LMFDB - Embedded newform 7120.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7120,2,Mod(1,7120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7120 = 2^{4} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7120.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:	$56.8534862392$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.49803$ of defining polynomial
Character	$\chi$	$=$	7120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82660 q^{3} +1.00000 q^{5} -0.0498231 q^{7} +4.98967 q^{9} +O(q^{10})$	$q+2.82660 q^{3} +1.00000 q^{5} -0.0498231 q^{7} +4.98967 q^{9} -4.45116 q^{11} -2.43229 q^{13} +2.82660 q^{15} -2.48065 q^{17} +5.16842 q^{19} -0.140830 q^{21} +4.99050 q^{23} +1.00000 q^{25} +5.62400 q^{27} +2.59152 q^{29} +7.31064 q^{31} -12.5817 q^{33} -0.0498231 q^{35} +5.13825 q^{37} -6.87512 q^{39} +9.11073 q^{41} +0.543007 q^{43} +4.98967 q^{45} +9.63395 q^{47} -6.99752 q^{49} -7.01180 q^{51} -10.7093 q^{53} -4.45116 q^{55} +14.6091 q^{57} +12.6337 q^{59} -2.47399 q^{61} -0.248601 q^{63} -2.43229 q^{65} +5.36128 q^{67} +14.1061 q^{69} -9.58146 q^{71} +2.86369 q^{73} +2.82660 q^{75} +0.221771 q^{77} -6.62473 q^{79} +0.927782 q^{81} +12.0061 q^{83} -2.48065 q^{85} +7.32520 q^{87} +1.00000 q^{89} +0.121184 q^{91} +20.6643 q^{93} +5.16842 q^{95} +4.82051 q^{97} -22.2098 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * 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
$3$	2.82660	1.63194	0.815969	−	0.578095i	$-0.196204\pi$
$3$	2.82660	1.63194	0.815969	+	0.578095i	$0.196204\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0498231	−0.0188314	−0.00941568	−	0.999956i	$-0.502997\pi$
$7$	−0.0498231	−0.0188314	−0.00941568	+	0.999956i	$0.502997\pi$
$8$	0	0
$9$	4.98967	1.66322
$10$	0	0
$11$	−4.45116	−1.34208	−0.671038	−	0.741423i	$-0.734151\pi$
$11$	−4.45116	−1.34208	−0.671038	+	0.741423i	$0.734151\pi$
$12$	0	0
$13$	−2.43229	−0.674596	−0.337298	−	0.941398i	$-0.609513\pi$
$13$	−2.43229	−0.674596	−0.337298	+	0.941398i	$0.609513\pi$
$14$	0	0
$15$	2.82660	0.729825
$16$	0	0
$17$	−2.48065	−0.601646	−0.300823	−	0.953680i	$-0.597261\pi$
$17$	−2.48065	−0.601646	−0.300823	+	0.953680i	$0.597261\pi$
$18$	0	0
$19$	5.16842	1.18572	0.592858	−	0.805307i	$-0.297999\pi$
$19$	5.16842	1.18572	0.592858	+	0.805307i	$0.297999\pi$
$20$	0	0
$21$	−0.140830	−0.0307316
$22$	0	0
$23$	4.99050	1.04059	0.520295	−	0.853986i	$-0.325822\pi$
$23$	4.99050	1.04059	0.520295	+	0.853986i	$0.325822\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.62400	1.08234
$28$	0	0
$29$	2.59152	0.481234	0.240617	−	0.970620i	$-0.422650\pi$
$29$	2.59152	0.481234	0.240617	+	0.970620i	$0.422650\pi$
$30$	0	0
$31$	7.31064	1.31303	0.656515	−	0.754313i	$-0.272030\pi$
$31$	7.31064	1.31303	0.656515	+	0.754313i	$0.272030\pi$
$32$	0	0
$33$	−12.5817	−2.19019
$34$	0	0
$35$	−0.0498231	−0.00842164
$36$	0	0
$37$	5.13825	0.844724	0.422362	−	0.906427i	$-0.361201\pi$
$37$	5.13825	0.844724	0.422362	+	0.906427i	$0.361201\pi$
$38$	0	0
$39$	−6.87512	−1.10090
$40$	0	0
$41$	9.11073	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$41$	9.11073	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$42$	0	0
$43$	0.543007	0.0828078	0.0414039	−	0.999142i	$-0.486817\pi$
$43$	0.543007	0.0828078	0.0414039	+	0.999142i	$0.486817\pi$
$44$	0	0
$45$	4.98967	0.743816
$46$	0	0
$47$	9.63395	1.40526	0.702628	−	0.711557i	$-0.252010\pi$
$47$	9.63395	1.40526	0.702628	+	0.711557i	$0.252010\pi$
$48$	0	0
$49$	−6.99752	−0.999645
$50$	0	0
$51$	−7.01180	−0.981849
$52$	0	0
$53$	−10.7093	−1.47104	−0.735519	−	0.677504i	$-0.763062\pi$
$53$	−10.7093	−1.47104	−0.735519	+	0.677504i	$0.763062\pi$
$54$	0	0
$55$	−4.45116	−0.600195
$56$	0	0
$57$	14.6091	1.93502
$58$	0	0
$59$	12.6337	1.64477	0.822383	−	0.568934i	$-0.192644\pi$
$59$	12.6337	1.64477	0.822383	+	0.568934i	$0.192644\pi$
$60$	0	0
$61$	−2.47399	−0.316762	−0.158381	−	0.987378i	$-0.550627\pi$
$61$	−2.47399	−0.316762	−0.158381	+	0.987378i	$0.550627\pi$
$62$	0	0
$63$	−0.248601	−0.0313207
$64$	0	0
$65$	−2.43229	−0.301689
$66$	0	0
$67$	5.36128	0.654984	0.327492	−	0.944854i	$-0.393797\pi$
$67$	5.36128	0.654984	0.327492	+	0.944854i	$0.393797\pi$
$68$	0	0
$69$	14.1061	1.69818
$70$	0	0
$71$	−9.58146	−1.13711	−0.568555	−	0.822645i	$-0.692497\pi$
$71$	−9.58146	−1.13711	−0.568555	+	0.822645i	$0.692497\pi$
$72$	0	0
$73$	2.86369	0.335169	0.167585	−	0.985858i	$-0.446403\pi$
$73$	2.86369	0.335169	0.167585	+	0.985858i	$0.446403\pi$
$74$	0	0
$75$	2.82660	0.326388
$76$	0	0
$77$	0.221771	0.0252731
$78$	0	0
