LMFDB - Embedded newform 8037.2.a.n.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 $ x^16 - 4x^15 - 13x^14 + 65x^13 + 47x^12 - 390x^11 + 4x^10 + 1115x^9 - 320x^8 - 1639x^7 + 618x^6 + 1250x^5 - 487x^4 - 456x^3 + 179x^2 + 62*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.640778$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.640778 q^{2} -1.58940 q^{4} -3.12262 q^{5} -2.58301 q^{7} -2.30001 q^{8} +O(q^{10})$	\(q+0.640778 q^{2} -1.58940 q^{4} -3.12262 q^{5} -2.58301 q^{7} -2.30001 q^{8} -2.00091 q^{10} +3.99875 q^{11} -5.23482 q^{13} -1.65513 q^{14} +1.70501 q^{16} +1.51727 q^{17} -1.00000 q^{19} +4.96310 q^{20} +2.56231 q^{22} +7.12352 q^{23} +4.75075 q^{25} -3.35436 q^{26} +4.10544 q^{28} -1.75629 q^{29} -4.89389 q^{31} +5.69256 q^{32} +0.972233 q^{34} +8.06575 q^{35} +10.0750 q^{37} -0.640778 q^{38} +7.18206 q^{40} -9.98601 q^{41} +4.83832 q^{43} -6.35562 q^{44} +4.56459 q^{46} +1.00000 q^{47} -0.328076 q^{49} +3.04418 q^{50} +8.32024 q^{52} +7.61820 q^{53} -12.4866 q^{55} +5.94095 q^{56} -1.12540 q^{58} -0.952829 q^{59} +4.27394 q^{61} -3.13590 q^{62} +0.237647 q^{64} +16.3464 q^{65} +1.95517 q^{67} -2.41155 q^{68} +5.16836 q^{70} +0.697878 q^{71} -9.39441 q^{73} +6.45587 q^{74} +1.58940 q^{76} -10.3288 q^{77} +13.9455 q^{79} -5.32410 q^{80} -6.39882 q^{82} +13.7285 q^{83} -4.73785 q^{85} +3.10029 q^{86} -9.19716 q^{88} +13.8951 q^{89} +13.5216 q^{91} -11.3221 q^{92} +0.640778 q^{94} +3.12262 q^{95} -1.09104 q^{97} -0.210224 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

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 $

$2$	0.640778	0.453099	0.226549	−	0.974000i	$-0.427256\pi$
$2$	0.640778	0.453099	0.226549	+	0.974000i	$0.427256\pi$
$3$	0	0
$3$	0	0
$4$	−1.58940	−0.794702
$5$	−3.12262	−1.39648	−0.698239	−	0.715865i	$-0.746033\pi$
$5$	−3.12262	−1.39648	−0.698239	+	0.715865i	$0.746033\pi$
$6$	0	0
$7$	−2.58301	−0.976285	−0.488142	−	0.872764i	$-0.662325\pi$
$7$	−2.58301	−0.976285	−0.488142	+	0.872764i	$0.662325\pi$
$8$	−2.30001	−0.813177
$9$	0	0
$10$	−2.00091	−0.632742
$11$	3.99875	1.20567	0.602834	−	0.797867i	$-0.294038\pi$
$11$	3.99875	1.20567	0.602834	+	0.797867i	$0.294038\pi$
$12$	0	0
$13$	−5.23482	−1.45188	−0.725939	−	0.687759i	$-0.758595\pi$
$13$	−5.23482	−1.45188	−0.725939	+	0.687759i	$0.758595\pi$
$14$	−1.65513	−0.442353
$15$	0	0
$16$	1.70501	0.426252
$17$	1.51727	0.367992	0.183996	−	0.982927i	$-0.441097\pi$
$17$	1.51727	0.367992	0.183996	+	0.982927i	$0.441097\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.96310	1.10978
$21$	0	0
$22$	2.56231	0.546286
$23$	7.12352	1.48536	0.742678	−	0.669649i	$-0.233555\pi$
$23$	7.12352	1.48536	0.742678	+	0.669649i	$0.233555\pi$
$24$	0	0
$25$	4.75075	0.950151
$26$	−3.35436	−0.657844
$27$	0	0
$28$	4.10544	0.775855
$29$	−1.75629	−0.326136	−0.163068	−	0.986615i	$-0.552139\pi$
$29$	−1.75629	−0.326136	−0.163068	+	0.986615i	$0.552139\pi$
$30$	0	0
$31$	−4.89389	−0.878968	−0.439484	−	0.898250i	$-0.644839\pi$
$31$	−4.89389	−0.878968	−0.439484	+	0.898250i	$0.644839\pi$
$32$	5.69256	1.00631
$33$	0	0
$34$	0.972233	0.166737
$35$	8.06575	1.36336
$36$	0	0
$37$	10.0750	1.65633	0.828164	−	0.560486i	$-0.189386\pi$
$37$	10.0750	1.65633	0.828164	+	0.560486i	$0.189386\pi$
$38$	−0.640778	−0.103948
$39$	0	0
$40$	7.18206	1.13558
$41$	−9.98601	−1.55955	−0.779776	−	0.626058i	$-0.784667\pi$
$41$	−9.98601	−1.55955	−0.779776	+	0.626058i	$0.784667\pi$
$42$	0	0
$43$	4.83832	0.737837	0.368919	−	0.929462i	$-0.379728\pi$
$43$	4.83832	0.737837	0.368919	+	0.929462i	$0.379728\pi$
$44$	−6.35562	−0.958146
$45$	0	0
$46$	4.56459	0.673013
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−0.328076	−0.0468680
$50$	3.04418	0.430512
$51$	0	0
$52$	8.32024	1.15381
$53$	7.61820	1.04644	0.523220	−	0.852197i	$-0.324730\pi$
$53$	7.61820	1.04644	0.523220	+	0.852197i	$0.324730\pi$
$54$	0	0
$55$	−12.4866	−1.68369
$56$	5.94095	0.793892
$57$	0	0
$58$	−1.12540	−0.147772
$59$	−0.952829	−0.124048	−0.0620239	−	0.998075i	$-0.519755\pi$
$59$	−0.952829	−0.124048	−0.0620239	+	0.998075i	$0.519755\pi$
$60$	0	0
$61$	4.27394	0.547222	0.273611	−	0.961840i	$-0.411782\pi$
$61$	4.27394	0.547222	0.273611	+	0.961840i	$0.411782\pi$
$62$	−3.13590	−0.398259
$63$	0	0
$64$	0.237647	0.0297058
$65$	16.3464	2.02752
$66$	0	0
$67$	1.95517	0.238863	0.119431	−	0.992842i	$-0.461893\pi$
$67$	1.95517	0.238863	0.119431	+	0.992842i	$0.461893\pi$
$68$	−2.41155	−0.292444
$69$	0	0
$70$	5.16836	0.617737
$71$	0.697878	0.0828229	0.0414115	−	0.999142i	$-0.486815\pi$
$71$	0.697878	0.0828229	0.0414115	+	0.999142i	$0.486815\pi$
$72$	0	0
$73$	−9.39441	−1.09953	−0.549766	−	0.835319i	$-0.685283\pi$
$73$	−9.39441	−1.09953	−0.549766	+	0.835319i	$0.685283\pi$
$74$	6.45587	0.750480
$75$	0	0
$76$	1.58940	0.182317
$77$	−10.3288	−1.17707
$78$	0	0
$79$	13.9455	1.56899	0.784493	−	0.620137i	$-0.212923\pi$
