LMFDB - Embedded newform 7623.2.a.co.1.1

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			1.1
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	$-0.673535\pi$
$2$	−1.46673	−1.03714	−0.518568	+	0.855036i	$0.673535\pi$
$3$	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	$-0.429581\pi$
$13$	1.58232	0.438856	0.219428	+	0.975629i	$0.429581\pi$
$14$	−1.46673	−0.392001
$15$	0	0
$16$	−4.27971	−1.06993
$17$	5.22732	1.26781	0.633906	−	0.773410i	$-0.281451\pi$
$17$	5.22732	1.26781	0.633906	+	0.773410i	$0.281451\pi$
$18$	0	0
$19$	−4.22192	−0.968575	−0.484287	−	0.874909i	$-0.660921\pi$
$19$	−4.22192	−0.968575	−0.484287	+	0.874909i	$0.660921\pi$
$20$	−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	$-0.418748\pi$
$29$	2.71947	0.504993	0.252496	+	0.967598i	$0.418748\pi$
$30$	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	$-0.525907\pi$
$41$	−1.04112	−0.162596	−0.0812980	+	0.996690i	$0.525907\pi$
$42$	0	0
$43$	−8.70820	−1.32799	−0.663994	−	0.747738i	$-0.731140\pi$
$43$	−8.70820	−1.32799	−0.663994	+	0.747738i	$0.731140\pi$
$44$	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	$-0.0703874\pi$
$61$	15.2401	1.95130	0.975651	+	0.219331i	$0.0703874\pi$
$62$	−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	$-0.215476\pi$
$73$	13.3200	1.55899	0.779495	+	0.626408i	$0.215476\pi$
$74$	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	$-0.564588\pi$
$79$	−3.58232	−0.403042	−0.201521	+	0.979484i	$0.564588\pi$
$80$	1.99748	0.223325
$81$	0	0
$82$	1.52705	0.168634
$83$	−17.2589	−1.89441	−0.947204	−	0.320631i	$-0.896105\pi$
$83$	−17.2589	−1.89441	−0.947204	+	0.320631i	$0.896105\pi$
$84$	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	$-0.497169\pi$
$101$	0.178781	0.0177894	0.00889469	+	0.999960i	$0.497169\pi$
$102$	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	$-0.769020\pi$
$107$	−15.4762	−1.49614	−0.748071	+	0.663618i	$0.769020\pi$
$108$	0	0
$109$	−11.0349	−1.05695	−0.528476	−	0.848948i	$-0.677236\pi$
$109$	−11.0349	−1.05695	−0.528476	+	0.848948i	$0.677236\pi$
$110$	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	$-0.376298\pi$
$127$	8.54023	0.757823	0.378911	+	0.925433i	$0.376298\pi$
$128$	−12.4252	−1.09824
$129$	0	0
$130$	1.08321	0.0950035
$131$	−9.66708	−0.844617	−0.422308	−	0.906452i	$-0.638780\pi$
$131$	−9.66708	−0.844617	−0.422308	+	0.906452i	$0.638780\pi$
$132$	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	$-0.633140\pi$
$139$	−9.57765	−0.812366	−0.406183	+	0.913792i	$0.633140\pi$
$140$	−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	$-0.290008\pi$
$149$	14.9625	1.22578	0.612888	+	0.790170i	$0.290008\pi$
$150$	0	0
$151$	2.87233	0.233747	0.116874	−	0.993147i	$-0.462713\pi$
$151$	2.87233	0.233747	0.116874	+	0.993147i	$0.462713\pi$
$152$	−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	$-0.421305\pi$
$167$	6.32491	0.489437	0.244718	+	0.969594i	$0.421305\pi$
$168$	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	$-0.483790\pi$
$173$	1.33906	0.101807	0.0509035	+	0.998704i	$0.483790\pi$
$174$	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	$-0.644029\pi$
$193$	−12.1475	−0.874393	−0.437197	+	0.899366i	$0.644029\pi$
$194$	3.96548	0.284705
$195$	0	0
$196$	0.151302	0.0108073
$197$	−2.30179	−0.163996	−0.0819978	−	0.996633i	$-0.526130\pi$
$197$	−2.30179	−0.163996	−0.0819978	+	0.996633i	$0.526130\pi$
$198$	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	$-0.559125\pi$