$79$	−6.62473	−0.745340	−0.372670	−	0.927964i	$-0.621558\pi$
$79$	−6.62473	−0.745340	−0.372670	+	0.927964i	$0.621558\pi$
$80$	0	0
$81$	0.927782	0.103087
$82$	0	0
$83$	12.0061	1.31784	0.658919	−	0.752214i	$-0.271014\pi$
$83$	12.0061	1.31784	0.658919	+	0.752214i	$0.271014\pi$
$84$	0	0
$85$	−2.48065	−0.269064
$86$	0	0
$87$	7.32520	0.785344
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	0.121184	0.0127036
$92$	0	0
$93$	20.6643	2.14278
$94$	0	0
$95$	5.16842	0.530269
$96$	0	0
$97$	4.82051	0.489449	0.244724	−	0.969593i	$-0.421303\pi$
$97$	4.82051	0.489449	0.244724	+	0.969593i	$0.421303\pi$
$98$	0	0
$99$	−22.2098	−2.23217
$100$	0	0
$101$	−11.5561	−1.14988	−0.574938	−	0.818197i	$-0.694974\pi$
$101$	−11.5561	−1.14988	−0.574938	+	0.818197i	$0.694974\pi$
$102$	0	0
$103$	2.92606	0.288314	0.144157	−	0.989555i	$-0.453953\pi$
$103$	2.92606	0.288314	0.144157	+	0.989555i	$0.453953\pi$
$104$	0	0
$105$	−0.140830	−0.0137436
$106$	0	0
$107$	3.97982	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$107$	3.97982	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$108$	0	0
$109$	5.30200	0.507840	0.253920	−	0.967225i	$-0.418280\pi$
$109$	5.30200	0.507840	0.253920	+	0.967225i	$0.418280\pi$
$110$	0	0
$111$	14.5238	1.37854
$112$	0	0
$113$	−10.3888	−0.977296	−0.488648	−	0.872481i	$-0.662510\pi$
$113$	−10.3888	−0.977296	−0.488648	+	0.872481i	$0.662510\pi$
$114$	0	0
$115$	4.99050	0.465366
$116$	0	0
$117$	−12.1363	−1.12200
$118$	0	0
$119$	0.123594	0.0113298
$120$	0	0
$121$	8.81287	0.801170
$122$	0	0
$123$	25.7524	2.32201
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.31244	−0.471403	−0.235701	−	0.971826i	$-0.575739\pi$
$127$	−5.31244	−0.471403	−0.235701	+	0.971826i	$0.575739\pi$
$128$	0	0
$129$	1.53486	0.135137
$130$	0	0
$131$	22.2268	1.94197	0.970984	−	0.239145i	$-0.0768671\pi$
$131$	22.2268	1.94197	0.970984	+	0.239145i	$0.0768671\pi$
$132$	0	0
$133$	−0.257507	−0.0223287
$134$	0	0
$135$	5.62400	0.484036
$136$	0	0
$137$	8.08077	0.690387	0.345193	−	0.938532i	$-0.387813\pi$
$137$	8.08077	0.690387	0.345193	+	0.938532i	$0.387813\pi$
$138$	0	0
$139$	−0.763404	−0.0647511	−0.0323755	−	0.999476i	$-0.510307\pi$
$139$	−0.763404	−0.0647511	−0.0323755	+	0.999476i	$0.510307\pi$
$140$	0	0
$141$	27.2313	2.29329
$142$	0	0
$143$	10.8265	0.905360
$144$	0	0
$145$	2.59152	0.215214
$146$	0	0
$147$	−19.7792	−1.63136
$148$	0	0
$149$	−0.645836	−0.0529089	−0.0264545	−	0.999650i	$-0.508422\pi$
$149$	−0.645836	−0.0529089	−0.0264545	+	0.999650i	$0.508422\pi$
$150$	0	0
$151$	16.4449	1.33827	0.669134	−	0.743142i	$-0.266665\pi$
$151$	16.4449	1.33827	0.669134	+	0.743142i	$0.266665\pi$
$152$	0	0
$153$	−12.3776	−1.00067
$154$	0	0
$155$	7.31064	0.587205
$156$	0	0
$157$	8.23439	0.657176	0.328588	−	0.944473i	$-0.393427\pi$
$157$	8.23439	0.657176	0.328588	+	0.944473i	$0.393427\pi$
$158$	0	0
$159$	−30.2710	−2.40064
$160$	0	0
$161$	−0.248642	−0.0195957
$162$	0	0
$163$	−1.57747	−0.123557	−0.0617786	−	0.998090i	$-0.519677\pi$
$163$	−1.57747	−0.123557	−0.0617786	+	0.998090i	$0.519677\pi$
$164$	0	0
$165$	−12.5817	−0.979481
$166$	0	0
$167$	−1.78625	−0.138224	−0.0691121	−	0.997609i	$-0.522017\pi$
$167$	−1.78625	−0.138224	−0.0691121	+	0.997609i	$0.522017\pi$
$168$	0	0
$169$	−7.08396	−0.544920
$170$	0	0
$171$	25.7887	1.97211
$172$	0	0
$173$	23.0808	1.75480	0.877399	−	0.479761i	$-0.159277\pi$
$173$	23.0808	1.75480	0.877399	+	0.479761i	$0.159277\pi$
$174$	0	0
$175$	−0.0498231	−0.00376627
$176$	0	0
$177$	35.7104	2.68416
$178$	0	0
$179$	9.63601	0.720229	0.360115	−	0.932908i	$-0.382738\pi$
$179$	9.63601	0.720229	0.360115	+	0.932908i	$0.382738\pi$
$180$	0	0
$181$	−7.10071	−0.527792	−0.263896	−	0.964551i	$-0.585008\pi$
$181$	−7.10071	−0.527792	−0.263896	+	0.964551i	$0.585008\pi$
$182$	0	0
$183$	−6.99298	−0.516936
$184$	0	0
$185$	5.13825	0.377772
$186$	0	0
$187$	11.0418	0.807455
$188$	0	0
$189$	−0.280205	−0.0203819
$190$	0	0
$191$	−9.47202	−0.685371	−0.342686	−	0.939450i	$-0.611337\pi$
$191$	−9.47202	−0.685371	−0.342686	+	0.939450i	$0.611337\pi$
$192$	0	0
$193$	11.7250	0.843985	0.421992	−	0.906599i	$-0.361331\pi$
$193$	11.7250	0.843985	0.421992	+	0.906599i	$0.361331\pi$
$194$	0	0
$195$	−6.87512	−0.492337
$196$	0	0
$197$	−18.2857	−1.30280	−0.651402	−	0.758733i	$-0.725819\pi$
$197$	−18.2857	−1.30280	−0.651402	+	0.758733i	$0.725819\pi$
$198$	0	0
$199$	−25.0469	−1.77553	−0.887763	−	0.460302i	$-0.847741\pi$
$199$	−25.0469	−1.77553	−0.887763	+	0.460302i	$0.847741\pi$
$200$	0	0
$201$	15.1542	1.06889
$202$	0	0
$203$	−0.129118	−0.00906229
$204$	0	0
