LMFDB - Embedded newform 3640.2.a.t.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.509552$ 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)

$f(q)$	$=$	$q+0.509552 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.74036 q^{9} +O(q^{10})$	$q+0.509552 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.74036 q^{9} -0.675104 q^{11} +1.00000 q^{13} -0.509552 q^{15} +0.415460 q^{17} +1.39636 q^{19} +0.509552 q^{21} -3.69421 q^{23} +1.00000 q^{25} -2.92501 q^{27} -3.83445 q^{29} +10.8212 q^{31} -0.344001 q^{33} -1.00000 q^{35} +0.840656 q^{37} +0.509552 q^{39} -6.10967 q^{41} +5.85003 q^{43} +2.74036 q^{45} -12.9867 q^{47} +1.00000 q^{49} +0.211699 q^{51} -2.36931 q^{53} +0.675104 q^{55} +0.711516 q^{57} -14.5154 q^{59} -8.00584 q^{61} -2.74036 q^{63} -1.00000 q^{65} +6.10967 q^{67} -1.88239 q^{69} -7.44250 q^{71} -1.13671 q^{73} +0.509552 q^{75} -0.675104 q^{77} +5.09056 q^{79} +6.73062 q^{81} -2.98090 q^{83} -0.415460 q^{85} -1.95385 q^{87} -5.31516 q^{89} +1.00000 q^{91} +5.51396 q^{93} -1.39636 q^{95} -14.6556 q^{97} +1.85003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 + 4 * q^7	$ 4 q - 4 q^{5} + 4 q^{7} - q^{11} + 4 q^{13} - 11 q^{17} - 3 q^{19} - 9 q^{23} + 4 q^{25} + 3 q^{27} - 15 q^{29} - 6 q^{31} + q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} - 3 q^{47} + 4 q^{49} - 4 q^{51} - 2 q^{53} + q^{55} - 28 q^{57} - 3 q^{59} + 21 q^{61} - 4 q^{65} + 6 q^{67} - 23 q^{69} - 16 q^{71} + 15 q^{73} - q^{77} + 6 q^{79} - 8 q^{81} - 16 q^{83} + 11 q^{85} - 13 q^{87} + q^{89} + 4 q^{91} - 2 q^{93} + 3 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^7 - q^11 + 4 * q^13 - 11 * q^17 - 3 * q^19 - 9 * q^23 + 4 * q^25 + 3 * q^27 - 15 * q^29 - 6 * q^31 + q^33 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 - 3 * q^47 + 4 * q^49 - 4 * q^51 - 2 * q^53 + q^55 - 28 * q^57 - 3 * q^59 + 21 * q^61 - 4 * q^65 + 6 * q^67 - 23 * q^69 - 16 * q^71 + 15 * q^73 - q^77 + 6 * q^79 - 8 * q^81 - 16 * q^83 + 11 * q^85 - 13 * q^87 + q^89 + 4 * q^91 - 2 * q^93 + 3 * q^95 - 9 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.509552	0.294190	0.147095	−	0.989122i	$-0.453008\pi$
$3$	0.509552	0.294190	0.147095	+	0.989122i	$0.453008\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.74036	−0.913452
$10$	0	0
$11$	−0.675104	−0.203552	−0.101776	−	0.994807i	$-0.532452\pi$
$11$	−0.675104	−0.203552	−0.101776	+	0.994807i	$0.532452\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.509552	−0.131566
$16$	0	0
$17$	0.415460	0.100764	0.0503820	−	0.998730i	$-0.483956\pi$
$17$	0.415460	0.100764	0.0503820	+	0.998730i	$0.483956\pi$
$18$	0	0
$19$	1.39636	0.320346	0.160173	−	0.987089i	$-0.448795\pi$
$19$	1.39636	0.320346	0.160173	+	0.987089i	$0.448795\pi$
$20$	0	0
$21$	0.509552	0.111193
$22$	0	0
$23$	−3.69421	−0.770296	−0.385148	−	0.922855i	$-0.625850\pi$
$23$	−3.69421	−0.770296	−0.385148	+	0.922855i	$0.625850\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.92501	−0.562919
$28$	0	0
$29$	−3.83445	−0.712039	−0.356020	−	0.934478i	$-0.615866\pi$
$29$	−3.83445	−0.712039	−0.356020	+	0.934478i	$0.615866\pi$
$30$	0	0
$31$	10.8212	1.94354	0.971771	−	0.235925i	$-0.0758121\pi$
$31$	10.8212	1.94354	0.971771	+	0.235925i	$0.0758121\pi$
$32$	0	0
$33$	−0.344001	−0.0598829
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.840656	0.138203	0.0691015	−	0.997610i	$-0.477987\pi$
$37$	0.840656	0.138203	0.0691015	+	0.997610i	$0.477987\pi$
$38$	0	0
$39$	0.509552	0.0815937
$40$	0	0
$41$	−6.10967	−0.954170	−0.477085	−	0.878857i	$-0.658307\pi$
$41$	−6.10967	−0.954170	−0.477085	+	0.878857i	$0.658307\pi$
$42$	0	0
$43$	5.85003	0.892121	0.446060	−	0.895003i	$-0.352827\pi$
$43$	5.85003	0.892121	0.446060	+	0.895003i	$0.352827\pi$
$44$	0	0
$45$	2.74036	0.408508
$46$	0	0
$47$	−12.9867	−1.89431	−0.947155	−	0.320776i	$-0.896056\pi$
$47$	−12.9867	−1.89431	−0.947155	+	0.320776i	$0.896056\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.211699	0.0296438
$52$	0	0
$53$	−2.36931	−0.325450	−0.162725	−	0.986671i	$-0.552028\pi$
$53$	−2.36931	−0.325450	−0.162725	+	0.986671i	$0.552028\pi$
$54$	0	0
$55$	0.675104	0.0910310
$56$	0	0
$57$	0.711516	0.0942426
$58$	0	0
$59$	−14.5154	−1.88974	−0.944872	−	0.327441i	$-0.893814\pi$
$59$	−14.5154	−1.88974	−0.944872	+	0.327441i	$0.893814\pi$
$60$	0	0
$61$	−8.00584	−1.02504	−0.512522	−	0.858674i	$-0.671289\pi$
$61$	−8.00584	−1.02504	−0.512522	+	0.858674i	$0.671289\pi$
$62$	0	0
$63$	−2.74036	−0.345252
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.10967	0.746415	0.373207	−	0.927748i	$-0.378258\pi$
$67$	6.10967	0.746415	0.373207	+	0.927748i	$0.378258\pi$
$68$	0	0
$69$	−1.88239	−0.226614
$70$	0	0
$71$	−7.44250	−0.883263	−0.441631	−	0.897197i	$-0.645600\pi$
