LMFDB - Embedded newform 7220.2.a.p.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.133593.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 2 $ x^4 - x^3 - 8x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.3
Root			$-0.353450$ 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)

$f(q)$	$=$	$q+0.353450 q^{3} +1.00000 q^{5} -4.30507 q^{7} -2.87507 q^{9} +O(q^{10})$	$q+0.353450 q^{3} +1.00000 q^{5} -4.30507 q^{7} -2.87507 q^{9} +6.01196 q^{11} +5.95162 q^{13} +0.353450 q^{15} +3.87507 q^{17} -1.52162 q^{21} +0.783442 q^{23} +1.00000 q^{25} -2.07654 q^{27} -7.96358 q^{29} +4.49034 q^{31} +2.12493 q^{33} -4.30507 q^{35} +0.988035 q^{37} +2.10360 q^{39} -6.30507 q^{41} -1.57001 q^{43} -2.87507 q^{45} -1.26182 q^{47} +11.5336 q^{49} +1.36964 q^{51} +8.14886 q^{53} +6.01196 q^{55} -5.25668 q^{59} +5.61013 q^{61} +12.3774 q^{63} +5.95162 q^{65} -7.04325 q^{67} +0.276907 q^{69} +5.81472 q^{71} -9.24049 q^{73} +0.353450 q^{75} -25.8819 q^{77} +13.9949 q^{79} +7.89127 q^{81} +6.58197 q^{83} +3.87507 q^{85} -2.81472 q^{87} +3.38473 q^{89} -25.6221 q^{91} +1.58711 q^{93} +7.39670 q^{97} -17.2848 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^9	$ 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47} + 10 q^{49} + 12 q^{51} + 5 q^{53} + 2 q^{55} + 11 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} - 3 q^{69} + 14 q^{71} + 4 q^{73} - q^{75} - 22 q^{77} + 13 q^{79} + 24 q^{81} + 5 q^{83} - q^{85} - 2 q^{87} + 5 q^{89} - 46 q^{91} + 28 q^{93} - q^{97} - 24 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 9 * q^13 - q^15 - q^17 + 8 * q^21 + 4 * q^25 - 10 * q^27 + 5 * q^29 + 10 * q^31 + 25 * q^33 + 26 * q^37 - 27 * q^39 - 8 * q^41 - 7 * q^43 + 5 * q^45 - 16 * q^47 + 10 * q^49 + 12 * q^51 + 5 * q^53 + 2 * q^55 + 11 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 - 3 * q^69 + 14 * q^71 + 4 * q^73 - q^75 - 22 * q^77 + 13 * q^79 + 24 * q^81 + 5 * q^83 - q^85 - 2 * q^87 + 5 * q^89 - 46 * q^91 + 28 * q^93 - q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.353450	0.204064	0.102032	−	0.994781i	$-0.467466\pi$
$3$	0.353450	0.204064	0.102032	+	0.994781i	$0.467466\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.30507	−1.62716	−0.813581	−	0.581452i	$-0.802485\pi$
$7$	−4.30507	−1.62716	−0.813581	+	0.581452i	$0.802485\pi$
$8$	0	0
$9$	−2.87507	−0.958358
$10$	0	0
$11$	6.01196	1.81268	0.906338	−	0.422554i	$-0.138866\pi$
$11$	6.01196	1.81268	0.906338	+	0.422554i	$0.138866\pi$
$12$	0	0
$13$	5.95162	1.65068	0.825341	−	0.564635i	$-0.190983\pi$
$13$	5.95162	1.65068	0.825341	+	0.564635i	$0.190983\pi$
$14$	0	0
$15$	0.353450	0.0912603
$16$	0	0
$17$	3.87507	0.939843	0.469922	−	0.882708i	$-0.344282\pi$
$17$	3.87507	0.939843	0.469922	+	0.882708i	$0.344282\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.52162	−0.332046
$22$	0	0
$23$	0.783442	0.163359	0.0816794	−	0.996659i	$-0.473972\pi$
$23$	0.783442	0.163359	0.0816794	+	0.996659i	$0.473972\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.07654	−0.399631
$28$	0	0
$29$	−7.96358	−1.47880	−0.739400	−	0.673267i	$-0.764891\pi$
$29$	−7.96358	−1.47880	−0.739400	+	0.673267i	$0.764891\pi$
$30$	0	0
$31$	4.49034	0.806489	0.403245	−	0.915092i	$-0.367882\pi$
$31$	4.49034	0.806489	0.403245	+	0.915092i	$0.367882\pi$
$32$	0	0
$33$	2.12493	0.369902
$34$	0	0
$35$	−4.30507	−0.727689
$36$	0	0
$37$	0.988035	0.162432	0.0812160	−	0.996697i	$-0.474120\pi$
$37$	0.988035	0.162432	0.0812160	+	0.996697i	$0.474120\pi$
$38$	0	0
$39$	2.10360	0.336845
$40$	0	0
$41$	−6.30507	−0.984686	−0.492343	−	0.870401i	$-0.663859\pi$
$41$	−6.30507	−0.984686	−0.492343	+	0.870401i	$0.663859\pi$
$42$	0	0
$43$	−1.57001	−0.239424	−0.119712	−	0.992809i	$-0.538197\pi$
$43$	−1.57001	−0.239424	−0.119712	+	0.992809i	$0.538197\pi$
$44$	0	0
$45$	−2.87507	−0.428591
$46$	0	0
$47$	−1.26182	−0.184055	−0.0920275	−	0.995756i	$-0.529335\pi$
$47$	−1.26182	−0.184055	−0.0920275	+	0.995756i	$0.529335\pi$
$48$	0	0
$49$	11.5336	1.64766
$50$	0	0
$51$	1.36964	0.191788
$52$	0	0
$53$	8.14886	1.11933	0.559666	−	0.828718i	$-0.310930\pi$
$53$	8.14886	1.11933	0.559666	+	0.828718i	$0.310930\pi$
$54$	0	0
$55$	6.01196	0.810653
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.25668	−0.684362	−0.342181	−	0.939634i	$-0.611166\pi$
$59$	−5.25668	−0.684362	−0.342181	+	0.939634i	$0.611166\pi$
$60$	0	0
$61$	5.61013	0.718304	0.359152	−	0.933279i	$-0.383066\pi$
$61$	5.61013	0.718304	0.359152	+	0.933279i	$0.383066\pi$
$62$	0	0
$63$	12.3774	1.55940
$64$	0	0
$65$	5.95162	0.738207
$66$	0	0
$67$	−7.04325	−0.860470	−0.430235	−	0.902717i	$-0.641569\pi$
$67$	−7.04325	−0.860470	−0.430235	+	0.902717i	$0.641569\pi$
$68$	0	0
$69$	0.276907	0.0333357
$70$	0	0
$71$	5.81472	0.690081	0.345040	−	0.938588i	$-0.387865\pi$