$205$	9.11073	0.636321
$206$	0	0
$207$	24.9009	1.73073
$208$	0	0
$209$	−23.0055	−1.59132
$210$	0	0
$211$	17.6282	1.21358	0.606789	−	0.794863i	$-0.292457\pi$
$211$	17.6282	1.21358	0.606789	+	0.794863i	$0.292457\pi$
$212$	0	0
$213$	−27.0830	−1.85569
$214$	0	0
$215$	0.543007	0.0370328
$216$	0	0
$217$	−0.364239	−0.0247261
$218$	0	0
$219$	8.09450	0.546976
$220$	0	0
$221$	6.03366	0.405868
$222$	0	0
$223$	−27.8359	−1.86403	−0.932014	−	0.362421i	$-0.881950\pi$
$223$	−27.8359	−1.86403	−0.932014	+	0.362421i	$0.881950\pi$
$224$	0	0
$225$	4.98967	0.332645
$226$	0	0
$227$	−10.7150	−0.711181	−0.355591	−	0.934642i	$-0.615720\pi$
$227$	−10.7150	−0.711181	−0.355591	+	0.934642i	$0.615720\pi$
$228$	0	0
$229$	−20.0837	−1.32717	−0.663584	−	0.748102i	$-0.730966\pi$
$229$	−20.0837	−1.32717	−0.663584	+	0.748102i	$0.730966\pi$
$230$	0	0
$231$	0.626857	0.0412442
$232$	0	0
$233$	1.64318	0.107648	0.0538242	−	0.998550i	$-0.482859\pi$
$233$	1.64318	0.107648	0.0538242	+	0.998550i	$0.482859\pi$
$234$	0	0
$235$	9.63395	0.628450
$236$	0	0
$237$	−18.7255	−1.21635
$238$	0	0
$239$	12.3117	0.796379	0.398189	−	0.917303i	$-0.369639\pi$
$239$	12.3117	0.796379	0.398189	+	0.917303i	$0.369639\pi$
$240$	0	0
$241$	−10.7917	−0.695152	−0.347576	−	0.937652i	$-0.612995\pi$
$241$	−10.7917	−0.695152	−0.347576	+	0.937652i	$0.612995\pi$
$242$	0	0
$243$	−14.2495	−0.914107
$244$	0	0
$245$	−6.99752	−0.447055
$246$	0	0
$247$	−12.5711	−0.799880
$248$	0	0
$249$	33.9364	2.15063
$250$	0	0
$251$	31.0270	1.95840	0.979202	−	0.202888i	$-0.0650326\pi$
$251$	31.0270	1.95840	0.979202	+	0.202888i	$0.0650326\pi$
$252$	0	0
$253$	−22.2135	−1.39655
$254$	0	0
$255$	−7.01180	−0.439096
$256$	0	0
$257$	16.5586	1.03290	0.516449	−	0.856318i	$-0.327253\pi$
$257$	16.5586	1.03290	0.516449	+	0.856318i	$0.327253\pi$
$258$	0	0
$259$	−0.256004	−0.0159073
$260$	0	0
$261$	12.9308	0.800399
$262$	0	0
$263$	12.0270	0.741619	0.370810	−	0.928709i	$-0.379080\pi$
$263$	12.0270	0.741619	0.370810	+	0.928709i	$0.379080\pi$
$264$	0	0
$265$	−10.7093	−0.657868
$266$	0	0
$267$	2.82660	0.172985
$268$	0	0
$269$	−22.0871	−1.34667	−0.673336	−	0.739336i	$-0.735139\pi$
$269$	−22.0871	−1.34667	−0.673336	+	0.739336i	$0.735139\pi$
$270$	0	0
$271$	−13.8002	−0.838302	−0.419151	−	0.907917i	$-0.637672\pi$
$271$	−13.8002	−0.838302	−0.419151	+	0.907917i	$0.637672\pi$
$272$	0	0
$273$	0.342539	0.0207314
$274$	0	0
$275$	−4.45116	−0.268415
$276$	0	0
$277$	3.04290	0.182830	0.0914151	−	0.995813i	$-0.470861\pi$
$277$	3.04290	0.182830	0.0914151	+	0.995813i	$0.470861\pi$
$278$	0	0
$279$	36.4777	2.18386
$280$	0	0
$281$	21.1218	1.26002	0.630011	−	0.776586i	$-0.283050\pi$
$281$	21.1218	1.26002	0.630011	+	0.776586i	$0.283050\pi$
$282$	0	0
$283$	−32.4871	−1.93116	−0.965579	−	0.260111i	$-0.916241\pi$
$283$	−32.4871	−1.93116	−0.965579	+	0.260111i	$0.916241\pi$
$284$	0	0
$285$	14.6091	0.865366
$286$	0	0
$287$	−0.453924	−0.0267943
$288$	0	0
$289$	−10.8464	−0.638022
$290$	0	0
$291$	13.6257	0.798750
$292$	0	0
$293$	32.0101	1.87005	0.935026	−	0.354578i	$-0.115376\pi$
$293$	32.0101	1.87005	0.935026	+	0.354578i	$0.115376\pi$
$294$	0	0
$295$	12.6337	0.735562
$296$	0	0
$297$	−25.0333	−1.45258
$298$	0	0
$299$	−12.1383	−0.701979
$300$	0	0
$301$	−0.0270543	−0.00155938
$302$	0	0
$303$	−32.6645	−1.87653
$304$	0	0
$305$	−2.47399	−0.141660
$306$	0	0
$307$	19.6188	1.11970	0.559852	−	0.828593i	$-0.310858\pi$
$307$	19.6188	1.11970	0.559852	+	0.828593i	$0.310858\pi$
$308$	0	0
$309$	8.27081	0.470510
$310$	0	0
$311$	−24.8863	−1.41117	−0.705586	−	0.708625i	$-0.749316\pi$
$311$	−24.8863	−1.41117	−0.705586	+	0.708625i	$0.749316\pi$
$312$	0	0
$313$	−6.17328	−0.348934	−0.174467	−	0.984663i	$-0.555820\pi$
$313$	−6.17328	−0.348934	−0.174467	+	0.984663i	$0.555820\pi$
$314$	0	0
$315$	−0.248601	−0.0140071
$316$	0	0
$317$	−32.9455	−1.85041	−0.925203	−	0.379473i	$-0.876105\pi$
$317$	−32.9455	−1.85041	−0.925203	+	0.379473i	$0.876105\pi$
$318$	0	0
$319$	−11.5353	−0.645853
$320$	0	0
$321$	11.2494	0.627878
$322$	0	0
$323$	−12.8210	−0.713381
$324$	0	0
$325$	−2.43229	−0.134919
$326$	0	0
$327$	14.9866	0.828763
$328$	0	0
$329$	−0.479993	−0.0264629
$330$	0	0
$331$	−30.9785	−1.70273	−0.851367	−	0.524570i	$-0.824226\pi$
$331$	−30.9785	−1.70273	−0.851367	+	0.524570i	$0.824226\pi$
$332$	0	0
$333$	25.6382	1.40496
$334$	0	0
$335$	5.36128	0.292918
$336$	0	0
$337$	−3.53709	−0.192678	−0.0963389	−	0.995349i	$-0.530713\pi$
$337$	−3.53709	−0.192678	−0.0963389	+	0.995349i	$0.530713\pi$
$338$	0	0
$339$	−29.3650	−1.59489
$340$	0	0
$341$	−32.5409	−1.76219
$342$	0	0
