LMFDB - Embedded newform 3825.2.a.bq.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3825,2,Mod(1,3825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3825.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3825 = 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,0,11,0,0,1,9,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.5427787731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.18219$ of defining polynomial
Character	$\chi$	$=$	3825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60242 q^{2} +4.77260 q^{4} -3.53650 q^{7} -7.21549 q^{8} -2.94609 q^{11} -4.01064 q^{13} +9.20348 q^{14} +9.23255 q^{16} -1.00000 q^{17} -6.97745 q^{19} +7.66698 q^{22} -6.12692 q^{23} +10.4374 q^{26} -16.8783 q^{28} -5.30040 q^{29} +6.49485 q^{31} -9.59601 q^{32} +2.60242 q^{34} -3.43224 q^{37} +18.1583 q^{38} -4.61307 q^{41} -10.2901 q^{43} -14.0605 q^{44} +15.9448 q^{46} +3.67705 q^{47} +5.50686 q^{49} -19.1412 q^{52} -6.77260 q^{53} +25.5176 q^{56} +13.7939 q^{58} -9.92573 q^{59} -2.36438 q^{61} -16.9024 q^{62} +6.50778 q^{64} +9.56650 q^{67} -4.77260 q^{68} -5.51248 q^{71} -2.00515 q^{73} +8.93214 q^{74} -33.3006 q^{76} +10.4189 q^{77} +10.5803 q^{79} +12.0052 q^{82} +9.07301 q^{83} +26.7792 q^{86} +21.2575 q^{88} -2.63321 q^{89} +14.1837 q^{91} -29.2414 q^{92} -9.56923 q^{94} -5.86816 q^{97} -14.3312 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 11 q^{4} + q^{7} + 9 q^{8} - 4 q^{11} - 3 q^{13} + 7 q^{14} + 27 q^{16} - 5 q^{17} + 6 q^{19} + 18 q^{22} - 4 q^{23} + 5 q^{26} - 15 q^{28} - 2 q^{29} + 21 q^{31} + 9 q^{32} - q^{34} - 2 q^{37}+ \cdots - 38 q^{98}+O(q^{100}) $ 5 * q + q^2 + 11 * q^4 + q^7 + 9 * q^8 - 4 * q^11 - 3 * q^13 + 7 * q^14 + 27 * q^16 - 5 * q^17 + 6 * q^19 + 18 * q^22 - 4 * q^23 + 5 * q^26 - 15 * q^28 - 2 * q^29 + 21 * q^31 + 9 * q^32 - q^34 - 2 * q^37 + 34 * q^38 + 8 * q^41 - 4 * q^43 + 4 * q^44 - 6 * q^46 + 2 * q^47 + 10 * q^49 - 13 * q^52 - 21 * q^53 + 29 * q^56 + 22 * q^58 - 12 * q^59 - 2 * q^61 - 3 * q^62 + 43 * q^64 - 12 * q^67 - 11 * q^68 - 21 * q^71 + 22 * q^73 + 50 * q^74 - 14 * q^76 + 32 * q^77 + 41 * q^79 + 28 * q^82 + 8 * q^83 + 36 * q^86 + 14 * q^88 + 10 * q^89 + 13 * q^91 - 56 * q^92 - 10 * q^94 - 20 * q^97 - 38 * q^98

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 $

$2$	−2.60242	−1.84019	−0.920095	−	0.391694i	$-0.871889\pi$
$2$	−2.60242	−1.84019	−0.920095	+	0.391694i	$0.871889\pi$
$3$	0	0
$3$	0	0
$4$	4.77260	2.38630
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.53650	−1.33667	−0.668337	−	0.743859i	$-0.732993\pi$
$7$	−3.53650	−1.33667	−0.668337	+	0.743859i	$0.732993\pi$
$8$	−7.21549	−2.55106
$9$	0	0
$10$	0	0
$11$	−2.94609	−0.888280	−0.444140	−	0.895957i	$-0.646491\pi$
$11$	−2.94609	−0.888280	−0.444140	+	0.895957i	$0.646491\pi$
$12$	0	0
$13$	−4.01064	−1.11235	−0.556176	−	0.831064i	$-0.687732\pi$
$13$	−4.01064	−1.11235	−0.556176	+	0.831064i	$0.687732\pi$
$14$	9.20348	2.45973
$15$	0	0
$16$	9.23255	2.30814
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.97745	−1.60074	−0.800368	−	0.599508i	$-0.795363\pi$
$19$	−6.97745	−1.60074	−0.800368	+	0.599508i	$0.795363\pi$
$20$	0	0
$21$	0	0
$22$	7.66698	1.63460
$23$	−6.12692	−1.27755	−0.638775	−	0.769393i	$-0.720559\pi$
$23$	−6.12692	−1.27755	−0.638775	+	0.769393i	$0.720559\pi$
$24$	0	0
$25$	0	0
$26$	10.4374	2.04694
$27$	0	0
$28$	−16.8783	−3.18971
$29$	−5.30040	−0.984260	−0.492130	−	0.870522i	$-0.663782\pi$
$29$	−5.30040	−0.984260	−0.492130	+	0.870522i	$0.663782\pi$
$30$	0	0
$31$	6.49485	1.16651	0.583255	−	0.812289i	$-0.301779\pi$
$31$	6.49485	1.16651	0.583255	+	0.812289i	$0.301779\pi$
$32$	−9.59601	−1.69635
$33$	0	0
$34$	2.60242	0.446312
$35$	0	0
$36$	0	0
$37$	−3.43224	−0.564257	−0.282128	−	0.959377i	$-0.591040\pi$
$37$	−3.43224	−0.564257	−0.282128	+	0.959377i	$0.591040\pi$
$38$	18.1583	2.94566
$39$	0	0
$40$	0	0
$41$	−4.61307	−0.720440	−0.360220	−	0.932867i	$-0.617298\pi$
$41$	−4.61307	−0.720440	−0.360220	+	0.932867i	$0.617298\pi$
$42$	0	0
$43$	−10.2901	−1.56923	−0.784614	−	0.619985i	$-0.787139\pi$
$43$	−10.2901	−1.56923	−0.784614	+	0.619985i	$0.787139\pi$
$44$	−14.0605	−2.11970
$45$	0	0
$46$	15.9448	2.35094
$47$	3.67705	0.536352	0.268176	−	0.963370i	$-0.413579\pi$
$47$	3.67705	0.536352	0.268176	+	0.963370i	$0.413579\pi$
$48$	0	0
$49$	5.50686	0.786695
$50$	0	0
$51$	0	0
$52$	−19.1412	−2.65441
$53$	−6.77260	−0.930289	−0.465144	−	0.885235i	$-0.653998\pi$
$53$	−6.77260	−0.930289	−0.465144	+	0.885235i	$0.653998\pi$
$54$	0	0
$55$	0	0
$56$	25.5176	3.40993
$57$	0	0
$58$	13.7939	1.81123
$59$	−9.92573	−1.29222	−0.646110	−	0.763244i	$-0.723605\pi$
$59$	−9.92573	−1.29222	−0.646110	+	0.763244i	$0.723605\pi$
$60$	0	0
$61$	−2.36438	−0.302728	−0.151364	−	0.988478i	$-0.548367\pi$
$61$	−2.36438	−0.302728	−0.151364	+	0.988478i	$0.548367\pi$
