LMFDB - Embedded newform 7623.2.a.ch.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.46673$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46673 q^{2} +0.151302 q^{4} -0.466732 q^{5} -1.00000 q^{7} -2.71154 q^{8} +O(q^{10})$	$q+1.46673 q^{2} +0.151302 q^{4} -0.466732 q^{5} -1.00000 q^{7} -2.71154 q^{8} -0.684570 q^{10} -1.58232 q^{13} -1.46673 q^{14} -4.27971 q^{16} -5.22732 q^{17} +4.22192 q^{19} -0.0706175 q^{20} +1.80505 q^{23} -4.78216 q^{25} -2.32083 q^{26} -0.151302 q^{28} -2.71947 q^{29} +1.29386 q^{31} -0.854102 q^{32} -7.66708 q^{34} +0.466732 q^{35} -1.94221 q^{37} +6.19242 q^{38} +1.26556 q^{40} +1.04112 q^{41} +8.70820 q^{43} +2.64753 q^{46} +6.39530 q^{47} +1.00000 q^{49} -7.01415 q^{50} -0.239408 q^{52} +13.2044 q^{53} +2.71154 q^{56} -3.98873 q^{58} +8.60389 q^{59} -15.2401 q^{61} +1.89775 q^{62} +7.30669 q^{64} +0.738517 q^{65} -4.67583 q^{67} -0.790906 q^{68} +0.684570 q^{70} -9.74310 q^{71} -13.3200 q^{73} -2.84870 q^{74} +0.638786 q^{76} +3.58232 q^{79} +1.99748 q^{80} +1.52705 q^{82} +17.2589 q^{83} +2.43976 q^{85} +12.7726 q^{86} +8.91982 q^{89} +1.58232 q^{91} +0.273109 q^{92} +9.38018 q^{94} -1.97050 q^{95} -2.70362 q^{97} +1.46673 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} + 6 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 + 6 * q^5 - 4 * q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} + 6 q^{5} - 4 q^{7} - 9 q^{8} - 14 q^{10} + 2 q^{14} - 4 q^{16} + 3 q^{17} + 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} - 4 q^{28} - 3 q^{29} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - 7 q^{37} + 20 q^{38} - 13 q^{40} - 4 q^{41} + 8 q^{43} - 3 q^{46} + 14 q^{47} + 4 q^{49} - 33 q^{50} - 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} - 19 q^{61} - 10 q^{62} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{70} + 7 q^{71} - 11 q^{73} - 8 q^{74} - 26 q^{76} + 8 q^{79} + 4 q^{80} + 3 q^{82} + q^{83} + 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} - 20 q^{94} - 17 q^{95} - 15 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 + 6 * q^5 - 4 * q^7 - 9 * q^8 - 14 * q^10 + 2 * q^14 - 4 * q^16 + 3 * q^17 + 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 - 4 * q^28 - 3 * q^29 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - 7 * q^37 + 20 * q^38 - 13 * q^40 - 4 * q^41 + 8 * q^43 - 3 * q^46 + 14 * q^47 + 4 * q^49 - 33 * q^50 - 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 - 19 * q^61 - 10 * q^62 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^70 + 7 * q^71 - 11 * q^73 - 8 * q^74 - 26 * q^76 + 8 * q^79 + 4 * q^80 + 3 * q^82 + q^83 + 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 - 20 * q^94 - 17 * q^95 - 15 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.46673	1.03714	0.518568	−	0.855036i	$-0.326465\pi$
$2$	1.46673	1.03714	0.518568	+	0.855036i	$0.326465\pi$
$3$	0	0
$3$	0	0
$4$	0.151302	0.0756511
$5$	−0.466732	−0.208729	−0.104364	−	0.994539i	$-0.533281\pi$
$5$	−0.466732	−0.208729	−0.104364	+	0.994539i	$0.533281\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.71154	−0.958676
$9$	0	0
$10$	−0.684570	−0.216480
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.58232	−0.438856	−0.219428	−	0.975629i	$-0.570419\pi$
$13$	−1.58232	−0.438856	−0.219428	+	0.975629i	$0.570419\pi$
$14$	−1.46673	−0.392001
$15$	0	0
$16$	−4.27971	−1.06993
$17$	−5.22732	−1.26781	−0.633906	−	0.773410i	$-0.718549\pi$
$17$	−5.22732	−1.26781	−0.633906	+	0.773410i	$0.718549\pi$
$18$	0	0
$19$	4.22192	0.968575	0.484287	−	0.874909i	$-0.339079\pi$
$19$	4.22192	0.968575	0.484287	+	0.874909i	$0.339079\pi$
$20$	−0.0706175	−0.0157906
$21$	0	0
$22$	0	0
$23$	1.80505	0.376380	0.188190	−	0.982133i	$-0.439738\pi$
$23$	1.80505	0.376380	0.188190	+	0.982133i	$0.439738\pi$
$24$	0	0
$25$	−4.78216	−0.956432
$26$	−2.32083	−0.455153
$27$	0	0
$28$	−0.151302	−0.0285934
$29$	−2.71947	−0.504993	−0.252496	−	0.967598i	$-0.581252\pi$
$29$	−2.71947	−0.504993	−0.252496	+	0.967598i	$0.581252\pi$
$30$	0	0
$31$	1.29386	0.232384	0.116192	−	0.993227i	$-0.462931\pi$
$31$	1.29386	0.232384	0.116192	+	0.993227i	$0.462931\pi$
$32$	−0.854102	−0.150985
$33$	0	0
$34$	−7.66708	−1.31489
$35$	0.466732	0.0788921
$36$	0	0
$37$	−1.94221	−0.319297	−0.159648	−	0.987174i	$-0.551036\pi$
$37$	−1.94221	−0.319297	−0.159648	+	0.987174i	$0.551036\pi$
$38$	6.19242	1.00454
$39$	0	0
$40$	1.26556	0.200103
$41$	1.04112	0.162596	0.0812980	−	0.996690i	$-0.474093\pi$
$41$	1.04112	0.162596	0.0812980	+	0.996690i	$0.474093\pi$
$42$	0	0
$43$	8.70820	1.32799	0.663994	−	0.747738i	$-0.268860\pi$
$43$	8.70820	1.32799	0.663994	+	0.747738i	$0.268860\pi$
$44$	0	0
$45$	0	0
$46$	2.64753	0.390357
$47$	6.39530	0.932850	0.466425	−	0.884561i	$-0.345542\pi$
$47$	6.39530	0.932850	0.466425	+	0.884561i	$0.345542\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.01415	−0.991950
$51$	0	0
$52$	−0.239408	−0.0331999
$53$	13.2044	1.81377	0.906884	−	0.421380i	$-0.138454\pi$
$53$	13.2044	1.81377	0.906884	+	0.421380i	$0.138454\pi$
$54$	0	0
$55$	0	0
$56$	2.71154	0.362345
$57$	0	0
$58$	−3.98873	−0.523746
$59$	8.60389	1.12013	0.560065	−	0.828448i	$-0.310776\pi$
$59$	8.60389	1.12013	0.560065	+	0.828448i	$0.310776\pi$
$60$	0	0
$61$	−15.2401	−1.95130	−0.975651	−	0.219331i	$-0.929613\pi$