$71$	5.81472	0.690081	0.345040	+	0.938588i	$0.387865\pi$
$72$	0	0
$73$	−9.24049	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$73$	−9.24049	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$74$	0	0
$75$	0.353450	0.0408128
$76$	0	0
$77$	−25.8819	−2.94952
$78$	0	0
$79$	13.9949	1.57455	0.787273	−	0.616605i	$-0.211492\pi$
$79$	13.9949	1.57455	0.787273	+	0.616605i	$0.211492\pi$
$80$	0	0
$81$	7.89127	0.876807
$82$	0	0
$83$	6.58197	0.722465	0.361233	−	0.932476i	$-0.382356\pi$
$83$	6.58197	0.722465	0.361233	+	0.932476i	$0.382356\pi$
$84$	0	0
$85$	3.87507	0.420311
$86$	0	0
$87$	−2.81472	−0.301770
$88$	0	0
$89$	3.38473	0.358781	0.179390	−	0.983778i	$-0.442587\pi$
$89$	3.38473	0.358781	0.179390	+	0.983778i	$0.442587\pi$
$90$	0	0
$91$	−25.6221	−2.68593
$92$	0	0
$93$	1.58711	0.164576
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.39670	0.751021	0.375510	−	0.926818i	$-0.377467\pi$
$97$	7.39670	0.751021	0.375510	+	0.926818i	$0.377467\pi$
$98$	0	0
$99$	−17.2848	−1.73719
$100$	0	0
$101$	−9.80737	−0.975870	−0.487935	−	0.872880i	$-0.662250\pi$
$101$	−9.80737	−0.975870	−0.487935	+	0.872880i	$0.662250\pi$
$102$	0	0
$103$	−14.4368	−1.42250	−0.711251	−	0.702938i	$-0.751871\pi$
$103$	−14.4368	−1.42250	−0.711251	+	0.702938i	$0.751871\pi$
$104$	0	0
$105$	−1.52162	−0.148495
$106$	0	0
$107$	9.49034	0.917466	0.458733	−	0.888574i	$-0.348303\pi$
$107$	9.49034	0.917466	0.458733	+	0.888574i	$0.348303\pi$
$108$	0	0
$109$	2.61839	0.250796	0.125398	−	0.992106i	$-0.459979\pi$
$109$	2.61839	0.250796	0.125398	+	0.992106i	$0.459979\pi$
$110$	0	0
$111$	0.349221	0.0331466
$112$	0	0
$113$	13.6705	1.28601	0.643005	−	0.765862i	$-0.277687\pi$
$113$	13.6705	1.28601	0.643005	+	0.765862i	$0.277687\pi$
$114$	0	0
$115$	0.783442	0.0730563
$116$	0	0
$117$	−17.1113	−1.58194
$118$	0	0
$119$	−16.6824	−1.52928
$120$	0	0
$121$	25.1437	2.28579
$122$	0	0
$123$	−2.22852	−0.200939
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.0672	−1.15952	−0.579762	−	0.814786i	$-0.696855\pi$
$127$	−13.0672	−1.15952	−0.579762	+	0.814786i	$0.696855\pi$
$128$	0	0
$129$	−0.554919	−0.0488579
$130$	0	0
$131$	−7.50140	−0.655400	−0.327700	−	0.944782i	$-0.606274\pi$
$131$	−7.50140	−0.655400	−0.327700	+	0.944782i	$0.606274\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.07654	−0.178720
$136$	0	0
$137$	−8.42999	−0.720223	−0.360111	−	0.932909i	$-0.617261\pi$
$137$	−8.42999	−0.720223	−0.360111	+	0.932909i	$0.617261\pi$
$138$	0	0
$139$	8.77921	0.744643	0.372322	−	0.928104i	$-0.378562\pi$
$139$	8.77921	0.744643	0.372322	+	0.928104i	$0.378562\pi$
$140$	0	0
$141$	−0.445989	−0.0375591
$142$	0	0
$143$	35.7809	2.99215
$144$	0	0
$145$	−7.96358	−0.661339
$146$	0	0
$147$	4.07654	0.336228
$148$	0	0
$149$	−1.83183	−0.150069	−0.0750345	−	0.997181i	$-0.523907\pi$
$149$	−1.83183	−0.150069	−0.0750345	+	0.997181i	$0.523907\pi$
$150$	0	0
$151$	0.389869	0.0317271	0.0158635	−	0.999874i	$-0.494950\pi$
$151$	0.389869	0.0317271	0.0158635	+	0.999874i	$0.494950\pi$
$152$	0	0
$153$	−11.1411	−0.900706
$154$	0	0
$155$	4.49034	0.360673
$156$	0	0
$157$	−2.75216	−0.219646	−0.109823	−	0.993951i	$-0.535028\pi$
$157$	−2.75216	−0.219646	−0.109823	+	0.993951i	$0.535028\pi$
$158$	0	0
$159$	2.88021	0.228416
$160$	0	0
$161$	−3.37277	−0.265811
$162$	0	0
$163$	−15.0953	−1.18236	−0.591179	−	0.806540i	$-0.701337\pi$
$163$	−15.0953	−1.18236	−0.591179	+	0.806540i	$0.701337\pi$
$164$	0	0
$165$	2.12493	0.165425
$166$	0	0
$167$	−2.98804	−0.231221	−0.115611	−	0.993295i	$-0.536882\pi$
$167$	−2.98804	−0.231221	−0.115611	+	0.993295i	$0.536882\pi$
$168$	0	0
$169$	22.4217	1.72475
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.1890	1.76303	0.881513	−	0.472160i	$-0.156526\pi$
$173$	23.1890	1.76303	0.881513	+	0.472160i	$0.156526\pi$
$174$	0	0
$175$	−4.30507	−0.325432
$176$	0	0
$177$	−1.85797	−0.139654
$178$	0	0
$179$	13.5091	1.00972	0.504860	−	0.863201i	$-0.331544\pi$
$179$	13.5091	1.00972	0.504860	+	0.863201i	$0.331544\pi$
$180$	0	0
$181$	20.3119	1.50977	0.754886	−	0.655857i	$-0.227692\pi$
$181$	20.3119	1.50977	0.754886	+	0.655857i	$0.227692\pi$
$182$	0	0
$183$	1.98290	0.146580
$184$	0	0
$185$	0.988035	0.0726418
$186$	0	0
$187$	23.2968	1.70363
$188$	0	0
$189$	8.93965	0.650264
$190$	0	0
$191$	−7.04838	−0.510003	−0.255002	−	0.966941i	$-0.582076\pi$
$191$	−7.04838	−0.510003	−0.255002	+	0.966941i	$0.582076\pi$
$192$	0	0
$193$	23.7783	1.71160	0.855800	−	0.517307i	$-0.173065\pi$
$193$	23.7783	1.71160	0.855800	+	0.517307i	$0.173065\pi$
$194$	0	0
$195$	2.10360	0.150642
$196$	0	0
$197$	23.7428	1.69160	0.845802	−	0.533497i	$-0.179122\pi$
$197$	23.7428	1.69160	0.845802	+	0.533497i	$0.179122\pi$
$198$	0	0
$199$	−22.9787	−1.62891	−0.814457	−	0.580223i	$-0.802965\pi$
