LMFDB - Embedded newform 5625.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53767$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53767 q^{2} +4.43979 q^{4} +1.04054 q^{7} -6.19138 q^{8} +O(q^{10})$	$q-2.53767 q^{2} +4.43979 q^{4} +1.04054 q^{7} -6.19138 q^{8} -2.97101 q^{11} +5.66922 q^{13} -2.64054 q^{14} +6.83213 q^{16} +5.08361 q^{17} -5.37156 q^{19} +7.53945 q^{22} -3.86039 q^{23} -14.3866 q^{26} +4.61976 q^{28} -0.679696 q^{29} +0.850111 q^{31} -4.95495 q^{32} -12.9006 q^{34} -1.61763 q^{37} +13.6313 q^{38} -1.16529 q^{41} +5.68601 q^{43} -13.1906 q^{44} +9.79640 q^{46} -3.28640 q^{47} -5.91729 q^{49} +25.1701 q^{52} -12.6861 q^{53} -6.44235 q^{56} +1.72485 q^{58} -3.21187 q^{59} -5.42093 q^{61} -2.15730 q^{62} -1.09021 q^{64} -0.929140 q^{67} +22.5702 q^{68} +1.41358 q^{71} +11.3234 q^{73} +4.10501 q^{74} -23.8486 q^{76} -3.09144 q^{77} -1.44707 q^{79} +2.95713 q^{82} +11.4756 q^{83} -14.4292 q^{86} +18.3946 q^{88} -9.07225 q^{89} +5.89903 q^{91} -17.1393 q^{92} +8.33982 q^{94} -6.02928 q^{97} +15.0161 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8	$ 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * 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 $

$2$	−2.53767	−1.79441	−0.897203	−	0.441618i	$-0.854405\pi$
$2$	−2.53767	−1.79441	−0.897203	+	0.441618i	$0.854405\pi$
$3$	0	0
$3$	0	0
$4$	4.43979	2.21989
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.04054	0.393285	0.196643	−	0.980475i	$-0.436996\pi$
$7$	1.04054	0.393285	0.196643	+	0.980475i	$0.436996\pi$
$8$	−6.19138	−2.18898
$9$	0	0
$10$	0	0
$11$	−2.97101	−0.895792	−0.447896	−	0.894086i	$-0.647827\pi$
$11$	−2.97101	−0.895792	−0.447896	+	0.894086i	$0.647827\pi$
$12$	0	0
$13$	5.66922	1.57236	0.786180	−	0.617998i	$-0.212056\pi$
$13$	5.66922	1.57236	0.786180	+	0.617998i	$0.212056\pi$
$14$	−2.64054	−0.705714
$15$	0	0
$16$	6.83213	1.70803
$17$	5.08361	1.23296	0.616479	−	0.787372i	$-0.288559\pi$
$17$	5.08361	1.23296	0.616479	+	0.787372i	$0.288559\pi$
$18$	0	0
$19$	−5.37156	−1.23232	−0.616160	−	0.787621i	$-0.711313\pi$
$19$	−5.37156	−1.23232	−0.616160	+	0.787621i	$0.711313\pi$
$20$	0	0
$21$	0	0
$22$	7.53945	1.60742
$23$	−3.86039	−0.804946	−0.402473	−	0.915432i	$-0.631849\pi$
$23$	−3.86039	−0.804946	−0.402473	+	0.915432i	$0.631849\pi$
$24$	0	0
$25$	0	0
$26$	−14.3866	−2.82145
$27$	0	0
$28$	4.61976	0.873052
$29$	−0.679696	−0.126216	−0.0631082	−	0.998007i	$-0.520101\pi$
$29$	−0.679696	−0.126216	−0.0631082	+	0.998007i	$0.520101\pi$
$30$	0	0
$31$	0.850111	0.152684	0.0763422	−	0.997082i	$-0.475676\pi$
$31$	0.850111	0.152684	0.0763422	+	0.997082i	$0.475676\pi$
$32$	−4.95495	−0.875921
$33$	0	0
$34$	−12.9006	−2.21243
$35$	0	0
$36$	0	0
$37$	−1.61763	−0.265936	−0.132968	−	0.991120i	$-0.542451\pi$
$37$	−1.61763	−0.265936	−0.132968	+	0.991120i	$0.542451\pi$
$38$	13.6313	2.21128
$39$	0	0
$40$	0	0
$41$	−1.16529	−0.181988	−0.0909939	−	0.995851i	$-0.529004\pi$
$41$	−1.16529	−0.181988	−0.0909939	+	0.995851i	$0.529004\pi$
$42$	0	0
$43$	5.68601	0.867109	0.433554	−	0.901127i	$-0.357259\pi$
$43$	5.68601	0.867109	0.433554	+	0.901127i	$0.357259\pi$
$44$	−13.1906	−1.98856
$45$	0	0
$46$	9.79640	1.44440
$47$	−3.28640	−0.479371	−0.239686	−	0.970851i	$-0.577044\pi$
$47$	−3.28640	−0.479371	−0.239686	+	0.970851i	$0.577044\pi$
$48$	0	0
$49$	−5.91729	−0.845327
$50$	0	0
$51$	0	0
$52$	25.1701	3.49047
$53$	−12.6861	−1.74257	−0.871287	−	0.490773i	$-0.836714\pi$
$53$	−12.6861	−1.74257	−0.871287	+	0.490773i	$0.836714\pi$
$54$	0	0
$55$	0	0
$56$	−6.44235	−0.860896
$57$	0	0
$58$	1.72485	0.226484
$59$	−3.21187	−0.418150	−0.209075	−	0.977900i	$-0.567045\pi$
$59$	−3.21187	−0.418150	−0.209075	+	0.977900i	$0.567045\pi$
$60$	0	0
$61$	−5.42093	−0.694079	−0.347039	−	0.937851i	$-0.612813\pi$
$61$	−5.42093	−0.694079	−0.347039	+	0.937851i	$0.612813\pi$
$62$	−2.15730	−0.273978
$63$	0	0
$64$	−1.09021	−0.136276
$65$	0	0
$66$	0	0
$67$	−0.929140	−0.113513	−0.0567563	−	0.998388i	$-0.518076\pi$
$67$	−0.929140	−0.113513	−0.0567563	+	0.998388i	$0.518076\pi$
$68$	22.5702	2.73703
$69$	0	0
$70$	0	0
$71$	1.41358	0.167761	0.0838807	−	0.996476i	$-0.473269\pi$
$71$	1.41358	0.167761	0.0838807	+	0.996476i	$0.473269\pi$
$72$	0	0
$73$	11.3234	1.32530	0.662650	−	0.748929i	$-0.269432\pi$
$73$	11.3234	1.32530	0.662650	+	0.748929i	$0.269432\pi$
$74$	4.10501	0.477198
$75$	0	0
$76$	−23.8486	−2.73562
$77$	−3.09144	−0.352302
$78$	0	0
$79$	−1.44707	−0.162809	−0.0814043	−	0.996681i	$-0.525941\pi$
$79$	−1.44707	−0.162809	−0.0814043	+	0.996681i	$0.525941\pi$
$80$	0	0
$81$	0	0
$82$	2.95713	0.326560
$83$	11.4756	1.25961	0.629806	−	0.776752i	$-0.283134\pi$
$83$	11.4756	1.25961	0.629806	+	0.776752i	$0.283134\pi$
