LMFDB - Embedded newform 3825.2.a.bq.1.4

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			1.4
Root			$-2.48887$ 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)

$f(q)$	$=$	$q+2.19447 q^{2} +2.81568 q^{4} +3.05725 q^{7} +1.78998 q^{8} +5.36180 q^{11} -4.59895 q^{13} +6.70903 q^{14} -1.70331 q^{16} -1.00000 q^{17} +4.57325 q^{19} +11.7663 q^{22} -1.24730 q^{23} -10.0922 q^{26} +8.60824 q^{28} +5.93018 q^{29} +9.84580 q^{31} -7.31781 q^{32} -2.19447 q^{34} +4.20461 q^{37} +10.0358 q^{38} -0.404485 q^{41} -5.76142 q^{43} +15.0971 q^{44} -2.73715 q^{46} +3.35693 q^{47} +2.34678 q^{49} -12.9492 q^{52} -4.81568 q^{53} +5.47242 q^{56} +13.0136 q^{58} -12.7392 q^{59} +4.97774 q^{61} +21.6063 q^{62} -12.6521 q^{64} +6.82926 q^{67} -2.81568 q^{68} -11.9408 q^{71} +10.8876 q^{73} +9.22687 q^{74} +12.8768 q^{76} +16.3924 q^{77} +16.7139 q^{79} -0.887628 q^{82} -4.11450 q^{83} -12.6432 q^{86} +9.59752 q^{88} +10.4142 q^{89} -14.0601 q^{91} -3.51199 q^{92} +7.36667 q^{94} -2.27443 q^{97} +5.14994 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.19447	1.55172	0.775861	−	0.630904i	$-0.217316\pi$
$2$	2.19447	1.55172	0.775861	+	0.630904i	$0.217316\pi$
$3$	0	0
$3$	0	0
$4$	2.81568	1.40784
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.05725	1.15553	0.577766	−	0.816202i	$-0.303925\pi$
$7$	3.05725	1.15553	0.577766	+	0.816202i	$0.303925\pi$
$8$	1.78998	0.632854
$9$	0	0
$10$	0	0
$11$	5.36180	1.61664	0.808322	−	0.588741i	$-0.200376\pi$
$11$	5.36180	1.61664	0.808322	+	0.588741i	$0.200376\pi$
$12$	0	0
$13$	−4.59895	−1.27552	−0.637760	−	0.770235i	$-0.720139\pi$
$13$	−4.59895	−1.27552	−0.637760	+	0.770235i	$0.720139\pi$
$14$	6.70903	1.79306
$15$	0	0
$16$	−1.70331	−0.425827
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.57325	1.04918	0.524588	−	0.851356i	$-0.324219\pi$
$19$	4.57325	1.04918	0.524588	+	0.851356i	$0.324219\pi$
$20$	0	0
$21$	0	0
$22$	11.7663	2.50858
$23$	−1.24730	−0.260079	−0.130040	−	0.991509i	$-0.541510\pi$
$23$	−1.24730	−0.260079	−0.130040	+	0.991509i	$0.541510\pi$
$24$	0	0
$25$	0	0
$26$	−10.0922	−1.97925
$27$	0	0
$28$	8.60824	1.62680
$29$	5.93018	1.10121	0.550604	−	0.834767i	$-0.314398\pi$
$29$	5.93018	1.10121	0.550604	+	0.834767i	$0.314398\pi$
$30$	0	0
$31$	9.84580	1.76836	0.884179	−	0.467149i	$-0.154719\pi$
$31$	9.84580	1.76836	0.884179	+	0.467149i	$0.154719\pi$
$32$	−7.31781	−1.29362
$33$	0	0
$34$	−2.19447	−0.376348
$35$	0	0
$36$	0	0
$37$	4.20461	0.691234	0.345617	−	0.938376i	$-0.387670\pi$
$37$	4.20461	0.691234	0.345617	+	0.938376i	$0.387670\pi$
$38$	10.0358	1.62803
$39$	0	0
$40$	0	0
$41$	−0.404485	−0.0631699	−0.0315850	−	0.999501i	$-0.510055\pi$
$41$	−0.404485	−0.0631699	−0.0315850	+	0.999501i	$0.510055\pi$
$42$	0	0
$43$	−5.76142	−0.878608	−0.439304	−	0.898339i	$-0.644775\pi$
$43$	−5.76142	−0.878608	−0.439304	+	0.898339i	$0.644775\pi$
$44$	15.0971	2.27597
$45$	0	0
$46$	−2.73715	−0.403571
$47$	3.35693	0.489659	0.244829	−	0.969566i	$-0.421268\pi$
$47$	3.35693	0.489659	0.244829	+	0.969566i	$0.421268\pi$
$48$	0	0
$49$	2.34678	0.335255
$50$	0	0
$51$	0	0
$52$	−12.9492	−1.79573
$53$	−4.81568	−0.661484	−0.330742	−	0.943721i	$-0.607299\pi$
$53$	−4.81568	−0.661484	−0.330742	+	0.943721i	$0.607299\pi$
$54$	0	0
$55$	0	0
$56$	5.47242	0.731283
$57$	0	0
$58$	13.0136	1.70877
$59$	−12.7392	−1.65850	−0.829248	−	0.558881i	$-0.811231\pi$
$59$	−12.7392	−1.65850	−0.829248	+	0.558881i	$0.811231\pi$
$60$	0	0
$61$	4.97774	0.637334	0.318667	−	0.947867i	$-0.396765\pi$
$61$	4.97774	0.637334	0.318667	+	0.947867i	$0.396765\pi$
$62$	21.6063	2.74400
$63$	0	0
$64$	−12.6521	−1.58151
$65$	0	0
$66$	0	0
$67$	6.82926	0.834327	0.417163	−	0.908831i	$-0.363024\pi$
