LMFDB - Embedded newform 5625.2.a.bc.1.4

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:	8.8.13366265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 $ x^8 - x^7 - 12x^6 + 10x^5 + 41x^4 - 20x^3 - 48x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.770071$ 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)

$f(q)$	$=$	$q-0.770071 q^{2} -1.40699 q^{4} -3.98808 q^{7} +2.62363 q^{8} +O(q^{10})$	$q-0.770071 q^{2} -1.40699 q^{4} -3.98808 q^{7} +2.62363 q^{8} +6.35014 q^{11} -5.35433 q^{13} +3.07111 q^{14} +0.793602 q^{16} -3.45746 q^{17} +2.32026 q^{19} -4.89006 q^{22} +0.955221 q^{23} +4.12322 q^{26} +5.61119 q^{28} -7.26630 q^{29} +6.40484 q^{31} -5.85838 q^{32} +2.66249 q^{34} +2.83348 q^{37} -1.78677 q^{38} +5.35164 q^{41} -3.93593 q^{43} -8.93458 q^{44} -0.735588 q^{46} +2.48348 q^{47} +8.90478 q^{49} +7.53350 q^{52} +3.95079 q^{53} -10.4632 q^{56} +5.59557 q^{58} -0.0941232 q^{59} -6.61174 q^{61} -4.93218 q^{62} +2.92417 q^{64} +10.2811 q^{67} +4.86461 q^{68} +5.81587 q^{71} -3.79897 q^{73} -2.18198 q^{74} -3.26459 q^{76} -25.3249 q^{77} -5.44480 q^{79} -4.12114 q^{82} +9.24696 q^{83} +3.03095 q^{86} +16.6604 q^{88} -10.1278 q^{89} +21.3535 q^{91} -1.34399 q^{92} -1.91245 q^{94} -7.89936 q^{97} -6.85731 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10}) $ 8 * q + q^2 + 9 * q^4 - 12 * q^7 + 3 * q^8	$ 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8} - 12 q^{11} - 14 q^{13} - 16 q^{14} + 15 q^{16} - q^{17} + 16 q^{19} - 18 q^{22} - 4 q^{23} + 34 q^{26} + 21 q^{28} - 2 q^{29} + 13 q^{31} - 18 q^{32} - 37 q^{34} + 8 q^{37} - 24 q^{38} + 12 q^{41} - 20 q^{43} - 47 q^{44} + 33 q^{46} - 15 q^{47} + 30 q^{49} + q^{52} - 4 q^{53} - 60 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} + 4 q^{62} + 41 q^{64} - 19 q^{67} - 33 q^{68} - 21 q^{71} + 19 q^{73} + 9 q^{74} - q^{76} - 11 q^{77} + 10 q^{79} - 24 q^{82} - 27 q^{83} - 42 q^{86} - 53 q^{88} + 9 q^{89} - 12 q^{91} - 63 q^{92} + 14 q^{94} - 24 q^{97} - 24 q^{98}+O(q^{100}) $ 8 * q + q^2 + 9 * q^4 - 12 * q^7 + 3 * q^8 - 12 * q^11 - 14 * q^13 - 16 * q^14 + 15 * q^16 - q^17 + 16 * q^19 - 18 * q^22 - 4 * q^23 + 34 * q^26 + 21 * q^28 - 2 * q^29 + 13 * q^31 - 18 * q^32 - 37 * q^34 + 8 * q^37 - 24 * q^38 + 12 * q^41 - 20 * q^43 - 47 * q^44 + 33 * q^46 - 15 * q^47 + 30 * q^49 + q^52 - 4 * q^53 - 60 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 + 4 * q^62 + 41 * q^64 - 19 * q^67 - 33 * q^68 - 21 * q^71 + 19 * q^73 + 9 * q^74 - q^76 - 11 * q^77 + 10 * q^79 - 24 * q^82 - 27 * q^83 - 42 * q^86 - 53 * q^88 + 9 * q^89 - 12 * q^91 - 63 * q^92 + 14 * q^94 - 24 * q^97 - 24 * 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 $

$2$	−0.770071	−0.544523	−0.272261	−	0.962223i	$-0.587772\pi$
$2$	−0.770071	−0.544523	−0.272261	+	0.962223i	$0.587772\pi$
$3$	0	0
$3$	0	0
$4$	−1.40699	−0.703495
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.98808	−1.50735	−0.753676	−	0.657246i	$-0.771721\pi$
$7$	−3.98808	−1.50735	−0.753676	+	0.657246i	$0.771721\pi$
$8$	2.62363	0.927592
$9$	0	0
$10$	0	0
$11$	6.35014	1.91464	0.957319	−	0.289033i	$-0.0933337\pi$
$11$	6.35014	1.91464	0.957319	+	0.289033i	$0.0933337\pi$
$12$	0	0
$13$	−5.35433	−1.48503	−0.742513	−	0.669832i	$-0.766366\pi$
$13$	−5.35433	−1.48503	−0.742513	+	0.669832i	$0.766366\pi$
$14$	3.07111	0.820787
$15$	0	0
$16$	0.793602	0.198400
$17$	−3.45746	−0.838557	−0.419279	−	0.907858i	$-0.637717\pi$
$17$	−3.45746	−0.838557	−0.419279	+	0.907858i	$0.637717\pi$
$18$	0	0
$19$	2.32026	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$19$	2.32026	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$20$	0	0
$21$	0	0
$22$	−4.89006	−1.04256
$23$	0.955221	0.199177	0.0995886	−	0.995029i	$-0.468247\pi$
$23$	0.955221	0.199177	0.0995886	+	0.995029i	$0.468247\pi$
$24$	0	0
$25$	0	0
$26$	4.12322	0.808630
$27$	0	0
$28$	5.61119	1.06042
$29$	−7.26630	−1.34932	−0.674659	−	0.738129i	$-0.735709\pi$
$29$	−7.26630	−1.34932	−0.674659	+	0.738129i	$0.735709\pi$
$30$	0	0
$31$	6.40484	1.15034	0.575172	−	0.818033i	$-0.304935\pi$
$31$	6.40484	1.15034	0.575172	+	0.818033i	$0.304935\pi$
$32$	−5.85838	−1.03563
$33$	0	0
$34$	2.66249	0.456613
$35$	0	0
$36$	0	0
$37$	2.83348	0.465821	0.232911	−	0.972498i	$-0.425175\pi$
$37$	2.83348	0.465821	0.232911	+	0.972498i	$0.425175\pi$
$38$	−1.78677	−0.289852
$39$	0	0
$40$	0	0
$41$	5.35164	0.835785	0.417893	−	0.908496i	$-0.362769\pi$
$41$	5.35164	0.835785	0.417893	+	0.908496i	$0.362769\pi$
$42$	0	0
$43$	−3.93593	−0.600224	−0.300112	−	0.953904i	$-0.597024\pi$
$43$	−3.93593	−0.600224	−0.300112	+	0.953904i	$0.597024\pi$
$44$	−8.93458	−1.34694
$45$	0	0
$46$	−0.735588	−0.108457
$47$	2.48348	0.362252	0.181126	−	0.983460i	$-0.442026\pi$
$47$	2.48348	0.362252	0.181126	+	0.983460i	$0.442026\pi$
$48$	0	0
$49$	8.90478	1.27211
$50$	0	0
$51$	0	0
$52$	7.53350	1.04471
$53$	3.95079	0.542682	0.271341	−	0.962483i	$-0.412533\pi$
