LMFDB - Embedded newform 3640.2.a.z.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2112217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - x^{2} + 10x + 4 $ x^5 - 10x^3 - x^2 + 10x + 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			1.5
Root			$-2.94326$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.94326 q^{3} +1.00000 q^{5} +1.00000 q^{7} +5.66276 q^{9} +O(q^{10})$	$q+2.94326 q^{3} +1.00000 q^{5} +1.00000 q^{7} +5.66276 q^{9} +4.38824 q^{11} -1.00000 q^{13} +2.94326 q^{15} +1.63356 q^{17} -4.89392 q^{19} +2.94326 q^{21} +3.49827 q^{23} +1.00000 q^{25} +7.83717 q^{27} -4.44153 q^{29} +4.94326 q^{31} +12.9157 q^{33} +1.00000 q^{35} +5.58422 q^{37} -2.94326 q^{39} -10.5493 q^{41} -7.79458 q^{43} +5.66276 q^{45} +1.74728 q^{47} +1.00000 q^{49} +4.80797 q^{51} +5.11003 q^{53} +4.38824 q^{55} -14.4041 q^{57} +1.19598 q^{59} +4.85731 q^{61} +5.66276 q^{63} -1.00000 q^{65} +9.03990 q^{67} +10.2963 q^{69} +6.15033 q^{71} -4.12442 q^{73} +2.94326 q^{75} +4.38824 q^{77} +0.113723 q^{79} +6.07854 q^{81} -8.94837 q^{83} +1.63356 q^{85} -13.0726 q^{87} -14.6570 q^{89} -1.00000 q^{91} +14.5493 q^{93} -4.89392 q^{95} -0.922622 q^{97} +24.8495 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.94326	1.69929	0.849645	−	0.527355i	$-0.176816\pi$
$3$	2.94326	1.69929	0.849645	+	0.527355i	$0.176816\pi$
$4$	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
$9$	5.66276	1.88759
$10$	0	0
$11$	4.38824	1.32310	0.661552	−	0.749899i	$-0.269898\pi$
$11$	4.38824	1.32310	0.661552	+	0.749899i	$0.269898\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.94326	0.759945
$16$	0	0
$17$	1.63356	0.396195	0.198098	−	0.980182i	$-0.436524\pi$
$17$	1.63356	0.396195	0.198098	+	0.980182i	$0.436524\pi$
$18$	0	0
$19$	−4.89392	−1.12274	−0.561371	−	0.827564i	$-0.689726\pi$
$19$	−4.89392	−1.12274	−0.561371	+	0.827564i	$0.689726\pi$
$20$	0	0
$21$	2.94326	0.642271
$22$	0	0
$23$	3.49827	0.729440	0.364720	−	0.931117i	$-0.381165\pi$
$23$	3.49827	0.729440	0.364720	+	0.931117i	$0.381165\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.83717	1.50826
$28$	0	0
$29$	−4.44153	−0.824771	−0.412386	−	0.911009i	$-0.635304\pi$
$29$	−4.44153	−0.824771	−0.412386	+	0.911009i	$0.635304\pi$
$30$	0	0
$31$	4.94326	0.887835	0.443917	−	0.896068i	$-0.353588\pi$
$31$	4.94326	0.887835	0.443917	+	0.896068i	$0.353588\pi$
$32$	0	0
$33$	12.9157	2.24834
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	5.58422	0.918040	0.459020	−	0.888426i	$-0.348201\pi$
$37$	5.58422	0.918040	0.459020	+	0.888426i	$0.348201\pi$
$38$	0	0
$39$	−2.94326	−0.471298
$40$	0	0
$41$	−10.5493	−1.64752	−0.823759	−	0.566939i	$-0.808127\pi$
$41$	−10.5493	−1.64752	−0.823759	+	0.566939i	$0.808127\pi$
$42$	0	0
$43$	−7.79458	−1.18866	−0.594332	−	0.804220i	$-0.702583\pi$
$43$	−7.79458	−1.18866	−0.594332	+	0.804220i	$0.702583\pi$
$44$	0	0
$45$	5.66276	0.844154
$46$	0	0
$47$	1.74728	0.254867	0.127433	−	0.991847i	$-0.459326\pi$
$47$	1.74728	0.254867	0.127433	+	0.991847i	$0.459326\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.80797	0.673251
$52$	0	0
$53$	5.11003	0.701917	0.350959	−	0.936391i	$-0.385856\pi$
$53$	5.11003	0.701917	0.350959	+	0.936391i	$0.385856\pi$
$54$	0	0
$55$	4.38824	0.591710
$56$	0	0
$57$	−14.4041	−1.90786
$58$	0	0
$59$	1.19598	0.155703	0.0778515	−	0.996965i	$-0.475194\pi$
$59$	1.19598	0.155703	0.0778515	+	0.996965i	$0.475194\pi$
$60$	0	0
$61$	4.85731	0.621915	0.310958	−	0.950424i	$-0.399350\pi$
$61$	4.85731	0.621915	0.310958	+	0.950424i	$0.399350\pi$
$62$	0	0
$63$	5.66276	0.713440
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	9.03990	1.10440	0.552200	−	0.833712i	$-0.313789\pi$
$67$	9.03990	1.10440	0.552200	+	0.833712i	$0.313789\pi$
$68$	0	0
$69$	10.2963	1.23953
$70$	0	0
$71$	6.15033	0.729910	0.364955	−	0.931025i	$-0.381084\pi$
$71$	6.15033	0.729910	0.364955	+	0.931025i	$0.381084\pi$
$72$	0	0
$73$	−4.12442	−0.482727	−0.241364	−	0.970435i	$-0.577595\pi$
$73$	−4.12442	−0.482727	−0.241364	+	0.970435i	$0.577595\pi$
$74$	0	0
$75$	2.94326	0.339858
$76$	0	0
$77$	4.38824	0.500086
$78$	0	0
$79$	0.113723	0.0127948	0.00639741	−	0.999980i	$-0.497964\pi$
$79$	0.113723	0.0127948	0.00639741	+	0.999980i	$0.497964\pi$
$80$	0	0
$81$	6.07854	0.675393
$82$	0	0
$83$	−8.94837	−0.982211	−0.491105	−	0.871100i	$-0.663407\pi$
$83$	−8.94837	−0.982211	−0.491105	+	0.871100i	$0.663407\pi$
$84$	0	0
$85$	1.63356	0.177184
$86$	0	0
$87$	−13.0726	−1.40153
$88$	0	0