$62$	−16.9024	−2.14660
$63$	0	0
$64$	6.50778	0.813473
$65$	0	0
$66$	0	0
$67$	9.56650	1.16873	0.584367	−	0.811490i	$-0.301343\pi$
$67$	9.56650	1.16873	0.584367	+	0.811490i	$0.301343\pi$
$68$	−4.77260	−0.578763
$69$	0	0
$70$	0	0
$71$	−5.51248	−0.654212	−0.327106	−	0.944988i	$-0.606073\pi$
$71$	−5.51248	−0.654212	−0.327106	+	0.944988i	$0.606073\pi$
$72$	0	0
$73$	−2.00515	−0.234685	−0.117343	−	0.993091i	$-0.537438\pi$
$73$	−2.00515	−0.234685	−0.117343	+	0.993091i	$0.537438\pi$
$74$	8.93214	1.03834
$75$	0	0
$76$	−33.3006	−3.81984
$77$	10.4189	1.18734
$78$	0	0
$79$	10.5803	1.19038	0.595191	−	0.803584i	$-0.297076\pi$
$79$	10.5803	1.19038	0.595191	+	0.803584i	$0.297076\pi$
$80$	0	0
$81$	0	0
$82$	12.0052	1.32575
$83$	9.07301	0.995892	0.497946	−	0.867208i	$-0.334088\pi$
$83$	9.07301	0.995892	0.497946	+	0.867208i	$0.334088\pi$
$84$	0	0
$85$	0	0
$86$	26.7792	2.88768
$87$	0	0
$88$	21.2575	2.26606
$89$	−2.63321	−0.279119	−0.139560	−	0.990214i	$-0.544569\pi$
$89$	−2.63321	−0.279119	−0.139560	+	0.990214i	$0.544569\pi$
$90$	0	0
$91$	14.1837	1.48685
$92$	−29.2414	−3.04862
$93$	0	0
$94$	−9.56923	−0.986991
$95$	0	0
$96$	0	0
$97$	−5.86816	−0.595822	−0.297911	−	0.954594i	$-0.596290\pi$
$97$	−5.86816	−0.595822	−0.297911	+	0.954594i	$0.596290\pi$
$98$	−14.3312	−1.44767
$99$	0	0
$100$	0	0
$101$	7.90283	0.786361	0.393180	−	0.919461i	$-0.371375\pi$
$101$	7.90283	0.786361	0.393180	+	0.919461i	$0.371375\pi$
$102$	0	0
$103$	6.36826	0.627483	0.313742	−	0.949508i	$-0.398417\pi$
$103$	6.36826	0.627483	0.313742	+	0.949508i	$0.398417\pi$
$104$	28.9388	2.83768
$105$	0	0
$106$	17.6252	1.71191
$107$	6.85432	0.662632	0.331316	−	0.943520i	$-0.392507\pi$
$107$	6.85432	0.662632	0.331316	+	0.943520i	$0.392507\pi$
$108$	0	0
$109$	−14.6758	−1.40569	−0.702843	−	0.711345i	$-0.748086\pi$
$109$	−14.6758	−1.40569	−0.702843	+	0.711345i	$0.748086\pi$
$110$	0	0
$111$	0	0
$112$	−32.6509	−3.08522
$113$	13.3994	1.26051	0.630255	−	0.776388i	$-0.282950\pi$
$113$	13.3994	1.26051	0.630255	+	0.776388i	$0.282950\pi$
$114$	0	0
$115$	0	0
$116$	−25.2967	−2.34874
$117$	0	0
$118$	25.8309	2.37793
$119$	3.53650	0.324191
$120$	0	0
$121$	−2.32055	−0.210959
$122$	6.15313	0.557078
$123$	0	0
$124$	30.9974	2.78365
$125$	0	0
$126$	0	0
$127$	−4.63321	−0.411131	−0.205565	−	0.978643i	$-0.565903\pi$
$127$	−4.63321	−0.411131	−0.205565	+	0.978643i	$0.565903\pi$
$128$	2.25602	0.199405
$129$	0	0
$130$	0	0
$131$	12.1496	1.06151	0.530757	−	0.847524i	$-0.321908\pi$
$131$	12.1496	1.06151	0.530757	+	0.847524i	$0.321908\pi$
$132$	0	0
$133$	24.6758	2.13966
$134$	−24.8961	−2.15069
$135$	0	0
$136$	7.21549	0.618723
$137$	−8.86852	−0.757689	−0.378844	−	0.925460i	$-0.623678\pi$
$137$	−8.86852	−0.757689	−0.378844	+	0.925460i	$0.623678\pi$
$138$	0	0
$139$	7.32306	0.621134	0.310567	−	0.950552i	$-0.399481\pi$
$139$	7.32306	0.621134	0.310567	+	0.950552i	$0.399481\pi$
$140$	0	0
$141$	0	0
$142$	14.3458	1.20387
$143$	11.8157	0.988081
$144$	0	0
$145$	0	0
$146$	5.21825	0.431866
$147$	0	0
$148$	−16.3807	−1.34649
$149$	−13.9059	−1.13922	−0.569608	−	0.821916i	$-0.692905\pi$
$149$	−13.9059	−1.13922	−0.569608	+	0.821916i	$0.692905\pi$
$150$	0	0
$151$	14.3884	1.17091	0.585456	−	0.810704i	$-0.300916\pi$
$151$	14.3884	1.17091	0.585456	+	0.810704i	$0.300916\pi$
$152$	50.3457	4.08358
$153$	0	0
$154$	−27.1143	−2.18493
$155$	0	0
$156$	0	0
$157$	8.68608	0.693224	0.346612	−	0.938009i	$-0.387332\pi$
$157$	8.68608	0.693224	0.346612	+	0.938009i	$0.387332\pi$
$158$	−27.5345	−2.19053
$159$	0	0
$160$	0	0
$161$	21.6679	1.70767
$162$	0	0
$163$	−8.95868	−0.701698	−0.350849	−	0.936432i	$-0.614107\pi$
$163$	−8.95868	−0.701698	−0.350849	+	0.936432i	$0.614107\pi$
$164$	−22.0163	−1.71919
$165$	0	0
$166$	−23.6118	−1.83263
$167$	−4.37318	−0.338407	−0.169203	−	0.985581i	$-0.554119\pi$
$167$	−4.37318	−0.338407	−0.169203	+	0.985581i	$0.554119\pi$
$168$	0	0
$169$	3.08527	0.237328
$170$	0	0
$171$	0	0
$172$	−49.1106	−3.74465
$173$	−8.82433	−0.670901	−0.335451	−	0.942058i	$-0.608889\pi$
$173$	−8.82433	−0.670901	−0.335451	+	0.942058i	$0.608889\pi$
$174$	0	0
$175$	0	0
$176$	−27.1999	−2.05027
$177$	0	0
$178$	6.85272	0.513633
$179$	−9.42951	−0.704795	−0.352397	−	0.935850i	$-0.614633\pi$
$179$	−9.42951	−0.704795	−0.352397	+	0.935850i	$0.614633\pi$
$180$	0	0
$181$	−11.7939	−0.876633	−0.438317	−	0.898821i	$-0.644425\pi$
$181$	−11.7939	−0.876633	−0.438317	+	0.898821i	$0.644425\pi$
$182$	−36.9119	−2.73609
$183$	0	0
$184$	44.2087	3.25911
$185$	0	0
$186$	0	0
$187$	2.94609	0.215440
$188$	17.5491	1.27990
$189$	0	0
$190$	0	0
$191$	−5.19969	−0.376237	−0.188118	−	0.982146i	$-0.560239\pi$
$191$	−5.19969	−0.376237	−0.188118	+	0.982146i	$0.560239\pi$
$192$	0	0
$193$	−14.3936	−1.03607	−0.518035	−	0.855359i	$-0.673336\pi$
$193$	−14.3936	−1.03607	−0.518035	+	0.855359i	$0.673336\pi$
$194$	15.2714	1.09643