$199$	−22.9787	−1.62891	−0.814457	+	0.580223i	$0.802965\pi$
$200$	0	0
$201$	−2.48943	−0.175591
$202$	0	0
$203$	34.2837	2.40625
$204$	0	0
$205$	−6.30507	−0.440365
$206$	0	0
$207$	−2.25245	−0.156556
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.38473	0.0953289	0.0476645	−	0.998863i	$-0.484822\pi$
$211$	1.38473	0.0953289	0.0476645	+	0.998863i	$0.484822\pi$
$212$	0	0
$213$	2.05521	0.140821
$214$	0	0
$215$	−1.57001	−0.107074
$216$	0	0
$217$	−19.3312	−1.31229
$218$	0	0
$219$	−3.26605	−0.220699
$220$	0	0
$221$	23.0629	1.55138
$222$	0	0
$223$	23.2999	1.56028	0.780139	−	0.625606i	$-0.215148\pi$
$223$	23.2999	1.56028	0.780139	+	0.625606i	$0.215148\pi$
$224$	0	0
$225$	−2.87507	−0.191672
$226$	0	0
$227$	4.48943	0.297974	0.148987	−	0.988839i	$-0.452399\pi$
$227$	4.48943	0.297974	0.148987	+	0.988839i	$0.452399\pi$
$228$	0	0
$229$	9.20830	0.608501	0.304251	−	0.952592i	$-0.401594\pi$
$229$	9.20830	0.608501	0.304251	+	0.952592i	$0.401594\pi$
$230$	0	0
$231$	−9.14795	−0.601891
$232$	0	0
$233$	−4.32529	−0.283359	−0.141680	−	0.989913i	$-0.545250\pi$
$233$	−4.32529	−0.283359	−0.141680	+	0.989913i	$0.545250\pi$
$234$	0	0
$235$	−1.26182	−0.0823119
$236$	0	0
$237$	4.94648	0.321308
$238$	0	0
$239$	−14.8267	−0.959059	−0.479529	−	0.877526i	$-0.659193\pi$
$239$	−14.8267	−0.959059	−0.479529	+	0.877526i	$0.659193\pi$
$240$	0	0
$241$	−2.88905	−0.186100	−0.0930500	−	0.995661i	$-0.529662\pi$
$241$	−2.88905	−0.186100	−0.0930500	+	0.995661i	$0.529662\pi$
$242$	0	0
$243$	9.01879	0.578556
$244$	0	0
$245$	11.5336	0.736854
$246$	0	0
$247$	0	0
$248$	0	0
$249$	2.32640	0.147429
$250$	0	0
$251$	15.3051	0.966047	0.483024	−	0.875607i	$-0.339539\pi$
$251$	15.3051	0.966047	0.483024	+	0.875607i	$0.339539\pi$
$252$	0	0
$253$	4.71002	0.296117
$254$	0	0
$255$	1.36964	0.0857704
$256$	0	0
$257$	−13.0843	−0.816175	−0.408087	−	0.912943i	$-0.633804\pi$
$257$	−13.0843	−0.816175	−0.408087	+	0.912943i	$0.633804\pi$
$258$	0	0
$259$	−4.25356	−0.264303
$260$	0	0
$261$	22.8959	1.41722
$262$	0	0
$263$	19.3796	1.19500	0.597499	−	0.801870i	$-0.296161\pi$
$263$	19.3796	1.19500	0.597499	+	0.801870i	$0.296161\pi$
$264$	0	0
$265$	8.14886	0.500580
$266$	0	0
$267$	1.19633	0.0732144
$268$	0	0
$269$	1.45705	0.0888377	0.0444188	−	0.999013i	$-0.485856\pi$
$269$	1.45705	0.0888377	0.0444188	+	0.999013i	$0.485856\pi$
$270$	0	0
$271$	12.5627	0.763127	0.381563	−	0.924343i	$-0.375386\pi$
$271$	12.5627	0.763127	0.381563	+	0.924343i	$0.375386\pi$
$272$	0	0
$273$	−9.05612	−0.548101
$274$	0	0
$275$	6.01196	0.362535
$276$	0	0
$277$	−4.39448	−0.264039	−0.132019	−	0.991247i	$-0.542146\pi$
$277$	−4.39448	−0.264039	−0.132019	+	0.991247i	$0.542146\pi$
$278$	0	0
$279$	−12.9101	−0.772905
$280$	0	0
$281$	−4.32529	−0.258025	−0.129013	−	0.991643i	$-0.541181\pi$
$281$	−4.32529	−0.258025	−0.129013	+	0.991643i	$0.541181\pi$
$282$	0	0
$283$	7.49769	0.445692	0.222846	−	0.974854i	$-0.428465\pi$
$283$	7.49769	0.445692	0.222846	+	0.974854i	$0.428465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.1437	1.60224
$288$	0	0
$289$	−1.98381	−0.116694
$290$	0	0
$291$	2.61436	0.153256
$292$	0	0
$293$	22.2837	1.30183	0.650915	−	0.759151i	$-0.274385\pi$
$293$	22.2837	1.30183	0.650915	+	0.759151i	$0.274385\pi$
$294$	0	0
$295$	−5.25668	−0.306056
$296$	0	0
$297$	−12.4841	−0.724401
$298$	0	0
$299$	4.66274	0.269653
$300$	0	0
$301$	6.75899	0.389582
$302$	0	0
$303$	−3.46641	−0.199140
$304$	0	0
$305$	5.61013	0.321235
$306$	0	0
$307$	30.2806	1.72821	0.864103	−	0.503315i	$-0.167887\pi$
$307$	30.2806	1.72821	0.864103	+	0.503315i	$0.167887\pi$
$308$	0	0
$309$	−5.10269	−0.290282
$310$	0	0
$311$	−5.37224	−0.304632	−0.152316	−	0.988332i	$-0.548673\pi$
$311$	−5.37224	−0.304632	−0.152316	+	0.988332i	$0.548673\pi$
$312$	0	0
$313$	−4.80737	−0.271729	−0.135864	−	0.990727i	$-0.543381\pi$
$313$	−4.80737	−0.271729	−0.135864	+	0.990727i	$0.543381\pi$
$314$	0	0
$315$	12.3774	0.697386
$316$	0	0
$317$	29.7710	1.67210	0.836052	−	0.548651i	$-0.184858\pi$
$317$	29.7710	1.67210	0.836052	+	0.548651i	$0.184858\pi$
$318$	0	0
$319$	−47.8768	−2.68058
$320$	0	0
$321$	3.35436	0.187222
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5.95162	0.330136
$326$	0	0
$327$	0.925470	0.0511786
$328$	0	0
$329$	5.43221	0.299487
$330$	0	0
$331$	−30.8042	−1.69315	−0.846577	−	0.532266i	$-0.821341\pi$
$331$	−30.8042	−1.69315	−0.846577	+	0.532266i	$0.821341\pi$
$332$	0	0
$333$	−2.84067	−0.155668
$334$	0	0
$335$	−7.04325	−0.384814
$336$	0	0
$337$	6.65058	0.362280	0.181140	−	0.983457i	$-0.442021\pi$
$337$	6.65058	0.362280	0.181140	+	0.983457i	$0.442021\pi$
$338$	0	0
$339$	4.83183	0.262429
$340$	0	0
$341$	26.9958	1.46190
$342$	0	0
$343$	−19.5174	−1.05384
$344$	0	0
$345$	0.276907	0.0149082
$346$	0	0
$347$	15.4107	0.827288	0.413644	−	0.910439i	$-0.364256\pi$
