LMFDB - Embedded newform 7220.2.a.x.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7220,2,Mod(1,7220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7220, 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("7220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7220 = 2^{2} \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7220.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:	$57.6519902594$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 12x^{7} + 39x^{6} + 21x^{5} - 99x^{4} + 8x^{3} + 48x^{2} - 9x - 3 $ x^9 - 3x^8 - 12x^7 + 39x^6 + 21x^5 - 99x^4 + 8x^3 + 48x^2 - 9x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.79921$ of defining polynomial
Character	$\chi$	$=$	7220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79921 q^{3} -1.00000 q^{5} +0.673555 q^{7} +0.237149 q^{9} +O(q^{10})$	$q+1.79921 q^{3} -1.00000 q^{5} +0.673555 q^{7} +0.237149 q^{9} +6.42552 q^{11} -4.55923 q^{13} -1.79921 q^{15} -1.29423 q^{17} +1.21187 q^{21} -1.35993 q^{23} +1.00000 q^{25} -4.97094 q^{27} -7.23947 q^{29} -5.94481 q^{31} +11.5608 q^{33} -0.673555 q^{35} -11.5704 q^{37} -8.20300 q^{39} +2.67024 q^{41} +9.55239 q^{43} -0.237149 q^{45} -11.7533 q^{47} -6.54632 q^{49} -2.32858 q^{51} +3.58501 q^{53} -6.42552 q^{55} +0.160090 q^{59} +7.24456 q^{61} +0.159733 q^{63} +4.55923 q^{65} -5.46468 q^{67} -2.44679 q^{69} -11.7576 q^{71} -3.70296 q^{73} +1.79921 q^{75} +4.32794 q^{77} +6.42489 q^{79} -9.65521 q^{81} +5.04502 q^{83} +1.29423 q^{85} -13.0253 q^{87} -7.25639 q^{89} -3.07089 q^{91} -10.6960 q^{93} +7.88351 q^{97} +1.52381 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} - 9 q^{5} + 6 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 - 9 * q^5 + 6 * q^9	$ 9 q + 3 q^{3} - 9 q^{5} + 6 q^{9} - 3 q^{13} - 3 q^{15} + 3 q^{17} - 12 q^{21} + 6 q^{23} + 9 q^{25} - 9 q^{29} - 18 q^{31} + 9 q^{33} - 30 q^{37} - 42 q^{39} - 6 q^{41} + 24 q^{43} - 6 q^{45} - 21 q^{47} + 15 q^{49} + 42 q^{51} - 9 q^{53} - 3 q^{59} - 24 q^{61} - 51 q^{63} + 3 q^{65} - 9 q^{69} - 18 q^{71} - 33 q^{73} + 3 q^{75} + 3 q^{79} + 33 q^{81} + 30 q^{83} - 3 q^{85} + 9 q^{87} + 3 q^{89} - 15 q^{91} - 15 q^{93} - 15 q^{97} - 6 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 - 9 * q^5 + 6 * q^9 - 3 * q^13 - 3 * q^15 + 3 * q^17 - 12 * q^21 + 6 * q^23 + 9 * q^25 - 9 * q^29 - 18 * q^31 + 9 * q^33 - 30 * q^37 - 42 * q^39 - 6 * q^41 + 24 * q^43 - 6 * q^45 - 21 * q^47 + 15 * q^49 + 42 * q^51 - 9 * q^53 - 3 * q^59 - 24 * q^61 - 51 * q^63 + 3 * q^65 - 9 * q^69 - 18 * q^71 - 33 * q^73 + 3 * q^75 + 3 * q^79 + 33 * q^81 + 30 * q^83 - 3 * q^85 + 9 * q^87 + 3 * q^89 - 15 * q^91 - 15 * q^93 - 15 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.79921	1.03877	0.519387	−	0.854539i	$-0.326160\pi$
$3$	1.79921	1.03877	0.519387	+	0.854539i	$0.326160\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.673555	0.254580	0.127290	−	0.991866i	$-0.459372\pi$
$7$	0.673555	0.254580	0.127290	+	0.991866i	$0.459372\pi$
$8$	0	0
$9$	0.237149	0.0790498
$10$	0	0
$11$	6.42552	1.93737	0.968683	−	0.248301i	$-0.0798722\pi$
$11$	6.42552	1.93737	0.968683	+	0.248301i	$0.0798722\pi$
$12$	0	0
$13$	−4.55923	−1.26450	−0.632251	−	0.774763i	$-0.717869\pi$
$13$	−4.55923	−1.26450	−0.632251	+	0.774763i	$0.717869\pi$
$14$	0	0
$15$	−1.79921	−0.464554
$16$	0	0
$17$	−1.29423	−0.313896	−0.156948	−	0.987607i	$-0.550166\pi$
$17$	−1.29423	−0.313896	−0.156948	+	0.987607i	$0.550166\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	1.21187	0.264451
$22$	0	0
$23$	−1.35993	−0.283564	−0.141782	−	0.989898i	$-0.545283\pi$
$23$	−1.35993	−0.283564	−0.141782	+	0.989898i	$0.545283\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.97094	−0.956658
$28$	0	0
$29$	−7.23947	−1.34434	−0.672168	−	0.740398i	$-0.734637\pi$
$29$	−7.23947	−1.34434	−0.672168	+	0.740398i	$0.734637\pi$
$30$	0	0
$31$	−5.94481	−1.06772	−0.533860	−	0.845573i	$-0.679259\pi$
$31$	−5.94481	−1.06772	−0.533860	+	0.845573i	$0.679259\pi$
$32$	0	0
$33$	11.5608	2.01248
$34$	0	0
$35$	−0.673555	−0.113852
$36$	0	0
$37$	−11.5704	−1.90216	−0.951079	−	0.308948i	$-0.900023\pi$
$37$	−11.5704	−1.90216	−0.951079	+	0.308948i	$0.900023\pi$
$38$	0	0
$39$	−8.20300	−1.31353
$40$	0	0
$41$	2.67024	0.417022	0.208511	−	0.978020i	$-0.433138\pi$
$41$	2.67024	0.417022	0.208511	+	0.978020i	$0.433138\pi$
$42$	0	0
$43$	9.55239	1.45673	0.728363	−	0.685191i	$-0.240281\pi$
$43$	9.55239	1.45673	0.728363	+	0.685191i	$0.240281\pi$
$44$	0	0
$45$	−0.237149	−0.0353521
$46$	0	0
$47$	−11.7533	−1.71440	−0.857198	−	0.514987i	$-0.827797\pi$
$47$	−11.7533	−1.71440	−0.857198	+	0.514987i	$0.827797\pi$
$48$	0	0
$49$	−6.54632	−0.935189
$50$	0	0
$51$	−2.32858	−0.326067
$52$	0	0
$53$	3.58501	0.492439	0.246220	−	0.969214i	$-0.420812\pi$
$53$	3.58501	0.492439	0.246220	+	0.969214i	$0.420812\pi$