$61$	−15.2401	−1.95130	−0.975651	+	0.219331i	$0.929613\pi$
$62$	1.89775	0.241014
$63$	0	0
$64$	7.30669	0.913336
$65$	0.738517	0.0916018
$66$	0	0
$67$	−4.67583	−0.571243	−0.285622	−	0.958342i	$-0.592200\pi$
$67$	−4.67583	−0.571243	−0.285622	+	0.958342i	$0.592200\pi$
$68$	−0.790906	−0.0959114
$69$	0	0
$70$	0.684570	0.0818218
$71$	−9.74310	−1.15629	−0.578147	−	0.815933i	$-0.696224\pi$
$71$	−9.74310	−1.15629	−0.578147	+	0.815933i	$0.696224\pi$
$72$	0	0
$73$	−13.3200	−1.55899	−0.779495	−	0.626408i	$-0.784524\pi$
$73$	−13.3200	−1.55899	−0.779495	+	0.626408i	$0.784524\pi$
$74$	−2.84870	−0.331154
$75$	0	0
$76$	0.638786	0.0732737
$77$	0	0
$78$	0	0
$79$	3.58232	0.403042	0.201521	−	0.979484i	$-0.435412\pi$
$79$	3.58232	0.403042	0.201521	+	0.979484i	$0.435412\pi$
$80$	1.99748	0.223325
$81$	0	0
$82$	1.52705	0.168634
$83$	17.2589	1.89441	0.947204	−	0.320631i	$-0.103895\pi$
$83$	17.2589	1.89441	0.947204	+	0.320631i	$0.103895\pi$
$84$	0	0
$85$	2.43976	0.264629
$86$	12.7726	1.37730
$87$	0	0
$88$	0	0
$89$	8.91982	0.945499	0.472750	−	0.881197i	$-0.343262\pi$
$89$	8.91982	0.945499	0.472750	+	0.881197i	$0.343262\pi$
$90$	0	0
$91$	1.58232	0.165872
$92$	0.273109	0.0284735
$93$	0	0
$94$	9.38018	0.967492
$95$	−1.97050	−0.202169
$96$	0	0
$97$	−2.70362	−0.274511	−0.137255	−	0.990536i	$-0.543828\pi$
$97$	−2.70362	−0.274511	−0.137255	+	0.990536i	$0.543828\pi$
$98$	1.46673	0.148162
$99$	0	0
$100$	−0.723551	−0.0723551
$101$	−0.178781	−0.0177894	−0.00889469	−	0.999960i	$-0.502831\pi$
$101$	−0.178781	−0.0177894	−0.00889469	+	0.999960i	$0.502831\pi$
$102$	0	0
$103$	16.8772	1.66296	0.831481	−	0.555553i	$-0.187493\pi$
$103$	16.8772	1.66296	0.831481	+	0.555553i	$0.187493\pi$
$104$	4.29052	0.420720
$105$	0	0
$106$	19.3674	1.88112
$107$	15.4762	1.49614	0.748071	−	0.663618i	$-0.230980\pi$
$107$	15.4762	1.49614	0.748071	+	0.663618i	$0.230980\pi$
$108$	0	0
$109$	11.0349	1.05695	0.528476	−	0.848948i	$-0.322764\pi$
$109$	11.0349	1.05695	0.528476	+	0.848948i	$0.322764\pi$
$110$	0	0
$111$	0	0
$112$	4.27971	0.404395
$113$	−1.77008	−0.166515	−0.0832574	−	0.996528i	$-0.526532\pi$
$113$	−1.77008	−0.166515	−0.0832574	+	0.996528i	$0.526532\pi$
$114$	0	0
$115$	−0.842476	−0.0785613
$116$	−0.411462	−0.0382033
$117$	0	0
$118$	12.6196	1.16173
$119$	5.22732	0.479188
$120$	0	0
$121$	0	0
$122$	−22.3532	−2.02376
$123$	0	0
$124$	0.195764	0.0175801
$125$	4.56565	0.408364
$126$	0	0
$127$	−8.54023	−0.757823	−0.378911	−	0.925433i	$-0.623702\pi$
$127$	−8.54023	−0.757823	−0.378911	+	0.925433i	$0.623702\pi$
$128$	12.4252	1.09824
$129$	0	0
$130$	1.08321	0.0950035
$131$	9.66708	0.844617	0.422308	−	0.906452i	$-0.361220\pi$
$131$	9.66708	0.844617	0.422308	+	0.906452i	$0.361220\pi$
$132$	0	0
$133$	−4.22192	−0.366087
$134$	−6.85818	−0.592457
$135$	0	0
$136$	14.1741	1.21542
$137$	14.0108	1.19702	0.598512	−	0.801114i	$-0.295759\pi$
$137$	14.0108	1.19702	0.598512	+	0.801114i	$0.295759\pi$
$138$	0	0
$139$	9.57765	0.812366	0.406183	−	0.913792i	$-0.366860\pi$
$139$	9.57765	0.812366	0.406183	+	0.913792i	$0.366860\pi$
$140$	0.0706175	0.00596827
$141$	0	0
$142$	−14.2905	−1.19923
$143$	0	0
$144$	0	0
$145$	1.26926	0.105407
$146$	−19.5369	−1.61688
$147$	0	0
$148$	−0.293860	−0.0241552
$149$	−14.9625	−1.22578	−0.612888	−	0.790170i	$-0.709992\pi$
$149$	−14.9625	−1.22578	−0.612888	+	0.790170i	$0.709992\pi$
$150$	0	0
$151$	−2.87233	−0.233747	−0.116874	−	0.993147i	$-0.537287\pi$
$151$	−2.87233	−0.233747	−0.116874	+	0.993147i	$0.537287\pi$
$152$	−11.4479	−0.928549
$153$	0	0
$154$	0	0
$155$	−0.603886	−0.0485053
$156$	0	0
$157$	−18.8823	−1.50697	−0.753487	−	0.657463i	$-0.771630\pi$
$157$	−18.8823	−1.50697	−0.753487	+	0.657463i	$0.771630\pi$
$158$	5.25430	0.418009
$159$	0	0
$160$	0.398637	0.0315150
$161$	−1.80505	−0.142258
$162$	0	0
$163$	11.5951	0.908202	0.454101	−	0.890950i	$-0.349961\pi$
$163$	11.5951	0.908202	0.454101	+	0.890950i	$0.349961\pi$
$164$	0.157524	0.0123006
$165$	0	0
$166$	25.3142	1.96476
$167$	−6.32491	−0.489437	−0.244718	−	0.969594i	$-0.578695\pi$
$167$	−6.32491	−0.489437	−0.244718	+	0.969594i	$0.578695\pi$
$168$	0	0
$169$	−10.4963	−0.807406
$170$	3.57847	0.274456
$171$	0	0
$172$	1.31757	0.100464
$173$	−1.33906	−0.101807	−0.0509035	−	0.998704i	$-0.516210\pi$
$173$	−1.33906	−0.101807	−0.0509035	+	0.998704i	$0.516210\pi$
$174$	0	0
$175$	4.78216	0.361497
$176$	0	0
$177$	0	0
$178$	13.0830	0.980611
$179$	−17.7888	−1.32960	−0.664799	−	0.747022i	$-0.731483\pi$
$179$	−17.7888	−1.32960	−0.664799	+	0.747022i	$0.731483\pi$
$180$	0	0
$181$	−0.963777	−0.0716370	−0.0358185	−	0.999358i	$-0.511404\pi$
$181$	−0.963777	−0.0716370	−0.0358185	+	0.999358i	$0.511404\pi$
$182$	2.32083	0.172032
$183$	0	0
$184$	−4.89448	−0.360826
$185$	0.906490	0.0666465
$186$	0	0
$187$	0	0
$188$	0.967622	0.0705711
$189$	0	0
$190$	−2.89020	−0.209677
$191$	16.0888	1.16415	0.582074	−	0.813136i	$-0.302241\pi$
$191$	16.0888	1.16415	0.582074	+	0.813136i	$0.302241\pi$
$192$	0	0
$193$	12.1475	0.874393	0.437197	−	0.899366i	$-0.355971\pi$
$193$	12.1475	0.874393	0.437197	+	0.899366i	$0.355971\pi$
$194$	−3.96548	−0.284705
$195$	0	0
$196$	0.151302	0.0108073
$197$	2.30179	0.163996	0.0819978	−	0.996633i	$-0.473870\pi$
$197$	2.30179	0.163996	0.0819978	+	0.996633i	$0.473870\pi$
$198$	0	0
$199$	20.2797	1.43759	0.718795	−	0.695222i	$-0.244694\pi$