$84$	0	0
$85$	0	0
$86$	−14.4292	−1.55594
$87$	0	0
$88$	18.3946	1.96087
$89$	−9.07225	−0.961657	−0.480828	−	0.876815i	$-0.659664\pi$
$89$	−9.07225	−0.961657	−0.480828	+	0.876815i	$0.659664\pi$
$90$	0	0
$91$	5.89903	0.618386
$92$	−17.1393	−1.78689
$93$	0	0
$94$	8.33982	0.860187
$95$	0	0
$96$	0	0
$97$	−6.02928	−0.612181	−0.306091	−	0.952002i	$-0.599021\pi$
$97$	−6.02928	−0.612181	−0.306091	+	0.952002i	$0.599021\pi$
$98$	15.0161	1.51686
$99$	0	0
$100$	0	0
$101$	15.3408	1.52647	0.763236	−	0.646120i	$-0.223610\pi$
$101$	15.3408	1.52647	0.763236	+	0.646120i	$0.223610\pi$
$102$	0	0
$103$	−12.0590	−1.18820	−0.594102	−	0.804390i	$-0.702493\pi$
$103$	−12.0590	−1.18820	−0.594102	+	0.804390i	$0.702493\pi$
$104$	−35.1003	−3.44187
$105$	0	0
$106$	32.1933	3.12689
$107$	−6.49787	−0.628173	−0.314086	−	0.949394i	$-0.601698\pi$
$107$	−6.49787	−0.628173	−0.314086	+	0.949394i	$0.601698\pi$
$108$	0	0
$109$	−2.31057	−0.221313	−0.110656	−	0.993859i	$-0.535295\pi$
$109$	−2.31057	−0.221313	−0.110656	+	0.993859i	$0.535295\pi$
$110$	0	0
$111$	0	0
$112$	7.10907	0.671744
$113$	3.87281	0.364323	0.182162	−	0.983269i	$-0.441691\pi$
$113$	3.87281	0.364323	0.182162	+	0.983269i	$0.441691\pi$
$114$	0	0
$115$	0	0
$116$	−3.01771	−0.280187
$117$	0	0
$118$	8.15067	0.750330
$119$	5.28968	0.484904
$120$	0	0
$121$	−2.17312	−0.197556
$122$	13.7565	1.24546
$123$	0	0
$124$	3.77431	0.338943
$125$	0	0
$126$	0	0
$127$	−11.6938	−1.03765	−0.518827	−	0.854879i	$-0.673631\pi$
$127$	−11.6938	−1.03765	−0.518827	+	0.854879i	$0.673631\pi$
$128$	12.6765	1.12045
$129$	0	0
$130$	0	0
$131$	7.96210	0.695652	0.347826	−	0.937559i	$-0.386920\pi$
$131$	7.96210	0.695652	0.347826	+	0.937559i	$0.386920\pi$
$132$	0	0
$133$	−5.58930	−0.484654
$134$	2.35785	0.203688
$135$	0	0
$136$	−31.4746	−2.69892
$137$	−11.0513	−0.944176	−0.472088	−	0.881551i	$-0.656500\pi$
$137$	−11.0513	−0.944176	−0.472088	+	0.881551i	$0.656500\pi$
$138$	0	0
$139$	−12.2698	−1.04071	−0.520357	−	0.853949i	$-0.674201\pi$
$139$	−12.2698	−1.04071	−0.520357	+	0.853949i	$0.674201\pi$
$140$	0	0
$141$	0	0
$142$	−3.58721	−0.301032
$143$	−16.8433	−1.40851
$144$	0	0
$145$	0	0
$146$	−28.7350	−2.37813
$147$	0	0
$148$	−7.18192	−0.590350
$149$	4.62832	0.379167	0.189584	−	0.981865i	$-0.439286\pi$
$149$	4.62832	0.379167	0.189584	+	0.981865i	$0.439286\pi$
$150$	0	0
$151$	−4.67249	−0.380242	−0.190121	−	0.981761i	$-0.560888\pi$
$151$	−4.67249	−0.380242	−0.190121	+	0.981761i	$0.560888\pi$
$152$	33.2574	2.69753
$153$	0	0
$154$	7.84506	0.632173
$155$	0	0
$156$	0	0
$157$	−14.9726	−1.19494	−0.597472	−	0.801890i	$-0.703828\pi$
$157$	−14.9726	−1.19494	−0.597472	+	0.801890i	$0.703828\pi$
$158$	3.67220	0.292145
$159$	0	0
$160$	0	0
$161$	−4.01687	−0.316574
$162$	0	0
$163$	−11.9112	−0.932958	−0.466479	−	0.884532i	$-0.654478\pi$
$163$	−11.9112	−0.932958	−0.466479	+	0.884532i	$0.654478\pi$
$164$	−5.17364	−0.403994
$165$	0	0
$166$	−29.1214	−2.26026
$167$	−10.6081	−0.820881	−0.410440	−	0.911887i	$-0.634625\pi$
$167$	−10.6081	−0.820881	−0.410440	+	0.911887i	$0.634625\pi$
$168$	0	0
$169$	19.1401	1.47231
$170$	0	0
$171$	0	0
$172$	25.2447	1.92489
$173$	−1.25338	−0.0952928	−0.0476464	−	0.998864i	$-0.515172\pi$
$173$	−1.25338	−0.0952928	−0.0476464	+	0.998864i	$0.515172\pi$
$174$	0	0
$175$	0	0
$176$	−20.2983	−1.53004
$177$	0	0
$178$	23.0224	1.72560
$179$	20.8113	1.55551	0.777755	−	0.628567i	$-0.216358\pi$
$179$	20.8113	1.55551	0.777755	+	0.628567i	$0.216358\pi$
$180$	0	0
$181$	−6.92706	−0.514884	−0.257442	−	0.966294i	$-0.582880\pi$
$181$	−6.92706	−0.514884	−0.257442	+	0.966294i	$0.582880\pi$
$182$	−14.9698	−1.10964
$183$	0	0
$184$	23.9011	1.76201
$185$	0	0
$186$	0	0
$187$	−15.1034	−1.10447
$188$	−14.5909	−1.06415
$189$	0	0
$190$	0	0
$191$	−8.16415	−0.590737	−0.295369	−	0.955383i	$-0.595442\pi$
$191$	−8.16415	−0.590737	−0.295369	+	0.955383i	$0.595442\pi$
$192$	0	0
$193$	13.9629	1.00507	0.502537	−	0.864556i	$-0.332400\pi$
$193$	13.9629	1.00507	0.502537	+	0.864556i	$0.332400\pi$
$194$	15.3004	1.09850
$195$	0	0
$196$	−26.2715	−1.87653
$197$	5.76250	0.410561	0.205281	−	0.978703i	$-0.434189\pi$
$197$	5.76250	0.410561	0.205281	+	0.978703i	$0.434189\pi$
$198$	0	0
$199$	−26.5748	−1.88384	−0.941919	−	0.335841i	$-0.890980\pi$
$199$	−26.5748	−1.88384	−0.941919	+	0.335841i	$0.890980\pi$
$200$	0	0
$201$	0	0
$202$	−38.9301	−2.73911
$203$	−0.707248	−0.0496391
$204$	0	0
$205$	0	0
$206$	30.6017	2.13212
$207$	0	0
$208$	38.7329	2.68564
$209$	15.9589	1.10390
$210$	0	0
$211$	−26.4594	−1.82154	−0.910771	−	0.412912i	$-0.864512\pi$
$211$	−26.4594	−1.82154	−0.910771	+	0.412912i	$0.864512\pi$
$212$	−56.3237	−3.86833
$213$	0	0