$54$	0	0
$55$	−6.42552	−0.866416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.160090	0.0208420	0.0104210	−	0.999946i	$-0.496683\pi$
$59$	0.160090	0.0208420	0.0104210	+	0.999946i	$0.496683\pi$
$60$	0	0
$61$	7.24456	0.927570	0.463785	−	0.885948i	$-0.346491\pi$
$61$	7.24456	0.927570	0.463785	+	0.885948i	$0.346491\pi$
$62$	0	0
$63$	0.159733	0.0201245
$64$	0	0
$65$	4.55923	0.565503
$66$	0	0
$67$	−5.46468	−0.667617	−0.333808	−	0.942641i	$-0.608334\pi$
$67$	−5.46468	−0.667617	−0.333808	+	0.942641i	$0.608334\pi$
$68$	0	0
$69$	−2.44679	−0.294559
$70$	0	0
$71$	−11.7576	−1.39537	−0.697685	−	0.716405i	$-0.745786\pi$
$71$	−11.7576	−1.39537	−0.697685	+	0.716405i	$0.745786\pi$
$72$	0	0
$73$	−3.70296	−0.433399	−0.216699	−	0.976238i	$-0.569529\pi$
$73$	−3.70296	−0.433399	−0.216699	+	0.976238i	$0.569529\pi$
$74$	0	0
$75$	1.79921	0.207755
$76$	0	0
$77$	4.32794	0.493214
$78$	0	0
$79$	6.42489	0.722857	0.361428	−	0.932400i	$-0.382289\pi$
$79$	6.42489	0.722857	0.361428	+	0.932400i	$0.382289\pi$
$80$	0	0
$81$	−9.65521	−1.07280
$82$	0	0
$83$	5.04502	0.553763	0.276882	−	0.960904i	$-0.410699\pi$
$83$	5.04502	0.553763	0.276882	+	0.960904i	$0.410699\pi$
$84$	0	0
$85$	1.29423	0.140379
$86$	0	0
$87$	−13.0253	−1.39646
$88$	0	0
$89$	−7.25639	−0.769176	−0.384588	−	0.923088i	$-0.625656\pi$
$89$	−7.25639	−0.769176	−0.384588	+	0.923088i	$0.625656\pi$
$90$	0	0
$91$	−3.07089	−0.321917
$92$	0	0
$93$	−10.6960	−1.10912
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.88351	0.800449	0.400224	−	0.916417i	$-0.368932\pi$
$97$	7.88351	0.800449	0.400224	+	0.916417i	$0.368932\pi$
$98$	0	0
$99$	1.52381	0.153148
$100$	0	0
$101$	−13.5880	−1.35206	−0.676029	−	0.736875i	$-0.736301\pi$
$101$	−13.5880	−1.35206	−0.676029	+	0.736875i	$0.736301\pi$
$102$	0	0
$103$	10.7821	1.06239	0.531197	−	0.847249i	$-0.321743\pi$
$103$	10.7821	1.06239	0.531197	+	0.847249i	$0.321743\pi$
$104$	0	0
$105$	−1.21187	−0.118266
$106$	0	0
$107$	−1.11035	−0.107342	−0.0536708	−	0.998559i	$-0.517092\pi$
$107$	−1.11035	−0.107342	−0.0536708	+	0.998559i	$0.517092\pi$
$108$	0	0
$109$	−2.50583	−0.240015	−0.120007	−	0.992773i	$-0.538292\pi$
$109$	−2.50583	−0.240015	−0.120007	+	0.992773i	$0.538292\pi$
$110$	0	0
$111$	−20.8175	−1.97591
$112$	0	0
$113$	14.8298	1.39507	0.697536	−	0.716550i	$-0.254280\pi$
$113$	14.8298	1.39507	0.697536	+	0.716550i	$0.254280\pi$
$114$	0	0
$115$	1.35993	0.126814
$116$	0	0
$117$	−1.08122	−0.0999586
$118$	0	0
$119$	−0.871733	−0.0799116
$120$	0	0
$121$	30.2873	2.75339
$122$	0	0
$123$	4.80432	0.433191
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.85293	0.519363	0.259682	−	0.965694i	$-0.416382\pi$
$127$	5.85293	0.519363	0.259682	+	0.965694i	$0.416382\pi$
$128$	0	0
$129$	17.1867	1.51321
$130$	0	0
$131$	−8.13674	−0.710910	−0.355455	−	0.934693i	$-0.615674\pi$
$131$	−8.13674	−0.710910	−0.355455	+	0.934693i	$0.615674\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.97094	0.427831
$136$	0	0
$137$	4.97827	0.425322	0.212661	−	0.977126i	$-0.431787\pi$
$137$	4.97827	0.425322	0.212661	+	0.977126i	$0.431787\pi$
$138$	0	0
$139$	2.09689	0.177856	0.0889279	−	0.996038i	$-0.471656\pi$
$139$	2.09689	0.177856	0.0889279	+	0.996038i	$0.471656\pi$
$140$	0	0
$141$	−21.1466	−1.78087
$142$	0	0
$143$	−29.2954	−2.44980
$144$	0	0
$145$	7.23947	0.601206
$146$	0	0
$147$	−11.7782	−0.971449
$148$	0	0
$149$	−8.60892	−0.705270	−0.352635	−	0.935761i	$-0.614714\pi$
$149$	−8.60892	−0.705270	−0.352635	+	0.935761i	$0.614714\pi$
$150$	0	0
$151$	−18.3265	−1.49139	−0.745693	−	0.666289i	$-0.767882\pi$
$151$	−18.3265	−1.49139	−0.745693	+	0.666289i	$0.767882\pi$
$152$	0	0
$153$	−0.306925	−0.0248134
$154$	0	0
$155$	5.94481	0.477499
$156$	0	0
$157$	13.3496	1.06541	0.532707	−	0.846300i	$-0.321175\pi$
$157$	13.3496	1.06541	0.532707	+	0.846300i	$0.321175\pi$
$158$	0	0
$159$	6.45018	0.511533
$160$	0	0
$161$	−0.915985	−0.0721898
$162$	0	0
$163$	−21.5315	−1.68648	−0.843240	−	0.537538i	$-0.819355\pi$
$163$	−21.5315	−1.68648	−0.843240	+	0.537538i	$0.819355\pi$
$164$	0	0
$165$	−11.5608	−0.900010
$166$	0	0
$167$	12.1757	0.942185	0.471093	−	0.882084i	$-0.343860\pi$
$167$	12.1757	0.942185	0.471093	+	0.882084i	$0.343860\pi$
$168$	0	0
$169$	7.78656	0.598966
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.16129	0.164320	0.0821600	−	0.996619i	$-0.473818\pi$
$173$	2.16129	0.164320	0.0821600	+	0.996619i	$0.473818\pi$
$174$	0	0
$175$	0.673555	0.0509160
$176$	0	0
$177$	0.288035	0.0216501
$178$	0	0
$179$	−18.1478	−1.35643	−0.678215	−	0.734864i	$-0.737246\pi$
$179$	−18.1478	−1.35643	−0.678215	+	0.734864i	$0.737246\pi$
$180$	0	0
$181$	7.47439	0.555567	0.277784	−	0.960644i	$-0.410400\pi$
$181$	7.47439	0.555567	0.277784	+	0.960644i	$0.410400\pi$
$182$	0	0
$183$	13.0345	0.963535
$184$	0	0