$71$	−7.44250	−0.883263	−0.441631	+	0.897197i	$0.645600\pi$
$72$	0	0
$73$	−1.13671	−0.133042	−0.0665210	−	0.997785i	$-0.521190\pi$
$73$	−1.13671	−0.133042	−0.0665210	+	0.997785i	$0.521190\pi$
$74$	0	0
$75$	0.509552	0.0588381
$76$	0	0
$77$	−0.675104	−0.0769352
$78$	0	0
$79$	5.09056	0.572733	0.286367	−	0.958120i	$-0.407552\pi$
$79$	5.09056	0.572733	0.286367	+	0.958120i	$0.407552\pi$
$80$	0	0
$81$	6.73062	0.747847
$82$	0	0
$83$	−2.98090	−0.327196	−0.163598	−	0.986527i	$-0.552310\pi$
$83$	−2.98090	−0.327196	−0.163598	+	0.986527i	$0.552310\pi$
$84$	0	0
$85$	−0.415460	−0.0450630
$86$	0	0
$87$	−1.95385	−0.209475
$88$	0	0
$89$	−5.31516	−0.563406	−0.281703	−	0.959502i	$-0.590899\pi$
$89$	−5.31516	−0.563406	−0.281703	+	0.959502i	$0.590899\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	5.51396	0.571771
$94$	0	0
$95$	−1.39636	−0.143263
$96$	0	0
$97$	−14.6556	−1.48805	−0.744027	−	0.668149i	$-0.767087\pi$
$97$	−14.6556	−1.48805	−0.744027	+	0.668149i	$0.767087\pi$
$98$	0	0
$99$	1.85003	0.185935
$100$	0	0
$101$	0.544600	0.0541897	0.0270948	−	0.999633i	$-0.491374\pi$
$101$	0.544600	0.0541897	0.0270948	+	0.999633i	$0.491374\pi$
$102$	0	0
$103$	−4.74656	−0.467693	−0.233846	−	0.972274i	$-0.575131\pi$
$103$	−4.74656	−0.467693	−0.233846	+	0.972274i	$0.575131\pi$
$104$	0	0
$105$	−0.509552	−0.0497272
$106$	0	0
$107$	7.88824	0.762585	0.381292	−	0.924455i	$-0.375479\pi$
$107$	7.88824	0.762585	0.381292	+	0.924455i	$0.375479\pi$
$108$	0	0
$109$	−12.3116	−1.17924	−0.589620	−	0.807681i	$-0.700723\pi$
$109$	−12.3116	−1.17924	−0.589620	+	0.807681i	$0.700723\pi$
$110$	0	0
$111$	0.428358	0.0406580
$112$	0	0
$113$	6.98674	0.657257	0.328628	−	0.944459i	$-0.393414\pi$
$113$	6.98674	0.657257	0.328628	+	0.944459i	$0.393414\pi$
$114$	0	0
$115$	3.69421	0.344487
$116$	0	0
$117$	−2.74036	−0.253346
$118$	0	0
$119$	0.415460	0.0380852
$120$	0	0
$121$	−10.5442	−0.958567
$122$	0	0
$123$	−3.11320	−0.280708
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.7865	−1.04588	−0.522942	−	0.852369i	$-0.675165\pi$
$127$	−11.7865	−1.04588	−0.522942	+	0.852369i	$0.675165\pi$
$128$	0	0
$129$	2.98090	0.262453
$130$	0	0
$131$	−2.03821	−0.178079	−0.0890396	−	0.996028i	$-0.528380\pi$
$131$	−2.03821	−0.178079	−0.0890396	+	0.996028i	$0.528380\pi$
$132$	0	0
$133$	1.39636	0.121079
$134$	0	0
$135$	2.92501	0.251745
$136$	0	0
$137$	2.70358	0.230982	0.115491	−	0.993309i	$-0.463156\pi$
$137$	2.70358	0.230982	0.115491	+	0.993309i	$0.463156\pi$
$138$	0	0
$139$	−17.2384	−1.46215	−0.731073	−	0.682299i	$-0.760980\pi$
$139$	−17.2384	−1.46215	−0.731073	+	0.682299i	$0.760980\pi$
$140$	0	0
$141$	−6.61742	−0.557288
$142$	0	0
$143$	−0.675104	−0.0564550
$144$	0	0
$145$	3.83445	0.318434
$146$	0	0
$147$	0.509552	0.0420272
$148$	0	0
$149$	−1.19439	−0.0978484	−0.0489242	−	0.998802i	$-0.515579\pi$
$149$	−1.19439	−0.0978484	−0.0489242	+	0.998802i	$0.515579\pi$
$150$	0	0
$151$	−16.9614	−1.38030	−0.690151	−	0.723666i	$-0.742456\pi$
$151$	−16.9614	−1.38030	−0.690151	+	0.723666i	$0.742456\pi$
$152$	0	0
$153$	−1.13851	−0.0920430
$154$	0	0
$155$	−10.8212	−0.869179
$156$	0	0
$157$	3.84735	0.307052	0.153526	−	0.988145i	$-0.450937\pi$
$157$	3.84735	0.307052	0.153526	+	0.988145i	$0.450937\pi$
$158$	0	0
$159$	−1.20729	−0.0957443
$160$	0	0
$161$	−3.69421	−0.291144
$162$	0	0
$163$	−15.4963	−1.21376	−0.606882	−	0.794792i	$-0.707580\pi$
$163$	−15.4963	−1.21376	−0.606882	+	0.794792i	$0.707580\pi$
$164$	0	0
$165$	0.344001	0.0267804
$166$	0	0
$167$	16.1999	1.25358	0.626792	−	0.779187i	$-0.284368\pi$
$167$	16.1999	1.25358	0.626792	+	0.779187i	$0.284368\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.82651	−0.292621
$172$	0	0
$173$	16.6965	1.26941	0.634707	−	0.772753i	$-0.281121\pi$
$173$	16.6965	1.26941	0.634707	+	0.772753i	$0.281121\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−7.39636	−0.555944
$178$	0	0
$179$	−4.27254	−0.319345	−0.159672	−	0.987170i	$-0.551044\pi$
$179$	−4.27254	−0.319345	−0.159672	+	0.987170i	$0.551044\pi$
$180$	0	0
$181$	22.7986	1.69460	0.847302	−	0.531112i	$-0.178226\pi$
$181$	22.7986	1.69460	0.847302	+	0.531112i	$0.178226\pi$
$182$	0	0
$183$	−4.07940	−0.301558
$184$	0	0
$185$	−0.840656	−0.0618062
$186$	0	0
$187$	−0.280479	−0.0205107
$188$	0	0
$189$	−2.92501	−0.212763
$190$	0	0
$191$	18.3654	1.32888	0.664438	−	0.747344i	$-0.268671\pi$
$191$	18.3654	1.32888	0.664438	+	0.747344i	$0.268671\pi$
$192$	0	0
$193$	19.2269	1.38398	0.691992	−	0.721905i	$-0.256733\pi$
$193$	19.2269	1.38398	0.691992	+	0.721905i	$0.256733\pi$
$194$	0	0
$195$	−0.509552	−0.0364898
$196$	0	0
$197$	−16.4137	−1.16943	−0.584713	−	0.811241i	$-0.698793\pi$
$197$	−16.4137	−1.16943	−0.584713	+	0.811241i	$0.698793\pi$