$89$	−14.6570	−1.55364	−0.776820	−	0.629723i	$-0.783168\pi$
$89$	−14.6570	−1.55364	−0.776820	+	0.629723i	$0.783168\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	14.5493	1.50869
$94$	0	0
$95$	−4.89392	−0.502105
$96$	0	0
$97$	−0.922622	−0.0936781	−0.0468390	−	0.998902i	$-0.514915\pi$
$97$	−0.922622	−0.0936781	−0.0468390	+	0.998902i	$0.514915\pi$
$98$	0	0
$99$	24.8495	2.49747
$100$	0	0
$101$	−9.53511	−0.948779	−0.474390	−	0.880315i	$-0.657331\pi$
$101$	−9.53511	−0.948779	−0.474390	+	0.880315i	$0.657331\pi$
$102$	0	0
$103$	−16.2194	−1.59815	−0.799074	−	0.601233i	$-0.794676\pi$
$103$	−16.2194	−1.59815	−0.799074	+	0.601233i	$0.794676\pi$
$104$	0	0
$105$	2.94326	0.287232
$106$	0	0
$107$	−9.26036	−0.895233	−0.447616	−	0.894226i	$-0.647727\pi$
$107$	−9.26036	−0.895233	−0.447616	+	0.894226i	$0.647727\pi$
$108$	0	0
$109$	14.0584	1.34655	0.673275	−	0.739392i	$-0.264887\pi$
$109$	14.0584	1.34655	0.673275	+	0.739392i	$0.264887\pi$
$110$	0	0
$111$	16.4358	1.56002
$112$	0	0
$113$	−14.0841	−1.32493	−0.662463	−	0.749095i	$-0.730489\pi$
$113$	−14.0841	−1.32493	−0.662463	+	0.749095i	$0.730489\pi$
$114$	0	0
$115$	3.49827	0.326216
$116$	0	0
$117$	−5.66276	−0.523522
$118$	0	0
$119$	1.63356	0.149748
$120$	0	0
$121$	8.25665	0.750604
$122$	0	0
$123$	−31.0492	−2.79961
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.82724	0.517084	0.258542	−	0.966000i	$-0.416758\pi$
$127$	5.82724	0.517084	0.258542	+	0.966000i	$0.416758\pi$
$128$	0	0
$129$	−22.9415	−2.01988
$130$	0	0
$131$	−2.53422	−0.221416	−0.110708	−	0.993853i	$-0.535312\pi$
$131$	−2.53422	−0.221416	−0.110708	+	0.993853i	$0.535312\pi$
$132$	0	0
$133$	−4.89392	−0.424356
$134$	0	0
$135$	7.83717	0.674516
$136$	0	0
$137$	12.2363	1.04542	0.522708	−	0.852512i	$-0.324922\pi$
$137$	12.2363	1.04542	0.522708	+	0.852512i	$0.324922\pi$
$138$	0	0
$139$	−6.15708	−0.522237	−0.261118	−	0.965307i	$-0.584091\pi$
$139$	−6.15708	−0.522237	−0.261118	+	0.965307i	$0.584091\pi$
$140$	0	0
$141$	5.14269	0.433092
$142$	0	0
$143$	−4.38824	−0.366963
$144$	0	0
$145$	−4.44153	−0.368849
$146$	0	0
$147$	2.94326	0.242756
$148$	0	0
$149$	−0.700233	−0.0573653	−0.0286827	−	0.999589i	$-0.509131\pi$
$149$	−0.700233	−0.0573653	−0.0286827	+	0.999589i	$0.509131\pi$
$150$	0	0
$151$	7.17518	0.583908	0.291954	−	0.956432i	$-0.405695\pi$
$151$	7.17518	0.583908	0.291954	+	0.956432i	$0.405695\pi$
$152$	0	0
$153$	9.25043	0.747853
$154$	0	0
$155$	4.94326	0.397052
$156$	0	0
$157$	11.3642	0.906958	0.453479	−	0.891267i	$-0.350183\pi$
$157$	11.3642	0.906958	0.453479	+	0.891267i	$0.350183\pi$
$158$	0	0
$159$	15.0401	1.19276
$160$	0	0
$161$	3.49827	0.275703
$162$	0	0
$163$	12.3934	0.970722	0.485361	−	0.874314i	$-0.338688\pi$
$163$	12.3934	0.970722	0.485361	+	0.874314i	$0.338688\pi$
$164$	0	0
$165$	12.9157	1.00549
$166$	0	0
$167$	22.9850	1.77864	0.889318	−	0.457289i	$-0.151179\pi$
$167$	22.9850	1.77864	0.889318	+	0.457289i	$0.151179\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−27.7131	−2.11927
$172$	0	0
$173$	5.17442	0.393404	0.196702	−	0.980463i	$-0.436977\pi$
$173$	5.17442	0.393404	0.196702	+	0.980463i	$0.436977\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	3.52007	0.264585
$178$	0	0
$179$	22.7617	1.70129	0.850645	−	0.525740i	$-0.176212\pi$
$179$	22.7617	1.70129	0.850645	+	0.525740i	$0.176212\pi$
$180$	0	0
$181$	−12.2412	−0.909883	−0.454942	−	0.890521i	$-0.650340\pi$
$181$	−12.2412	−0.909883	−0.454942	+	0.890521i	$0.650340\pi$
$182$	0	0
$183$	14.2963	1.05681
$184$	0	0
$185$	5.58422	0.410560
$186$	0	0
$187$	7.16843	0.524208
$188$	0	0
$189$	7.83717	0.570070
$190$	0	0
$191$	10.9906	0.795249	0.397625	−	0.917548i	$-0.369835\pi$
$191$	10.9906	0.795249	0.397625	+	0.917548i	$0.369835\pi$
$192$	0	0
$193$	−20.9040	−1.50470	−0.752351	−	0.658763i	$-0.771080\pi$
$193$	−20.9040	−1.50470	−0.752351	+	0.658763i	$0.771080\pi$
$194$	0	0
$195$	−2.94326	−0.210771
$196$	0	0
$197$	−21.1956	−1.51012	−0.755061	−	0.655654i	$-0.772393\pi$
$197$	−21.1956	−1.51012	−0.755061	+	0.655654i	$0.772393\pi$
$198$	0	0
$199$	−2.24226	−0.158949	−0.0794747	−	0.996837i	$-0.525324\pi$
$199$	−2.24226	−0.158949	−0.0794747	+	0.996837i	$0.525324\pi$
$200$	0	0
$201$	26.6067	1.87669
$202$	0	0
$203$	−4.44153	−0.311734
$204$	0	0
$205$	−10.5493	−0.736793
$206$	0	0
$207$	19.8099	1.37688
$208$	0	0
$209$	−21.4757	−1.48550
$210$	0	0
$211$	−9.95561	−0.685373	−0.342686	−	0.939450i	$-0.611337\pi$
$211$	−9.95561	−0.685373	−0.342686	+	0.939450i	$0.611337\pi$