$185$	11.5704	0.850671
$186$	0	0
$187$	−8.31607	−0.608132
$188$	0	0
$189$	−3.34820	−0.243546
$190$	0	0
$191$	−21.9662	−1.58942	−0.794710	−	0.606990i	$-0.792377\pi$
$191$	−21.9662	−1.58942	−0.794710	+	0.606990i	$0.792377\pi$
$192$	0	0
$193$	11.9457	0.859873	0.429937	−	0.902859i	$-0.358536\pi$
$193$	11.9457	0.859873	0.429937	+	0.902859i	$0.358536\pi$
$194$	0	0
$195$	8.20300	0.587429
$196$	0	0
$197$	−7.75300	−0.552379	−0.276189	−	0.961103i	$-0.589072\pi$
$197$	−7.75300	−0.552379	−0.276189	+	0.961103i	$0.589072\pi$
$198$	0	0
$199$	−9.21437	−0.653189	−0.326595	−	0.945165i	$-0.605901\pi$
$199$	−9.21437	−0.653189	−0.326595	+	0.945165i	$0.605901\pi$
$200$	0	0
$201$	−9.83209	−0.693502
$202$	0	0
$203$	−4.87618	−0.342241
$204$	0	0
$205$	−2.67024	−0.186498
$206$	0	0
$207$	−0.322506	−0.0224157
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.6419	−1.42105	−0.710523	−	0.703674i	$-0.751542\pi$
$211$	−20.6419	−1.42105	−0.710523	+	0.703674i	$0.751542\pi$
$212$	0	0
$213$	−21.1544	−1.44947
$214$	0	0
$215$	−9.55239	−0.651468
$216$	0	0
$217$	−4.00416	−0.271820
$218$	0	0
$219$	−6.66240	−0.450203
$220$	0	0
$221$	5.90067	0.396922
$222$	0	0
$223$	2.36069	0.158084	0.0790418	−	0.996871i	$-0.474814\pi$
$223$	2.36069	0.158084	0.0790418	+	0.996871i	$0.474814\pi$
$224$	0	0
$225$	0.237149	0.0158100
$226$	0	0
$227$	15.3448	1.01847	0.509236	−	0.860627i	$-0.329928\pi$
$227$	15.3448	1.01847	0.509236	+	0.860627i	$0.329928\pi$
$228$	0	0
$229$	9.54861	0.630990	0.315495	−	0.948927i	$-0.397829\pi$
$229$	9.54861	0.630990	0.315495	+	0.948927i	$0.397829\pi$
$230$	0	0
$231$	7.78686	0.512338
$232$	0	0
$233$	25.9077	1.69727	0.848634	−	0.528981i	$-0.177426\pi$
$233$	25.9077	1.69727	0.848634	+	0.528981i	$0.177426\pi$
$234$	0	0
$235$	11.7533	0.766701
$236$	0	0
$237$	11.5597	0.750884
$238$	0	0
$239$	14.8462	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$239$	14.8462	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$240$	0	0
$241$	−19.2948	−1.24288	−0.621442	−	0.783460i	$-0.713453\pi$
$241$	−19.2948	−1.24288	−0.621442	+	0.783460i	$0.713453\pi$
$242$	0	0
$243$	−2.45890	−0.157738
$244$	0	0
$245$	6.54632	0.418229
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.07704	0.575234
$250$	0	0
$251$	19.0239	1.20078	0.600388	−	0.799709i	$-0.295013\pi$
$251$	19.0239	1.20078	0.600388	+	0.799709i	$0.295013\pi$
$252$	0	0
$253$	−8.73823	−0.549368
$254$	0	0
$255$	2.32858	0.145822
$256$	0	0
$257$	−11.8474	−0.739021	−0.369511	−	0.929226i	$-0.620475\pi$
$257$	−11.8474	−0.739021	−0.369511	+	0.929226i	$0.620475\pi$
$258$	0	0
$259$	−7.79328	−0.484251
$260$	0	0
$261$	−1.71684	−0.106270
$262$	0	0
$263$	4.06243	0.250500	0.125250	−	0.992125i	$-0.460027\pi$
$263$	4.06243	0.250500	0.125250	+	0.992125i	$0.460027\pi$
$264$	0	0
$265$	−3.58501	−0.220225
$266$	0	0
$267$	−13.0558	−0.798999
$268$	0	0
$269$	−5.08283	−0.309905	−0.154953	−	0.987922i	$-0.549523\pi$
$269$	−5.08283	−0.309905	−0.154953	+	0.987922i	$0.549523\pi$
$270$	0	0
$271$	−14.9515	−0.908238	−0.454119	−	0.890941i	$-0.650046\pi$
$271$	−14.9515	−0.908238	−0.454119	+	0.890941i	$0.650046\pi$
$272$	0	0
$273$	−5.52517	−0.334399
$274$	0	0
$275$	6.42552	0.387473
$276$	0	0
$277$	12.0950	0.726721	0.363361	−	0.931649i	$-0.381629\pi$
$277$	12.0950	0.726721	0.363361	+	0.931649i	$0.381629\pi$
$278$	0	0
$279$	−1.40981	−0.0844030
$280$	0	0
$281$	−21.8102	−1.30109	−0.650543	−	0.759469i	$-0.725459\pi$
$281$	−21.8102	−1.30109	−0.650543	+	0.759469i	$0.725459\pi$
$282$	0	0
$283$	−16.0147	−0.951972	−0.475986	−	0.879453i	$-0.657909\pi$
$283$	−16.0147	−0.951972	−0.475986	+	0.879453i	$0.657909\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.79856	0.106165
$288$	0	0
$289$	−15.3250	−0.901469
$290$	0	0
$291$	14.1841	0.831485
$292$	0	0
$293$	−6.31847	−0.369129	−0.184565	−	0.982820i	$-0.559087\pi$
$293$	−6.31847	−0.369129	−0.184565	+	0.982820i	$0.559087\pi$
$294$	0	0
$295$	−0.160090	−0.00932081
$296$	0	0
$297$	−31.9409	−1.85340
$298$	0	0
$299$	6.20022	0.358568
$300$	0	0
$301$	6.43406	0.370853
$302$	0	0
$303$	−24.4477	−1.40448
$304$	0	0
$305$	−7.24456	−0.414822
$306$	0	0
$307$	0.802552	0.0458041	0.0229020	−	0.999738i	$-0.492709\pi$
$307$	0.802552	0.0458041	0.0229020	+	0.999738i	$0.492709\pi$
$308$	0	0
$309$	19.3993	1.10359
$310$	0	0
$311$	14.0438	0.796353	0.398176	−	0.917309i	$-0.369643\pi$
$311$	14.0438	0.796353	0.398176	+	0.917309i	$0.369643\pi$
$312$	0	0
$313$	21.8835	1.23693	0.618464	−	0.785814i	$-0.287755\pi$
$313$	21.8835	1.23693	0.618464	+	0.785814i	$0.287755\pi$
$314$	0	0
$315$	−0.159733	−0.00899994
$316$	0	0
$317$	−31.2029	−1.75253	−0.876264	−	0.481832i	$-0.839972\pi$
$317$	−31.2029	−1.75253	−0.876264	+	0.481832i	$0.839972\pi$
$318$	0	0
$319$	−46.5174	−2.60447
$320$	0	0
$321$	−1.99775	−0.111504
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.55923	−0.252900