$198$	0	0
$199$	−16.9738	−1.20324	−0.601622	−	0.798781i	$-0.705479\pi$
$199$	−16.9738	−1.20324	−0.601622	+	0.798781i	$0.705479\pi$
$200$	0	0
$201$	3.11320	0.219588
$202$	0	0
$203$	−3.83445	−0.269126
$204$	0	0
$205$	6.10967	0.426718
$206$	0	0
$207$	10.1234	0.703628
$208$	0	0
$209$	−0.942685	−0.0652069
$210$	0	0
$211$	−24.1155	−1.66018	−0.830090	−	0.557629i	$-0.811711\pi$
$211$	−24.1155	−1.66018	−0.830090	+	0.557629i	$0.811711\pi$
$212$	0	0
$213$	−3.79235	−0.259847
$214$	0	0
$215$	−5.85003	−0.398968
$216$	0	0
$217$	10.8212	0.734590
$218$	0	0
$219$	−0.579214	−0.0391397
$220$	0	0
$221$	0.415460	0.0279469
$222$	0	0
$223$	−27.8052	−1.86198	−0.930988	−	0.365049i	$-0.881052\pi$
$223$	−27.8052	−1.86198	−0.930988	+	0.365049i	$0.881052\pi$
$224$	0	0
$225$	−2.74036	−0.182690
$226$	0	0
$227$	8.71916	0.578711	0.289355	−	0.957222i	$-0.406559\pi$
$227$	8.71916	0.578711	0.289355	+	0.957222i	$0.406559\pi$
$228$	0	0
$229$	25.5962	1.69145	0.845723	−	0.533622i	$-0.179170\pi$
$229$	25.5962	1.69145	0.845723	+	0.533622i	$0.179170\pi$
$230$	0	0
$231$	−0.344001	−0.0226336
$232$	0	0
$233$	−14.5633	−0.954076	−0.477038	−	0.878883i	$-0.658290\pi$
$233$	−14.5633	−0.954076	−0.477038	+	0.878883i	$0.658290\pi$
$234$	0	0
$235$	12.9867	0.847161
$236$	0	0
$237$	2.59391	0.168493
$238$	0	0
$239$	−12.7004	−0.821522	−0.410761	−	0.911743i	$-0.634737\pi$
$239$	−12.7004	−0.821522	−0.410761	+	0.911743i	$0.634737\pi$
$240$	0	0
$241$	26.5007	1.70706	0.853530	−	0.521044i	$-0.174457\pi$
$241$	26.5007	1.70706	0.853530	+	0.521044i	$0.174457\pi$
$242$	0	0
$243$	12.2046	0.782928
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	1.39636	0.0888480
$248$	0	0
$249$	−1.51892	−0.0962578
$250$	0	0
$251$	−16.1558	−1.01975	−0.509873	−	0.860250i	$-0.670308\pi$
$251$	−16.1558	−1.01975	−0.509873	+	0.860250i	$0.670308\pi$
$252$	0	0
$253$	2.49398	0.156795
$254$	0	0
$255$	−0.211699	−0.0132571
$256$	0	0
$257$	−22.5380	−1.40588	−0.702942	−	0.711247i	$-0.748131\pi$
$257$	−22.5380	−1.40588	−0.702942	+	0.711247i	$0.748131\pi$
$258$	0	0
$259$	0.840656	0.0522358
$260$	0	0
$261$	10.5078	0.650414
$262$	0	0
$263$	−20.5887	−1.26955	−0.634775	−	0.772697i	$-0.718907\pi$
$263$	−20.5887	−1.26955	−0.634775	+	0.772697i	$0.718907\pi$
$264$	0	0
$265$	2.36931	0.145546
$266$	0	0
$267$	−2.70835	−0.165749
$268$	0	0
$269$	17.9011	1.09145	0.545726	−	0.837964i	$-0.316254\pi$
$269$	17.9011	1.09145	0.545726	+	0.837964i	$0.316254\pi$
$270$	0	0
$271$	23.8052	1.44607	0.723033	−	0.690814i	$-0.242748\pi$
$271$	23.8052	1.44607	0.723033	+	0.690814i	$0.242748\pi$
$272$	0	0
$273$	0.509552	0.0308395
$274$	0	0
$275$	−0.675104	−0.0407103
$276$	0	0
$277$	−23.2473	−1.39679	−0.698396	−	0.715711i	$-0.746103\pi$
$277$	−23.2473	−1.39679	−0.698396	+	0.715711i	$0.746103\pi$
$278$	0	0
$279$	−29.6539	−1.77533
$280$	0	0
$281$	24.2381	1.44592	0.722961	−	0.690889i	$-0.242780\pi$
$281$	24.2381	1.44592	0.722961	+	0.690889i	$0.242780\pi$
$282$	0	0
$283$	8.44077	0.501752	0.250876	−	0.968019i	$-0.419281\pi$
$283$	8.44077	0.501752	0.250876	+	0.968019i	$0.419281\pi$
$284$	0	0
$285$	−0.711516	−0.0421466
$286$	0	0
$287$	−6.10967	−0.360642
$288$	0	0
$289$	−16.8274	−0.989847
$290$	0	0
$291$	−7.46781	−0.437771
$292$	0	0
$293$	2.11176	0.123371	0.0616853	−	0.998096i	$-0.480352\pi$
$293$	2.11176	0.123371	0.0616853	+	0.998096i	$0.480352\pi$
$294$	0	0
$295$	14.5154	0.845119
$296$	0	0
$297$	1.97469	0.114583
$298$	0	0
$299$	−3.69421	−0.213642
$300$	0	0
$301$	5.85003	0.337190
$302$	0	0
$303$	0.277502	0.0159421
$304$	0	0
$305$	8.00584	0.458413
$306$	0	0
$307$	−8.75450	−0.499646	−0.249823	−	0.968292i	$-0.580372\pi$
$307$	−8.75450	−0.499646	−0.249823	+	0.968292i	$0.580372\pi$
$308$	0	0
$309$	−2.41862	−0.137591
$310$	0	0
$311$	32.6074	1.84900	0.924498	−	0.381187i	$-0.124485\pi$
$311$	32.6074	1.84900	0.924498	+	0.381187i	$0.124485\pi$
$312$	0	0
$313$	−32.5465	−1.83964	−0.919820	−	0.392341i	$-0.871665\pi$
$313$	−32.5465	−1.83964	−0.919820	+	0.392341i	$0.871665\pi$
$314$	0	0
$315$	2.74036	0.154402
$316$	0	0
$317$	−4.57040	−0.256699	−0.128349	−	0.991729i	$-0.540968\pi$
$317$	−4.57040	−0.256699	−0.128349	+	0.991729i	$0.540968\pi$
$318$	0	0
$319$	2.58865	0.144937
$320$	0	0
$321$	4.01947	0.224345
$322$	0	0
$323$	0.580130	0.0322793
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−6.27342	−0.346921
$328$	0	0
$329$	−12.9867	−0.715982
$330$	0	0
$331$	−2.30579	−0.126738	−0.0633689	−	0.997990i	$-0.520184\pi$
$331$	−2.30579	−0.126738	−0.0633689	+	0.997990i	$0.520184\pi$
$332$	0	0
$333$	−2.30370	−0.126242
$334$	0	0
$335$	−6.10967	−0.333807
$336$	0	0
$337$	−8.72919	−0.475509	−0.237755	−	0.971325i	$-0.576411\pi$
$337$	−8.72919	−0.475509	−0.237755	+	0.971325i	$0.576411\pi$
$338$	0	0