$212$	0	0
$213$	18.1020	1.24033
$214$	0	0
$215$	−7.79458	−0.531586
$216$	0	0
$217$	4.94326	0.335570
$218$	0	0
$219$	−12.1392	−0.820293
$220$	0	0
$221$	−1.63356	−0.109885
$222$	0	0
$223$	−24.2894	−1.62654	−0.813269	−	0.581887i	$-0.802314\pi$
$223$	−24.2894	−1.62654	−0.813269	+	0.581887i	$0.802314\pi$
$224$	0	0
$225$	5.66276	0.377517
$226$	0	0
$227$	−13.3885	−0.888626	−0.444313	−	0.895872i	$-0.646552\pi$
$227$	−13.3885	−0.888626	−0.444313	+	0.895872i	$0.646552\pi$
$228$	0	0
$229$	8.98869	0.593989	0.296995	−	0.954879i	$-0.404016\pi$
$229$	8.98869	0.593989	0.296995	+	0.954879i	$0.404016\pi$
$230$	0	0
$231$	12.9157	0.849791
$232$	0	0
$233$	14.6883	0.962260	0.481130	−	0.876649i	$-0.340226\pi$
$233$	14.6883	0.962260	0.481130	+	0.876649i	$0.340226\pi$
$234$	0	0
$235$	1.74728	0.113980
$236$	0	0
$237$	0.334715	0.0217421
$238$	0	0
$239$	−17.4689	−1.12997	−0.564986	−	0.825101i	$-0.691118\pi$
$239$	−17.4689	−1.12997	−0.564986	+	0.825101i	$0.691118\pi$
$240$	0	0
$241$	19.5637	1.26021	0.630103	−	0.776512i	$-0.283013\pi$
$241$	19.5637	1.26021	0.630103	+	0.776512i	$0.283013\pi$
$242$	0	0
$243$	−5.62082	−0.360576
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	4.89392	0.311392
$248$	0	0
$249$	−26.3373	−1.66906
$250$	0	0
$251$	4.23791	0.267495	0.133747	−	0.991015i	$-0.457299\pi$
$251$	4.23791	0.267495	0.133747	+	0.991015i	$0.457299\pi$
$252$	0	0
$253$	15.3513	0.965125
$254$	0	0
$255$	4.80797	0.301087
$256$	0	0
$257$	28.4643	1.77556	0.887778	−	0.460272i	$-0.152248\pi$
$257$	28.4643	1.77556	0.887778	+	0.460272i	$0.152248\pi$
$258$	0	0
$259$	5.58422	0.346986
$260$	0	0
$261$	−25.1513	−1.55683
$262$	0	0
$263$	−22.0656	−1.36062	−0.680312	−	0.732922i	$-0.738156\pi$
$263$	−22.0656	−1.36062	−0.680312	+	0.732922i	$0.738156\pi$
$264$	0	0
$265$	5.11003	0.313907
$266$	0	0
$267$	−43.1393	−2.64008
$268$	0	0
$269$	−12.9009	−0.786582	−0.393291	−	0.919414i	$-0.628663\pi$
$269$	−12.9009	−0.786582	−0.393291	+	0.919414i	$0.628663\pi$
$270$	0	0
$271$	−5.89761	−0.358254	−0.179127	−	0.983826i	$-0.557327\pi$
$271$	−5.89761	−0.358254	−0.179127	+	0.983826i	$0.557327\pi$
$272$	0	0
$273$	−2.94326	−0.178134
$274$	0	0
$275$	4.38824	0.264621
$276$	0	0
$277$	25.0118	1.50281	0.751407	−	0.659838i	$-0.229375\pi$
$277$	25.0118	1.50281	0.751407	+	0.659838i	$0.229375\pi$
$278$	0	0
$279$	27.9925	1.67586
$280$	0	0
$281$	−19.7031	−1.17539	−0.587695	−	0.809083i	$-0.699964\pi$
$281$	−19.7031	−1.17539	−0.587695	+	0.809083i	$0.699964\pi$
$282$	0	0
$283$	−9.18739	−0.546133	−0.273067	−	0.961995i	$-0.588038\pi$
$283$	−9.18739	−0.546133	−0.273067	+	0.961995i	$0.588038\pi$
$284$	0	0
$285$	−14.4041	−0.853222
$286$	0	0
$287$	−10.5493	−0.622704
$288$	0	0
$289$	−14.3315	−0.843029
$290$	0	0
$291$	−2.71551	−0.159186
$292$	0	0
$293$	19.4678	1.13732	0.568660	−	0.822573i	$-0.307462\pi$
$293$	19.4678	1.13732	0.568660	+	0.822573i	$0.307462\pi$
$294$	0	0
$295$	1.19598	0.0696325
$296$	0	0
$297$	34.3914	1.99559
$298$	0	0
$299$	−3.49827	−0.202310
$300$	0	0
$301$	−7.79458	−0.449272
$302$	0	0
$303$	−28.0643	−1.61225
$304$	0	0
$305$	4.85731	0.278129
$306$	0	0
$307$	3.22399	0.184003	0.0920014	−	0.995759i	$-0.470674\pi$
$307$	3.22399	0.184003	0.0920014	+	0.995759i	$0.470674\pi$
$308$	0	0
$309$	−47.7379	−2.71572
$310$	0	0
$311$	−21.3770	−1.21218	−0.606089	−	0.795397i	$-0.707263\pi$
$311$	−21.3770	−1.21218	−0.606089	+	0.795397i	$0.707263\pi$
$312$	0	0
$313$	−7.61687	−0.430531	−0.215266	−	0.976556i	$-0.569062\pi$
$313$	−7.61687	−0.430531	−0.215266	+	0.976556i	$0.569062\pi$
$314$	0	0
$315$	5.66276	0.319060
$316$	0	0
$317$	−7.35817	−0.413276	−0.206638	−	0.978417i	$-0.566252\pi$
$317$	−7.35817	−0.413276	−0.206638	+	0.978417i	$0.566252\pi$
$318$	0	0
$319$	−19.4905	−1.09126
$320$	0	0
$321$	−27.2556	−1.52126
$322$	0	0
$323$	−7.99449	−0.444825
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	41.3775	2.28818
$328$	0	0
$329$	1.74728	0.0963306
$330$	0	0
$331$	15.8486	0.871120	0.435560	−	0.900160i	$-0.356550\pi$
$331$	15.8486	0.871120	0.435560	+	0.900160i	$0.356550\pi$
$332$	0	0
$333$	31.6221	1.73288
$334$	0	0
$335$	9.03990	0.493903
$336$	0	0
$337$	−1.23446	−0.0672451	−0.0336226	−	0.999435i	$-0.510704\pi$
$337$	−1.23446	−0.0672451	−0.0336226	+	0.999435i	$0.510704\pi$
$338$	0	0
$339$	−41.4532	−2.25143
$340$	0	0
$341$	21.6922	1.17470
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	10.2963	0.554335
$346$	0	0
$347$	14.0249	0.752895	0.376447	−	0.926438i	$-0.377146\pi$