$67$	6.82926	0.834327	0.417163	+	0.908831i	$0.363024\pi$
$68$	−2.81568	−0.341451
$69$	0	0
$70$	0	0
$71$	−11.9408	−1.41711	−0.708555	−	0.705656i	$-0.750652\pi$
$71$	−11.9408	−1.41711	−0.708555	+	0.705656i	$0.750652\pi$
$72$	0	0
$73$	10.8876	1.27430	0.637150	−	0.770740i	$-0.280113\pi$
$73$	10.8876	1.27430	0.637150	+	0.770740i	$0.280113\pi$
$74$	9.22687	1.07260
$75$	0	0
$76$	12.8768	1.47707
$77$	16.3924	1.86808
$78$	0	0
$79$	16.7139	1.88046	0.940230	−	0.340539i	$-0.110609\pi$
$79$	16.7139	1.88046	0.940230	+	0.340539i	$0.110609\pi$
$80$	0	0
$81$	0	0
$82$	−0.887628	−0.0980221
$83$	−4.11450	−0.451625	−0.225813	−	0.974171i	$-0.572504\pi$
$83$	−4.11450	−0.451625	−0.225813	+	0.974171i	$0.572504\pi$
$84$	0	0
$85$	0	0
$86$	−12.6432	−1.36335
$87$	0	0
$88$	9.59752	1.02310
$89$	10.4142	1.10391	0.551953	−	0.833875i	$-0.313883\pi$
$89$	10.4142	1.10391	0.551953	+	0.833875i	$0.313883\pi$
$90$	0	0
$91$	−14.0601	−1.47390
$92$	−3.51199	−0.366150
$93$	0	0
$94$	7.36667	0.759814
$95$	0	0
$96$	0	0
$97$	−2.27443	−0.230933	−0.115467	−	0.993311i	$-0.536836\pi$
$97$	−2.27443	−0.230933	−0.115467	+	0.993311i	$0.536836\pi$
$98$	5.14994	0.520222
$99$	0	0
$100$	0	0
$101$	−8.12465	−0.808433	−0.404216	−	0.914663i	$-0.632456\pi$
$101$	−8.12465	−0.808433	−0.404216	+	0.914663i	$0.632456\pi$
$102$	0	0
$103$	−5.15706	−0.508140	−0.254070	−	0.967186i	$-0.581769\pi$
$103$	−5.15706	−0.508140	−0.254070	+	0.967186i	$0.581769\pi$
$104$	−8.23203	−0.807217
$105$	0	0
$106$	−10.5678	−1.02644
$107$	−5.61021	−0.542360	−0.271180	−	0.962529i	$-0.587414\pi$
$107$	−5.61021	−0.542360	−0.271180	+	0.962529i	$0.587414\pi$
$108$	0	0
$109$	−3.98158	−0.381366	−0.190683	−	0.981652i	$-0.561070\pi$
$109$	−3.98158	−0.381366	−0.190683	+	0.981652i	$0.561070\pi$
$110$	0	0
$111$	0	0
$112$	−5.20744	−0.492057
$113$	0.913734	0.0859569	0.0429785	−	0.999076i	$-0.486315\pi$
$113$	0.913734	0.0859569	0.0429785	+	0.999076i	$0.486315\pi$
$114$	0	0
$115$	0	0
$116$	16.6975	1.55032
$117$	0	0
$118$	−27.9556	−2.57352
$119$	−3.05725	−0.280258
$120$	0	0
$121$	17.7489	1.61354
$122$	10.9235	0.988965
$123$	0	0
$124$	27.7226	2.48956
$125$	0	0
$126$	0	0
$127$	8.41422	0.746642	0.373321	−	0.927702i	$-0.378219\pi$
$127$	8.41422	0.746642	0.373321	+	0.927702i	$0.378219\pi$
$128$	−13.1289	−1.16044
$129$	0	0
$130$	0	0
$131$	1.34723	0.117708	0.0588542	−	0.998267i	$-0.481255\pi$
$131$	1.34723	0.117708	0.0588542	+	0.998267i	$0.481255\pi$
$132$	0	0
$133$	13.9816	1.21236
$134$	14.9866	1.29464
$135$	0	0
$136$	−1.78998	−0.153490
$137$	−12.5650	−1.07350	−0.536749	−	0.843742i	$-0.680348\pi$
$137$	−12.5650	−1.07350	−0.536749	+	0.843742i	$0.680348\pi$
$138$	0	0
$139$	−9.83024	−0.833790	−0.416895	−	0.908955i	$-0.636882\pi$
$139$	−9.83024	−0.833790	−0.416895	+	0.908955i	$0.636882\pi$
$140$	0	0
$141$	0	0
$142$	−26.2036	−2.19896
$143$	−24.6586	−2.06206
$144$	0	0
$145$	0	0
$146$	23.8925	1.97736
$147$	0	0
$148$	11.8388	0.973146
$149$	9.03003	0.739769	0.369884	−	0.929078i	$-0.379397\pi$
$149$	9.03003	0.739769	0.369884	+	0.929078i	$0.379397\pi$
$150$	0	0
$151$	−5.97576	−0.486301	−0.243150	−	0.969989i	$-0.578181\pi$
$151$	−5.97576	−0.486301	−0.243150	+	0.969989i	$0.578181\pi$
$152$	8.18603	0.663975
$153$	0	0
$154$	35.9725	2.89875
$155$	0	0
$156$	0	0
$157$	−8.71002	−0.695135	−0.347568	−	0.937655i	$-0.612992\pi$
$157$	−8.71002	−0.695135	−0.347568	+	0.937655i	$0.612992\pi$
$158$	36.6781	2.91795
$159$	0	0
$160$	0	0
$161$	−3.81330	−0.300530
$162$	0	0
$163$	0.852508	0.0667736	0.0333868	−	0.999443i	$-0.489371\pi$
$163$	0.852508	0.0667736	0.0333868	+	0.999443i	$0.489371\pi$
$164$	−1.13890	−0.0889331
$165$	0	0
$166$	−9.02913	−0.700797
$167$	−1.32122	−0.102239	−0.0511195	−	0.998693i	$-0.516279\pi$
$167$	−1.32122	−0.102239	−0.0511195	+	0.998693i	$0.516279\pi$