$195$	0	0
$196$	26.2821	1.87729
$197$	−16.0840	−1.14594	−0.572969	−	0.819577i	$-0.694208\pi$
$197$	−16.0840	−1.14594	−0.572969	+	0.819577i	$0.694208\pi$
$198$	0	0
$199$	18.1750	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$199$	18.1750	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$200$	0	0
$201$	0	0
$202$	−20.5665	−1.44705
$203$	18.7449	1.31563
$204$	0	0
$205$	0	0
$206$	−16.5729	−1.15469
$207$	0	0
$208$	−37.0285	−2.56746
$209$	20.5562	1.42190
$210$	0	0
$211$	−8.40614	−0.578702	−0.289351	−	0.957223i	$-0.593440\pi$
$211$	−8.40614	−0.578702	−0.289351	+	0.957223i	$0.593440\pi$
$212$	−32.3230	−2.21995
$213$	0	0
$214$	−17.8378	−1.21937
$215$	0	0
$216$	0	0
$217$	−22.9691	−1.55924
$218$	38.1926	2.58673
$219$	0	0
$220$	0	0
$221$	4.01064	0.269785
$222$	0	0
$223$	2.90591	0.194594	0.0972971	−	0.995255i	$-0.468980\pi$
$223$	2.90591	0.194594	0.0972971	+	0.995255i	$0.468980\pi$
$224$	33.9363	2.26747
$225$	0	0
$226$	−34.8709	−2.31958
$227$	−15.8127	−1.04952	−0.524762	−	0.851249i	$-0.675846\pi$
$227$	−15.8127	−1.04952	−0.524762	+	0.851249i	$0.675846\pi$
$228$	0	0
$229$	−23.1302	−1.52849	−0.764244	−	0.644927i	$-0.776888\pi$
$229$	−23.1302	−1.52849	−0.764244	+	0.644927i	$0.776888\pi$
$230$	0	0
$231$	0	0
$232$	38.2450	2.51091
$233$	−14.5265	−0.951665	−0.475833	−	0.879536i	$-0.657853\pi$
$233$	−14.5265	−0.951665	−0.475833	+	0.879536i	$0.657853\pi$
$234$	0	0
$235$	0	0
$236$	−47.3716	−3.08363
$237$	0	0
$238$	−9.20348	−0.596573
$239$	−3.56923	−0.230874	−0.115437	−	0.993315i	$-0.536827\pi$
$239$	−3.56923	−0.230874	−0.115437	+	0.993315i	$0.536827\pi$
$240$	0	0
$241$	−17.7990	−1.14654	−0.573269	−	0.819367i	$-0.694325\pi$
$241$	−17.7990	−1.14654	−0.573269	+	0.819367i	$0.694325\pi$
$242$	6.03904	0.388204
$243$	0	0
$244$	−11.2843	−0.722401
$245$	0	0
$246$	0	0
$247$	27.9841	1.78058
$248$	−46.8636	−2.97584
$249$	0	0
$250$	0	0
$251$	−7.45480	−0.470543	−0.235271	−	0.971930i	$-0.575598\pi$
$251$	−7.45480	−0.470543	−0.235271	+	0.971930i	$0.575598\pi$
$252$	0	0
$253$	18.0505	1.13482
$254$	12.0576	0.756559
$255$	0	0
$256$	−18.8867	−1.18042
$257$	26.4740	1.65140	0.825702	−	0.564106i	$-0.190779\pi$
$257$	26.4740	1.65140	0.825702	+	0.564106i	$0.190779\pi$
$258$	0	0
$259$	12.1381	0.754227
$260$	0	0
$261$	0	0
$262$	−31.6183	−1.95339
$263$	12.4974	0.770621	0.385310	−	0.922787i	$-0.374094\pi$
$263$	12.4974	0.770621	0.385310	+	0.922787i	$0.374094\pi$
$264$	0	0
$265$	0	0
$266$	−64.2168	−3.93739
$267$	0	0
$268$	45.6571	2.78895
$269$	−2.83773	−0.173020	−0.0865098	−	0.996251i	$-0.527571\pi$
$269$	−2.83773	−0.173020	−0.0865098	+	0.996251i	$0.527571\pi$
$270$	0	0
$271$	−7.60005	−0.461670	−0.230835	−	0.972993i	$-0.574146\pi$
$271$	−7.60005	−0.461670	−0.230835	+	0.972993i	$0.574146\pi$
$272$	−9.23255	−0.559805
$273$	0	0
$274$	23.0796	1.39429
$275$	0	0
$276$	0	0
$277$	30.4187	1.82768	0.913842	−	0.406070i	$-0.133101\pi$
$277$	30.4187	1.82768	0.913842	+	0.406070i	$0.133101\pi$
$278$	−19.0577	−1.14300
$279$	0	0
$280$	0	0
$281$	20.5944	1.22856	0.614279	−	0.789089i	$-0.289447\pi$
$281$	20.5944	1.22856	0.614279	+	0.789089i	$0.289447\pi$
$282$	0	0
$283$	4.14433	0.246355	0.123177	−	0.992385i	$-0.460692\pi$
$283$	4.14433	0.246355	0.123177	+	0.992385i	$0.460692\pi$
$284$	−26.3089	−1.56115
$285$	0	0
$286$	−30.7495	−1.81826
$287$	16.3141	0.962993
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	−9.56980	−0.560030
$293$	−7.85031	−0.458620	−0.229310	−	0.973353i	$-0.573647\pi$
$293$	−7.85031	−0.458620	−0.229310	+	0.973353i	$0.573647\pi$
$294$	0	0
$295$	0	0
$296$	24.7653	1.43945
$297$	0	0
$298$	36.1891	2.09638
$299$	24.5729	1.42109
$300$	0	0
$301$	36.3910	2.09754
$302$	−37.4447	−2.15470
$303$	0	0
$304$	−64.4196	−3.69472
$305$	0	0
$306$	0	0
$307$	−0.473348	−0.0270154	−0.0135077	−	0.999909i	$-0.504300\pi$
$307$	−0.473348	−0.0270154	−0.0135077	+	0.999909i	$0.504300\pi$
$308$	49.7251	2.83335
$309$	0	0
$310$	0	0
$311$	−18.3062	−1.03805	−0.519023	−	0.854760i	$-0.673704\pi$
$311$	−18.3062	−1.03805	−0.519023	+	0.854760i	$0.673704\pi$
$312$	0	0
$313$	3.84193	0.217159	0.108579	−	0.994088i	$-0.465370\pi$
$313$	3.84193	0.217159	0.108579	+	0.994088i	$0.465370\pi$
$314$	−22.6048	−1.27567
$315$	0	0
$316$	50.4958	2.84061
$317$	4.52266	0.254018	0.127009	−	0.991902i	$-0.459462\pi$
$317$	4.52266	0.254018	0.127009	+	0.991902i	$0.459462\pi$
$318$	0	0
$319$	15.6155	0.874299
$320$	0	0
$321$	0	0
$322$	−56.3890	−3.14243
$323$	6.97745	0.388236
$324$	0	0
$325$	0	0
$326$	23.3143	1.29126
$327$	0	0
$328$	33.2855	1.83789
$329$	−13.0039	−0.716928
$330$	0	0
$331$	17.4347	0.958296	0.479148	−	0.877734i	$-0.340946\pi$
$331$	17.4347	0.958296	0.479148	+	0.877734i	$0.340946\pi$
$332$	43.3019	2.37650
$333$	0	0
$334$	11.3809	0.622733
$335$	0	0
$336$	0	0
$337$	−0.943903	−0.0514177	−0.0257088	−	0.999669i	$-0.508184\pi$