$347$	14.0249	0.752895	0.376447	+	0.926438i	$0.377146\pi$
$348$	0	0
$349$	3.21469	0.172079	0.0860393	−	0.996292i	$-0.472579\pi$
$349$	3.21469	0.172079	0.0860393	+	0.996292i	$0.472579\pi$
$350$	0	0
$351$	−7.83717	−0.418317
$352$	0	0
$353$	19.0955	1.01635	0.508176	−	0.861254i	$-0.330320\pi$
$353$	19.0955	1.01635	0.508176	+	0.861254i	$0.330320\pi$
$354$	0	0
$355$	6.15033	0.326426
$356$	0	0
$357$	4.80797	0.254465
$358$	0	0
$359$	13.4179	0.708170	0.354085	−	0.935213i	$-0.384792\pi$
$359$	13.4179	0.708170	0.354085	+	0.935213i	$0.384792\pi$
$360$	0	0
$361$	4.95043	0.260549
$362$	0	0
$363$	24.3014	1.27549
$364$	0	0
$365$	−4.12442	−0.215882
$366$	0	0
$367$	−19.8933	−1.03842	−0.519210	−	0.854647i	$-0.673774\pi$
$367$	−19.8933	−1.03842	−0.519210	+	0.854647i	$0.673774\pi$
$368$	0	0
$369$	−59.7379	−3.10983
$370$	0	0
$371$	5.11003	0.265300
$372$	0	0
$373$	−28.4286	−1.47198	−0.735989	−	0.676993i	$-0.763283\pi$
$373$	−28.4286	−1.47198	−0.735989	+	0.676993i	$0.763283\pi$
$374$	0	0
$375$	2.94326	0.151989
$376$	0	0
$377$	4.44153	0.228750
$378$	0	0
$379$	−21.2822	−1.09319	−0.546597	−	0.837396i	$-0.684077\pi$
$379$	−21.2822	−1.09319	−0.546597	+	0.837396i	$0.684077\pi$
$380$	0	0
$381$	17.1511	0.878675
$382$	0	0
$383$	32.6633	1.66902	0.834508	−	0.550996i	$-0.185752\pi$
$383$	32.6633	1.66902	0.834508	+	0.550996i	$0.185752\pi$
$384$	0	0
$385$	4.38824	0.223645
$386$	0	0
$387$	−44.1388	−2.24370
$388$	0	0
$389$	13.3482	0.676780	0.338390	−	0.941006i	$-0.390118\pi$
$389$	13.3482	0.676780	0.338390	+	0.941006i	$0.390118\pi$
$390$	0	0
$391$	5.71462	0.289001
$392$	0	0
$393$	−7.45887	−0.376250
$394$	0	0
$395$	0.113723	0.00572202
$396$	0	0
$397$	−1.63076	−0.0818453	−0.0409227	−	0.999162i	$-0.513030\pi$
$397$	−1.63076	−0.0818453	−0.0409227	+	0.999162i	$0.513030\pi$
$398$	0	0
$399$	−14.4041	−0.721105
$400$	0	0
$401$	35.2273	1.75917	0.879584	−	0.475744i	$-0.157821\pi$
$401$	35.2273	1.75917	0.879584	+	0.475744i	$0.157821\pi$
$402$	0	0
$403$	−4.94326	−0.246241
$404$	0	0
$405$	6.07854	0.302045
$406$	0	0
$407$	24.5049	1.21466
$408$	0	0
$409$	−19.9013	−0.984057	−0.492029	−	0.870579i	$-0.663744\pi$
$409$	−19.9013	−0.984057	−0.492029	+	0.870579i	$0.663744\pi$
$410$	0	0
$411$	36.0145	1.77646
$412$	0	0
$413$	1.19598	0.0588502
$414$	0	0
$415$	−8.94837	−0.439258
$416$	0	0
$417$	−18.1219	−0.887431
$418$	0	0
$419$	−35.1243	−1.71593	−0.857967	−	0.513705i	$-0.828272\pi$
$419$	−35.1243	−1.71593	−0.857967	+	0.513705i	$0.828272\pi$
$420$	0	0
$421$	−1.45685	−0.0710023	−0.0355011	−	0.999370i	$-0.511303\pi$
$421$	−1.45685	−0.0710023	−0.0355011	+	0.999370i	$0.511303\pi$
$422$	0	0
$423$	9.89441	0.481083
$424$	0	0
$425$	1.63356	0.0792391
$426$	0	0
$427$	4.85731	0.235062
$428$	0	0
$429$	−12.9157	−0.623576
$430$	0	0
$431$	−8.66283	−0.417274	−0.208637	−	0.977993i	$-0.566903\pi$
$431$	−8.66283	−0.417274	−0.208637	+	0.977993i	$0.566903\pi$
$432$	0	0
$433$	31.5165	1.51458	0.757292	−	0.653076i	$-0.226522\pi$
$433$	31.5165	1.51458	0.757292	+	0.653076i	$0.226522\pi$
$434$	0	0
$435$	−13.0726	−0.626781
$436$	0	0
$437$	−17.1203	−0.818973
$438$	0	0
$439$	−38.9197	−1.85754	−0.928769	−	0.370659i	$-0.879132\pi$
$439$	−38.9197	−1.85754	−0.928769	+	0.370659i	$0.879132\pi$
$440$	0	0
$441$	5.66276	0.269655
$442$	0	0
$443$	−39.8897	−1.89522	−0.947608	−	0.319437i	$-0.896506\pi$
$443$	−39.8897	−1.89522	−0.947608	+	0.319437i	$0.896506\pi$
$444$	0	0
$445$	−14.6570	−0.694809
$446$	0	0
$447$	−2.06097	−0.0974803
$448$	0	0
$449$	25.8894	1.22179	0.610897	−	0.791710i	$-0.290809\pi$
$449$	25.8894	1.22179	0.610897	+	0.791710i	$0.290809\pi$
$450$	0	0
$451$	−46.2927	−2.17984
$452$	0	0
$453$	21.1184	0.992229
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−19.5368	−0.913891	−0.456945	−	0.889495i	$-0.651057\pi$
$457$	−19.5368	−0.913891	−0.456945	+	0.889495i	$0.651057\pi$
$458$	0	0
$459$	12.8025	0.597568
$460$	0	0
$461$	27.7387	1.29192	0.645961	−	0.763371i	$-0.276457\pi$
$461$	27.7387	1.29192	0.645961	+	0.763371i	$0.276457\pi$
$462$	0	0
$463$	−10.0194	−0.465641	−0.232820	−	0.972520i	$-0.574795\pi$
$463$	−10.0194	−0.465641	−0.232820	+	0.972520i	$0.574795\pi$
$464$	0	0
$465$	14.5493	0.674706
$466$	0	0
$467$	−31.9443	−1.47820	−0.739102	−	0.673593i	$-0.764750\pi$
$467$	−31.9443	−1.47820	−0.739102	+	0.673593i	$0.764750\pi$
$468$	0	0
$469$	9.03990	0.417424
$470$	0	0
$471$	33.4476	1.54118
$472$	0	0
$473$	−34.2045	−1.57272
$474$	0	0
$475$	−4.89392	−0.224548
$476$	0	0
$477$	28.9369	1.32493
$478$	0	0