$53$	3.95079	0.542682	0.271341	+	0.962483i	$0.412533\pi$
$54$	0	0
$55$	0	0
$56$	−10.4632	−1.39821
$57$	0	0
$58$	5.59557	0.734735
$59$	−0.0941232	−0.0122538	−0.00612690	−	0.999981i	$-0.501950\pi$
$59$	−0.0941232	−0.0122538	−0.00612690	+	0.999981i	$0.501950\pi$
$60$	0	0
$61$	−6.61174	−0.846547	−0.423273	−	0.906002i	$-0.639119\pi$
$61$	−6.61174	−0.846547	−0.423273	+	0.906002i	$0.639119\pi$
$62$	−4.93218	−0.626388
$63$	0	0
$64$	2.92417	0.365521
$65$	0	0
$66$	0	0
$67$	10.2811	1.25603	0.628016	−	0.778201i	$-0.283867\pi$
$67$	10.2811	1.25603	0.628016	+	0.778201i	$0.283867\pi$
$68$	4.86461	0.589921
$69$	0	0
$70$	0	0
$71$	5.81587	0.690217	0.345109	−	0.938563i	$-0.387842\pi$
$71$	5.81587	0.690217	0.345109	+	0.938563i	$0.387842\pi$
$72$	0	0
$73$	−3.79897	−0.444635	−0.222318	−	0.974974i	$-0.571362\pi$
$73$	−3.79897	−0.444635	−0.222318	+	0.974974i	$0.571362\pi$
$74$	−2.18198	−0.253650
$75$	0	0
$76$	−3.26459	−0.374474
$77$	−25.3249	−2.88603
$78$	0	0
$79$	−5.44480	−0.612587	−0.306294	−	0.951937i	$-0.599089\pi$
$79$	−5.44480	−0.612587	−0.306294	+	0.951937i	$0.599089\pi$
$80$	0	0
$81$	0	0
$82$	−4.12114	−0.455104
$83$	9.24696	1.01499	0.507493	−	0.861656i	$-0.330572\pi$
$83$	9.24696	1.01499	0.507493	+	0.861656i	$0.330572\pi$
$84$	0	0
$85$	0	0
$86$	3.03095	0.326836
$87$	0	0
$88$	16.6604	1.77600
$89$	−10.1278	−1.07355	−0.536773	−	0.843727i	$-0.680357\pi$
$89$	−10.1278	−1.07355	−0.536773	+	0.843727i	$0.680357\pi$
$90$	0	0
$91$	21.3535	2.23846
$92$	−1.34399	−0.140120
$93$	0	0
$94$	−1.91245	−0.197255
$95$	0	0
$96$	0	0
$97$	−7.89936	−0.802058	−0.401029	−	0.916065i	$-0.631347\pi$
$97$	−7.89936	−0.802058	−0.401029	+	0.916065i	$0.631347\pi$
$98$	−6.85731	−0.692693
$99$	0	0
$100$	0	0
$101$	−8.72974	−0.868641	−0.434321	−	0.900758i	$-0.643011\pi$
$101$	−8.72974	−0.868641	−0.434321	+	0.900758i	$0.643011\pi$
$102$	0	0
$103$	3.82390	0.376780	0.188390	−	0.982094i	$-0.439673\pi$
$103$	3.82390	0.376780	0.188390	+	0.982094i	$0.439673\pi$
$104$	−14.0478	−1.37750
$105$	0	0
$106$	−3.04239	−0.295503
$107$	14.1662	1.36950	0.684748	−	0.728780i	$-0.259912\pi$
$107$	14.1662	1.36950	0.684748	+	0.728780i	$0.259912\pi$
$108$	0	0
$109$	14.0557	1.34629	0.673145	−	0.739511i	$-0.264943\pi$
$109$	14.0557	1.34629	0.673145	+	0.739511i	$0.264943\pi$
$110$	0	0
$111$	0	0
$112$	−3.16495	−0.299059
$113$	9.26557	0.871632	0.435816	−	0.900036i	$-0.356460\pi$
$113$	9.26557	0.871632	0.435816	+	0.900036i	$0.356460\pi$
$114$	0	0
$115$	0	0
$116$	10.2236	0.949239
$117$	0	0
$118$	0.0724815	0.00667247
$119$	13.7886	1.26400
$120$	0	0
$121$	29.3242	2.66584
$122$	5.09151	0.460964
$123$	0	0
$124$	−9.01155	−0.809261
$125$	0	0
$126$	0	0
$127$	−6.91919	−0.613979	−0.306989	−	0.951713i	$-0.599322\pi$
$127$	−6.91919	−0.613979	−0.306989	+	0.951713i	$0.599322\pi$
$128$	9.46494	0.836591
$129$	0	0
$130$	0	0
$131$	−22.0453	−1.92611	−0.963055	−	0.269305i	$-0.913206\pi$
$131$	−22.0453	−1.92611	−0.963055	+	0.269305i	$0.913206\pi$
$132$	0	0
$133$	−9.25339	−0.802371
$134$	−7.91715	−0.683938
$135$	0	0
$136$	−9.07108	−0.777839
$137$	1.29992	0.111060	0.0555299	−	0.998457i	$-0.482315\pi$
$137$	1.29992	0.111060	0.0555299	+	0.998457i	$0.482315\pi$
$138$	0	0
$139$	3.84419	0.326060	0.163030	−	0.986621i	$-0.447873\pi$
$139$	3.84419	0.326060	0.163030	+	0.986621i	$0.447873\pi$
$140$	0	0
$141$	0	0
$142$	−4.47864	−0.375839
$143$	−34.0008	−2.84329
$144$	0	0
$145$	0	0
$146$	2.92547	0.242114
$147$	0	0
$148$	−3.98668	−0.327703
$149$	−16.1035	−1.31925	−0.659623	−	0.751596i	$-0.729284\pi$
$149$	−16.1035	−1.31925	−0.659623	+	0.751596i	$0.729284\pi$
$150$	0	0
$151$	−14.8215	−1.20616	−0.603079	−	0.797682i	$-0.706059\pi$
$151$	−14.8215	−1.20616	−0.603079	+	0.797682i	$0.706059\pi$
$152$	6.08750	0.493761
$153$	0	0
$154$	19.5019	1.57151
$155$	0	0
$156$	0	0
$157$	5.29762	0.422796	0.211398	−	0.977400i	$-0.432198\pi$
$157$	5.29762	0.422796	0.211398	+	0.977400i	$0.432198\pi$
$158$	4.19288	0.333568
$159$	0	0
$160$	0	0
$161$	−3.80950	−0.300230
$162$	0	0
$163$	2.86408	0.224332	0.112166	−	0.993689i	$-0.464221\pi$
$163$	2.86408	0.224332	0.112166	+	0.993689i	$0.464221\pi$
$164$	−7.52970	−0.587971
$165$	0	0
$166$	−7.12082	−0.552683
$167$	−2.57868	−0.199544	−0.0997720	−	0.995010i	$-0.531811\pi$
$167$	−2.57868	−0.199544	−0.0997720	+	0.995010i	$0.531811\pi$
$168$	0	0
$169$	15.6689	1.20530
$170$	0	0
$171$	0	0
$172$	5.53782	0.422255
$173$	−20.2272	−1.53784	−0.768921	−	0.639343i	$-0.779206\pi$
$173$	−20.2272	−1.53784	−0.768921	+	0.639343i	$0.779206\pi$
$174$	0	0
$175$	0	0
$176$	5.03948	0.379865
$177$	0	0
$178$	7.79913	0.584570
$179$	26.2365	1.96101	0.980506	−	0.196491i	$-0.0629546\pi$
$179$	26.2365	1.96101	0.980506	+	0.196491i	$0.0629546\pi$
$180$	0	0
$181$	2.37843	0.176787	0.0883936	−	0.996086i	$-0.471827\pi$
$181$	2.37843	0.176787	0.0883936	+	0.996086i	$0.471827\pi$
$182$	−16.4437	−1.21889
$183$	0	0
$184$	2.50614	0.184755
$185$	0	0
$186$	0	0
$187$	−21.9553	−1.60553
$188$	−3.49423	−0.254843