$326$	0	0
$327$	−4.50850	−0.249321
$328$	0	0
$329$	−7.91650	−0.436451
$330$	0	0
$331$	−11.7732	−0.647111	−0.323556	−	0.946209i	$-0.604878\pi$
$331$	−11.7732	−0.647111	−0.323556	+	0.946209i	$0.604878\pi$
$332$	0	0
$333$	−2.74391	−0.150365
$334$	0	0
$335$	5.46468	0.298567
$336$	0	0
$337$	−9.49124	−0.517021	−0.258510	−	0.966008i	$-0.583232\pi$
$337$	−9.49124	−0.517021	−0.258510	+	0.966008i	$0.583232\pi$
$338$	0	0
$339$	26.6819	1.44916
$340$	0	0
$341$	−38.1985	−2.06856
$342$	0	0
$343$	−9.12419	−0.492660
$344$	0	0
$345$	2.44679	0.131731
$346$	0	0
$347$	11.7622	0.631427	0.315714	−	0.948855i	$-0.397756\pi$
$347$	11.7622	0.631427	0.315714	+	0.948855i	$0.397756\pi$
$348$	0	0
$349$	12.4240	0.665042	0.332521	−	0.943096i	$-0.392101\pi$
$349$	12.4240	0.665042	0.332521	+	0.943096i	$0.392101\pi$
$350$	0	0
$351$	22.6637	1.20970
$352$	0	0
$353$	11.4645	0.610192	0.305096	−	0.952322i	$-0.401311\pi$
$353$	11.4645	0.610192	0.305096	+	0.952322i	$0.401311\pi$
$354$	0	0
$355$	11.7576	0.624028
$356$	0	0
$357$	−1.56843	−0.0830100
$358$	0	0
$359$	−30.9829	−1.63522	−0.817608	−	0.575775i	$-0.804700\pi$
$359$	−30.9829	−1.63522	−0.817608	+	0.575775i	$0.804700\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	54.4931	2.86014
$364$	0	0
$365$	3.70296	0.193822
$366$	0	0
$367$	−4.50371	−0.235092	−0.117546	−	0.993067i	$-0.537503\pi$
$367$	−4.50371	−0.235092	−0.117546	+	0.993067i	$0.537503\pi$
$368$	0	0
$369$	0.633246	0.0329655
$370$	0	0
$371$	2.41470	0.125365
$372$	0	0
$373$	19.1223	0.990116	0.495058	−	0.868860i	$-0.335147\pi$
$373$	19.1223	0.990116	0.495058	+	0.868860i	$0.335147\pi$
$374$	0	0
$375$	−1.79921	−0.0929107
$376$	0	0
$377$	33.0064	1.69992
$378$	0	0
$379$	1.21152	0.0622314	0.0311157	−	0.999516i	$-0.490094\pi$
$379$	1.21152	0.0622314	0.0311157	+	0.999516i	$0.490094\pi$
$380$	0	0
$381$	10.5306	0.539501
$382$	0	0
$383$	−6.13901	−0.313689	−0.156844	−	0.987623i	$-0.550132\pi$
$383$	−6.13901	−0.313689	−0.156844	+	0.987623i	$0.550132\pi$
$384$	0	0
$385$	−4.32794	−0.220572
$386$	0	0
$387$	2.26534	0.115154
$388$	0	0
$389$	−0.119150	−0.00604113	−0.00302056	−	0.999995i	$-0.500961\pi$
$389$	−0.119150	−0.00604113	−0.00302056	+	0.999995i	$0.500961\pi$
$390$	0	0
$391$	1.76005	0.0890097
$392$	0	0
$393$	−14.6397	−0.738474
$394$	0	0
$395$	−6.42489	−0.323271
$396$	0	0
$397$	−2.75490	−0.138264	−0.0691322	−	0.997608i	$-0.522023\pi$
$397$	−2.75490	−0.138264	−0.0691322	+	0.997608i	$0.522023\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.8073	−1.08900	−0.544502	−	0.838759i	$-0.683281\pi$
$401$	−21.8073	−1.08900	−0.544502	+	0.838759i	$0.683281\pi$
$402$	0	0
$403$	27.1038	1.35013
$404$	0	0
$405$	9.65521	0.479771
$406$	0	0
$407$	−74.3456	−3.68518
$408$	0	0
$409$	−15.5732	−0.770045	−0.385022	−	0.922907i	$-0.625806\pi$
$409$	−15.5732	−0.770045	−0.385022	+	0.922907i	$0.625806\pi$
$410$	0	0
$411$	8.95694	0.441813
$412$	0	0
$413$	0.107829	0.00530594
$414$	0	0
$415$	−5.04502	−0.247650
$416$	0	0
$417$	3.77274	0.184752
$418$	0	0
$419$	−15.2596	−0.745479	−0.372739	−	0.927936i	$-0.621581\pi$
$419$	−15.2596	−0.745479	−0.372739	+	0.927936i	$0.621581\pi$
$420$	0	0
$421$	−8.20065	−0.399675	−0.199838	−	0.979829i	$-0.564041\pi$
$421$	−8.20065	−0.399675	−0.199838	+	0.979829i	$0.564041\pi$
$422$	0	0
$423$	−2.78729	−0.135523
$424$	0	0
$425$	−1.29423	−0.0627792
$426$	0	0
$427$	4.87961	0.236141
$428$	0	0
$429$	−52.7085	−2.54479
$430$	0	0
$431$	8.32667	0.401082	0.200541	−	0.979685i	$-0.435730\pi$
$431$	8.32667	0.401082	0.200541	+	0.979685i	$0.435730\pi$
$432$	0	0
$433$	−33.9569	−1.63187	−0.815933	−	0.578146i	$-0.803776\pi$
$433$	−33.9569	−1.63187	−0.815933	+	0.578146i	$0.803776\pi$
$434$	0	0
$435$	13.0253	0.624516
$436$	0	0
$437$	0	0
$438$	0	0
$439$	12.5160	0.597356	0.298678	−	0.954354i	$-0.403454\pi$
$439$	12.5160	0.597356	0.298678	+	0.954354i	$0.403454\pi$
$440$	0	0
$441$	−1.55246	−0.0739265
$442$	0	0
$443$	−23.3168	−1.10781	−0.553907	−	0.832579i	$-0.686864\pi$
$443$	−23.3168	−1.10781	−0.553907	+	0.832579i	$0.686864\pi$
$444$	0	0
$445$	7.25639	0.343986
$446$	0	0
$447$	−15.4892	−0.732616
$448$	0	0
$449$	21.5239	1.01577	0.507887	−	0.861424i	$-0.330427\pi$
$449$	21.5239	1.01577	0.507887	+	0.861424i	$0.330427\pi$
$450$	0	0
$451$	17.1577	0.807924
$452$	0	0
$453$	−32.9731	−1.54921
$454$	0	0
$455$	3.07089	0.143966
$456$	0	0
$457$	−30.0765	−1.40692	−0.703459	−	0.710735i	$-0.748362\pi$
$457$	−30.0765	−1.40692	−0.703459	+	0.710735i	$0.748362\pi$
$458$	0	0
$459$	6.43353	0.300291
$460$	0	0
$461$	19.0513	0.887308	0.443654	−	0.896198i	$-0.353682\pi$
$461$	19.0513	0.887308	0.443654	+	0.896198i	$0.353682\pi$
$462$	0	0
$463$	−20.5551	−0.955275	−0.477637	−	0.878557i	$-0.658507\pi$
$463$	−20.5551	−0.955275	−0.477637	+	0.878557i	$0.658507\pi$
$464$	0	0
$465$	10.6960	0.496013