$199$	20.2797	1.43759	0.718795	+	0.695222i	$0.244694\pi$
$200$	12.9670	0.916908
$201$	0	0
$202$	−0.262224	−0.0184500
$203$	2.71947	0.190869
$204$	0	0
$205$	−0.485925	−0.0339384
$206$	24.7544	1.72472
$207$	0	0
$208$	6.77186	0.469544
$209$	0	0
$210$	0	0
$211$	5.36530	0.369362	0.184681	−	0.982799i	$-0.440875\pi$
$211$	5.36530	0.369362	0.184681	+	0.982799i	$0.440875\pi$
$212$	1.99786	0.137214
$213$	0	0
$214$	22.6995	1.55170
$215$	−4.06440	−0.277189
$216$	0	0
$217$	−1.29386	−0.0878330
$218$	16.1852	1.09620
$219$	0	0
$220$	0	0
$221$	8.27128	0.556387
$222$	0	0
$223$	−25.4230	−1.70245	−0.851225	−	0.524800i	$-0.824140\pi$
$223$	−25.4230	−1.70245	−0.851225	+	0.524800i	$0.824140\pi$
$224$	0.854102	0.0570671
$225$	0	0
$226$	−2.59623	−0.172699
$227$	21.7098	1.44093	0.720464	−	0.693493i	$-0.243929\pi$
$227$	21.7098	1.44093	0.720464	+	0.693493i	$0.243929\pi$
$228$	0	0
$229$	20.5307	1.35670	0.678352	−	0.734737i	$-0.262694\pi$
$229$	20.5307	1.35670	0.678352	+	0.734737i	$0.262694\pi$
$230$	−1.23569	−0.0814787
$231$	0	0
$232$	7.37396	0.484124
$233$	−0.694056	−0.0454691	−0.0227345	−	0.999742i	$-0.507237\pi$
$233$	−0.694056	−0.0454691	−0.0227345	+	0.999742i	$0.507237\pi$
$234$	0	0
$235$	−2.98489	−0.194713
$236$	1.30179	0.0847391
$237$	0	0
$238$	7.66708	0.496983
$239$	−0.346561	−0.0224171	−0.0112086	−	0.999937i	$-0.503568\pi$
$239$	−0.346561	−0.0224171	−0.0112086	+	0.999937i	$0.503568\pi$
$240$	0	0
$241$	−10.4372	−0.672317	−0.336158	−	0.941806i	$-0.609128\pi$
$241$	−10.4372	−0.672317	−0.336158	+	0.941806i	$0.609128\pi$
$242$	0	0
$243$	0	0
$244$	−2.30587	−0.147618
$245$	−0.466732	−0.0298184
$246$	0	0
$247$	−6.68041	−0.425064
$248$	−3.50836	−0.222781
$249$	0	0
$250$	6.69658	0.423529
$251$	6.99502	0.441522	0.220761	−	0.975328i	$-0.429146\pi$
$251$	6.99502	0.441522	0.220761	+	0.975328i	$0.429146\pi$
$252$	0	0
$253$	0	0
$254$	−12.5262	−0.785965
$255$	0	0
$256$	3.61099	0.225687
$257$	9.94843	0.620566	0.310283	−	0.950644i	$-0.399576\pi$
$257$	9.94843	0.620566	0.310283	+	0.950644i	$0.399576\pi$
$258$	0	0
$259$	1.94221	0.120683
$260$	0.111739	0.00692978
$261$	0	0
$262$	14.1790	0.875983
$263$	−14.1803	−0.874397	−0.437199	−	0.899365i	$-0.644029\pi$
$263$	−14.1803	−0.874397	−0.437199	+	0.899365i	$0.644029\pi$
$264$	0	0
$265$	−6.16293	−0.378586
$266$	−6.19242	−0.379682
$267$	0	0
$268$	−0.707463	−0.0432152
$269$	18.4031	1.12206	0.561028	−	0.827797i	$-0.310406\pi$
$269$	18.4031	1.12206	0.561028	+	0.827797i	$0.310406\pi$
$270$	0	0
$271$	0.730591	0.0443802	0.0221901	−	0.999754i	$-0.492936\pi$
$271$	0.730591	0.0443802	0.0221901	+	0.999754i	$0.492936\pi$
$272$	22.3714	1.35647
$273$	0	0
$274$	20.5501	1.24148
$275$	0	0
$276$	0	0
$277$	15.0644	0.905132	0.452566	−	0.891731i	$-0.350509\pi$
$277$	15.0644	0.905132	0.452566	+	0.891731i	$0.350509\pi$
$278$	14.0478	0.842534
$279$	0	0
$280$	−1.26556	−0.0756319
$281$	10.6961	0.638077	0.319039	−	0.947742i	$-0.396640\pi$
$281$	10.6961	0.638077	0.319039	+	0.947742i	$0.396640\pi$
$282$	0	0
$283$	9.10890	0.541468	0.270734	−	0.962654i	$-0.412734\pi$
$283$	9.10890	0.541468	0.270734	+	0.962654i	$0.412734\pi$
$284$	−1.47415	−0.0874749
$285$	0	0
$286$	0	0
$287$	−1.04112	−0.0614555
$288$	0	0
$289$	10.3249	0.607348
$290$	1.86167	0.109321
$291$	0	0
$292$	−2.01535	−0.117939
$293$	−11.8890	−0.694563	−0.347281	−	0.937761i	$-0.612895\pi$
$293$	−11.8890	−0.694563	−0.347281	+	0.937761i	$0.612895\pi$
$294$	0	0
$295$	−4.01571	−0.233804
$296$	5.26638	0.306102
$297$	0	0
$298$	−21.9460	−1.27130
$299$	−2.85617	−0.165176
$300$	0	0
$301$	−8.70820	−0.501933
$302$	−4.21294	−0.242427
$303$	0	0
$304$	−18.0686	−1.03631
$305$	7.11306	0.407293
$306$	0	0
$307$	2.22072	0.126743	0.0633716	−	0.997990i	$-0.479815\pi$
$307$	2.22072	0.126743	0.0633716	+	0.997990i	$0.479815\pi$
$308$	0	0
$309$	0	0
$310$	−0.885738	−0.0503066
$311$	21.4126	1.21420	0.607098	−	0.794627i	$-0.292334\pi$
$311$	21.4126	1.21420	0.607098	+	0.794627i	$0.292334\pi$
$312$	0	0
$313$	−31.5548	−1.78358	−0.891790	−	0.452449i	$-0.850550\pi$
$313$	−31.5548	−1.78358	−0.891790	+	0.452449i	$0.850550\pi$
$314$	−27.6953	−1.56294
$315$	0	0
$316$	0.542012	0.0304906
$317$	13.2007	0.741423	0.370712	−	0.928748i	$-0.379114\pi$
$317$	13.2007	0.741423	0.370712	+	0.928748i	$0.379114\pi$
$318$	0	0
$319$	0	0
$320$	−3.41026	−0.190639
$321$	0	0
$322$	−2.64753	−0.147541
$323$	−22.0693	−1.22797
$324$	0	0
$325$	7.56689	0.419736
$326$	17.0070	0.941929
$327$	0	0
$328$	−2.82305	−0.155877
$329$	−6.39530	−0.352584
$330$	0	0
$331$	9.47653	0.520877	0.260439	−	0.965490i	$-0.416133\pi$
$331$	9.47653	0.520877	0.260439	+	0.965490i	$0.416133\pi$
$332$	2.61131	0.143314
$333$	0	0
$334$	−9.27695	−0.507612
$335$	2.18236	0.119235
$336$	0	0
$337$	−19.2011	−1.04595	−0.522975	−	0.852348i	$-0.675178\pi$
$337$	−19.2011	−1.04595	−0.522975	+	0.852348i	$0.675178\pi$
$338$	−15.3952	−0.837390
$339$	0	0
$340$	0.369141	0.0200195
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−23.6127	−1.27311
$345$	0	0
$346$	−1.96405	−0.105588
$347$	−3.04831	−0.163642	−0.0818208	−	0.996647i	$-0.526074\pi$
$347$	−3.04831	−0.163642	−0.0818208	+	0.996647i	$0.526074\pi$
$348$	0	0
$349$	19.3961	1.03825	0.519125	−	0.854698i	$-0.326258\pi$
$349$	19.3961	1.03825	0.519125	+	0.854698i	$0.326258\pi$
$350$	7.01415	0.374922