$189$	0	0
$190$	0	0
$191$	−5.71292	−0.413372	−0.206686	−	0.978407i	$-0.566268\pi$
$191$	−5.71292	−0.413372	−0.206686	+	0.978407i	$0.566268\pi$
$192$	0	0
$193$	−4.18017	−0.300895	−0.150448	−	0.988618i	$-0.548071\pi$
$193$	−4.18017	−0.300895	−0.150448	+	0.988618i	$0.548071\pi$
$194$	6.08307	0.436739
$195$	0	0
$196$	−12.5289	−0.894924
$197$	−14.6732	−1.04542	−0.522710	−	0.852511i	$-0.675079\pi$
$197$	−14.6732	−1.04542	−0.522710	+	0.852511i	$0.675079\pi$
$198$	0	0
$199$	8.77623	0.622130	0.311065	−	0.950389i	$-0.399314\pi$
$199$	8.77623	0.622130	0.311065	+	0.950389i	$0.399314\pi$
$200$	0	0
$201$	0	0
$202$	6.72252	0.472995
$203$	28.9786	2.03390
$204$	0	0
$205$	0	0
$206$	−2.94467	−0.205165
$207$	0	0
$208$	−4.24921	−0.294630
$209$	14.7340	1.01917
$210$	0	0
$211$	18.7411	1.29019	0.645095	−	0.764103i	$-0.276818\pi$
$211$	18.7411	1.29019	0.645095	+	0.764103i	$0.276818\pi$
$212$	−5.55872	−0.381774
$213$	0	0
$214$	−10.9090	−0.745722
$215$	0	0
$216$	0	0
$217$	−25.5430	−1.73397
$218$	−10.8239	−0.733085
$219$	0	0
$220$	0	0
$221$	18.5124	1.24528
$222$	0	0
$223$	−16.5574	−1.10877	−0.554383	−	0.832262i	$-0.687046\pi$
$223$	−16.5574	−1.10877	−0.554383	+	0.832262i	$0.687046\pi$
$224$	23.3637	1.56105
$225$	0	0
$226$	−7.13515	−0.474623
$227$	−5.07350	−0.336740	−0.168370	−	0.985724i	$-0.553850\pi$
$227$	−5.07350	−0.336740	−0.168370	+	0.985724i	$0.553850\pi$
$228$	0	0
$229$	−23.9323	−1.58149	−0.790744	−	0.612147i	$-0.790306\pi$
$229$	−23.9323	−1.58149	−0.790744	+	0.612147i	$0.790306\pi$
$230$	0	0
$231$	0	0
$232$	−19.0641	−1.25162
$233$	−18.5301	−1.21395	−0.606974	−	0.794722i	$-0.707617\pi$
$233$	−18.5301	−1.21395	−0.606974	+	0.794722i	$0.707617\pi$
$234$	0	0
$235$	0	0
$236$	0.132430	0.00862048
$237$	0	0
$238$	−10.6182	−0.688277
$239$	14.5812	0.943179	0.471590	−	0.881818i	$-0.343680\pi$
$239$	14.5812	0.943179	0.471590	+	0.881818i	$0.343680\pi$
$240$	0	0
$241$	0.0849630	0.00547295	0.00273647	−	0.999996i	$-0.499129\pi$
$241$	0.0849630	0.00547295	0.00273647	+	0.999996i	$0.499129\pi$
$242$	−22.5818	−1.45161
$243$	0	0
$244$	9.30266	0.595541
$245$	0	0
$246$	0	0
$247$	−12.4235	−0.790486
$248$	16.8039	1.06705
$249$	0	0
$250$	0	0
$251$	−20.6190	−1.30146	−0.650729	−	0.759310i	$-0.725537\pi$
$251$	−20.6190	−1.30146	−0.650729	+	0.759310i	$0.725537\pi$
$252$	0	0
$253$	6.06578	0.381352
$254$	5.32827	0.334325
$255$	0	0
$256$	−13.1370	−0.821063
$257$	−12.1228	−0.756199	−0.378100	−	0.925765i	$-0.623422\pi$
$257$	−12.1228	−0.756199	−0.378100	+	0.925765i	$0.623422\pi$
$258$	0	0
$259$	−11.3001	−0.702157
$260$	0	0
$261$	0	0
$262$	16.9765	1.04881
$263$	−7.25319	−0.447251	−0.223625	−	0.974675i	$-0.571789\pi$
$263$	−7.25319	−0.447251	−0.223625	+	0.974675i	$0.571789\pi$
$264$	0	0
$265$	0	0
$266$	7.12577	0.436909
$267$	0	0
$268$	−14.4654	−0.883612
$269$	−10.3778	−0.632744	−0.316372	−	0.948635i	$-0.602465\pi$
$269$	−10.3778	−0.632744	−0.316372	+	0.948635i	$0.602465\pi$
$270$	0	0
$271$	−29.4681	−1.79006	−0.895031	−	0.446004i	$-0.852847\pi$
$271$	−29.4681	−1.79006	−0.895031	+	0.446004i	$0.852847\pi$
$272$	−2.74385	−0.166370
$273$	0	0
$274$	−1.00103	−0.0604745
$275$	0	0
$276$	0	0
$277$	−17.4446	−1.04815	−0.524073	−	0.851673i	$-0.675588\pi$
$277$	−17.4446	−1.04815	−0.524073	+	0.851673i	$0.675588\pi$
$278$	−2.96030	−0.177547
$279$	0	0
$280$	0	0
$281$	−8.95212	−0.534039	−0.267019	−	0.963691i	$-0.586039\pi$
$281$	−8.95212	−0.534039	−0.267019	+	0.963691i	$0.586039\pi$
$282$	0	0
$283$	26.1417	1.55396	0.776982	−	0.629523i	$-0.216750\pi$
$283$	26.1417	1.55396	0.776982	+	0.629523i	$0.216750\pi$
$284$	−8.18288	−0.485564
$285$	0	0
$286$	26.1830	1.54823
$287$	−21.3428	−1.25982
$288$	0	0
$289$	−5.04597	−0.296822
$290$	0	0
$291$	0	0
$292$	5.34511	0.312799
$293$	−18.9520	−1.10719	−0.553594	−	0.832787i	$-0.686744\pi$
$293$	−18.9520	−1.10719	−0.553594	+	0.832787i	$0.686744\pi$
$294$	0	0
$295$	0	0
$296$	7.43399	0.432092
$297$	0	0
$298$	12.4008	0.718360
$299$	−5.11457	−0.295783
$300$	0	0
$301$	15.6968	0.904749
$302$	11.4136	0.656780
$303$	0	0
$304$	1.84137	0.105610
$305$	0	0
$306$	0	0
$307$	6.75782	0.385689	0.192845	−	0.981229i	$-0.438229\pi$
$307$	6.75782	0.385689	0.192845	+	0.981229i	$0.438229\pi$
$308$	35.6318	2.03031
$309$	0	0
$310$	0	0
$311$	−13.0403	−0.739445	−0.369723	−	0.929142i	$-0.620547\pi$
$311$	−13.0403	−0.739445	−0.369723	+	0.929142i	$0.620547\pi$
$312$	0	0
$313$	−20.1274	−1.13767	−0.568833	−	0.822453i	$-0.692605\pi$
$313$	−20.1274	−1.13767	−0.568833	+	0.822453i	$0.692605\pi$
$314$	−4.07955	−0.230222
$315$	0	0
$316$	7.66078	0.430952
$317$	−12.7288	−0.714921	−0.357460	−	0.933928i	$-0.616357\pi$
$317$	−12.7288	−0.714921	−0.357460	+	0.933928i	$0.616357\pi$
$318$	0	0
$319$	−46.1420	−2.58346
$320$	0	0
$321$	0	0
$322$	2.93358	0.163482
$323$	−8.02222	−0.446368
$324$	0	0
$325$	0	0
$326$	−2.20554	−0.122154
$327$	0	0
$328$	14.0407	0.775267
$329$	−9.90430	−0.546042
$330$	0	0
$331$	−16.3131	−0.896649	−0.448324	−	0.893871i	$-0.647979\pi$