$339$	3.56011	0.193359
$340$	0	0
$341$	−7.30543	−0.395611
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.88239	0.101345
$346$	0	0
$347$	−34.1928	−1.83557	−0.917783	−	0.397082i	$-0.870023\pi$
$347$	−34.1928	−1.83557	−0.917783	+	0.397082i	$0.870023\pi$
$348$	0	0
$349$	15.4581	0.827452	0.413726	−	0.910401i	$-0.364227\pi$
$349$	15.4581	0.827452	0.413726	+	0.910401i	$0.364227\pi$
$350$	0	0
$351$	−2.92501	−0.156126
$352$	0	0
$353$	1.23260	0.0656048	0.0328024	−	0.999462i	$-0.489557\pi$
$353$	1.23260	0.0656048	0.0328024	+	0.999462i	$0.489557\pi$
$354$	0	0
$355$	7.44250	0.395007
$356$	0	0
$357$	0.211699	0.0112043
$358$	0	0
$359$	−17.6078	−0.929302	−0.464651	−	0.885494i	$-0.653820\pi$
$359$	−17.6078	−0.929302	−0.464651	+	0.885494i	$0.653820\pi$
$360$	0	0
$361$	−17.0502	−0.897379
$362$	0	0
$363$	−5.37284	−0.282001
$364$	0	0
$365$	1.13671	0.0594982
$366$	0	0
$367$	−0.0764198	−0.00398908	−0.00199454	−	0.999998i	$-0.500635\pi$
$367$	−0.0764198	−0.00398908	−0.00199454	+	0.999998i	$0.500635\pi$
$368$	0	0
$369$	16.7427	0.871589
$370$	0	0
$371$	−2.36931	−0.123009
$372$	0	0
$373$	4.92681	0.255101	0.127550	−	0.991832i	$-0.459289\pi$
$373$	4.92681	0.255101	0.127550	+	0.991832i	$0.459289\pi$
$374$	0	0
$375$	−0.509552	−0.0263132
$376$	0	0
$377$	−3.83445	−0.197484
$378$	0	0
$379$	17.4230	0.894961	0.447481	−	0.894294i	$-0.352321\pi$
$379$	17.4230	0.894961	0.447481	+	0.894294i	$0.352321\pi$
$380$	0	0
$381$	−6.00584	−0.307689
$382$	0	0
$383$	15.0249	0.767739	0.383869	−	0.923387i	$-0.374591\pi$
$383$	15.0249	0.767739	0.383869	+	0.923387i	$0.374591\pi$
$384$	0	0
$385$	0.675104	0.0344065
$386$	0	0
$387$	−16.0312	−0.814909
$388$	0	0
$389$	23.2722	1.17995	0.589973	−	0.807423i	$-0.299138\pi$
$389$	23.2722	1.17995	0.589973	+	0.807423i	$0.299138\pi$
$390$	0	0
$391$	−1.53480	−0.0776180
$392$	0	0
$393$	−1.03857	−0.0523892
$394$	0	0
$395$	−5.09056	−0.256134
$396$	0	0
$397$	13.6689	0.686022	0.343011	−	0.939331i	$-0.388553\pi$
$397$	13.6689	0.686022	0.343011	+	0.939331i	$0.388553\pi$
$398$	0	0
$399$	0.711516	0.0356204
$400$	0	0
$401$	−6.07282	−0.303262	−0.151631	−	0.988437i	$-0.548453\pi$
$401$	−6.07282	−0.303262	−0.151631	+	0.988437i	$0.548453\pi$
$402$	0	0
$403$	10.8212	0.539042
$404$	0	0
$405$	−6.73062	−0.334447
$406$	0	0
$407$	−0.567530	−0.0281314
$408$	0	0
$409$	30.2123	1.49390	0.746951	−	0.664880i	$-0.231517\pi$
$409$	30.2123	1.49390	0.746951	+	0.664880i	$0.231517\pi$
$410$	0	0
$411$	1.37762	0.0679528
$412$	0	0
$413$	−14.5154	−0.714256
$414$	0	0
$415$	2.98090	0.146326
$416$	0	0
$417$	−8.78389	−0.430149
$418$	0	0
$419$	−14.5704	−0.711810	−0.355905	−	0.934522i	$-0.615827\pi$
$419$	−14.5704	−0.711810	−0.355905	+	0.934522i	$0.615827\pi$
$420$	0	0
$421$	37.6553	1.83521	0.917603	−	0.397499i	$-0.130122\pi$
$421$	37.6553	1.83521	0.917603	+	0.397499i	$0.130122\pi$
$422$	0	0
$423$	35.5883	1.73036
$424$	0	0
$425$	0.415460	0.0201528
$426$	0	0
$427$	−8.00584	−0.387430
$428$	0	0
$429$	−0.344001	−0.0166085
$430$	0	0
$431$	3.35021	0.161374	0.0806869	−	0.996739i	$-0.474289\pi$
$431$	3.35021	0.161374	0.0806869	+	0.996739i	$0.474289\pi$
$432$	0	0
$433$	21.7774	1.04655	0.523276	−	0.852163i	$-0.324710\pi$
$433$	21.7774	1.04655	0.523276	+	0.852163i	$0.324710\pi$
$434$	0	0
$435$	1.95385	0.0936801
$436$	0	0
$437$	−5.15843	−0.246761
$438$	0	0
$439$	−34.9261	−1.66693	−0.833465	−	0.552572i	$-0.813647\pi$
$439$	−34.9261	−1.66693	−0.833465	+	0.552572i	$0.813647\pi$
$440$	0	0
$441$	−2.74036	−0.130493
$442$	0	0
$443$	3.26673	0.155207	0.0776036	−	0.996984i	$-0.475273\pi$
$443$	3.26673	0.155207	0.0776036	+	0.996984i	$0.475273\pi$
$444$	0	0
$445$	5.31516	0.251963
$446$	0	0
$447$	−0.608605	−0.0287860
$448$	0	0
$449$	−6.21934	−0.293509	−0.146754	−	0.989173i	$-0.546883\pi$
$449$	−6.21934	−0.293509	−0.146754	+	0.989173i	$0.546883\pi$
$450$	0	0
$451$	4.12466	0.194223
$452$	0	0
$453$	−8.64274	−0.406071
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	29.2886	1.37006	0.685032	−	0.728513i	$-0.259788\pi$
$457$	29.2886	1.37006	0.685032	+	0.728513i	$0.259788\pi$
$458$	0	0
$459$	−1.21523	−0.0567219
$460$	0	0
$461$	20.8536	0.971247	0.485623	−	0.874168i	$-0.338593\pi$
$461$	20.8536	0.971247	0.485623	+	0.874168i	$0.338593\pi$
$462$	0	0
$463$	−18.7341	−0.870650	−0.435325	−	0.900273i	$-0.643366\pi$
$463$	−18.7341	−0.870650	−0.435325	+	0.900273i	$0.643366\pi$
$464$	0	0
$465$	−5.51396	−0.255704
$466$	0	0
$467$	13.4692	0.623279	0.311640	−	0.950200i	$-0.399122\pi$
$467$	13.4692	0.623279	0.311640	+	0.950200i	$0.399122\pi$
$468$	0	0
$469$	6.10967	0.282118
$470$	0	0
$471$	1.96042	0.0903316
$472$	0	0
$473$	−3.94938	−0.181592
$474$	0	0
$475$	1.39636	0.0640692
$476$	0	0
$477$	6.49276	0.297283
$478$	0	0