$347$	15.4107	0.827288	0.413644	+	0.910439i	$0.364256\pi$
$348$	0	0
$349$	−27.0157	−1.44612	−0.723058	−	0.690788i	$-0.757264\pi$
$349$	−27.0157	−1.44612	−0.723058	+	0.690788i	$0.757264\pi$
$350$	0	0
$351$	−12.3588	−0.659663
$352$	0	0
$353$	19.7783	1.05269	0.526346	−	0.850270i	$-0.323561\pi$
$353$	19.7783	1.05269	0.526346	+	0.850270i	$0.323561\pi$
$354$	0	0
$355$	5.81472	0.308614
$356$	0	0
$357$	−5.89640	−0.312071
$358$	0	0
$359$	35.6815	1.88320	0.941600	−	0.336734i	$-0.109322\pi$
$359$	35.6815	1.88320	0.941600	+	0.336734i	$0.109322\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	8.88704	0.466449
$364$	0	0
$365$	−9.24049	−0.483669
$366$	0	0
$367$	−2.10470	−0.109864	−0.0549322	−	0.998490i	$-0.517494\pi$
$367$	−2.10470	−0.109864	−0.0549322	+	0.998490i	$0.517494\pi$
$368$	0	0
$369$	18.1275	0.943681
$370$	0	0
$371$	−35.0814	−1.82133
$372$	0	0
$373$	32.0208	1.65797	0.828987	−	0.559268i	$-0.188918\pi$
$373$	32.0208	1.65797	0.828987	+	0.559268i	$0.188918\pi$
$374$	0	0
$375$	0.353450	0.0182521
$376$	0	0
$377$	−47.3962	−2.44103
$378$	0	0
$379$	2.24784	0.115464	0.0577319	−	0.998332i	$-0.481613\pi$
$379$	2.24784	0.115464	0.0577319	+	0.998332i	$0.481613\pi$
$380$	0	0
$381$	−4.61859	−0.236617
$382$	0	0
$383$	−2.66957	−0.136409	−0.0682044	−	0.997671i	$-0.521727\pi$
$383$	−2.66957	−0.136409	−0.0682044	+	0.997671i	$0.521727\pi$
$384$	0	0
$385$	−25.8819	−1.31906
$386$	0	0
$387$	4.51389	0.229454
$388$	0	0
$389$	15.5207	0.786932	0.393466	−	0.919339i	$-0.371276\pi$
$389$	15.5207	0.786932	0.393466	+	0.919339i	$0.371276\pi$
$390$	0	0
$391$	3.03589	0.153532
$392$	0	0
$393$	−2.65137	−0.133744
$394$	0	0
$395$	13.9949	0.704158
$396$	0	0
$397$	2.49679	0.125310	0.0626551	−	0.998035i	$-0.480043\pi$
$397$	2.49679	0.125310	0.0626551	+	0.998035i	$0.480043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.7180	1.08454	0.542271	−	0.840203i	$-0.317564\pi$
$401$	21.7180	1.08454	0.542271	+	0.840203i	$0.317564\pi$
$402$	0	0
$403$	26.7248	1.33126
$404$	0	0
$405$	7.89127	0.392120
$406$	0	0
$407$	5.94003	0.294437
$408$	0	0
$409$	−24.2400	−1.19859	−0.599294	−	0.800529i	$-0.704552\pi$
$409$	−24.2400	−1.19859	−0.599294	+	0.800529i	$0.704552\pi$
$410$	0	0
$411$	−2.97958	−0.146972
$412$	0	0
$413$	22.6304	1.11357
$414$	0	0
$415$	6.58197	0.323096
$416$	0	0
$417$	3.10301	0.151955
$418$	0	0
$419$	−37.6543	−1.83953	−0.919766	−	0.392467i	$-0.871622\pi$
$419$	−37.6543	−1.83953	−0.919766	+	0.392467i	$0.871622\pi$
$420$	0	0
$421$	37.3769	1.82164	0.910818	−	0.412808i	$-0.135452\pi$
$421$	37.3769	1.82164	0.910818	+	0.412808i	$0.135452\pi$
$422$	0	0
$423$	3.62782	0.176391
$424$	0	0
$425$	3.87507	0.187969
$426$	0	0
$427$	−24.1520	−1.16880
$428$	0	0
$429$	12.6467	0.610591
$430$	0	0
$431$	−10.8298	−0.521654	−0.260827	−	0.965386i	$-0.583995\pi$
$431$	−10.8298	−0.521654	−0.260827	+	0.965386i	$0.583995\pi$
$432$	0	0
$433$	−11.4206	−0.548840	−0.274420	−	0.961610i	$-0.588486\pi$
$433$	−11.4206	−0.548840	−0.274420	+	0.961610i	$0.588486\pi$
$434$	0	0
$435$	−2.81472	−0.134956
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−30.1224	−1.43766	−0.718832	−	0.695184i	$-0.755323\pi$
$439$	−30.1224	−1.43766	−0.718832	+	0.695184i	$0.755323\pi$
$440$	0	0
$441$	−33.1599	−1.57904
$442$	0	0
$443$	−20.6090	−0.979164	−0.489582	−	0.871957i	$-0.662851\pi$
$443$	−20.6090	−0.979164	−0.489582	+	0.871957i	$0.662851\pi$
$444$	0	0
$445$	3.38473	0.160452
$446$	0	0
$447$	−0.647458	−0.0306237
$448$	0	0
$449$	3.61436	0.170572	0.0852861	−	0.996357i	$-0.472820\pi$
$449$	3.61436	0.170572	0.0852861	+	0.996357i	$0.472820\pi$
$450$	0	0
$451$	−37.9058	−1.78492
$452$	0	0
$453$	0.137799	0.00647437
$454$	0	0
$455$	−25.6221	−1.20118
$456$	0	0
$457$	5.50452	0.257491	0.128745	−	0.991678i	$-0.458905\pi$
$457$	5.50452	0.257491	0.128745	+	0.991678i	$0.458905\pi$
$458$	0	0
$459$	−8.04675	−0.375590
$460$	0	0
$461$	19.3211	0.899872	0.449936	−	0.893061i	$-0.351447\pi$
$461$	19.3211	0.899872	0.449936	+	0.893061i	$0.351447\pi$
$462$	0	0
$463$	1.05722	0.0491334	0.0245667	−	0.999698i	$-0.492179\pi$
$463$	1.05722	0.0491334	0.0245667	+	0.999698i	$0.492179\pi$
$464$	0	0
$465$	1.58711	0.0736004
$466$	0	0
$467$	21.9413	1.01532	0.507662	−	0.861556i	$-0.330510\pi$
$467$	21.9413	1.01532	0.507662	+	0.861556i	$0.330510\pi$
$468$	0	0
$469$	30.3216	1.40012
$470$	0	0
$471$	−0.972750	−0.0448219
$472$	0	0
$473$	−9.43883	−0.433998
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−23.4286	−1.07272
$478$	0	0
$479$	−13.2041	−0.603309	−0.301655	−	0.953417i	$-0.597539\pi$
$479$	−13.2041	−0.603309	−0.301655	+	0.953417i	$0.597539\pi$
$480$	0	0
$481$	5.88041	0.268123
$482$	0	0
$483$	−1.19210	−0.0542426
$484$	0	0
$485$	7.39670	0.335867
$486$	0	0
$487$	−15.5627	−0.705211	−0.352606	−	0.935772i	$-0.614704\pi$