$79$	13.9455	1.56899	0.784493	+	0.620137i	$0.212923\pi$
$80$	−5.32410	−0.595252
$81$	0	0
$82$	−6.39882	−0.706631
$83$	13.7285	1.50690	0.753451	−	0.657504i	$-0.228388\pi$
$83$	13.7285	1.50690	0.753451	+	0.657504i	$0.228388\pi$
$84$	0	0
$85$	−4.73785	−0.513892
$86$	3.10029	0.334313
$87$	0	0
$88$	−9.19716	−0.980421
$89$	13.8951	1.47288	0.736440	−	0.676503i	$-0.236505\pi$
$89$	13.8951	1.47288	0.736440	+	0.676503i	$0.236505\pi$
$90$	0	0
$91$	13.5216	1.41745
$92$	−11.3221	−1.18041
$93$	0	0
$94$	0.640778	0.0660912
$95$	3.12262	0.320374
$96$	0	0
$97$	−1.09104	−0.110779	−0.0553893	−	0.998465i	$-0.517640\pi$
$97$	−1.09104	−0.110779	−0.0553893	+	0.998465i	$0.517640\pi$
$98$	−0.210224	−0.0212358
$99$	0	0
$100$	−7.55087	−0.755087
$101$	1.25099	0.124478	0.0622390	−	0.998061i	$-0.480176\pi$
$101$	1.25099	0.124478	0.0622390	+	0.998061i	$0.480176\pi$
$102$	0	0
$103$	−13.8824	−1.36788	−0.683938	−	0.729540i	$-0.739734\pi$
$103$	−13.8824	−1.36788	−0.683938	+	0.729540i	$0.739734\pi$
$104$	12.0401	1.18063
$105$	0	0
$106$	4.88158	0.474141
$107$	0.452007	0.0436972	0.0218486	−	0.999761i	$-0.493045\pi$
$107$	0.452007	0.0436972	0.0218486	+	0.999761i	$0.493045\pi$
$108$	0	0
$109$	2.03229	0.194658	0.0973290	−	0.995252i	$-0.468970\pi$
$109$	2.03229	0.194658	0.0973290	+	0.995252i	$0.468970\pi$
$110$	−8.00112	−0.762877
$111$	0	0
$112$	−4.40405	−0.416144
$113$	−9.79873	−0.921787	−0.460893	−	0.887455i	$-0.652471\pi$
$113$	−9.79873	−0.921787	−0.460893	+	0.887455i	$0.652471\pi$
$114$	0	0
$115$	−22.2440	−2.07427
$116$	2.79146	0.259181
$117$	0	0
$118$	−0.610552	−0.0562059
$119$	−3.91912	−0.359265
$120$	0	0
$121$	4.98997	0.453633
$122$	2.73865	0.247946
$123$	0	0
$124$	7.77836	0.698517
$125$	0.778298	0.0696131
$126$	0	0
$127$	20.1324	1.78646	0.893230	−	0.449601i	$-0.148434\pi$
$127$	20.1324	1.78646	0.893230	+	0.449601i	$0.148434\pi$
$128$	−11.2328	−0.992852
$129$	0	0
$130$	10.4744	0.918665
$131$	7.34254	0.641521	0.320761	−	0.947160i	$-0.396062\pi$
$131$	7.34254	0.641521	0.320761	+	0.947160i	$0.396062\pi$
$132$	0	0
$133$	2.58301	0.223975
$134$	1.25283	0.108228
$135$	0	0
$136$	−3.48974	−0.299242
$137$	−10.1117	−0.863902	−0.431951	−	0.901897i	$-0.642175\pi$
$137$	−10.1117	−0.863902	−0.431951	+	0.901897i	$0.642175\pi$
$138$	0	0
$139$	−15.5396	−1.31805	−0.659025	−	0.752121i	$-0.729031\pi$
$139$	−15.5396	−1.31805	−0.659025	+	0.752121i	$0.729031\pi$
$140$	−12.8197	−1.08346
$141$	0	0
$142$	0.447185	0.0375269
$143$	−20.9327	−1.75048
$144$	0	0
$145$	5.48424	0.455441
$146$	−6.01973	−0.498196
$147$	0	0
$148$	−16.0133	−1.31629
$149$	−14.0144	−1.14811	−0.574053	−	0.818818i	$-0.694630\pi$
$149$	−14.0144	−1.14811	−0.574053	+	0.818818i	$0.694630\pi$
$150$	0	0
$151$	−14.6807	−1.19470	−0.597348	−	0.801982i	$-0.703779\pi$
$151$	−14.6807	−1.19470	−0.597348	+	0.801982i	$0.703779\pi$
$152$	2.30001	0.186556
$153$	0	0
$154$	−6.61846	−0.533331
$155$	15.2817	1.22746
$156$	0	0
$157$	−8.91129	−0.711199	−0.355599	−	0.934638i	$-0.615723\pi$
$157$	−8.91129	−0.711199	−0.355599	+	0.934638i	$0.615723\pi$
$158$	8.93595	0.710906
$159$	0	0
$160$	−17.7757	−1.40529
$161$	−18.4001	−1.45013
$162$	0	0
$163$	−1.68358	−0.131868	−0.0659340	−	0.997824i	$-0.521003\pi$
$163$	−1.68358	−0.131868	−0.0659340	+	0.997824i	$0.521003\pi$
$164$	15.8718	1.23938
$165$	0	0
$166$	8.79695	0.682776
$167$	11.8413	0.916308	0.458154	−	0.888873i	$-0.348511\pi$
$167$	11.8413	0.916308	0.458154	+	0.888873i	$0.348511\pi$
$168$	0	0
$169$	14.4033	1.10795
$170$	−3.03591	−0.232844
$171$	0	0
$172$	−7.69005	−0.586360
$173$	−2.79928	−0.212825	−0.106413	−	0.994322i	$-0.533936\pi$
$173$	−2.79928	−0.212825	−0.106413	+	0.994322i	$0.533936\pi$
$174$	0	0
$175$	−12.2712	−0.927618
$176$	6.81790	0.513919
$177$	0	0
$178$	8.90369	0.667360
$179$	−7.53083	−0.562881	−0.281441	−	0.959579i	$-0.590812\pi$
$179$	−7.53083	−0.562881	−0.281441	+	0.959579i	$0.590812\pi$
$180$	0	0
$181$	1.71353	0.127365	0.0636827	−	0.997970i	$-0.479715\pi$
$181$	1.71353	0.127365	0.0636827	+	0.997970i	$0.479715\pi$
$182$	8.66433	0.642243
$183$	0	0
$184$	−16.3842	−1.20786
$185$	−31.4605	−2.31303
$186$	0	0
$187$	6.06717	0.443676
$188$	−1.58940	−0.115919
$189$	0	0
$190$	2.00091	0.145161
$191$	−15.6274	−1.13076	−0.565378	−	0.824832i	$-0.691270\pi$
$191$	−15.6274	−1.13076	−0.565378	+	0.824832i	$0.691270\pi$
$192$	0	0
$193$	−11.8691	−0.854354	−0.427177	−	0.904168i	$-0.640492\pi$
$193$	−11.8691	−0.854354	−0.427177	+	0.904168i	$0.640492\pi$
$194$	−0.699116	−0.0501936
$195$	0	0
$196$	0.521445	0.0372460
$197$	−15.5990	−1.11138	−0.555692	−	0.831388i	$-0.687547\pi$
$197$	−15.5990	−1.11138	−0.555692	+	0.831388i	$0.687547\pi$
$198$	0	0
$199$	−7.61541	−0.539842	−0.269921	−	0.962882i	$-0.586998\pi$
$199$	−7.61541	−0.539842	−0.269921	+	0.962882i	$0.586998\pi$
$200$	−10.9268	−0.772641
$201$	0	0
$202$	0.801607	0.0564008
$203$	4.53652	0.318401