$479$	9.66573	0.441639	0.220819	−	0.975315i	$-0.429127\pi$
$479$	9.66573	0.441639	0.220819	+	0.975315i	$0.429127\pi$
$480$	0	0
$481$	0.840656	0.0383306
$482$	0	0
$483$	−1.88239	−0.0856519
$484$	0	0
$485$	14.6556	0.665478
$486$	0	0
$487$	−23.4199	−1.06126	−0.530628	−	0.847605i	$-0.678044\pi$
$487$	−23.4199	−1.06126	−0.530628	+	0.847605i	$0.678044\pi$
$488$	0	0
$489$	−7.89617	−0.357077
$490$	0	0
$491$	28.6233	1.29175	0.645875	−	0.763443i	$-0.276493\pi$
$491$	28.6233	1.29175	0.645875	+	0.763443i	$0.276493\pi$
$492$	0	0
$493$	−1.59306	−0.0717479
$494$	0	0
$495$	−1.85003	−0.0831525
$496$	0	0
$497$	−7.44250	−0.333842
$498$	0	0
$499$	18.9291	0.847381	0.423690	−	0.905807i	$-0.360734\pi$
$499$	18.9291	0.847381	0.423690	+	0.905807i	$0.360734\pi$
$500$	0	0
$501$	8.25468	0.368792
$502$	0	0
$503$	−19.8846	−0.886612	−0.443306	−	0.896370i	$-0.646195\pi$
$503$	−19.8846	−0.886612	−0.443306	+	0.896370i	$0.646195\pi$
$504$	0	0
$505$	−0.544600	−0.0242344
$506$	0	0
$507$	0.509552	0.0226300
$508$	0	0
$509$	−8.15372	−0.361407	−0.180704	−	0.983538i	$-0.557837\pi$
$509$	−8.15372	−0.361407	−0.180704	+	0.983538i	$0.557837\pi$
$510$	0	0
$511$	−1.13671	−0.0502852
$512$	0	0
$513$	−4.08436	−0.180329
$514$	0	0
$515$	4.74656	0.209159
$516$	0	0
$517$	8.76740	0.385590
$518$	0	0
$519$	8.50775	0.373449
$520$	0	0
$521$	−9.36895	−0.410461	−0.205231	−	0.978714i	$-0.565794\pi$
$521$	−9.36895	−0.410461	−0.205231	+	0.978714i	$0.565794\pi$
$522$	0	0
$523$	−18.0183	−0.787883	−0.393942	−	0.919135i	$-0.628889\pi$
$523$	−18.0183	−0.787883	−0.393942	+	0.919135i	$0.628889\pi$
$524$	0	0
$525$	0.509552	0.0222387
$526$	0	0
$527$	4.49577	0.195839
$528$	0	0
$529$	−9.35282	−0.406644
$530$	0	0
$531$	39.7774	1.72619
$532$	0	0
$533$	−6.10967	−0.264639
$534$	0	0
$535$	−7.88824	−0.341038
$536$	0	0
$537$	−2.17708	−0.0939481
$538$	0	0
$539$	−0.675104	−0.0290788
$540$	0	0
$541$	−13.1267	−0.564360	−0.282180	−	0.959361i	$-0.591058\pi$
$541$	−13.1267	−0.564360	−0.282180	+	0.959361i	$0.591058\pi$
$542$	0	0
$543$	11.6171	0.498536
$544$	0	0
$545$	12.3116	0.527372
$546$	0	0
$547$	5.79940	0.247964	0.123982	−	0.992284i	$-0.460433\pi$
$547$	5.79940	0.247964	0.123982	+	0.992284i	$0.460433\pi$
$548$	0	0
$549$	21.9389	0.936328
$550$	0	0
$551$	−5.35425	−0.228099
$552$	0	0
$553$	5.09056	0.216473
$554$	0	0
$555$	−0.428358	−0.0181828
$556$	0	0
$557$	11.8097	0.500394	0.250197	−	0.968195i	$-0.419505\pi$
$557$	11.8097	0.500394	0.250197	+	0.968195i	$0.419505\pi$
$558$	0	0
$559$	5.85003	0.247430
$560$	0	0
$561$	−0.142919	−0.00603403
$562$	0	0
$563$	−28.0902	−1.18386	−0.591930	−	0.805989i	$-0.701634\pi$
$563$	−28.0902	−1.18386	−0.591930	+	0.805989i	$0.701634\pi$
$564$	0	0
$565$	−6.98674	−0.293934
$566$	0	0
$567$	6.73062	0.282660
$568$	0	0
$569$	−8.13678	−0.341112	−0.170556	−	0.985348i	$-0.554556\pi$
$569$	−8.13678	−0.341112	−0.170556	+	0.985348i	$0.554556\pi$
$570$	0	0
$571$	−4.40363	−0.184286	−0.0921431	−	0.995746i	$-0.529372\pi$
$571$	−4.40363	−0.184286	−0.0921431	+	0.995746i	$0.529372\pi$
$572$	0	0
$573$	9.35815	0.390942
$574$	0	0
$575$	−3.69421	−0.154059
$576$	0	0
$577$	−12.9162	−0.537707	−0.268853	−	0.963181i	$-0.586645\pi$
$577$	−12.9162	−0.537707	−0.268853	+	0.963181i	$0.586645\pi$
$578$	0	0
$579$	9.79712	0.407154
$580$	0	0
$581$	−2.98090	−0.123668
$582$	0	0
$583$	1.59953	0.0662459
$584$	0	0
$585$	2.74036	0.113300
$586$	0	0
$587$	−5.39547	−0.222695	−0.111348	−	0.993782i	$-0.535517\pi$
$587$	−5.39547	−0.222695	−0.111348	+	0.993782i	$0.535517\pi$
$588$	0	0
$589$	15.1102	0.622606
$590$	0	0
$591$	−8.36362	−0.344033
$592$	0	0
$593$	−9.09266	−0.373391	−0.186695	−	0.982418i	$-0.559778\pi$
$593$	−9.09266	−0.373391	−0.186695	+	0.982418i	$0.559778\pi$
$594$	0	0
$595$	−0.415460	−0.0170322
$596$	0	0
$597$	−8.64906	−0.353983
$598$	0	0
$599$	−20.1816	−0.824598	−0.412299	−	0.911049i	$-0.635274\pi$
$599$	−20.1816	−0.824598	−0.412299	+	0.911049i	$0.635274\pi$
$600$	0	0
$601$	2.95856	0.120682	0.0603411	−	0.998178i	$-0.480781\pi$
$601$	2.95856	0.120682	0.0603411	+	0.998178i	$0.480781\pi$
$602$	0	0
$603$	−16.7427	−0.681814
$604$	0	0
$605$	10.5442	0.428684
$606$	0	0
$607$	41.0116	1.66461	0.832305	−	0.554318i	$-0.187021\pi$
$607$	41.0116	1.66461	0.832305	+	0.554318i	$0.187021\pi$
$608$	0	0
$609$	−1.95385	−0.0791741
$610$	0	0
$611$	−12.9867	−0.525387
$612$	0	0
$613$	27.7412	1.12046	0.560229	−	0.828338i	$-0.310713\pi$
$613$	27.7412	1.12046	0.560229	+	0.828338i	$0.310713\pi$
$614$	0	0
$615$	3.11320	0.125536
$616$	0	0
$617$	−49.3157	−1.98538	−0.992689	−	0.120704i	$-0.961485\pi$
$617$	−49.3157	−1.98538	−0.992689	+	0.120704i	$0.961485\pi$
$618$	0	0
$619$	36.1320	1.45227	0.726133	−	0.687554i	$-0.241316\pi$
$619$	36.1320	1.45227	0.726133	+	0.687554i	$0.241316\pi$
$620$	0	0