$343$	0.697399	0.0376560
$344$	0	0
$345$	14.1061	0.759449
$346$	0	0
$347$	13.0198	0.698939	0.349470	−	0.936948i	$-0.386362\pi$
$347$	13.0198	0.698939	0.349470	+	0.936948i	$0.386362\pi$
$348$	0	0
$349$	27.6562	1.48040	0.740201	−	0.672385i	$-0.234730\pi$
$349$	27.6562	1.48040	0.740201	+	0.672385i	$0.234730\pi$
$350$	0	0
$351$	−13.6792	−0.730142
$352$	0	0
$353$	−26.3090	−1.40029	−0.700144	−	0.714002i	$-0.746881\pi$
$353$	−26.3090	−1.40029	−0.700144	+	0.714002i	$0.746881\pi$
$354$	0	0
$355$	−9.58146	−0.508531
$356$	0	0
$357$	0.349350	0.0184895
$358$	0	0
$359$	−14.0008	−0.738934	−0.369467	−	0.929244i	$-0.620460\pi$
$359$	−14.0008	−0.738934	−0.369467	+	0.929244i	$0.620460\pi$
$360$	0	0
$361$	7.71256	0.405924
$362$	0	0
$363$	24.9105	1.30746
$364$	0	0
$365$	2.86369	0.149892
$366$	0	0
$367$	−22.0304	−1.14998	−0.574989	−	0.818161i	$-0.694994\pi$
$367$	−22.0304	−1.14998	−0.574989	+	0.818161i	$0.694994\pi$
$368$	0	0
$369$	45.4595	2.36653
$370$	0	0
$371$	0.533571	0.0277016
$372$	0	0
$373$	13.6705	0.707830	0.353915	−	0.935278i	$-0.384850\pi$
$373$	13.6705	0.707830	0.353915	+	0.935278i	$0.384850\pi$
$374$	0	0
$375$	2.82660	0.145965
$376$	0	0
$377$	−6.30334	−0.324639
$378$	0	0
$379$	15.1907	0.780292	0.390146	−	0.920753i	$-0.372425\pi$
$379$	15.1907	0.780292	0.390146	+	0.920753i	$0.372425\pi$
$380$	0	0
$381$	−15.0161	−0.769300
$382$	0	0
$383$	0.297784	0.0152160	0.00760802	−	0.999971i	$-0.497578\pi$
$383$	0.297784	0.0152160	0.00760802	+	0.999971i	$0.497578\pi$
$384$	0	0
$385$	0.221771	0.0113025
$386$	0	0
$387$	2.70943	0.137728
$388$	0	0
$389$	−2.25933	−0.114553	−0.0572763	−	0.998358i	$-0.518242\pi$
$389$	−2.25933	−0.114553	−0.0572763	+	0.998358i	$0.518242\pi$
$390$	0	0
$391$	−12.3797	−0.626067
$392$	0	0
$393$	62.8264	3.16917
$394$	0	0
$395$	−6.62473	−0.333326
$396$	0	0
$397$	−23.8373	−1.19636	−0.598180	−	0.801361i	$-0.704109\pi$
$397$	−23.8373	−1.19636	−0.598180	+	0.801361i	$0.704109\pi$
$398$	0	0
$399$	−0.727868	−0.0364390
$400$	0	0
$401$	26.1681	1.30677	0.653386	−	0.757025i	$-0.273348\pi$
$401$	26.1681	1.30677	0.653386	+	0.757025i	$0.273348\pi$
$402$	0	0
$403$	−17.7816	−0.885766
$404$	0	0
$405$	0.927782	0.0461018
$406$	0	0
$407$	−22.8712	−1.13368
$408$	0	0
$409$	−22.9348	−1.13406	−0.567028	−	0.823699i	$-0.691907\pi$
$409$	−22.9348	−1.13406	−0.567028	+	0.823699i	$0.691907\pi$
$410$	0	0
$411$	22.8411	1.12667
$412$	0	0
$413$	−0.629449	−0.0309732
$414$	0	0
$415$	12.0061	0.589355
$416$	0	0
$417$	−2.15784	−0.105670
$418$	0	0
$419$	−5.25412	−0.256681	−0.128340	−	0.991730i	$-0.540965\pi$
$419$	−5.25412	−0.256681	−0.128340	+	0.991730i	$0.540965\pi$
$420$	0	0
$421$	18.0800	0.881166	0.440583	−	0.897712i	$-0.354772\pi$
$421$	18.0800	0.881166	0.440583	+	0.897712i	$0.354772\pi$
$422$	0	0
$423$	48.0702	2.33725
$424$	0	0
$425$	−2.48065	−0.120329
$426$	0	0
$427$	0.123262	0.00596506
$428$	0	0
$429$	30.6023	1.47749
$430$	0	0
$431$	−5.55910	−0.267773	−0.133886	−	0.990997i	$-0.542746\pi$
$431$	−5.55910	−0.267773	−0.133886	+	0.990997i	$0.542746\pi$
$432$	0	0
$433$	20.5795	0.988986	0.494493	−	0.869182i	$-0.335354\pi$
$433$	20.5795	0.988986	0.494493	+	0.869182i	$0.335354\pi$
$434$	0	0
$435$	7.32520	0.351217
$436$	0	0
$437$	25.7930	1.23385
$438$	0	0
$439$	−26.6104	−1.27004	−0.635022	−	0.772494i	$-0.719009\pi$
$439$	−26.6104	−1.27004	−0.635022	+	0.772494i	$0.719009\pi$
$440$	0	0
$441$	−34.9153	−1.66263
$442$	0	0
$443$	−20.4411	−0.971188	−0.485594	−	0.874185i	$-0.661397\pi$
$443$	−20.4411	−0.971188	−0.485594	+	0.874185i	$0.661397\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	−1.82552	−0.0863441
$448$	0	0
$449$	5.61381	0.264932	0.132466	−	0.991188i	$-0.457710\pi$
$449$	5.61381	0.264932	0.132466	+	0.991188i	$0.457710\pi$
$450$	0	0
$451$	−40.5533	−1.90958
$452$	0	0
$453$	46.4832	2.18397
$454$	0	0
$455$	0.121184	0.00568121
$456$	0	0
$457$	−7.13528	−0.333774	−0.166887	−	0.985976i	$-0.553372\pi$
$457$	−7.13528	−0.333774	−0.166887	+	0.985976i	$0.553372\pi$
$458$	0	0
$459$	−13.9512	−0.651184
$460$	0	0
$461$	−24.0844	−1.12172	−0.560860	−	0.827911i	$-0.689529\pi$
$461$	−24.0844	−1.12172	−0.560860	+	0.827911i	$0.689529\pi$
$462$	0	0
$463$	22.2301	1.03312	0.516561	−	0.856250i	$-0.327212\pi$
$463$	22.2301	1.03312	0.516561	+	0.856250i	$0.327212\pi$
$464$	0	0
$465$	20.6643	0.958282
$466$	0	0
$467$	−10.4176	−0.482069	−0.241034	−	0.970517i	$-0.577487\pi$
$467$	−10.4176	−0.482069	−0.241034	+	0.970517i	$0.577487\pi$
$468$	0	0
$469$	−0.267115	−0.0123342
$470$	0	0
$471$	23.2753	1.07247
$472$	0	0
$473$	−2.41701	−0.111134
$474$	0	0
