LMFDB - Embedded newform 3640.2.a.s.1.4

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:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.75660$ 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)

$f(q)$	$=$	$q+1.75660 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.0856374 q^{9} +O(q^{10})$	$q+1.75660 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.0856374 q^{9} -3.23607 q^{11} +1.00000 q^{13} +1.75660 q^{15} -4.59883 q^{17} +0.150431 q^{19} -1.75660 q^{21} +1.40734 q^{23} +1.00000 q^{25} -5.11936 q^{27} -8.67714 q^{29} -4.96893 q^{31} -5.68447 q^{33} -1.00000 q^{35} -0.520530 q^{37} +1.75660 q^{39} +3.55777 q^{41} +0.277129 q^{43} +0.0856374 q^{45} -11.1155 q^{47} +1.00000 q^{49} -8.07830 q^{51} +0.171275 q^{53} -3.23607 q^{55} +0.264246 q^{57} +7.78362 q^{59} +3.68447 q^{61} -0.0856374 q^{63} +1.00000 q^{65} -1.95542 q^{67} +2.47214 q^{69} -1.55160 q^{71} +5.85575 q^{73} +1.75660 q^{75} +3.23607 q^{77} -5.00352 q^{79} -9.24958 q^{81} -10.3279 q^{83} -4.59883 q^{85} -15.2422 q^{87} -15.4918 q^{89} -1.00000 q^{91} -8.72842 q^{93} +0.150431 q^{95} +0.869786 q^{97} -0.277129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * 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
$3$	1.75660	1.01417	0.507086	−	0.861895i	$-0.330723\pi$
$3$	1.75660	1.01417	0.507086	+	0.861895i	$0.330723\pi$
$4$	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
$9$	0.0856374	0.0285458
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.75660	0.453552
$16$	0	0
$17$	−4.59883	−1.11538	−0.557691	−	0.830049i	$-0.688312\pi$
$17$	−4.59883	−1.11538	−0.557691	+	0.830049i	$0.688312\pi$
$18$	0	0
$19$	0.150431	0.0345111	0.0172556	−	0.999851i	$-0.494507\pi$
$19$	0.150431	0.0345111	0.0172556	+	0.999851i	$0.494507\pi$
$20$	0	0
$21$	−1.75660	−0.383321
$22$	0	0
$23$	1.40734	0.293451	0.146726	−	0.989177i	$-0.453127\pi$
$23$	1.40734	0.293451	0.146726	+	0.989177i	$0.453127\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.11936	−0.985222
$28$	0	0
$29$	−8.67714	−1.61130	−0.805652	−	0.592389i	$-0.798185\pi$
$29$	−8.67714	−1.61130	−0.805652	+	0.592389i	$0.798185\pi$
$30$	0	0
$31$	−4.96893	−0.892447	−0.446223	−	0.894922i	$-0.647231\pi$
$31$	−4.96893	−0.892447	−0.446223	+	0.894922i	$0.647231\pi$
$32$	0	0
$33$	−5.68447	−0.989539
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.520530	−0.0855747	−0.0427873	−	0.999084i	$-0.513624\pi$
$37$	−0.520530	−0.0855747	−0.0427873	+	0.999084i	$0.513624\pi$
$38$	0	0
$39$	1.75660	0.281281
$40$	0	0
$41$	3.55777	0.555631	0.277815	−	0.960634i	$-0.410390\pi$
$41$	3.55777	0.555631	0.277815	+	0.960634i	$0.410390\pi$
$42$	0	0
$43$	0.277129	0.0422617	0.0211309	−	0.999777i	$-0.493273\pi$
$43$	0.277129	0.0422617	0.0211309	+	0.999777i	$0.493273\pi$
$44$	0	0
$45$	0.0856374	0.0127661
$46$	0	0
$47$	−11.1155	−1.62137	−0.810685	−	0.585483i	$-0.800905\pi$
$47$	−11.1155	−1.62137	−0.810685	+	0.585483i	$0.800905\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.07830	−1.13119
$52$	0	0
$53$	0.171275	0.0235264	0.0117632	−	0.999931i	$-0.496256\pi$
$53$	0.171275	0.0235264	0.0117632	+	0.999931i	$0.496256\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	0	0
$57$	0.264246	0.0350002
$58$	0	0
$59$	7.78362	1.01334	0.506670	−	0.862140i	$-0.330876\pi$
$59$	7.78362	1.01334	0.506670	+	0.862140i	$0.330876\pi$
$60$	0	0
$61$	3.68447	0.471748	0.235874	−	0.971784i	$-0.424205\pi$
$61$	3.68447	0.471748	0.235874	+	0.971784i	$0.424205\pi$
$62$	0	0
$63$	−0.0856374	−0.0107893
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.95542	−0.238893	−0.119446	−	0.992841i	$-0.538112\pi$
$67$	−1.95542	−0.238893	−0.119446	+	0.992841i	$0.538112\pi$
$68$	0	0
$69$	2.47214	0.297610
$70$	0	0
$71$	−1.55160	−0.184141	−0.0920703	−	0.995753i	$-0.529348\pi$
$71$	−1.55160	−0.184141	−0.0920703	+	0.995753i	$0.529348\pi$
$72$	0	0
$73$	5.85575	0.685363	0.342682	−	0.939452i	$-0.388665\pi$
$73$	5.85575	0.685363	0.342682	+	0.939452i	$0.388665\pi$
$74$	0	0
$75$	1.75660	0.202834
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	−5.00352	−0.562940	−0.281470	−	0.959570i	$-0.590822\pi$
$79$	−5.00352	−0.562940	−0.281470	+	0.959570i	$0.590822\pi$
$80$	0	0
$81$	−9.24958	−1.02773
$82$	0	0
$83$	−10.3279	−1.13363	−0.566816	−	0.823844i	$-0.691825\pi$
$83$	−10.3279	−1.13363	−0.566816	+	0.823844i	$0.691825\pi$
$84$	0	0
$85$	−4.59883	−0.498814
$86$	0	0
$87$	−15.2422	−1.63414
$88$	0	0
$89$	−15.4918	−1.64213	−0.821065	−	0.570835i	$-0.806620\pi$