$337$	−0.943903	−0.0514177	−0.0257088	+	0.999669i	$0.508184\pi$
$338$	−8.02917	−0.436729
$339$	0	0
$340$	0	0
$341$	−19.1344	−1.03619
$342$	0	0
$343$	5.28048	0.285119
$344$	74.2482	4.00320
$345$	0	0
$346$	22.9646	1.23459
$347$	−19.2108	−1.03129	−0.515645	−	0.856802i	$-0.672448\pi$
$347$	−19.2108	−1.03129	−0.515645	+	0.856802i	$0.672448\pi$
$348$	0	0
$349$	−31.7831	−1.70131	−0.850655	−	0.525724i	$-0.823795\pi$
$349$	−31.7831	−1.70131	−0.850655	+	0.525724i	$0.823795\pi$
$350$	0	0
$351$	0	0
$352$	28.2707	1.50683
$353$	15.4511	0.822380	0.411190	−	0.911550i	$-0.365113\pi$
$353$	15.4511	0.822380	0.411190	+	0.911550i	$0.365113\pi$
$354$	0	0
$355$	0	0
$356$	−12.5673	−0.666064
$357$	0	0
$358$	24.5396	1.29696
$359$	27.9639	1.47588	0.737940	−	0.674866i	$-0.235799\pi$
$359$	27.9639	1.47588	0.737940	+	0.674866i	$0.235799\pi$
$360$	0	0
$361$	29.6848	1.56236
$362$	30.6927	1.61317
$363$	0	0
$364$	67.6930	3.54808
$365$	0	0
$366$	0	0
$367$	22.7225	1.18610	0.593051	−	0.805165i	$-0.297923\pi$
$367$	22.7225	1.18610	0.593051	+	0.805165i	$0.297923\pi$
$368$	−56.5671	−2.94876
$369$	0	0
$370$	0	0
$371$	23.9513	1.24349
$372$	0	0
$373$	−35.5230	−1.83931	−0.919656	−	0.392725i	$-0.871532\pi$
$373$	−35.5230	−1.83931	−0.919656	+	0.392725i	$0.871532\pi$
$374$	−7.66698	−0.396450
$375$	0	0
$376$	−26.5317	−1.36827
$377$	21.2580	1.09484
$378$	0	0
$379$	−30.4727	−1.56528	−0.782640	−	0.622475i	$-0.786127\pi$
$379$	−30.4727	−1.56528	−0.782640	+	0.622475i	$0.786127\pi$
$380$	0	0
$381$	0	0
$382$	13.5318	0.692347
$383$	25.9667	1.32683	0.663417	−	0.748250i	$-0.269105\pi$
$383$	25.9667	1.32683	0.663417	+	0.748250i	$0.269105\pi$
$384$	0	0
$385$	0	0
$386$	37.4581	1.90657
$387$	0	0
$388$	−28.0064	−1.42181
$389$	−6.37729	−0.323342	−0.161671	−	0.986845i	$-0.551688\pi$
$389$	−6.37729	−0.323342	−0.161671	+	0.986845i	$0.551688\pi$
$390$	0	0
$391$	6.12692	0.309852
$392$	−39.7347	−2.00691
$393$	0	0
$394$	41.8574	2.10874
$395$	0	0
$396$	0	0
$397$	−13.1874	−0.661858	−0.330929	−	0.943656i	$-0.607362\pi$
$397$	−13.1874	−0.661858	−0.330929	+	0.943656i	$0.607362\pi$
$398$	−47.2989	−2.37088
$399$	0	0
$400$	0	0
$401$	28.2411	1.41030	0.705148	−	0.709061i	$-0.250881\pi$
$401$	28.2411	1.41030	0.705148	+	0.709061i	$0.250881\pi$
$402$	0	0
$403$	−26.0486	−1.29757
$404$	37.7171	1.87649
$405$	0	0
$406$	−48.7822	−2.42102
$407$	10.1117	0.501218
$408$	0	0
$409$	−21.0374	−1.04023	−0.520117	−	0.854095i	$-0.674112\pi$
$409$	−21.0374	−1.04023	−0.520117	+	0.854095i	$0.674112\pi$
$410$	0	0
$411$	0	0
$412$	30.3932	1.49737
$413$	35.1024	1.72728
$414$	0	0
$415$	0	0
$416$	38.4862	1.88694
$417$	0	0
$418$	−53.4959	−2.61657
$419$	28.1482	1.37513	0.687565	−	0.726123i	$-0.258680\pi$
$419$	28.1482	1.37513	0.687565	+	0.726123i	$0.258680\pi$
$420$	0	0
$421$	−16.6639	−0.812147	−0.406074	−	0.913840i	$-0.633102\pi$
$421$	−16.6639	−0.812147	−0.406074	+	0.913840i	$0.633102\pi$
$422$	21.8763	1.06492
$423$	0	0
$424$	48.8677	2.37322
$425$	0	0
$426$	0	0
$427$	8.36165	0.404649
$428$	32.7130	1.58124
$429$	0	0
$430$	0	0
$431$	11.1833	0.538682	0.269341	−	0.963045i	$-0.413194\pi$
$431$	11.1833	0.538682	0.269341	+	0.963045i	$0.413194\pi$
$432$	0	0
$433$	11.0440	0.530743	0.265372	−	0.964146i	$-0.414505\pi$
$433$	11.0440	0.530743	0.265372	+	0.964146i	$0.414505\pi$
$434$	59.7753	2.86930
$435$	0	0
$436$	−70.0417	−3.35439
$437$	42.7503	2.04502
$438$	0	0
$439$	−5.34654	−0.255176	−0.127588	−	0.991827i	$-0.540724\pi$
$439$	−5.34654	−0.255176	−0.127588	+	0.991827i	$0.540724\pi$
$440$	0	0
$441$	0	0
$442$	−10.4374	−0.496456
$443$	−16.5863	−0.788040	−0.394020	−	0.919102i	$-0.628916\pi$
$443$	−16.5863	−0.788040	−0.394020	+	0.919102i	$0.628916\pi$
$444$	0	0
$445$	0	0
$446$	−7.56241	−0.358091
$447$	0	0
$448$	−23.0148	−1.08735
$449$	−35.2901	−1.66544	−0.832722	−	0.553691i	$-0.813219\pi$
$449$	−35.2901	−1.66544	−0.832722	+	0.553691i	$0.813219\pi$
$450$	0	0
$451$	13.5905	0.639952
$452$	63.9500	3.00796
$453$	0	0
$454$	41.1513	1.93132
$455$	0	0
$456$	0	0
$457$	9.73794	0.455522	0.227761	−	0.973717i	$-0.426860\pi$
$457$	9.73794	0.455522	0.227761	+	0.973717i	$0.426860\pi$
$458$	60.1947	2.81271
$459$	0	0
$460$	0	0
$461$	16.4097	0.764276	0.382138	−	0.924105i	$-0.375188\pi$
$461$	16.4097	0.764276	0.382138	+	0.924105i	$0.375188\pi$
$462$	0	0
$463$	16.9720	0.788755	0.394378	−	0.918948i	$-0.370960\pi$
$463$	16.9720	0.788755	0.394378	+	0.918948i	$0.370960\pi$
$464$	−48.9362	−2.27181
$465$	0	0
$466$	37.8042	1.75125
$467$	23.2884	1.07766	0.538828	−	0.842416i	$-0.318867\pi$
$467$	23.2884	1.07766	0.538828	+	0.842416i	$0.318867\pi$
$468$	0	0
$469$	−33.8320	−1.56221
$470$	0	0
$471$	0	0
$472$	71.6190	3.29653
$473$	30.3156	1.39391
$474$	0	0
$475$	0	0
$476$	16.8783	0.773617
$477$	0	0
$478$	9.28864	0.424853
$479$	−2.30406	−0.105275	−0.0526376	−	0.998614i	$-0.516763\pi$