$487$	−15.5627	−0.705211	−0.352606	+	0.935772i	$0.614704\pi$
$488$	0	0
$489$	−5.33544	−0.241277
$490$	0	0
$491$	−29.5956	−1.33563	−0.667816	−	0.744326i	$-0.732771\pi$
$491$	−29.5956	−1.33563	−0.667816	+	0.744326i	$0.732771\pi$
$492$	0	0
$493$	−30.8595	−1.38984
$494$	0	0
$495$	−17.2848	−0.776896
$496$	0	0
$497$	−25.0328	−1.12287
$498$	0	0
$499$	−5.84379	−0.261604	−0.130802	−	0.991409i	$-0.541755\pi$
$499$	−5.84379	−0.261604	−0.130802	+	0.991409i	$0.541755\pi$
$500$	0	0
$501$	−1.05612	−0.0471840
$502$	0	0
$503$	6.29088	0.280497	0.140248	−	0.990116i	$-0.455210\pi$
$503$	6.29088	0.280497	0.140248	+	0.990116i	$0.455210\pi$
$504$	0	0
$505$	−9.80737	−0.436422
$506$	0	0
$507$	7.92495	0.351959
$508$	0	0
$509$	3.69181	0.163637	0.0818183	−	0.996647i	$-0.473927\pi$
$509$	3.69181	0.163637	0.0818183	+	0.996647i	$0.473927\pi$
$510$	0	0
$511$	39.7809	1.75980
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.4368	−0.636162
$516$	0	0
$517$	−7.58600	−0.333632
$518$	0	0
$519$	8.19614	0.359770
$520$	0	0
$521$	−29.0510	−1.27275	−0.636373	−	0.771381i	$-0.719566\pi$
$521$	−29.0510	−1.27275	−0.636373	+	0.771381i	$0.719566\pi$
$522$	0	0
$523$	−18.4337	−0.806049	−0.403025	−	0.915189i	$-0.632041\pi$
$523$	−18.4337	−0.806049	−0.403025	+	0.915189i	$0.632041\pi$
$524$	0	0
$525$	−1.52162	−0.0664091
$526$	0	0
$527$	17.4004	0.757973
$528$	0	0
$529$	−22.3862	−0.973314
$530$	0	0
$531$	15.1133	0.655863
$532$	0	0
$533$	−37.5253	−1.62540
$534$	0	0
$535$	9.49034	0.410303
$536$	0	0
$537$	4.77480	0.206048
$538$	0	0
$539$	69.3395	2.98666
$540$	0	0
$541$	−42.1750	−1.81324	−0.906622	−	0.421943i	$-0.861348\pi$
$541$	−42.1750	−1.81324	−0.906622	+	0.421943i	$0.861348\pi$
$542$	0	0
$543$	7.17923	0.308090
$544$	0	0
$545$	2.61839	0.112160
$546$	0	0
$547$	−15.7544	−0.673608	−0.336804	−	0.941575i	$-0.609346\pi$
$547$	−15.7544	−0.673608	−0.336804	+	0.941575i	$0.609346\pi$
$548$	0	0
$549$	−16.1295	−0.688392
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−60.2488	−2.56204
$554$	0	0
$555$	0.349221	0.0148236
$556$	0	0
$557$	23.1848	0.982369	0.491185	−	0.871055i	$-0.336564\pi$
$557$	23.1848	0.982369	0.491185	+	0.871055i	$0.336564\pi$
$558$	0	0
$559$	−9.34408	−0.395213
$560$	0	0
$561$	8.23425	0.347650
$562$	0	0
$563$	0.970934	0.0409200	0.0204600	−	0.999791i	$-0.493487\pi$
$563$	0.970934	0.0409200	0.0204600	+	0.999791i	$0.493487\pi$
$564$	0	0
$565$	13.6705	0.575121
$566$	0	0
$567$	−33.9724	−1.42671
$568$	0	0
$569$	30.1395	1.26351	0.631757	−	0.775167i	$-0.282334\pi$
$569$	30.1395	1.26351	0.631757	+	0.775167i	$0.282334\pi$
$570$	0	0
$571$	−31.8976	−1.33487	−0.667436	−	0.744667i	$-0.732608\pi$
$571$	−31.8976	−1.33487	−0.667436	+	0.744667i	$0.732608\pi$
$572$	0	0
$573$	−2.49125	−0.104073
$574$	0	0
$575$	0.783442	0.0326718
$576$	0	0
$577$	23.2889	0.969528	0.484764	−	0.874645i	$-0.338905\pi$
$577$	23.2889	0.969528	0.484764	+	0.874645i	$0.338905\pi$
$578$	0	0
$579$	8.40443	0.349276
$580$	0	0
$581$	−28.3358	−1.17557
$582$	0	0
$583$	48.9906	2.02898
$584$	0	0
$585$	−17.1113	−0.707467
$586$	0	0
$587$	9.48117	0.391330	0.195665	−	0.980671i	$-0.437314\pi$
$587$	9.48117	0.391330	0.195665	+	0.980671i	$0.437314\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	8.39188	0.345196
$592$	0	0
$593$	−42.4468	−1.74308	−0.871540	−	0.490324i	$-0.836878\pi$
$593$	−42.4468	−1.74308	−0.871540	+	0.490324i	$0.836878\pi$
$594$	0	0
$595$	−16.6824	−0.683914
$596$	0	0
$597$	−8.12180	−0.332403
$598$	0	0
$599$	25.7923	1.05384	0.526922	−	0.849914i	$-0.323346\pi$
$599$	25.7923	1.05384	0.526922	+	0.849914i	$0.323346\pi$
$600$	0	0
$601$	−0.206080	−0.00840619	−0.00420310	−	0.999991i	$-0.501338\pi$
$601$	−0.206080	−0.00840619	−0.00420310	+	0.999991i	$0.501338\pi$
$602$	0	0
$603$	20.2499	0.824638
$604$	0	0
$605$	25.1437	1.02224
$606$	0	0
$607$	−0.100472	−0.00407802	−0.00203901	−	0.999998i	$-0.500649\pi$
$607$	−0.100472	−0.00407802	−0.00203901	+	0.999998i	$0.500649\pi$
$608$	0	0
$609$	12.1176	0.491029
$610$	0	0
$611$	−7.50986	−0.303816
$612$	0	0
$613$	−39.4801	−1.59458	−0.797292	−	0.603593i	$-0.793735\pi$
$613$	−39.4801	−1.59458	−0.797292	+	0.603593i	$0.793735\pi$
$614$	0	0
$615$	−2.22852	−0.0898627
$616$	0	0
$617$	−28.3316	−1.14059	−0.570294	−	0.821441i	$-0.693171\pi$
$617$	−28.3316	−1.14059	−0.570294	+	0.821441i	$0.693171\pi$
$618$	0	0
$619$	−1.39670	−0.0561380	−0.0280690	−	0.999606i	$-0.508936\pi$
$619$	−1.39670	−0.0561380	−0.0280690	+	0.999606i	$0.508936\pi$
$620$	0	0
$621$	−1.62685	−0.0652832
$622$	0	0
$623$	−14.5715	−0.583795
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.82871	0.152661
$630$	0	0
$631$	8.14795	0.324365	0.162182	−	0.986761i	$-0.448147\pi$
$631$	8.14795	0.324365	0.162182	+	0.986761i	$0.448147\pi$
$632$	0	0
$633$	0.489433	0.0194532
$634$	0	0