$331$	−16.3131	−0.896649	−0.448324	+	0.893871i	$0.647979\pi$
$332$	−13.0104	−0.714038
$333$	0	0
$334$	1.98576	0.108656
$335$	0	0
$336$	0	0
$337$	−11.0359	−0.601166	−0.300583	−	0.953756i	$-0.597181\pi$
$337$	−11.0359	−0.601166	−0.300583	+	0.953756i	$0.597181\pi$
$338$	−12.0662	−0.656313
$339$	0	0
$340$	0	0
$341$	40.6716	2.20249
$342$	0	0
$343$	−7.59641	−0.410168
$344$	−10.3264	−0.556763
$345$	0	0
$346$	15.5764	0.837390
$347$	−30.3961	−1.63175	−0.815875	−	0.578229i	$-0.803744\pi$
$347$	−30.3961	−1.63175	−0.815875	+	0.578229i	$0.803744\pi$
$348$	0	0
$349$	−6.45842	−0.345711	−0.172856	−	0.984947i	$-0.555299\pi$
$349$	−6.45842	−0.345711	−0.172856	+	0.984947i	$0.555299\pi$
$350$	0	0
$351$	0	0
$352$	−37.2015	−1.98285
$353$	4.25227	0.226325	0.113163	−	0.993576i	$-0.463902\pi$
$353$	4.25227	0.226325	0.113163	+	0.993576i	$0.463902\pi$
$354$	0	0
$355$	0	0
$356$	14.2497	0.755234
$357$	0	0
$358$	−20.2040	−1.06782
$359$	8.14893	0.430084	0.215042	−	0.976605i	$-0.431011\pi$
$359$	8.14893	0.430084	0.215042	+	0.976605i	$0.431011\pi$
$360$	0	0
$361$	−13.6164	−0.716652
$362$	−1.83156	−0.0962646
$363$	0	0
$364$	−30.0442	−1.57474
$365$	0	0
$366$	0	0
$367$	−7.18365	−0.374984	−0.187492	−	0.982266i	$-0.560036\pi$
$367$	−7.18365	−0.374984	−0.187492	+	0.982266i	$0.560036\pi$
$368$	0.758065	0.0395169
$369$	0	0
$370$	0	0
$371$	−15.7561	−0.818013
$372$	0	0
$373$	7.23599	0.374666	0.187333	−	0.982297i	$-0.440016\pi$
$373$	7.23599	0.374666	0.187333	+	0.982297i	$0.440016\pi$
$374$	16.9072	0.874250
$375$	0	0
$376$	6.51571	0.336022
$377$	38.9062	2.00377
$378$	0	0
$379$	1.57210	0.0807535	0.0403767	−	0.999185i	$-0.487144\pi$
$379$	1.57210	0.0807535	0.0403767	+	0.999185i	$0.487144\pi$
$380$	0	0
$381$	0	0
$382$	4.39935	0.225090
$383$	28.4834	1.45543	0.727717	−	0.685878i	$-0.240581\pi$
$383$	28.4834	1.45543	0.727717	+	0.685878i	$0.240581\pi$
$384$	0	0
$385$	0	0
$386$	3.21903	0.163844
$387$	0	0
$388$	11.1143	0.564244
$389$	−5.45707	−0.276684	−0.138342	−	0.990385i	$-0.544177\pi$
$389$	−5.45707	−0.276684	−0.138342	+	0.990385i	$0.544177\pi$
$390$	0	0
$391$	−3.30264	−0.167022
$392$	23.3628	1.18000
$393$	0	0
$394$	11.2994	0.569255
$395$	0	0
$396$	0	0
$397$	13.6918	0.687172	0.343586	−	0.939121i	$-0.388358\pi$
$397$	13.6918	0.687172	0.343586	+	0.939121i	$0.388358\pi$
$398$	−6.75832	−0.338764
$399$	0	0
$400$	0	0
$401$	−10.6255	−0.530614	−0.265307	−	0.964164i	$-0.585473\pi$
$401$	−10.6255	−0.530614	−0.265307	+	0.964164i	$0.585473\pi$
$402$	0	0
$403$	−34.2937	−1.70829
$404$	12.2827	0.611085
$405$	0	0
$406$	−22.3156	−1.10750
$407$	17.9930	0.891879
$408$	0	0
$409$	−20.5777	−1.01750	−0.508750	−	0.860914i	$-0.669892\pi$
$409$	−20.5777	−1.01750	−0.508750	+	0.860914i	$0.669892\pi$
$410$	0	0
$411$	0	0
$412$	−5.38018	−0.265063
$413$	0.375371	0.0184708
$414$	0	0
$415$	0	0
$416$	31.3677	1.53793
$417$	0	0
$418$	−11.3462	−0.554962
$419$	−7.44362	−0.363644	−0.181822	−	0.983331i	$-0.558200\pi$
$419$	−7.44362	−0.363644	−0.181822	+	0.983331i	$0.558200\pi$
$420$	0	0
$421$	26.2168	1.27773	0.638863	−	0.769320i	$-0.279405\pi$
$421$	26.2168	1.27773	0.638863	+	0.769320i	$0.279405\pi$
$422$	−14.4320	−0.702537
$423$	0	0
$424$	10.3654	0.503387
$425$	0	0
$426$	0	0
$427$	26.3682	1.27604
$428$	−19.9317	−0.963434
$429$	0	0
$430$	0	0
$431$	22.9859	1.10719	0.553597	−	0.832785i	$-0.313255\pi$
$431$	22.9859	1.10719	0.553597	+	0.832785i	$0.313255\pi$
$432$	0	0
$433$	29.5347	1.41934	0.709672	−	0.704532i	$-0.248843\pi$
$433$	29.5347	1.41934	0.709672	+	0.704532i	$0.248843\pi$
$434$	19.6699	0.944188
$435$	0	0
$436$	−19.7762	−0.947108
$437$	2.21636	0.106023
$438$	0	0
$439$	23.4898	1.12111	0.560553	−	0.828119i	$-0.310589\pi$
$439$	23.4898	1.12111	0.560553	+	0.828119i	$0.310589\pi$
$440$	0	0
$441$	0	0
$442$	−14.2559	−0.678082
$443$	19.5349	0.928130	0.464065	−	0.885801i	$-0.346390\pi$
$443$	19.5349	0.928130	0.464065	+	0.885801i	$0.346390\pi$
$444$	0	0
$445$	0	0
$446$	12.7504	0.603748
$447$	0	0
$448$	−11.6618	−0.550969
$449$	−1.18687	−0.0560118	−0.0280059	−	0.999608i	$-0.508916\pi$
$449$	−1.18687	−0.0560118	−0.0280059	+	0.999608i	$0.508916\pi$
$450$	0	0
$451$	33.9836	1.60023
$452$	−13.0366	−0.613189
$453$	0	0
$454$	3.90696	0.183363
$455$	0	0
$456$	0	0
$457$	23.4359	1.09629	0.548144	−	0.836384i	$-0.315335\pi$
$457$	23.4359	1.09629	0.548144	+	0.836384i	$0.315335\pi$
$458$	18.4295	0.861156
$459$	0	0
$460$	0	0
$461$	−35.4249	−1.64990	−0.824950	−	0.565205i	$-0.808797\pi$
$461$	−35.4249	−1.64990	−0.824950	+	0.565205i	$0.808797\pi$
$462$	0	0
$463$	−9.44996	−0.439177	−0.219588	−	0.975593i	$-0.570471\pi$
$463$	−9.44996	−0.439177	−0.219588	+	0.975593i	$0.570471\pi$
$464$	−5.76655	−0.267706
$465$	0	0
$466$	14.2695	0.661022
$467$	−0.0277734	−0.00128520	−0.000642599	−	1.00000i	$-0.500205\pi$
$467$	−0.0277734	−0.00128520	−0.000642599		1.00000i	$0.500205\pi$
$468$	0	0
$469$	−41.0017	−1.89328
$470$	0	0
$471$	0	0
$472$	−0.246944	−0.0113665