$351$	0	0
$352$	0	0
$353$	10.7585	0.572619	0.286309	−	0.958137i	$-0.407572\pi$
$353$	10.7585	0.572619	0.286309	+	0.958137i	$0.407572\pi$
$354$	0	0
$355$	4.54742	0.241352
$356$	1.34959	0.0715280
$357$	0	0
$358$	−26.0914	−1.37897
$359$	−0.607226	−0.0320481	−0.0160241	−	0.999872i	$-0.505101\pi$
$359$	−0.607226	−0.0320481	−0.0160241	+	0.999872i	$0.505101\pi$
$360$	0	0
$361$	−1.17539	−0.0618628
$362$	−1.41360	−0.0742973
$363$	0	0
$364$	0.239408	0.0125484
$365$	6.21688	0.325406
$366$	0	0
$367$	27.6628	1.44399	0.721994	−	0.691899i	$-0.243226\pi$
$367$	27.6628	1.44399	0.721994	+	0.691899i	$0.243226\pi$
$368$	−7.72511	−0.402699
$369$	0	0
$370$	1.32958	0.0691215
$371$	−13.2044	−0.685540
$372$	0	0
$373$	−29.4513	−1.52493	−0.762465	−	0.647029i	$-0.776011\pi$
$373$	−29.4513	−1.52493	−0.762465	+	0.647029i	$0.776011\pi$
$374$	0	0
$375$	0	0
$376$	−17.3411	−0.894300
$377$	4.30306	0.221619
$378$	0	0
$379$	25.3436	1.30182	0.650908	−	0.759157i	$-0.274388\pi$
$379$	25.3436	1.30182	0.650908	+	0.759157i	$0.274388\pi$
$380$	−0.298142	−0.0152943
$381$	0	0
$382$	23.5980	1.20738
$383$	31.9322	1.63166	0.815829	−	0.578293i	$-0.196281\pi$
$383$	31.9322	1.63166	0.815829	+	0.578293i	$0.196281\pi$
$384$	0	0
$385$	0	0
$386$	17.8171	0.906865
$387$	0	0
$388$	−0.409063	−0.0207670
$389$	−17.7517	−0.900047	−0.450024	−	0.893017i	$-0.648584\pi$
$389$	−17.7517	−0.900047	−0.450024	+	0.893017i	$0.648584\pi$
$390$	0	0
$391$	−9.43560	−0.477179
$392$	−2.71154	−0.136954
$393$	0	0
$394$	3.37610	0.170086
$395$	−1.67198	−0.0841265
$396$	0	0
$397$	−13.3047	−0.667742	−0.333871	−	0.942619i	$-0.608355\pi$
$397$	−13.3047	−0.667742	−0.333871	+	0.942619i	$0.608355\pi$
$398$	29.7449	1.49098
$399$	0	0
$400$	20.4663	1.02331
$401$	3.48962	0.174264	0.0871318	−	0.996197i	$-0.472230\pi$
$401$	3.48962	0.174264	0.0871318	+	0.996197i	$0.472230\pi$
$402$	0	0
$403$	−2.04730	−0.101983
$404$	−0.0270500	−0.00134579
$405$	0	0
$406$	3.98873	0.197958
$407$	0	0
$408$	0	0
$409$	−29.6255	−1.46489	−0.732443	−	0.680828i	$-0.761620\pi$
$409$	−29.6255	−1.46489	−0.732443	+	0.680828i	$0.761620\pi$
$410$	−0.712721	−0.0351988
$411$	0	0
$412$	2.55356	0.125805
$413$	−8.60389	−0.423370
$414$	0	0
$415$	−8.05527	−0.395418
$416$	1.35146	0.0662608
$417$	0	0
$418$	0	0
$419$	11.6452	0.568907	0.284454	−	0.958690i	$-0.408188\pi$
$419$	11.6452	0.568907	0.284454	+	0.958690i	$0.408188\pi$
$420$	0	0
$421$	19.8848	0.969128	0.484564	−	0.874756i	$-0.338978\pi$
$421$	19.8848	0.969128	0.484564	+	0.874756i	$0.338978\pi$
$422$	7.86945	0.383079
$423$	0	0
$424$	−35.8044	−1.73882
$425$	24.9979	1.21258
$426$	0	0
$427$	15.2401	0.737523
$428$	2.34159	0.113185
$429$	0	0
$430$	−5.96138	−0.287483
$431$	−30.2464	−1.45692	−0.728458	−	0.685090i	$-0.759763\pi$
$431$	−30.2464	−1.45692	−0.728458	+	0.685090i	$0.759763\pi$
$432$	0	0
$433$	−5.70719	−0.274270	−0.137135	−	0.990552i	$-0.543789\pi$
$433$	−5.70719	−0.274270	−0.137135	+	0.990552i	$0.543789\pi$
$434$	−1.89775	−0.0910947
$435$	0	0
$436$	1.66960	0.0799596
$437$	7.62079	0.364552
$438$	0	0
$439$	6.84875	0.326873	0.163436	−	0.986554i	$-0.447742\pi$
$439$	6.84875	0.326873	0.163436	+	0.986554i	$0.447742\pi$
$440$	0	0
$441$	0	0
$442$	12.1317	0.577048
$443$	−0.100695	−0.00478417	−0.00239209	−	0.999997i	$-0.500761\pi$
$443$	−0.100695	−0.00478417	−0.00239209	+	0.999997i	$0.500761\pi$
$444$	0	0
$445$	−4.16316	−0.197353
$446$	−37.2887	−1.76567
$447$	0	0
$448$	−7.30669	−0.345208
$449$	30.8047	1.45377	0.726883	−	0.686762i	$-0.240968\pi$
$449$	30.8047	1.45377	0.726883	+	0.686762i	$0.240968\pi$
$450$	0	0
$451$	0	0
$452$	−0.267817	−0.0125970
$453$	0	0
$454$	31.8424	1.49444
$455$	−0.738517	−0.0346222
$456$	0	0
$457$	23.1356	1.08224	0.541118	−	0.840947i	$-0.318001\pi$
$457$	23.1356	1.08224	0.541118	+	0.840947i	$0.318001\pi$
$458$	30.1130	1.40709
$459$	0	0
$460$	−0.127468	−0.00594325
$461$	2.77839	0.129403	0.0647013	−	0.997905i	$-0.479391\pi$
$461$	2.77839	0.129403	0.0647013	+	0.997905i	$0.479391\pi$
$462$	0	0
$463$	−26.0950	−1.21274	−0.606369	−	0.795184i	$-0.707374\pi$
$463$	−26.0950	−1.21274	−0.606369	+	0.795184i	$0.707374\pi$
$464$	11.6385	0.540306
$465$	0	0
$466$	−1.01799	−0.0471576
$467$	2.65829	0.123011	0.0615055	−	0.998107i	$-0.480410\pi$
$467$	2.65829	0.123011	0.0615055	+	0.998107i	$0.480410\pi$
$468$	0	0
$469$	4.67583	0.215910
$470$	−4.37803	−0.201943
$471$	0	0
$472$	−23.3298	−1.07384
$473$	0	0
$474$	0	0
$475$	−20.1899	−0.926376
$476$	0.790906	0.0362511
$477$	0	0
$478$	−0.508312	−0.0232496
$479$	8.28223	0.378425	0.189212	−	0.981936i	$-0.439407\pi$
$479$	8.28223	0.378425	0.189212	+	0.981936i	$0.439407\pi$
$480$	0	0
$481$	3.07319	0.140125
$482$	−15.3085	−0.697284
$483$	0	0
$484$	0	0
$485$	1.26186	0.0572983
$486$	0	0
$487$	−19.5956	−0.887960	−0.443980	−	0.896037i	$-0.646434\pi$
$487$	−19.5956	−0.887960	−0.443980	+	0.896037i	$0.646434\pi$
$488$	41.3243	1.87066
$489$	0	0
$490$	−0.684570	−0.0309257
$491$	28.6817	1.29439	0.647193	−	0.762327i	$-0.275943\pi$
$491$	28.6817	1.29439	0.647193	+	0.762327i	$0.275943\pi$
$492$	0	0
$493$	14.2156	0.640236
$494$	−9.79837	−0.440850
$495$	0	0
$496$	−5.53735	−0.248634
$497$	9.74310	0.437038
$498$	0	0
$499$	27.9499	1.25121	0.625605	−	0.780140i	$-0.284852\pi$
$499$	27.9499	1.25121	0.625605	+	0.780140i	$0.284852\pi$
$500$	0.690792	0.0308932