$635$	−13.0672	−0.518555
$636$	0	0
$637$	68.6435	2.71975
$638$	0	0
$639$	−16.7178	−0.661344
$640$	0	0
$641$	−19.5273	−0.771284	−0.385642	−	0.922649i	$-0.626020\pi$
$641$	−19.5273	−0.771284	−0.385642	+	0.922649i	$0.626020\pi$
$642$	0	0
$643$	43.1890	1.70321	0.851604	−	0.524186i	$-0.175631\pi$
$643$	43.1890	1.70321	0.851604	+	0.524186i	$0.175631\pi$
$644$	0	0
$645$	−0.554919	−0.0218499
$646$	0	0
$647$	27.4390	1.07874	0.539370	−	0.842069i	$-0.318662\pi$
$647$	27.4390	1.07874	0.539370	+	0.842069i	$0.318662\pi$
$648$	0	0
$649$	−31.6030	−1.24053
$650$	0	0
$651$	−6.83261	−0.267791
$652$	0	0
$653$	−3.40515	−0.133254	−0.0666270	−	0.997778i	$-0.521224\pi$
$653$	−3.40515	−0.133254	−0.0666270	+	0.997778i	$0.521224\pi$
$654$	0	0
$655$	−7.50140	−0.293104
$656$	0	0
$657$	26.5671	1.03648
$658$	0	0
$659$	−12.1972	−0.475137	−0.237569	−	0.971371i	$-0.576350\pi$
$659$	−12.1972	−0.475137	−0.237569	+	0.971371i	$0.576350\pi$
$660$	0	0
$661$	38.1366	1.48334	0.741670	−	0.670765i	$-0.234034\pi$
$661$	38.1366	1.48334	0.741670	+	0.670765i	$0.234034\pi$
$662$	0	0
$663$	8.15159	0.316582
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.23900	−0.241575
$668$	0	0
$669$	8.23535	0.318397
$670$	0	0
$671$	33.7279	1.30205
$672$	0	0
$673$	37.9505	1.46288	0.731442	−	0.681903i	$-0.238848\pi$
$673$	37.9505	1.46288	0.731442	+	0.681903i	$0.238848\pi$
$674$	0	0
$675$	−2.07654	−0.0799262
$676$	0	0
$677$	32.4499	1.24715	0.623575	−	0.781763i	$-0.285679\pi$
$677$	32.4499	1.24715	0.623575	+	0.781763i	$0.285679\pi$
$678$	0	0
$679$	−31.8433	−1.22203
$680$	0	0
$681$	1.58679	0.0608059
$682$	0	0
$683$	−40.5283	−1.55077	−0.775385	−	0.631488i	$-0.782444\pi$
$683$	−40.5283	−1.55077	−0.775385	+	0.631488i	$0.782444\pi$
$684$	0	0
$685$	−8.42999	−0.322093
$686$	0	0
$687$	3.25467	0.124173
$688$	0	0
$689$	48.4989	1.84766
$690$	0	0
$691$	1.78657	0.0679642	0.0339821	−	0.999422i	$-0.489181\pi$
$691$	1.78657	0.0679642	0.0339821	+	0.999422i	$0.489181\pi$
$692$	0	0
$693$	74.4124	2.82669
$694$	0	0
$695$	8.77921	0.333015
$696$	0	0
$697$	−24.4326	−0.925450
$698$	0	0
$699$	−1.52877	−0.0578235
$700$	0	0
$701$	−43.7228	−1.65139	−0.825693	−	0.564120i	$-0.809216\pi$
$701$	−43.7228	−1.65139	−0.825693	+	0.564120i	$0.809216\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.445989	−0.0167969
$706$	0	0
$707$	42.2214	1.58790
$708$	0	0
$709$	14.6403	0.549828	0.274914	−	0.961469i	$-0.411351\pi$
$709$	14.6403	0.549828	0.274914	+	0.961469i	$0.411351\pi$
$710$	0	0
$711$	−40.2363	−1.50898
$712$	0	0
$713$	3.51792	0.131747
$714$	0	0
$715$	35.7809	1.33813
$716$	0	0
$717$	−5.24049	−0.195710
$718$	0	0
$719$	−22.3557	−0.833726	−0.416863	−	0.908969i	$-0.636870\pi$
$719$	−22.3557	−0.833726	−0.416863	+	0.908969i	$0.636870\pi$
$720$	0	0
$721$	62.1515	2.31464
$722$	0	0
$723$	−1.02113	−0.0379764
$724$	0	0
$725$	−7.96358	−0.295760
$726$	0	0
$727$	8.30084	0.307861	0.153930	−	0.988082i	$-0.450807\pi$
$727$	8.30084	0.307861	0.153930	+	0.988082i	$0.450807\pi$
$728$	0	0
$729$	−20.4861	−0.758745
$730$	0	0
$731$	−6.08390	−0.225021
$732$	0	0
$733$	35.7912	1.32198	0.660989	−	0.750396i	$-0.270137\pi$
$733$	35.7912	1.32198	0.660989	+	0.750396i	$0.270137\pi$
$734$	0	0
$735$	4.07654	0.150366
$736$	0	0
$737$	−42.3438	−1.55975
$738$	0	0
$739$	−33.2433	−1.22287	−0.611437	−	0.791293i	$-0.709408\pi$
$739$	−33.2433	−1.22287	−0.611437	+	0.791293i	$0.709408\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.5489	0.680492	0.340246	−	0.940336i	$-0.389490\pi$
$743$	18.5489	0.680492	0.340246	+	0.940336i	$0.389490\pi$
$744$	0	0
$745$	−1.83183	−0.0671129
$746$	0	0
$747$	−18.9237	−0.692380
$748$	0	0
$749$	−40.8565	−1.49287
$750$	0	0
$751$	−43.1496	−1.57455	−0.787276	−	0.616600i	$-0.788509\pi$
$751$	−43.1496	−1.57455	−0.787276	+	0.616600i	$0.788509\pi$
$752$	0	0
$753$	5.40957	0.197136
$754$	0	0
$755$	0.389869	0.0141888
$756$	0	0
$757$	9.54868	0.347053	0.173526	−	0.984829i	$-0.444484\pi$
$757$	9.54868	0.347053	0.173526	+	0.984829i	$0.444484\pi$
$758$	0	0
$759$	1.66476	0.0604268
$760$	0	0
$761$	35.8512	1.29960	0.649802	−	0.760103i	$-0.274852\pi$
$761$	35.8512	1.29960	0.649802	+	0.760103i	$0.274852\pi$
$762$	0	0
$763$	−11.2723	−0.408086
$764$	0	0
$765$	−11.1411	−0.402808
$766$	0	0
$767$	−31.2857	−1.12966
$768$	0	0
$769$	−17.8740	−0.644552	−0.322276	−	0.946646i	$-0.604448\pi$
$769$	−17.8740	−0.644552	−0.322276	+	0.946646i	$0.604448\pi$
$770$	0	0
$771$	−4.62463	−0.166552
$772$	0	0
$773$	−1.85354	−0.0666671	−0.0333336	−	0.999444i	$-0.510612\pi$
$773$	−1.85354	−0.0666671	−0.0333336	+	0.999444i	$0.510612\pi$
$774$	0	0
$775$	4.49034	0.161298
$776$	0	0
$777$	−1.50342	−0.0539348
$778$	0	0
$779$	0	0
$780$	0	0
$781$	34.9579	1.25089
$782$	0	0
$783$	16.5367	0.590974
$784$	0	0
$785$	−2.75216	−0.0982288
$786$	0	0