$204$	0	0
$205$	31.1825	2.17788
$206$	−8.89555	−0.619782
$207$	0	0
$208$	−8.92542	−0.618866
$209$	−3.99875	−0.276599
$210$	0	0
$211$	9.44805	0.650431	0.325215	−	0.945640i	$-0.394563\pi$
$211$	9.44805	0.650431	0.325215	+	0.945640i	$0.394563\pi$
$212$	−12.1084	−0.831608
$213$	0	0
$214$	0.289636	0.0197991
$215$	−15.1082	−1.03037
$216$	0	0
$217$	12.6409	0.858123
$218$	1.30225	0.0881993
$219$	0	0
$220$	19.8462	1.33803
$221$	−7.94263	−0.534279
$222$	0	0
$223$	−4.92756	−0.329974	−0.164987	−	0.986296i	$-0.552758\pi$
$223$	−4.92756	−0.329974	−0.164987	+	0.986296i	$0.552758\pi$
$224$	−14.7039	−0.982446
$225$	0	0
$226$	−6.27881	−0.417660
$227$	−20.1567	−1.33785	−0.668924	−	0.743331i	$-0.733245\pi$
$227$	−20.1567	−1.33785	−0.668924	+	0.743331i	$0.733245\pi$
$228$	0	0
$229$	−16.0553	−1.06096	−0.530481	−	0.847697i	$-0.677989\pi$
$229$	−16.0553	−1.06096	−0.530481	+	0.847697i	$0.677989\pi$
$230$	−14.2535	−0.939847
$231$	0	0
$232$	4.03950	0.265206
$233$	−16.6969	−1.09385	−0.546925	−	0.837181i	$-0.684202\pi$
$233$	−16.6969	−1.09385	−0.546925	+	0.837181i	$0.684202\pi$
$234$	0	0
$235$	−3.12262	−0.203697
$236$	1.51443	0.0985809
$237$	0	0
$238$	−2.51128	−0.162782
$239$	5.37121	0.347435	0.173717	−	0.984796i	$-0.444422\pi$
$239$	5.37121	0.347435	0.173717	+	0.984796i	$0.444422\pi$
$240$	0	0
$241$	−14.7919	−0.952829	−0.476415	−	0.879221i	$-0.658064\pi$
$241$	−14.7919	−0.952829	−0.476415	+	0.879221i	$0.658064\pi$
$242$	3.19746	0.205541
$243$	0	0
$244$	−6.79302	−0.434879
$245$	1.02446	0.0654501
$246$	0	0
$247$	5.23482	0.333084
$248$	11.2560	0.714756
$249$	0	0
$250$	0.498717	0.0315416
$251$	−1.61867	−0.102169	−0.0510846	−	0.998694i	$-0.516268\pi$
$251$	−1.61867	−0.102169	−0.0510846	+	0.998694i	$0.516268\pi$
$252$	0	0
$253$	28.4851	1.79084
$254$	12.9004	0.809442
$255$	0	0
$256$	−7.67305	−0.479565
$257$	−16.9833	−1.05939	−0.529694	−	0.848189i	$-0.677693\pi$
$257$	−16.9833	−1.05939	−0.529694	+	0.848189i	$0.677693\pi$
$258$	0	0
$259$	−26.0239	−1.61705
$260$	−25.9809	−1.61127
$261$	0	0
$262$	4.70494	0.290672
$263$	21.1711	1.30547	0.652733	−	0.757588i	$-0.273623\pi$
$263$	21.1711	1.30547	0.652733	+	0.757588i	$0.273623\pi$
$264$	0	0
$265$	−23.7888	−1.46133
$266$	1.65513	0.101483
$267$	0	0
$268$	−3.10756	−0.189824
$269$	21.6473	1.31986	0.659929	−	0.751328i	$-0.270587\pi$
$269$	21.6473	1.31986	0.659929	+	0.751328i	$0.270587\pi$
$270$	0	0
$271$	22.6575	1.37635	0.688174	−	0.725546i	$-0.258413\pi$
$271$	22.6575	1.37635	0.688174	+	0.725546i	$0.258413\pi$
$272$	2.58696	0.156857
$273$	0	0
$274$	−6.47936	−0.391433
$275$	18.9971	1.14557
$276$	0	0
$277$	10.9954	0.660652	0.330326	−	0.943867i	$-0.392841\pi$
$277$	10.9954	0.660652	0.330326	+	0.943867i	$0.392841\pi$
$278$	−9.95742	−0.597206
$279$	0	0
$280$	−18.5513	−1.10865
$281$	−26.4656	−1.57881	−0.789404	−	0.613875i	$-0.789610\pi$
$281$	−26.4656	−1.57881	−0.789404	+	0.613875i	$0.789610\pi$
$282$	0	0
$283$	12.2059	0.725564	0.362782	−	0.931874i	$-0.381827\pi$
$283$	12.2059	0.725564	0.362782	+	0.931874i	$0.381827\pi$
$284$	−1.10921	−0.0658195
$285$	0	0
$286$	−13.4132	−0.793141
$287$	25.7939	1.52257
$288$	0	0
$289$	−14.6979	−0.864582
$290$	3.51418	0.206360
$291$	0	0
$292$	14.9315	0.873800
$293$	14.8045	0.864886	0.432443	−	0.901661i	$-0.357652\pi$
$293$	14.8045	0.864886	0.432443	+	0.901661i	$0.357652\pi$
$294$	0	0
$295$	2.97532	0.173230
$296$	−23.1727	−1.34689
$297$	0	0
$298$	−8.98014	−0.520206
$299$	−37.2903	−2.15656
$300$	0	0
$301$	−12.4974	−0.720339
$302$	−9.40705	−0.541315
$303$	0	0
$304$	−1.70501	−0.0977890
$305$	−13.3459	−0.764184
$306$	0	0
$307$	−9.46801	−0.540368	−0.270184	−	0.962809i	$-0.587085\pi$
$307$	−9.46801	−0.540368	−0.270184	+	0.962809i	$0.587085\pi$
$308$	16.4166	0.935423
$309$	0	0
$310$	9.79221	0.556160
$311$	7.23198	0.410088	0.205044	−	0.978753i	$-0.434266\pi$
$311$	7.23198	0.410088	0.205044	+	0.978753i	$0.434266\pi$
$312$	0	0
$313$	−24.7381	−1.39828	−0.699140	−	0.714985i	$-0.746434\pi$
$313$	−24.7381	−1.39828	−0.699140	+	0.714985i	$0.746434\pi$
$314$	−5.71016	−0.322243
$315$	0	0
$316$	−22.1650	−1.24688
$317$	31.6782	1.77923	0.889613	−	0.456714i	$-0.150974\pi$
$317$	31.6782	1.77923	0.889613	+	0.456714i	$0.150974\pi$
$318$	0	0
$319$	−7.02297	−0.393211
$320$	−0.742080	−0.0414835
$321$	0	0
$322$	−11.7904	−0.657052
$323$	−1.51727	−0.0844231
$324$	0	0
$325$	−24.8693	−1.37950
$326$	−1.07880	−0.0597492
$327$	0	0
$328$	22.9679	1.26819
$329$	−2.58301	−0.142406
$330$	0	0
$331$	−2.70560	−0.148713	−0.0743566	−	0.997232i	$-0.523690\pi$
$331$	−2.70560	−0.148713	−0.0743566	+	0.997232i	$0.523690\pi$
$332$	−21.8202	−1.19754
$333$	0	0
$334$	7.58765	0.415178
$335$	−6.10527	−0.333566
$336$	0	0
$337$	−0.736543	−0.0401221	−0.0200610	−	0.999799i	$-0.506386\pi$
$337$	−0.736543	−0.0401221	−0.0200610	+	0.999799i	$0.506386\pi$
$338$	9.22935	0.502010
$339$	0	0
$340$	7.53036	0.408391