$501$	0	0
$502$	10.2598	0.457918
$503$	8.09736	0.361043	0.180522	−	0.983571i	$-0.442221\pi$
$503$	8.09736	0.361043	0.180522	+	0.983571i	$0.442221\pi$
$504$	0	0
$505$	0.0834428	0.00371316
$506$	0	0
$507$	0	0
$508$	−1.29216	−0.0573301
$509$	−16.3002	−0.722492	−0.361246	−	0.932471i	$-0.617648\pi$
$509$	−16.3002	−0.722492	−0.361246	+	0.932471i	$0.617648\pi$
$510$	0	0
$511$	13.3200	0.589243
$512$	−19.5539	−0.864170
$513$	0	0
$514$	14.5917	0.643611
$515$	−7.87714	−0.347108
$516$	0	0
$517$	0	0
$518$	2.84870	0.125165
$519$	0	0
$520$	−2.00252	−0.0878164
$521$	−7.68605	−0.336732	−0.168366	−	0.985725i	$-0.553849\pi$
$521$	−7.68605	−0.336732	−0.168366	+	0.985725i	$0.553849\pi$
$522$	0	0
$523$	26.1229	1.14228	0.571138	−	0.820854i	$-0.306502\pi$
$523$	26.1229	1.14228	0.571138	+	0.820854i	$0.306502\pi$
$524$	1.46265	0.0638962
$525$	0	0
$526$	−20.7988	−0.906869
$527$	−6.76343	−0.294619
$528$	0	0
$529$	−19.7418	−0.858338
$530$	−9.03936	−0.392645
$531$	0	0
$532$	−0.638786	−0.0276949
$533$	−1.64738	−0.0713561
$534$	0	0
$535$	−7.22324	−0.312288
$536$	12.6787	0.547637
$537$	0	0
$538$	26.9924	1.16372
$539$	0	0
$540$	0	0
$541$	−25.8777	−1.11257	−0.556284	−	0.830992i	$-0.687773\pi$
$541$	−25.8777	−1.11257	−0.556284	+	0.830992i	$0.687773\pi$
$542$	1.07158	0.0460283
$543$	0	0
$544$	4.46467	0.191421
$545$	−5.15034	−0.220616
$546$	0	0
$547$	−38.0968	−1.62890	−0.814451	−	0.580232i	$-0.802962\pi$
$547$	−38.0968	−1.62890	−0.814451	+	0.580232i	$0.802962\pi$
$548$	2.11987	0.0905562
$549$	0	0
$550$	0	0
$551$	−11.4814	−0.489123
$552$	0	0
$553$	−3.58232	−0.152336
$554$	22.0954	0.938745
$555$	0	0
$556$	1.44912	0.0614564
$557$	−34.5422	−1.46360	−0.731799	−	0.681520i	$-0.761319\pi$
$557$	−34.5422	−1.46360	−0.731799	+	0.681520i	$0.761319\pi$
$558$	0	0
$559$	−13.7791	−0.582795
$560$	−1.99748	−0.0844088
$561$	0	0
$562$	15.6883	0.661773
$563$	−19.4819	−0.821066	−0.410533	−	0.911846i	$-0.634657\pi$
$563$	−19.4819	−0.821066	−0.410533	+	0.911846i	$0.634657\pi$
$564$	0	0
$565$	0.826151	0.0347565
$566$	13.3603	0.561576
$567$	0	0
$568$	26.4189	1.10851
$569$	−17.1288	−0.718075	−0.359038	−	0.933323i	$-0.616895\pi$
$569$	−17.1288	−0.718075	−0.359038	+	0.933323i	$0.616895\pi$
$570$	0	0
$571$	−3.85581	−0.161360	−0.0806802	−	0.996740i	$-0.525709\pi$
$571$	−3.85581	−0.161360	−0.0806802	+	0.996740i	$0.525709\pi$
$572$	0	0
$573$	0	0
$574$	−1.52705	−0.0637377
$575$	−8.63206	−0.359982
$576$	0	0
$577$	−9.78185	−0.407224	−0.203612	−	0.979052i	$-0.565268\pi$
$577$	−9.78185	−0.407224	−0.203612	+	0.979052i	$0.565268\pi$
$578$	15.1439	0.629902
$579$	0	0
$580$	0.192042	0.00797412
$581$	−17.2589	−0.716019
$582$	0	0
$583$	0	0
$584$	36.1178	1.49457
$585$	0	0
$586$	−17.4380	−0.720356
$587$	6.09891	0.251729	0.125865	−	0.992047i	$-0.459830\pi$
$587$	6.09891	0.251729	0.125865	+	0.992047i	$0.459830\pi$
$588$	0	0
$589$	5.46257	0.225081
$590$	−5.88997	−0.242486
$591$	0	0
$592$	8.31209	0.341625
$593$	−13.2330	−0.543413	−0.271706	−	0.962380i	$-0.587588\pi$
$593$	−13.2330	−0.543413	−0.271706	+	0.962380i	$0.587588\pi$
$594$	0	0
$595$	−2.43976	−0.100020
$596$	−2.26386	−0.0927313
$597$	0	0
$598$	−4.18923	−0.171310
$599$	5.92515	0.242095	0.121048	−	0.992647i	$-0.461375\pi$
$599$	5.92515	0.242095	0.121048	+	0.992647i	$0.461375\pi$
$600$	0	0
$601$	12.7408	0.519708	0.259854	−	0.965648i	$-0.416326\pi$
$601$	12.7408	0.519708	0.259854	+	0.965648i	$0.416326\pi$
$602$	−12.7726	−0.520572
$603$	0	0
$604$	−0.434590	−0.0176832
$605$	0	0
$606$	0	0
$607$	8.36141	0.339379	0.169690	−	0.985498i	$-0.445724\pi$
$607$	8.36141	0.339379	0.169690	+	0.985498i	$0.445724\pi$
$608$	−3.60595	−0.146241
$609$	0	0
$610$	10.4330	0.422418
$611$	−10.1194	−0.409386
$612$	0	0
$613$	−6.68294	−0.269921	−0.134961	−	0.990851i	$-0.543091\pi$
$613$	−6.68294	−0.269921	−0.134961	+	0.990851i	$0.543091\pi$
$614$	3.25720	0.131450
$615$	0	0
$616$	0	0
$617$	−11.8669	−0.477741	−0.238871	−	0.971051i	$-0.576777\pi$
$617$	−11.8669	−0.477741	−0.238871	+	0.971051i	$0.576777\pi$
$618$	0	0
$619$	20.6206	0.828814	0.414407	−	0.910092i	$-0.363989\pi$
$619$	20.6206	0.828814	0.414407	+	0.910092i	$0.363989\pi$
$620$	−0.0913692	−0.00366948
$621$	0	0
$622$	31.4065	1.25929
$623$	−8.91982	−0.357365
$624$	0	0
$625$	21.7799	0.871195
$626$	−46.2824	−1.84982
$627$	0	0
$628$	−2.85694	−0.114004
$629$	10.1525	0.404809
$630$	0	0
$631$	−15.4795	−0.616228	−0.308114	−	0.951349i	$-0.599698\pi$
$631$	−15.4795	−0.616228	−0.308114	+	0.951349i	$0.599698\pi$
$632$	−9.71361	−0.386387
$633$	0	0
$634$	19.3618	0.768957
$635$	3.98600	0.158179
$636$	0	0
$637$	−1.58232	−0.0626937
$638$	0	0
$639$	0	0
$640$	−5.79921	−0.229234
$641$	23.5785	0.931294	0.465647	−	0.884971i	$-0.345822\pi$
$641$	23.5785	0.931294	0.465647	+	0.884971i	$0.345822\pi$
$642$	0	0
$643$	28.6806	1.13105	0.565527	−	0.824730i	$-0.308673\pi$
$643$	28.6806	1.13105	0.565527	+	0.824730i	$0.308673\pi$
$644$	−0.273109	−0.0107620
$645$	0	0
$646$	−32.3698	−1.27357
$647$	−5.42763	−0.213382	−0.106691	−	0.994292i	$-0.534026\pi$
$647$	−5.42763	−0.213382	−0.106691	+	0.994292i	$0.534026\pi$
$648$	0	0
$649$	0	0
$650$	11.0986	0.435323
$651$	0	0
$652$	1.75437	0.0687064
$653$	12.3421	0.482983	0.241492	−	0.970403i	$-0.422363\pi$
$653$	12.3421	0.482983	0.241492	+	0.970403i	$0.422363\pi$
$654$	0	0
$655$	−4.51193	−0.176296