$787$	53.8501	1.91955	0.959774	−	0.280773i	$-0.0905907\pi$
$787$	53.8501	1.91955	0.959774	+	0.280773i	$0.0905907\pi$
$788$	0	0
$789$	6.84971	0.243856
$790$	0	0
$791$	−58.8523	−2.09255
$792$	0	0
$793$	33.3893	1.18569
$794$	0	0
$795$	2.88021	0.102151
$796$	0	0
$797$	44.2766	1.56836	0.784179	−	0.620535i	$-0.213085\pi$
$797$	44.2766	1.56836	0.784179	+	0.620535i	$0.213085\pi$
$798$	0	0
$799$	−4.88964	−0.172983
$800$	0	0
$801$	−9.73135	−0.343840
$802$	0	0
$803$	−55.5535	−1.96044
$804$	0	0
$805$	−3.37277	−0.118874
$806$	0	0
$807$	0.514992	0.0181286
$808$	0	0
$809$	−16.7641	−0.589395	−0.294698	−	0.955591i	$-0.595219\pi$
$809$	−16.7641	−0.589395	−0.294698	+	0.955591i	$0.595219\pi$
$810$	0	0
$811$	27.4405	0.963567	0.481784	−	0.876290i	$-0.339989\pi$
$811$	27.4405	0.963567	0.481784	+	0.876290i	$0.339989\pi$
$812$	0	0
$813$	4.44027	0.155727
$814$	0	0
$815$	−15.0953	−0.528767
$816$	0	0
$817$	0	0
$818$	0	0
$819$	73.6654	2.57408
$820$	0	0
$821$	−23.8808	−0.833445	−0.416723	−	0.909034i	$-0.636821\pi$
$821$	−23.8808	−0.833445	−0.416723	+	0.909034i	$0.636821\pi$
$822$	0	0
$823$	−22.6819	−0.790642	−0.395321	−	0.918543i	$-0.629367\pi$
$823$	−22.6819	−0.790642	−0.395321	+	0.918543i	$0.629367\pi$
$824$	0	0
$825$	2.12493	0.0739804
$826$	0	0
$827$	−32.2243	−1.12055	−0.560274	−	0.828307i	$-0.689304\pi$
$827$	−32.2243	−1.12055	−0.560274	+	0.828307i	$0.689304\pi$
$828$	0	0
$829$	0.593350	0.0206079	0.0103040	−	0.999947i	$-0.496720\pi$
$829$	0.593350	0.0206079	0.0103040	+	0.999947i	$0.496720\pi$
$830$	0	0
$831$	−1.55323	−0.0538809
$832$	0	0
$833$	44.6935	1.54854
$834$	0	0
$835$	−2.98804	−0.103405
$836$	0	0
$837$	−9.32438	−0.322298
$838$	0	0
$839$	21.9553	0.757982	0.378991	−	0.925400i	$-0.376271\pi$
$839$	21.9553	0.757982	0.378991	+	0.925400i	$0.376271\pi$
$840$	0	0
$841$	34.4186	1.18685
$842$	0	0
$843$	−1.52877	−0.0526537
$844$	0	0
$845$	22.4217	0.771331
$846$	0	0
$847$	−108.245	−3.71935
$848$	0	0
$849$	2.65006	0.0909497
$850$	0	0
$851$	0.774068	0.0265347
$852$	0	0
$853$	19.5279	0.668623	0.334312	−	0.942463i	$-0.391496\pi$
$853$	19.5279	0.668623	0.334312	+	0.942463i	$0.391496\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.843408	0.0288103	0.0144051	−	0.999896i	$-0.495415\pi$
$857$	0.843408	0.0288103	0.0144051	+	0.999896i	$0.495415\pi$
$858$	0	0
$859$	33.4295	1.14060	0.570299	−	0.821437i	$-0.306827\pi$
$859$	33.4295	1.14060	0.570299	+	0.821437i	$0.306827\pi$
$860$	0	0
$861$	9.59394	0.326961
$862$	0	0
$863$	7.13064	0.242730	0.121365	−	0.992608i	$-0.461273\pi$
$863$	7.13064	0.242730	0.121365	+	0.992608i	$0.461273\pi$
$864$	0	0
$865$	23.1890	0.788449
$866$	0	0
$867$	−0.701176	−0.0238132
$868$	0	0
$869$	84.1366	2.85414
$870$	0	0
$871$	−41.9187	−1.42036
$872$	0	0
$873$	−21.2660	−0.719747
$874$	0	0
$875$	−4.30507	−0.145538
$876$	0	0
$877$	−16.0313	−0.541338	−0.270669	−	0.962672i	$-0.587245\pi$
$877$	−16.0313	−0.541338	−0.270669	+	0.962672i	$0.587245\pi$
$878$	0	0
$879$	7.87618	0.265657
$880$	0	0
$881$	−23.0319	−0.775963	−0.387982	−	0.921667i	$-0.626828\pi$
$881$	−23.0319	−0.775963	−0.387982	+	0.921667i	$0.626828\pi$
$882$	0	0
$883$	41.0600	1.38178	0.690890	−	0.722960i	$-0.257219\pi$
$883$	41.0600	1.38178	0.690890	+	0.722960i	$0.257219\pi$
$884$	0	0
$885$	−1.85797	−0.0624550
$886$	0	0
$887$	−10.1380	−0.340401	−0.170200	−	0.985409i	$-0.554442\pi$
$887$	−10.1380	−0.340401	−0.170200	+	0.985409i	$0.554442\pi$
$888$	0	0
$889$	56.2551	1.88673
$890$	0	0
$891$	47.4420	1.58937
$892$	0	0
$893$	0	0
$894$	0	0
$895$	13.5091	0.451561
$896$	0	0
$897$	1.64805	0.0550266
$898$	0	0
$899$	−35.7592	−1.19264
$900$	0	0
$901$	31.5774	1.05200
$902$	0	0
$903$	2.38896	0.0794997
$904$	0	0
$905$	20.3119	0.675190
$906$	0	0
$907$	−38.1833	−1.26785	−0.633927	−	0.773393i	$-0.718558\pi$
$907$	−38.1833	−1.26785	−0.633927	+	0.773393i	$0.718558\pi$
$908$	0	0
$909$	28.1969	0.935233
$910$	0	0
$911$	14.4138	0.477550	0.238775	−	0.971075i	$-0.423254\pi$
$911$	14.4138	0.477550	0.238775	+	0.971075i	$0.423254\pi$
$912$	0	0
$913$	39.5706	1.30960
$914$	0	0
$915$	1.98290	0.0655526
$916$	0	0
$917$	32.2940	1.06644
$918$	0	0
$919$	5.20920	0.171836	0.0859179	−	0.996302i	$-0.472618\pi$
$919$	5.20920	0.171836	0.0859179	+	0.996302i	$0.472618\pi$
$920$	0	0
$921$	10.7027	0.352665
$922$	0	0
$923$	34.6070	1.13910
$924$	0	0
$925$	0.988035	0.0324864
$926$	0	0
$927$	41.5069	1.36327
$928$	0	0
$929$	−38.3538	−1.25835	−0.629174	−	0.777264i	$-0.716607\pi$
$929$	−38.3538	−1.25835	−0.629174	+	0.777264i	$0.716607\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.89882	−0.0621645
$934$	0	0
$935$	23.2968	0.761887
$936$	0	0
$937$	60.4578	1.97507	0.987536	−	0.157396i	$-0.0503098\pi$
$937$	60.4578	1.97507	0.987536	+	0.157396i	$0.0503098\pi$
$938$	0	0
$939$	−1.69916	−0.0554501
$940$	0	0