$475$	5.16842	0.237143
$476$	0	0
$477$	−53.4360	−2.44666
$478$	0	0
$479$	20.6366	0.942912	0.471456	−	0.881889i	$-0.343729\pi$
$479$	20.6366	0.942912	0.471456	+	0.881889i	$0.343729\pi$
$480$	0	0
$481$	−12.4977	−0.569848
$482$	0	0
$483$	−0.702811	−0.0319790
$484$	0	0
$485$	4.82051	0.218888
$486$	0	0
$487$	29.5496	1.33902	0.669510	−	0.742803i	$-0.266504\pi$
$487$	29.5496	1.33902	0.669510	+	0.742803i	$0.266504\pi$
$488$	0	0
$489$	−4.45888	−0.201638
$490$	0	0
$491$	−37.8108	−1.70638	−0.853189	−	0.521602i	$-0.825335\pi$
$491$	−37.8108	−1.70638	−0.853189	+	0.521602i	$0.825335\pi$
$492$	0	0
$493$	−6.42866	−0.289532
$494$	0	0
$495$	−22.2098	−0.998258
$496$	0	0
$497$	0.477378	0.0214133
$498$	0	0
$499$	−7.09756	−0.317730	−0.158865	−	0.987300i	$-0.550783\pi$
$499$	−7.09756	−0.317730	−0.158865	+	0.987300i	$0.550783\pi$
$500$	0	0
$501$	−5.04902	−0.225573
$502$	0	0
$503$	25.0615	1.11744	0.558718	−	0.829358i	$-0.311294\pi$
$503$	25.0615	1.11744	0.558718	+	0.829358i	$0.311294\pi$
$504$	0	0
$505$	−11.5561	−0.514240
$506$	0	0
$507$	−20.0235	−0.889275
$508$	0	0
$509$	−1.59934	−0.0708895	−0.0354447	−	0.999372i	$-0.511285\pi$
$509$	−1.59934	−0.0708895	−0.0354447	+	0.999372i	$0.511285\pi$
$510$	0	0
$511$	−0.142678	−0.00631169
$512$	0	0
$513$	29.0672	1.28335
$514$	0	0
$515$	2.92606	0.128938
$516$	0	0
$517$	−42.8823	−1.88596
$518$	0	0
$519$	65.2401	2.86372
$520$	0	0
$521$	25.2660	1.10693	0.553463	−	0.832874i	$-0.313306\pi$
$521$	25.2660	1.10693	0.553463	+	0.832874i	$0.313306\pi$
$522$	0	0
$523$	1.75283	0.0766458	0.0383229	−	0.999265i	$-0.487798\pi$
$523$	1.75283	0.0766458	0.0383229	+	0.999265i	$0.487798\pi$
$524$	0	0
$525$	−0.140830	−0.00614632
$526$	0	0
$527$	−18.1351	−0.789979
$528$	0	0
$529$	1.90507	0.0828292
$530$	0	0
$531$	63.0379	2.73561
$532$	0	0
$533$	−22.1599	−0.959854
$534$	0	0
$535$	3.97982	0.172063
$536$	0	0
$537$	27.2372	1.17537
$538$	0	0
$539$	31.1471	1.34160
$540$	0	0
$541$	5.15589	0.221669	0.110835	−	0.993839i	$-0.464648\pi$
$541$	5.15589	0.221669	0.110835	+	0.993839i	$0.464648\pi$
$542$	0	0
$543$	−20.0709	−0.861323
$544$	0	0
$545$	5.30200	0.227113
$546$	0	0
$547$	−13.3060	−0.568925	−0.284462	−	0.958687i	$-0.591815\pi$
$547$	−13.3060	−0.568925	−0.284462	+	0.958687i	$0.591815\pi$
$548$	0	0
$549$	−12.3444	−0.526846
$550$	0	0
$551$	13.3941	0.570607
$552$	0	0
$553$	0.330064	0.0140358
$554$	0	0
$555$	14.5238	0.616501
$556$	0	0
$557$	−32.2911	−1.36822	−0.684109	−	0.729380i	$-0.739809\pi$
$557$	−32.2911	−1.36822	−0.684109	+	0.729380i	$0.739809\pi$
$558$	0	0
$559$	−1.32075	−0.0558619
$560$	0	0
$561$	31.2107	1.31772
$562$	0	0
$563$	1.17102	0.0493525	0.0246762	−	0.999695i	$-0.492145\pi$
$563$	1.17102	0.0493525	0.0246762	+	0.999695i	$0.492145\pi$
$564$	0	0
$565$	−10.3888	−0.437060
$566$	0	0
$567$	−0.0462249	−0.00194126
$568$	0	0
$569$	20.9355	0.877662	0.438831	−	0.898570i	$-0.355393\pi$
$569$	20.9355	0.877662	0.438831	+	0.898570i	$0.355393\pi$
$570$	0	0
$571$	44.5105	1.86271	0.931354	−	0.364116i	$-0.118629\pi$
$571$	44.5105	1.86271	0.931354	+	0.364116i	$0.118629\pi$
$572$	0	0
$573$	−26.7736	−1.11848
$574$	0	0
$575$	4.99050	0.208118
$576$	0	0
$577$	−44.7453	−1.86277	−0.931386	−	0.364034i	$-0.881399\pi$
$577$	−44.7453	−1.86277	−0.931386	+	0.364034i	$0.881399\pi$
$578$	0	0
$579$	33.1419	1.37733
$580$	0	0
$581$	−0.598180	−0.0248167
$582$	0	0
$583$	47.6690	1.97425
$584$	0	0
$585$	−12.1363	−0.501775
$586$	0	0
$587$	−14.8194	−0.611663	−0.305832	−	0.952086i	$-0.598934\pi$
$587$	−14.8194	−0.611663	−0.305832	+	0.952086i	$0.598934\pi$
$588$	0	0
$589$	37.7845	1.55688
$590$	0	0
$591$	−51.6864	−2.12609
$592$	0	0
$593$	−23.3625	−0.959384	−0.479692	−	0.877437i	$-0.659252\pi$
$593$	−23.3625	−0.959384	−0.479692	+	0.877437i	$0.659252\pi$
$594$	0	0
$595$	0.123594	0.00506684
$596$	0	0
$597$	−70.7975	−2.89755
$598$	0	0
$599$	2.73683	0.111824	0.0559120	−	0.998436i	$-0.482193\pi$
$599$	2.73683	0.111824	0.0559120	+	0.998436i	$0.482193\pi$
$600$	0	0
$601$	−14.7244	−0.600622	−0.300311	−	0.953841i	$-0.597090\pi$
$601$	−14.7244	−0.600622	−0.300311	+	0.953841i	$0.597090\pi$
$602$	0	0
$603$	26.7510	1.08938
$604$	0	0
$605$	8.81287	0.358294
$606$	0	0
$607$	7.93966	0.322261	0.161131	−	0.986933i	$-0.448486\pi$
$607$	7.93966	0.322261	0.161131	+	0.986933i	$0.448486\pi$
$608$	0	0
$609$	−0.364964	−0.0147891
$610$	0	0
$611$	−23.4326	−0.947981
$612$	0	0
$613$	−5.63827	−0.227728	−0.113864	−	0.993496i	$-0.536323\pi$
$613$	−5.63827	−0.227728	−0.113864	+	0.993496i	$0.536323\pi$
$614$	0	0
$615$	25.7524	1.03844
$616$	0	0