$89$	−15.4918	−1.64213	−0.821065	+	0.570835i	$0.806620\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−8.72842	−0.905095
$94$	0	0
$95$	0.150431	0.0154338
$96$	0	0
$97$	0.869786	0.0883134	0.0441567	−	0.999025i	$-0.485940\pi$
$97$	0.869786	0.0883134	0.0441567	+	0.999025i	$0.485940\pi$
$98$	0	0
$99$	−0.277129	−0.0278525
$100$	0	0
$101$	−4.08212	−0.406186	−0.203093	−	0.979159i	$-0.565099\pi$
$101$	−4.08212	−0.406186	−0.203093	+	0.979159i	$0.565099\pi$
$102$	0	0
$103$	3.41023	0.336020	0.168010	−	0.985785i	$-0.446266\pi$
$103$	3.41023	0.336020	0.168010	+	0.985785i	$0.446266\pi$
$104$	0	0
$105$	−1.75660	−0.171426
$106$	0	0
$107$	−10.4067	−1.00606	−0.503028	−	0.864270i	$-0.667781\pi$
$107$	−10.4067	−1.00606	−0.503028	+	0.864270i	$0.667781\pi$
$108$	0	0
$109$	10.1566	0.972827	0.486413	−	0.873729i	$-0.338305\pi$
$109$	10.1566	0.972827	0.486413	+	0.873729i	$0.338305\pi$
$110$	0	0
$111$	−0.914363	−0.0867875
$112$	0	0
$113$	7.19767	0.677100	0.338550	−	0.940948i	$-0.390064\pi$
$113$	7.19767	0.677100	0.338550	+	0.940948i	$0.390064\pi$
$114$	0	0
$115$	1.40734	0.131235
$116$	0	0
$117$	0.0856374	0.00791718
$118$	0	0
$119$	4.59883	0.421574
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	6.24958	0.563506
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.23607	−0.287155	−0.143577	−	0.989639i	$-0.545861\pi$
$127$	−3.23607	−0.287155	−0.143577	+	0.989639i	$0.545861\pi$
$128$	0	0
$129$	0.486803	0.0428607
$130$	0	0
$131$	12.3132	1.07581	0.537905	−	0.843005i	$-0.319216\pi$
$131$	12.3132	1.07581	0.537905	+	0.843005i	$0.319216\pi$
$132$	0	0
$133$	−0.150431	−0.0130440
$134$	0	0
$135$	−5.11936	−0.440605
$136$	0	0
$137$	0.308823	0.0263845	0.0131923	−	0.999913i	$-0.495801\pi$
$137$	0.308823	0.0263845	0.0131923	+	0.999913i	$0.495801\pi$
$138$	0	0
$139$	−0.740199	−0.0627829	−0.0313914	−	0.999507i	$-0.509994\pi$
$139$	−0.740199	−0.0627829	−0.0313914	+	0.999507i	$0.509994\pi$
$140$	0	0
$141$	−19.5256	−1.64435
$142$	0	0
$143$	−3.23607	−0.270614
$144$	0	0
$145$	−8.67714	−0.720597
$146$	0	0
$147$	1.75660	0.144882
$148$	0	0
$149$	−5.04809	−0.413556	−0.206778	−	0.978388i	$-0.566298\pi$
$149$	−5.04809	−0.413556	−0.206778	+	0.978388i	$0.566298\pi$
$150$	0	0
$151$	7.43436	0.605000	0.302500	−	0.953149i	$-0.402179\pi$
$151$	7.43436	0.605000	0.302500	+	0.953149i	$0.402179\pi$
$152$	0	0
$153$	−0.393832	−0.0318395
$154$	0	0
$155$	−4.96893	−0.399114
$156$	0	0
$157$	−19.3059	−1.54078	−0.770389	−	0.637575i	$-0.779938\pi$
$157$	−19.3059	−1.54078	−0.770389	+	0.637575i	$0.779938\pi$
$158$	0	0
$159$	0.300861	0.0238598
$160$	0	0
$161$	−1.40734	−0.110914
$162$	0	0
$163$	−17.6361	−1.38136	−0.690682	−	0.723159i	$-0.742690\pi$
$163$	−17.6361	−1.38136	−0.690682	+	0.723159i	$0.742690\pi$
$164$	0	0
$165$	−5.68447	−0.442535
$166$	0	0
$167$	19.5256	1.51093	0.755466	−	0.655188i	$-0.227410\pi$
$167$	19.5256	1.51093	0.755466	+	0.655188i	$0.227410\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.0128825	0.000985148 0
$172$	0	0
$173$	1.07479	0.0817146	0.0408573	−	0.999165i	$-0.486991\pi$
$173$	1.07479	0.0817146	0.0408573	+	0.999165i	$0.486991\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	13.6727	1.02770
$178$	0	0
$179$	−4.07160	−0.304326	−0.152163	−	0.988355i	$-0.548624\pi$
$179$	−4.07160	−0.304326	−0.152163	+	0.988355i	$0.548624\pi$
$180$	0	0
$181$	25.9907	1.93187	0.965935	−	0.258785i	$-0.0833220\pi$
$181$	25.9907	1.93187	0.965935	+	0.258785i	$0.0833220\pi$
$182$	0	0
$183$	6.47214	0.478434
$184$	0	0
$185$	−0.520530	−0.0382701
$186$	0	0
$187$	14.8821	1.08829
$188$	0	0
$189$	5.11936	0.372379
$190$	0	0
$191$	−16.9927	−1.22955	−0.614773	−	0.788704i	$-0.710752\pi$
$191$	−16.9927	−1.22955	−0.614773	+	0.788704i	$0.710752\pi$
$192$	0	0
$193$	22.4874	1.61868	0.809338	−	0.587343i	$-0.199826\pi$
$193$	22.4874	1.61868	0.809338	+	0.587343i	$0.199826\pi$
$194$	0	0
$195$	1.75660	0.125793
$196$	0	0
$197$	−2.11585	−0.150748	−0.0753740	−	0.997155i	$-0.524015\pi$
$197$	−2.11585	−0.150748	−0.0753740	+	0.997155i	$0.524015\pi$
$198$	0	0
$199$	8.13491	0.576668	0.288334	−	0.957530i	$-0.406899\pi$
$199$	8.13491	0.576668	0.288334	+	0.957530i	$0.406899\pi$
$200$	0	0
$201$	−3.43489	−0.242279
$202$	0	0
$203$	8.67714	0.609016
$204$	0	0
$205$	3.55777	0.248486
$206$	0	0
$207$	0.120521	0.00837680
$208$	0	0
$209$	−0.486803	−0.0336729
$210$	0	0
$211$	−18.6663	−1.28504	−0.642520	−	0.766269i	$-0.722111\pi$
$211$	−18.6663	−1.28504	−0.642520	+	0.766269i	$0.722111\pi$