$341$	−19.5694	−1.05974
$342$	0	0
$343$	18.9285	1.02204
$344$	−11.1282	−0.599992
$345$	0	0
$346$	−1.79372	−0.0964309
$347$	3.75039	0.201331	0.100666	−	0.994920i	$-0.467903\pi$
$347$	3.75039	0.201331	0.100666	+	0.994920i	$0.467903\pi$
$348$	0	0
$349$	32.3045	1.72922	0.864610	−	0.502443i	$-0.167565\pi$
$349$	32.3045	1.72922	0.864610	+	0.502443i	$0.167565\pi$
$350$	−7.86314	−0.420302
$351$	0	0
$352$	22.7631	1.21328
$353$	−29.6267	−1.57687	−0.788435	−	0.615118i	$-0.789108\pi$
$353$	−29.6267	−1.57687	−0.788435	+	0.615118i	$0.789108\pi$
$354$	0	0
$355$	−2.17921	−0.115660
$356$	−22.0850	−1.17050
$357$	0	0
$358$	−4.82559	−0.255041
$359$	32.9422	1.73862	0.869310	−	0.494267i	$-0.164563\pi$
$359$	32.9422	1.73862	0.869310	+	0.494267i	$0.164563\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.09799	0.0577091
$363$	0	0
$364$	−21.4912	−1.12645
$365$	29.3352	1.53547
$366$	0	0
$367$	−36.2079	−1.89003	−0.945017	−	0.327020i	$-0.893955\pi$
$367$	−36.2079	−1.89003	−0.945017	+	0.327020i	$0.893955\pi$
$368$	12.1457	0.633136
$369$	0	0
$370$	−20.1592	−1.04803
$371$	−19.6779	−1.02162
$372$	0	0
$373$	−14.6177	−0.756874	−0.378437	−	0.925627i	$-0.623538\pi$
$373$	−14.6177	−0.756874	−0.378437	+	0.925627i	$0.623538\pi$
$374$	3.88771	0.201029
$375$	0	0
$376$	−2.30001	−0.118614
$377$	9.19388	0.473509
$378$	0	0
$379$	3.39755	0.174521	0.0872603	−	0.996186i	$-0.472189\pi$
$379$	3.39755	0.174521	0.0872603	+	0.996186i	$0.472189\pi$
$380$	−4.96310	−0.254602
$381$	0	0
$382$	−10.0137	−0.512344
$383$	22.2677	1.13783	0.568913	−	0.822398i	$-0.307364\pi$
$383$	22.2677	1.13783	0.568913	+	0.822398i	$0.307364\pi$
$384$	0	0
$385$	32.2529	1.64376
$386$	−7.60544	−0.387107
$387$	0	0
$388$	1.73411	0.0880359
$389$	−8.97048	−0.454821	−0.227411	−	0.973799i	$-0.573026\pi$
$389$	−8.97048	−0.454821	−0.227411	+	0.973799i	$0.573026\pi$
$390$	0	0
$391$	10.8083	0.546599
$392$	0.754578	0.0381119
$393$	0	0
$394$	−9.99551	−0.503566
$395$	−43.5464	−2.19106
$396$	0	0
$397$	13.4324	0.674153	0.337077	−	0.941477i	$-0.390562\pi$
$397$	13.4324	0.674153	0.337077	+	0.941477i	$0.390562\pi$
$398$	−4.87979	−0.244602
$399$	0	0
$400$	8.10008	0.405004
$401$	24.1611	1.20655	0.603275	−	0.797533i	$-0.293862\pi$
$401$	24.1611	1.20655	0.603275	+	0.797533i	$0.293862\pi$
$402$	0	0
$403$	25.6186	1.27615
$404$	−1.98833	−0.0989229
$405$	0	0
$406$	2.90690	0.144267
$407$	40.2876	1.99698
$408$	0	0
$409$	−32.5916	−1.61155	−0.805774	−	0.592223i	$-0.798251\pi$
$409$	−32.5916	−1.61155	−0.805774	+	0.592223i	$0.798251\pi$
$410$	19.9811	0.986795
$411$	0	0
$412$	22.0648	1.08705
$413$	2.46116	0.121106
$414$	0	0
$415$	−42.8690	−2.10436
$416$	−29.7995	−1.46104
$417$	0	0
$418$	−2.56231	−0.125327
$419$	−19.0237	−0.929366	−0.464683	−	0.885477i	$-0.653832\pi$
$419$	−19.0237	−0.929366	−0.464683	+	0.885477i	$0.653832\pi$
$420$	0	0
$421$	15.4843	0.754659	0.377330	−	0.926079i	$-0.376842\pi$
$421$	15.4843	0.754659	0.377330	+	0.926079i	$0.376842\pi$
$422$	6.05411	0.294709
$423$	0	0
$424$	−17.5220	−0.850941
$425$	7.20817	0.349648
$426$	0	0
$427$	−11.0396	−0.534245
$428$	−0.718422	−0.0347262
$429$	0	0
$430$	−9.68103	−0.466861
$431$	13.0982	0.630919	0.315460	−	0.948939i	$-0.397841\pi$
$431$	13.0982	0.630919	0.315460	+	0.948939i	$0.397841\pi$
$432$	0	0
$433$	−27.6604	−1.32927	−0.664637	−	0.747167i	$-0.731414\pi$
$433$	−27.6604	−1.32927	−0.664637	+	0.747167i	$0.731414\pi$
$434$	8.10004	0.388814
$435$	0	0
$436$	−3.23013	−0.154695
$437$	−7.12352	−0.340764
$438$	0	0
$439$	10.1486	0.484364	0.242182	−	0.970231i	$-0.422137\pi$
$439$	10.1486	0.484364	0.242182	+	0.970231i	$0.422137\pi$
$440$	28.7192	1.36914
$441$	0	0
$442$	−5.08946	−0.242081
$443$	8.24953	0.391947	0.195973	−	0.980609i	$-0.437213\pi$
$443$	8.24953	0.391947	0.195973	+	0.980609i	$0.437213\pi$
$444$	0	0
$445$	−43.3892	−2.05684
$446$	−3.15747	−0.149511
$447$	0	0
$448$	−0.613843	−0.0290013
$449$	29.0578	1.37132	0.685662	−	0.727920i	$-0.259513\pi$
$449$	29.0578	1.37132	0.685662	+	0.727920i	$0.259513\pi$
$450$	0	0
$451$	−39.9315	−1.88030
$452$	15.5741	0.732546
$453$	0	0
$454$	−12.9160	−0.606177
$455$	−42.2227	−1.97943
$456$	0	0
$457$	26.4409	1.23685	0.618427	−	0.785842i	$-0.287770\pi$
$457$	26.4409	1.23685	0.618427	+	0.785842i	$0.287770\pi$
$458$	−10.2879	−0.480721
$459$	0	0
$460$	35.3547	1.64842
$461$	−31.3465	−1.45995	−0.729977	−	0.683472i	$-0.760469\pi$
$461$	−31.3465	−1.45995	−0.729977	+	0.683472i	$0.760469\pi$
$462$	0	0
$463$	6.99639	0.325150	0.162575	−	0.986696i	$-0.448020\pi$
$463$	6.99639	0.325150	0.162575	+	0.986696i	$0.448020\pi$
$464$	−2.99450	−0.139016
$465$	0	0
$466$	−10.6990	−0.495622
$467$	17.6270	0.815682	0.407841	−	0.913053i	$-0.366282\pi$
$467$	17.6270	0.815682	0.407841	+	0.913053i	$0.366282\pi$
$468$	0	0
$469$	−5.05023	−0.233198
$470$	−2.00091	−0.0922949
$471$	0	0
$472$	2.19152	0.100873
$473$	19.3472	0.889586
$474$	0	0