$621$	10.8056	0.433614
$622$	0	0
$623$	−5.31516	−0.212947
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.480348	−0.0191832
$628$	0	0
$629$	0.349259	0.0139259
$630$	0	0
$631$	2.56991	0.102307	0.0511533	−	0.998691i	$-0.483710\pi$
$631$	2.56991	0.102307	0.0511533	+	0.998691i	$0.483710\pi$
$632$	0	0
$633$	−12.2881	−0.488409
$634$	0	0
$635$	11.7865	0.467733
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	20.3951	0.806818
$640$	0	0
$641$	10.3907	0.410409	0.205205	−	0.978719i	$-0.434214\pi$
$641$	10.3907	0.410409	0.205205	+	0.978719i	$0.434214\pi$
$642$	0	0
$643$	−5.22676	−0.206123	−0.103062	−	0.994675i	$-0.532864\pi$
$643$	−5.22676	−0.206123	−0.103062	+	0.994675i	$0.532864\pi$
$644$	0	0
$645$	−2.98090	−0.117373
$646$	0	0
$647$	23.4004	0.919965	0.459982	−	0.887928i	$-0.347856\pi$
$647$	23.4004	0.919965	0.459982	+	0.887928i	$0.347856\pi$
$648$	0	0
$649$	9.79940	0.384660
$650$	0	0
$651$	5.51396	0.216109
$652$	0	0
$653$	21.2348	0.830984	0.415492	−	0.909597i	$-0.363609\pi$
$653$	21.2348	0.830984	0.415492	+	0.909597i	$0.363609\pi$
$654$	0	0
$655$	2.03821	0.0796395
$656$	0	0
$657$	3.11499	0.121528
$658$	0	0
$659$	−14.7653	−0.575175	−0.287587	−	0.957754i	$-0.592853\pi$
$659$	−14.7653	−0.575175	−0.287587	+	0.957754i	$0.592853\pi$
$660$	0	0
$661$	35.7525	1.39061	0.695306	−	0.718714i	$-0.255269\pi$
$661$	35.7525	1.39061	0.695306	+	0.718714i	$0.255269\pi$
$662$	0	0
$663$	0.211699	0.00822170
$664$	0	0
$665$	−1.39636	−0.0541483
$666$	0	0
$667$	14.1653	0.548481
$668$	0	0
$669$	−14.1682	−0.547775
$670$	0	0
$671$	5.40478	0.208649
$672$	0	0
$673$	−42.5244	−1.63920	−0.819598	−	0.572940i	$-0.805803\pi$
$673$	−42.5244	−1.63920	−0.819598	+	0.572940i	$0.805803\pi$
$674$	0	0
$675$	−2.92501	−0.112584
$676$	0	0
$677$	32.5173	1.24974	0.624872	−	0.780728i	$-0.285151\pi$
$677$	32.5173	1.24974	0.624872	+	0.780728i	$0.285151\pi$
$678$	0	0
$679$	−14.6556	−0.562432
$680$	0	0
$681$	4.44287	0.170251
$682$	0	0
$683$	−7.81318	−0.298963	−0.149482	−	0.988765i	$-0.547760\pi$
$683$	−7.81318	−0.298963	−0.149482	+	0.988765i	$0.547760\pi$
$684$	0	0
$685$	−2.70358	−0.103298
$686$	0	0
$687$	13.0426	0.497607
$688$	0	0
$689$	−2.36931	−0.0902636
$690$	0	0
$691$	18.2296	0.693485	0.346743	−	0.937960i	$-0.387288\pi$
$691$	18.2296	0.693485	0.346743	+	0.937960i	$0.387288\pi$
$692$	0	0
$693$	1.85003	0.0702767
$694$	0	0
$695$	17.2384	0.653891
$696$	0	0
$697$	−2.53833	−0.0961459
$698$	0	0
$699$	−7.42079	−0.280680
$700$	0	0
$701$	−0.947162	−0.0357738	−0.0178869	−	0.999840i	$-0.505694\pi$
$701$	−0.947162	−0.0357738	−0.0178869	+	0.999840i	$0.505694\pi$
$702$	0	0
$703$	1.17385	0.0442727
$704$	0	0
$705$	6.61742	0.249227
$706$	0	0
$707$	0.544600	0.0204818
$708$	0	0
$709$	36.7027	1.37840	0.689199	−	0.724572i	$-0.257963\pi$
$709$	36.7027	1.37840	0.689199	+	0.724572i	$0.257963\pi$
$710$	0	0
$711$	−13.9500	−0.523164
$712$	0	0
$713$	−39.9757	−1.49710
$714$	0	0
$715$	0.675104	0.0252475
$716$	0	0
$717$	−6.47153	−0.241684
$718$	0	0
$719$	−25.9993	−0.969609	−0.484805	−	0.874623i	$-0.661109\pi$
$719$	−25.9993	−0.969609	−0.484805	+	0.874623i	$0.661109\pi$
$720$	0	0
$721$	−4.74656	−0.176771
$722$	0	0
$723$	13.5035	0.502200
$724$	0	0
$725$	−3.83445	−0.142408
$726$	0	0
$727$	−21.5150	−0.797948	−0.398974	−	0.916962i	$-0.630634\pi$
$727$	−21.5150	−0.797948	−0.398974	+	0.916962i	$0.630634\pi$
$728$	0	0
$729$	−13.9730	−0.517517
$730$	0	0
$731$	2.43045	0.0898936
$732$	0	0
$733$	40.8497	1.50882	0.754408	−	0.656405i	$-0.227924\pi$
$733$	40.8497	1.50882	0.754408	+	0.656405i	$0.227924\pi$
$734$	0	0
$735$	−0.509552	−0.0187951
$736$	0	0
$737$	−4.12466	−0.151934
$738$	0	0
$739$	−19.2473	−0.708022	−0.354011	−	0.935241i	$-0.615182\pi$
$739$	−19.2473	−0.708022	−0.354011	+	0.935241i	$0.615182\pi$
$740$	0	0
$741$	0.711516	0.0261382
$742$	0	0
$743$	15.3157	0.561880	0.280940	−	0.959725i	$-0.409354\pi$
$743$	15.3157	0.561880	0.280940	+	0.959725i	$0.409354\pi$
$744$	0	0
$745$	1.19439	0.0437591
$746$	0	0
$747$	8.16871	0.298878
$748$	0	0
$749$	7.88824	0.288230
$750$	0	0
$751$	41.7631	1.52396	0.761979	−	0.647601i	$-0.224228\pi$
$751$	41.7631	1.52396	0.761979	+	0.647601i	$0.224228\pi$
$752$	0	0
$753$	−8.23224	−0.299999
$754$	0	0
$755$	16.9614	0.617289
$756$	0	0
$757$	−6.35403	−0.230941	−0.115471	−	0.993311i	$-0.536838\pi$
$757$	−6.35403	−0.230941	−0.115471	+	0.993311i	$0.536838\pi$
$758$	0	0
$759$	1.27081	0.0461275
$760$	0	0
$761$	29.3751	1.06485	0.532423	−	0.846478i	$-0.321282\pi$
$761$	29.3751	1.06485	0.532423	+	0.846478i	$0.321282\pi$
$762$	0	0
$763$	−12.3116	−0.445711
$764$	0	0
$765$	1.13851	0.0411629
$766$	0	0
$767$	−14.5154	−0.524121
$768$	0	0
$769$	−29.7823	−1.07398	−0.536989	−	0.843589i	$-0.680438\pi$