$479$	−2.30406	−0.105275	−0.0526376	+	0.998614i	$0.516763\pi$
$480$	0	0
$481$	13.7655	0.627653
$482$	46.3206	2.10985
$483$	0	0
$484$	−11.0750	−0.503411
$485$	0	0
$486$	0	0
$487$	−14.0889	−0.638430	−0.319215	−	0.947682i	$-0.603419\pi$
$487$	−14.0889	−0.638430	−0.319215	+	0.947682i	$0.603419\pi$
$488$	17.0602	0.772278
$489$	0	0
$490$	0	0
$491$	−21.0485	−0.949905	−0.474953	−	0.880011i	$-0.657535\pi$
$491$	−21.0485	−0.949905	−0.474953	+	0.880011i	$0.657535\pi$
$492$	0	0
$493$	5.30040	0.238718
$494$	−72.8264	−3.27661
$495$	0	0
$496$	59.9640	2.69247
$497$	19.4949	0.874467
$498$	0	0
$499$	27.6747	1.23889	0.619446	−	0.785039i	$-0.287357\pi$
$499$	27.6747	1.23889	0.619446	+	0.785039i	$0.287357\pi$
$500$	0	0
$501$	0	0
$502$	19.4005	0.865889
$503$	−31.0855	−1.38603	−0.693017	−	0.720921i	$-0.743719\pi$
$503$	−31.0855	−1.38603	−0.693017	+	0.720921i	$0.743719\pi$
$504$	0	0
$505$	0	0
$506$	−46.9749	−2.08829
$507$	0	0
$508$	−22.1125	−0.981082
$509$	20.5481	0.910781	0.455390	−	0.890292i	$-0.349500\pi$
$509$	20.5481	0.910781	0.455390	+	0.890292i	$0.349500\pi$
$510$	0	0
$511$	7.09123	0.313697
$512$	44.6391	1.97279
$513$	0	0
$514$	−68.8966	−3.03890
$515$	0	0
$516$	0	0
$517$	−10.8329	−0.476431
$518$	−31.5886	−1.38792
$519$	0	0
$520$	0	0
$521$	27.9505	1.22453	0.612267	−	0.790651i	$-0.290258\pi$
$521$	27.9505	1.22453	0.612267	+	0.790651i	$0.290258\pi$
$522$	0	0
$523$	−0.0826499	−0.00361403	−0.00180701	−	0.999998i	$-0.500575\pi$
$523$	−0.0826499	−0.00361403	−0.00180701	+	0.999998i	$0.500575\pi$
$524$	57.9851	2.53309
$525$	0	0
$526$	−32.5234	−1.41809
$527$	−6.49485	−0.282920
$528$	0	0
$529$	14.5391	0.632136
$530$	0	0
$531$	0	0
$532$	117.768	5.10588
$533$	18.5014	0.801383
$534$	0	0
$535$	0	0
$536$	−69.0270	−2.98151
$537$	0	0
$538$	7.38498	0.318389
$539$	−16.2237	−0.698805
$540$	0	0
$541$	−10.7378	−0.461654	−0.230827	−	0.972995i	$-0.574143\pi$
$541$	−10.7378	−0.461654	−0.230827	+	0.972995i	$0.574143\pi$
$542$	19.7785	0.849561
$543$	0	0
$544$	9.59601	0.411426
$545$	0	0
$546$	0	0
$547$	36.8538	1.57576	0.787878	−	0.615832i	$-0.211180\pi$
$547$	36.8538	1.57576	0.787878	+	0.615832i	$0.211180\pi$
$548$	−42.3259	−1.80807
$549$	0	0
$550$	0	0
$551$	36.9833	1.57554
$552$	0	0
$553$	−37.4174	−1.59115
$554$	−79.1624	−3.36329
$555$	0	0
$556$	34.9501	1.48221
$557$	2.41625	0.102380	0.0511899	−	0.998689i	$-0.483699\pi$
$557$	2.41625	0.102380	0.0511899	+	0.998689i	$0.483699\pi$
$558$	0	0
$559$	41.2700	1.74553
$560$	0	0
$561$	0	0
$562$	−53.5953	−2.26078
$563$	4.74857	0.200129	0.100064	−	0.994981i	$-0.468095\pi$
$563$	4.74857	0.200129	0.100064	+	0.994981i	$0.468095\pi$
$564$	0	0
$565$	0	0
$566$	−10.7853	−0.453340
$567$	0	0
$568$	39.7753	1.66893
$569$	−46.0041	−1.92859	−0.964296	−	0.264827i	$-0.914685\pi$
$569$	−46.0041	−1.92859	−0.964296	+	0.264827i	$0.914685\pi$
$570$	0	0
$571$	9.95720	0.416696	0.208348	−	0.978055i	$-0.433191\pi$
$571$	9.95720	0.416696	0.208348	+	0.978055i	$0.433191\pi$
$572$	56.3918	2.35786
$573$	0	0
$574$	−42.4563	−1.77209
$575$	0	0
$576$	0	0
$577$	−18.8669	−0.785439	−0.392720	−	0.919658i	$-0.628466\pi$
$577$	−18.8669	−0.785439	−0.392720	+	0.919658i	$0.628466\pi$
$578$	−2.60242	−0.108247
$579$	0	0
$580$	0	0
$581$	−32.0867	−1.33118
$582$	0	0
$583$	19.9527	0.826357
$584$	14.4682	0.598696
$585$	0	0
$586$	20.4298	0.843949
$587$	−23.8851	−0.985843	−0.492921	−	0.870074i	$-0.664071\pi$
$587$	−23.8851	−0.985843	−0.492921	+	0.870074i	$0.664071\pi$
$588$	0	0
$589$	−45.3175	−1.86728
$590$	0	0
$591$	0	0
$592$	−31.6883	−1.30238
$593$	−37.9019	−1.55644	−0.778222	−	0.627989i	$-0.783878\pi$
$593$	−37.9019	−1.55644	−0.778222	+	0.627989i	$0.783878\pi$
$594$	0	0
$595$	0	0
$596$	−66.3674	−2.71852
$597$	0	0
$598$	−63.9490	−2.61507
$599$	−17.8049	−0.727488	−0.363744	−	0.931499i	$-0.618502\pi$
$599$	−17.8049	−0.727488	−0.363744	+	0.931499i	$0.618502\pi$
$600$	0	0
$601$	33.6532	1.37274	0.686372	−	0.727251i	$-0.259202\pi$
$601$	33.6532	1.37274	0.686372	+	0.727251i	$0.259202\pi$
$602$	−94.7049	−3.85988
$603$	0	0
$604$	68.6702	2.79415
$605$	0	0
$606$	0	0
$607$	−5.89699	−0.239351	−0.119676	−	0.992813i	$-0.538185\pi$
$607$	−5.89699	−0.239351	−0.119676	+	0.992813i	$0.538185\pi$
$608$	66.9557	2.71541
$609$	0	0
$610$	0	0
$611$	−14.7473	−0.596613
$612$	0	0
$613$	−7.04143	−0.284401	−0.142200	−	0.989838i	$-0.545418\pi$
$613$	−7.04143	−0.284401	−0.142200	+	0.989838i	$0.545418\pi$
$614$	1.23185	0.0497135
$615$	0	0
$616$	−75.1772	−3.02898
$617$	6.04823	0.243492	0.121746	−	0.992561i	$-0.461151\pi$
$617$	6.04823	0.243492	0.121746	+	0.992561i	$0.461151\pi$
$618$	0	0
$619$	−34.0992	−1.37056	−0.685282	−	0.728278i	$-0.740321\pi$
$619$	−34.0992	−1.37056	−0.685282	+	0.728278i	$0.740321\pi$
$620$	0	0
$621$	0	0
$622$	47.6404	1.91020
$623$	9.31235	0.373092
$624$	0	0
$625$	0	0
$626$	−9.99833	−0.399614
$627$	0	0