$473$	−24.9937	−1.14921
$474$	0	0
$475$	0	0
$476$	−19.4005	−0.889219
$477$	0	0
$478$	−11.2286	−0.513583
$479$	−9.30016	−0.424935	−0.212467	−	0.977168i	$-0.568150\pi$
$479$	−9.30016	−0.424935	−0.212467	+	0.977168i	$0.568150\pi$
$480$	0	0
$481$	−15.1714	−0.691756
$482$	−0.0654276	−0.00298014
$483$	0	0
$484$	−41.2589	−1.87541
$485$	0	0
$486$	0	0
$487$	32.5963	1.47708	0.738539	−	0.674211i	$-0.235516\pi$
$487$	32.5963	1.47708	0.738539	+	0.674211i	$0.235516\pi$
$488$	−17.3467	−0.785250
$489$	0	0
$490$	0	0
$491$	1.08119	0.0487932	0.0243966	−	0.999702i	$-0.492234\pi$
$491$	1.08119	0.0487932	0.0243966	+	0.999702i	$0.492234\pi$
$492$	0	0
$493$	25.1230	1.13148
$494$	9.56695	0.430437
$495$	0	0
$496$	5.08290	0.228229
$497$	−23.1942	−1.04040
$498$	0	0
$499$	−43.9536	−1.96763	−0.983816	−	0.179181i	$-0.942655\pi$
$499$	−43.9536	−1.96763	−0.983816	+	0.179181i	$0.942655\pi$
$500$	0	0
$501$	0	0
$502$	15.8781	0.708673
$503$	−29.0142	−1.29368	−0.646841	−	0.762625i	$-0.723910\pi$
$503$	−29.0142	−1.29368	−0.646841	+	0.762625i	$0.723910\pi$
$504$	0	0
$505$	0	0
$506$	−4.67108	−0.207655
$507$	0	0
$508$	9.73523	0.431931
$509$	−5.13104	−0.227429	−0.113715	−	0.993513i	$-0.536275\pi$
$509$	−5.13104	−0.227429	−0.113715	+	0.993513i	$0.536275\pi$
$510$	0	0
$511$	15.1506	0.670222
$512$	−8.81345	−0.389503
$513$	0	0
$514$	9.33541	0.411767
$515$	0	0
$516$	0	0
$517$	15.7704	0.693582
$518$	8.70191	0.382340
$519$	0	0
$520$	0	0
$521$	8.20068	0.359278	0.179639	−	0.983733i	$-0.442507\pi$
$521$	8.20068	0.359278	0.179639	+	0.983733i	$0.442507\pi$
$522$	0	0
$523$	−5.23806	−0.229044	−0.114522	−	0.993421i	$-0.536534\pi$
$523$	−5.23806	−0.229044	−0.114522	+	0.993421i	$0.536534\pi$
$524$	31.0176	1.35501
$525$	0	0
$526$	5.58547	0.243538
$527$	−22.1445	−0.964629
$528$	0	0
$529$	−22.0876	−0.960328
$530$	0	0
$531$	0	0
$532$	13.0194	0.564464
$533$	−28.6545	−1.24116
$534$	0	0
$535$	0	0
$536$	26.9737	1.16508
$537$	0	0
$538$	7.99163	0.344544
$539$	56.5466	2.43563
$540$	0	0
$541$	13.2888	0.571329	0.285664	−	0.958330i	$-0.407786\pi$
$541$	13.2888	0.571329	0.285664	+	0.958330i	$0.407786\pi$
$542$	22.6926	0.974729
$543$	0	0
$544$	20.2551	0.868431
$545$	0	0
$546$	0	0
$547$	40.1311	1.71588	0.857940	−	0.513750i	$-0.171744\pi$
$547$	40.1311	1.71588	0.857940	+	0.513750i	$0.171744\pi$
$548$	−1.82898	−0.0781300
$549$	0	0
$550$	0	0
$551$	−16.8597	−0.718249
$552$	0	0
$553$	21.7143	0.923385
$554$	13.4336	0.570740
$555$	0	0
$556$	−5.40874	−0.229382
$557$	−45.6945	−1.93614	−0.968069	−	0.250686i	$-0.919344\pi$
$557$	−45.6945	−1.93614	−0.968069	+	0.250686i	$0.919344\pi$
$558$	0	0
$559$	21.0743	0.891348
$560$	0	0
$561$	0	0
$562$	6.89377	0.290796
$563$	0.600201	0.0252955	0.0126477	−	0.999920i	$-0.495974\pi$
$563$	0.600201	0.0252955	0.0126477	+	0.999920i	$0.495974\pi$
$564$	0	0
$565$	0	0
$566$	−20.1310	−0.846168
$567$	0	0
$568$	15.2587	0.640240
$569$	11.9725	0.501914	0.250957	−	0.967998i	$-0.419255\pi$
$569$	11.9725	0.501914	0.250957	+	0.967998i	$0.419255\pi$
$570$	0	0
$571$	−19.4133	−0.812422	−0.406211	−	0.913779i	$-0.633150\pi$
$571$	−19.4133	−0.812422	−0.406211	+	0.913779i	$0.633150\pi$
$572$	47.8387	2.00024
$573$	0	0
$574$	16.4354	0.686002
$575$	0	0
$576$	0	0
$577$	−26.0641	−1.08506	−0.542532	−	0.840035i	$-0.682534\pi$
$577$	−26.0641	−1.08506	−0.542532	+	0.840035i	$0.682534\pi$
$578$	3.88575	0.161626
$579$	0	0
$580$	0	0
$581$	−36.8776	−1.52994
$582$	0	0
$583$	25.0880	1.03904
$584$	−9.96706	−0.412440
$585$	0	0
$586$	14.5944	0.602888
$587$	2.53968	0.104824	0.0524120	−	0.998626i	$-0.483309\pi$
$587$	2.53968	0.104824	0.0524120	+	0.998626i	$0.483309\pi$
$588$	0	0
$589$	14.8609	0.612333
$590$	0	0
$591$	0	0
$592$	2.24865	0.0924191
$593$	4.27995	0.175757	0.0878783	−	0.996131i	$-0.471991\pi$
$593$	4.27995	0.175757	0.0878783	+	0.996131i	$0.471991\pi$
$594$	0	0
$595$	0	0
$596$	22.6574	0.928084
$597$	0	0
$598$	3.93858	0.161061
$599$	−27.1909	−1.11099	−0.555495	−	0.831520i	$-0.687471\pi$
$599$	−27.1909	−1.11099	−0.555495	+	0.831520i	$0.687471\pi$
$600$	0	0
$601$	−44.6772	−1.82242	−0.911211	−	0.411940i	$-0.864851\pi$
$601$	−44.6772	−1.82242	−0.911211	+	0.411940i	$0.864851\pi$
$602$	−12.0877	−0.492656
$603$	0	0
$604$	20.8537	0.848526
$605$	0	0
$606$	0	0
$607$	−3.48433	−0.141425	−0.0707123	−	0.997497i	$-0.522527\pi$
$607$	−3.48433	−0.141425	−0.0707123	+	0.997497i	$0.522527\pi$
$608$	−13.5930	−0.551268
$609$	0	0
$610$	0	0
$611$	−13.2974	−0.537954
$612$	0	0
$613$	11.1118	0.448803	0.224402	−	0.974497i	$-0.427957\pi$
$613$	11.1118	0.448803	0.224402	+	0.974497i	$0.427957\pi$
$614$	−5.20400	−0.210016
$615$	0	0
$616$	−66.4429	−2.67706
$617$	−33.0355	−1.32996	−0.664980	−	0.746861i	$-0.731560\pi$
$617$	−33.0355	−1.32996	−0.664980	+	0.746861i	$0.731560\pi$
$618$	0	0
$619$	17.0417	0.684964	0.342482	−	0.939524i	$-0.388732\pi$
$619$	17.0417	0.684964	0.342482	+	0.939524i	$0.388732\pi$
$620$	0	0
$621$	0	0
$622$	10.0419	0.402645
$623$	40.3905	1.61821