$617$	24.0408	0.967848	0.483924	−	0.875110i	$-0.339211\pi$
$617$	24.0408	0.967848	0.483924	+	0.875110i	$0.339211\pi$
$618$	0	0
$619$	−18.0743	−0.726469	−0.363234	−	0.931698i	$-0.618328\pi$
$619$	−18.0743	−0.726469	−0.363234	+	0.931698i	$0.618328\pi$
$620$	0	0
$621$	28.0665	1.12627
$622$	0	0
$623$	−0.0498231	−0.00199612
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−65.0273	−2.59694
$628$	0	0
$629$	−12.7462	−0.508224
$630$	0	0
$631$	3.58541	0.142733	0.0713665	−	0.997450i	$-0.477264\pi$
$631$	3.58541	0.142733	0.0713665	+	0.997450i	$0.477264\pi$
$632$	0	0
$633$	49.8279	1.98048
$634$	0	0
$635$	−5.31244	−0.210818
$636$	0	0
$637$	17.0200	0.674357
$638$	0	0
$639$	−47.8083	−1.89127
$640$	0	0
$641$	−19.3761	−0.765311	−0.382656	−	0.923891i	$-0.624990\pi$
$641$	−19.3761	−0.765311	−0.382656	+	0.923891i	$0.624990\pi$
$642$	0	0
$643$	−26.7275	−1.05403	−0.527015	−	0.849856i	$-0.676689\pi$
$643$	−26.7275	−1.05403	−0.527015	+	0.849856i	$0.676689\pi$
$644$	0	0
$645$	1.53486	0.0604352
$646$	0	0
$647$	−20.6868	−0.813281	−0.406641	−	0.913588i	$-0.633300\pi$
$647$	−20.6868	−0.813281	−0.406641	+	0.913588i	$0.633300\pi$
$648$	0	0
$649$	−56.2346	−2.20740
$650$	0	0
$651$	−1.02956	−0.0403515
$652$	0	0
$653$	−33.7364	−1.32021	−0.660104	−	0.751174i	$-0.729488\pi$
$653$	−33.7364	−1.32021	−0.660104	+	0.751174i	$0.729488\pi$
$654$	0	0
$655$	22.2268	0.868474
$656$	0	0
$657$	14.2888	0.557461
$658$	0	0
$659$	9.44930	0.368092	0.184046	−	0.982918i	$-0.441080\pi$
$659$	9.44930	0.368092	0.184046	+	0.982918i	$0.441080\pi$
$660$	0	0
$661$	−36.6486	−1.42547	−0.712733	−	0.701435i	$-0.752543\pi$
$661$	−36.6486	−1.42547	−0.712733	+	0.701435i	$0.752543\pi$
$662$	0	0
$663$	17.0547	0.662352
$664$	0	0
$665$	−0.257507	−0.00998568
$666$	0	0
$667$	12.9330	0.500768
$668$	0	0
$669$	−78.6809	−3.04198
$670$	0	0
$671$	11.0121	0.425119
$672$	0	0
$673$	42.5711	1.64099	0.820497	−	0.571651i	$-0.193697\pi$
$673$	42.5711	1.64099	0.820497	+	0.571651i	$0.193697\pi$
$674$	0	0
$675$	5.62400	0.216468
$676$	0	0
$677$	−29.6301	−1.13878	−0.569389	−	0.822068i	$-0.692820\pi$
$677$	−29.6301	−1.13878	−0.569389	+	0.822068i	$0.692820\pi$
$678$	0	0
$679$	−0.240173	−0.00921698
$680$	0	0
$681$	−30.2871	−1.16060
$682$	0	0
$683$	−18.4389	−0.705545	−0.352772	−	0.935709i	$-0.614761\pi$
$683$	−18.4389	−0.705545	−0.352772	+	0.935709i	$0.614761\pi$
$684$	0	0
$685$	8.08077	0.308750
$686$	0	0
$687$	−56.7686	−2.16586
$688$	0	0
$689$	26.0482	0.992357
$690$	0	0
$691$	39.7294	1.51138	0.755689	−	0.654930i	$-0.227302\pi$
$691$	39.7294	1.51138	0.755689	+	0.654930i	$0.227302\pi$
$692$	0	0
$693$	1.10656	0.0420348
$694$	0	0
$695$	−0.763404	−0.0289576
$696$	0	0
$697$	−22.6005	−0.856056
$698$	0	0
$699$	4.64462	0.175676
$700$	0	0
$701$	−8.13101	−0.307104	−0.153552	−	0.988141i	$-0.549071\pi$
$701$	−8.13101	−0.307104	−0.153552	+	0.988141i	$0.549071\pi$
$702$	0	0
$703$	26.5567	1.00160
$704$	0	0
$705$	27.2313	1.02559
$706$	0	0
$707$	0.575761	0.0216537
$708$	0	0
$709$	24.2930	0.912343	0.456171	−	0.889892i	$-0.349220\pi$
$709$	24.2930	0.912343	0.456171	+	0.889892i	$0.349220\pi$
$710$	0	0
$711$	−33.0552	−1.23967
$712$	0	0
$713$	36.4838	1.36633
$714$	0	0
$715$	10.8265	0.404889
$716$	0	0
$717$	34.8003	1.29964
$718$	0	0
$719$	−35.3355	−1.31779	−0.658896	−	0.752234i	$-0.728976\pi$
$719$	−35.3355	−1.31779	−0.658896	+	0.752234i	$0.728976\pi$
$720$	0	0
$721$	−0.145785	−0.00542933
$722$	0	0
$723$	−30.5037	−1.13445
$724$	0	0
$725$	2.59152	0.0962468
$726$	0	0
$727$	27.6135	1.02413	0.512065	−	0.858947i	$-0.328881\pi$
$727$	27.6135	1.02413	0.512065	+	0.858947i	$0.328881\pi$
$728$	0	0
$729$	−43.0610	−1.59485
$730$	0	0
$731$	−1.34701	−0.0498210
$732$	0	0
$733$	−19.3580	−0.715005	−0.357503	−	0.933912i	$-0.616372\pi$
$733$	−19.3580	−0.715005	−0.357503	+	0.933912i	$0.616372\pi$
$734$	0	0
$735$	−19.7792	−0.729566
$736$	0	0
$737$	−23.8639	−0.879039
$738$	0	0
$739$	−28.8545	−1.06143	−0.530715	−	0.847550i	$-0.678077\pi$
$739$	−28.8545	−1.06143	−0.530715	+	0.847550i	$0.678077\pi$
$740$	0	0
$741$	−35.5335	−1.30536
$742$	0	0
$743$	−25.5760	−0.938292	−0.469146	−	0.883121i	$-0.655438\pi$
$743$	−25.5760	−0.938292	−0.469146	+	0.883121i	$0.655438\pi$
$744$	0	0
$745$	−0.645836	−0.0236616
$746$	0	0
$747$	59.9063	2.19186
$748$	0	0
$749$	−0.198287	−0.00724525
$750$	0	0
$751$	26.6217	0.971440	0.485720	−	0.874114i	$-0.338557\pi$
$751$	26.6217	0.971440	0.485720	+	0.874114i	$0.338557\pi$
$752$	0	0
$753$	87.7008	3.19599
$754$	0	0
$755$	16.4449	0.598492
$756$	0	0
$757$	16.8066	0.610848	0.305424	−	0.952216i	$-0.401202\pi$