$628$	41.4552	1.65424
$629$	3.43224	0.136852
$630$	0	0
$631$	18.9841	0.755743	0.377872	−	0.925858i	$-0.376656\pi$
$631$	18.9841	0.755743	0.377872	+	0.925858i	$0.376656\pi$
$632$	−76.3424	−3.03674
$633$	0	0
$634$	−11.7699	−0.467441
$635$	0	0
$636$	0	0
$637$	−22.0861	−0.875082
$638$	−40.6381	−1.60888
$639$	0	0
$640$	0	0
$641$	16.3869	0.647245	0.323623	−	0.946186i	$-0.395099\pi$
$641$	16.3869	0.647245	0.323623	+	0.946186i	$0.395099\pi$
$642$	0	0
$643$	30.8332	1.21594	0.607972	−	0.793958i	$-0.291983\pi$
$643$	30.8332	1.21594	0.607972	+	0.793958i	$0.291983\pi$
$644$	103.412	4.07501
$645$	0	0
$646$	−18.1583	−0.714428
$647$	15.8642	0.623684	0.311842	−	0.950134i	$-0.399054\pi$
$647$	15.8642	0.623684	0.311842	+	0.950134i	$0.399054\pi$
$648$	0	0
$649$	29.2421	1.14785
$650$	0	0
$651$	0	0
$652$	−42.7562	−1.67446
$653$	7.51669	0.294151	0.147075	−	0.989125i	$-0.453014\pi$
$653$	7.51669	0.294151	0.147075	+	0.989125i	$0.453014\pi$
$654$	0	0
$655$	0	0
$656$	−42.5904	−1.66287
$657$	0	0
$658$	33.8416	1.31928
$659$	3.89783	0.151838	0.0759190	−	0.997114i	$-0.475811\pi$
$659$	3.89783	0.151838	0.0759190	+	0.997114i	$0.475811\pi$
$660$	0	0
$661$	−16.6761	−0.648625	−0.324313	−	0.945950i	$-0.605133\pi$
$661$	−16.6761	−0.648625	−0.324313	+	0.945950i	$0.605133\pi$
$662$	−45.3724	−1.76345
$663$	0	0
$664$	−65.4662	−2.54058
$665$	0	0
$666$	0	0
$667$	32.4751	1.25744
$668$	−20.8715	−0.807541
$669$	0	0
$670$	0	0
$671$	6.96569	0.268907
$672$	0	0
$673$	−39.2465	−1.51284	−0.756420	−	0.654086i	$-0.773054\pi$
$673$	−39.2465	−1.51284	−0.756420	+	0.654086i	$0.773054\pi$
$674$	2.45644	0.0946184
$675$	0	0
$676$	14.7248	0.566337
$677$	15.3340	0.589332	0.294666	−	0.955600i	$-0.404792\pi$
$677$	15.3340	0.589332	0.294666	+	0.955600i	$0.404792\pi$
$678$	0	0
$679$	20.7528	0.796419
$680$	0	0
$681$	0	0
$682$	49.7959	1.90678
$683$	−15.0386	−0.575435	−0.287717	−	0.957715i	$-0.592896\pi$
$683$	−15.0386	−0.575435	−0.287717	+	0.957715i	$0.592896\pi$
$684$	0	0
$685$	0	0
$686$	−13.7420	−0.524674
$687$	0	0
$688$	−95.0040	−3.62199
$689$	27.1625	1.03481
$690$	0	0
$691$	0.735679	0.0279866	0.0139933	−	0.999902i	$-0.495546\pi$
$691$	0.735679	0.0279866	0.0139933	+	0.999902i	$0.495546\pi$
$692$	−42.1150	−1.60097
$693$	0	0
$694$	49.9947	1.89777
$695$	0	0
$696$	0	0
$697$	4.61307	0.174732
$698$	82.7131	3.13074
$699$	0	0
$700$	0	0
$701$	16.8115	0.634962	0.317481	−	0.948265i	$-0.397163\pi$
$701$	16.8115	0.634962	0.317481	+	0.948265i	$0.397163\pi$
$702$	0	0
$703$	23.9483	0.903227
$704$	−19.1725	−0.722592
$705$	0	0
$706$	−40.2104	−1.51334
$707$	−27.9484	−1.05111
$708$	0	0
$709$	15.0110	0.563750	0.281875	−	0.959451i	$-0.409044\pi$
$709$	15.0110	0.563750	0.281875	+	0.959451i	$0.409044\pi$
$710$	0	0
$711$	0	0
$712$	18.9999	0.712051
$713$	−39.7934	−1.49028
$714$	0	0
$715$	0	0
$716$	−45.0033	−1.68185
$717$	0	0
$718$	−72.7740	−2.71590
$719$	−37.4098	−1.39515	−0.697575	−	0.716512i	$-0.745738\pi$
$719$	−37.4098	−1.39515	−0.697575	+	0.716512i	$0.745738\pi$
$720$	0	0
$721$	−22.5214	−0.838740
$722$	−77.2524	−2.87504
$723$	0	0
$724$	−56.2876	−2.09191
$725$	0	0
$726$	0	0
$727$	6.76798	0.251011	0.125505	−	0.992093i	$-0.459945\pi$
$727$	6.76798	0.251011	0.125505	+	0.992093i	$0.459945\pi$
$728$	−102.342	−3.79305
$729$	0	0
$730$	0	0
$731$	10.2901	0.380594
$732$	0	0
$733$	18.3230	0.676774	0.338387	−	0.941007i	$-0.390119\pi$
$733$	18.3230	0.676774	0.338387	+	0.941007i	$0.390119\pi$
$734$	−59.1334	−2.18266
$735$	0	0
$736$	58.7940	2.16717
$737$	−28.1838	−1.03816
$738$	0	0
$739$	−0.240801	−0.00885802	−0.00442901	−	0.999990i	$-0.501410\pi$
$739$	−0.240801	−0.00885802	−0.00442901	+	0.999990i	$0.501410\pi$
$740$	0	0
$741$	0	0
$742$	−62.3315	−2.28826
$743$	−38.0128	−1.39455	−0.697277	−	0.716801i	$-0.745605\pi$
$743$	−38.0128	−1.39455	−0.697277	+	0.716801i	$0.745605\pi$
$744$	0	0
$745$	0	0
$746$	92.4459	3.38468
$747$	0	0
$748$	14.0605	0.514104
$749$	−24.2403	−0.885722
$750$	0	0
$751$	51.5263	1.88022	0.940111	−	0.340868i	$-0.110721\pi$
$751$	51.5263	1.88022	0.940111	+	0.340868i	$0.110721\pi$
$752$	33.9485	1.23797
$753$	0	0
$754$	−55.3224	−2.01472
$755$	0	0
$756$	0	0
$757$	0.984558	0.0357844	0.0178922	−	0.999840i	$-0.494304\pi$
$757$	0.984558	0.0357844	0.0178922	+	0.999840i	$0.494304\pi$
$758$	79.3029	2.88041
$759$	0	0
$760$	0	0
$761$	25.8638	0.937563	0.468781	−	0.883314i	$-0.344693\pi$
$761$	25.8638	0.937563	0.468781	+	0.883314i	$0.344693\pi$
$762$	0	0
$763$	51.9010	1.87894
$764$	−24.8161	−0.897814
$765$	0	0
$766$	−67.5762	−2.44163
$767$	39.8086	1.43740
$768$	0	0
$769$	47.2099	1.70243	0.851216	−	0.524816i	$-0.175866\pi$
$769$	47.2099	1.70243	0.851216	+	0.524816i	$0.175866\pi$
$770$	0	0
$771$	0	0
$772$	−68.6947	−2.47238
$773$	−1.18874	−0.0427560	−0.0213780	−	0.999771i	$-0.506805\pi$
$773$	−1.18874	−0.0427560	−0.0213780	+	0.999771i	$0.506805\pi$