$479$	1.64349	0.0750929	0.0375465	−	0.999295i	$-0.488046\pi$
$479$	1.64349	0.0750929	0.0375465	+	0.999295i	$0.488046\pi$
$480$	0	0
$481$	−5.58422	−0.254618
$482$	0	0
$483$	10.2963	0.468498
$484$	0	0
$485$	−0.922622	−0.0418941
$486$	0	0
$487$	−22.4415	−1.01692	−0.508461	−	0.861085i	$-0.669786\pi$
$487$	−22.4415	−1.01692	−0.508461	+	0.861085i	$0.669786\pi$
$488$	0	0
$489$	36.4768	1.64954
$490$	0	0
$491$	−5.55296	−0.250601	−0.125301	−	0.992119i	$-0.539990\pi$
$491$	−5.55296	−0.250601	−0.125301	+	0.992119i	$0.539990\pi$
$492$	0	0
$493$	−7.25548	−0.326771
$494$	0	0
$495$	24.8495	1.11690
$496$	0	0
$497$	6.15033	0.275880
$498$	0	0
$499$	6.89874	0.308830	0.154415	−	0.988006i	$-0.450651\pi$
$499$	6.89874	0.308830	0.154415	+	0.988006i	$0.450651\pi$
$500$	0	0
$501$	67.6509	3.02242
$502$	0	0
$503$	17.7900	0.793217	0.396608	−	0.917988i	$-0.370187\pi$
$503$	17.7900	0.793217	0.396608	+	0.917988i	$0.370187\pi$
$504$	0	0
$505$	−9.53511	−0.424307
$506$	0	0
$507$	2.94326	0.130715
$508$	0	0
$509$	22.7905	1.01017	0.505085	−	0.863069i	$-0.331461\pi$
$509$	22.7905	1.01017	0.505085	+	0.863069i	$0.331461\pi$
$510$	0	0
$511$	−4.12442	−0.182454
$512$	0	0
$513$	−38.3545	−1.69339
$514$	0	0
$515$	−16.2194	−0.714713
$516$	0	0
$517$	7.66748	0.337215
$518$	0	0
$519$	15.2296	0.668507
$520$	0	0
$521$	3.86432	0.169299	0.0846495	−	0.996411i	$-0.473023\pi$
$521$	3.86432	0.169299	0.0846495	+	0.996411i	$0.473023\pi$
$522$	0	0
$523$	−17.2782	−0.755523	−0.377762	−	0.925903i	$-0.623306\pi$
$523$	−17.2782	−0.755523	−0.377762	+	0.925903i	$0.623306\pi$
$524$	0	0
$525$	2.94326	0.128454
$526$	0	0
$527$	8.07508	0.351756
$528$	0	0
$529$	−10.7621	−0.467917
$530$	0	0
$531$	6.77253	0.293903
$532$	0	0
$533$	10.5493	0.456940
$534$	0	0
$535$	−9.26036	−0.400360
$536$	0	0
$537$	66.9935	2.89099
$538$	0	0
$539$	4.38824	0.189015
$540$	0	0
$541$	−12.7630	−0.548723	−0.274362	−	0.961627i	$-0.588467\pi$
$541$	−12.7630	−0.548723	−0.274362	+	0.961627i	$0.588467\pi$
$542$	0	0
$543$	−36.0291	−1.54615
$544$	0	0
$545$	14.0584	0.602196
$546$	0	0
$547$	1.14182	0.0488206	0.0244103	−	0.999702i	$-0.492229\pi$
$547$	1.14182	0.0488206	0.0244103	+	0.999702i	$0.492229\pi$
$548$	0	0
$549$	27.5058	1.17392
$550$	0	0
$551$	21.7365	0.926005
$552$	0	0
$553$	0.113723	0.00483599
$554$	0	0
$555$	16.4358	0.697660
$556$	0	0
$557$	36.9899	1.56731	0.783656	−	0.621196i	$-0.213353\pi$
$557$	36.9899	1.56731	0.783656	+	0.621196i	$0.213353\pi$
$558$	0	0
$559$	7.79458	0.329676
$560$	0	0
$561$	21.0985	0.890781
$562$	0	0
$563$	1.41262	0.0595348	0.0297674	−	0.999557i	$-0.490523\pi$
$563$	1.41262	0.0595348	0.0297674	+	0.999557i	$0.490523\pi$
$564$	0	0
$565$	−14.0841	−0.592524
$566$	0	0
$567$	6.07854	0.255275
$568$	0	0
$569$	−8.35332	−0.350189	−0.175095	−	0.984552i	$-0.556023\pi$
$569$	−8.35332	−0.350189	−0.175095	+	0.984552i	$0.556023\pi$
$570$	0	0
$571$	8.62823	0.361080	0.180540	−	0.983568i	$-0.442215\pi$
$571$	8.62823	0.361080	0.180540	+	0.983568i	$0.442215\pi$
$572$	0	0
$573$	32.3480	1.35136
$574$	0	0
$575$	3.49827	0.145888
$576$	0	0
$577$	8.43602	0.351196	0.175598	−	0.984462i	$-0.443814\pi$
$577$	8.43602	0.351196	0.175598	+	0.984462i	$0.443814\pi$
$578$	0	0
$579$	−61.5258	−2.55692
$580$	0	0
$581$	−8.94837	−0.371241
$582$	0	0
$583$	22.4240	0.928709
$584$	0	0
$585$	−5.66276	−0.234126
$586$	0	0
$587$	−21.8434	−0.901573	−0.450787	−	0.892632i	$-0.648856\pi$
$587$	−21.8434	−0.901573	−0.450787	+	0.892632i	$0.648856\pi$
$588$	0	0
$589$	−24.1919	−0.996809
$590$	0	0
$591$	−62.3840	−2.56614
$592$	0	0
$593$	35.3474	1.45154	0.725772	−	0.687936i	$-0.241483\pi$
$593$	35.3474	1.45154	0.725772	+	0.687936i	$0.241483\pi$
$594$	0	0
$595$	1.63356	0.0669693
$596$	0	0
$597$	−6.59953	−0.270101
$598$	0	0
$599$	−1.04401	−0.0426571	−0.0213286	−	0.999773i	$-0.506790\pi$
$599$	−1.04401	−0.0426571	−0.0213286	+	0.999773i	$0.506790\pi$
$600$	0	0
$601$	42.1053	1.71751	0.858756	−	0.512385i	$-0.171238\pi$
$601$	42.1053	1.71751	0.858756	+	0.512385i	$0.171238\pi$
$602$	0	0
$603$	51.1908	2.08465
$604$	0	0
$605$	8.25665	0.335680
$606$	0	0
$607$	−12.3833	−0.502622	−0.251311	−	0.967906i	$-0.580862\pi$
$607$	−12.3833	−0.502622	−0.251311	+	0.967906i	$0.580862\pi$
$608$	0	0
$609$	−13.0726	−0.529727
$610$	0	0
$611$	−1.74728	−0.0706873
$612$	0	0
$613$	−11.6366	−0.469999	−0.235000	−	0.971995i	$-0.575509\pi$
$613$	−11.6366	−0.469999	−0.235000	+	0.971995i	$0.575509\pi$
$614$	0	0
$615$	−31.0492	−1.25202
$616$	0	0