$168$	0	0
$169$	8.15035	0.626950
$170$	0	0
$171$	0	0
$172$	−16.2223	−1.23694
$173$	7.49672	0.569965	0.284983	−	0.958533i	$-0.408012\pi$
$173$	7.49672	0.569965	0.284983	+	0.958533i	$0.408012\pi$
$174$	0	0
$175$	0	0
$176$	−9.13279	−0.688410
$177$	0	0
$178$	22.8537	1.71295
$179$	−15.9913	−1.19525	−0.597624	−	0.801777i	$-0.703888\pi$
$179$	−15.9913	−1.19525	−0.597624	+	0.801777i	$0.703888\pi$
$180$	0	0
$181$	−11.0136	−0.818633	−0.409317	−	0.912392i	$-0.634233\pi$
$181$	−11.0136	−0.818633	−0.409317	+	0.912392i	$0.634233\pi$
$182$	−30.8545	−2.28709
$183$	0	0
$184$	−2.23264	−0.164592
$185$	0	0
$186$	0	0
$187$	−5.36180	−0.392094
$188$	9.45204	0.689361
$189$	0	0
$190$	0	0
$191$	−8.49870	−0.614944	−0.307472	−	0.951557i	$-0.599483\pi$
$191$	−8.49870	−0.614944	−0.307472	+	0.951557i	$0.599483\pi$
$192$	0	0
$193$	18.8634	1.35782	0.678908	−	0.734223i	$-0.262453\pi$
$193$	18.8634	1.35782	0.678908	+	0.734223i	$0.262453\pi$
$194$	−4.99116	−0.358344
$195$	0	0
$196$	6.60779	0.471985
$197$	−10.7750	−0.767687	−0.383843	−	0.923398i	$-0.625400\pi$
$197$	−10.7750	−0.767687	−0.383843	+	0.923398i	$0.625400\pi$
$198$	0	0
$199$	−12.7488	−0.903735	−0.451868	−	0.892085i	$-0.649242\pi$
$199$	−12.7488	−0.903735	−0.451868	+	0.892085i	$0.649242\pi$
$200$	0	0
$201$	0	0
$202$	−17.8293	−1.25446
$203$	18.1301	1.27248
$204$	0	0
$205$	0	0
$206$	−11.3170	−0.788492
$207$	0	0
$208$	7.83343	0.543151
$209$	24.5209	1.69614
$210$	0	0
$211$	26.3541	1.81429	0.907146	−	0.420817i	$-0.138257\pi$
$211$	26.3541	1.81429	0.907146	+	0.420817i	$0.138257\pi$
$212$	−13.5594	−0.931264
$213$	0	0
$214$	−12.3114	−0.841591
$215$	0	0
$216$	0	0
$217$	30.1011	2.04339
$218$	−8.73744	−0.591774
$219$	0	0
$220$	0	0
$221$	4.59895	0.309359
$222$	0	0
$223$	−20.0300	−1.34131	−0.670655	−	0.741769i	$-0.733987\pi$
$223$	−20.0300	−1.34131	−0.670655	+	0.741769i	$0.733987\pi$
$224$	−22.3724	−1.49482
$225$	0	0
$226$	2.00516	0.133381
$227$	−13.2928	−0.882277	−0.441138	−	0.897439i	$-0.645425\pi$
$227$	−13.2928	−0.882277	−0.441138	+	0.897439i	$0.645425\pi$
$228$	0	0
$229$	10.0240	0.662404	0.331202	−	0.943560i	$-0.392546\pi$
$229$	10.0240	0.662404	0.331202	+	0.943560i	$0.392546\pi$
$230$	0	0
$231$	0	0
$232$	10.6149	0.696903
$233$	5.12121	0.335502	0.167751	−	0.985829i	$-0.446350\pi$
$233$	5.12121	0.335502	0.167751	+	0.985829i	$0.446350\pi$
$234$	0	0
$235$	0	0
$236$	−35.8694	−2.33490
$237$	0	0
$238$	−6.70903	−0.434882
$239$	13.3667	0.864618	0.432309	−	0.901726i	$-0.357699\pi$
$239$	13.3667	0.864618	0.432309	+	0.901726i	$0.357699\pi$
$240$	0	0
$241$	−4.12595	−0.265776	−0.132888	−	0.991131i	$-0.542425\pi$
$241$	−4.12595	−0.265776	−0.132888	+	0.991131i	$0.542425\pi$
$242$	38.9493	2.50376
$243$	0	0
$244$	14.0157	0.897264
$245$	0	0
$246$	0	0
$247$	−21.0322	−1.33824
$248$	17.6238	1.11911
$249$	0	0
$250$	0	0
$251$	7.71502	0.486968	0.243484	−	0.969905i	$-0.421710\pi$
$251$	7.71502	0.486968	0.243484	+	0.969905i	$0.421710\pi$
$252$	0	0
$253$	−6.68775	−0.420456
$254$	18.4647	1.15858
$255$	0	0
$256$	−3.50680	−0.219175
$257$	18.4651	1.15182	0.575912	−	0.817512i	$-0.304647\pi$
$257$	18.4651	1.15182	0.575912	+	0.817512i	$0.304647\pi$
$258$	0	0
$259$	12.8546	0.798743
$260$	0	0
$261$	0	0
$262$	2.95646	0.182651
$263$	18.7644	1.15707	0.578533	−	0.815659i	$-0.303626\pi$
$263$	18.7644	1.15707	0.578533	+	0.815659i	$0.303626\pi$
$264$	0	0
$265$	0	0
$266$	30.6821	1.88124
$267$	0	0
$268$	19.2290	1.17460
$269$	−14.7847	−0.901441	−0.450721	−	0.892665i	$-0.648833\pi$
$269$	−14.7847	−0.901441	−0.450721	+	0.892665i	$0.648833\pi$
$270$	0	0
$271$	−24.2920	−1.47563	−0.737817	−	0.675001i	$-0.764143\pi$
$271$	−24.2920	−1.47563	−0.737817	+	0.675001i	$0.764143\pi$