$624$	0	0
$625$	0	0
$626$	15.4995	0.619485
$627$	0	0
$628$	−7.45370	−0.297435
$629$	−9.79664	−0.390618
$630$	0	0
$631$	7.72790	0.307643	0.153821	−	0.988099i	$-0.450842\pi$
$631$	7.72790	0.307643	0.153821	+	0.988099i	$0.450842\pi$
$632$	−14.2851	−0.568231
$633$	0	0
$634$	9.80208	0.389290
$635$	0	0
$636$	0	0
$637$	−47.6792	−1.88912
$638$	35.5327	1.40675
$639$	0	0
$640$	0	0
$641$	−14.4910	−0.572361	−0.286181	−	0.958176i	$-0.592386\pi$
$641$	−14.4910	−0.572361	−0.286181	+	0.958176i	$0.592386\pi$
$642$	0	0
$643$	−42.5134	−1.67656	−0.838282	−	0.545238i	$-0.816439\pi$
$643$	−42.5134	−1.67656	−0.838282	+	0.545238i	$0.816439\pi$
$644$	5.35992	0.211211
$645$	0	0
$646$	6.17768	0.243058
$647$	−26.3343	−1.03531	−0.517655	−	0.855590i	$-0.673195\pi$
$647$	−26.3343	−1.03531	−0.517655	+	0.855590i	$0.673195\pi$
$648$	0	0
$649$	−0.597695	−0.0234616
$650$	0	0
$651$	0	0
$652$	−4.02973	−0.157816
$653$	4.46840	0.174862	0.0874310	−	0.996171i	$-0.472134\pi$
$653$	4.46840	0.174862	0.0874310	+	0.996171i	$0.472134\pi$
$654$	0	0
$655$	0	0
$656$	4.24707	0.165820
$657$	0	0
$658$	7.62702	0.297332
$659$	−40.2092	−1.56633	−0.783165	−	0.621814i	$-0.786396\pi$
$659$	−40.2092	−1.56633	−0.783165	+	0.621814i	$0.786396\pi$
$660$	0	0
$661$	−16.0012	−0.622374	−0.311187	−	0.950349i	$-0.600727\pi$
$661$	−16.0012	−0.622374	−0.311187	+	0.950349i	$0.600727\pi$
$662$	12.5622	0.488246
$663$	0	0
$664$	24.2606	0.941492
$665$	0	0
$666$	0	0
$667$	−6.94092	−0.268754
$668$	3.62817	0.140378
$669$	0	0
$670$	0	0
$671$	−41.9855	−1.62083
$672$	0	0
$673$	10.6585	0.410854	0.205427	−	0.978672i	$-0.434142\pi$
$673$	10.6585	0.410854	0.205427	+	0.978672i	$0.434142\pi$
$674$	8.49846	0.327349
$675$	0	0
$676$	−22.0460	−0.847922
$677$	15.9650	0.613585	0.306793	−	0.951776i	$-0.400744\pi$
$677$	15.9650	0.613585	0.306793	+	0.951776i	$0.400744\pi$
$678$	0	0
$679$	31.5033	1.20898
$680$	0	0
$681$	0	0
$682$	−31.3201	−1.19931
$683$	1.13294	0.0433508	0.0216754	−	0.999765i	$-0.493100\pi$
$683$	1.13294	0.0433508	0.0216754	+	0.999765i	$0.493100\pi$
$684$	0	0
$685$	0	0
$686$	5.84978	0.223346
$687$	0	0
$688$	−3.12356	−0.119085
$689$	−21.1538	−0.805897
$690$	0	0
$691$	−11.4786	−0.436668	−0.218334	−	0.975874i	$-0.570062\pi$
$691$	−11.4786	−0.436668	−0.218334	+	0.975874i	$0.570062\pi$
$692$	28.4594	1.08186
$693$	0	0
$694$	23.4072	0.888525
$695$	0	0
$696$	0	0
$697$	−18.5031	−0.700854
$698$	4.97344	0.188248
$699$	0	0
$700$	0	0
$701$	27.9382	1.05521	0.527605	−	0.849490i	$-0.323090\pi$
$701$	27.9382	1.05521	0.527605	+	0.849490i	$0.323090\pi$
$702$	0	0
$703$	6.57442	0.247959
$704$	18.5689	0.699840
$705$	0	0
$706$	−3.27455	−0.123239
$707$	34.8149	1.30935
$708$	0	0
$709$	26.2614	0.986266	0.493133	−	0.869954i	$-0.335852\pi$
$709$	26.2614	0.986266	0.493133	+	0.869954i	$0.335852\pi$
$710$	0	0
$711$	0	0
$712$	−26.5716	−0.995812
$713$	6.11804	0.229122
$714$	0	0
$715$	0	0
$716$	−36.9146	−1.37956
$717$	0	0
$718$	−6.27526	−0.234191
$719$	47.8778	1.78554	0.892771	−	0.450510i	$-0.148758\pi$
$719$	47.8778	1.78554	0.892771	+	0.450510i	$0.148758\pi$
$720$	0	0
$721$	−15.2500	−0.567940
$722$	10.4856	0.390233
$723$	0	0
$724$	−3.34642	−0.124369
$725$	0	0
$726$	0	0
$727$	6.74514	0.250163	0.125082	−	0.992146i	$-0.460081\pi$
$727$	6.74514	0.250163	0.125082	+	0.992146i	$0.460081\pi$
$728$	56.0236	2.07637
$729$	0	0
$730$	0	0
$731$	13.6083	0.503322
$732$	0	0
$733$	−42.9974	−1.58815	−0.794073	−	0.607822i	$-0.792043\pi$
$733$	−42.9974	−1.58815	−0.794073	+	0.607822i	$0.792043\pi$
$734$	5.53192	0.204187
$735$	0	0
$736$	−5.59605	−0.206273
$737$	65.2862	2.40485
$738$	0	0
$739$	−17.0432	−0.626943	−0.313472	−	0.949598i	$-0.601492\pi$
$739$	−17.0432	−0.626943	−0.313472	+	0.949598i	$0.601492\pi$
$740$	0	0
$741$	0	0
$742$	12.1333	0.445427
$743$	−50.4068	−1.84925	−0.924623	−	0.380884i	$-0.875620\pi$
$743$	−50.4068	−1.84925	−0.924623	+	0.380884i	$0.875620\pi$
$744$	0	0
$745$	0	0
$746$	−5.57223	−0.204014
$747$	0	0
$748$	30.8910	1.12949
$749$	−56.4958	−2.06431
$750$	0	0
$751$	−3.00535	−0.109667	−0.0548334	−	0.998496i	$-0.517463\pi$
$751$	−3.00535	−0.109667	−0.0548334	+	0.998496i	$0.517463\pi$
$752$	1.97089	0.0718710
$753$	0	0
$754$	−29.9606	−1.09110
$755$	0	0
$756$	0	0
$757$	8.74406	0.317808	0.158904	−	0.987294i	$-0.449204\pi$
$757$	8.74406	0.317808	0.158904	+	0.987294i	$0.449204\pi$
$758$	−1.21063	−0.0439721
$759$	0	0
$760$	0	0
$761$	31.8086	1.15306	0.576530	−	0.817076i	$-0.304406\pi$
$761$	31.8086	1.15306	0.576530	+	0.817076i	$0.304406\pi$
$762$	0	0
$763$	−56.0551	−2.02933
$764$	8.03802	0.290805
$765$	0	0
$766$	−21.9342	−0.792517
$767$	0.503967	0.0181972
$768$	0	0
$769$	28.7776	1.03775	0.518874	−	0.854851i	$-0.326351\pi$
$769$	28.7776	1.03775	0.518874	+	0.854851i	$0.326351\pi$
$770$	0	0
$771$	0	0
$772$	5.88146	0.211678
$773$	34.6320	1.24563	0.622814	−	0.782370i	$-0.285989\pi$
$773$	34.6320	1.24563	0.622814	+	0.782370i	$0.285989\pi$
$774$	0	0
$775$	0	0
$776$	−20.7250	−0.743983