$475$	−4.75075	−0.217980
$476$	6.22906	0.285508
$477$	0	0
$478$	3.44175	0.157422
$479$	41.1625	1.88076	0.940381	−	0.340124i	$-0.110469\pi$
$479$	41.1625	1.88076	0.940381	+	0.340124i	$0.110469\pi$
$480$	0	0
$481$	−52.7411	−2.40479
$482$	−9.47832	−0.431726
$483$	0	0
$484$	−7.93107	−0.360503
$485$	3.40691	0.154700
$486$	0	0
$487$	0.395641	0.0179282	0.00896411	−	0.999960i	$-0.497147\pi$
$487$	0.395641	0.0179282	0.00896411	+	0.999960i	$0.497147\pi$
$488$	−9.83012	−0.444989
$489$	0	0
$490$	0.656449	0.0296553
$491$	5.99373	0.270493	0.135247	−	0.990812i	$-0.456817\pi$
$491$	5.99373	0.270493	0.135247	+	0.990812i	$0.456817\pi$
$492$	0	0
$493$	−2.66477	−0.120015
$494$	3.35436	0.150920
$495$	0	0
$496$	−8.34412	−0.374662
$497$	−1.80262	−0.0808588
$498$	0	0
$499$	−1.10589	−0.0495064	−0.0247532	−	0.999694i	$-0.507880\pi$
$499$	−1.10589	−0.0495064	−0.0247532	+	0.999694i	$0.507880\pi$
$500$	−1.23703	−0.0553217
$501$	0	0
$502$	−1.03721	−0.0462928
$503$	−28.0346	−1.25000	−0.625001	−	0.780624i	$-0.714902\pi$
$503$	−28.0346	−1.25000	−0.625001	+	0.780624i	$0.714902\pi$
$504$	0	0
$505$	−3.90636	−0.173831
$506$	18.2526	0.811429
$507$	0	0
$508$	−31.9985	−1.41970
$509$	−19.9490	−0.884223	−0.442111	−	0.896960i	$-0.645770\pi$
$509$	−19.9490	−0.884223	−0.442111	+	0.896960i	$0.645770\pi$
$510$	0	0
$511$	24.2658	1.07346
$512$	17.5489	0.775561
$513$	0	0
$514$	−10.8825	−0.480007
$515$	43.3495	1.91021
$516$	0	0
$517$	3.99875	0.175865
$518$	−16.6756	−0.732682
$519$	0	0
$520$	−37.5968	−1.64873
$521$	−2.70656	−0.118577	−0.0592883	−	0.998241i	$-0.518883\pi$
$521$	−2.70656	−0.118577	−0.0592883	+	0.998241i	$0.518883\pi$
$522$	0	0
$523$	−30.9304	−1.35249	−0.676247	−	0.736675i	$-0.736395\pi$
$523$	−30.9304	−1.35249	−0.676247	+	0.736675i	$0.736395\pi$
$524$	−11.6703	−0.509818
$525$	0	0
$526$	13.5660	0.591504
$527$	−7.42534	−0.323453
$528$	0	0
$529$	27.7445	1.20628
$530$	−15.2433	−0.662127
$531$	0	0
$532$	−4.10544	−0.177993
$533$	52.2750	2.26428
$534$	0	0
$535$	−1.41145	−0.0610222
$536$	−4.49692	−0.194237
$537$	0	0
$538$	13.8711	0.598025
$539$	−1.31189	−0.0565072
$540$	0	0
$541$	−11.5017	−0.494497	−0.247248	−	0.968952i	$-0.579526\pi$
$541$	−11.5017	−0.494497	−0.247248	+	0.968952i	$0.579526\pi$
$542$	14.5185	0.623621
$543$	0	0
$544$	8.63714	0.370314
$545$	−6.34607	−0.271836
$546$	0	0
$547$	−32.9353	−1.40821	−0.704106	−	0.710095i	$-0.748652\pi$
$547$	−32.9353	−1.40821	−0.704106	+	0.710095i	$0.748652\pi$
$548$	16.0716	0.686544
$549$	0	0
$550$	12.1729	0.519054
$551$	1.75629	0.0748207
$552$	0	0
$553$	−36.0212	−1.53178
$554$	7.04564	0.299340
$555$	0	0
$556$	24.6986	1.04746
$557$	−19.2739	−0.816661	−0.408330	−	0.912834i	$-0.633889\pi$
$557$	−19.2739	−0.816661	−0.408330	+	0.912834i	$0.633889\pi$
$558$	0	0
$559$	−25.3277	−1.07125
$560$	13.7522	0.581136
$561$	0	0
$562$	−16.9586	−0.715355
$563$	14.8806	0.627142	0.313571	−	0.949565i	$-0.398475\pi$
$563$	14.8806	0.627142	0.313571	+	0.949565i	$0.398475\pi$
$564$	0	0
$565$	30.5977	1.28726
$566$	7.82126	0.328752
$567$	0	0
$568$	−1.60513	−0.0673497
$569$	−26.1519	−1.09635	−0.548173	−	0.836365i	$-0.684676\pi$
$569$	−26.1519	−1.09635	−0.548173	+	0.836365i	$0.684676\pi$
$570$	0	0
$571$	−6.05075	−0.253216	−0.126608	−	0.991953i	$-0.540409\pi$
$571$	−6.05075	−0.253216	−0.126608	+	0.991953i	$0.540409\pi$
$572$	33.2705	1.39111
$573$	0	0
$574$	16.5282	0.689873
$575$	33.8421	1.41131
$576$	0	0
$577$	−4.88514	−0.203371	−0.101686	−	0.994817i	$-0.532424\pi$
$577$	−4.88514	−0.203371	−0.101686	+	0.994817i	$0.532424\pi$
$578$	−9.41809	−0.391741
$579$	0	0
$580$	−8.71667	−0.361940
$581$	−35.4609	−1.47117
$582$	0	0
$583$	30.4633	1.26166
$584$	21.6072	0.894114
$585$	0	0
$586$	9.48638	0.391879
$587$	27.0377	1.11596	0.557982	−	0.829853i	$-0.311576\pi$
$587$	27.0377	1.11596	0.557982	+	0.829853i	$0.311576\pi$
$588$	0	0
$589$	4.89389	0.201649
$590$	1.90652	0.0784902
$591$	0	0
$592$	17.1781	0.706014
$593$	17.6708	0.725653	0.362827	−	0.931857i	$-0.381812\pi$
$593$	17.6708	0.725653	0.362827	+	0.931857i	$0.381812\pi$
$594$	0	0
$595$	12.2379	0.501705
$596$	22.2746	0.912402
$597$	0	0
$598$	−23.8948	−0.977132
$599$	−5.51025	−0.225143	−0.112571	−	0.993644i	$-0.535909\pi$
$599$	−5.51025	−0.225143	−0.112571	+	0.993644i	$0.535909\pi$
$600$	0	0
$601$	−21.6954	−0.884975	−0.442487	−	0.896775i	$-0.645904\pi$
$601$	−21.6954	−0.884975	−0.442487	+	0.896775i	$0.645904\pi$
$602$	−8.00807	−0.326385
$603$	0	0
$604$	23.3335	0.949426
$605$	−15.5818	−0.633489
$606$	0	0
$607$	9.48233	0.384876	0.192438	−	0.981309i	$-0.438361\pi$
$607$	9.48233	0.384876	0.192438	+	0.981309i	$0.438361\pi$
$608$	−5.69256	−0.230864
$609$	0	0
$610$	−8.55176	−0.346251
$611$	−5.23482	−0.211778
$612$	0	0
$613$	−24.6306	−0.994820	−0.497410	−	0.867515i	$-0.665716\pi$
$613$	−24.6306	−0.994820	−0.497410	+	0.867515i	$0.665716\pi$
$614$	−6.06689	−0.244840
$615$	0	0