$617$	−17.9959	−0.724487	−0.362244	−	0.932083i	$-0.617989\pi$
$617$	−17.9959	−0.724487	−0.362244	+	0.932083i	$0.617989\pi$
$618$	0	0
$619$	−6.68722	−0.268782	−0.134391	−	0.990928i	$-0.542908\pi$
$619$	−6.68722	−0.268782	−0.134391	+	0.990928i	$0.542908\pi$
$620$	0	0
$621$	27.4166	1.10019
$622$	0	0
$623$	−14.6570	−0.587221
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−63.2084	−2.52430
$628$	0	0
$629$	9.12213	0.363723
$630$	0	0
$631$	−30.8301	−1.22733	−0.613664	−	0.789567i	$-0.710305\pi$
$631$	−30.8301	−1.22733	−0.613664	+	0.789567i	$0.710305\pi$
$632$	0	0
$633$	−29.3019	−1.16465
$634$	0	0
$635$	5.82724	0.231247
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	34.8278	1.37777
$640$	0	0
$641$	36.0416	1.42356	0.711779	−	0.702403i	$-0.247890\pi$
$641$	36.0416	1.42356	0.711779	+	0.702403i	$0.247890\pi$
$642$	0	0
$643$	42.6395	1.68154	0.840769	−	0.541393i	$-0.182103\pi$
$643$	42.6395	1.68154	0.840769	+	0.541393i	$0.182103\pi$
$644$	0	0
$645$	−22.9415	−0.903319
$646$	0	0
$647$	−30.6117	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$647$	−30.6117	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$648$	0	0
$649$	5.24824	0.206011
$650$	0	0
$651$	14.5493	0.570231
$652$	0	0
$653$	−35.0406	−1.37124	−0.685622	−	0.727957i	$-0.740470\pi$
$653$	−35.0406	−1.37124	−0.685622	+	0.727957i	$0.740470\pi$
$654$	0	0
$655$	−2.53422	−0.0990203
$656$	0	0
$657$	−23.3556	−0.911189
$658$	0	0
$659$	7.53203	0.293406	0.146703	−	0.989181i	$-0.453134\pi$
$659$	7.53203	0.293406	0.146703	+	0.989181i	$0.453134\pi$
$660$	0	0
$661$	35.6641	1.38717	0.693587	−	0.720373i	$-0.256029\pi$
$661$	35.6641	1.38717	0.693587	+	0.720373i	$0.256029\pi$
$662$	0	0
$663$	−4.80797	−0.186726
$664$	0	0
$665$	−4.89392	−0.189778
$666$	0	0
$667$	−15.5377	−0.601621
$668$	0	0
$669$	−71.4899	−2.76396
$670$	0	0
$671$	21.3150	0.822858
$672$	0	0
$673$	3.61291	0.139267	0.0696337	−	0.997573i	$-0.477817\pi$
$673$	3.61291	0.139267	0.0696337	+	0.997573i	$0.477817\pi$
$674$	0	0
$675$	7.83717	0.301653
$676$	0	0
$677$	31.1608	1.19761	0.598804	−	0.800896i	$-0.295643\pi$
$677$	31.1608	1.19761	0.598804	+	0.800896i	$0.295643\pi$
$678$	0	0
$679$	−0.922622	−0.0354070
$680$	0	0
$681$	−39.4058	−1.51003
$682$	0	0
$683$	−36.4164	−1.39343	−0.696717	−	0.717346i	$-0.745357\pi$
$683$	−36.4164	−1.39343	−0.696717	+	0.717346i	$0.745357\pi$
$684$	0	0
$685$	12.2363	0.467524
$686$	0	0
$687$	26.4560	1.00936
$688$	0	0
$689$	−5.11003	−0.194677
$690$	0	0
$691$	−33.5542	−1.27646	−0.638231	−	0.769845i	$-0.720333\pi$
$691$	−33.5542	−1.27646	−0.638231	+	0.769845i	$0.720333\pi$
$692$	0	0
$693$	24.8495	0.943956
$694$	0	0
$695$	−6.15708	−0.233551
$696$	0	0
$697$	−17.2328	−0.652740
$698$	0	0
$699$	43.2313	1.63516
$700$	0	0
$701$	40.4536	1.52791	0.763956	−	0.645268i	$-0.223254\pi$
$701$	40.4536	1.52791	0.763956	+	0.645268i	$0.223254\pi$
$702$	0	0
$703$	−27.3287	−1.03072
$704$	0	0
$705$	5.14269	0.193685
$706$	0	0
$707$	−9.53511	−0.358605
$708$	0	0
$709$	20.6197	0.774388	0.387194	−	0.921998i	$-0.373444\pi$
$709$	20.6197	0.774388	0.387194	+	0.921998i	$0.373444\pi$
$710$	0	0
$711$	0.643985	0.0241513
$712$	0	0
$713$	17.2929	0.647623
$714$	0	0
$715$	−4.38824	−0.164111
$716$	0	0
$717$	−51.4155	−1.92015
$718$	0	0
$719$	0.754289	0.0281302	0.0140651	−	0.999901i	$-0.495523\pi$
$719$	0.754289	0.0281302	0.0140651	+	0.999901i	$0.495523\pi$
$720$	0	0
$721$	−16.2194	−0.604043
$722$	0	0
$723$	57.5809	2.14146
$724$	0	0
$725$	−4.44153	−0.164954
$726$	0	0
$727$	−17.4323	−0.646530	−0.323265	−	0.946309i	$-0.604780\pi$
$727$	−17.4323	−0.646530	−0.323265	+	0.946309i	$0.604780\pi$
$728$	0	0
$729$	−34.7791	−1.28812
$730$	0	0
$731$	−12.7329	−0.470943
$732$	0	0
$733$	5.50399	0.203295	0.101647	−	0.994821i	$-0.467589\pi$
$733$	5.50399	0.203295	0.101647	+	0.994821i	$0.467589\pi$
$734$	0	0
$735$	2.94326	0.108564
$736$	0	0
$737$	39.6692	1.46124
$738$	0	0
$739$	53.3360	1.96200	0.980999	−	0.194012i	$-0.0621502\pi$
$739$	53.3360	1.96200	0.980999	+	0.194012i	$0.0621502\pi$
$740$	0	0
$741$	14.4041	0.529146
$742$	0	0
$743$	39.4752	1.44820	0.724102	−	0.689692i	$-0.242254\pi$
$743$	39.4752	1.44820	0.724102	+	0.689692i	$0.242254\pi$
$744$	0	0
$745$	−0.700233	−0.0256546
$746$	0	0
$747$	−50.6724	−1.85401
$748$	0	0
$749$	−9.26036	−0.338366
$750$	0	0
$751$	−17.2216	−0.628424	−0.314212	−	0.949353i	$-0.601740\pi$
$751$	−17.2216	−0.628424	−0.314212	+	0.949353i	$0.601740\pi$
$752$	0	0
$753$	12.4733	0.454551
$754$	0	0
$755$	7.17518	0.261132
$756$	0	0