$777$	0	0
$778$	4.20233	0.150661
$779$	12.4172	0.444893
$780$	0	0
$781$	36.9316	1.32152
$782$	2.54327	0.0909470
$783$	0	0
$784$	7.06685	0.252388
$785$	0	0
$786$	0	0
$787$	52.9281	1.88668	0.943342	−	0.331821i	$-0.107663\pi$
$787$	52.9281	1.88668	0.943342	+	0.331821i	$0.107663\pi$
$788$	20.6450	0.735448
$789$	0	0
$790$	0	0
$791$	−36.9518	−1.31386
$792$	0	0
$793$	35.4015	1.25714
$794$	−10.5437	−0.374181
$795$	0	0
$796$	−12.3481	−0.437666
$797$	32.9585	1.16745	0.583724	−	0.811952i	$-0.301595\pi$
$797$	32.9585	1.16745	0.583724	+	0.811952i	$0.301595\pi$
$798$	0	0
$799$	−8.58652	−0.303769
$800$	0	0
$801$	0	0
$802$	8.18242	0.288931
$803$	−24.1240	−0.851316
$804$	0	0
$805$	0	0
$806$	26.4086	0.930202
$807$	0	0
$808$	−22.9036	−0.805744
$809$	33.7389	1.18620	0.593098	−	0.805130i	$-0.297905\pi$
$809$	33.7389	1.18620	0.593098	+	0.805130i	$0.297905\pi$
$810$	0	0
$811$	−23.3207	−0.818902	−0.409451	−	0.912332i	$-0.634280\pi$
$811$	−23.3207	−0.818902	−0.409451	+	0.912332i	$0.634280\pi$
$812$	−40.7726	−1.43084
$813$	0	0
$814$	−13.8559	−0.485648
$815$	0	0
$816$	0	0
$817$	−9.13240	−0.319502
$818$	15.8463	0.554052
$819$	0	0
$820$	0	0
$821$	−16.9918	−0.593018	−0.296509	−	0.955030i	$-0.595822\pi$
$821$	−16.9918	−0.593018	−0.296509	+	0.955030i	$0.595822\pi$
$822$	0	0
$823$	29.8674	1.04111	0.520555	−	0.853828i	$-0.325725\pi$
$823$	29.8674	1.04111	0.520555	+	0.853828i	$0.325725\pi$
$824$	10.0325	0.349498
$825$	0	0
$826$	−0.289062	−0.0100578
$827$	−34.1803	−1.18857	−0.594283	−	0.804256i	$-0.702564\pi$
$827$	−34.1803	−1.18857	−0.594283	+	0.804256i	$0.702564\pi$
$828$	0	0
$829$	45.5662	1.58258	0.791289	−	0.611442i	$-0.209410\pi$
$829$	45.5662	1.58258	0.791289	+	0.611442i	$0.209410\pi$
$830$	0	0
$831$	0	0
$832$	−15.6570	−0.542808
$833$	−30.7879	−1.06674
$834$	0	0
$835$	0	0
$836$	−20.7306	−0.716982
$837$	0	0
$838$	5.73212	0.198013
$839$	5.10480	0.176237	0.0881186	−	0.996110i	$-0.471915\pi$
$839$	5.10480	0.176237	0.0881186	+	0.996110i	$0.471915\pi$
$840$	0	0
$841$	23.7992	0.820661
$842$	−20.1888	−0.695751
$843$	0	0
$844$	−26.3685	−0.907642
$845$	0	0
$846$	0	0
$847$	−116.947	−4.01836
$848$	3.13535	0.107668
$849$	0	0
$850$	0	0
$851$	2.70660	0.0927810
$852$	0	0
$853$	−57.1027	−1.95516	−0.977580	−	0.210564i	$-0.932470\pi$
$853$	−57.1027	−1.95516	−0.977580	+	0.210564i	$0.932470\pi$
$854$	−20.3054	−0.694835
$855$	0	0
$856$	37.1667	1.27033
$857$	46.0057	1.57153	0.785763	−	0.618528i	$-0.212271\pi$
$857$	46.0057	1.57153	0.785763	+	0.618528i	$0.212271\pi$
$858$	0	0
$859$	−39.1766	−1.33669	−0.668343	−	0.743853i	$-0.732996\pi$
$859$	−39.1766	−1.33669	−0.668343	+	0.743853i	$0.732996\pi$
$860$	0	0
$861$	0	0
$862$	−17.7008	−0.602892
$863$	−34.8513	−1.18635	−0.593175	−	0.805073i	$-0.702126\pi$
$863$	−34.8513	−1.18635	−0.593175	+	0.805073i	$0.702126\pi$
$864$	0	0
$865$	0	0
$866$	−22.7438	−0.772865
$867$	0	0
$868$	35.9388	1.21984
$869$	−34.5752	−1.17288
$870$	0	0
$871$	−55.0482	−1.86524
$872$	36.8768	1.24881
$873$	0	0
$874$	−1.70676	−0.0577319
$875$	0	0
$876$	0	0
$877$	−24.0922	−0.813537	−0.406768	−	0.913531i	$-0.633344\pi$
$877$	−24.0922	−0.813537	−0.406768	+	0.913531i	$0.633344\pi$
$878$	−18.0888	−0.610467
$879$	0	0
$880$	0	0
$881$	2.41163	0.0812499	0.0406250	−	0.999174i	$-0.487065\pi$
$881$	2.41163	0.0812499	0.0406250	+	0.999174i	$0.487065\pi$
$882$	0	0
$883$	1.48748	0.0500578	0.0250289	−	0.999687i	$-0.492032\pi$
$883$	1.48748	0.0500578	0.0250289	+	0.999687i	$0.492032\pi$
$884$	−26.0468	−0.876047
$885$	0	0
$886$	−15.0432	−0.505388
$887$	−20.1745	−0.677392	−0.338696	−	0.940896i	$-0.609986\pi$
$887$	−20.1745	−0.677392	−0.338696	+	0.940896i	$0.609986\pi$
$888$	0	0
$889$	27.5943	0.925482
$890$	0	0
$891$	0	0
$892$	23.2961	0.780011
$893$	5.76232	0.192829
$894$	0	0
$895$	0	0
$896$	−37.7470	−1.26104
$897$	0	0
$898$	0.913974	0.0304997
$899$	−46.5395	−1.55218
$900$	0	0
$901$	−13.6597	−0.455070
$902$	−26.1698	−0.871360
$903$	0	0
$904$	24.3094	0.808518
$905$	0	0
$906$	0	0
$907$	−26.2850	−0.872778	−0.436389	−	0.899758i	$-0.643743\pi$
$907$	−26.2850	−0.872778	−0.436389	+	0.899758i	$0.643743\pi$
$908$	7.13837	0.236895
$909$	0	0
$910$	0	0
$911$	22.8586	0.757340	0.378670	−	0.925532i	$-0.376381\pi$
$911$	22.8586	0.757340	0.378670	+	0.925532i	$0.376381\pi$
$912$	0	0
$913$	58.7195	1.94333
$914$	−18.0473	−0.596953
$915$	0	0
$916$	33.6724	1.11257
$917$	87.9186	2.90333
$918$	0	0
$919$	−15.7960	−0.521061	−0.260530	−	0.965466i	$-0.583897\pi$
$919$	−15.7960	−0.521061	−0.260530	+	0.965466i	$0.583897\pi$
$920$	0	0
$921$	0	0
$922$	27.2797	0.898408
$923$	−31.1401	−1.02499
$924$	0	0
$925$	0	0
$926$	7.27714	0.239142
$927$	0	0
$928$	42.5688	1.39739
$929$	−48.5302	−1.59222	−0.796111	−	0.605150i	$-0.793113\pi$
$929$	−48.5302	−1.59222	−0.796111	+	0.605150i	$0.793113\pi$
$930$	0	0
$931$	20.6614	0.677151
$932$	26.0717	0.854006
$933$	0	0
$934$	0.0213875	0.000699819 0
$935$	0	0
$936$	0	0
$937$	51.4725	1.68153	0.840767	−	0.541397i	$-0.182104\pi$