$616$	23.7563	0.957170
$617$	47.1167	1.89685	0.948423	−	0.317008i	$-0.102678\pi$
$617$	47.1167	1.89685	0.948423	+	0.317008i	$0.102678\pi$
$618$	0	0
$619$	−4.41007	−0.177255	−0.0886277	−	0.996065i	$-0.528248\pi$
$619$	−4.41007	−0.177255	−0.0886277	+	0.996065i	$0.528248\pi$
$620$	−24.2889	−0.975464
$621$	0	0
$622$	4.63410	0.185810
$623$	−35.8912	−1.43795
$624$	0	0
$625$	−26.1841	−1.04736
$626$	−15.8516	−0.633559
$627$	0	0
$628$	14.1636	0.565191
$629$	15.2866	0.609515
$630$	0	0
$631$	11.8948	0.473525	0.236762	−	0.971568i	$-0.423914\pi$
$631$	11.8948	0.473525	0.236762	+	0.971568i	$0.423914\pi$
$632$	−32.0747	−1.27586
$633$	0	0
$634$	20.2987	0.806165
$635$	−62.8657	−2.49475
$636$	0	0
$637$	1.71742	0.0680466
$638$	−4.50017	−0.178163
$639$	0	0
$640$	35.0759	1.38650
$641$	−4.38818	−0.173323	−0.0866614	−	0.996238i	$-0.527620\pi$
$641$	−4.38818	−0.173323	−0.0866614	+	0.996238i	$0.527620\pi$
$642$	0	0
$643$	−39.1388	−1.54348	−0.771742	−	0.635936i	$-0.780614\pi$
$643$	−39.1388	−1.54348	−0.771742	+	0.635936i	$0.780614\pi$
$644$	29.2452	1.15242
$645$	0	0
$646$	−0.972233	−0.0382520
$647$	35.9942	1.41508	0.707538	−	0.706675i	$-0.249806\pi$
$647$	35.9942	1.41508	0.707538	+	0.706675i	$0.249806\pi$
$648$	0	0
$649$	−3.81012	−0.149560
$650$	−15.9357	−0.625051
$651$	0	0
$652$	2.67589	0.104796
$653$	−18.5379	−0.725445	−0.362723	−	0.931897i	$-0.618153\pi$
$653$	−18.5379	−0.725445	−0.362723	+	0.931897i	$0.618153\pi$
$654$	0	0
$655$	−22.9280	−0.895870
$656$	−17.0262	−0.664763
$657$	0	0
$658$	−1.65513	−0.0645239
$659$	−13.2333	−0.515498	−0.257749	−	0.966212i	$-0.582981\pi$
$659$	−13.2333	−0.515498	−0.257749	+	0.966212i	$0.582981\pi$
$660$	0	0
$661$	−33.2017	−1.29140	−0.645698	−	0.763593i	$-0.723433\pi$
$661$	−33.2017	−1.29140	−0.645698	+	0.763593i	$0.723433\pi$
$662$	−1.73369	−0.0673818
$663$	0	0
$664$	−31.5758	−1.22538
$665$	−8.06575	−0.312776
$666$	0	0
$667$	−12.5110	−0.484427
$668$	−18.8206	−0.728191
$669$	0	0
$670$	−3.91212	−0.151138
$671$	17.0904	0.659768
$672$	0	0
$673$	24.5884	0.947812	0.473906	−	0.880576i	$-0.342844\pi$
$673$	24.5884	0.947812	0.473906	+	0.880576i	$0.342844\pi$
$674$	−0.471961	−0.0181793
$675$	0	0
$676$	−22.8927	−0.880489
$677$	−35.7456	−1.37382	−0.686908	−	0.726745i	$-0.741032\pi$
$677$	−35.7456	−1.37382	−0.686908	+	0.726745i	$0.741032\pi$
$678$	0	0
$679$	2.81817	0.108151
$680$	10.8971	0.417885
$681$	0	0
$682$	−12.5397	−0.480168
$683$	−20.5007	−0.784436	−0.392218	−	0.919872i	$-0.628292\pi$
$683$	−20.5007	−0.784436	−0.392218	+	0.919872i	$0.628292\pi$
$684$	0	0
$685$	31.5750	1.20642
$686$	12.1290	0.463085
$687$	0	0
$688$	8.24939	0.314505
$689$	−39.8799	−1.51930
$690$	0	0
$691$	46.1634	1.75614	0.878070	−	0.478532i	$-0.158831\pi$
$691$	46.1634	1.75614	0.878070	+	0.478532i	$0.158831\pi$
$692$	4.44919	0.169133
$693$	0	0
$694$	2.40317	0.0912229
$695$	48.5242	1.84063
$696$	0	0
$697$	−15.1515	−0.573903
$698$	20.7000	0.783507
$699$	0	0
$700$	19.5039	0.737179
$701$	−39.4973	−1.49179	−0.745896	−	0.666062i	$-0.767978\pi$
$701$	−39.4973	−1.49179	−0.745896	+	0.666062i	$0.767978\pi$
$702$	0	0
$703$	−10.0750	−0.379988
$704$	0.950288	0.0358153
$705$	0	0
$706$	−18.9841	−0.714478
$707$	−3.23131	−0.121526
$708$	0	0
$709$	46.8790	1.76058	0.880290	−	0.474436i	$-0.157348\pi$
$709$	46.8790	1.76058	0.880290	+	0.474436i	$0.157348\pi$
$710$	−1.39639	−0.0524056
$711$	0	0
$712$	−31.9589	−1.19771
$713$	−34.8617	−1.30558
$714$	0	0
$715$	65.3649	2.44451
$716$	11.9695	0.447322
$717$	0	0
$718$	21.1086	0.787767
$719$	−23.1264	−0.862468	−0.431234	−	0.902240i	$-0.641922\pi$
$719$	−23.1264	−0.862468	−0.431234	+	0.902240i	$0.641922\pi$
$720$	0	0
$721$	35.8584	1.33544
$722$	0.640778	0.0238473
$723$	0	0
$724$	−2.72348	−0.101217
$725$	−8.34372	−0.309878
$726$	0	0
$727$	53.0551	1.96771	0.983853	−	0.178979i	$-0.0572795\pi$
$727$	53.0551	1.96771	0.983853	+	0.178979i	$0.0572795\pi$
$728$	−31.0998	−1.15263
$729$	0	0
$730$	18.7973	0.695720
$731$	7.34104	0.271518
$732$	0	0
$733$	45.9764	1.69818	0.849089	−	0.528250i	$-0.177152\pi$
$733$	45.9764	1.69818	0.849089	+	0.528250i	$0.177152\pi$
$734$	−23.2012	−0.856372
$735$	0	0
$736$	40.5510	1.49473
$737$	7.81824	0.287989
$738$	0	0
$739$	33.8420	1.24490	0.622449	−	0.782660i	$-0.286138\pi$
$739$	33.8420	1.24490	0.622449	+	0.782660i	$0.286138\pi$
$740$	50.0035	1.83817
$741$	0	0
$742$	−12.6092	−0.462897
$743$	−1.82677	−0.0670179	−0.0335089	−	0.999438i	$-0.510668\pi$
$743$	−1.82677	−0.0670179	−0.0335089	+	0.999438i	$0.510668\pi$
$744$	0	0
$745$	43.7617	1.60331
$746$	−9.36668	−0.342939
$747$	0	0
$748$	−9.64319	−0.352590
$749$	−1.16754	−0.0426609
$750$	0	0
$751$	−29.8272	−1.08841	−0.544204	−	0.838953i	$-0.683168\pi$
$751$	−29.8272	−1.08841	−0.544204	+	0.838953i	$0.683168\pi$
$752$	1.70501	0.0621753
$753$	0	0
$754$	5.89124	0.214546
$755$	45.8421	1.66837
$756$	0	0