$212$	0	0
$213$	−2.72553	−0.186750
$214$	0	0
$215$	0.277129	0.0189000
$216$	0	0
$217$	4.96893	0.337313
$218$	0	0
$219$	10.2862	0.695077
$220$	0	0
$221$	−4.59883	−0.309351
$222$	0	0
$223$	17.8885	1.19791	0.598953	−	0.800784i	$-0.295584\pi$
$223$	17.8885	1.19791	0.598953	+	0.800784i	$0.295584\pi$
$224$	0	0
$225$	0.0856374	0.00570916
$226$	0	0
$227$	−9.59532	−0.636864	−0.318432	−	0.947946i	$-0.603156\pi$
$227$	−9.59532	−0.636864	−0.318432	+	0.947946i	$0.603156\pi$
$228$	0	0
$229$	14.9267	0.986385	0.493193	−	0.869920i	$-0.335830\pi$
$229$	14.9267	0.986385	0.493193	+	0.869920i	$0.335830\pi$
$230$	0	0
$231$	5.68447	0.374011
$232$	0	0
$233$	−1.64404	−0.107705	−0.0538523	−	0.998549i	$-0.517150\pi$
$233$	−1.64404	−0.107705	−0.0538523	+	0.998549i	$0.517150\pi$
$234$	0	0
$235$	−11.1155	−0.725098
$236$	0	0
$237$	−8.78917	−0.570918
$238$	0	0
$239$	8.32459	0.538473	0.269237	−	0.963074i	$-0.413229\pi$
$239$	8.32459	0.538473	0.269237	+	0.963074i	$0.413229\pi$
$240$	0	0
$241$	−16.8446	−1.08506	−0.542528	−	0.840038i	$-0.682533\pi$
$241$	−16.8446	−1.08506	−0.542528	+	0.840038i	$0.682533\pi$
$242$	0	0
$243$	−0.889701	−0.0570743
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	0.150431	0.00957167
$248$	0	0
$249$	−18.1419	−1.14970
$250$	0	0
$251$	5.66277	0.357431	0.178715	−	0.983901i	$-0.442806\pi$
$251$	5.66277	0.357431	0.178715	+	0.983901i	$0.442806\pi$
$252$	0	0
$253$	−4.55426	−0.286324
$254$	0	0
$255$	−8.07830	−0.505883
$256$	0	0
$257$	15.2868	0.953566	0.476783	−	0.879021i	$-0.341803\pi$
$257$	15.2868	0.953566	0.476783	+	0.879021i	$0.341803\pi$
$258$	0	0
$259$	0.520530	0.0323442
$260$	0	0
$261$	−0.743088	−0.0459960
$262$	0	0
$263$	−10.6942	−0.659430	−0.329715	−	0.944080i	$-0.606953\pi$
$263$	−10.6942	−0.659430	−0.329715	+	0.944080i	$0.606953\pi$
$264$	0	0
$265$	0.171275	0.0105213
$266$	0	0
$267$	−27.2129	−1.66540
$268$	0	0
$269$	−0.521459	−0.0317939	−0.0158970	−	0.999874i	$-0.505060\pi$
$269$	−0.521459	−0.0317939	−0.0158970	+	0.999874i	$0.505060\pi$
$270$	0	0
$271$	−4.29883	−0.261135	−0.130568	−	0.991439i	$-0.541680\pi$
$271$	−4.29883	−0.261135	−0.130568	+	0.991439i	$0.541680\pi$
$272$	0	0
$273$	−1.75660	−0.106314
$274$	0	0
$275$	−3.23607	−0.195142
$276$	0	0
$277$	−4.58128	−0.275262	−0.137631	−	0.990484i	$-0.543949\pi$
$277$	−4.58128	−0.275262	−0.137631	+	0.990484i	$0.543949\pi$
$278$	0	0
$279$	−0.425527	−0.0254756
$280$	0	0
$281$	−5.04809	−0.301144	−0.150572	−	0.988599i	$-0.548112\pi$
$281$	−5.04809	−0.301144	−0.150572	+	0.988599i	$0.548112\pi$
$282$	0	0
$283$	−28.2777	−1.68093	−0.840467	−	0.541862i	$-0.817720\pi$
$283$	−28.2777	−1.68093	−0.840467	+	0.541862i	$0.817720\pi$
$284$	0	0
$285$	0.264246	0.0156526
$286$	0	0
$287$	−3.55777	−0.210009
$288$	0	0
$289$	4.14927	0.244075
$290$	0	0
$291$	1.52786	0.0895650
$292$	0	0
$293$	10.7109	0.625735	0.312868	−	0.949797i	$-0.398710\pi$
$293$	10.7109	0.625735	0.312868	+	0.949797i	$0.398710\pi$
$294$	0	0
$295$	7.78362	0.453180
$296$	0	0
$297$	16.5666	0.961292
$298$	0	0
$299$	1.40734	0.0813887
$300$	0	0
$301$	−0.277129	−0.0159734
$302$	0	0
$303$	−7.17065	−0.411943
$304$	0	0
$305$	3.68447	0.210972
$306$	0	0
$307$	3.74660	0.213830	0.106915	−	0.994268i	$-0.465903\pi$
$307$	3.74660	0.213830	0.106915	+	0.994268i	$0.465903\pi$
$308$	0	0
$309$	5.99041	0.340782
$310$	0	0
$311$	−7.01404	−0.397730	−0.198865	−	0.980027i	$-0.563725\pi$
$311$	−7.01404	−0.397730	−0.198865	+	0.980027i	$0.563725\pi$
$312$	0	0
$313$	−11.6778	−0.660066	−0.330033	−	0.943969i	$-0.607060\pi$
$313$	−11.6778	−0.660066	−0.330033	+	0.943969i	$0.607060\pi$
$314$	0	0
$315$	−0.0856374	−0.00482512
$316$	0	0
$317$	32.6070	1.83139	0.915697	−	0.401869i	$-0.131639\pi$
$317$	32.6070	1.83139	0.915697	+	0.401869i	$0.131639\pi$
$318$	0	0
$319$	28.0798	1.57217
$320$	0	0
$321$	−18.2804	−1.02031
$322$	0	0
$323$	−0.691805	−0.0384931
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	17.8411	0.986614
$328$	0	0
$329$	11.1155	0.612820
$330$	0	0
$331$	−23.2561	−1.27827	−0.639134	−	0.769095i	$-0.720707\pi$
$331$	−23.2561	−1.27827	−0.639134	+	0.769095i	$0.720707\pi$
$332$	0	0
$333$	−0.0445769	−0.00244280
$334$	0	0
$335$	−1.95542	−0.106836
$336$	0	0
$337$	13.6575	0.743969	0.371984	−	0.928239i	$-0.378677\pi$
$337$	13.6575	0.743969	0.371984	+	0.928239i	$0.378677\pi$
$338$	0	0
$339$	12.6434	0.686696
$340$	0	0
$341$	16.0798	0.870770
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.47214	0.133095
$346$	0	0