$466$	0	0
$467$	−40.5709	−1.87740	−0.938699	−	0.344738i	$-0.887968\pi$
$467$	−40.5709	−1.87740	−0.938699	+	0.344738i	$0.887968\pi$
$468$	0	0
$469$	−3.68076	−0.169962
$470$	0	0
$471$	24.0187	1.10672
$472$	0	0
$473$	61.3790	2.82221
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.850183	0.0389272
$478$	0	0
$479$	12.7780	0.583843	0.291921	−	0.956442i	$-0.405705\pi$
$479$	12.7780	0.583843	0.291921	+	0.956442i	$0.405705\pi$
$480$	0	0
$481$	52.7520	2.40528
$482$	0	0
$483$	−1.64805	−0.0749888
$484$	0	0
$485$	−7.88351	−0.357972
$486$	0	0
$487$	1.50978	0.0684147	0.0342074	−	0.999415i	$-0.489109\pi$
$487$	1.50978	0.0684147	0.0342074	+	0.999415i	$0.489109\pi$
$488$	0	0
$489$	−38.7397	−1.75187
$490$	0	0
$491$	29.4119	1.32734	0.663669	−	0.748026i	$-0.268998\pi$
$491$	29.4119	1.32734	0.663669	+	0.748026i	$0.268998\pi$
$492$	0	0
$493$	9.36952	0.421982
$494$	0	0
$495$	−1.52381	−0.0684900
$496$	0	0
$497$	−7.91939	−0.355233
$498$	0	0
$499$	17.9498	0.803545	0.401773	−	0.915740i	$-0.368394\pi$
$499$	17.9498	0.803545	0.401773	+	0.915740i	$0.368394\pi$
$500$	0	0
$501$	21.9066	0.978717
$502$	0	0
$503$	7.05818	0.314709	0.157354	−	0.987542i	$-0.449703\pi$
$503$	7.05818	0.314709	0.157354	+	0.987542i	$0.449703\pi$
$504$	0	0
$505$	13.5880	0.604659
$506$	0	0
$507$	14.0096	0.622190
$508$	0	0
$509$	6.93064	0.307195	0.153598	−	0.988133i	$-0.450914\pi$
$509$	6.93064	0.307195	0.153598	+	0.988133i	$0.450914\pi$
$510$	0	0
$511$	−2.49415	−0.110335
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.7821	−0.475117
$516$	0	0
$517$	−75.5210	−3.32141
$518$	0	0
$519$	3.88861	0.170691
$520$	0	0
$521$	36.7103	1.60831	0.804155	−	0.594420i	$-0.202618\pi$
$521$	36.7103	1.60831	0.804155	+	0.594420i	$0.202618\pi$
$522$	0	0
$523$	4.08461	0.178607	0.0893037	−	0.996004i	$-0.471536\pi$
$523$	4.08461	0.178607	0.0893037	+	0.996004i	$0.471536\pi$
$524$	0	0
$525$	1.21187	0.0528901
$526$	0	0
$527$	7.69393	0.335153
$528$	0	0
$529$	−21.1506	−0.919591
$530$	0	0
$531$	0.0379653	0.00164755
$532$	0	0
$533$	−12.1742	−0.527325
$534$	0	0
$535$	1.11035	0.0480047
$536$	0	0
$537$	−32.6516	−1.40902
$538$	0	0
$539$	−42.0635	−1.81180
$540$	0	0
$541$	34.8685	1.49911	0.749557	−	0.661940i	$-0.230267\pi$
$541$	34.8685	1.49911	0.749557	+	0.661940i	$0.230267\pi$
$542$	0	0
$543$	13.4480	0.577108
$544$	0	0
$545$	2.50583	0.107338
$546$	0	0
$547$	15.8928	0.679527	0.339763	−	0.940511i	$-0.389653\pi$
$547$	15.8928	0.679527	0.339763	+	0.940511i	$0.389653\pi$
$548$	0	0
$549$	1.71804	0.0733242
$550$	0	0
$551$	0	0
$552$	0	0
$553$	4.32752	0.184025
$554$	0	0
$555$	20.8175	0.883654
$556$	0	0
$557$	−10.5004	−0.444915	−0.222458	−	0.974942i	$-0.571408\pi$
$557$	−10.5004	−0.444915	−0.222458	+	0.974942i	$0.571408\pi$
$558$	0	0
$559$	−43.5515	−1.84203
$560$	0	0
$561$	−14.9623	−0.631711
$562$	0	0
$563$	−6.97702	−0.294046	−0.147023	−	0.989133i	$-0.546969\pi$
$563$	−6.97702	−0.294046	−0.147023	+	0.989133i	$0.546969\pi$
$564$	0	0
$565$	−14.8298	−0.623895
$566$	0	0
$567$	−6.50331	−0.273113
$568$	0	0
$569$	25.5455	1.07092	0.535461	−	0.844560i	$-0.320138\pi$
$569$	25.5455	1.07092	0.535461	+	0.844560i	$0.320138\pi$
$570$	0	0
$571$	−1.23193	−0.0515548	−0.0257774	−	0.999668i	$-0.508206\pi$
$571$	−1.23193	−0.0515548	−0.0257774	+	0.999668i	$0.508206\pi$
$572$	0	0
$573$	−39.5218	−1.65105
$574$	0	0
$575$	−1.35993	−0.0567129
$576$	0	0
$577$	5.06779	0.210975	0.105487	−	0.994421i	$-0.466360\pi$
$577$	5.06779	0.210975	0.105487	+	0.994421i	$0.466360\pi$
$578$	0	0
$579$	21.4929	0.893213
$580$	0	0
$581$	3.39810	0.140977
$582$	0	0
$583$	23.0355	0.954035
$584$	0	0
$585$	1.08122	0.0447029
$586$	0	0
$587$	−24.6014	−1.01541	−0.507703	−	0.861532i	$-0.669505\pi$
$587$	−24.6014	−1.01541	−0.507703	+	0.861532i	$0.669505\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−13.9493	−0.573796
$592$	0	0
$593$	21.7070	0.891400	0.445700	−	0.895182i	$-0.352955\pi$
$593$	21.7070	0.891400	0.445700	+	0.895182i	$0.352955\pi$
$594$	0	0
$595$	0.871733	0.0357376
$596$	0	0
$597$	−16.5786	−0.678516
$598$	0	0
$599$	37.5492	1.53422	0.767109	−	0.641516i	$-0.221694\pi$
$599$	37.5492	1.53422	0.767109	+	0.641516i	$0.221694\pi$
$600$	0	0
$601$	−18.6326	−0.760038	−0.380019	−	0.924979i	$-0.624083\pi$
$601$	−18.6326	−0.760038	−0.380019	+	0.924979i	$0.624083\pi$
$602$	0	0
$603$	−1.29594	−0.0527750
$604$	0	0
$605$	−30.2873	−1.23135
$606$	0	0
$607$	43.4925	1.76531	0.882653	−	0.470025i	$-0.155755\pi$
$607$	43.4925	1.76531	0.882653	+	0.470025i	$0.155755\pi$
$608$	0	0
$609$	−8.77327	−0.355511
$610$	0	0
$611$	53.5860	2.16786
$612$	0	0
$613$	48.4303	1.95608	0.978040	−	0.208419i	$-0.0668317\pi$
$613$	48.4303	1.95608	0.978040	+	0.208419i	$0.0668317\pi$
$614$	0	0
$615$	−4.80432	−0.193729
$616$	0	0
$617$	−23.2076	−0.934303	−0.467151	−	0.884177i	$-0.654720\pi$