$757$	−1.96329	−0.0713570	−0.0356785	−	0.999363i	$-0.511359\pi$
$757$	−1.96329	−0.0713570	−0.0356785	+	0.999363i	$0.511359\pi$
$758$	2.17708	0.0790750
$759$	0	0
$760$	−7.18206	−0.260521
$761$	41.6760	1.51075	0.755377	−	0.655291i	$-0.227454\pi$
$761$	41.6760	1.51075	0.755377	+	0.655291i	$0.227454\pi$
$762$	0	0
$763$	−5.24942	−0.190042
$764$	24.8382	0.898614
$765$	0	0
$766$	14.2687	0.515547
$767$	4.98789	0.180102
$768$	0	0
$769$	14.2985	0.515618	0.257809	−	0.966196i	$-0.416999\pi$
$769$	14.2985	0.515618	0.257809	+	0.966196i	$0.416999\pi$
$770$	20.6669	0.744785
$771$	0	0
$772$	18.8647	0.678957
$773$	−31.6586	−1.13868	−0.569340	−	0.822102i	$-0.692801\pi$
$773$	−31.6586	−1.13868	−0.569340	+	0.822102i	$0.692801\pi$
$774$	0	0
$775$	−23.2497	−0.835152
$776$	2.50941	0.0900826
$777$	0	0
$778$	−5.74809	−0.206079
$779$	9.98601	0.357786
$780$	0	0
$781$	2.79064	0.0998569
$782$	6.92572	0.247663
$783$	0	0
$784$	−0.559372	−0.0199776
$785$	27.8266	0.993173
$786$	0	0
$787$	−13.3586	−0.476183	−0.238092	−	0.971243i	$-0.576522\pi$
$787$	−13.3586	−0.476183	−0.238092	+	0.971243i	$0.576522\pi$
$788$	24.7931	0.883219
$789$	0	0
$790$	−27.9036	−0.992764
$791$	25.3102	0.899927
$792$	0	0
$793$	−22.3733	−0.794500
$794$	8.60720	0.305458
$795$	0	0
$796$	12.1040	0.429013
$797$	30.8006	1.09101	0.545507	−	0.838106i	$-0.316337\pi$
$797$	30.8006	1.09101	0.545507	+	0.838106i	$0.316337\pi$
$798$	0	0
$799$	1.51727	0.0536771
$800$	27.0439	0.956147
$801$	0	0
$802$	15.4819	0.546686
$803$	−37.5658	−1.32567
$804$	0	0
$805$	57.4565	2.02508
$806$	16.4159	0.578224
$807$	0	0
$808$	−2.87729	−0.101223
$809$	−20.5457	−0.722347	−0.361173	−	0.932499i	$-0.617624\pi$
$809$	−20.5457	−0.722347	−0.361173	+	0.932499i	$0.617624\pi$
$810$	0	0
$811$	43.3073	1.52072	0.760362	−	0.649499i	$-0.225021\pi$
$811$	43.3073	1.52072	0.760362	+	0.649499i	$0.225021\pi$
$812$	−7.21036	−0.253034
$813$	0	0
$814$	25.8154	0.904829
$815$	5.25718	0.184151
$816$	0	0
$817$	−4.83832	−0.169271
$818$	−20.8840	−0.730191
$819$	0	0
$820$	−49.5616	−1.73077
$821$	−6.72169	−0.234589	−0.117294	−	0.993097i	$-0.537422\pi$
$821$	−6.72169	−0.234589	−0.117294	+	0.993097i	$0.537422\pi$
$822$	0	0
$823$	7.90050	0.275394	0.137697	−	0.990474i	$-0.456030\pi$
$823$	7.90050	0.275394	0.137697	+	0.990474i	$0.456030\pi$
$824$	31.9297	1.11232
$825$	0	0
$826$	1.57706	0.0548729
$827$	42.6083	1.48164	0.740818	−	0.671706i	$-0.234438\pi$
$827$	42.6083	1.48164	0.740818	+	0.671706i	$0.234438\pi$
$828$	0	0
$829$	−25.3147	−0.879216	−0.439608	−	0.898190i	$-0.644883\pi$
$829$	−25.3147	−0.879216	−0.439608	+	0.898190i	$0.644883\pi$
$830$	−27.4695	−0.953481
$831$	0	0
$832$	−1.24404	−0.0431292
$833$	−0.497779	−0.0172470
$834$	0	0
$835$	−36.9759	−1.27960
$836$	6.35562	0.219814
$837$	0	0
$838$	−12.1899	−0.421095
$839$	48.6795	1.68060	0.840301	−	0.542120i	$-0.182378\pi$
$839$	48.6795	1.68060	0.840301	+	0.542120i	$0.182378\pi$
$840$	0	0
$841$	−25.9154	−0.893636
$842$	9.92201	0.341935
$843$	0	0
$844$	−15.0168	−0.516899
$845$	−44.9762	−1.54723
$846$	0	0
$847$	−12.8891	−0.442875
$848$	12.9891	0.446048
$849$	0	0
$850$	4.61884	0.158425
$851$	71.7698	2.46024
$852$	0	0
$853$	−42.8721	−1.46791	−0.733957	−	0.679196i	$-0.762329\pi$
$853$	−42.8721	−1.46791	−0.733957	+	0.679196i	$0.762329\pi$
$854$	−7.07395	−0.242066
$855$	0	0
$856$	−1.03962	−0.0355335
$857$	21.4211	0.731730	0.365865	−	0.930668i	$-0.380773\pi$
$857$	21.4211	0.731730	0.365865	+	0.930668i	$0.380773\pi$
$858$	0	0
$859$	37.3852	1.27557	0.637784	−	0.770215i	$-0.279851\pi$
$859$	37.3852	1.27557	0.637784	+	0.770215i	$0.279851\pi$
$860$	24.0131	0.818840
$861$	0	0
$862$	8.39306	0.285869
$863$	−38.5262	−1.31145	−0.655724	−	0.755001i	$-0.727636\pi$
$863$	−38.5262	−1.31145	−0.655724	+	0.755001i	$0.727636\pi$
$864$	0	0
$865$	8.74109	0.297206
$866$	−17.7242	−0.602292
$867$	0	0
$868$	−20.0916	−0.681952
$869$	55.7644	1.89168
$870$	0	0
$871$	−10.2350	−0.346799
$872$	−4.67429	−0.158291
$873$	0	0
$874$	−4.56459	−0.154400
$875$	−2.01035	−0.0679622
$876$	0	0
$877$	22.4836	0.759217	0.379609	−	0.925147i	$-0.376059\pi$
$877$	22.4836	0.759217	0.379609	+	0.925147i	$0.376059\pi$
$878$	6.50297	0.219465
$879$	0	0
$880$	−21.2897	−0.717676
$881$	−38.6326	−1.30156	−0.650782	−	0.759265i	$-0.725559\pi$
$881$	−38.6326	−1.30156	−0.650782	+	0.759265i	$0.725559\pi$
$882$	0	0
$883$	−54.9733	−1.85000	−0.924999	−	0.379970i	$-0.875934\pi$
$883$	−54.9733	−1.85000	−0.924999	+	0.379970i	$0.875934\pi$
$884$	12.6240	0.424593
$885$	0	0
$886$	5.28612	0.177591
$887$	−13.3181	−0.447179	−0.223589	−	0.974683i	$-0.571777\pi$
$887$	−13.3181	−0.447179	−0.223589	+	0.974683i	$0.571777\pi$
$888$	0	0
$889$	−52.0021	−1.74409
$890$	−27.8028	−0.931953
$891$	0	0
$892$	7.83188	0.262231
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	23.5159	0.786051
$896$	29.0145	0.969306
$897$	0	0
$898$	18.6196	0.621345
$899$	8.59511	0.286663
$900$	0	0