$941$	24.1719	0.787981	0.393990	−	0.919115i	$-0.371094\pi$
$941$	24.1719	0.787981	0.393990	+	0.919115i	$0.371094\pi$
$942$	0	0
$943$	−4.93965	−0.160857
$944$	0	0
$945$	8.93965	0.290807
$946$	0	0
$947$	9.82779	0.319360	0.159680	−	0.987169i	$-0.448954\pi$
$947$	9.82779	0.319360	0.159680	+	0.987169i	$0.448954\pi$
$948$	0	0
$949$	−54.9958	−1.78524
$950$	0	0
$951$	10.5225	0.341216
$952$	0	0
$953$	2.09735	0.0679397	0.0339699	−	0.999423i	$-0.489185\pi$
$953$	2.09735	0.0679397	0.0339699	+	0.999423i	$0.489185\pi$
$954$	0	0
$955$	−7.04838	−0.228080
$956$	0	0
$957$	−16.9220	−0.547011
$958$	0	0
$959$	36.2917	1.17192
$960$	0	0
$961$	−10.8368	−0.349575
$962$	0	0
$963$	−27.2854	−0.879261
$964$	0	0
$965$	23.7783	0.765451
$966$	0	0
$967$	−38.3540	−1.23338	−0.616691	−	0.787205i	$-0.711527\pi$
$967$	−38.3540	−1.23338	−0.616691	+	0.787205i	$0.711527\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.1364	−0.614115	−0.307058	−	0.951691i	$-0.599344\pi$
$971$	−19.1364	−0.614115	−0.307058	+	0.951691i	$0.599344\pi$
$972$	0	0
$973$	−37.7951	−1.21165
$974$	0	0
$975$	2.10360	0.0673690
$976$	0	0
$977$	−19.8444	−0.634878	−0.317439	−	0.948279i	$-0.602823\pi$
$977$	−19.8444	−0.634878	−0.317439	+	0.948279i	$0.602823\pi$
$978$	0	0
$979$	20.3489	0.650353
$980$	0	0
$981$	−7.52807	−0.240353
$982$	0	0
$983$	−41.6161	−1.32735	−0.663673	−	0.748023i	$-0.731003\pi$
$983$	−41.6161	−1.32735	−0.663673	+	0.748023i	$0.731003\pi$
$984$	0	0
$985$	23.7428	0.756508
$986$	0	0
$987$	1.92001	0.0611147
$988$	0	0
$989$	−1.23001	−0.0391120
$990$	0	0
$991$	11.9374	0.379205	0.189603	−	0.981861i	$-0.439280\pi$
$991$	11.9374	0.379205	0.189603	+	0.981861i	$0.439280\pi$
$992$	0	0
$993$	−10.8877	−0.345512
$994$	0	0
$995$	−22.9787	−0.728473
$996$	0	0
$997$	−19.2749	−0.610442	−0.305221	−	0.952282i	$-0.598730\pi$
$997$	−19.2749	−0.610442	−0.305221	+	0.952282i	$0.598730\pi$
$998$	0	0
$999$	−2.05170	−0.0649128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.p.1.3		4
19.8	odd	6		380.2.i.c.121.3	✓	8
19.12	odd	6		380.2.i.c.201.3	yes	8
19.18	odd	2		7220.2.a.r.1.2		4
57.8	even	6		3420.2.t.w.1261.1		8
57.50	even	6		3420.2.t.w.3241.1		8
76.27	even	6		1520.2.q.m.881.2		8
76.31	even	6		1520.2.q.m.961.2		8
95.8	even	12		1900.2.s.d.349.4		16
95.12	even	12		1900.2.s.d.49.4		16
95.27	even	12		1900.2.s.d.349.5		16
95.69	odd	6		1900.2.i.d.201.2		8
95.84	odd	6		1900.2.i.d.501.2		8
95.88	even	12		1900.2.s.d.49.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.c.121.3	✓	8	19.8	odd	6
380.2.i.c.201.3	yes	8	19.12	odd	6
1520.2.q.m.881.2		8	76.27	even	6
1520.2.q.m.961.2		8	76.31	even	6
1900.2.i.d.201.2		8	95.69	odd	6
1900.2.i.d.501.2		8	95.84	odd	6
1900.2.s.d.49.4		16	95.12	even	12
1900.2.s.d.49.5		16	95.88	even	12
1900.2.s.d.349.4		16	95.8	even	12
1900.2.s.d.349.5		16	95.27	even	12
3420.2.t.w.1261.1		8	57.8	even	6
3420.2.t.w.3241.1		8	57.50	even	6
7220.2.a.p.1.3		4	1.1	even	1	trivial
7220.2.a.r.1.2		4	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

\(f(q)\)	\(=\)	\(q+0.353450 q^{3} +1.00000 q^{5} -4.30507 q^{7} -2.87507 q^{9} +O(q^{10})\)	\(q+0.353450 q^{3} +1.00000 q^{5} -4.30507 q^{7} -2.87507 q^{9} +6.01196 q^{11} +5.95162 q^{13} +0.353450 q^{15} +3.87507 q^{17} -1.52162 q^{21} +0.783442 q^{23} +1.00000 q^{25} -2.07654 q^{27} -7.96358 q^{29} +4.49034 q^{31} +2.12493 q^{33} -4.30507 q^{35} +0.988035 q^{37} +2.10360 q^{39} -6.30507 q^{41} -1.57001 q^{43} -2.87507 q^{45} -1.26182 q^{47} +11.5336 q^{49} +1.36964 q^{51} +8.14886 q^{53} +6.01196 q^{55} -5.25668 q^{59} +5.61013 q^{61} +12.3774 q^{63} +5.95162 q^{65} -7.04325 q^{67} +0.276907 q^{69} +5.81472 q^{71} -9.24049 q^{73} +0.353450 q^{75} -25.8819 q^{77} +13.9949 q^{79} +7.89127 q^{81} +6.58197 q^{83} +3.87507 q^{85} -2.81472 q^{87} +3.38473 q^{89} -25.6221 q^{91} +1.58711 q^{93} +7.39670 q^{97} -17.2848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^9	\( 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47} + 10 q^{49} + 12 q^{51} + 5 q^{53} + 2 q^{55} + 11 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} - 3 q^{69} + 14 q^{71} + 4 q^{73} - q^{75} - 22 q^{77} + 13 q^{79} + 24 q^{81} + 5 q^{83} - q^{85} - 2 q^{87} + 5 q^{89} - 46 q^{91} + 28 q^{93} - q^{97} - 24 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 9 * q^13 - q^15 - q^17 + 8 * q^21 + 4 * q^25 - 10 * q^27 + 5 * q^29 + 10 * q^31 + 25 * q^33 + 26 * q^37 - 27 * q^39 - 8 * q^41 - 7 * q^43 + 5 * q^45 - 16 * q^47 + 10 * q^49 + 12 * q^51 + 5 * q^53 + 2 * q^55 + 11 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 - 3 * q^69 + 14 * q^71 + 4 * q^73 - q^75 - 22 * q^77 + 13 * q^79 + 24 * q^81 + 5 * q^83 - q^85 - 2 * q^87 + 5 * q^89 - 46 * q^91 + 28 * q^93 - q^97 - 24 * q^99

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