$214$	16.4895	1.12720
$215$	0	0
$216$	0	0
$217$	0.884570	0.0600486
$218$	5.86348	0.397125
$219$	0	0
$220$	0	0
$221$	28.8201	1.93865
$222$	0	0
$223$	27.4456	1.83789	0.918946	−	0.394383i	$-0.129042\pi$
$223$	27.4456	1.83789	0.918946	+	0.394383i	$0.129042\pi$
$224$	−5.15581	−0.344487
$225$	0	0
$226$	−9.82792	−0.653744
$227$	0.130161	0.00863907	0.00431954	−	0.999991i	$-0.498625\pi$
$227$	0.130161	0.00863907	0.00431954	+	0.999991i	$0.498625\pi$
$228$	0	0
$229$	2.82530	0.186701	0.0933504	−	0.995633i	$-0.470242\pi$
$229$	2.82530	0.186701	0.0933504	+	0.995633i	$0.470242\pi$
$230$	0	0
$231$	0	0
$232$	4.20826	0.276286
$233$	7.98709	0.523252	0.261626	−	0.965169i	$-0.415741\pi$
$233$	7.98709	0.523252	0.261626	+	0.965169i	$0.415741\pi$
$234$	0	0
$235$	0	0
$236$	−14.2600	−0.928247
$237$	0	0
$238$	−13.4235	−0.870115
$239$	19.0619	1.23301	0.616506	−	0.787350i	$-0.288548\pi$
$239$	19.0619	1.23301	0.616506	+	0.787350i	$0.288548\pi$
$240$	0	0
$241$	21.1199	1.36045	0.680226	−	0.733002i	$-0.261882\pi$
$241$	21.1199	1.36045	0.680226	+	0.733002i	$0.261882\pi$
$242$	5.51466	0.354496
$243$	0	0
$244$	−24.0678	−1.54078
$245$	0	0
$246$	0	0
$247$	−30.4526	−1.93765
$248$	−5.26336	−0.334224
$249$	0	0
$250$	0	0
$251$	−30.2224	−1.90762	−0.953811	−	0.300408i	$-0.902877\pi$
$251$	−30.2224	−1.90762	−0.953811	+	0.300408i	$0.902877\pi$
$252$	0	0
$253$	11.4692	0.721065
$254$	29.6750	1.86197
$255$	0	0
$256$	−29.9884	−1.87427
$257$	5.10215	0.318263	0.159132	−	0.987257i	$-0.449131\pi$
$257$	5.10215	0.318263	0.159132	+	0.987257i	$0.449131\pi$
$258$	0	0
$259$	−1.68320	−0.104589
$260$	0	0
$261$	0	0
$262$	−20.2052	−1.24828
$263$	6.41540	0.395591	0.197795	−	0.980243i	$-0.436622\pi$
$263$	6.41540	0.395591	0.197795	+	0.980243i	$0.436622\pi$
$264$	0	0
$265$	0	0
$266$	14.1838	0.869666
$267$	0	0
$268$	−4.12518	−0.251986
$269$	17.4592	1.06450	0.532252	−	0.846586i	$-0.321346\pi$
$269$	17.4592	1.06450	0.532252	+	0.846586i	$0.321346\pi$
$270$	0	0
$271$	−0.0951857	−0.00578212	−0.00289106	−	0.999996i	$-0.500920\pi$
$271$	−0.0951857	−0.00578212	−0.00289106	+	0.999996i	$0.500920\pi$
$272$	34.7319	2.10593
$273$	0	0
$274$	28.0446	1.69424
$275$	0	0
$276$	0	0
$277$	18.4007	1.10559	0.552796	−	0.833316i	$-0.313561\pi$
$277$	18.4007	1.10559	0.552796	+	0.833316i	$0.313561\pi$
$278$	31.1369	1.86747
$279$	0	0
$280$	0	0
$281$	−17.9361	−1.06998	−0.534988	−	0.844860i	$-0.679684\pi$
$281$	−17.9361	−1.06998	−0.534988	+	0.844860i	$0.679684\pi$
$282$	0	0
$283$	22.2399	1.32203	0.661014	−	0.750374i	$-0.270126\pi$
$283$	22.2399	1.32203	0.661014	+	0.750374i	$0.270126\pi$
$284$	6.27601	0.372413
$285$	0	0
$286$	42.7428	2.52743
$287$	−1.21253	−0.0715732
$288$	0	0
$289$	8.84312	0.520184
$290$	0	0
$291$	0	0
$292$	50.2734	2.94203
$293$	−18.7316	−1.09431	−0.547155	−	0.837031i	$-0.684289\pi$
$293$	−18.7316	−1.09431	−0.547155	+	0.837031i	$0.684289\pi$
$294$	0	0
$295$	0	0
$296$	10.0154	0.582130
$297$	0	0
$298$	−11.7452	−0.680380
$299$	−21.8854	−1.26566
$300$	0	0
$301$	5.91650	0.341021
$302$	11.8573	0.682309
$303$	0	0
$304$	−36.6992	−2.10484
$305$	0	0
$306$	0	0
$307$	7.03850	0.401708	0.200854	−	0.979621i	$-0.435628\pi$
$307$	7.03850	0.401708	0.200854	+	0.979621i	$0.435628\pi$
$308$	−13.7253	−0.782073
$309$	0	0
$310$	0	0
$311$	29.2790	1.66026	0.830130	−	0.557570i	$-0.188266\pi$
$311$	29.2790	1.66026	0.830130	+	0.557570i	$0.188266\pi$
$312$	0	0
$313$	15.6354	0.883764	0.441882	−	0.897073i	$-0.354311\pi$
$313$	15.6354	0.883764	0.441882	+	0.897073i	$0.354311\pi$
$314$	37.9956	2.14422
$315$	0	0
$316$	−6.42470	−0.361418
$317$	−6.24144	−0.350554	−0.175277	−	0.984519i	$-0.556082\pi$
$317$	−6.24144	−0.350554	−0.175277	+	0.984519i	$0.556082\pi$
$318$	0	0
$319$	2.01938	0.113064
$320$	0	0
$321$	0	0
$322$	10.1935	0.568062
$323$	−27.3069	−1.51940
$324$	0	0
$325$	0	0
$326$	30.2268	1.67411
$327$	0	0
$328$	7.21476	0.398369
$329$	−3.41962	−0.188530
$330$	0	0
$331$	21.4575	1.17941	0.589705	−	0.807619i	$-0.299244\pi$
$331$	21.4575	1.17941	0.589705	+	0.807619i	$0.299244\pi$
$332$	50.9493	2.79621
$333$	0	0
$334$	26.9199	1.47299
$335$	0	0
$336$	0	0
$337$	34.7511	1.89301	0.946507	−	0.322683i	$-0.104585\pi$
$337$	34.7511	1.89301	0.946507	+	0.322683i	$0.104585\pi$
$338$	−48.5712	−2.64193
$339$	0	0
$340$	0	0
$341$	−2.52569	−0.136774
$342$	0	0
$343$	−13.4409	−0.725740
$344$	−35.2043	−1.89809
$345$	0	0
$346$	3.18067	0.170994
$347$	−28.7241	−1.54199	−0.770995	−	0.636841i	$-0.780241\pi$
$347$	−28.7241	−1.54199	−0.770995	+	0.636841i	$0.780241\pi$
$348$	0	0
$349$	−12.2834	−0.657515	−0.328758	−	0.944414i	$-0.606630\pi$
$349$	−12.2834	−0.657515	−0.328758	+	0.944414i	$0.606630\pi$
$350$	0	0
$351$	0	0
$352$	14.7212	0.784643