$901$	11.5589	0.385082
$902$	−25.5872	−0.851962
$903$	0	0
$904$	22.5372	0.749576
$905$	−5.35069	−0.177863
$906$	0	0
$907$	−51.1905	−1.69975	−0.849876	−	0.526982i	$-0.823323\pi$
$907$	−51.1905	−1.69975	−0.849876	+	0.526982i	$0.823323\pi$
$908$	32.0371	1.06319
$909$	0	0
$910$	−27.0554	−0.896878
$911$	−27.1070	−0.898095	−0.449048	−	0.893508i	$-0.648237\pi$
$911$	−27.1070	−0.898095	−0.449048	+	0.893508i	$0.648237\pi$
$912$	0	0
$913$	54.8969	1.81682
$914$	16.9428	0.560417
$915$	0	0
$916$	25.5183	0.843149
$917$	−18.9658	−0.626307
$918$	0	0
$919$	39.9949	1.31931	0.659656	−	0.751568i	$-0.270702\pi$
$919$	39.9949	1.31931	0.659656	+	0.751568i	$0.270702\pi$
$920$	51.1615	1.68675
$921$	0	0
$922$	−20.0862	−0.661503
$923$	−3.65327	−0.120249
$924$	0	0
$925$	47.8641	1.57376
$926$	4.48313	0.147325
$927$	0	0
$928$	−9.99780	−0.328194
$929$	−7.39735	−0.242699	−0.121350	−	0.992610i	$-0.538722\pi$
$929$	−7.39735	−0.242699	−0.121350	+	0.992610i	$0.538722\pi$
$930$	0	0
$931$	0.328076	0.0107522
$932$	26.5381	0.869285
$933$	0	0
$934$	11.2950	0.369584
$935$	−18.9455	−0.619583
$936$	0	0
$937$	−31.2000	−1.01926	−0.509631	−	0.860393i	$-0.670218\pi$
$937$	−31.2000	−1.01926	−0.509631	+	0.860393i	$0.670218\pi$
$938$	−3.23608	−0.105662
$939$	0	0
$940$	4.96310	0.161879
$941$	−42.4403	−1.38351	−0.691757	−	0.722131i	$-0.743163\pi$
$941$	−42.4403	−1.38351	−0.691757	+	0.722131i	$0.743163\pi$
$942$	0	0
$943$	−71.1355	−2.31649
$944$	−1.62458	−0.0528756
$945$	0	0
$946$	12.3973	0.403070
$947$	5.17028	0.168011	0.0840057	−	0.996465i	$-0.473229\pi$
$947$	5.17028	0.168011	0.0840057	+	0.996465i	$0.473229\pi$
$948$	0	0
$949$	49.1780	1.59639
$950$	−3.04418	−0.0987662
$951$	0	0
$952$	9.01401	0.292146
$953$	33.7211	1.09233	0.546167	−	0.837676i	$-0.316086\pi$
$953$	33.7211	1.09233	0.546167	+	0.837676i	$0.316086\pi$
$954$	0	0
$955$	48.7983	1.57908
$956$	−8.53702	−0.276107
$957$	0	0
$958$	26.3760	0.852170
$959$	26.1186	0.843414
$960$	0	0
$961$	−7.04987	−0.227415
$962$	−33.7953	−1.08961
$963$	0	0
$964$	23.5103	0.757215
$965$	37.0626	1.19309
$966$	0	0
$967$	−28.7255	−0.923748	−0.461874	−	0.886945i	$-0.652823\pi$
$967$	−28.7255	−0.923748	−0.461874	+	0.886945i	$0.652823\pi$
$968$	−11.4770	−0.368884
$969$	0	0
$970$	2.18307	0.0700943
$971$	15.3229	0.491735	0.245868	−	0.969303i	$-0.420927\pi$
$971$	15.3229	0.491735	0.245868	+	0.969303i	$0.420927\pi$
$972$	0	0
$973$	40.1388	1.28679
$974$	0.253518	0.00812325
$975$	0	0
$976$	7.28712	0.233255
$977$	−6.70497	−0.214511	−0.107256	−	0.994231i	$-0.534206\pi$
$977$	−6.70497	−0.214511	−0.107256	+	0.994231i	$0.534206\pi$
$978$	0	0
$979$	55.5631	1.77580
$980$	−1.62827	−0.0520133
$981$	0	0
$982$	3.84065	0.122560
$983$	20.9575	0.668440	0.334220	−	0.942495i	$-0.391527\pi$
$983$	20.9575	0.668440	0.334220	+	0.942495i	$0.391527\pi$
$984$	0	0
$985$	48.7098	1.55202
$986$	−1.70753	−0.0543787
$987$	0	0
$988$	−8.32024	−0.264702
$989$	34.4659	1.09595
$990$	0	0
$991$	28.9003	0.918047	0.459024	−	0.888424i	$-0.348199\pi$
$991$	28.9003	0.918047	0.459024	+	0.888424i	$0.348199\pi$
$992$	−27.8587	−0.884515
$993$	0	0
$994$	−1.15508	−0.0366370
$995$	23.7800	0.753877
$996$	0	0
$997$	54.3212	1.72037	0.860185	−	0.509983i	$-0.170348\pi$
$997$	54.3212	1.72037	0.860185	+	0.509983i	$0.170348\pi$
$998$	−0.708630	−0.0224313
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.n.1.10		16
3.2	odd	2		893.2.a.b.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.b.1.7	✓	16	3.2	odd	2
8037.2.a.n.1.10		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.640778 q^{2} -1.58940 q^{4} -3.12262 q^{5} -2.58301 q^{7} -2.30001 q^{8} +O(q^{10})\)	\(q+0.640778 q^{2} -1.58940 q^{4} -3.12262 q^{5} -2.58301 q^{7} -2.30001 q^{8} -2.00091 q^{10} +3.99875 q^{11} -5.23482 q^{13} -1.65513 q^{14} +1.70501 q^{16} +1.51727 q^{17} -1.00000 q^{19} +4.96310 q^{20} +2.56231 q^{22} +7.12352 q^{23} +4.75075 q^{25} -3.35436 q^{26} +4.10544 q^{28} -1.75629 q^{29} -4.89389 q^{31} +5.69256 q^{32} +0.972233 q^{34} +8.06575 q^{35} +10.0750 q^{37} -0.640778 q^{38} +7.18206 q^{40} -9.98601 q^{41} +4.83832 q^{43} -6.35562 q^{44} +4.56459 q^{46} +1.00000 q^{47} -0.328076 q^{49} +3.04418 q^{50} +8.32024 q^{52} +7.61820 q^{53} -12.4866 q^{55} +5.94095 q^{56} -1.12540 q^{58} -0.952829 q^{59} +4.27394 q^{61} -3.13590 q^{62} +0.237647 q^{64} +16.3464 q^{65} +1.95517 q^{67} -2.41155 q^{68} +5.16836 q^{70} +0.697878 q^{71} -9.39441 q^{73} +6.45587 q^{74} +1.58940 q^{76} -10.3288 q^{77} +13.9455 q^{79} -5.32410 q^{80} -6.39882 q^{82} +13.7285 q^{83} -4.73785 q^{85} +3.10029 q^{86} -9.19716 q^{88} +13.8951 q^{89} +13.5216 q^{91} -11.3221 q^{92} +0.640778 q^{94} +3.12262 q^{95} -1.09104 q^{97} -0.210224 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

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Database

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