$757$	16.8066	0.610848	0.305424	+	0.952216i	$0.401202\pi$
$758$	0	0
$759$	−62.7888	−2.27909
$760$	0	0
$761$	−23.6421	−0.857026	−0.428513	−	0.903536i	$-0.640962\pi$
$761$	−23.6421	−0.857026	−0.428513	+	0.903536i	$0.640962\pi$
$762$	0	0
$763$	−0.264162	−0.00956331
$764$	0	0
$765$	−12.3776	−0.447514
$766$	0	0
$767$	−30.7288	−1.10955
$768$	0	0
$769$	20.1859	0.727921	0.363960	−	0.931414i	$-0.381424\pi$
$769$	20.1859	0.727921	0.363960	+	0.931414i	$0.381424\pi$
$770$	0	0
$771$	46.8046	1.68563
$772$	0	0
$773$	−14.8504	−0.534132	−0.267066	−	0.963678i	$-0.586054\pi$
$773$	−14.8504	−0.534132	−0.267066	+	0.963678i	$0.586054\pi$
$774$	0	0
$775$	7.31064	0.262606
$776$	0	0
$777$	−0.723620	−0.0259597
$778$	0	0
$779$	47.0881	1.68710
$780$	0	0
$781$	42.6487	1.52609
$782$	0	0
$783$	14.5747	0.520858
$784$	0	0
$785$	8.23439	0.293898
$786$	0	0
$787$	−22.9126	−0.816746	−0.408373	−	0.912815i	$-0.633904\pi$
$787$	−22.9126	−0.816746	−0.408373	+	0.912815i	$0.633904\pi$
$788$	0	0
$789$	33.9956	1.21028
$790$	0	0
$791$	0.517602	0.0184038
$792$	0	0
$793$	6.01747	0.213687
$794$	0	0
$795$	−30.2710	−1.07360
$796$	0	0
$797$	32.7951	1.16166	0.580831	−	0.814024i	$-0.302728\pi$
$797$	32.7951	1.16166	0.580831	+	0.814024i	$0.302728\pi$
$798$	0	0
$799$	−23.8985	−0.845467
$800$	0	0
$801$	4.98967	0.176301
$802$	0	0
$803$	−12.7467	−0.449823
$804$	0	0
$805$	−0.248642	−0.00876348
$806$	0	0
$807$	−62.4313	−2.19769
$808$	0	0
$809$	−25.8540	−0.908978	−0.454489	−	0.890752i	$-0.650178\pi$
$809$	−25.8540	−0.908978	−0.454489	+	0.890752i	$0.650178\pi$
$810$	0	0
$811$	5.12583	0.179992	0.0899962	−	0.995942i	$-0.471315\pi$
$811$	5.12583	0.179992	0.0899962	+	0.995942i	$0.471315\pi$
$812$	0	0
$813$	−39.0076	−1.36806
$814$	0	0
$815$	−1.57747	−0.0552565
$816$	0	0
$817$	2.80649	0.0981866
$818$	0	0
$819$	0.604669	0.0211289
$820$	0	0
$821$	−15.9798	−0.557701	−0.278850	−	0.960335i	$-0.589953\pi$
$821$	−15.9798	−0.557701	−0.278850	+	0.960335i	$0.589953\pi$
$822$	0	0
$823$	−37.5936	−1.31043	−0.655216	−	0.755442i	$-0.727422\pi$
$823$	−37.5936	−1.31043	−0.655216	+	0.755442i	$0.727422\pi$
$824$	0	0
$825$	−12.5817	−0.438037
$826$	0	0
$827$	53.0600	1.84508	0.922538	−	0.385906i	$-0.126111\pi$
$827$	53.0600	1.84508	0.922538	+	0.385906i	$0.126111\pi$
$828$	0	0
$829$	7.43536	0.258241	0.129120	−	0.991629i	$-0.458785\pi$
$829$	7.43536	0.258241	0.129120	+	0.991629i	$0.458785\pi$
$830$	0	0
$831$	8.60106	0.298368
$832$	0	0
$833$	17.3584	0.601432
$834$	0	0
$835$	−1.78625	−0.0618157
$836$	0	0
$837$	41.1150	1.42114
$838$	0	0
$839$	38.6435	1.33412	0.667061	−	0.745003i	$-0.267552\pi$
$839$	38.6435	1.33412	0.667061	+	0.745003i	$0.267552\pi$
$840$	0	0
$841$	−22.2840	−0.768414
$842$	0	0
$843$	59.7029	2.05628
$844$	0	0
$845$	−7.08396	−0.243695
$846$	0	0
$847$	−0.439084	−0.0150871
$848$	0	0
$849$	−91.8280	−3.15153
$850$	0	0
$851$	25.6424	0.879012
$852$	0	0
$853$	34.8883	1.19455	0.597277	−	0.802035i	$-0.296249\pi$
$853$	34.8883	1.19455	0.597277	+	0.802035i	$0.296249\pi$
$854$	0	0
$855$	25.7887	0.881955
$856$	0	0
$857$	−5.99133	−0.204660	−0.102330	−	0.994751i	$-0.532630\pi$
$857$	−5.99133	−0.204660	−0.102330	+	0.994751i	$0.532630\pi$
$858$	0	0
$859$	−36.8422	−1.25704	−0.628519	−	0.777794i	$-0.716339\pi$
$859$	−36.8422	−1.25704	−0.628519	+	0.777794i	$0.716339\pi$
$860$	0	0
$861$	−1.28306	−0.0437267
$862$	0	0
$863$	−51.2726	−1.74534	−0.872669	−	0.488312i	$-0.837613\pi$
$863$	−51.2726	−1.74534	−0.872669	+	0.488312i	$0.837613\pi$
$864$	0	0
$865$	23.0808	0.784770
$866$	0	0
$867$	−30.6584	−1.04121
$868$	0	0
$869$	29.4877	1.00030
$870$	0	0
$871$	−13.0402	−0.441850
$872$	0	0
$873$	24.0528	0.814062
$874$	0	0
$875$	−0.0498231	−0.00168433
$876$	0	0
$877$	−5.47506	−0.184880	−0.0924399	−	0.995718i	$-0.529467\pi$
$877$	−5.47506	−0.184880	−0.0924399	+	0.995718i	$0.529467\pi$
$878$	0	0
$879$	90.4799	3.05181
$880$	0	0
$881$	−16.6965	−0.562521	−0.281260	−	0.959632i	$-0.590752\pi$
$881$	−16.6965	−0.562521	−0.281260	+	0.959632i	$0.590752\pi$
$882$	0	0
$883$	20.7620	0.698696	0.349348	−	0.936993i	$-0.386403\pi$
$883$	20.7620	0.698696	0.349348	+	0.936993i	$0.386403\pi$
$884$	0	0
$885$	35.7104	1.20039
$886$	0	0
$887$	30.8479	1.03577	0.517886	−	0.855450i	$-0.326719\pi$
$887$	30.8479	1.03577	0.517886	+	0.855450i	$0.326719\pi$
$888$	0	0
$889$	0.264682	0.00887715
$890$	0	0
$891$	−4.12971	−0.138350
$892$	0	0
$893$	49.7923	1.66624
$894$	0	0
$895$	9.63601	0.322096
$896$	0	0
$897$	−34.3103	−1.14559
$898$	0	0
$899$	18.9457	0.631875
$900$	0	0
$901$	26.5661	0.885044
$902$	0	0
$903$	−0.0764717	−0.00254482