$353$	26.9779	1.43589	0.717945	−	0.696100i	$-0.245083\pi$
$353$	26.9779	1.43589	0.717945	+	0.696100i	$0.245083\pi$
$354$	0	0
$355$	0	0
$356$	−40.2789	−2.13478
$357$	0	0
$358$	−52.8123	−2.79122
$359$	−16.4243	−0.866839	−0.433419	−	0.901192i	$-0.642693\pi$
$359$	−16.4243	−0.866839	−0.433419	+	0.901192i	$0.642693\pi$
$360$	0	0
$361$	9.85366	0.518614
$362$	17.5786	0.923912
$363$	0	0
$364$	26.1904	1.37275
$365$	0	0
$366$	0	0
$367$	−29.8953	−1.56052	−0.780262	−	0.625453i	$-0.784914\pi$
$367$	−29.8953	−1.56052	−0.780262	+	0.625453i	$0.784914\pi$
$368$	−26.3747	−1.37487
$369$	0	0
$370$	0	0
$371$	−13.2004	−0.685329
$372$	0	0
$373$	−24.8307	−1.28568	−0.642842	−	0.765999i	$-0.722245\pi$
$373$	−24.8307	−1.28568	−0.642842	+	0.765999i	$0.722245\pi$
$374$	38.3276	1.98187
$375$	0	0
$376$	20.3474	1.04934
$377$	−3.85335	−0.198458
$378$	0	0
$379$	19.3966	0.996338	0.498169	−	0.867080i	$-0.334006\pi$
$379$	19.3966	0.996338	0.498169	+	0.867080i	$0.334006\pi$
$380$	0	0
$381$	0	0
$382$	20.7179	1.06002
$383$	11.2508	0.574891	0.287446	−	0.957797i	$-0.407194\pi$
$383$	11.2508	0.574891	0.287446	+	0.957797i	$0.407194\pi$
$384$	0	0
$385$	0	0
$386$	−35.4334	−1.80351
$387$	0	0
$388$	−26.7687	−1.35898
$389$	−14.9017	−0.755549	−0.377774	−	0.925898i	$-0.623310\pi$
$389$	−14.9017	−0.755549	−0.377774	+	0.925898i	$0.623310\pi$
$390$	0	0
$391$	−19.6247	−0.992464
$392$	36.6362	1.85041
$393$	0	0
$394$	−14.6233	−0.736713
$395$	0	0
$396$	0	0
$397$	24.0966	1.20937	0.604687	−	0.796463i	$-0.293298\pi$
$397$	24.0966	1.20937	0.604687	+	0.796463i	$0.293298\pi$
$398$	67.4382	3.38037
$399$	0	0
$400$	0	0
$401$	−13.4580	−0.672059	−0.336030	−	0.941851i	$-0.609084\pi$
$401$	−13.4580	−0.672059	−0.336030	+	0.941851i	$0.609084\pi$
$402$	0	0
$403$	4.81947	0.240075
$404$	68.1101	3.38860
$405$	0	0
$406$	1.79476	0.0890727
$407$	4.80598	0.238224
$408$	0	0
$409$	−35.4737	−1.75406	−0.877030	−	0.480435i	$-0.840479\pi$
$409$	−35.4737	−1.75406	−0.877030	+	0.480435i	$0.840479\pi$
$410$	0	0
$411$	0	0
$412$	−53.5392	−2.63769
$413$	−3.34206	−0.164452
$414$	0	0
$415$	0	0
$416$	−28.0907	−1.37726
$417$	0	0
$418$	−40.4986	−1.98085
$419$	−15.8120	−0.772466	−0.386233	−	0.922401i	$-0.626224\pi$
$419$	−15.8120	−0.772466	−0.386233	+	0.922401i	$0.626224\pi$
$420$	0	0
$421$	11.9813	0.583935	0.291967	−	0.956428i	$-0.405690\pi$
$421$	11.9813	0.583935	0.291967	+	0.956428i	$0.405690\pi$
$422$	67.1454	3.26859
$423$	0	0
$424$	78.5447	3.81447
$425$	0	0
$426$	0	0
$427$	−5.64067	−0.272971
$428$	−28.8492	−1.39448
$429$	0	0
$430$	0	0
$431$	−14.6428	−0.705317	−0.352659	−	0.935752i	$-0.614722\pi$
$431$	−14.6428	−0.705317	−0.352659	+	0.935752i	$0.614722\pi$
$432$	0	0
$433$	−4.27293	−0.205344	−0.102672	−	0.994715i	$-0.532739\pi$
$433$	−4.27293	−0.205344	−0.102672	+	0.994715i	$0.532739\pi$
$434$	−2.24475	−0.107751
$435$	0	0
$436$	−10.2584	−0.491291
$437$	20.7363	0.991952
$438$	0	0
$439$	−20.3073	−0.969214	−0.484607	−	0.874732i	$-0.661037\pi$
$439$	−20.3073	−0.969214	−0.484607	+	0.874732i	$0.661037\pi$
$440$	0	0
$441$	0	0
$442$	−73.1361	−3.47873
$443$	19.0543	0.905299	0.452649	−	0.891689i	$-0.350479\pi$
$443$	19.0543	0.905299	0.452649	+	0.891689i	$0.350479\pi$
$444$	0	0
$445$	0	0
$446$	−69.6479	−3.29793
$447$	0	0
$448$	−1.13440	−0.0535952
$449$	−29.4793	−1.39122	−0.695608	−	0.718421i	$-0.744865\pi$
$449$	−29.4793	−1.39122	−0.695608	+	0.718421i	$0.744865\pi$
$450$	0	0
$451$	3.46209	0.163023
$452$	17.1944	0.808759
$453$	0	0
$454$	−0.330305	−0.0155020
$455$	0	0
$456$	0	0
$457$	−28.3015	−1.32389	−0.661945	−	0.749553i	$-0.730269\pi$
$457$	−28.3015	−1.32389	−0.661945	+	0.749553i	$0.730269\pi$
$458$	−7.16968	−0.335017
$459$	0	0
$460$	0	0
$461$	16.5575	0.771157	0.385579	−	0.922675i	$-0.374002\pi$
$461$	16.5575	0.771157	0.385579	+	0.922675i	$0.374002\pi$
$462$	0	0
$463$	−9.20933	−0.427994	−0.213997	−	0.976834i	$-0.568648\pi$
$463$	−9.20933	−0.427994	−0.213997	+	0.976834i	$0.568648\pi$
$464$	−4.64377	−0.215582
$465$	0	0
$466$	−20.2686	−0.938926
$467$	10.6123	0.491078	0.245539	−	0.969387i	$-0.421035\pi$
$467$	10.6123	0.491078	0.245539	+	0.969387i	$0.421035\pi$
$468$	0	0
$469$	−0.966803	−0.0446428
$470$	0	0
$471$	0	0
$472$	19.8859	0.915323
$473$	−16.8932	−0.776749
$474$	0	0
$475$	0	0
$476$	23.4850	1.07644
$477$	0	0
$478$	−48.3729	−2.21253
$479$	−33.6061	−1.53550	−0.767751	−	0.640749i	$-0.778624\pi$
$479$	−33.6061	−1.53550	−0.767751	+	0.640749i	$0.778624\pi$
$480$	0	0
$481$	−9.17069	−0.418147
$482$	−53.5954	−2.44120
$483$	0	0
$484$	−9.64818	−0.438554
$485$	0	0
$486$	0	0
$487$	−29.2486	−1.32538	−0.662690	−	0.748894i	$-0.730585\pi$
$487$	−29.2486	−1.32538	−0.662690	+	0.748894i	$0.730585\pi$