$656$	−4.45570	−0.173966
$657$	0	0
$658$	−9.38018	−0.365678
$659$	−16.2115	−0.631512	−0.315756	−	0.948840i	$-0.602258\pi$
$659$	−16.2115	−0.631512	−0.315756	+	0.948840i	$0.602258\pi$
$660$	0	0
$661$	43.7050	1.69993	0.849964	−	0.526840i	$-0.176623\pi$
$661$	43.7050	1.69993	0.849964	+	0.526840i	$0.176623\pi$
$662$	13.8995	0.540220
$663$	0	0
$664$	−46.7982	−1.81612
$665$	1.97050	0.0764129
$666$	0	0
$667$	−4.90879	−0.190069
$668$	−0.956973	−0.0370264
$669$	0	0
$670$	3.20093	0.123663
$671$	0	0
$672$	0	0
$673$	−5.86102	−0.225926	−0.112963	−	0.993599i	$-0.536034\pi$
$673$	−5.86102	−0.225926	−0.112963	+	0.993599i	$0.536034\pi$
$674$	−28.1629	−1.08479
$675$	0	0
$676$	−1.58811	−0.0610811
$677$	20.5279	0.788952	0.394476	−	0.918906i	$-0.370926\pi$
$677$	20.5279	0.788952	0.394476	+	0.918906i	$0.370926\pi$
$678$	0	0
$679$	2.70362	0.103755
$680$	−6.61551	−0.253693
$681$	0	0
$682$	0	0
$683$	−38.7055	−1.48103	−0.740513	−	0.672042i	$-0.765417\pi$
$683$	−38.7055	−1.48103	−0.740513	+	0.672042i	$0.765417\pi$
$684$	0	0
$685$	−6.53929	−0.249853
$686$	−1.46673	−0.0560001
$687$	0	0
$688$	−37.2686	−1.42085
$689$	−20.8936	−0.795982
$690$	0	0
$691$	−23.1300	−0.879907	−0.439954	−	0.898020i	$-0.645005\pi$
$691$	−23.1300	−0.879907	−0.439954	+	0.898020i	$0.645005\pi$
$692$	−0.202603	−0.00770182
$693$	0	0
$694$	−4.47105	−0.169719
$695$	−4.47020	−0.169564
$696$	0	0
$697$	−5.44228	−0.206141
$698$	28.4489	1.07681
$699$	0	0
$700$	0.723551	0.0273477
$701$	−35.5107	−1.34122	−0.670610	−	0.741810i	$-0.733968\pi$
$701$	−35.5107	−1.34122	−0.670610	+	0.741810i	$0.733968\pi$
$702$	0	0
$703$	−8.19985	−0.309263
$704$	0	0
$705$	0	0
$706$	15.7799	0.593883
$707$	0.178781	0.00672375
$708$	0	0
$709$	43.8045	1.64511	0.822556	−	0.568684i	$-0.192547\pi$
$709$	43.8045	1.64511	0.822556	+	0.568684i	$0.192547\pi$
$710$	6.66984	0.250315
$711$	0	0
$712$	−24.1865	−0.906427
$713$	2.33549	0.0874647
$714$	0	0
$715$	0	0
$716$	−2.69149	−0.100586
$717$	0	0
$718$	−0.890637	−0.0332383
$719$	−15.8605	−0.591496	−0.295748	−	0.955266i	$-0.595569\pi$
$719$	−15.8605	−0.591496	−0.295748	+	0.955266i	$0.595569\pi$
$720$	0	0
$721$	−16.8772	−0.628541
$722$	−1.72399	−0.0641602
$723$	0	0
$724$	−0.145822	−0.00541942
$725$	13.0049	0.482992
$726$	0	0
$727$	13.7719	0.510770	0.255385	−	0.966839i	$-0.417798\pi$
$727$	13.7719	0.510770	0.255385	+	0.966839i	$0.417798\pi$
$728$	−4.29052	−0.159017
$729$	0	0
$730$	9.11849	0.337490
$731$	−45.5206	−1.68364
$732$	0	0
$733$	−19.5677	−0.722750	−0.361375	−	0.932421i	$-0.617693\pi$
$733$	−19.5677	−0.722750	−0.361375	+	0.932421i	$0.617693\pi$
$734$	40.5740	1.49761
$735$	0	0
$736$	−1.54170	−0.0568278
$737$	0	0
$738$	0	0
$739$	−11.1542	−0.410313	−0.205157	−	0.978729i	$-0.565770\pi$
$739$	−11.1542	−0.410313	−0.205157	+	0.978729i	$0.565770\pi$
$740$	0.137154	0.00504188
$741$	0	0
$742$	−19.3674	−0.710998
$743$	−20.6550	−0.757757	−0.378878	−	0.925446i	$-0.623690\pi$
$743$	−20.6550	−0.757757	−0.378878	+	0.925446i	$0.623690\pi$
$744$	0	0
$745$	6.98348	0.255855
$746$	−43.1972	−1.58156
$747$	0	0
$748$	0	0
$749$	−15.4762	−0.565489
$750$	0	0
$751$	26.5991	0.970614	0.485307	−	0.874344i	$-0.338708\pi$
$751$	26.5991	0.970614	0.485307	+	0.874344i	$0.338708\pi$
$752$	−27.3700	−0.998082
$753$	0	0
$754$	6.31144	0.229849
$755$	1.34061	0.0487897
$756$	0	0
$757$	21.0999	0.766890	0.383445	−	0.923564i	$-0.374738\pi$
$757$	21.0999	0.766890	0.383445	+	0.923564i	$0.374738\pi$
$758$	37.1723	1.35016
$759$	0	0
$760$	5.34311	0.193815
$761$	7.99743	0.289907	0.144953	−	0.989438i	$-0.453697\pi$
$761$	7.99743	0.289907	0.144953	+	0.989438i	$0.453697\pi$
$762$	0	0
$763$	−11.0349	−0.399490
$764$	2.43428	0.0880691
$765$	0	0
$766$	46.8360	1.69225
$767$	−13.6141	−0.491576
$768$	0	0
$769$	−52.0476	−1.87689	−0.938443	−	0.345435i	$-0.887731\pi$
$769$	−52.0476	−1.87689	−0.938443	+	0.345435i	$0.887731\pi$
$770$	0	0
$771$	0	0
$772$	1.83794	0.0661488
$773$	1.58099	0.0568644	0.0284322	−	0.999596i	$-0.490949\pi$
$773$	1.58099	0.0568644	0.0284322	+	0.999596i	$0.490949\pi$
$774$	0	0
$775$	−6.18745	−0.222260
$776$	7.33098	0.263167
$777$	0	0
$778$	−26.0370	−0.933471
$779$	4.39553	0.157486
$780$	0	0
$781$	0	0
$782$	−13.8395	−0.494899
$783$	0	0
$784$	−4.27971	−0.152847
$785$	8.81298	0.314549
$786$	0	0
$787$	23.3907	0.833789	0.416894	−	0.908955i	$-0.363118\pi$
$787$	23.3907	0.833789	0.416894	+	0.908955i	$0.363118\pi$
$788$	0.348265	0.0124064
$789$	0	0
$790$	−2.45235	−0.0872506
$791$	1.77008	0.0629367
$792$	0	0
$793$	24.1147	0.856339
$794$	−19.5144	−0.692539
$795$	0	0
$796$	3.06836	0.108755
$797$	46.1518	1.63478	0.817391	−	0.576084i	$-0.195420\pi$
$797$	46.1518	1.63478	0.817391	+	0.576084i	$0.195420\pi$
$798$	0	0
$799$	−33.4303	−1.18268
$800$	4.08445	0.144407
$801$	0	0
$802$	5.11834	0.180735
$803$	0	0
$804$	0	0
$805$	0.842476	0.0296934
$806$	−3.00283	−0.105770
$807$	0	0
$808$	0.484773	0.0170542
$809$	−33.7501	−1.18659	−0.593295	−	0.804985i	$-0.702173\pi$
$809$	−33.7501	−1.18659	−0.593295	+	0.804985i	$0.702173\pi$
$810$	0	0
$811$	−30.7650	−1.08030	−0.540152	−	0.841567i	$-0.681633\pi$
$811$	−30.7650	−1.08030	−0.540152	+	0.841567i	$0.681633\pi$
$812$	0.411462	0.0144395
$813$	0	0
$814$	0	0
$815$	−5.41182	−0.189568
$816$	0	0
$817$	36.7653	1.28626
$818$	−43.4527	−1.51929
$819$	0	0