$617$	−23.2076	−0.934303	−0.467151	+	0.884177i	$0.654720\pi$
$618$	0	0
$619$	−4.66322	−0.187431	−0.0937154	−	0.995599i	$-0.529874\pi$
$619$	−4.66322	−0.187431	−0.0937154	+	0.995599i	$0.529874\pi$
$620$	0	0
$621$	6.76012	0.271274
$622$	0	0
$623$	−4.88758	−0.195817
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.9747	0.597080
$630$	0	0
$631$	9.04130	0.359928	0.179964	−	0.983673i	$-0.442402\pi$
$631$	9.04130	0.359928	0.179964	+	0.983673i	$0.442402\pi$
$632$	0	0
$633$	−37.1391	−1.47615
$634$	0	0
$635$	−5.85293	−0.232266
$636$	0	0
$637$	29.8462	1.18255
$638$	0	0
$639$	−2.78831	−0.110304
$640$	0	0
$641$	−6.38002	−0.251996	−0.125998	−	0.992031i	$-0.540213\pi$
$641$	−6.38002	−0.251996	−0.125998	+	0.992031i	$0.540213\pi$
$642$	0	0
$643$	−41.9423	−1.65404	−0.827021	−	0.562171i	$-0.809966\pi$
$643$	−41.9423	−1.65404	−0.827021	+	0.562171i	$0.809966\pi$
$644$	0	0
$645$	−17.1867	−0.676727
$646$	0	0
$647$	28.4481	1.11841	0.559205	−	0.829030i	$-0.311107\pi$
$647$	28.4481	1.11841	0.559205	+	0.829030i	$0.311107\pi$
$648$	0	0
$649$	1.02866	0.0403785
$650$	0	0
$651$	−7.20431	−0.282359
$652$	0	0
$653$	29.1382	1.14027	0.570134	−	0.821552i	$-0.306891\pi$
$653$	29.1382	1.14027	0.570134	+	0.821552i	$0.306891\pi$
$654$	0	0
$655$	8.13674	0.317929
$656$	0	0
$657$	−0.878155	−0.0342601
$658$	0	0
$659$	10.2706	0.400085	0.200042	−	0.979787i	$-0.435892\pi$
$659$	10.2706	0.400085	0.200042	+	0.979787i	$0.435892\pi$
$660$	0	0
$661$	−17.5414	−0.682281	−0.341141	−	0.940012i	$-0.610813\pi$
$661$	−17.5414	−0.682281	−0.341141	+	0.940012i	$0.610813\pi$
$662$	0	0
$663$	10.6165	0.412312
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.84515	0.381206
$668$	0	0
$669$	4.24738	0.164213
$670$	0	0
$671$	46.5500	1.79704
$672$	0	0
$673$	−37.6114	−1.44981	−0.724906	−	0.688847i	$-0.758117\pi$
$673$	−37.6114	−1.44981	−0.724906	+	0.688847i	$0.758117\pi$
$674$	0	0
$675$	−4.97094	−0.191332
$676$	0	0
$677$	5.47921	0.210583	0.105291	−	0.994441i	$-0.466422\pi$
$677$	5.47921	0.210583	0.105291	+	0.994441i	$0.466422\pi$
$678$	0	0
$679$	5.30998	0.203778
$680$	0	0
$681$	27.6086	1.05796
$682$	0	0
$683$	7.75347	0.296678	0.148339	−	0.988937i	$-0.452607\pi$
$683$	7.75347	0.296678	0.148339	+	0.988937i	$0.452607\pi$
$684$	0	0
$685$	−4.97827	−0.190210
$686$	0	0
$687$	17.1799	0.655455
$688$	0	0
$689$	−16.3449	−0.622691
$690$	0	0
$691$	45.7436	1.74017	0.870084	−	0.492904i	$-0.164065\pi$
$691$	45.7436	1.74017	0.870084	+	0.492904i	$0.164065\pi$
$692$	0	0
$693$	1.02637	0.0389885
$694$	0	0
$695$	−2.09689	−0.0795396
$696$	0	0
$697$	−3.45590	−0.130902
$698$	0	0
$699$	46.6133	1.76308
$700$	0	0
$701$	−9.76421	−0.368789	−0.184394	−	0.982852i	$-0.559032\pi$
$701$	−9.76421	−0.368789	−0.184394	+	0.982852i	$0.559032\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	21.1466	0.796429
$706$	0	0
$707$	−9.15228	−0.344207
$708$	0	0
$709$	−31.6671	−1.18928	−0.594641	−	0.803991i	$-0.702706\pi$
$709$	−31.6671	−1.18928	−0.594641	+	0.803991i	$0.702706\pi$
$710$	0	0
$711$	1.52366	0.0571417
$712$	0	0
$713$	8.08451	0.302767
$714$	0	0
$715$	29.2954	1.09559
$716$	0	0
$717$	26.7115	0.997559
$718$	0	0
$719$	32.6317	1.21696	0.608479	−	0.793570i	$-0.291780\pi$
$719$	32.6317	1.21696	0.608479	+	0.793570i	$0.291780\pi$
$720$	0	0
$721$	7.26235	0.270464
$722$	0	0
$723$	−34.7153	−1.29108
$724$	0	0
$725$	−7.23947	−0.268867
$726$	0	0
$727$	33.1888	1.23090	0.615452	−	0.788174i	$-0.288973\pi$
$727$	33.1888	1.23090	0.615452	+	0.788174i	$0.288973\pi$
$728$	0	0
$729$	24.5416	0.908946
$730$	0	0
$731$	−12.3630	−0.457261
$732$	0	0
$733$	−7.20880	−0.266263	−0.133132	−	0.991098i	$-0.542503\pi$
$733$	−7.20880	−0.266263	−0.133132	+	0.991098i	$0.542503\pi$
$734$	0	0
$735$	11.7782	0.434445
$736$	0	0
$737$	−35.1134	−1.29342
$738$	0	0
$739$	24.8227	0.913119	0.456559	−	0.889693i	$-0.349082\pi$
$739$	24.8227	0.913119	0.456559	+	0.889693i	$0.349082\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.4377	1.07996	0.539982	−	0.841676i	$-0.318431\pi$
$743$	29.4377	1.07996	0.539982	+	0.841676i	$0.318431\pi$
$744$	0	0
$745$	8.60892	0.315407
$746$	0	0
$747$	1.19642	0.0437749
$748$	0	0
$749$	−0.747882	−0.0273270
$750$	0	0
$751$	−32.6676	−1.19206	−0.596028	−	0.802964i	$-0.703255\pi$
$751$	−32.6676	−1.19206	−0.596028	+	0.802964i	$0.703255\pi$
$752$	0	0
$753$	34.2279	1.24733
$754$	0	0
$755$	18.3265	0.666968
$756$	0	0
$757$	51.8379	1.88408	0.942040	−	0.335499i	$-0.108905\pi$
$757$	51.8379	1.88408	0.942040	+	0.335499i	$0.108905\pi$
$758$	0	0
$759$	−15.7219	−0.570669
$760$	0	0
$761$	29.7963	1.08011	0.540057	−	0.841628i	$-0.318403\pi$
$761$	29.7963	1.08011	0.540057	+	0.841628i	$0.318403\pi$
$762$	0	0
$763$	−1.68781	−0.0611029
$764$	0	0
$765$	0.306925	0.0110969
$766$	0	0
$767$	−0.729887	−0.0263547
$768$	0	0
$769$	−12.4635	−0.449447	−0.224723	−	0.974423i	$-0.572148\pi$