$272$	1.70331	0.103278
$273$	0	0
$274$	−27.5734	−1.66577
$275$	0	0
$276$	0	0
$277$	−5.84481	−0.351181	−0.175590	−	0.984463i	$-0.556183\pi$
$277$	−5.84481	−0.351181	−0.175590	+	0.984463i	$0.556183\pi$
$278$	−21.5721	−1.29381
$279$	0	0
$280$	0	0
$281$	0.651914	0.0388899	0.0194450	−	0.999811i	$-0.493810\pi$
$281$	0.651914	0.0388899	0.0194450	+	0.999811i	$0.493810\pi$
$282$	0	0
$283$	4.62452	0.274899	0.137450	−	0.990509i	$-0.456110\pi$
$283$	4.62452	0.274899	0.137450	+	0.990509i	$0.456110\pi$
$284$	−33.6214	−1.99506
$285$	0	0
$286$	−54.1126	−3.19974
$287$	−1.23661	−0.0729949
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	30.6561	1.79401
$293$	−3.46169	−0.202234	−0.101117	−	0.994875i	$-0.532242\pi$
$293$	−3.46169	−0.202234	−0.101117	+	0.994875i	$0.532242\pi$
$294$	0	0
$295$	0	0
$296$	7.52617	0.437450
$297$	0	0
$298$	19.8161	1.14792
$299$	5.73626	0.331736
$300$	0	0
$301$	−17.6141	−1.01526
$302$	−13.1136	−0.754603
$303$	0	0
$304$	−7.78966	−0.446767
$305$	0	0
$306$	0	0
$307$	−19.7625	−1.12790	−0.563952	−	0.825808i	$-0.690720\pi$
$307$	−19.7625	−1.12790	−0.563952	+	0.825808i	$0.690720\pi$
$308$	46.1557	2.62996
$309$	0	0
$310$	0	0
$311$	−26.2366	−1.48774	−0.743871	−	0.668324i	$-0.767012\pi$
$311$	−26.2366	−1.48774	−0.743871	+	0.668324i	$0.767012\pi$
$312$	0	0
$313$	−22.9825	−1.29905	−0.649523	−	0.760342i	$-0.725031\pi$
$313$	−22.9825	−1.29905	−0.649523	+	0.760342i	$0.725031\pi$
$314$	−19.1138	−1.07866
$315$	0	0
$316$	47.0610	2.64739
$317$	−10.9419	−0.614558	−0.307279	−	0.951619i	$-0.599418\pi$
$317$	−10.9419	−0.614558	−0.307279	+	0.951619i	$0.599418\pi$
$318$	0	0
$319$	31.7964	1.78026
$320$	0	0
$321$	0	0
$322$	−8.36815	−0.466339
$323$	−4.57325	−0.254463
$324$	0	0
$325$	0	0
$326$	1.87080	0.103614
$327$	0	0
$328$	−0.724020	−0.0399773
$329$	10.2630	0.565816
$330$	0	0
$331$	11.1037	0.610314	0.305157	−	0.952302i	$-0.401291\pi$
$331$	11.1037	0.610314	0.305157	+	0.952302i	$0.401291\pi$
$332$	−11.5851	−0.635816
$333$	0	0
$334$	−2.89937	−0.158646
$335$	0	0
$336$	0	0
$337$	30.0360	1.63617	0.818083	−	0.575101i	$-0.195037\pi$
$337$	30.0360	1.63617	0.818083	+	0.575101i	$0.195037\pi$
$338$	17.8857	0.972851
$339$	0	0
$340$	0	0
$341$	52.7912	2.85880
$342$	0	0
$343$	−14.2260	−0.768134
$344$	−10.3128	−0.556030
$345$	0	0
$346$	16.4513	0.884428
$347$	−26.4956	−1.42236	−0.711179	−	0.703011i	$-0.751839\pi$
$347$	−26.4956	−1.42236	−0.711179	+	0.703011i	$0.751839\pi$
$348$	0	0
$349$	11.8225	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$349$	11.8225	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$350$	0	0
$351$	0	0
$352$	−39.2366	−2.09132
$353$	−11.3987	−0.606690	−0.303345	−	0.952881i	$-0.598103\pi$
$353$	−11.3987	−0.606690	−0.303345	+	0.952881i	$0.598103\pi$
$354$	0	0
$355$	0	0
$356$	29.3231	1.55412
$357$	0	0
$358$	−35.0924	−1.85469
$359$	−12.2135	−0.644601	−0.322301	−	0.946637i	$-0.604456\pi$
$359$	−12.2135	−0.644601	−0.322301	+	0.946637i	$0.604456\pi$
$360$	0	0
$361$	1.91463	0.100770
$362$	−24.1689	−1.27029
$363$	0	0
$364$	−39.5889	−2.07502
$365$	0	0
$366$	0	0
$367$	−6.91651	−0.361039	−0.180519	−	0.983571i	$-0.557778\pi$
$367$	−6.91651	−0.361039	−0.180519	+	0.983571i	$0.557778\pi$
$368$	2.12453	0.110749
$369$	0	0
$370$	0	0
$371$	−14.7227	−0.764367
$372$	0	0
$373$	−27.3487	−1.41606	−0.708030	−	0.706183i	$-0.750416\pi$
$373$	−27.3487	−1.41606	−0.708030	+	0.706183i	$0.750416\pi$
$374$	−11.7663	−0.608420
$375$	0	0
$376$	6.00884	0.309882
$377$	−27.2726	−1.40461
$378$	0	0
$379$	14.9563	0.768255	0.384127	−	0.923280i	$-0.374502\pi$
$379$	14.9563	0.768255	0.384127	+	0.923280i	$0.374502\pi$
$380$	0	0
$381$	0	0
$382$	−18.6501	−0.954222
$383$	−28.4094	−1.45165	−0.725826	−	0.687879i	$-0.758542\pi$