$820$	−0.0735215	−0.00256748
$821$	−12.0784	−0.421538	−0.210769	−	0.977536i	$-0.567597\pi$
$821$	−12.0784	−0.421538	−0.210769	+	0.977536i	$0.567597\pi$
$822$	0	0
$823$	−24.8187	−0.865124	−0.432562	−	0.901604i	$-0.642390\pi$
$823$	−24.8187	−0.865124	−0.432562	+	0.901604i	$0.642390\pi$
$824$	−45.7634	−1.59424
$825$	0	0
$826$	−12.6196	−0.439092
$827$	−5.17330	−0.179893	−0.0899466	−	0.995947i	$-0.528670\pi$
$827$	−5.17330	−0.179893	−0.0899466	+	0.995947i	$0.528670\pi$
$828$	0	0
$829$	31.5535	1.09590	0.547949	−	0.836511i	$-0.315409\pi$
$829$	31.5535	1.09590	0.547949	+	0.836511i	$0.315409\pi$
$830$	−11.8149	−0.410102
$831$	0	0
$832$	−11.5615	−0.400822
$833$	−5.22732	−0.181116
$834$	0	0
$835$	2.95204	0.102160
$836$	0	0
$837$	0	0
$838$	17.0804	0.590034
$839$	5.83642	0.201496	0.100748	−	0.994912i	$-0.467876\pi$
$839$	5.83642	0.201496	0.100748	+	0.994912i	$0.467876\pi$
$840$	0	0
$841$	−21.6045	−0.744982
$842$	29.1657	1.00512
$843$	0	0
$844$	0.811781	0.0279427
$845$	4.89895	0.168529
$846$	0	0
$847$	0	0
$848$	−56.5112	−1.94060
$849$	0	0
$850$	36.6652	1.25761
$851$	−3.50579	−0.120177
$852$	0	0
$853$	−20.3462	−0.696640	−0.348320	−	0.937376i	$-0.613248\pi$
$853$	−20.3462	−0.696640	−0.348320	+	0.937376i	$0.613248\pi$
$854$	22.3532	0.764911
$855$	0	0
$856$	−41.9644	−1.43432
$857$	15.1087	0.516104	0.258052	−	0.966131i	$-0.416919\pi$
$857$	15.1087	0.516104	0.258052	+	0.966131i	$0.416919\pi$
$858$	0	0
$859$	−33.9641	−1.15884	−0.579420	−	0.815029i	$-0.696721\pi$
$859$	−33.9641	−1.15884	−0.579420	+	0.815029i	$0.696721\pi$
$860$	−0.614952	−0.0209697
$861$	0	0
$862$	−44.3633	−1.51102
$863$	2.77734	0.0945417	0.0472709	−	0.998882i	$-0.484948\pi$
$863$	2.77734	0.0945417	0.0472709	+	0.998882i	$0.484948\pi$
$864$	0	0
$865$	0.624983	0.0212501
$866$	−8.37092	−0.284456
$867$	0	0
$868$	−0.195764	−0.00664466
$869$	0	0
$870$	0	0
$871$	7.39864	0.250693
$872$	−29.9216	−1.01327
$873$	0	0
$874$	11.1777	0.378090
$875$	−4.56565	−0.154347
$876$	0	0
$877$	49.6783	1.67752	0.838759	−	0.544503i	$-0.183282\pi$
$877$	49.6783	1.67752	0.838759	+	0.544503i	$0.183282\pi$
$878$	10.0453	0.339012
$879$	0	0
$880$	0	0
$881$	27.3064	0.919975	0.459988	−	0.887925i	$-0.347854\pi$
$881$	27.3064	0.919975	0.459988	+	0.887925i	$0.347854\pi$
$882$	0	0
$883$	−17.8109	−0.599386	−0.299693	−	0.954036i	$-0.596884\pi$
$883$	−17.8109	−0.599386	−0.299693	+	0.954036i	$0.596884\pi$
$884$	1.25146	0.0420912
$885$	0	0
$886$	−0.147693	−0.00496184
$887$	16.4729	0.553105	0.276553	−	0.960999i	$-0.410808\pi$
$887$	16.4729	0.553105	0.276553	+	0.960999i	$0.410808\pi$
$888$	0	0
$889$	8.54023	0.286430
$890$	−6.10625	−0.204682
$891$	0	0
$892$	−3.84656	−0.128792
$893$	27.0004	0.903535
$894$	0	0
$895$	8.30260	0.277525
$896$	−12.4252	−0.415095
$897$	0	0
$898$	45.1823	1.50775
$899$	−3.51861	−0.117352
$900$	0	0
$901$	−69.0238	−2.29952
$902$	0	0
$903$	0	0
$904$	4.79964	0.159634
$905$	0.449825	0.0149527
$906$	0	0
$907$	28.4877	0.945918	0.472959	−	0.881084i	$-0.343186\pi$
$907$	28.4877	0.945918	0.472959	+	0.881084i	$0.343186\pi$
$908$	3.28473	0.109008
$909$	0	0
$910$	−1.08321	−0.0359080
$911$	11.2353	0.372242	0.186121	−	0.982527i	$-0.440408\pi$
$911$	11.2353	0.372242	0.186121	+	0.982527i	$0.440408\pi$
$912$	0	0
$913$	0	0
$914$	33.9337	1.12243
$915$	0	0
$916$	3.10634	0.102636
$917$	−9.66708	−0.319235
$918$	0	0
$919$	0.780457	0.0257449	0.0128724	−	0.999917i	$-0.495902\pi$
$919$	0.780457	0.0257449	0.0128724	+	0.999917i	$0.495902\pi$
$920$	2.28441	0.0753148
$921$	0	0
$922$	4.07516	0.134208
$923$	15.4167	0.507446
$924$	0	0
$925$	9.28795	0.305386
$926$	−38.2744	−1.25777
$927$	0	0
$928$	2.32270	0.0762465
$929$	−26.5963	−0.872597	−0.436298	−	0.899802i	$-0.643711\pi$
$929$	−26.5963	−0.872597	−0.436298	+	0.899802i	$0.643711\pi$
$930$	0	0
$931$	4.22192	0.138368
$932$	−0.105012	−0.00343979
$933$	0	0
$934$	3.89900	0.127579
$935$	0	0
$936$	0	0
$937$	41.9697	1.37109	0.685544	−	0.728031i	$-0.259564\pi$
$937$	41.9697	1.37109	0.685544	+	0.728031i	$0.259564\pi$
$938$	6.85818	0.223928
$939$	0	0
$940$	−0.451620	−0.0147302
$941$	49.0330	1.59843	0.799215	−	0.601046i	$-0.205249\pi$
$941$	49.0330	1.59843	0.799215	+	0.601046i	$0.205249\pi$
$942$	0	0
$943$	1.87928	0.0611978
$944$	−36.8222	−1.19846
$945$	0	0
$946$	0	0
$947$	27.2953	0.886978	0.443489	−	0.896280i	$-0.353741\pi$
$947$	27.2953	0.886978	0.443489	+	0.896280i	$0.353741\pi$
$948$	0	0
$949$	21.0765	0.684171
$950$	−29.6132	−0.960778
$951$	0	0
$952$	−14.1741	−0.459386
$953$	−19.7408	−0.639466	−0.319733	−	0.947508i	$-0.603593\pi$
$953$	−19.7408	−0.639466	−0.319733	+	0.947508i	$0.603593\pi$
$954$	0	0
$955$	−7.50918	−0.242991
$956$	−0.0524354	−0.00169588
$957$	0	0
$958$	12.1478	0.392478
$959$	−14.0108	−0.452433
$960$	0	0
$961$	−29.3259	−0.945998
$962$	4.50754	0.145329
$963$	0	0
$964$	−1.57917	−0.0508615
$965$	−5.66960	−0.182511
$966$	0	0
$967$	−12.6734	−0.407551	−0.203775	−	0.979018i	$-0.565321\pi$
$967$	−12.6734	−0.407551	−0.203775	+	0.979018i	$0.565321\pi$
$968$	0	0
$969$	0	0
$970$	1.85082	0.0594261
$971$	−16.7036	−0.536045	−0.268022	−	0.963413i	$-0.586370\pi$
$971$	−16.7036	−0.536045	−0.268022	+	0.963413i	$0.586370\pi$
$972$	0	0
$973$	−9.57765	−0.307045
$974$	−28.7414	−0.920935
$975$	0	0
$976$	65.2234	2.08775
$977$	−27.3452	−0.874851	−0.437425	−	0.899255i	$-0.644110\pi$