$769$	−29.7823	−1.07398	−0.536989	+	0.843589i	$0.680438\pi$
$770$	0	0
$771$	−11.4843	−0.413597
$772$	0	0
$773$	33.2066	1.19436	0.597178	−	0.802108i	$-0.296288\pi$
$773$	33.2066	1.19436	0.597178	+	0.802108i	$0.296288\pi$
$774$	0	0
$775$	10.8212	0.388708
$776$	0	0
$777$	0.428358	0.0153673
$778$	0	0
$779$	−8.53127	−0.305664
$780$	0	0
$781$	5.02446	0.179789
$782$	0	0
$783$	11.2158	0.400820
$784$	0	0
$785$	−3.84735	−0.137318
$786$	0	0
$787$	−16.6739	−0.594360	−0.297180	−	0.954821i	$-0.596046\pi$
$787$	−16.6739	−0.594360	−0.297180	+	0.954821i	$0.596046\pi$
$788$	0	0
$789$	−10.4910	−0.373489
$790$	0	0
$791$	6.98674	0.248420
$792$	0	0
$793$	−8.00584	−0.284296
$794$	0	0
$795$	1.20729	0.0428181
$796$	0	0
$797$	−26.9553	−0.954805	−0.477403	−	0.878685i	$-0.658422\pi$
$797$	−26.9553	−0.954805	−0.477403	+	0.878685i	$0.658422\pi$
$798$	0	0
$799$	−5.39547	−0.190878
$800$	0	0
$801$	14.5654	0.514644
$802$	0	0
$803$	0.767399	0.0270809
$804$	0	0
$805$	3.69421	0.130204
$806$	0	0
$807$	9.12157	0.321094
$808$	0	0
$809$	13.9567	0.490692	0.245346	−	0.969436i	$-0.421098\pi$
$809$	13.9567	0.490692	0.245346	+	0.969436i	$0.421098\pi$
$810$	0	0
$811$	−19.3972	−0.681129	−0.340565	−	0.940221i	$-0.610618\pi$
$811$	−19.3972	−0.681129	−0.340565	+	0.940221i	$0.610618\pi$
$812$	0	0
$813$	12.1300	0.425418
$814$	0	0
$815$	15.4963	0.542811
$816$	0	0
$817$	8.16871	0.285787
$818$	0	0
$819$	−2.74036	−0.0957558
$820$	0	0
$821$	−30.1733	−1.05306	−0.526528	−	0.850158i	$-0.676507\pi$
$821$	−30.1733	−1.05306	−0.526528	+	0.850158i	$0.676507\pi$
$822$	0	0
$823$	5.09777	0.177697	0.0888486	−	0.996045i	$-0.471681\pi$
$823$	5.09777	0.177697	0.0888486	+	0.996045i	$0.471681\pi$
$824$	0	0
$825$	−0.344001	−0.0119766
$826$	0	0
$827$	−11.3378	−0.394254	−0.197127	−	0.980378i	$-0.563161\pi$
$827$	−11.3378	−0.394254	−0.197127	+	0.980378i	$0.563161\pi$
$828$	0	0
$829$	34.1234	1.18516	0.592578	−	0.805513i	$-0.298110\pi$
$829$	34.1234	1.18516	0.592578	+	0.805513i	$0.298110\pi$
$830$	0	0
$831$	−11.8457	−0.410923
$832$	0	0
$833$	0.415460	0.0143948
$834$	0	0
$835$	−16.1999	−0.560620
$836$	0	0
$837$	−31.6521	−1.09406
$838$	0	0
$839$	13.8293	0.477442	0.238721	−	0.971088i	$-0.423272\pi$
$839$	13.8293	0.477442	0.238721	+	0.971088i	$0.423272\pi$
$840$	0	0
$841$	−14.2970	−0.493000
$842$	0	0
$843$	12.3506	0.425376
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−10.5442	−0.362304
$848$	0	0
$849$	4.30102	0.147611
$850$	0	0
$851$	−3.10556	−0.106457
$852$	0	0
$853$	24.8949	0.852386	0.426193	−	0.904632i	$-0.359854\pi$
$853$	24.8949	0.852386	0.426193	+	0.904632i	$0.359854\pi$
$854$	0	0
$855$	3.82651	0.130864
$856$	0	0
$857$	36.3742	1.24252	0.621260	−	0.783604i	$-0.286621\pi$
$857$	36.3742	1.24252	0.621260	+	0.783604i	$0.286621\pi$
$858$	0	0
$859$	−35.1289	−1.19858	−0.599292	−	0.800531i	$-0.704551\pi$
$859$	−35.1289	−1.19858	−0.599292	+	0.800531i	$0.704551\pi$
$860$	0	0
$861$	−3.11320	−0.106097
$862$	0	0
$863$	47.5926	1.62007	0.810036	−	0.586380i	$-0.199448\pi$
$863$	47.5926	1.62007	0.810036	+	0.586380i	$0.199448\pi$
$864$	0	0
$865$	−16.6965	−0.567699
$866$	0	0
$867$	−8.57444	−0.291203
$868$	0	0
$869$	−3.43666	−0.116581
$870$	0	0
$871$	6.10967	0.207018
$872$	0	0
$873$	40.1617	1.35927
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−53.8854	−1.81958	−0.909791	−	0.415068i	$-0.863758\pi$
$877$	−53.8854	−1.81958	−0.909791	+	0.415068i	$0.863758\pi$
$878$	0	0
$879$	1.07605	0.0362944
$880$	0	0
$881$	37.0690	1.24889	0.624443	−	0.781070i	$-0.285326\pi$
$881$	37.0690	1.24889	0.624443	+	0.781070i	$0.285326\pi$
$882$	0	0
$883$	30.3340	1.02082	0.510410	−	0.859931i	$-0.329494\pi$
$883$	30.3340	1.02082	0.510410	+	0.859931i	$0.329494\pi$
$884$	0	0
$885$	7.39636	0.248626
$886$	0	0
$887$	−15.7067	−0.527379	−0.263689	−	0.964608i	$-0.584939\pi$
$887$	−15.7067	−0.527379	−0.263689	+	0.964608i	$0.584939\pi$
$888$	0	0
$889$	−11.7865	−0.395307
$890$	0	0
$891$	−4.54387	−0.152225
$892$	0	0
$893$	−18.1341	−0.606835
$894$	0	0
$895$	4.27254	0.142815
$896$	0	0
$897$	−1.88239	−0.0628513
$898$	0	0
$899$	−41.4933	−1.38388
$900$	0	0
$901$	−0.984355	−0.0327936
$902$	0	0
$903$	2.98090	0.0991980
$904$	0	0
$905$	−22.7986	−0.757850
$906$	0	0
$907$	−28.8462	−0.957822	−0.478911	−	0.877863i	$-0.658968\pi$
$907$	−28.8462	−0.957822	−0.478911	+	0.877863i	$0.658968\pi$
$908$	0	0
$909$	−1.49240	−0.0494997
$910$	0	0
$911$	8.48020	0.280961	0.140481	−	0.990083i	$-0.455135\pi$
$911$	8.48020	0.280961	0.140481	+	0.990083i	$0.455135\pi$
$912$	0	0
$913$	2.01241	0.0666012
$914$	0	0
$915$	4.07940	0.134861
$916$	0	0
$917$	−2.03821	−0.0673076
$918$	0	0
$919$	−18.7253	−0.617691	−0.308845	−	0.951112i	$-0.599943\pi$
$919$	−18.7253	−0.617691	−0.308845	+	0.951112i	$0.599943\pi$
$920$	0	0