$383$	−28.4094	−1.45165	−0.725826	+	0.687879i	$0.758542\pi$
$384$	0	0
$385$	0	0
$386$	41.3951	2.10695
$387$	0	0
$388$	−6.40406	−0.325117
$389$	22.2240	1.12680	0.563401	−	0.826184i	$-0.309493\pi$
$389$	22.2240	1.12680	0.563401	+	0.826184i	$0.309493\pi$
$390$	0	0
$391$	1.24730	0.0630785
$392$	4.20070	0.212167
$393$	0	0
$394$	−23.6454	−1.19124
$395$	0	0
$396$	0	0
$397$	1.76615	0.0886406	0.0443203	−	0.999017i	$-0.485888\pi$
$397$	1.76615	0.0886406	0.0443203	+	0.999017i	$0.485888\pi$
$398$	−27.9767	−1.40235
$399$	0	0
$400$	0	0
$401$	4.79423	0.239412	0.119706	−	0.992809i	$-0.461805\pi$
$401$	4.79423	0.239412	0.119706	+	0.992809i	$0.461805\pi$
$402$	0	0
$403$	−45.2803	−2.25557
$404$	−22.8764	−1.13814
$405$	0	0
$406$	39.7858	1.97454
$407$	22.5443	1.11748
$408$	0	0
$409$	−36.2834	−1.79410	−0.897050	−	0.441929i	$-0.854294\pi$
$409$	−36.2834	−1.79410	−0.897050	+	0.441929i	$0.854294\pi$
$410$	0	0
$411$	0	0
$412$	−14.5206	−0.715379
$413$	−38.9468	−1.91645
$414$	0	0
$415$	0	0
$416$	33.6543	1.65004
$417$	0	0
$418$	53.8102	2.63194
$419$	24.4452	1.19423	0.597113	−	0.802157i	$-0.296314\pi$
$419$	24.4452	1.19423	0.597113	+	0.802157i	$0.296314\pi$
$420$	0	0
$421$	−14.0909	−0.686750	−0.343375	−	0.939198i	$-0.611570\pi$
$421$	−14.0909	−0.686750	−0.343375	+	0.939198i	$0.611570\pi$
$422$	57.8332	2.81527
$423$	0	0
$424$	−8.61997	−0.418623
$425$	0	0
$426$	0	0
$427$	15.2182	0.736460
$428$	−15.7966	−0.763556
$429$	0	0
$430$	0	0
$431$	17.5277	0.844282	0.422141	−	0.906530i	$-0.361279\pi$
$431$	17.5277	0.844282	0.422141	+	0.906530i	$0.361279\pi$
$432$	0	0
$433$	−11.1755	−0.537059	−0.268530	−	0.963271i	$-0.586538\pi$
$433$	−11.1755	−0.537059	−0.268530	+	0.963271i	$0.586538\pi$
$434$	66.0558	3.17078
$435$	0	0
$436$	−11.2109	−0.536902
$437$	−5.70420	−0.272869
$438$	0	0
$439$	−12.0419	−0.574727	−0.287363	−	0.957822i	$-0.592779\pi$
$439$	−12.0419	−0.574727	−0.287363	+	0.957822i	$0.592779\pi$
$440$	0	0
$441$	0	0
$442$	10.0922	0.480039
$443$	−39.5984	−1.88138	−0.940689	−	0.339269i	$-0.889820\pi$
$443$	−39.5984	−1.88138	−0.940689	+	0.339269i	$0.889820\pi$
$444$	0	0
$445$	0	0
$446$	−43.9552	−2.08134
$447$	0	0
$448$	−38.6806	−1.82748
$449$	−11.6778	−0.551107	−0.275554	−	0.961286i	$-0.588861\pi$
$449$	−11.6778	−0.551107	−0.275554	+	0.961286i	$0.588861\pi$
$450$	0	0
$451$	−2.16877	−0.102123
$452$	2.57278	0.121014
$453$	0	0
$454$	−29.1707	−1.36905
$455$	0	0
$456$	0	0
$457$	20.2148	0.945606	0.472803	−	0.881168i	$-0.343242\pi$
$457$	20.2148	0.945606	0.472803	+	0.881168i	$0.343242\pi$
$458$	21.9973	1.02787
$459$	0	0
$460$	0	0
$461$	−2.77786	−0.129378	−0.0646890	−	0.997905i	$-0.520606\pi$
$461$	−2.77786	−0.129378	−0.0646890	+	0.997905i	$0.520606\pi$
$462$	0	0
$463$	33.8186	1.57168	0.785842	−	0.618427i	$-0.212230\pi$
$463$	33.8186	1.57168	0.785842	+	0.618427i	$0.212230\pi$
$464$	−10.1009	−0.468924
$465$	0	0
$466$	11.2383	0.520605
$467$	−41.1417	−1.90381	−0.951904	−	0.306395i	$-0.900877\pi$
$467$	−41.1417	−1.90381	−0.951904	+	0.306395i	$0.900877\pi$
$468$	0	0
$469$	20.8788	0.964092
$470$	0	0
$471$	0	0
$472$	−22.8028	−1.04959
$473$	−30.8916	−1.42039
$474$	0	0
$475$	0	0
$476$	−8.60824	−0.394558
$477$	0	0
$478$	29.3327	1.34165
$479$	10.8273	0.494710	0.247355	−	0.968925i	$-0.420439\pi$
$479$	10.8273	0.494710	0.247355	+	0.968925i	$0.420439\pi$
$480$	0	0
$481$	−19.3368	−0.881682
$482$	−9.05426	−0.412410
$483$	0	0
$484$	49.9752	2.27160
$485$	0	0
$486$	0	0
$487$	−17.2533	−0.781820	−0.390910	−	0.920429i	$-0.627840\pi$
$487$	−17.2533	−0.781820	−0.390910	+	0.920429i	$0.627840\pi$
$488$	8.91005	0.403339
$489$	0	0
$490$	0	0
$491$	28.6442	1.29269	0.646347	−	0.763043i	$-0.276296\pi$
$491$	28.6442	1.29269	0.646347	+	0.763043i	$0.276296\pi$