$211$	−5.36530	−0.369362	−0.184681	+	0.982799i	$0.559125\pi$
$212$	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	$-0.756071\pi$
$227$	−21.7098	−1.44093	−0.720464	+	0.693493i	$0.756071\pi$
$228$	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	$-0.492763\pi$
$233$	0.694056	0.0454691	0.0227345	+	0.999742i	$0.492763\pi$
$234$	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	$-0.496432\pi$
$239$	0.346561	0.0224171	0.0112086	+	0.999937i	$0.496432\pi$
$240$	0	0
$241$	10.4372	0.672317	0.336158	−	0.941806i	$-0.390872\pi$
$241$	10.4372	0.672317	0.336158	+	0.941806i	$0.390872\pi$
$242$	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	$-0.355971\pi$
$263$	14.1803	0.874397	0.437199	+	0.899365i	$0.355971\pi$
$264$	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	$-0.507064\pi$
$271$	−0.730591	−0.0443802	−0.0221901	+	0.999754i	$0.507064\pi$
$272$	−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	$-0.649491\pi$
$277$	−15.0644	−0.905132	−0.452566	+	0.891731i	$0.649491\pi$
$278$	14.0478	0.842534
$279$	0	0
$280$	−1.26556	−0.0756319
$281$	−10.6961	−0.638077	−0.319039	−	0.947742i	$-0.603360\pi$
$281$	−10.6961	−0.638077	−0.319039	+	0.947742i	$0.603360\pi$
$282$	0	0
$283$	−9.10890	−0.541468	−0.270734	−	0.962654i	$-0.587266\pi$
$283$	−9.10890	−0.541468	−0.270734	+	0.962654i	$0.587266\pi$
$284$	−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	$-0.387105\pi$
$293$	11.8890	0.694563	0.347281	+	0.937761i	$0.387105\pi$
$294$	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	$-0.520185\pi$
$307$	−2.22072	−0.126743	−0.0633716	+	0.997990i	$0.520185\pi$
$308$	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	$-0.324822\pi$
$337$	19.2011	1.04595	0.522975	+	0.852348i	$0.324822\pi$
$338$	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	$-0.473926\pi$
$347$	3.04831	0.163642	0.0818208	+	0.996647i	$0.473926\pi$
$348$	0	0
$349$	−19.3961	−1.03825	−0.519125	−	0.854698i	$-0.673742\pi$
$349$	−19.3961	−1.03825	−0.519125	+	0.854698i	$0.673742\pi$
$350$	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	$-0.494899\pi$
$359$	0.607226	0.0320481	0.0160241	+	0.999872i	$0.494899\pi$
$360$	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	$-0.223989\pi$
$373$	29.4513	1.52493	0.762465	+	0.647029i	$0.223989\pi$
$374$	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	$-0.238380\pi$
$409$	29.6255	1.46489	0.732443	+	0.680828i	$0.238380\pi$
$410$	−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	$-0.240237\pi$
$431$	30.2464	1.45692	0.728458	+	0.685090i	$0.240237\pi$
$432$	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	$-0.552258\pi$
$439$	−6.84875	−0.326873	−0.163436	+	0.986554i	$0.552258\pi$
$440$	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	$-0.681999\pi$
$457$	−23.1356	−1.08224	−0.541118	+	0.840947i	$0.681999\pi$
$458$	−30.1130	−1.40709
$459$	0	0
$460$	−0.127468	−0.00594325
$461$	−2.77839	−0.129403	−0.0647013	−	0.997905i	$-0.520609\pi$
$461$	−2.77839	−0.129403	−0.0647013	+	0.997905i	$0.520609\pi$
$462$	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	$-0.560593\pi$
$479$	−8.28223	−0.378425	−0.189212	+	0.981936i	$0.560593\pi$
$480$	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	$-0.724057\pi$
$491$	−28.6817	−1.29439	−0.647193	+	0.762327i	$0.724057\pi$
$492$	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	$-0.557779\pi$
$503$	−8.09736	−0.361043	−0.180522	+	0.983571i	$0.557779\pi$
