989$	−20.2748	−0.644702
$990$	0	0
$991$	−14.8100	−0.470455	−0.235228	−	0.971940i	$-0.575584\pi$
$991$	−14.8100	−0.470455	−0.235228	+	0.971940i	$0.575584\pi$
$992$	0	0
$993$	58.7861	1.86552
$994$	0	0
$995$	0.744376	0.0235983
$996$	0	0
$997$	−0.308201	−0.00976082	−0.00488041	−	0.999988i	$-0.501553\pi$
$997$	−0.308201	−0.00976082	−0.00488041	+	0.999988i	$0.501553\pi$
$998$	0	0
$999$	96.5324	3.05415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.q.1.3	✓	3
4.3	odd	2		7280.2.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.q.1.3	✓	3	1.1	even	1	trivial
7280.2.a.bm.1.1		3	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)

\(f(q)\)	\(=\)	\(q+3.06871 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.41697 q^{9} +O(q^{10})\)	\(q+3.06871 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.41697 q^{9} +4.00000 q^{11} -1.00000 q^{13} +3.06871 q^{15} +0.720448 q^{17} +1.27955 q^{19} -3.06871 q^{21} -2.00000 q^{23} +1.00000 q^{25} +10.4857 q^{27} -3.06871 q^{29} -5.76523 q^{31} +12.2748 q^{33} -1.00000 q^{35} +9.20612 q^{37} -3.06871 q^{39} +3.27955 q^{41} +10.1374 q^{43} +6.41697 q^{45} +0.696520 q^{47} +1.00000 q^{49} +2.21084 q^{51} -4.69652 q^{53} +4.00000 q^{55} +3.92657 q^{57} +11.0687 q^{59} -2.00000 q^{61} -6.41697 q^{63} -1.00000 q^{65} +1.41697 q^{67} -6.13742 q^{69} -0.559104 q^{71} -14.9714 q^{73} +3.06871 q^{75} -4.00000 q^{77} -5.41697 q^{79} +12.9266 q^{81} -8.97135 q^{83} +0.720448 q^{85} -9.41697 q^{87} +14.0401 q^{89} +1.00000 q^{91} -17.6918 q^{93} +1.27955 q^{95} -0.137416 q^{97} +25.6679 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9	\( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9} + 12 q^{11} - 3 q^{13} + q^{15} - q^{17} + 7 q^{19} - q^{21} - 6 q^{23} + 3 q^{25} + 10 q^{27} - q^{29} + q^{31} + 4 q^{33} - 3 q^{35} + 3 q^{37} - q^{39} + 13 q^{41} + 14 q^{43} + 6 q^{45} - 8 q^{47} + 3 q^{49} + 18 q^{51} - 4 q^{53} + 12 q^{55} - 16 q^{57} + 25 q^{59} - 6 q^{61} - 6 q^{63} - 3 q^{65} - 9 q^{67} - 2 q^{69} - 8 q^{71} - 2 q^{73} + q^{75} - 12 q^{77} - 3 q^{79} + 11 q^{81} + 16 q^{83} - q^{85} - 15 q^{87} - 9 q^{89} + 3 q^{91} - 7 q^{93} + 7 q^{95} + 16 q^{97} + 24 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9 + 12 * q^11 - 3 * q^13 + q^15 - q^17 + 7 * q^19 - q^21 - 6 * q^23 + 3 * q^25 + 10 * q^27 - q^29 + q^31 + 4 * q^33 - 3 * q^35 + 3 * q^37 - q^39 + 13 * q^41 + 14 * q^43 + 6 * q^45 - 8 * q^47 + 3 * q^49 + 18 * q^51 - 4 * q^53 + 12 * q^55 - 16 * q^57 + 25 * q^59 - 6 * q^61 - 6 * q^63 - 3 * q^65 - 9 * q^67 - 2 * q^69 - 8 * q^71 - 2 * q^73 + q^75 - 12 * q^77 - 3 * q^79 + 11 * q^81 + 16 * q^83 - q^85 - 15 * q^87 - 9 * q^89 + 3 * q^91 - 7 * q^93 + 7 * q^95 + 16 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

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