$904$	0	0
$905$	−7.10071	−0.236036
$906$	0	0
$907$	−30.8544	−1.02450	−0.512251	−	0.858836i	$-0.671188\pi$
$907$	−30.8544	−1.02450	−0.512251	+	0.858836i	$0.671188\pi$
$908$	0	0
$909$	−57.6611	−1.91250
$910$	0	0
$911$	−34.6247	−1.14717	−0.573584	−	0.819147i	$-0.694447\pi$
$911$	−34.6247	−1.14717	−0.573584	+	0.819147i	$0.694447\pi$
$912$	0	0
$913$	−53.4410	−1.76864
$914$	0	0
$915$	−6.99298	−0.231181
$916$	0	0
$917$	−1.10741	−0.0365699
$918$	0	0
$919$	3.03636	0.100160	0.0500802	−	0.998745i	$-0.484052\pi$
$919$	3.03636	0.100160	0.0500802	+	0.998745i	$0.484052\pi$
$920$	0	0
$921$	55.4545	1.82729
$922$	0	0
$923$	23.3049	0.767090
$924$	0	0
$925$	5.13825	0.168945
$926$	0	0
$927$	14.6001	0.479530
$928$	0	0
$929$	−17.2687	−0.566568	−0.283284	−	0.959036i	$-0.591424\pi$
$929$	−17.2687	−0.566568	−0.283284	+	0.959036i	$0.591424\pi$
$930$	0	0
$931$	−36.1661	−1.18530
$932$	0	0
$933$	−70.3435	−2.30294
$934$	0	0
$935$	11.0418	0.361105
$936$	0	0
$937$	41.8282	1.36647	0.683234	−	0.730199i	$-0.260573\pi$
$937$	41.8282	1.36647	0.683234	+	0.730199i	$0.260573\pi$
$938$	0	0
$939$	−17.4494	−0.569439
$940$	0	0
$941$	55.0411	1.79429	0.897145	−	0.441736i	$-0.145637\pi$
$941$	55.0411	1.79429	0.897145	+	0.441736i	$0.145637\pi$
$942$	0	0
$943$	45.4671	1.48061
$944$	0	0
$945$	−0.280205	−0.00911506
$946$	0	0
$947$	−5.20266	−0.169064	−0.0845318	−	0.996421i	$-0.526939\pi$
$947$	−5.20266	−0.169064	−0.0845318	+	0.996421i	$0.526939\pi$
$948$	0	0
$949$	−6.96532	−0.226104
$950$	0	0
$951$	−93.1238	−3.01975
$952$	0	0
$953$	23.9232	0.774950	0.387475	−	0.921880i	$-0.373347\pi$
$953$	23.9232	0.774950	0.387475	+	0.921880i	$0.373347\pi$
$954$	0	0
$955$	−9.47202	−0.306507
$956$	0	0
$957$	−32.6057	−1.05399
$958$	0	0
$959$	−0.402609	−0.0130009
$960$	0	0
$961$	22.4455	0.724049
$962$	0	0
$963$	19.8580	0.639915
$964$	0	0
$965$	11.7250	0.377441
$966$	0	0
$967$	−47.9852	−1.54310	−0.771550	−	0.636168i	$-0.780518\pi$
$967$	−47.9852	−1.54310	−0.771550	+	0.636168i	$0.780518\pi$
$968$	0	0
$969$	−36.2399	−1.16419
$970$	0	0
$971$	−4.06546	−0.130467	−0.0652334	−	0.997870i	$-0.520779\pi$
$971$	−4.06546	−0.130467	−0.0652334	+	0.997870i	$0.520779\pi$
$972$	0	0
$973$	0.0380351	0.00121935
$974$	0	0
$975$	−6.87512	−0.220180
$976$	0	0
$977$	4.40304	0.140866	0.0704328	−	0.997517i	$-0.477562\pi$
$977$	4.40304	0.140866	0.0704328	+	0.997517i	$0.477562\pi$
$978$	0	0
$979$	−4.45116	−0.142260
$980$	0	0
$981$	26.4552	0.844651
$982$	0	0
$983$	8.02508	0.255960	0.127980	−	0.991777i	$-0.459151\pi$
$983$	8.02508	0.255960	0.127980	+	0.991777i	$0.459151\pi$
$984$	0	0
$985$	−18.2857	−0.582631
$986$	0	0
$987$	−1.35675	−0.0431858
$988$	0	0
$989$	2.70988	0.0861691
$990$	0	0
$991$	−28.4040	−0.902282	−0.451141	−	0.892453i	$-0.648983\pi$
$991$	−28.4040	−0.902282	−0.451141	+	0.892453i	$0.648983\pi$
$992$	0	0
$993$	−87.5639	−2.77876
$994$	0	0
$995$	−25.0469	−0.794039
$996$	0	0
$997$	−6.03403	−0.191100	−0.0955499	−	0.995425i	$-0.530461\pi$
$997$	−6.03403	−0.191100	−0.0955499	+	0.995425i	$0.530461\pi$
$998$	0	0
$999$	28.8975	0.914277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bj.1.6		7
4.3	odd	2		445.2.a.f.1.4	✓	7
12.11	even	2		4005.2.a.o.1.4		7
20.19	odd	2		2225.2.a.k.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.4	✓	7	4.3	odd	2
2225.2.a.k.1.4		7	20.19	odd	2
4005.2.a.o.1.4		7	12.11	even	2
7120.2.a.bj.1.6		7	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:	\( N \)	\(=\)	\( 7120 = 2^{4} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7120.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82660 q^{3} +1.00000 q^{5} -0.0498231 q^{7} +4.98967 q^{9} +O(q^{10})\)	\(q+2.82660 q^{3} +1.00000 q^{5} -0.0498231 q^{7} +4.98967 q^{9} -4.45116 q^{11} -2.43229 q^{13} +2.82660 q^{15} -2.48065 q^{17} +5.16842 q^{19} -0.140830 q^{21} +4.99050 q^{23} +1.00000 q^{25} +5.62400 q^{27} +2.59152 q^{29} +7.31064 q^{31} -12.5817 q^{33} -0.0498231 q^{35} +5.13825 q^{37} -6.87512 q^{39} +9.11073 q^{41} +0.543007 q^{43} +4.98967 q^{45} +9.63395 q^{47} -6.99752 q^{49} -7.01180 q^{51} -10.7093 q^{53} -4.45116 q^{55} +14.6091 q^{57} +12.6337 q^{59} -2.47399 q^{61} -0.248601 q^{63} -2.43229 q^{65} +5.36128 q^{67} +14.1061 q^{69} -9.58146 q^{71} +2.86369 q^{73} +2.82660 q^{75} +0.221771 q^{77} -6.62473 q^{79} +0.927782 q^{81} +12.0061 q^{83} -2.48065 q^{85} +7.32520 q^{87} +1.00000 q^{89} +0.121184 q^{91} +20.6643 q^{93} +5.16842 q^{95} +4.82051 q^{97} -22.2098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * q^99

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