$769$	−12.4635	−0.449447	−0.224723	+	0.974423i	$0.572148\pi$
$770$	0	0
$771$	−21.3160	−0.767676
$772$	0	0
$773$	29.2590	1.05237	0.526187	−	0.850369i	$-0.323621\pi$
$773$	29.2590	1.05237	0.526187	+	0.850369i	$0.323621\pi$
$774$	0	0
$775$	−5.94481	−0.213544
$776$	0	0
$777$	−14.0217	−0.503027
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−75.5486	−2.70334
$782$	0	0
$783$	35.9870	1.28607
$784$	0	0
$785$	−13.3496	−0.476468
$786$	0	0
$787$	32.2439	1.14937	0.574686	−	0.818374i	$-0.305124\pi$
$787$	32.2439	1.14937	0.574686	+	0.818374i	$0.305124\pi$
$788$	0	0
$789$	7.30915	0.260213
$790$	0	0
$791$	9.98870	0.355157
$792$	0	0
$793$	−33.0296	−1.17291
$794$	0	0
$795$	−6.45018	−0.228764
$796$	0	0
$797$	−2.58966	−0.0917304	−0.0458652	−	0.998948i	$-0.514604\pi$
$797$	−2.58966	−0.0917304	−0.0458652	+	0.998948i	$0.514604\pi$
$798$	0	0
$799$	15.2114	0.538142
$800$	0	0
$801$	−1.72085	−0.0608032
$802$	0	0
$803$	−23.7934	−0.839652
$804$	0	0
$805$	0.915985	0.0322842
$806$	0	0
$807$	−9.14506	−0.321921
$808$	0	0
$809$	20.6174	0.724871	0.362435	−	0.932009i	$-0.381945\pi$
$809$	20.6174	0.724871	0.362435	+	0.932009i	$0.381945\pi$
$810$	0	0
$811$	−12.1457	−0.426494	−0.213247	−	0.976998i	$-0.568404\pi$
$811$	−12.1457	−0.426494	−0.213247	+	0.976998i	$0.568404\pi$
$812$	0	0
$813$	−26.9008	−0.943453
$814$	0	0
$815$	21.5315	0.754217
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−0.728260	−0.0254475
$820$	0	0
$821$	3.98204	0.138974	0.0694871	−	0.997583i	$-0.477864\pi$
$821$	3.98204	0.138974	0.0694871	+	0.997583i	$0.477864\pi$
$822$	0	0
$823$	1.31539	0.0458517	0.0229259	−	0.999737i	$-0.492702\pi$
$823$	1.31539	0.0458517	0.0229259	+	0.999737i	$0.492702\pi$
$824$	0	0
$825$	11.5608	0.402497
$826$	0	0
$827$	3.06034	0.106418	0.0532092	−	0.998583i	$-0.483055\pi$
$827$	3.06034	0.106418	0.0532092	+	0.998583i	$0.483055\pi$
$828$	0	0
$829$	−51.2943	−1.78153	−0.890763	−	0.454468i	$-0.849829\pi$
$829$	−51.2943	−1.78153	−0.890763	+	0.454468i	$0.849829\pi$
$830$	0	0
$831$	21.7615	0.754898
$832$	0	0
$833$	8.47243	0.293552
$834$	0	0
$835$	−12.1757	−0.421358
$836$	0	0
$837$	29.5513	1.02144
$838$	0	0
$839$	−13.7195	−0.473648	−0.236824	−	0.971553i	$-0.576107\pi$
$839$	−13.7195	−0.473648	−0.236824	+	0.971553i	$0.576107\pi$
$840$	0	0
$841$	23.4100	0.807241
$842$	0	0
$843$	−39.2411	−1.35153
$844$	0	0
$845$	−7.78656	−0.267866
$846$	0	0
$847$	20.4001	0.700957
$848$	0	0
$849$	−28.8137	−0.988883
$850$	0	0
$851$	15.7349	0.539384
$852$	0	0
$853$	49.4406	1.69281	0.846407	−	0.532537i	$-0.178761\pi$
$853$	49.4406	1.69281	0.846407	+	0.532537i	$0.178761\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.1197	0.379840	0.189920	−	0.981800i	$-0.439177\pi$
$857$	11.1197	0.379840	0.189920	+	0.981800i	$0.439177\pi$
$858$	0	0
$859$	−38.5236	−1.31441	−0.657204	−	0.753712i	$-0.728261\pi$
$859$	−38.5236	−1.31441	−0.657204	+	0.753712i	$0.728261\pi$
$860$	0	0
$861$	3.23597	0.110282
$862$	0	0
$863$	7.85921	0.267531	0.133765	−	0.991013i	$-0.457293\pi$
$863$	7.85921	0.267531	0.133765	+	0.991013i	$0.457293\pi$
$864$	0	0
$865$	−2.16129	−0.0734861
$866$	0	0
$867$	−27.5728	−0.936422
$868$	0	0
$869$	41.2832	1.40044
$870$	0	0
$871$	24.9147	0.844203
$872$	0	0
$873$	1.86957	0.0632753
$874$	0	0
$875$	−0.673555	−0.0227703
$876$	0	0
$877$	−40.4609	−1.36627	−0.683134	−	0.730293i	$-0.739383\pi$
$877$	−40.4609	−1.36627	−0.683134	+	0.730293i	$0.739383\pi$
$878$	0	0
$879$	−11.3682	−0.383441
$880$	0	0
$881$	−0.497997	−0.0167780	−0.00838898	−	0.999965i	$-0.502670\pi$
$881$	−0.497997	−0.0167780	−0.00838898	+	0.999965i	$0.502670\pi$
$882$	0	0
$883$	−44.5976	−1.50083	−0.750415	−	0.660967i	$-0.770146\pi$
$883$	−44.5976	−1.50083	−0.750415	+	0.660967i	$0.770146\pi$
$884$	0	0
$885$	−0.288035	−0.00968220
$886$	0	0
$887$	31.4772	1.05690	0.528451	−	0.848964i	$-0.322773\pi$
$887$	31.4772	1.05690	0.528451	+	0.848964i	$0.322773\pi$
$888$	0	0
$889$	3.94227	0.132219
$890$	0	0
$891$	−62.0397	−2.07841
$892$	0	0
$893$	0	0
$894$	0	0
$895$	18.1478	0.606614
$896$	0	0
$897$	11.1555	0.372471
$898$	0	0
$899$	43.0373	1.43537
$900$	0	0
$901$	−4.63982	−0.154575
$902$	0	0
$903$	11.5762	0.385232
$904$	0	0
$905$	−7.47439	−0.248457
$906$	0	0
$907$	−38.3608	−1.27375	−0.636874	−	0.770968i	$-0.719773\pi$
$907$	−38.3608	−1.27375	−0.636874	+	0.770968i	$0.719773\pi$
$908$	0	0
$909$	−3.22239	−0.106880
$910$	0	0
$911$	−48.1083	−1.59390	−0.796949	−	0.604046i	$-0.793554\pi$
$911$	−48.1083	−1.59390	−0.796949	+	0.604046i	$0.793554\pi$
$912$	0	0
$913$	32.4169	1.07284
$914$	0	0
$915$	−13.0345	−0.430906
$916$	0	0
$917$	−5.48054	−0.180983
$918$	0	0
$919$	−8.60107	−0.283723	−0.141861	−	0.989887i	$-0.545309\pi$
$919$	−8.60107	−0.283723	−0.141861	+	0.989887i	$0.545309\pi$
$920$	0	0
$921$	1.44396	0.0475801
$922$	0	0
$923$	53.6056	1.76445