$504$	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	$-0.693498\pi$
$523$	−26.1229	−1.14228	−0.571138	+	0.820854i	$0.693498\pi$
$524$	−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	$-0.312227\pi$
$541$	25.8777	1.11257	0.556284	+	0.830992i	$0.312227\pi$
$542$	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	$-0.197038\pi$
$547$	38.0968	1.62890	0.814451	+	0.580232i	$0.197038\pi$
$548$	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	$-0.238681\pi$
$557$	34.5422	1.46360	0.731799	+	0.681520i	$0.238681\pi$
$558$	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	$-0.365343\pi$
$563$	19.4819	0.821066	0.410533	+	0.911846i	$0.365343\pi$
$564$	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	$-0.383105\pi$
$569$	17.1288	0.718075	0.359038	+	0.933323i	$0.383105\pi$
$570$	0	0
$571$	3.85581	0.161360	0.0806802	−	0.996740i	$-0.474291\pi$
$571$	3.85581	0.161360	0.0806802	+	0.996740i	$0.474291\pi$
$572$	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	$-0.412412\pi$
$593$	13.2330	0.543413	0.271706	+	0.962380i	$0.412412\pi$
$594$	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	$-0.583674\pi$
$601$	−12.7408	−0.519708	−0.259854	+	0.965648i	$0.583674\pi$
$602$	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	$-0.554276\pi$
$607$	−8.36141	−0.339379	−0.169690	+	0.985498i	$0.554276\pi$
$608$	−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	$-0.456909\pi$
$613$	6.68294	0.269921	0.134961	+	0.990851i	$0.456909\pi$
$614$	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	$-0.397742\pi$
$659$	16.2115	0.631512	0.315756	+	0.948840i	$0.397742\pi$
$660$	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	$-0.463966\pi$
$673$	5.86102	0.225926	0.112963	+	0.993599i	$0.463966\pi$
$674$	−28.1629	−1.08479
$675$	0	0
$676$	−1.58811	−0.0610811
$677$	−20.5279	−0.788952	−0.394476	−	0.918906i	$-0.629074\pi$
$677$	−20.5279	−0.788952	−0.394476	+	0.918906i	$0.629074\pi$
$678$	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	$-0.266032\pi$
$701$	35.5107	1.34122	0.670610	+	0.741810i	$0.266032\pi$
$702$	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	$-0.382307\pi$
$733$	19.5677	0.722750	0.361375	+	0.932421i	$0.382307\pi$
$734$	−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	$-0.434230\pi$
$739$	11.1542	0.410313	0.205157	+	0.978729i	$0.434230\pi$
$740$	0.137154	0.00504188
$741$	0	0
$742$	−19.3674	−0.710998
$743$	20.6550	0.757757	0.378878	−	0.925446i	$-0.376310\pi$
$743$	20.6550	0.757757	0.378878	+	0.925446i	$0.376310\pi$
$744$	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	$-0.546303\pi$
$761$	−7.99743	−0.289907	−0.144953	+	0.989438i	$0.546303\pi$
$762$	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	$-0.112269\pi$
$769$	52.0476	1.87689	0.938443	+	0.345435i	$0.112269\pi$
$770$	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	$-0.636882\pi$
$787$	−23.3907	−0.833789	−0.416894	+	0.908955i	$0.636882\pi$
$788$	−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	$-0.297827\pi$
$809$	33.7501	1.18659	0.593295	+	0.804985i	$0.297827\pi$
$810$	0	0
$811$	30.7650	1.08030	0.540152	−	0.841567i	$-0.318367\pi$
$811$	30.7650	1.08030	0.540152	+	0.841567i	$0.318367\pi$
$812$	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	$-0.432403\pi$
$821$	12.0784	0.421538	0.210769	+	0.977536i	$0.432403\pi$
$822$	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	$-0.471330\pi$
$827$	5.17330	0.179893	0.0899466	+	0.995947i	$0.471330\pi$