$492$	0	0
$493$	−5.93018	−0.267082
$494$	−46.1544	−2.07658
$495$	0	0
$496$	−16.7704	−0.753014
$497$	−36.5060	−1.63752
$498$	0	0
$499$	−10.5083	−0.470418	−0.235209	−	0.971945i	$-0.575577\pi$
$499$	−10.5083	−0.470418	−0.235209	+	0.971945i	$0.575577\pi$
$500$	0	0
$501$	0	0
$502$	16.9303	0.755638
$503$	−14.7844	−0.659206	−0.329603	−	0.944120i	$-0.606915\pi$
$503$	−14.7844	−0.659206	−0.329603	+	0.944120i	$0.606915\pi$
$504$	0	0
$505$	0	0
$506$	−14.6760	−0.652430
$507$	0	0
$508$	23.6918	1.05115
$509$	−32.5112	−1.44103	−0.720517	−	0.693438i	$-0.756095\pi$
$509$	−32.5112	−1.44103	−0.720517	+	0.693438i	$0.756095\pi$
$510$	0	0
$511$	33.2862	1.47250
$512$	18.5623	0.820344
$513$	0	0
$514$	40.5211	1.78731
$515$	0	0
$516$	0	0
$517$	17.9992	0.791603
$518$	28.2089	1.23943
$519$	0	0
$520$	0	0
$521$	−40.4949	−1.77411	−0.887057	−	0.461660i	$-0.847254\pi$
$521$	−40.4949	−1.77411	−0.887057	+	0.461660i	$0.847254\pi$
$522$	0	0
$523$	−19.7050	−0.861640	−0.430820	−	0.902438i	$-0.641776\pi$
$523$	−19.7050	−0.861640	−0.430820	+	0.902438i	$0.641776\pi$
$524$	3.79338	0.165715
$525$	0	0
$526$	41.1779	1.79544
$527$	−9.84580	−0.428890
$528$	0	0
$529$	−21.4443	−0.932359
$530$	0	0
$531$	0	0
$532$	39.3676	1.70680
$533$	1.86021	0.0805745
$534$	0	0
$535$	0	0
$536$	12.2242	0.528007
$537$	0	0
$538$	−32.4446	−1.39879
$539$	12.5830	0.541988
$540$	0	0
$541$	21.0224	0.903824	0.451912	−	0.892062i	$-0.350742\pi$
$541$	21.0224	0.903824	0.451912	+	0.892062i	$0.350742\pi$
$542$	−53.3080	−2.28977
$543$	0	0
$544$	7.31781	0.313749
$545$	0	0
$546$	0	0
$547$	−25.1379	−1.07482	−0.537410	−	0.843321i	$-0.680597\pi$
$547$	−25.1379	−1.07482	−0.537410	+	0.843321i	$0.680597\pi$
$548$	−35.3789	−1.51131
$549$	0	0
$550$	0	0
$551$	27.1202	1.15536
$552$	0	0
$553$	51.0986	2.17293
$554$	−12.8262	−0.544935
$555$	0	0
$556$	−27.6788	−1.17384
$557$	22.9470	0.972297	0.486149	−	0.873876i	$-0.338401\pi$
$557$	22.9470	0.972297	0.486149	+	0.873876i	$0.338401\pi$
$558$	0	0
$559$	26.4965	1.12068
$560$	0	0
$561$	0	0
$562$	1.43060	0.0603463
$563$	34.8974	1.47075	0.735374	−	0.677661i	$-0.237006\pi$
$563$	34.8974	1.47075	0.735374	+	0.677661i	$0.237006\pi$
$564$	0	0
$565$	0	0
$566$	10.1483	0.426567
$567$	0	0
$568$	−21.3738	−0.896823
$569$	−6.87405	−0.288175	−0.144088	−	0.989565i	$-0.546025\pi$
$569$	−6.87405	−0.288175	−0.144088	+	0.989565i	$0.546025\pi$
$570$	0	0
$571$	24.7188	1.03445	0.517224	−	0.855850i	$-0.326965\pi$
$571$	24.7188	1.03445	0.517224	+	0.855850i	$0.326965\pi$
$572$	−69.4309	−2.90305
$573$	0	0
$574$	−2.71370	−0.113268
$575$	0	0
$576$	0	0
$577$	−4.89908	−0.203951	−0.101976	−	0.994787i	$-0.532516\pi$
$577$	−4.89908	−0.203951	−0.101976	+	0.994787i	$0.532516\pi$
$578$	2.19447	0.0912777
$579$	0	0
$580$	0	0
$581$	−12.5791	−0.521868
$582$	0	0
$583$	−25.8207	−1.06938
$584$	19.4886	0.806446
$585$	0	0
$586$	−7.59657	−0.313811
$587$	23.8761	0.985471	0.492736	−	0.870179i	$-0.335997\pi$
$587$	23.8761	0.985471	0.492736	+	0.870179i	$0.335997\pi$
$588$	0	0
$589$	45.0273	1.85532
$590$	0	0
$591$	0	0
$592$	−7.16175	−0.294346
$593$	−37.8742	−1.55531	−0.777654	−	0.628693i	$-0.783590\pi$
$593$	−37.8742	−1.55531	−0.777654	+	0.628693i	$0.783590\pi$
$594$	0	0
$595$	0	0
$596$	25.4257	1.04148
$597$	0	0
$598$	12.5880	0.514762
$599$	−43.9867	−1.79725	−0.898625	−	0.438718i	$-0.855433\pi$
$599$	−43.9867	−1.79725	−0.898625	+	0.438718i	$0.855433\pi$
$600$	0	0
$601$	11.4083	0.465355	0.232678	−	0.972554i	$-0.425251\pi$
$601$	11.4083	0.465355	0.232678	+	0.972554i	$0.425251\pi$
$602$	−38.6535	−1.57540
$603$	0	0
$604$	−16.8258	−0.684634
$605$	0	0
$606$	0	0
$607$	17.4364	0.707721	0.353861	−	0.935298i	$-0.384869\pi$
$607$	17.4364	0.707721	0.353861	+	0.935298i	$0.384869\pi$