$347$	−5.87245	−0.315249	−0.157625	−	0.987499i	$-0.550384\pi$
$347$	−5.87245	−0.315249	−0.157625	+	0.987499i	$0.550384\pi$
$348$	0	0
$349$	14.9576	0.800660	0.400330	−	0.916371i	$-0.368896\pi$
$349$	14.9576	0.800660	0.400330	+	0.916371i	$0.368896\pi$
$350$	0	0
$351$	−5.11936	−0.273251
$352$	0	0
$353$	−19.3343	−1.02906	−0.514530	−	0.857472i	$-0.672034\pi$
$353$	−19.3343	−1.02906	−0.514530	+	0.857472i	$0.672034\pi$
$354$	0	0
$355$	−1.55160	−0.0823502
$356$	0	0
$357$	8.07830	0.427549
$358$	0	0
$359$	21.1863	1.11817	0.559085	−	0.829110i	$-0.311153\pi$
$359$	21.1863	1.11817	0.559085	+	0.829110i	$0.311153\pi$
$360$	0	0
$361$	−18.9774	−0.998809
$362$	0	0
$363$	−0.927245	−0.0486677
$364$	0	0
$365$	5.85575	0.306504
$366$	0	0
$367$	−13.2765	−0.693028	−0.346514	−	0.938045i	$-0.612635\pi$
$367$	−13.2765	−0.693028	−0.346514	+	0.938045i	$0.612635\pi$
$368$	0	0
$369$	0.304679	0.0158609
$370$	0	0
$371$	−0.171275	−0.00889215
$372$	0	0
$373$	17.1613	0.888578	0.444289	−	0.895883i	$-0.353456\pi$
$373$	17.1613	0.888578	0.444289	+	0.895883i	$0.353456\pi$
$374$	0	0
$375$	1.75660	0.0907103
$376$	0	0
$377$	−8.67714	−0.446895
$378$	0	0
$379$	−23.6167	−1.21311	−0.606555	−	0.795041i	$-0.707449\pi$
$379$	−23.6167	−1.21311	−0.606555	+	0.795041i	$0.707449\pi$
$380$	0	0
$381$	−5.68447	−0.291224
$382$	0	0
$383$	7.88151	0.402726	0.201363	−	0.979517i	$-0.435463\pi$
$383$	7.88151	0.402726	0.201363	+	0.979517i	$0.435463\pi$
$384$	0	0
$385$	3.23607	0.164925
$386$	0	0
$387$	0.0237326	0.00120639
$388$	0	0
$389$	13.3865	0.678722	0.339361	−	0.940656i	$-0.389789\pi$
$389$	13.3865	0.678722	0.339361	+	0.940656i	$0.389789\pi$
$390$	0	0
$391$	−6.47214	−0.327310
$392$	0	0
$393$	21.6294	1.09106
$394$	0	0
$395$	−5.00352	−0.251754
$396$	0	0
$397$	−8.25108	−0.414110	−0.207055	−	0.978329i	$-0.566388\pi$
$397$	−8.25108	−0.414110	−0.207055	+	0.978329i	$0.566388\pi$
$398$	0	0
$399$	−0.264246	−0.0132288
$400$	0	0
$401$	29.8388	1.49008	0.745038	−	0.667022i	$-0.232431\pi$
$401$	29.8388	1.49008	0.745038	+	0.667022i	$0.232431\pi$
$402$	0	0
$403$	−4.96893	−0.247520
$404$	0	0
$405$	−9.24958	−0.459615
$406$	0	0
$407$	1.68447	0.0834961
$408$	0	0
$409$	−19.1085	−0.944855	−0.472428	−	0.881369i	$-0.656622\pi$
$409$	−19.1085	−0.944855	−0.472428	+	0.881369i	$0.656622\pi$
$410$	0	0
$411$	0.542478	0.0267585
$412$	0	0
$413$	−7.78362	−0.383007
$414$	0	0
$415$	−10.3279	−0.506976
$416$	0	0
$417$	−1.30023	−0.0636727
$418$	0	0
$419$	−13.1560	−0.642712	−0.321356	−	0.946959i	$-0.604139\pi$
$419$	−13.1560	−0.642712	−0.321356	+	0.946959i	$0.604139\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	−0.951907	−0.0462833
$424$	0	0
$425$	−4.59883	−0.223076
$426$	0	0
$427$	−3.68447	−0.178304
$428$	0	0
$429$	−5.68447	−0.274449
$430$	0	0
$431$	−25.7463	−1.24016	−0.620078	−	0.784540i	$-0.712899\pi$
$431$	−25.7463	−1.24016	−0.620078	+	0.784540i	$0.712899\pi$
$432$	0	0
$433$	31.2182	1.50025	0.750126	−	0.661295i	$-0.229993\pi$
$433$	31.2182	1.50025	0.750126	+	0.661295i	$0.229993\pi$
$434$	0	0
$435$	−15.2422	−0.730810
$436$	0	0
$437$	0.211707	0.0101273
$438$	0	0
$439$	2.70554	0.129129	0.0645643	−	0.997914i	$-0.479434\pi$
$439$	2.70554	0.129129	0.0645643	+	0.997914i	$0.479434\pi$
$440$	0	0
$441$	0.0856374	0.00407797
$442$	0	0
$443$	−19.2144	−0.912902	−0.456451	−	0.889749i	$-0.650880\pi$
$443$	−19.2144	−0.912902	−0.456451	+	0.889749i	$0.650880\pi$
$444$	0	0
$445$	−15.4918	−0.734383
$446$	0	0
$447$	−8.86747	−0.419417
$448$	0	0
$449$	11.8287	0.558232	0.279116	−	0.960257i	$-0.409959\pi$
$449$	11.8287	0.558232	0.279116	+	0.960257i	$0.409959\pi$
$450$	0	0
$451$	−11.5132	−0.542135
$452$	0	0
$453$	13.0592	0.613574
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	3.96627	0.185534	0.0927672	−	0.995688i	$-0.470429\pi$
$457$	3.96627	0.185534	0.0927672	+	0.995688i	$0.470429\pi$
$458$	0	0
$459$	23.5431	1.09890
$460$	0	0
$461$	0.724603	0.0337481	0.0168741	−	0.999858i	$-0.494629\pi$
$461$	0.724603	0.0337481	0.0168741	+	0.999858i	$0.494629\pi$
$462$	0	0
$463$	−10.6112	−0.493144	−0.246572	−	0.969125i	$-0.579304\pi$
$463$	−10.6112	−0.493144	−0.246572	+	0.969125i	$0.579304\pi$
$464$	0	0
$465$	−8.72842	−0.404771
$466$	0	0
$467$	−42.4613	−1.96488	−0.982438	−	0.186589i	$-0.940257\pi$
$467$	−42.4613	−1.96488	−0.982438	+	0.186589i	$0.940257\pi$
$468$	0	0
$469$	1.95542	0.0902931
$470$	0	0
$471$	−33.9127	−1.56261
$472$	0	0
$473$	−0.896807	−0.0412352
$474$	0	0
$475$	0.150431	0.00690223
$476$	0	0
$477$	0.0146675	0.000671580 0
$478$	0	0