$488$	33.5630	1.51933
$489$	0	0
$490$	0	0
$491$	−13.9549	−0.629778	−0.314889	−	0.949129i	$-0.601967\pi$
$491$	−13.9549	−0.629778	−0.314889	+	0.949129i	$0.601967\pi$
$492$	0	0
$493$	−3.45531	−0.155619
$494$	77.2787	3.47693
$495$	0	0
$496$	5.80807	0.260790
$497$	1.47088	0.0659782
$498$	0	0
$499$	−35.7533	−1.60054	−0.800268	−	0.599642i	$-0.795310\pi$
$499$	−35.7533	−1.60054	−0.800268	+	0.599642i	$0.795310\pi$
$500$	0	0
$501$	0	0
$502$	76.6946	3.42305
$503$	−11.6443	−0.519193	−0.259596	−	0.965717i	$-0.583590\pi$
$503$	−11.6443	−0.519193	−0.259596	+	0.965717i	$0.583590\pi$
$504$	0	0
$505$	0	0
$506$	−29.1052	−1.29388
$507$	0	0
$508$	−51.9178	−2.30348
$509$	8.53362	0.378246	0.189123	−	0.981953i	$-0.439436\pi$
$509$	8.53362	0.378246	0.189123	+	0.981953i	$0.439436\pi$
$510$	0	0
$511$	11.7824	0.521222
$512$	50.7478	2.24276
$513$	0	0
$514$	−12.9476	−0.571094
$515$	0	0
$516$	0	0
$517$	9.76393	0.429417
$518$	4.27141	0.187675
$519$	0	0
$520$	0	0
$521$	−1.26530	−0.0554336	−0.0277168	−	0.999616i	$-0.508824\pi$
$521$	−1.26530	−0.0554336	−0.0277168	+	0.999616i	$0.508824\pi$
$522$	0	0
$523$	−27.2204	−1.19027	−0.595133	−	0.803627i	$-0.702901\pi$
$523$	−27.2204	−1.19027	−0.595133	+	0.803627i	$0.702901\pi$
$524$	35.3500	1.54427
$525$	0	0
$526$	−16.2802	−0.709850
$527$	4.32163	0.188253
$528$	0	0
$529$	−8.09742	−0.352062
$530$	0	0
$531$	0	0
$532$	−24.8153	−1.07588
$533$	−6.60629	−0.286150
$534$	0	0
$535$	0	0
$536$	5.75266	0.248477
$537$	0	0
$538$	−44.3057	−1.91015
$539$	17.5803	0.757237
$540$	0	0
$541$	30.3830	1.30627	0.653134	−	0.757242i	$-0.273454\pi$
$541$	30.3830	1.30627	0.653134	+	0.757242i	$0.273454\pi$
$542$	0.241550	0.0103755
$543$	0	0
$544$	−25.1891	−1.07997
$545$	0	0
$546$	0	0
$547$	−11.3768	−0.486439	−0.243219	−	0.969971i	$-0.578203\pi$
$547$	−11.3768	−0.486439	−0.243219	+	0.969971i	$0.578203\pi$
$548$	−49.0654	−2.09597
$549$	0	0
$550$	0	0
$551$	3.65103	0.155539
$552$	0	0
$553$	−1.50573	−0.0640303
$554$	−46.6950	−1.98388
$555$	0	0
$556$	−54.4755	−2.31028
$557$	−45.7532	−1.93862	−0.969312	−	0.245833i	$-0.920938\pi$
$557$	−45.7532	−1.93862	−0.969312	+	0.245833i	$0.920938\pi$
$558$	0	0
$559$	32.2353	1.36341
$560$	0	0
$561$	0	0
$562$	45.5158	1.91997
$563$	8.94289	0.376898	0.188449	−	0.982083i	$-0.439654\pi$
$563$	8.94289	0.376898	0.188449	+	0.982083i	$0.439654\pi$
$564$	0	0
$565$	0	0
$566$	−56.4377	−2.37225
$567$	0	0
$568$	−8.75203	−0.367227
$569$	2.28908	0.0959632	0.0479816	−	0.998848i	$-0.484721\pi$
$569$	2.28908	0.0959632	0.0479816	+	0.998848i	$0.484721\pi$
$570$	0	0
$571$	−30.7595	−1.28725	−0.643623	−	0.765342i	$-0.722570\pi$
$571$	−30.7595	−1.28725	−0.643623	+	0.765342i	$0.722570\pi$
$572$	−74.7806	−3.12674
$573$	0	0
$574$	3.07700	0.128431
$575$	0	0
$576$	0	0
$577$	−6.59227	−0.274440	−0.137220	−	0.990541i	$-0.543817\pi$
$577$	−6.59227	−0.274440	−0.137220	+	0.990541i	$0.543817\pi$
$578$	−22.4410	−0.933421
$579$	0	0
$580$	0	0
$581$	11.9408	0.495387
$582$	0	0
$583$	37.6906	1.56098
$584$	−70.1073	−2.90106
$585$	0	0
$586$	47.5346	1.96364
$587$	5.53771	0.228566	0.114283	−	0.993448i	$-0.463543\pi$
$587$	5.53771	0.228566	0.114283	+	0.993448i	$0.463543\pi$
$588$	0	0
$589$	−4.56642	−0.188156
$590$	0	0
$591$	0	0
$592$	−11.0518	−0.454228
$593$	1.88122	0.0772524	0.0386262	−	0.999254i	$-0.487702\pi$
$593$	1.88122	0.0772524	0.0386262	+	0.999254i	$0.487702\pi$
$594$	0	0
$595$	0	0
$596$	20.5488	0.841710
$597$	0	0
$598$	55.5380	2.27112
$599$	−17.8272	−0.728400	−0.364200	−	0.931321i	$-0.618658\pi$
$599$	−17.8272	−0.728400	−0.364200	+	0.931321i	$0.618658\pi$
$600$	0	0
$601$	33.0994	1.35015	0.675077	−	0.737747i	$-0.264110\pi$
$601$	33.0994	1.35015	0.675077	+	0.737747i	$0.264110\pi$
$602$	−15.0141	−0.611931
$603$	0	0
$604$	−20.7449	−0.844097
$605$	0	0
$606$	0	0
$607$	−23.4603	−0.952226	−0.476113	−	0.879384i	$-0.657955\pi$
$607$	−23.4603	−0.952226	−0.476113	+	0.879384i	$0.657955\pi$
$608$	26.6158	1.07941
$609$	0	0
$610$	0	0
$611$	−18.6314	−0.753744
$612$	0	0
$613$	47.2281	1.90752	0.953762	−	0.300564i	$-0.0971750\pi$
$613$	47.2281	1.90752	0.953762	+	0.300564i	$0.0971750\pi$
$614$	−17.8614	−0.720828
$615$	0	0
$616$	19.1403	0.771184
$617$	−16.2698	−0.654999	−0.327499	−	0.944851i	$-0.606206\pi$
$617$	−16.2698	−0.654999	−0.327499	+	0.944851i	$0.606206\pi$
$618$	0	0
$619$	35.4599	1.42525	0.712627	−	0.701543i	$-0.247505\pi$
$619$	35.4599	1.42525	0.712627	+	0.701543i	$0.247505\pi$
$620$	0	0
$621$	0	0
$622$	−74.3005	−2.97918
$623$	−9.44000	−0.378206
$624$	0	0
$625$	0	0
$626$	−39.6775	−1.58583
$627$	0	0
$628$	−66.4752	−2.65265
$629$	−8.22339	−0.327888
$630$	0	0
$631$	−29.2364	−1.16388	−0.581941	−	0.813231i	$-0.697706\pi$