$828$	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	$-0.386752\pi$
$853$	20.3462	0.696640	0.348320	+	0.937376i	$0.386752\pi$
$854$	−22.3532	−0.764911
$855$	0	0
$856$	−41.9644	−1.43432
$857$	−15.1087	−0.516104	−0.258052	−	0.966131i	$-0.583081\pi$
$857$	−15.1087	−0.516104	−0.258052	+	0.966131i	$0.583081\pi$
$858$	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	$-0.816718\pi$
$877$	−49.6783	−1.67752	−0.838759	+	0.544503i	$0.816718\pi$
$878$	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	$-0.589192\pi$
$887$	−16.4729	−0.553105	−0.276553	+	0.960999i	$0.589192\pi$
$888$	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	$-0.504098\pi$
$919$	−0.780457	−0.0257449	−0.0128724	+	0.999917i	$0.504098\pi$
$920$	−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	$-0.740436\pi$
$937$	−41.9697	−1.37109	−0.685544	+	0.728031i	$0.740436\pi$
$938$	6.85818	0.223928
$939$	0	0
$940$	−0.451620	−0.0147302
$941$	−49.0330	−1.59843	−0.799215	−	0.601046i	$-0.794751\pi$
$941$	−49.0330	−1.59843	−0.799215	+	0.601046i	$0.794751\pi$
$942$	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	$-0.396407\pi$
$953$	19.7408	0.639466	0.319733	+	0.947508i	$0.396407\pi$
$954$	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	$-0.434679\pi$
$967$	12.6734	0.407551	0.203775	+	0.979018i	$0.434679\pi$
$968$	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	$-0.673574\pi$
$997$	−32.7546	−1.03735	−0.518674	+	0.854972i	$0.673574\pi$
$998$	−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.co.1.1		4
3.2	odd	2		847.2.a.k.1.4		4
11.2	odd	10		693.2.m.g.631.1		8
11.6	odd	10		693.2.m.g.190.1		8
11.10	odd	2		7623.2.a.ch.1.4		4
21.20	even	2		5929.2.a.bb.1.4		4
33.2	even	10		77.2.f.a.15.2	✓	8
33.5	odd	10		847.2.f.q.729.1		8
33.8	even	10		847.2.f.p.372.1		8
33.14	odd	10		847.2.f.s.372.2		8
33.17	even	10		77.2.f.a.36.2	yes	8
33.20	odd	10		847.2.f.q.323.1		8
33.26	odd	10		847.2.f.s.148.2		8
33.29	even	10		847.2.f.p.148.1		8
33.32	even	2		847.2.a.l.1.1		4
231.2	even	30		539.2.q.c.312.2		16
231.17	odd	30		539.2.q.b.520.2		16
231.68	odd	30		539.2.q.b.312.2		16
231.83	odd	10		539.2.f.d.344.2		8
231.101	odd	30		539.2.q.b.422.1		16
231.116	even	30		539.2.q.c.520.2		16
231.149	even	30		539.2.q.c.410.1		16
231.167	odd	10		539.2.f.d.246.2		8
231.200	even	30		539.2.q.c.422.1		16
231.215	odd	30		539.2.q.b.410.1		16
231.230	odd	2		5929.2.a.bi.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.2	✓	8	33.2	even	10
77.2.f.a.36.2	yes	8	33.17	even	10
539.2.f.d.246.2		8	231.167	odd	10
539.2.f.d.344.2		8	231.83	odd	10
539.2.q.b.312.2		16	231.68	odd	30
539.2.q.b.410.1		16	231.215	odd	30
539.2.q.b.422.1		16	231.101	odd	30
539.2.q.b.520.2		16	231.17	odd	30
539.2.q.c.312.2		16	231.2	even	30
539.2.q.c.410.1		16	231.149	even	30
539.2.q.c.422.1		16	231.200	even	30
539.2.q.c.520.2		16	231.116	even	30
693.2.m.g.190.1		8	11.6	odd	10
693.2.m.g.631.1		8	11.2	odd	10
847.2.a.k.1.4		4	3.2	odd	2
847.2.a.l.1.1		4	33.32	even	2
847.2.f.p.148.1		8	33.29	even	10
847.2.f.p.372.1		8	33.8	even	10
847.2.f.q.323.1		8	33.20	odd	10
847.2.f.q.729.1		8	33.5	odd	10
847.2.f.s.148.2		8	33.26	odd	10
847.2.f.s.372.2		8	33.14	odd	10
5929.2.a.bb.1.4		4	21.20	even	2
5929.2.a.bi.1.1		4	231.230	odd	2
7623.2.a.ch.1.4		4	11.10	odd	2
7623.2.a.co.1.1		4	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
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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