$921$	−4.46088	−0.146991
$922$	0	0
$923$	−7.44250	−0.244973
$924$	0	0
$925$	0.840656	0.0276406
$926$	0	0
$927$	13.0073	0.427215
$928$	0	0
$929$	−44.7182	−1.46716	−0.733579	−	0.679604i	$-0.762152\pi$
$929$	−44.7182	−1.46716	−0.733579	+	0.679604i	$0.762152\pi$
$930$	0	0
$931$	1.39636	0.0457637
$932$	0	0
$933$	16.6152	0.543956
$934$	0	0
$935$	0.280479	0.00917264
$936$	0	0
$937$	55.8272	1.82380	0.911898	−	0.410416i	$-0.134617\pi$
$937$	55.8272	1.82380	0.911898	+	0.410416i	$0.134617\pi$
$938$	0	0
$939$	−16.5842	−0.541204
$940$	0	0
$941$	46.9722	1.53125	0.765625	−	0.643287i	$-0.222430\pi$
$941$	46.9722	1.53125	0.765625	+	0.643287i	$0.222430\pi$
$942$	0	0
$943$	22.5704	0.734993
$944$	0	0
$945$	2.92501	0.0951507
$946$	0	0
$947$	38.3578	1.24646	0.623230	−	0.782038i	$-0.285820\pi$
$947$	38.3578	1.24646	0.623230	+	0.782038i	$0.285820\pi$
$948$	0	0
$949$	−1.13671	−0.0368992
$950$	0	0
$951$	−2.32886	−0.0755183
$952$	0	0
$953$	−40.5692	−1.31416	−0.657082	−	0.753819i	$-0.728210\pi$
$953$	−40.5692	−1.31416	−0.657082	+	0.753819i	$0.728210\pi$
$954$	0	0
$955$	−18.3654	−0.594291
$956$	0	0
$957$	1.31905	0.0426390
$958$	0	0
$959$	2.70358	0.0873031
$960$	0	0
$961$	86.0981	2.77736
$962$	0	0
$963$	−21.6166	−0.696584
$964$	0	0
$965$	−19.2269	−0.618936
$966$	0	0
$967$	−9.01881	−0.290025	−0.145013	−	0.989430i	$-0.546322\pi$
$967$	−9.01881	−0.290025	−0.145013	+	0.989430i	$0.546322\pi$
$968$	0	0
$969$	0.295607	0.00949626
$970$	0	0
$971$	39.5718	1.26992	0.634960	−	0.772545i	$-0.281017\pi$
$971$	39.5718	1.26992	0.634960	+	0.772545i	$0.281017\pi$
$972$	0	0
$973$	−17.2384	−0.552639
$974$	0	0
$975$	0.509552	0.0163187
$976$	0	0
$977$	−49.2741	−1.57642	−0.788209	−	0.615408i	$-0.788991\pi$
$977$	−49.2741	−1.57642	−0.788209	+	0.615408i	$0.788991\pi$
$978$	0	0
$979$	3.58829	0.114682
$980$	0	0
$981$	33.7383	1.07718
$982$	0	0
$983$	−22.8080	−0.727462	−0.363731	−	0.931504i	$-0.618497\pi$
$983$	−22.8080	−0.727462	−0.363731	+	0.931504i	$0.618497\pi$
$984$	0	0
$985$	16.4137	0.522983
$986$	0	0
$987$	−6.61742	−0.210635
$988$	0	0
$989$	−21.6112	−0.687197
$990$	0	0
$991$	52.2989	1.66133	0.830664	−	0.556773i	$-0.187961\pi$
$991$	52.2989	1.66133	0.830664	+	0.556773i	$0.187961\pi$
$992$	0	0
$993$	−1.17492	−0.0372850
$994$	0	0
$995$	16.9738	0.538107
$996$	0	0
$997$	−10.1380	−0.321072	−0.160536	−	0.987030i	$-0.551322\pi$
$997$	−10.1380	−0.321072	−0.160536	+	0.987030i	$0.551322\pi$
$998$	0	0
$999$	−2.45893	−0.0777970

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.t.1.3	✓	4
4.3	odd	2		7280.2.a.bv.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.t.1.3	✓	4	1.1	even	1	trivial
7280.2.a.bv.1.2		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)

\(f(q)\)	\(=\)	\(q+0.509552 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.74036 q^{9} +O(q^{10})\)	\(q+0.509552 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.74036 q^{9} -0.675104 q^{11} +1.00000 q^{13} -0.509552 q^{15} +0.415460 q^{17} +1.39636 q^{19} +0.509552 q^{21} -3.69421 q^{23} +1.00000 q^{25} -2.92501 q^{27} -3.83445 q^{29} +10.8212 q^{31} -0.344001 q^{33} -1.00000 q^{35} +0.840656 q^{37} +0.509552 q^{39} -6.10967 q^{41} +5.85003 q^{43} +2.74036 q^{45} -12.9867 q^{47} +1.00000 q^{49} +0.211699 q^{51} -2.36931 q^{53} +0.675104 q^{55} +0.711516 q^{57} -14.5154 q^{59} -8.00584 q^{61} -2.74036 q^{63} -1.00000 q^{65} +6.10967 q^{67} -1.88239 q^{69} -7.44250 q^{71} -1.13671 q^{73} +0.509552 q^{75} -0.675104 q^{77} +5.09056 q^{79} +6.73062 q^{81} -2.98090 q^{83} -0.415460 q^{85} -1.95385 q^{87} -5.31516 q^{89} +1.00000 q^{91} +5.51396 q^{93} -1.39636 q^{95} -14.6556 q^{97} +1.85003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 + 4 * q^7	\( 4 q - 4 q^{5} + 4 q^{7} - q^{11} + 4 q^{13} - 11 q^{17} - 3 q^{19} - 9 q^{23} + 4 q^{25} + 3 q^{27} - 15 q^{29} - 6 q^{31} + q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} - 3 q^{47} + 4 q^{49} - 4 q^{51} - 2 q^{53} + q^{55} - 28 q^{57} - 3 q^{59} + 21 q^{61} - 4 q^{65} + 6 q^{67} - 23 q^{69} - 16 q^{71} + 15 q^{73} - q^{77} + 6 q^{79} - 8 q^{81} - 16 q^{83} + 11 q^{85} - 13 q^{87} + q^{89} + 4 q^{91} - 2 q^{93} + 3 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^7 - q^11 + 4 * q^13 - 11 * q^17 - 3 * q^19 - 9 * q^23 + 4 * q^25 + 3 * q^27 - 15 * q^29 - 6 * q^31 + q^33 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 - 3 * q^47 + 4 * q^49 - 4 * q^51 - 2 * q^53 + q^55 - 28 * q^57 - 3 * q^59 + 21 * q^61 - 4 * q^65 + 6 * q^67 - 23 * q^69 - 16 * q^71 + 15 * q^73 - q^77 + 6 * q^79 - 8 * q^81 - 16 * q^83 + 11 * q^85 - 13 * q^87 + q^89 + 4 * q^91 - 2 * q^93 + 3 * q^95 - 9 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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