$479$	22.5349	1.02965	0.514823	−	0.857296i	$-0.327858\pi$
$479$	22.5349	1.02965	0.514823	+	0.857296i	$0.327858\pi$
$480$	0	0
$481$	−0.520530	−0.0237341
$482$	0	0
$483$	−2.47214	−0.112486
$484$	0	0
$485$	0.869786	0.0394949
$486$	0	0
$487$	8.41671	0.381397	0.190699	−	0.981649i	$-0.438925\pi$
$487$	8.41671	0.381397	0.190699	+	0.981649i	$0.438925\pi$
$488$	0	0
$489$	−30.9795	−1.40094
$490$	0	0
$491$	−37.2909	−1.68291	−0.841457	−	0.540324i	$-0.818302\pi$
$491$	−37.2909	−1.68291	−0.841457	+	0.540324i	$0.818302\pi$
$492$	0	0
$493$	39.9047	1.79722
$494$	0	0
$495$	−0.277129	−0.0124560
$496$	0	0
$497$	1.55160	0.0695986
$498$	0	0
$499$	18.6250	0.833769	0.416885	−	0.908959i	$-0.363122\pi$
$499$	18.6250	0.833769	0.416885	+	0.908959i	$0.363122\pi$
$500$	0	0
$501$	34.2985	1.53235
$502$	0	0
$503$	−31.0074	−1.38255	−0.691275	−	0.722591i	$-0.742951\pi$
$503$	−31.0074	−1.38255	−0.691275	+	0.722591i	$0.742951\pi$
$504$	0	0
$505$	−4.08212	−0.181652
$506$	0	0
$507$	1.75660	0.0780133
$508$	0	0
$509$	−6.15246	−0.272703	−0.136352	−	0.990661i	$-0.543538\pi$
$509$	−6.15246	−0.272703	−0.136352	+	0.990661i	$0.543538\pi$
$510$	0	0
$511$	−5.85575	−0.259043
$512$	0	0
$513$	−0.770109	−0.0340011
$514$	0	0
$515$	3.41023	0.150273
$516$	0	0
$517$	35.9707	1.58199
$518$	0	0
$519$	1.88797	0.0828727
$520$	0	0
$521$	−42.1690	−1.84746	−0.923728	−	0.383049i	$-0.874874\pi$
$521$	−42.1690	−1.84746	−0.923728	+	0.383049i	$0.874874\pi$
$522$	0	0
$523$	22.9599	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$523$	22.9599	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$524$	0	0
$525$	−1.75660	−0.0766642
$526$	0	0
$527$	22.8513	0.995418
$528$	0	0
$529$	−21.0194	−0.913886
$530$	0	0
$531$	0.666569	0.0289266
$532$	0	0
$533$	3.55777	0.154104
$534$	0	0
$535$	−10.4067	−0.449922
$536$	0	0
$537$	−7.15216	−0.308639
$538$	0	0
$539$	−3.23607	−0.139387
$540$	0	0
$541$	9.97830	0.429001	0.214500	−	0.976724i	$-0.431188\pi$
$541$	9.97830	0.429001	0.214500	+	0.976724i	$0.431188\pi$
$542$	0	0
$543$	45.6551	1.95925
$544$	0	0
$545$	10.1566	0.435061
$546$	0	0
$547$	−21.3088	−0.911100	−0.455550	−	0.890210i	$-0.650557\pi$
$547$	−21.3088	−0.911100	−0.455550	+	0.890210i	$0.650557\pi$
$548$	0	0
$549$	0.315529	0.0134664
$550$	0	0
$551$	−1.30531	−0.0556079
$552$	0	0
$553$	5.00352	0.212771
$554$	0	0
$555$	−0.914363	−0.0388125
$556$	0	0
$557$	13.0582	0.553292	0.276646	−	0.960972i	$-0.410777\pi$
$557$	13.0582	0.553292	0.276646	+	0.960972i	$0.410777\pi$
$558$	0	0
$559$	0.277129	0.0117213
$560$	0	0
$561$	26.1419	1.10371
$562$	0	0
$563$	17.6413	0.743493	0.371747	−	0.928334i	$-0.378759\pi$
$563$	17.6413	0.743493	0.371747	+	0.928334i	$0.378759\pi$
$564$	0	0
$565$	7.19767	0.302808
$566$	0	0
$567$	9.24958	0.388446
$568$	0	0
$569$	−13.3756	−0.560736	−0.280368	−	0.959893i	$-0.590457\pi$
$569$	−13.3756	−0.560736	−0.280368	+	0.959893i	$0.590457\pi$
$570$	0	0
$571$	7.20563	0.301546	0.150773	−	0.988568i	$-0.451824\pi$
$571$	7.20563	0.301546	0.150773	+	0.988568i	$0.451824\pi$
$572$	0	0
$573$	−29.8493	−1.24697
$574$	0	0
$575$	1.40734	0.0586903
$576$	0	0
$577$	29.6622	1.23485	0.617426	−	0.786629i	$-0.288175\pi$
$577$	29.6622	1.23485	0.617426	+	0.786629i	$0.288175\pi$
$578$	0	0
$579$	39.5013	1.64162
$580$	0	0
$581$	10.3279	0.428473
$582$	0	0
$583$	−0.554257	−0.0229550
$584$	0	0
$585$	0.0856374	0.00354067
$586$	0	0
$587$	29.5660	1.22032	0.610159	−	0.792279i	$-0.291105\pi$
$587$	29.5660	1.22032	0.610159	+	0.792279i	$0.291105\pi$
$588$	0	0
$589$	−0.747479	−0.0307994
$590$	0	0
$591$	−3.71669	−0.152884
$592$	0	0
$593$	48.3309	1.98471	0.992356	−	0.123409i	$-0.0393828\pi$
$593$	48.3309	1.98471	0.992356	+	0.123409i	$0.0393828\pi$
$594$	0	0
$595$	4.59883	0.188534
$596$	0	0
$597$	14.2898	0.584841
$598$	0	0
$599$	31.0739	1.26964	0.634822	−	0.772659i	$-0.281074\pi$
$599$	31.0739	1.26964	0.634822	+	0.772659i	$0.281074\pi$
$600$	0	0
$601$	−10.9660	−0.447311	−0.223656	−	0.974668i	$-0.571799\pi$
$601$	−10.9660	−0.447311	−0.223656	+	0.974668i	$0.571799\pi$
$602$	0	0
$603$	−0.167457	−0.00681939
$604$	0	0
$605$	−0.527864	−0.0214607
$606$	0	0
$607$	19.5977	0.795445	0.397723	−	0.917506i	$-0.369801\pi$
$607$	19.5977	0.795445	0.397723	+	0.917506i	$0.369801\pi$
$608$	0	0
$609$	15.2422	0.617647
$610$	0	0
$611$	−11.1155	−0.449687
$612$	0	0
$613$	−30.7907	−1.24362	−0.621812	−	0.783167i	$-0.713603\pi$
$613$	−30.7907	−1.24362	−0.621812	+	0.783167i	$0.713603\pi$
$614$	0	0
$615$	6.24958	0.252007
$616$	0	0