$631$	−29.2364	−1.16388	−0.581941	+	0.813231i	$0.697706\pi$
$632$	8.95939	0.356385
$633$	0	0
$634$	15.8387	0.629037
$635$	0	0
$636$	0	0
$637$	−33.5464	−1.32916
$638$	−5.12453	−0.202882
$639$	0	0
$640$	0	0
$641$	15.7160	0.620745	0.310373	−	0.950615i	$-0.399546\pi$
$641$	15.7160	0.620745	0.310373	+	0.950615i	$0.399546\pi$
$642$	0	0
$643$	8.08055	0.318666	0.159333	−	0.987225i	$-0.449066\pi$
$643$	8.08055	0.318666	0.159333	+	0.987225i	$0.449066\pi$
$644$	−17.8340	−0.702760
$645$	0	0
$646$	69.2961	2.72642
$647$	11.0193	0.433214	0.216607	−	0.976259i	$-0.430501\pi$
$647$	11.0193	0.433214	0.216607	+	0.976259i	$0.430501\pi$
$648$	0	0
$649$	9.54248	0.374575
$650$	0	0
$651$	0	0
$652$	−52.8832	−2.07107
$653$	−31.0861	−1.21649	−0.608246	−	0.793748i	$-0.708127\pi$
$653$	−31.0861	−1.21649	−0.608246	+	0.793748i	$0.708127\pi$
$654$	0	0
$655$	0	0
$656$	−7.96142	−0.310841
$657$	0	0
$658$	8.67788	0.338299
$659$	−32.4252	−1.26311	−0.631553	−	0.775333i	$-0.717582\pi$
$659$	−32.4252	−1.26311	−0.631553	+	0.775333i	$0.717582\pi$
$660$	0	0
$661$	15.6076	0.607067	0.303534	−	0.952821i	$-0.401834\pi$
$661$	15.6076	0.607067	0.303534	+	0.952821i	$0.401834\pi$
$662$	−54.4521	−2.11634
$663$	0	0
$664$	−71.0499	−2.75727
$665$	0	0
$666$	0	0
$667$	2.62389	0.101597
$668$	−47.0978	−1.82227
$669$	0	0
$670$	0	0
$671$	16.1056	0.621750
$672$	0	0
$673$	−31.0271	−1.19601	−0.598003	−	0.801494i	$-0.704039\pi$
$673$	−31.0271	−1.19601	−0.598003	+	0.801494i	$0.704039\pi$
$674$	−88.1870	−3.39684
$675$	0	0
$676$	84.9778	3.26838
$677$	−23.0893	−0.887394	−0.443697	−	0.896177i	$-0.646333\pi$
$677$	−23.0893	−0.887394	−0.443697	+	0.896177i	$0.646333\pi$
$678$	0	0
$679$	−6.27369	−0.240762
$680$	0	0
$681$	0	0
$682$	6.40936	0.245427
$683$	−41.3541	−1.58237	−0.791185	−	0.611577i	$-0.790536\pi$
$683$	−41.3541	−1.58237	−0.791185	+	0.611577i	$0.790536\pi$
$684$	0	0
$685$	0	0
$686$	34.1086	1.30227
$687$	0	0
$688$	38.8476	1.48105
$689$	−71.9205	−2.73995
$690$	0	0
$691$	4.46909	0.170012	0.0850061	−	0.996380i	$-0.472909\pi$
$691$	4.46909	0.170012	0.0850061	+	0.996380i	$0.472909\pi$
$692$	−5.56475	−0.211540
$693$	0	0
$694$	72.8924	2.76696
$695$	0	0
$696$	0	0
$697$	−5.92389	−0.224383
$698$	31.1713	1.17985
$699$	0	0
$700$	0	0
$701$	25.2265	0.952791	0.476396	−	0.879231i	$-0.341943\pi$
$701$	25.2265	0.952791	0.476396	+	0.879231i	$0.341943\pi$
$702$	0	0
$703$	8.68919	0.327719
$704$	3.23901	0.122075
$705$	0	0
$706$	−68.4611	−2.57657
$707$	15.9627	0.600339
$708$	0	0
$709$	1.55990	0.0585834	0.0292917	−	0.999571i	$-0.490675\pi$
$709$	1.55990	0.0585834	0.0292917	+	0.999571i	$0.490675\pi$
$710$	0	0
$711$	0	0
$712$	56.1698	2.10505
$713$	−3.28176	−0.122903
$714$	0	0
$715$	0	0
$716$	92.3978	3.45307
$717$	0	0
$718$	41.6794	1.55546
$719$	−28.9403	−1.07929	−0.539645	−	0.841893i	$-0.681442\pi$
$719$	−28.9403	−1.07929	−0.539645	+	0.841893i	$0.681442\pi$
$720$	0	0
$721$	−12.5478	−0.467304
$722$	−25.0054	−0.930604
$723$	0	0
$724$	−30.7547	−1.14299
$725$	0	0
$726$	0	0
$727$	−36.9595	−1.37075	−0.685375	−	0.728190i	$-0.740362\pi$
$727$	−36.9595	−1.37075	−0.685375	+	0.728190i	$0.740362\pi$
$728$	−36.5231	−1.35364
$729$	0	0
$730$	0	0
$731$	28.9055	1.06911
$732$	0	0
$733$	−4.16161	−0.153713	−0.0768564	−	0.997042i	$-0.524488\pi$
$733$	−4.16161	−0.153713	−0.0768564	+	0.997042i	$0.524488\pi$
$734$	75.8646	2.80021
$735$	0	0
$736$	19.1280	0.705069
$737$	2.76048	0.101684
$738$	0	0
$739$	18.9577	0.697370	0.348685	−	0.937240i	$-0.386628\pi$
$739$	18.9577	0.697370	0.348685	+	0.937240i	$0.386628\pi$
$740$	0	0
$741$	0	0
$742$	33.4982	1.22976
$743$	11.7060	0.429452	0.214726	−	0.976674i	$-0.431114\pi$
$743$	11.7060	0.429452	0.214726	+	0.976674i	$0.431114\pi$
$744$	0	0
$745$	0	0
$746$	63.0122	2.30704
$747$	0	0
$748$	−67.0561	−2.45181
$749$	−6.76126	−0.247051
$750$	0	0
$751$	−4.95672	−0.180873	−0.0904367	−	0.995902i	$-0.528826\pi$
$751$	−4.95672	−0.180873	−0.0904367	+	0.995902i	$0.528826\pi$
$752$	−22.4531	−0.818782
$753$	0	0
$754$	9.77854	0.356113
$755$	0	0
$756$	0	0
$757$	18.6020	0.676101	0.338051	−	0.941128i	$-0.390232\pi$
$757$	18.6020	0.676101	0.338051	+	0.941128i	$0.390232\pi$
$758$	−49.2223	−1.78784
$759$	0	0
$760$	0	0
$761$	11.3585	0.411747	0.205873	−	0.978579i	$-0.433997\pi$
$761$	11.3585	0.411747	0.205873	+	0.978579i	$0.433997\pi$
$762$	0	0
$763$	−2.40423	−0.0870391
$764$	−36.2471	−1.31137
$765$	0	0
$766$	−28.5510	−1.03159
$767$	−18.2088	−0.657481
$768$	0	0
$769$	7.27477	0.262335	0.131168	−	0.991360i	$-0.458127\pi$
$769$	7.27477	0.262335	0.131168	+	0.991360i	$0.458127\pi$
$770$	0	0
$771$	0	0
$772$	61.9925	2.23116
$773$	−9.32288	−0.335321	−0.167660	−	0.985845i	$-0.553621\pi$