$757$	−28.4096	−1.03256	−0.516281	−	0.856419i	$-0.672684\pi$
$757$	−28.4096	−1.03256	−0.516281	+	0.856419i	$0.672684\pi$
$758$	0	0
$759$	45.1827	1.64003
$760$	0	0
$761$	−10.9677	−0.397577	−0.198789	−	0.980042i	$-0.563701\pi$
$761$	−10.9677	−0.397577	−0.198789	+	0.980042i	$0.563701\pi$
$762$	0	0
$763$	14.0584	0.508948
$764$	0	0
$765$	9.25043	0.334450
$766$	0	0
$767$	−1.19598	−0.0431842
$768$	0	0
$769$	12.7750	0.460677	0.230339	−	0.973111i	$-0.426017\pi$
$769$	12.7750	0.460677	0.230339	+	0.973111i	$0.426017\pi$
$770$	0	0
$771$	83.7778	3.01718
$772$	0	0
$773$	−46.9234	−1.68772	−0.843858	−	0.536566i	$-0.819721\pi$
$773$	−46.9234	−1.68772	−0.843858	+	0.536566i	$0.819721\pi$
$774$	0	0
$775$	4.94326	0.177567
$776$	0	0
$777$	16.4358	0.589630
$778$	0	0
$779$	51.6272	1.84974
$780$	0	0
$781$	26.9891	0.965746
$782$	0	0
$783$	−34.8090	−1.24397
$784$	0	0
$785$	11.3642	0.405604
$786$	0	0
$787$	−39.9787	−1.42509	−0.712544	−	0.701627i	$-0.752457\pi$
$787$	−39.9787	−1.42509	−0.712544	+	0.701627i	$0.752457\pi$
$788$	0	0
$789$	−64.9448	−2.31210
$790$	0	0
$791$	−14.0841	−0.500775
$792$	0	0
$793$	−4.85731	−0.172488
$794$	0	0
$795$	15.0401	0.533419
$796$	0	0
$797$	−28.8041	−1.02030	−0.510148	−	0.860087i	$-0.670409\pi$
$797$	−28.8041	−1.02030	−0.510148	+	0.860087i	$0.670409\pi$
$798$	0	0
$799$	2.85428	0.100977
$800$	0	0
$801$	−82.9991	−2.93263
$802$	0	0
$803$	−18.0990	−0.638698
$804$	0	0
$805$	3.49827	0.123298
$806$	0	0
$807$	−37.9707	−1.33663
$808$	0	0
$809$	9.92683	0.349009	0.174504	−	0.984656i	$-0.444168\pi$
$809$	9.92683	0.349009	0.174504	+	0.984656i	$0.444168\pi$
$810$	0	0
$811$	10.6896	0.375363	0.187681	−	0.982230i	$-0.439903\pi$
$811$	10.6896	0.375363	0.187681	+	0.982230i	$0.439903\pi$
$812$	0	0
$813$	−17.3582	−0.608778
$814$	0	0
$815$	12.3934	0.434120
$816$	0	0
$817$	38.1461	1.33456
$818$	0	0
$819$	−5.66276	−0.197873
$820$	0	0
$821$	−26.8665	−0.937647	−0.468824	−	0.883292i	$-0.655322\pi$
$821$	−26.8665	−0.937647	−0.468824	+	0.883292i	$0.655322\pi$
$822$	0	0
$823$	−3.90957	−0.136279	−0.0681396	−	0.997676i	$-0.521706\pi$
$823$	−3.90957	−0.136279	−0.0681396	+	0.997676i	$0.521706\pi$
$824$	0	0
$825$	12.9157	0.449667
$826$	0	0
$827$	−9.77993	−0.340082	−0.170041	−	0.985437i	$-0.554390\pi$
$827$	−9.77993	−0.340082	−0.170041	+	0.985437i	$0.554390\pi$
$828$	0	0
$829$	−13.7100	−0.476169	−0.238085	−	0.971244i	$-0.576520\pi$
$829$	−13.7100	−0.476169	−0.238085	+	0.971244i	$0.576520\pi$
$830$	0	0
$831$	73.6162	2.55372
$832$	0	0
$833$	1.63356	0.0565994
$834$	0	0
$835$	22.9850	0.795430
$836$	0	0
$837$	38.7412	1.33909
$838$	0	0
$839$	32.4833	1.12145	0.560724	−	0.828003i	$-0.310523\pi$
$839$	32.4833	1.12145	0.560724	+	0.828003i	$0.310523\pi$
$840$	0	0
$841$	−9.27282	−0.319753
$842$	0	0
$843$	−57.9913	−1.99733
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	8.25665	0.283702
$848$	0	0
$849$	−27.0408	−0.928039
$850$	0	0
$851$	19.5351	0.669655
$852$	0	0
$853$	−32.6946	−1.11944	−0.559721	−	0.828681i	$-0.689092\pi$
$853$	−32.6946	−1.11944	−0.559721	+	0.828681i	$0.689092\pi$
$854$	0	0
$855$	−27.7131	−0.947767
$856$	0	0
$857$	−39.8519	−1.36131	−0.680657	−	0.732602i	$-0.738306\pi$
$857$	−39.8519	−1.36131	−0.680657	+	0.732602i	$0.738306\pi$
$858$	0	0
$859$	−22.9129	−0.781777	−0.390888	−	0.920438i	$-0.627832\pi$
$859$	−22.9129	−0.781777	−0.390888	+	0.920438i	$0.627832\pi$
$860$	0	0
$861$	−31.0492	−1.05815
$862$	0	0
$863$	−12.0992	−0.411861	−0.205931	−	0.978567i	$-0.566022\pi$
$863$	−12.0992	−0.411861	−0.205931	+	0.978567i	$0.566022\pi$
$864$	0	0
$865$	5.17442	0.175935
$866$	0	0
$867$	−42.1813	−1.43255
$868$	0	0
$869$	0.499043	0.0169289
$870$	0	0
$871$	−9.03990	−0.306305
$872$	0	0
$873$	−5.22458	−0.176825
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−2.68262	−0.0905857	−0.0452929	−	0.998974i	$-0.514422\pi$
$877$	−2.68262	−0.0905857	−0.0452929	+	0.998974i	$0.514422\pi$
$878$	0	0
$879$	57.2987	1.93264
$880$	0	0
$881$	−40.4121	−1.36152	−0.680759	−	0.732508i	$-0.738350\pi$
$881$	−40.4121	−1.36152	−0.680759	+	0.732508i	$0.738350\pi$
$882$	0	0
$883$	−0.0967751	−0.00325674	−0.00162837	−	0.999999i	$-0.500518\pi$
$883$	−0.0967751	−0.00325674	−0.00162837	+	0.999999i	$0.500518\pi$
$884$	0	0
$885$	3.52007	0.118326
$886$	0	0
$887$	−2.62522	−0.0881464	−0.0440732	−	0.999028i	$-0.514033\pi$
$887$	−2.62522	−0.0881464	−0.0440732	+	0.999028i	$0.514033\pi$
$888$	0	0
$889$	5.82724	0.195439
$890$	0	0
$891$	26.6741	0.893616
$892$	0	0