$617$	−19.1725	−0.771855	−0.385927	−	0.922529i	$-0.626118\pi$
$617$	−19.1725	−0.771855	−0.385927	+	0.922529i	$0.626118\pi$
$618$	0	0
$619$	−22.1590	−0.890644	−0.445322	−	0.895370i	$-0.646911\pi$
$619$	−22.1590	−0.890644	−0.445322	+	0.895370i	$0.646911\pi$
$620$	0	0
$621$	−7.20470	−0.289115
$622$	0	0
$623$	15.4918	0.620667
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.855118	−0.0341501
$628$	0	0
$629$	2.39383	0.0954483
$630$	0	0
$631$	7.97627	0.317530	0.158765	−	0.987316i	$-0.449249\pi$
$631$	7.97627	0.317530	0.158765	+	0.987316i	$0.449249\pi$
$632$	0	0
$633$	−32.7892	−1.30325
$634$	0	0
$635$	−3.23607	−0.128419
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−0.132875	−0.00525644
$640$	0	0
$641$	−3.27540	−0.129370	−0.0646852	−	0.997906i	$-0.520604\pi$
$641$	−3.27540	−0.129370	−0.0646852	+	0.997906i	$0.520604\pi$
$642$	0	0
$643$	9.97956	0.393555	0.196778	−	0.980448i	$-0.436952\pi$
$643$	9.97956	0.393555	0.196778	+	0.980448i	$0.436952\pi$
$644$	0	0
$645$	0.486803	0.0191679
$646$	0	0
$647$	−41.3238	−1.62461	−0.812304	−	0.583235i	$-0.801787\pi$
$647$	−41.3238	−1.62461	−0.812304	+	0.583235i	$0.801787\pi$
$648$	0	0
$649$	−25.1883	−0.988728
$650$	0	0
$651$	8.72842	0.342094
$652$	0	0
$653$	17.4715	0.683713	0.341856	−	0.939752i	$-0.388944\pi$
$653$	17.4715	0.683713	0.341856	+	0.939752i	$0.388944\pi$
$654$	0	0
$655$	12.3132	0.481117
$656$	0	0
$657$	0.501471	0.0195642
$658$	0	0
$659$	−6.08270	−0.236948	−0.118474	−	0.992957i	$-0.537800\pi$
$659$	−6.08270	−0.236948	−0.118474	+	0.992957i	$0.537800\pi$
$660$	0	0
$661$	−31.5825	−1.22842	−0.614208	−	0.789144i	$-0.710524\pi$
$661$	−31.5825	−1.22842	−0.614208	+	0.789144i	$0.710524\pi$
$662$	0	0
$663$	−8.07830	−0.313735
$664$	0	0
$665$	−0.150431	−0.00583345
$666$	0	0
$667$	−12.2117	−0.472839
$668$	0	0
$669$	31.4230	1.21488
$670$	0	0
$671$	−11.9232	−0.460290
$672$	0	0
$673$	9.72490	0.374867	0.187434	−	0.982277i	$-0.439983\pi$
$673$	9.72490	0.374867	0.187434	+	0.982277i	$0.439983\pi$
$674$	0	0
$675$	−5.11936	−0.197044
$676$	0	0
$677$	10.2592	0.394292	0.197146	−	0.980374i	$-0.436833\pi$
$677$	10.2592	0.394292	0.197146	+	0.980374i	$0.436833\pi$
$678$	0	0
$679$	−0.869786	−0.0333793
$680$	0	0
$681$	−16.8551	−0.645890
$682$	0	0
$683$	−21.6376	−0.827939	−0.413970	−	0.910291i	$-0.635858\pi$
$683$	−21.6376	−0.827939	−0.413970	+	0.910291i	$0.635858\pi$
$684$	0	0
$685$	0.308823	0.0117995
$686$	0	0
$687$	26.2202	1.00036
$688$	0	0
$689$	0.171275	0.00652505
$690$	0	0
$691$	−12.6804	−0.482386	−0.241193	−	0.970477i	$-0.577539\pi$
$691$	−12.6804	−0.482386	−0.241193	+	0.970477i	$0.577539\pi$
$692$	0	0
$693$	0.277129	0.0105272
$694$	0	0
$695$	−0.740199	−0.0280774
$696$	0	0
$697$	−16.3616	−0.619740
$698$	0	0
$699$	−2.88792	−0.109231
$700$	0	0
$701$	36.8446	1.39160	0.695801	−	0.718235i	$-0.255050\pi$
$701$	36.8446	1.39160	0.695801	+	0.718235i	$0.255050\pi$
$702$	0	0
$703$	−0.0783037	−0.00295328
$704$	0	0
$705$	−19.5256	−0.735375
$706$	0	0
$707$	4.08212	0.153524
$708$	0	0
$709$	−1.89743	−0.0712597	−0.0356298	−	0.999365i	$-0.511344\pi$
$709$	−1.89743	−0.0712597	−0.0356298	+	0.999365i	$0.511344\pi$
$710$	0	0
$711$	−0.428488	−0.0160696
$712$	0	0
$713$	−6.99299	−0.261890
$714$	0	0
$715$	−3.23607	−0.121022
$716$	0	0
$717$	14.6230	0.546105
$718$	0	0
$719$	−28.6558	−1.06868	−0.534340	−	0.845270i	$-0.679440\pi$
$719$	−28.6558	−1.06868	−0.534340	+	0.845270i	$0.679440\pi$
$720$	0	0
$721$	−3.41023	−0.127004
$722$	0	0
$723$	−29.5892	−1.10043
$724$	0	0
$725$	−8.67714	−0.322261
$726$	0	0
$727$	36.8504	1.36671	0.683354	−	0.730087i	$-0.260521\pi$
$727$	36.8504	1.36671	0.683354	+	0.730087i	$0.260521\pi$
$728$	0	0
$729$	26.1859	0.969848
$730$	0	0
$731$	−1.27447	−0.0471379
$732$	0	0
$733$	31.2651	1.15480	0.577402	−	0.816460i	$-0.304067\pi$
$733$	31.2651	1.15480	0.577402	+	0.816460i	$0.304067\pi$
$734$	0	0
$735$	1.75660	0.0647931
$736$	0	0
$737$	6.32788	0.233091
$738$	0	0
$739$	3.09287	0.113773	0.0568866	−	0.998381i	$-0.481883\pi$
$739$	3.09287	0.113773	0.0568866	+	0.998381i	$0.481883\pi$
$740$	0	0
$741$	0.264246	0.00970732
$742$	0	0
$743$	−9.97355	−0.365894	−0.182947	−	0.983123i	$-0.558564\pi$
$743$	−9.97355	−0.365894	−0.182947	+	0.983123i	$0.558564\pi$
$744$	0	0
$745$	−5.04809	−0.184948
$746$	0	0
$747$	−0.884453	−0.0323605
$748$	0	0
$749$	10.4067	0.380253
$750$	0	0
$751$	−27.6672	−1.00959	−0.504795	−	0.863240i	$-0.668432\pi$
$751$	−27.6672	−1.00959	−0.504795	+	0.863240i	$0.668432\pi$
$752$	0	0
$753$	9.94721	0.362497