$977$	−27.3452	−0.874851	−0.437425	+	0.899255i	$0.644110\pi$
$978$	0	0
$979$	0	0
$980$	−0.0706175	−0.00225579
$981$	0	0
$982$	42.0683	1.34245
$983$	55.0065	1.75443	0.877217	−	0.480093i	$-0.159397\pi$
$983$	55.0065	1.75443	0.877217	+	0.480093i	$0.159397\pi$
$984$	0	0
$985$	−1.07432	−0.0342306
$986$	20.8504	0.664012
$987$	0	0
$988$	−1.01076	−0.0321566
$989$	15.7188	0.499828
$990$	0	0
$991$	53.2327	1.69099	0.845497	−	0.533980i	$-0.179304\pi$
$991$	53.2327	1.69099	0.845497	+	0.533980i	$0.179304\pi$
$992$	−1.10509	−0.0350866
$993$	0	0
$994$	14.2905	0.453268
$995$	−9.46519	−0.300067
$996$	0	0
$997$	32.7546	1.03735	0.518674	−	0.854972i	$-0.326426\pi$
$997$	32.7546	1.03735	0.518674	+	0.854972i	$0.326426\pi$
$998$	40.9950	1.29767
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ch.1.4		4
3.2	odd	2		847.2.a.l.1.1		4
11.5	even	5		693.2.m.g.190.1		8
11.9	even	5		693.2.m.g.631.1		8
11.10	odd	2		7623.2.a.co.1.1		4
21.20	even	2		5929.2.a.bi.1.1		4
33.2	even	10		847.2.f.q.323.1		8
33.5	odd	10		77.2.f.a.36.2	yes	8
33.8	even	10		847.2.f.s.372.2		8
33.14	odd	10		847.2.f.p.372.1		8
33.17	even	10		847.2.f.q.729.1		8
33.20	odd	10		77.2.f.a.15.2	✓	8
33.26	odd	10		847.2.f.p.148.1		8
33.29	even	10		847.2.f.s.148.2		8
33.32	even	2		847.2.a.k.1.4		4
231.5	even	30		539.2.q.b.410.1		16
231.20	even	10		539.2.f.d.246.2		8
231.38	even	30		539.2.q.b.520.2		16
231.53	odd	30		539.2.q.c.422.1		16
231.86	odd	30		539.2.q.c.312.2		16
231.104	even	10		539.2.f.d.344.2		8
231.137	odd	30		539.2.q.c.520.2		16
231.152	even	30		539.2.q.b.312.2		16
231.170	odd	30		539.2.q.c.410.1		16
231.185	even	30		539.2.q.b.422.1		16
231.230	odd	2		5929.2.a.bb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.2	✓	8	33.20	odd	10
77.2.f.a.36.2	yes	8	33.5	odd	10
539.2.f.d.246.2		8	231.20	even	10
539.2.f.d.344.2		8	231.104	even	10
539.2.q.b.312.2		16	231.152	even	30
539.2.q.b.410.1		16	231.5	even	30
539.2.q.b.422.1		16	231.185	even	30
539.2.q.b.520.2		16	231.38	even	30
539.2.q.c.312.2		16	231.86	odd	30
539.2.q.c.410.1		16	231.170	odd	30
539.2.q.c.422.1		16	231.53	odd	30
539.2.q.c.520.2		16	231.137	odd	30
693.2.m.g.190.1		8	11.5	even	5
693.2.m.g.631.1		8	11.9	even	5
847.2.a.k.1.4		4	33.32	even	2
847.2.a.l.1.1		4	3.2	odd	2
847.2.f.p.148.1		8	33.26	odd	10
847.2.f.p.372.1		8	33.14	odd	10
847.2.f.q.323.1		8	33.2	even	10
847.2.f.q.729.1		8	33.17	even	10
847.2.f.s.148.2		8	33.29	even	10
847.2.f.s.372.2		8	33.8	even	10
5929.2.a.bb.1.4		4	231.230	odd	2
5929.2.a.bi.1.1		4	21.20	even	2
7623.2.a.ch.1.4		4	1.1	even	1	trivial
7623.2.a.co.1.1		4	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46673 q^{2} +0.151302 q^{4} -0.466732 q^{5} -1.00000 q^{7} -2.71154 q^{8} +O(q^{10})\)	\(q+1.46673 q^{2} +0.151302 q^{4} -0.466732 q^{5} -1.00000 q^{7} -2.71154 q^{8} -0.684570 q^{10} -1.58232 q^{13} -1.46673 q^{14} -4.27971 q^{16} -5.22732 q^{17} +4.22192 q^{19} -0.0706175 q^{20} +1.80505 q^{23} -4.78216 q^{25} -2.32083 q^{26} -0.151302 q^{28} -2.71947 q^{29} +1.29386 q^{31} -0.854102 q^{32} -7.66708 q^{34} +0.466732 q^{35} -1.94221 q^{37} +6.19242 q^{38} +1.26556 q^{40} +1.04112 q^{41} +8.70820 q^{43} +2.64753 q^{46} +6.39530 q^{47} +1.00000 q^{49} -7.01415 q^{50} -0.239408 q^{52} +13.2044 q^{53} +2.71154 q^{56} -3.98873 q^{58} +8.60389 q^{59} -15.2401 q^{61} +1.89775 q^{62} +7.30669 q^{64} +0.738517 q^{65} -4.67583 q^{67} -0.790906 q^{68} +0.684570 q^{70} -9.74310 q^{71} -13.3200 q^{73} -2.84870 q^{74} +0.638786 q^{76} +3.58232 q^{79} +1.99748 q^{80} +1.52705 q^{82} +17.2589 q^{83} +2.43976 q^{85} +12.7726 q^{86} +8.91982 q^{89} +1.58232 q^{91} +0.273109 q^{92} +9.38018 q^{94} -1.97050 q^{95} -2.70362 q^{97} +1.46673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} + 6 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 + 6 * q^5 - 4 * q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} + 6 q^{5} - 4 q^{7} - 9 q^{8} - 14 q^{10} + 2 q^{14} - 4 q^{16} + 3 q^{17} + 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} - 4 q^{28} - 3 q^{29} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - 7 q^{37} + 20 q^{38} - 13 q^{40} - 4 q^{41} + 8 q^{43} - 3 q^{46} + 14 q^{47} + 4 q^{49} - 33 q^{50} - 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} - 19 q^{61} - 10 q^{62} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{70} + 7 q^{71} - 11 q^{73} - 8 q^{74} - 26 q^{76} + 8 q^{79} + 4 q^{80} + 3 q^{82} + q^{83} + 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} - 20 q^{94} - 17 q^{95} - 15 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 + 6 * q^5 - 4 * q^7 - 9 * q^8 - 14 * q^10 + 2 * q^14 - 4 * q^16 + 3 * q^17 + 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 - 4 * q^28 - 3 * q^29 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - 7 * q^37 + 20 * q^38 - 13 * q^40 - 4 * q^41 + 8 * q^43 - 3 * q^46 + 14 * q^47 + 4 * q^49 - 33 * q^50 - 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 - 19 * q^61 - 10 * q^62 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^70 + 7 * q^71 - 11 * q^73 - 8 * q^74 - 26 * q^76 + 8 * q^79 + 4 * q^80 + 3 * q^82 + q^83 + 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 - 20 * q^94 - 17 * q^95 - 15 * q^97 - 2 * q^98

Introduction

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