$937$	51.4725	1.68153	0.840767	+	0.541397i	$0.182104\pi$
$938$	31.5742	1.03094
$939$	0	0
$940$	0	0
$941$	19.4724	0.634782	0.317391	−	0.948295i	$-0.397193\pi$
$941$	19.4724	0.634782	0.317391	+	0.948295i	$0.397193\pi$
$942$	0	0
$943$	5.11199	0.166469
$944$	−0.0746963	−0.00243116
$945$	0	0
$946$	19.2469	0.625772
$947$	38.9572	1.26594	0.632970	−	0.774177i	$-0.281836\pi$
$947$	38.9572	1.26594	0.632970	+	0.774177i	$0.281836\pi$
$948$	0	0
$949$	20.3409	0.660295
$950$	0	0
$951$	0	0
$952$	36.1762	1.17248
$953$	0.532270	0.0172419	0.00862095	−	0.999963i	$-0.497256\pi$
$953$	0.532270	0.0172419	0.00862095	+	0.999963i	$0.497256\pi$
$954$	0	0
$955$	0	0
$956$	−20.5156	−0.663522
$957$	0	0
$958$	7.16178	0.231387
$959$	−5.18419	−0.167406
$960$	0	0
$961$	10.0220	0.323290
$962$	11.6831	0.376677
$963$	0	0
$964$	−0.119542	−0.00385019
$965$	0	0
$966$	0	0
$967$	36.8517	1.18507	0.592536	−	0.805544i	$-0.298127\pi$
$967$	36.8517	1.18507	0.592536	+	0.805544i	$0.298127\pi$
$968$	76.9358	2.47281
$969$	0	0
$970$	0	0
$971$	−4.25064	−0.136410	−0.0682048	−	0.997671i	$-0.521727\pi$
$971$	−4.25064	−0.136410	−0.0682048	+	0.997671i	$0.521727\pi$
$972$	0	0
$973$	−15.3310	−0.491488
$974$	−25.1014	−0.804302
$975$	0	0
$976$	−5.24709	−0.167955
$977$	31.6086	1.01125	0.505625	−	0.862754i	$-0.331262\pi$
$977$	31.6086	1.01125	0.505625	+	0.862754i	$0.331262\pi$
$978$	0	0
$979$	−64.3130	−2.05545
$980$	0	0
$981$	0	0
$982$	−0.832590	−0.0265690
$983$	−45.3330	−1.44590	−0.722949	−	0.690901i	$-0.757214\pi$
$983$	−45.3330	−1.44590	−0.722949	+	0.690901i	$0.757214\pi$
$984$	0	0
$985$	0	0
$986$	−19.3465	−0.616117
$987$	0	0
$988$	17.4797	0.556103
$989$	−3.75968	−0.119551
$990$	0	0
$991$	9.20124	0.292287	0.146143	−	0.989263i	$-0.453314\pi$
$991$	9.20124	0.292287	0.146143	+	0.989263i	$0.453314\pi$
$992$	−37.5220	−1.19132
$993$	0	0
$994$	17.8612	0.566522
$995$	0	0
$996$	0	0
$997$	13.1014	0.414924	0.207462	−	0.978243i	$-0.433480\pi$
$997$	13.1014	0.414924	0.207462	+	0.978243i	$0.433480\pi$
$998$	33.8474	1.07142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bc.1.4		8
3.2	odd	2		1875.2.a.n.1.5	✓	8
5.4	even	2		5625.2.a.u.1.5		8
15.2	even	4		1875.2.b.g.1249.10		16
15.8	even	4		1875.2.b.g.1249.7		16
15.14	odd	2		1875.2.a.o.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.5	✓	8	3.2	odd	2
1875.2.a.o.1.4	yes	8	15.14	odd	2
1875.2.b.g.1249.7		16	15.8	even	4
1875.2.b.g.1249.10		16	15.2	even	4
5625.2.a.u.1.5		8	5.4	even	2
5625.2.a.bc.1.4		8	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.770071 q^{2} -1.40699 q^{4} -3.98808 q^{7} +2.62363 q^{8} +O(q^{10})\)	\(q-0.770071 q^{2} -1.40699 q^{4} -3.98808 q^{7} +2.62363 q^{8} +6.35014 q^{11} -5.35433 q^{13} +3.07111 q^{14} +0.793602 q^{16} -3.45746 q^{17} +2.32026 q^{19} -4.89006 q^{22} +0.955221 q^{23} +4.12322 q^{26} +5.61119 q^{28} -7.26630 q^{29} +6.40484 q^{31} -5.85838 q^{32} +2.66249 q^{34} +2.83348 q^{37} -1.78677 q^{38} +5.35164 q^{41} -3.93593 q^{43} -8.93458 q^{44} -0.735588 q^{46} +2.48348 q^{47} +8.90478 q^{49} +7.53350 q^{52} +3.95079 q^{53} -10.4632 q^{56} +5.59557 q^{58} -0.0941232 q^{59} -6.61174 q^{61} -4.93218 q^{62} +2.92417 q^{64} +10.2811 q^{67} +4.86461 q^{68} +5.81587 q^{71} -3.79897 q^{73} -2.18198 q^{74} -3.26459 q^{76} -25.3249 q^{77} -5.44480 q^{79} -4.12114 q^{82} +9.24696 q^{83} +3.03095 q^{86} +16.6604 q^{88} -10.1278 q^{89} +21.3535 q^{91} -1.34399 q^{92} -1.91245 q^{94} -7.89936 q^{97} -6.85731 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10}) \) 8 * q + q^2 + 9 * q^4 - 12 * q^7 + 3 * q^8	\( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8} - 12 q^{11} - 14 q^{13} - 16 q^{14} + 15 q^{16} - q^{17} + 16 q^{19} - 18 q^{22} - 4 q^{23} + 34 q^{26} + 21 q^{28} - 2 q^{29} + 13 q^{31} - 18 q^{32} - 37 q^{34} + 8 q^{37} - 24 q^{38} + 12 q^{41} - 20 q^{43} - 47 q^{44} + 33 q^{46} - 15 q^{47} + 30 q^{49} + q^{52} - 4 q^{53} - 60 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} + 4 q^{62} + 41 q^{64} - 19 q^{67} - 33 q^{68} - 21 q^{71} + 19 q^{73} + 9 q^{74} - q^{76} - 11 q^{77} + 10 q^{79} - 24 q^{82} - 27 q^{83} - 42 q^{86} - 53 q^{88} + 9 q^{89} - 12 q^{91} - 63 q^{92} + 14 q^{94} - 24 q^{97} - 24 q^{98}+O(q^{100}) \) 8 * q + q^2 + 9 * q^4 - 12 * q^7 + 3 * q^8 - 12 * q^11 - 14 * q^13 - 16 * q^14 + 15 * q^16 - q^17 + 16 * q^19 - 18 * q^22 - 4 * q^23 + 34 * q^26 + 21 * q^28 - 2 * q^29 + 13 * q^31 - 18 * q^32 - 37 * q^34 + 8 * q^37 - 24 * q^38 + 12 * q^41 - 20 * q^43 - 47 * q^44 + 33 * q^46 - 15 * q^47 + 30 * q^49 + q^52 - 4 * q^53 - 60 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 + 4 * q^62 + 41 * q^64 - 19 * q^67 - 33 * q^68 - 21 * q^71 + 19 * q^73 + 9 * q^74 - q^76 - 11 * q^77 + 10 * q^79 - 24 * q^82 - 27 * q^83 - 42 * q^86 - 53 * q^88 + 9 * q^89 - 12 * q^91 - 63 * q^92 + 14 * q^94 - 24 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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