$774$	0	0
$775$	0	0
$776$	42.3417	1.51998
$777$	0	0
$778$	16.5964	0.595010
$779$	32.1874	1.15323
$780$	0	0
$781$	16.2403	0.581123
$782$	−15.9448	−0.570186
$783$	0	0
$784$	50.8424	1.81580
$785$	0	0
$786$	0	0
$787$	−38.6087	−1.37625	−0.688125	−	0.725592i	$-0.741566\pi$
$787$	−38.6087	−1.37625	−0.688125	+	0.725592i	$0.741566\pi$
$788$	−76.7626	−2.73455
$789$	0	0
$790$	0	0
$791$	−47.3870	−1.68489
$792$	0	0
$793$	9.48270	0.336741
$794$	34.3193	1.21795
$795$	0	0
$796$	86.7419	3.07448
$797$	7.77134	0.275275	0.137638	−	0.990483i	$-0.456049\pi$
$797$	7.77134	0.275275	0.137638	+	0.990483i	$0.456049\pi$
$798$	0	0
$799$	−3.67705	−0.130085
$800$	0	0
$801$	0	0
$802$	−73.4954	−2.59521
$803$	5.90736	0.208466
$804$	0	0
$805$	0	0
$806$	67.7893	2.38778
$807$	0	0
$808$	−57.0228	−2.00605
$809$	52.6740	1.85192	0.925960	−	0.377621i	$-0.123258\pi$
$809$	52.6740	1.85192	0.925960	+	0.377621i	$0.123258\pi$
$810$	0	0
$811$	−38.0502	−1.33612	−0.668062	−	0.744105i	$-0.732876\pi$
$811$	−38.0502	−1.33612	−0.668062	+	0.744105i	$0.732876\pi$
$812$	89.4620	3.13950
$813$	0	0
$814$	−26.3149	−0.922337
$815$	0	0
$816$	0	0
$817$	71.7988	2.51192
$818$	54.7483	1.91423
$819$	0	0
$820$	0	0
$821$	−22.8093	−0.796051	−0.398026	−	0.917374i	$-0.630305\pi$
$821$	−22.8093	−0.796051	−0.398026	+	0.917374i	$0.630305\pi$
$822$	0	0
$823$	−26.6089	−0.927530	−0.463765	−	0.885958i	$-0.653502\pi$
$823$	−26.6089	−0.927530	−0.463765	+	0.885958i	$0.653502\pi$
$824$	−45.9501	−1.60075
$825$	0	0
$826$	−91.3513	−3.17852
$827$	−0.707585	−0.0246052	−0.0123026	−	0.999924i	$-0.503916\pi$
$827$	−0.707585	−0.0246052	−0.0123026	+	0.999924i	$0.503916\pi$
$828$	0	0
$829$	39.7559	1.38078	0.690390	−	0.723437i	$-0.257439\pi$
$829$	39.7559	1.38078	0.690390	+	0.723437i	$0.257439\pi$
$830$	0	0
$831$	0	0
$832$	−26.1004	−0.904869
$833$	−5.50686	−0.190802
$834$	0	0
$835$	0	0
$836$	98.1067	3.39309
$837$	0	0
$838$	−73.2535	−2.53050
$839$	−15.4254	−0.532542	−0.266271	−	0.963898i	$-0.585792\pi$
$839$	−15.4254	−0.532542	−0.266271	+	0.963898i	$0.585792\pi$
$840$	0	0
$841$	−0.905714	−0.0312315
$842$	43.3664	1.49451
$843$	0	0
$844$	−40.1192	−1.38096
$845$	0	0
$846$	0	0
$847$	8.20662	0.281983
$848$	−62.5284	−2.14723
$849$	0	0
$850$	0	0
$851$	21.0291	0.720867
$852$	0	0
$853$	50.0605	1.71404	0.857020	−	0.515284i	$-0.172314\pi$
$853$	50.0605	1.71404	0.857020	+	0.515284i	$0.172314\pi$
$854$	−21.7606	−0.744631
$855$	0	0
$856$	−49.4573	−1.69041
$857$	−46.0441	−1.57284	−0.786418	−	0.617695i	$-0.788067\pi$
$857$	−46.0441	−1.57284	−0.786418	+	0.617695i	$0.788067\pi$
$858$	0	0
$859$	−37.1872	−1.26881	−0.634406	−	0.773000i	$-0.718755\pi$
$859$	−37.1872	−1.26881	−0.634406	+	0.773000i	$0.718755\pi$
$860$	0	0
$861$	0	0
$862$	−29.1038	−0.991279
$863$	48.2016	1.64080	0.820401	−	0.571788i	$-0.193750\pi$
$863$	48.2016	1.64080	0.820401	+	0.571788i	$0.193750\pi$
$864$	0	0
$865$	0	0
$866$	−28.7413	−0.976668
$867$	0	0
$868$	−109.622	−3.72083
$869$	−31.1707	−1.05739
$870$	0	0
$871$	−38.3678	−1.30004
$872$	105.893	3.58599
$873$	0	0
$874$	−111.254	−3.76323
$875$	0	0
$876$	0	0
$877$	−7.97840	−0.269411	−0.134706	−	0.990886i	$-0.543009\pi$
$877$	−7.97840	−0.269411	−0.134706	+	0.990886i	$0.543009\pi$
$878$	13.9139	0.469573
$879$	0	0
$880$	0	0
$881$	−48.5755	−1.63655	−0.818276	−	0.574826i	$-0.805070\pi$
$881$	−48.5755	−1.63655	−0.818276	+	0.574826i	$0.805070\pi$
$882$	0	0
$883$	−42.6792	−1.43627	−0.718134	−	0.695905i	$-0.755004\pi$
$883$	−42.6792	−1.43627	−0.718134	+	0.695905i	$0.755004\pi$
$884$	19.1412	0.643789
$885$	0	0
$886$	43.1646	1.45014
$887$	−53.1721	−1.78535	−0.892673	−	0.450706i	$-0.851172\pi$
$887$	−53.1721	−1.78535	−0.892673	+	0.450706i	$0.851172\pi$
$888$	0	0
$889$	16.3854	0.549547
$890$	0	0
$891$	0	0
$892$	13.8688	0.464361
$893$	−25.6564	−0.858559
$894$	0	0
$895$	0	0
$896$	−7.97841	−0.266540
$897$	0	0
$898$	91.8398	3.06473
$899$	−34.4254	−1.14815
$900$	0	0
$901$	6.77260	0.225628
$902$	−35.3683	−1.17763
$903$	0	0
$904$	−96.6832	−3.21564
$905$	0	0
$906$	0	0
$907$	−42.0949	−1.39774	−0.698868	−	0.715250i	$-0.746313\pi$
$907$	−42.0949	−1.39774	−0.698868	+	0.715250i	$0.746313\pi$
$908$	−75.4676	−2.50448
$909$	0	0
$910$	0	0
$911$	32.3227	1.07090	0.535450	−	0.844567i	$-0.320142\pi$
$911$	32.3227	1.07090	0.535450	+	0.844567i	$0.320142\pi$
$912$	0	0
$913$	−26.7299	−0.884631
$914$	−25.3422	−0.838247
$915$	0	0
$916$	−110.391	−3.64744
$917$	−42.9670	−1.41890
$918$	0	0
$919$	20.8713	0.688480	0.344240	−	0.938882i	$-0.388137\pi$
$919$	20.8713	0.688480	0.344240	+	0.938882i	$0.388137\pi$
$920$	0	0
$921$	0	0
$922$	−42.7050	−1.40641
$923$	22.1086	0.727714
$924$	0	0
$925$	0	0
$926$	−44.1683	−1.45146
$927$	0	0
$928$	50.8627	1.66965
$929$	−41.3293	−1.35597	−0.677985	−	0.735076i	$-0.737146\pi$
$929$	−41.3293	−1.35597	−0.677985	+	0.735076i	$0.737146\pi$
$930$	0	0