$924$	0	0
$925$	−11.5704	−0.380432
$926$	0	0
$927$	2.55697	0.0839820
$928$	0	0
$929$	−40.9234	−1.34265	−0.671326	−	0.741162i	$-0.734275\pi$
$929$	−40.9234	−1.34265	−0.671326	+	0.741162i	$0.734275\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	25.2678	0.827230
$934$	0	0
$935$	8.31607	0.271965
$936$	0	0
$937$	29.1571	0.952522	0.476261	−	0.879304i	$-0.341992\pi$
$937$	29.1571	0.952522	0.476261	+	0.879304i	$0.341992\pi$
$938$	0	0
$939$	39.3729	1.28489
$940$	0	0
$941$	42.1632	1.37448	0.687241	−	0.726430i	$-0.258822\pi$
$941$	42.1632	1.37448	0.687241	+	0.726430i	$0.258822\pi$
$942$	0	0
$943$	−3.63133	−0.118253
$944$	0	0
$945$	3.34820	0.108917
$946$	0	0
$947$	−21.8231	−0.709156	−0.354578	−	0.935026i	$-0.615376\pi$
$947$	−21.8231	−0.709156	−0.354578	+	0.935026i	$0.615376\pi$
$948$	0	0
$949$	16.8826	0.548034
$950$	0	0
$951$	−56.1404	−1.82048
$952$	0	0
$953$	43.1612	1.39813	0.699065	−	0.715059i	$-0.253600\pi$
$953$	43.1612	1.39813	0.699065	+	0.715059i	$0.253600\pi$
$954$	0	0
$955$	21.9662	0.710810
$956$	0	0
$957$	−83.6944	−2.70546
$958$	0	0
$959$	3.35314	0.108278
$960$	0	0
$961$	4.34080	0.140026
$962$	0	0
$963$	−0.263319	−0.00848534
$964$	0	0
$965$	−11.9457	−0.384547
$966$	0	0
$967$	−57.5365	−1.85025	−0.925125	−	0.379662i	$-0.876040\pi$
$967$	−57.5365	−1.85025	−0.925125	+	0.379662i	$0.876040\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.8643	−0.765841	−0.382920	−	0.923781i	$-0.625082\pi$
$971$	−23.8643	−0.765841	−0.382920	+	0.923781i	$0.625082\pi$
$972$	0	0
$973$	1.41237	0.0452785
$974$	0	0
$975$	−8.20300	−0.262706
$976$	0	0
$977$	38.6338	1.23600	0.618002	−	0.786176i	$-0.287942\pi$
$977$	38.6338	1.23600	0.618002	+	0.786176i	$0.287942\pi$
$978$	0	0
$979$	−46.6261	−1.49018
$980$	0	0
$981$	−0.594255	−0.0189731
$982$	0	0
$983$	15.2807	0.487379	0.243689	−	0.969853i	$-0.421642\pi$
$983$	15.2807	0.487379	0.243689	+	0.969853i	$0.421642\pi$
$984$	0	0
$985$	7.75300	0.247031
$986$	0	0
$987$	−14.2434	−0.453373
$988$	0	0
$989$	−12.9906	−0.413076
$990$	0	0
$991$	−5.51547	−0.175205	−0.0876023	−	0.996156i	$-0.527920\pi$
$991$	−5.51547	−0.175205	−0.0876023	+	0.996156i	$0.527920\pi$
$992$	0	0
$993$	−21.1824	−0.672202
$994$	0	0
$995$	9.21437	0.292115
$996$	0	0
$997$	−29.9070	−0.947165	−0.473582	−	0.880750i	$-0.657039\pi$
$997$	−29.9070	−0.947165	−0.473582	+	0.880750i	$0.657039\pi$
$998$	0	0
$999$	57.5157	1.81972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.x.1.7		9
19.2	odd	18		380.2.u.a.61.1	✓	18
19.10	odd	18		380.2.u.a.81.1	yes	18
19.18	odd	2		7220.2.a.v.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.u.a.61.1	✓	18	19.2	odd	18
380.2.u.a.81.1	yes	18	19.10	odd	18
7220.2.a.v.1.3		9	19.18	odd	2
7220.2.a.x.1.7		9	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:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.79921 q^{3} -1.00000 q^{5} +0.673555 q^{7} +0.237149 q^{9} +O(q^{10})\)	\(q+1.79921 q^{3} -1.00000 q^{5} +0.673555 q^{7} +0.237149 q^{9} +6.42552 q^{11} -4.55923 q^{13} -1.79921 q^{15} -1.29423 q^{17} +1.21187 q^{21} -1.35993 q^{23} +1.00000 q^{25} -4.97094 q^{27} -7.23947 q^{29} -5.94481 q^{31} +11.5608 q^{33} -0.673555 q^{35} -11.5704 q^{37} -8.20300 q^{39} +2.67024 q^{41} +9.55239 q^{43} -0.237149 q^{45} -11.7533 q^{47} -6.54632 q^{49} -2.32858 q^{51} +3.58501 q^{53} -6.42552 q^{55} +0.160090 q^{59} +7.24456 q^{61} +0.159733 q^{63} +4.55923 q^{65} -5.46468 q^{67} -2.44679 q^{69} -11.7576 q^{71} -3.70296 q^{73} +1.79921 q^{75} +4.32794 q^{77} +6.42489 q^{79} -9.65521 q^{81} +5.04502 q^{83} +1.29423 q^{85} -13.0253 q^{87} -7.25639 q^{89} -3.07089 q^{91} -10.6960 q^{93} +7.88351 q^{97} +1.52381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} - 9 q^{5} + 6 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 - 9 * q^5 + 6 * q^9	\( 9 q + 3 q^{3} - 9 q^{5} + 6 q^{9} - 3 q^{13} - 3 q^{15} + 3 q^{17} - 12 q^{21} + 6 q^{23} + 9 q^{25} - 9 q^{29} - 18 q^{31} + 9 q^{33} - 30 q^{37} - 42 q^{39} - 6 q^{41} + 24 q^{43} - 6 q^{45} - 21 q^{47} + 15 q^{49} + 42 q^{51} - 9 q^{53} - 3 q^{59} - 24 q^{61} - 51 q^{63} + 3 q^{65} - 9 q^{69} - 18 q^{71} - 33 q^{73} + 3 q^{75} + 3 q^{79} + 33 q^{81} + 30 q^{83} - 3 q^{85} + 9 q^{87} + 3 q^{89} - 15 q^{91} - 15 q^{93} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 - 9 * q^5 + 6 * q^9 - 3 * q^13 - 3 * q^15 + 3 * q^17 - 12 * q^21 + 6 * q^23 + 9 * q^25 - 9 * q^29 - 18 * q^31 + 9 * q^33 - 30 * q^37 - 42 * q^39 - 6 * q^41 + 24 * q^43 - 6 * q^45 - 21 * q^47 + 15 * q^49 + 42 * q^51 - 9 * q^53 - 3 * q^59 - 24 * q^61 - 51 * q^63 + 3 * q^65 - 9 * q^69 - 18 * q^71 - 33 * q^73 + 3 * q^75 + 3 * q^79 + 33 * q^81 + 30 * q^83 - 3 * q^85 + 9 * q^87 + 3 * q^89 - 15 * q^91 - 15 * q^93 - 15 * q^97 - 6 * q^99

Introduction

L-functions

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