$608$	−33.4662	−1.35723
$609$	0	0
$610$	0	0
$611$	−15.4384	−0.624569
$612$	0	0
$613$	0.620807	0.0250741	0.0125371	−	0.999921i	$-0.496009\pi$
$613$	0.620807	0.0250741	0.0125371	+	0.999921i	$0.496009\pi$
$614$	−43.3681	−1.75019
$615$	0	0
$616$	29.3420	1.18222
$617$	45.0359	1.81308	0.906539	−	0.422123i	$-0.138715\pi$
$617$	45.0359	1.81308	0.906539	+	0.422123i	$0.138715\pi$
$618$	0	0
$619$	−11.4780	−0.461340	−0.230670	−	0.973032i	$-0.574092\pi$
$619$	−11.4780	−0.461340	−0.230670	+	0.973032i	$0.574092\pi$
$620$	0	0
$621$	0	0
$622$	−57.5753	−2.30856
$623$	31.8389	1.27560
$624$	0	0
$625$	0	0
$626$	−50.4342	−2.01576
$627$	0	0
$628$	−24.5246	−0.978639
$629$	−4.20461	−0.167649
$630$	0	0
$631$	−10.9485	−0.435853	−0.217926	−	0.975965i	$-0.569929\pi$
$631$	−10.9485	−0.435853	−0.217926	+	0.975965i	$0.569929\pi$
$632$	29.9176	1.19006
$633$	0	0
$634$	−24.0116	−0.953623
$635$	0	0
$636$	0	0
$637$	−10.7927	−0.427624
$638$	69.7762	2.76247
$639$	0	0
$640$	0	0
$641$	1.51186	0.0597147	0.0298574	−	0.999554i	$-0.490495\pi$
$641$	1.51186	0.0597147	0.0298574	+	0.999554i	$0.490495\pi$
$642$	0	0
$643$	−17.7547	−0.700179	−0.350089	−	0.936716i	$-0.613849\pi$
$643$	−17.7547	−0.700179	−0.350089	+	0.936716i	$0.613849\pi$
$644$	−10.7370	−0.423098
$645$	0	0
$646$	−10.0358	−0.394855
$647$	35.1787	1.38302	0.691508	−	0.722369i	$-0.256947\pi$
$647$	35.1787	1.38302	0.691508	+	0.722369i	$0.256947\pi$
$648$	0	0
$649$	−68.3048	−2.68120
$650$	0	0
$651$	0	0
$652$	2.40039	0.0940065
$653$	−9.99410	−0.391099	−0.195550	−	0.980694i	$-0.562649\pi$
$653$	−9.99410	−0.391099	−0.195550	+	0.980694i	$0.562649\pi$
$654$	0	0
$655$	0	0
$656$	0.688962	0.0268995
$657$	0	0
$658$	22.5218	0.877989
$659$	23.9165	0.931655	0.465827	−	0.884876i	$-0.345757\pi$
$659$	23.9165	0.931655	0.465827	+	0.884876i	$0.345757\pi$
$660$	0	0
$661$	28.6063	1.11266	0.556328	−	0.830963i	$-0.312210\pi$
$661$	28.6063	1.11266	0.556328	+	0.830963i	$0.312210\pi$
$662$	24.3667	0.947037
$663$	0	0
$664$	−7.36488	−0.285813
$665$	0	0
$666$	0	0
$667$	−7.39670	−0.286401
$668$	−3.72013	−0.143936
$669$	0	0
$670$	0	0
$671$	26.6896	1.03034
$672$	0	0
$673$	12.9564	0.499431	0.249716	−	0.968319i	$-0.419663\pi$
$673$	12.9564	0.499431	0.249716	+	0.968319i	$0.419663\pi$
$674$	65.9130	2.53887
$675$	0	0
$676$	22.9488	0.882645
$677$	23.5326	0.904430	0.452215	−	0.891909i	$-0.350634\pi$
$677$	23.5326	0.904430	0.452215	+	0.891909i	$0.350634\pi$
$678$	0	0
$679$	−6.95350	−0.266851
$680$	0	0
$681$	0	0
$682$	115.848	4.43607
$683$	−25.4677	−0.974495	−0.487247	−	0.873264i	$-0.661999\pi$
$683$	−25.4677	−0.974495	−0.487247	+	0.873264i	$0.661999\pi$
$684$	0	0
$685$	0	0
$686$	−31.2186	−1.19193
$687$	0	0
$688$	9.81346	0.374135
$689$	22.1471	0.843736
$690$	0	0
$691$	47.8923	1.82191	0.910955	−	0.412507i	$-0.135347\pi$
$691$	47.8923	1.82191	0.910955	+	0.412507i	$0.135347\pi$
$692$	21.1084	0.802420
$693$	0	0
$694$	−58.1437	−2.20710
$695$	0	0
$696$	0	0
$697$	0.404485	0.0153210
$698$	25.9442	0.982001
$699$	0	0
$700$	0	0
$701$	5.52783	0.208783	0.104392	−	0.994536i	$-0.466710\pi$
$701$	5.52783	0.208783	0.104392	+	0.994536i	$0.466710\pi$
$702$	0	0
$703$	19.2287	0.725226
$704$	−67.8379	−2.55674
$705$	0	0
$706$	−25.0140	−0.941414
$707$	−24.8391	−0.934170
$708$	0	0
$709$	22.8895	0.859633	0.429817	−	0.902916i	$-0.358578\pi$
$709$	22.8895	0.859633	0.429817	+	0.902916i	$0.358578\pi$
$710$	0	0
$711$	0	0
$712$	18.6413	0.698611
$713$	−12.2806	−0.459913
$714$	0	0
$715$	0	0
$716$	−45.0264	−1.68272
$717$	0	0
$718$	−26.8020	−1.00024
$719$	−0.748823	−0.0279264	−0.0139632	−	0.999903i	$-0.504445\pi$
$719$	−0.748823	−0.0279264	−0.0139632	+	0.999903i	$0.504445\pi$
$720$	0	0
$721$	−15.7664	−0.587172
$722$	4.20159	0.156367
$723$	0	0