$931$	−38.4239	−1.25929
$932$	−69.3294	−2.27096
$933$	0	0
$934$	−60.6061	−1.98309
$935$	0	0
$936$	0	0
$937$	24.3395	0.795138	0.397569	−	0.917572i	$-0.369854\pi$
$937$	24.3395	0.795138	0.397569	+	0.917572i	$0.369854\pi$
$938$	88.0451	2.87477
$939$	0	0
$940$	0	0
$941$	33.4007	1.08883	0.544415	−	0.838816i	$-0.316752\pi$
$941$	33.4007	1.08883	0.544415	+	0.838816i	$0.316752\pi$
$942$	0	0
$943$	28.2639	0.920399
$944$	−91.6398	−2.98262
$945$	0	0
$946$	−78.8940	−2.56507
$947$	−14.7087	−0.477969	−0.238985	−	0.971023i	$-0.576815\pi$
$947$	−14.7087	−0.477969	−0.238985	+	0.971023i	$0.576815\pi$
$948$	0	0
$949$	8.04195	0.261053
$950$	0	0
$951$	0	0
$952$	−25.5176	−0.827031
$953$	−20.9652	−0.679128	−0.339564	−	0.940583i	$-0.610280\pi$
$953$	−20.9652	−0.679128	−0.339564	+	0.940583i	$0.610280\pi$
$954$	0	0
$955$	0	0
$956$	−17.0345	−0.550936
$957$	0	0
$958$	5.99614	0.193727
$959$	31.3635	1.01278
$960$	0	0
$961$	11.1831	0.360746
$962$	−35.8236	−1.15500
$963$	0	0
$964$	−84.9478	−2.73598
$965$	0	0
$966$	0	0
$967$	−31.1916	−1.00306	−0.501528	−	0.865141i	$-0.667229\pi$
$967$	−31.1916	−1.00306	−0.501528	+	0.865141i	$0.667229\pi$
$968$	16.7439	0.538168
$969$	0	0
$970$	0	0
$971$	−3.55989	−0.114242	−0.0571211	−	0.998367i	$-0.518192\pi$
$971$	−3.55989	−0.114242	−0.0571211	+	0.998367i	$0.518192\pi$
$972$	0	0
$973$	−25.8980	−0.830253
$974$	36.6653	1.17483
$975$	0	0
$976$	−21.8293	−0.698738
$977$	−51.0404	−1.63293	−0.816463	−	0.577397i	$-0.804068\pi$
$977$	−51.0404	−1.63293	−0.816463	+	0.577397i	$0.804068\pi$
$978$	0	0
$979$	7.75767	0.247936
$980$	0	0
$981$	0	0
$982$	54.7771	1.74801
$983$	−43.4903	−1.38712	−0.693562	−	0.720397i	$-0.743960\pi$
$983$	−43.4903	−1.38712	−0.693562	+	0.720397i	$0.743960\pi$
$984$	0	0
$985$	0	0
$986$	−13.7939	−0.439287
$987$	0	0
$988$	133.557	4.24901
$989$	63.0467	2.00477
$990$	0	0
$991$	−20.5455	−0.652650	−0.326325	−	0.945258i	$-0.605810\pi$
$991$	−20.5455	−0.652650	−0.326325	+	0.945258i	$0.605810\pi$
$992$	−62.3247	−1.97881
$993$	0	0
$994$	−50.7340	−1.60919
$995$	0	0
$996$	0	0
$997$	48.6107	1.53952	0.769758	−	0.638336i	$-0.220377\pi$
$997$	48.6107	1.53952	0.769758	+	0.638336i	$0.220377\pi$
$998$	−72.0214	−2.27980
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.bq.1.1		5
3.2	odd	2		425.2.a.i.1.5	✓	5
5.4	even	2		3825.2.a.bl.1.5		5
12.11	even	2		6800.2.a.bz.1.3		5
15.2	even	4		425.2.b.f.324.9		10
15.8	even	4		425.2.b.f.324.2		10
15.14	odd	2		425.2.a.j.1.1	yes	5
51.50	odd	2		7225.2.a.x.1.5		5
60.59	even	2		6800.2.a.cd.1.3		5
255.254	odd	2		7225.2.a.y.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.5	✓	5	3.2	odd	2
425.2.a.j.1.1	yes	5	15.14	odd	2
425.2.b.f.324.2		10	15.8	even	4
425.2.b.f.324.9		10	15.2	even	4
3825.2.a.bl.1.5		5	5.4	even	2
3825.2.a.bq.1.1		5	1.1	even	1	trivial
6800.2.a.bz.1.3		5	12.11	even	2
6800.2.a.cd.1.3		5	60.59	even	2
7225.2.a.x.1.5		5	51.50	odd	2
7225.2.a.y.1.1		5	255.254	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3825.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60242 q^{2} +4.77260 q^{4} -3.53650 q^{7} -7.21549 q^{8} -2.94609 q^{11} -4.01064 q^{13} +9.20348 q^{14} +9.23255 q^{16} -1.00000 q^{17} -6.97745 q^{19} +7.66698 q^{22} -6.12692 q^{23} +10.4374 q^{26} -16.8783 q^{28} -5.30040 q^{29} +6.49485 q^{31} -9.59601 q^{32} +2.60242 q^{34} -3.43224 q^{37} +18.1583 q^{38} -4.61307 q^{41} -10.2901 q^{43} -14.0605 q^{44} +15.9448 q^{46} +3.67705 q^{47} +5.50686 q^{49} -19.1412 q^{52} -6.77260 q^{53} +25.5176 q^{56} +13.7939 q^{58} -9.92573 q^{59} -2.36438 q^{61} -16.9024 q^{62} +6.50778 q^{64} +9.56650 q^{67} -4.77260 q^{68} -5.51248 q^{71} -2.00515 q^{73} +8.93214 q^{74} -33.3006 q^{76} +10.4189 q^{77} +10.5803 q^{79} +12.0052 q^{82} +9.07301 q^{83} +26.7792 q^{86} +21.2575 q^{88} -2.63321 q^{89} +14.1837 q^{91} -29.2414 q^{92} -9.56923 q^{94} -5.86816 q^{97} -14.3312 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 11 q^{4} + q^{7} + 9 q^{8} - 4 q^{11} - 3 q^{13} + 7 q^{14} + 27 q^{16} - 5 q^{17} + 6 q^{19} + 18 q^{22} - 4 q^{23} + 5 q^{26} - 15 q^{28} - 2 q^{29} + 21 q^{31} + 9 q^{32} - q^{34} - 2 q^{37}+ \cdots - 38 q^{98}+O(q^{100}) \) 5 * q + q^2 + 11 * q^4 + q^7 + 9 * q^8 - 4 * q^11 - 3 * q^13 + 7 * q^14 + 27 * q^16 - 5 * q^17 + 6 * q^19 + 18 * q^22 - 4 * q^23 + 5 * q^26 - 15 * q^28 - 2 * q^29 + 21 * q^31 + 9 * q^32 - q^34 - 2 * q^37 + 34 * q^38 + 8 * q^41 - 4 * q^43 + 4 * q^44 - 6 * q^46 + 2 * q^47 + 10 * q^49 - 13 * q^52 - 21 * q^53 + 29 * q^56 + 22 * q^58 - 12 * q^59 - 2 * q^61 - 3 * q^62 + 43 * q^64 - 12 * q^67 - 11 * q^68 - 21 * q^71 + 22 * q^73 + 50 * q^74 - 14 * q^76 + 32 * q^77 + 41 * q^79 + 28 * q^82 + 8 * q^83 + 36 * q^86 + 14 * q^88 + 10 * q^89 + 13 * q^91 - 56 * q^92 - 10 * q^94 - 20 * q^97 - 38 * q^98

Introduction

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