$893$	−8.55104	−0.286150
$894$	0	0
$895$	22.7617	0.760840
$896$	0	0
$897$	−10.2963	−0.343784
$898$	0	0
$899$	−21.9556	−0.732261
$900$	0	0
$901$	8.34752	0.278096
$902$	0	0
$903$	−22.9415	−0.763444
$904$	0	0
$905$	−12.2412	−0.406912
$906$	0	0
$907$	−31.7510	−1.05427	−0.527137	−	0.849780i	$-0.676735\pi$
$907$	−31.7510	−1.05427	−0.527137	+	0.849780i	$0.676735\pi$
$908$	0	0
$909$	−53.9950	−1.79090
$910$	0	0
$911$	−20.0219	−0.663356	−0.331678	−	0.943393i	$-0.607615\pi$
$911$	−20.0219	−0.663356	−0.331678	+	0.943393i	$0.607615\pi$
$912$	0	0
$913$	−39.2676	−1.29957
$914$	0	0
$915$	14.2963	0.472621
$916$	0	0
$917$	−2.53422	−0.0836874
$918$	0	0
$919$	14.5182	0.478913	0.239456	−	0.970907i	$-0.423031\pi$
$919$	14.5182	0.478913	0.239456	+	0.970907i	$0.423031\pi$
$920$	0	0
$921$	9.48903	0.312674
$922$	0	0
$923$	−6.15033	−0.202441
$924$	0	0
$925$	5.58422	0.183608
$926$	0	0
$927$	−91.8467	−3.01664
$928$	0	0
$929$	−22.9620	−0.753358	−0.376679	−	0.926344i	$-0.622934\pi$
$929$	−22.9620	−0.753358	−0.376679	+	0.926344i	$0.622934\pi$
$930$	0	0
$931$	−4.89392	−0.160392
$932$	0	0
$933$	−62.9180	−2.05984
$934$	0	0
$935$	7.16843	0.234433
$936$	0	0
$937$	29.5443	0.965171	0.482585	−	0.875849i	$-0.339698\pi$
$937$	29.5443	0.965171	0.482585	+	0.875849i	$0.339698\pi$
$938$	0	0
$939$	−22.4184	−0.731597
$940$	0	0
$941$	5.94033	0.193649	0.0968247	−	0.995301i	$-0.469131\pi$
$941$	5.94033	0.193649	0.0968247	+	0.995301i	$0.469131\pi$
$942$	0	0
$943$	−36.9042	−1.20177
$944$	0	0
$945$	7.83717	0.254943
$946$	0	0
$947$	33.9658	1.10374	0.551871	−	0.833930i	$-0.313914\pi$
$947$	33.9658	1.10374	0.551871	+	0.833930i	$0.313914\pi$
$948$	0	0
$949$	4.12442	0.133884
$950$	0	0
$951$	−21.6570	−0.702275
$952$	0	0
$953$	−21.2829	−0.689421	−0.344711	−	0.938709i	$-0.612023\pi$
$953$	−21.2829	−0.689421	−0.344711	+	0.938709i	$0.612023\pi$
$954$	0	0
$955$	10.9906	0.355646
$956$	0	0
$957$	−57.3655	−1.85436
$958$	0	0
$959$	12.2363	0.395130
$960$	0	0
$961$	−6.56422	−0.211749
$962$	0	0
$963$	−52.4392	−1.68983
$964$	0	0
$965$	−20.9040	−0.672923
$966$	0	0
$967$	−1.14796	−0.0369159	−0.0184579	−	0.999830i	$-0.505876\pi$
$967$	−1.14796	−0.0369159	−0.0184579	+	0.999830i	$0.505876\pi$
$968$	0	0
$969$	−23.5298	−0.755887
$970$	0	0
$971$	−5.99644	−0.192435	−0.0962175	−	0.995360i	$-0.530674\pi$
$971$	−5.99644	−0.192435	−0.0962175	+	0.995360i	$0.530674\pi$
$972$	0	0
$973$	−6.15708	−0.197387
$974$	0	0
$975$	−2.94326	−0.0942596
$976$	0	0
$977$	13.1231	0.419844	0.209922	−	0.977718i	$-0.432679\pi$
$977$	13.1231	0.419844	0.209922	+	0.977718i	$0.432679\pi$
$978$	0	0
$979$	−64.3185	−2.05563
$980$	0	0
$981$	79.6093	2.54173
$982$	0	0
$983$	29.7248	0.948075	0.474038	−	0.880505i	$-0.342796\pi$
$983$	29.7248	0.948075	0.474038	+	0.880505i	$0.342796\pi$
$984$	0	0
$985$	−21.1956	−0.675347
$986$	0	0
$987$	5.14269	0.163694
$988$	0	0
$989$	−27.2676	−0.867059
$990$	0	0
$991$	−3.03099	−0.0962825	−0.0481412	−	0.998841i	$-0.515330\pi$
$991$	−3.03099	−0.0962825	−0.0481412	+	0.998841i	$0.515330\pi$
$992$	0	0
$993$	46.6466	1.48029
$994$	0	0
$995$	−2.24226	−0.0710843
$996$	0	0
$997$	13.8090	0.437335	0.218668	−	0.975799i	$-0.429829\pi$
$997$	13.8090	0.437335	0.218668	+	0.975799i	$0.429829\pi$
$998$	0	0
$999$	43.7645	1.38465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.z.1.5	✓	5
4.3	odd	2		7280.2.a.cd.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.z.1.5	✓	5	1.1	even	1	trivial
7280.2.a.cd.1.1		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.94326 q^{3} +1.00000 q^{5} +1.00000 q^{7} +5.66276 q^{9} +O(q^{10})\)	\(q+2.94326 q^{3} +1.00000 q^{5} +1.00000 q^{7} +5.66276 q^{9} +4.38824 q^{11} -1.00000 q^{13} +2.94326 q^{15} +1.63356 q^{17} -4.89392 q^{19} +2.94326 q^{21} +3.49827 q^{23} +1.00000 q^{25} +7.83717 q^{27} -4.44153 q^{29} +4.94326 q^{31} +12.9157 q^{33} +1.00000 q^{35} +5.58422 q^{37} -2.94326 q^{39} -10.5493 q^{41} -7.79458 q^{43} +5.66276 q^{45} +1.74728 q^{47} +1.00000 q^{49} +4.80797 q^{51} +5.11003 q^{53} +4.38824 q^{55} -14.4041 q^{57} +1.19598 q^{59} +4.85731 q^{61} +5.66276 q^{63} -1.00000 q^{65} +9.03990 q^{67} +10.2963 q^{69} +6.15033 q^{71} -4.12442 q^{73} +2.94326 q^{75} +4.38824 q^{77} +0.113723 q^{79} +6.07854 q^{81} -8.94837 q^{83} +1.63356 q^{85} -13.0726 q^{87} -14.6570 q^{89} -1.00000 q^{91} +14.5493 q^{93} -4.89392 q^{95} -0.922622 q^{97} +24.8495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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