$754$	0	0
$755$	7.43436	0.270564
$756$	0	0
$757$	16.6164	0.603933	0.301966	−	0.953319i	$-0.402357\pi$
$757$	16.6164	0.603933	0.301966	+	0.953319i	$0.402357\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	22.1623	0.803381	0.401691	−	0.915775i	$-0.368423\pi$
$761$	22.1623	0.803381	0.401691	+	0.915775i	$0.368423\pi$
$762$	0	0
$763$	−10.1566	−0.367694
$764$	0	0
$765$	−0.393832	−0.0142390
$766$	0	0
$767$	7.78362	0.281050
$768$	0	0
$769$	10.9443	0.394661	0.197330	−	0.980337i	$-0.436773\pi$
$769$	10.9443	0.394661	0.197330	+	0.980337i	$0.436773\pi$
$770$	0	0
$771$	26.8528	0.967080
$772$	0	0
$773$	−43.6685	−1.57065	−0.785324	−	0.619085i	$-0.787504\pi$
$773$	−43.6685	−1.57065	−0.785324	+	0.619085i	$0.787504\pi$
$774$	0	0
$775$	−4.96893	−0.178489
$776$	0	0
$777$	0.914363	0.0328026
$778$	0	0
$779$	0.535198	0.0191755
$780$	0	0
$781$	5.02107	0.179668
$782$	0	0
$783$	44.4214	1.58749
$784$	0	0
$785$	−19.3059	−0.689056
$786$	0	0
$787$	−3.71149	−0.132300	−0.0661502	−	0.997810i	$-0.521072\pi$
$787$	−3.71149	−0.132300	−0.0661502	+	0.997810i	$0.521072\pi$
$788$	0	0
$789$	−18.7854	−0.668776
$790$	0	0
$791$	−7.19767	−0.255920
$792$	0	0
$793$	3.68447	0.130839
$794$	0	0
$795$	0.300861	0.0106704
$796$	0	0
$797$	29.9223	1.05990	0.529951	−	0.848029i	$-0.322210\pi$
$797$	29.9223	1.05990	0.529951	+	0.848029i	$0.322210\pi$
$798$	0	0
$799$	51.1186	1.80844
$800$	0	0
$801$	−1.32668	−0.0468759
$802$	0	0
$803$	−18.9496	−0.668717
$804$	0	0
$805$	−1.40734	−0.0496023
$806$	0	0
$807$	−0.915995	−0.0322445
$808$	0	0
$809$	26.0546	0.916032	0.458016	−	0.888944i	$-0.348560\pi$
$809$	26.0546	0.916032	0.458016	+	0.888944i	$0.348560\pi$
$810$	0	0
$811$	10.2167	0.358757	0.179379	−	0.983780i	$-0.442591\pi$
$811$	10.2167	0.358757	0.179379	+	0.983780i	$0.442591\pi$
$812$	0	0
$813$	−7.55132	−0.264836
$814$	0	0
$815$	−17.6361	−0.617765
$816$	0	0
$817$	0.0416886	0.00145850
$818$	0	0
$819$	−0.0856374	−0.00299241
$820$	0	0
$821$	−28.8328	−1.00627	−0.503136	−	0.864207i	$-0.667821\pi$
$821$	−28.8328	−1.00627	−0.503136	+	0.864207i	$0.667821\pi$
$822$	0	0
$823$	−32.6244	−1.13721	−0.568607	−	0.822609i	$-0.692518\pi$
$823$	−32.6244	−1.13721	−0.568607	+	0.822609i	$0.692518\pi$
$824$	0	0
$825$	−5.68447	−0.197908
$826$	0	0
$827$	36.3379	1.26359	0.631797	−	0.775134i	$-0.282318\pi$
$827$	36.3379	1.26359	0.631797	+	0.775134i	$0.282318\pi$
$828$	0	0
$829$	−31.7092	−1.10131	−0.550653	−	0.834734i	$-0.685621\pi$
$829$	−31.7092	−1.10131	−0.550653	+	0.834734i	$0.685621\pi$
$830$	0	0
$831$	−8.04747	−0.279163
$832$	0	0
$833$	−4.59883	−0.159340
$834$	0	0
$835$	19.5256	0.675710
$836$	0	0
$837$	25.4378	0.879258
$838$	0	0
$839$	−21.0127	−0.725439	−0.362719	−	0.931898i	$-0.618152\pi$
$839$	−21.0127	−0.725439	−0.362719	+	0.931898i	$0.618152\pi$
$840$	0	0
$841$	46.2927	1.59630
$842$	0	0
$843$	−8.86747	−0.305412
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	0.527864	0.0181376
$848$	0	0
$849$	−49.6726	−1.70476
$850$	0	0
$851$	−0.732565	−0.0251120
$852$	0	0
$853$	29.5864	1.01302	0.506510	−	0.862234i	$-0.330935\pi$
$853$	29.5864	1.01302	0.506510	+	0.862234i	$0.330935\pi$
$854$	0	0
$855$	0.0128825	0.000440572 0
$856$	0	0
$857$	−22.9633	−0.784412	−0.392206	−	0.919877i	$-0.628288\pi$
$857$	−22.9633	−0.784412	−0.392206	+	0.919877i	$0.628288\pi$
$858$	0	0
$859$	−32.2364	−1.09989	−0.549946	−	0.835200i	$-0.685352\pi$
$859$	−32.2364	−1.09989	−0.549946	+	0.835200i	$0.685352\pi$
$860$	0	0
$861$	−6.24958	−0.212985
$862$	0	0
$863$	8.48160	0.288717	0.144359	−	0.989525i	$-0.453888\pi$
$863$	8.48160	0.288717	0.144359	+	0.989525i	$0.453888\pi$
$864$	0	0
$865$	1.07479	0.0365439
$866$	0	0
$867$	7.28861	0.247534
$868$	0	0
$869$	16.1917	0.549266
$870$	0	0
$871$	−1.95542	−0.0662570
$872$	0	0
$873$	0.0744862	0.00252098
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	55.5067	1.87433	0.937165	−	0.348887i	$-0.113440\pi$
$877$	55.5067	1.87433	0.937165	+	0.348887i	$0.113440\pi$
$878$	0	0
$879$	18.8147	0.634604
$880$	0	0
$881$	−28.2264	−0.950971	−0.475485	−	0.879724i	$-0.657728\pi$
$881$	−28.2264	−0.950971	−0.475485	+	0.879724i	$0.657728\pi$
$882$	0	0
$883$	36.2344	1.21938	0.609692	−	0.792638i	$-0.291293\pi$
$883$	36.2344	1.21938	0.609692	+	0.792638i	$0.291293\pi$
$884$	0	0
$885$	13.6727	0.459603
$886$	0	0
$887$	−50.4798	−1.69495	−0.847473	−	0.530838i	$-0.821877\pi$
$887$	−50.4798	−1.69495	−0.847473	+	0.530838i	$0.821877\pi$
$888$	0	0
$889$	3.23607	0.108534
$890$	0	0
$891$	29.9323	1.00277