$773$	−9.32288	−0.335321	−0.167660	+	0.985845i	$0.553621\pi$
$774$	0	0
$775$	0	0
$776$	37.3296	1.34005
$777$	0	0
$778$	37.8158	1.35576
$779$	6.25943	0.224267
$780$	0	0
$781$	−4.19977	−0.150279
$782$	49.8011	1.78088
$783$	0	0
$784$	−40.4277	−1.44385
$785$	0	0
$786$	0	0
$787$	−14.8658	−0.529907	−0.264953	−	0.964261i	$-0.585357\pi$
$787$	−14.8658	−0.529907	−0.264953	+	0.964261i	$0.585357\pi$
$788$	25.5843	0.911402
$789$	0	0
$790$	0	0
$791$	4.02979	0.143283
$792$	0	0
$793$	−30.7324	−1.09134
$794$	−61.1493	−2.17011
$795$	0	0
$796$	−117.986	−4.18192
$797$	−9.57546	−0.339180	−0.169590	−	0.985515i	$-0.554244\pi$
$797$	−9.57546	−0.339180	−0.169590	+	0.985515i	$0.554244\pi$
$798$	0	0
$799$	−16.7068	−0.591044
$800$	0	0
$801$	0	0
$802$	34.1520	1.20595
$803$	−33.6418	−1.18719
$804$	0	0
$805$	0	0
$806$	−12.2302	−0.430792
$807$	0	0
$808$	−94.9810	−3.34142
$809$	−41.1436	−1.44653	−0.723266	−	0.690569i	$-0.757360\pi$
$809$	−41.1436	−1.44653	−0.723266	+	0.690569i	$0.757360\pi$
$810$	0	0
$811$	43.4398	1.52538	0.762688	−	0.646766i	$-0.223879\pi$
$811$	43.4398	1.52538	0.762688	+	0.646766i	$0.223879\pi$
$812$	−3.14003	−0.110193
$813$	0	0
$814$	−12.1960	−0.427470
$815$	0	0
$816$	0	0
$817$	−30.5428	−1.06856
$818$	90.0206	3.14750
$819$	0	0
$820$	0	0
$821$	−17.4617	−0.609416	−0.304708	−	0.952446i	$-0.598559\pi$
$821$	−17.4617	−0.609416	−0.304708	+	0.952446i	$0.598559\pi$
$822$	0	0
$823$	2.22456	0.0775432	0.0387716	−	0.999248i	$-0.487656\pi$
$823$	2.22456	0.0775432	0.0387716	+	0.999248i	$0.487656\pi$
$824$	74.6616	2.60096
$825$	0	0
$826$	8.48106	0.295094
$827$	31.4550	1.09380	0.546898	−	0.837199i	$-0.315808\pi$
$827$	31.4550	1.09380	0.546898	+	0.837199i	$0.315808\pi$
$828$	0	0
$829$	53.2947	1.85100	0.925500	−	0.378748i	$-0.123645\pi$
$829$	53.2947	1.85100	0.925500	+	0.378748i	$0.123645\pi$
$830$	0	0
$831$	0	0
$832$	−6.18061	−0.214274
$833$	−30.0812	−1.04225
$834$	0	0
$835$	0	0
$836$	70.8543	2.45055
$837$	0	0
$838$	40.1256	1.38612
$839$	8.37331	0.289079	0.144539	−	0.989499i	$-0.453830\pi$
$839$	8.37331	0.289079	0.144539	+	0.989499i	$0.453830\pi$
$840$	0	0
$841$	−28.5380	−0.984069
$842$	−30.4047	−1.04782
$843$	0	0
$844$	−117.474	−4.04363
$845$	0	0
$846$	0	0
$847$	−2.26121	−0.0776960
$848$	−86.6733	−2.97637
$849$	0	0
$850$	0	0
$851$	6.24467	0.214064
$852$	0	0
$853$	1.22370	0.0418989	0.0209494	−	0.999781i	$-0.493331\pi$
$853$	1.22370	0.0418989	0.0209494	+	0.999781i	$0.493331\pi$
$854$	14.3142	0.489821
$855$	0	0
$856$	40.2308	1.37506
$857$	−14.3684	−0.490816	−0.245408	−	0.969420i	$-0.578922\pi$
$857$	−14.3684	−0.490816	−0.245408	+	0.969420i	$0.578922\pi$
$858$	0	0
$859$	−10.3090	−0.351738	−0.175869	−	0.984414i	$-0.556274\pi$
$859$	−10.3090	−0.351738	−0.175869	+	0.984414i	$0.556274\pi$
$860$	0	0
$861$	0	0
$862$	37.1586	1.26563
$863$	30.2724	1.03048	0.515242	−	0.857045i	$-0.327702\pi$
$863$	30.2724	1.03048	0.515242	+	0.857045i	$0.327702\pi$
$864$	0	0
$865$	0	0
$866$	10.8433	0.368470
$867$	0	0
$868$	3.92730	0.133301
$869$	4.29927	0.145843
$870$	0	0
$871$	−5.26750	−0.178482
$872$	14.3056	0.484450
$873$	0	0
$874$	−52.6220	−1.77996
$875$	0	0
$876$	0	0
$877$	−30.6345	−1.03445	−0.517227	−	0.855848i	$-0.673036\pi$
$877$	−30.6345	−1.03445	−0.517227	+	0.855848i	$0.673036\pi$
$878$	51.5333	1.73916
$879$	0	0
$880$	0	0
$881$	−15.2211	−0.512813	−0.256406	−	0.966569i	$-0.582539\pi$
$881$	−15.2211	−0.512813	−0.256406	+	0.966569i	$0.582539\pi$
$882$	0	0
$883$	−40.1985	−1.35279	−0.676393	−	0.736541i	$-0.736458\pi$
$883$	−40.1985	−1.35279	−0.676393	+	0.736541i	$0.736458\pi$
$884$	127.955	4.30360
$885$	0	0
$886$	−48.3537	−1.62447
$887$	−31.8432	−1.06919	−0.534594	−	0.845109i	$-0.679536\pi$
$887$	−31.8432	−1.06919	−0.534594	+	0.845109i	$0.679536\pi$
$888$	0	0
$889$	−12.1678	−0.408094
$890$	0	0
$891$	0	0
$892$	121.853	4.07993
$893$	17.6531	0.590739
$894$	0	0
$895$	0	0
$896$	13.1903	0.440658
$897$	0	0
$898$	74.8089	2.49641
$899$	−0.577817	−0.0192713
$900$	0	0
$901$	−64.4914	−2.14852
$902$	−8.78565	−0.292530
$903$	0	0
$904$	−23.9780	−0.797498
$905$	0	0
$906$	0	0
$907$	28.8507	0.957970	0.478985	−	0.877823i	$-0.341005\pi$
$907$	28.8507	0.957970	0.478985	+	0.877823i	$0.341005\pi$
$908$	0.577886	0.0191778
$909$	0	0
$910$	0	0
$911$	−48.3760	−1.60277	−0.801385	−	0.598149i	$-0.795903\pi$
$911$	−48.3760	−1.60277	−0.801385	+	0.598149i	$0.795903\pi$
$912$	0	0
$913$	−34.0941	−1.12835
$914$	71.8200	2.37560
$915$	0	0
$916$	12.5437	0.414456
$917$	8.28485	0.273590
$918$	0	0
$919$	−8.38022	−0.276438	−0.138219	−	0.990402i	$-0.544138\pi$
$919$	−8.38022	−0.276438	−0.138219	+	0.990402i	$0.544138\pi$
$920$	0	0
$921$	0	0
$922$	−42.0174	−1.38377
$923$	8.01392	0.263781
$924$	0	0
$925$	0	0
$926$	23.3703	0.767995
$927$	0	0
$928$	3.36786	0.110556
$929$	−14.3126	−0.469580	−0.234790	−	0.972046i	$-0.575440\pi$
$929$	−14.3126	−0.469580	−0.234790	+	0.972046i	$0.575440\pi$
$930$	0	0
$931$	31.7851	1.04171
$932$	35.4610	1.16156
$933$	0	0
$934$	−26.9305	−0.881194
$935$	0	0
$936$	0	0
$937$	21.9244	0.716240	0.358120	−	0.933675i	$-0.383418\pi$
$937$	21.9244	0.716240	0.358120	+	0.933675i	$0.383418\pi$
$938$	2.45343	0.0801074
$939$	0	0
$940$	0	0
$941$	−22.3934	−0.730005	−0.365003	−	0.931007i	$-0.618932\pi$
$941$	−22.3934	−0.730005	−0.365003	+	0.931007i	$0.618932\pi$
$942$	0	0
$943$	4.49847	0.146490
$944$	−21.9439	−0.714213
$945$	0	0
$946$	42.8694	1.39380
$947$	43.5638	1.41563	0.707816	−	0.706397i	$-0.249681\pi$
$947$	43.5638	1.41563	0.707816	+	0.706397i	$0.249681\pi$
$948$	0	0
$949$	64.1947	2.08385
$950$	0	0
$951$	0	0
$952$	−32.7504	−1.06145
$953$	2.37169	0.0768266	0.0384133	−	0.999262i	$-0.487770\pi$
$953$	2.37169	0.0768266	0.0384133	+	0.999262i	$0.487770\pi$
$954$	0	0
$955$	0	0
$956$	84.6308	2.73716
$957$	0	0
$958$	85.2813	2.75531
$959$	−11.4993	−0.371331
$960$	0	0
$961$	−30.2773	−0.976687
$962$	23.2722	0.750326
$963$	0	0
$964$	93.7678	3.02006
$965$	0	0
$966$	0	0
$967$	19.7857	0.636266	0.318133	−	0.948046i	$-0.396944\pi$
$967$	19.7857	0.636266	0.318133	+	0.948046i	$0.396944\pi$
$968$	13.4546	0.432447
$969$	0	0
$970$	0	0
$971$	−49.3838	−1.58480	−0.792401	−	0.610000i	$-0.791169\pi$
$971$	−49.3838	−1.58480	−0.792401	+	0.610000i	$0.791169\pi$
$972$	0	0
$973$	−12.7672	−0.409298
$974$	74.2234	2.37827
$975$	0	0
$976$	−37.0365	−1.18551
$977$	−47.7652	−1.52814	−0.764072	−	0.645130i	$-0.776803\pi$
$977$	−47.7652	−1.52814	−0.764072	+	0.645130i	$0.776803\pi$
$978$	0	0
$979$	26.9537	0.861445
$980$	0	0
$981$	0	0
$982$	35.4131	1.13008
$983$	51.9282	1.65625	0.828127	−	0.560541i	$-0.189407\pi$
$983$	51.9282	1.65625	0.828127	+	0.560541i	$0.189407\pi$
$984$	0	0
$985$	0	0
$986$	8.76846	0.279245
$987$	0	0
$988$	−135.203	−4.30138
$989$	−21.9502	−0.697976
$990$	0	0
$991$	−16.3790	−0.520297	−0.260148	−	0.965569i	$-0.583771\pi$
$991$	−16.3790	−0.520297	−0.260148	+	0.965569i	$0.583771\pi$
$992$	−4.21226	−0.133739
$993$	0	0
$994$	−3.73262	−0.118392
$995$	0	0
$996$	0	0
$997$	22.2515	0.704713	0.352357	−	0.935866i	$-0.385380\pi$
$997$	22.2515	0.704713	0.352357	+	0.935866i	$0.385380\pi$
$998$	90.7302	2.87201
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.t.1.1		8
3.2	odd	2		1875.2.a.p.1.8		8
5.4	even	2		5625.2.a.bd.1.8		8
15.2	even	4		1875.2.b.h.1249.16		16
15.8	even	4		1875.2.b.h.1249.1		16
15.14	odd	2		1875.2.a.m.1.1		8
25.8	odd	20		225.2.m.b.64.1		16
25.22	odd	20		225.2.m.b.109.1		16
75.8	even	20		75.2.i.a.64.4	yes	16
75.17	even	20		375.2.i.c.199.1		16
75.29	odd	10		375.2.g.e.76.4		16
75.44	odd	10		375.2.g.e.301.4		16
75.47	even	20		75.2.i.a.34.4	✓	16
75.53	even	20		375.2.i.c.49.1		16
75.56	odd	10		375.2.g.d.301.1		16
75.71	odd	10		375.2.g.d.76.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.4	✓	16	75.47	even	20
75.2.i.a.64.4	yes	16	75.8	even	20
225.2.m.b.64.1		16	25.8	odd	20
225.2.m.b.109.1		16	25.22	odd	20
375.2.g.d.76.1		16	75.71	odd	10
375.2.g.d.301.1		16	75.56	odd	10
375.2.g.e.76.4		16	75.29	odd	10
375.2.g.e.301.4		16	75.44	odd	10
375.2.i.c.49.1		16	75.53	even	20
375.2.i.c.199.1		16	75.17	even	20
1875.2.a.m.1.1		8	15.14	odd	2
1875.2.a.p.1.8		8	3.2	odd	2
1875.2.b.h.1249.1		16	15.8	even	4
1875.2.b.h.1249.16		16	15.2	even	4
5625.2.a.t.1.1		8	1.1	even	1	trivial
5625.2.a.bd.1.8		8	5.4	even	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53767 q^{2} +4.43979 q^{4} +1.04054 q^{7} -6.19138 q^{8} +O(q^{10})\)	\(q-2.53767 q^{2} +4.43979 q^{4} +1.04054 q^{7} -6.19138 q^{8} -2.97101 q^{11} +5.66922 q^{13} -2.64054 q^{14} +6.83213 q^{16} +5.08361 q^{17} -5.37156 q^{19} +7.53945 q^{22} -3.86039 q^{23} -14.3866 q^{26} +4.61976 q^{28} -0.679696 q^{29} +0.850111 q^{31} -4.95495 q^{32} -12.9006 q^{34} -1.61763 q^{37} +13.6313 q^{38} -1.16529 q^{41} +5.68601 q^{43} -13.1906 q^{44} +9.79640 q^{46} -3.28640 q^{47} -5.91729 q^{49} +25.1701 q^{52} -12.6861 q^{53} -6.44235 q^{56} +1.72485 q^{58} -3.21187 q^{59} -5.42093 q^{61} -2.15730 q^{62} -1.09021 q^{64} -0.929140 q^{67} +22.5702 q^{68} +1.41358 q^{71} +11.3234 q^{73} +4.10501 q^{74} -23.8486 q^{76} -3.09144 q^{77} -1.44707 q^{79} +2.95713 q^{82} +11.4756 q^{83} -14.4292 q^{86} +18.3946 q^{88} -9.07225 q^{89} +5.89903 q^{91} -17.1393 q^{92} +8.33982 q^{94} -6.02928 q^{97} +15.0161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8	\( 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * q^98

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