$892$	0	0
$893$	−1.67212	−0.0559553
$894$	0	0
$895$	−4.07160	−0.136099
$896$	0	0
$897$	2.47214	0.0825422
$898$	0	0
$899$	43.1161	1.43800
$900$	0	0
$901$	−0.787665	−0.0262409
$902$	0	0
$903$	−0.486803	−0.0161998
$904$	0	0
$905$	25.9907	0.863959
$906$	0	0
$907$	−6.25483	−0.207688	−0.103844	−	0.994594i	$-0.533114\pi$
$907$	−6.25483	−0.207688	−0.103844	+	0.994594i	$0.533114\pi$
$908$	0	0
$909$	−0.349582	−0.0115949
$910$	0	0
$911$	−8.07160	−0.267424	−0.133712	−	0.991020i	$-0.542690\pi$
$911$	−8.07160	−0.267424	−0.133712	+	0.991020i	$0.542690\pi$
$912$	0	0
$913$	33.4217	1.10610
$914$	0	0
$915$	6.47214	0.213962
$916$	0	0
$917$	−12.3132	−0.406618
$918$	0	0
$919$	11.1088	0.366447	0.183223	−	0.983071i	$-0.441347\pi$
$919$	11.1088	0.366447	0.183223	+	0.983071i	$0.441347\pi$
$920$	0	0
$921$	6.58128	0.216860
$922$	0	0
$923$	−1.55160	−0.0510714
$924$	0	0
$925$	−0.520530	−0.0171149
$926$	0	0
$927$	0.292043	0.00959196
$928$	0	0
$929$	−3.25947	−0.106940	−0.0534699	−	0.998569i	$-0.517028\pi$
$929$	−3.25947	−0.106940	−0.0534699	+	0.998569i	$0.517028\pi$
$930$	0	0
$931$	0.150431	0.00493016
$932$	0	0
$933$	−12.3208	−0.403367
$934$	0	0
$935$	14.8821	0.486698
$936$	0	0
$937$	21.8799	0.714784	0.357392	−	0.933954i	$-0.383666\pi$
$937$	21.8799	0.714784	0.357392	+	0.933954i	$0.383666\pi$
$938$	0	0
$939$	−20.5131	−0.669421
$940$	0	0
$941$	−31.4393	−1.02489	−0.512446	−	0.858720i	$-0.671260\pi$
$941$	−31.4393	−1.02489	−0.512446	+	0.858720i	$0.671260\pi$
$942$	0	0
$943$	5.00701	0.163051
$944$	0	0
$945$	5.11936	0.166533
$946$	0	0
$947$	15.7425	0.511561	0.255781	−	0.966735i	$-0.417668\pi$
$947$	15.7425	0.511561	0.255781	+	0.966735i	$0.417668\pi$
$948$	0	0
$949$	5.85575	0.190086
$950$	0	0
$951$	57.2775	1.85735
$952$	0	0
$953$	−10.5959	−0.343236	−0.171618	−	0.985164i	$-0.554900\pi$
$953$	−10.5959	−0.343236	−0.171618	+	0.985164i	$0.554900\pi$
$954$	0	0
$955$	−16.9927	−0.549870
$956$	0	0
$957$	49.3249	1.59445
$958$	0	0
$959$	−0.308823	−0.00997241
$960$	0	0
$961$	−6.30970	−0.203539
$962$	0	0
$963$	−0.891204	−0.0287187
$964$	0	0
$965$	22.4874	0.723894
$966$	0	0
$967$	10.4020	0.334507	0.167254	−	0.985914i	$-0.446510\pi$
$967$	10.4020	0.334507	0.167254	+	0.985914i	$0.446510\pi$
$968$	0	0
$969$	−1.21522	−0.0390386
$970$	0	0
$971$	6.73211	0.216044	0.108022	−	0.994149i	$-0.465548\pi$
$971$	6.73211	0.216044	0.108022	+	0.994149i	$0.465548\pi$
$972$	0	0
$973$	0.740199	0.0237297
$974$	0	0
$975$	1.75660	0.0562562
$976$	0	0
$977$	15.2059	0.486479	0.243240	−	0.969966i	$-0.421790\pi$
$977$	15.2059	0.486479	0.243240	+	0.969966i	$0.421790\pi$
$978$	0	0
$979$	50.1326	1.60224
$980$	0	0
$981$	0.869786	0.0277701
$982$	0	0
$983$	−17.1694	−0.547619	−0.273809	−	0.961784i	$-0.588284\pi$
$983$	−17.1694	−0.547619	−0.273809	+	0.961784i	$0.588284\pi$
$984$	0	0
$985$	−2.11585	−0.0674165
$986$	0	0
$987$	19.5256	0.621505
$988$	0	0
$989$	0.390015	0.0124018
$990$	0	0
$991$	7.56481	0.240304	0.120152	−	0.992756i	$-0.461662\pi$
$991$	7.56481	0.240304	0.120152	+	0.992756i	$0.461662\pi$
$992$	0	0
$993$	−40.8515	−1.29638
$994$	0	0
$995$	8.13491	0.257894
$996$	0	0
$997$	22.1931	0.702862	0.351431	−	0.936214i	$-0.385695\pi$
$997$	22.1931	0.702862	0.351431	+	0.936214i	$0.385695\pi$
$998$	0	0
$999$	2.66478	0.0843100

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.s.1.4	✓	4	1.1	even	1	trivial
7280.2.a.ca.1.1		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)

\(f(q)\)	\(=\)	\(q+1.75660 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.0856374 q^{9} +O(q^{10})\)	\(q+1.75660 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.0856374 q^{9} -3.23607 q^{11} +1.00000 q^{13} +1.75660 q^{15} -4.59883 q^{17} +0.150431 q^{19} -1.75660 q^{21} +1.40734 q^{23} +1.00000 q^{25} -5.11936 q^{27} -8.67714 q^{29} -4.96893 q^{31} -5.68447 q^{33} -1.00000 q^{35} -0.520530 q^{37} +1.75660 q^{39} +3.55777 q^{41} +0.277129 q^{43} +0.0856374 q^{45} -11.1155 q^{47} +1.00000 q^{49} -8.07830 q^{51} +0.171275 q^{53} -3.23607 q^{55} +0.264246 q^{57} +7.78362 q^{59} +3.68447 q^{61} -0.0856374 q^{63} +1.00000 q^{65} -1.95542 q^{67} +2.47214 q^{69} -1.55160 q^{71} +5.85575 q^{73} +1.75660 q^{75} +3.23607 q^{77} -5.00352 q^{79} -9.24958 q^{81} -10.3279 q^{83} -4.59883 q^{85} -15.2422 q^{87} -15.4918 q^{89} -1.00000 q^{91} -8.72842 q^{93} +0.150431 q^{95} +0.869786 q^{97} -0.277129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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