$724$	−31.0107	−1.15250
$725$	0	0
$726$	0	0
$727$	−51.8312	−1.92231	−0.961157	−	0.276004i	$-0.910990\pi$
$727$	−51.8312	−1.92231	−0.961157	+	0.276004i	$0.910990\pi$
$728$	−25.1674	−0.932766
$729$	0	0
$730$	0	0
$731$	5.76142	0.213094
$732$	0	0
$733$	−0.440590	−0.0162735	−0.00813677	−	0.999967i	$-0.502590\pi$
$733$	−0.440590	−0.0162735	−0.00813677	+	0.999967i	$0.502590\pi$
$734$	−15.1780	−0.560232
$735$	0	0
$736$	9.12748	0.336443
$737$	36.6171	1.34881
$738$	0	0
$739$	−30.4658	−1.12070	−0.560351	−	0.828255i	$-0.689334\pi$
$739$	−30.4658	−1.12070	−0.560351	+	0.828255i	$0.689334\pi$
$740$	0	0
$741$	0	0
$742$	−32.3086	−1.18608
$743$	−15.1434	−0.555557	−0.277779	−	0.960645i	$-0.589598\pi$
$743$	−15.1434	−0.555557	−0.277779	+	0.960645i	$0.589598\pi$
$744$	0	0
$745$	0	0
$746$	−60.0157	−2.19733
$747$	0	0
$748$	−15.0971	−0.552005
$749$	−17.1518	−0.626714
$750$	0	0
$751$	34.1610	1.24655	0.623277	−	0.782001i	$-0.285801\pi$
$751$	34.1610	1.24655	0.623277	+	0.782001i	$0.285801\pi$
$752$	−5.71789	−0.208510
$753$	0	0
$754$	−59.8488	−2.17957
$755$	0	0
$756$	0	0
$757$	20.5792	0.747965	0.373982	−	0.927436i	$-0.377992\pi$
$757$	20.5792	0.747965	0.373982	+	0.927436i	$0.377992\pi$
$758$	32.8211	1.19212
$759$	0	0
$760$	0	0
$761$	18.8045	0.681661	0.340831	−	0.940125i	$-0.389292\pi$
$761$	18.8045	0.681661	0.340831	+	0.940125i	$0.389292\pi$
$762$	0	0
$763$	−12.1727	−0.440681
$764$	−23.9296	−0.865743
$765$	0	0
$766$	−62.3434	−2.25256
$767$	58.5867	2.11544
$768$	0	0
$769$	−36.5974	−1.31974	−0.659868	−	0.751382i	$-0.729388\pi$
$769$	−36.5974	−1.31974	−0.659868	+	0.751382i	$0.729388\pi$
$770$	0	0
$771$	0	0
$772$	53.1133	1.91159
$773$	−19.4040	−0.697912	−0.348956	−	0.937139i	$-0.613464\pi$
$773$	−19.4040	−0.697912	−0.348956	+	0.937139i	$0.613464\pi$
$774$	0	0
$775$	0	0
$776$	−4.07118	−0.146147
$777$	0	0
$778$	48.7698	1.74848
$779$	−1.84981	−0.0662764
$780$	0	0
$781$	−64.0240	−2.29096
$782$	2.73715	0.0978803
$783$	0	0
$784$	−3.99730	−0.142761
$785$	0	0
$786$	0	0
$787$	28.7760	1.02575	0.512876	−	0.858462i	$-0.328580\pi$
$787$	28.7760	1.02575	0.512876	+	0.858462i	$0.328580\pi$
$788$	−30.3389	−1.08078
$789$	0	0
$790$	0	0
$791$	2.79352	0.0993260
$792$	0	0
$793$	−22.8924	−0.812932
$794$	3.87576	0.137546
$795$	0	0
$796$	−35.8964	−1.27231
$797$	−4.55967	−0.161512	−0.0807559	−	0.996734i	$-0.525733\pi$
$797$	−4.55967	−0.161512	−0.0807559	+	0.996734i	$0.525733\pi$
$798$	0	0
$799$	−3.35693	−0.118760
$800$	0	0
$801$	0	0
$802$	10.5208	0.371501
$803$	58.3773	2.06009
$804$	0	0
$805$	0	0
$806$	−99.3662	−3.50002
$807$	0	0
$808$	−14.5430	−0.511620
$809$	−37.0052	−1.30103	−0.650516	−	0.759493i	$-0.725447\pi$
$809$	−37.0052	−1.30103	−0.650516	+	0.759493i	$0.725447\pi$
$810$	0	0
$811$	−30.4268	−1.06843	−0.534215	−	0.845349i	$-0.679393\pi$
$811$	−30.4268	−1.06843	−0.534215	+	0.845349i	$0.679393\pi$
$812$	51.0484	1.79145
$813$	0	0
$814$	49.4726	1.73402
$815$	0	0
$816$	0	0
$817$	−26.3484	−0.921814
$818$	−79.6227	−2.78394
$819$	0	0
$820$	0	0
$821$	−2.43436	−0.0849596	−0.0424798	−	0.999097i	$-0.513526\pi$
$821$	−2.43436	−0.0849596	−0.0424798	+	0.999097i	$0.513526\pi$
$822$	0	0
$823$	−30.1177	−1.04984	−0.524918	−	0.851153i	$-0.675904\pi$
$823$	−30.1177	−1.04984	−0.524918	+	0.851153i	$0.675904\pi$
$824$	−9.23103	−0.321578
$825$	0	0
$826$	−85.4674	−2.97379
$827$	17.7944	0.618771	0.309385	−	0.950937i	$-0.399877\pi$
$827$	17.7944	0.618771	0.309385	+	0.950937i	$0.399877\pi$
$828$	0	0
$829$	27.9359	0.970254	0.485127	−	0.874444i	$-0.338773\pi$
$829$	27.9359	0.970254	0.485127	+	0.874444i	$0.338773\pi$
$830$	0	0
$831$	0	0
$832$	58.1863	2.01725
$833$	−2.34678	−0.0813113
$834$	0	0
$835$	0	0
$836$	69.0429	2.38790
$837$	0	0
$838$	53.6441	1.85311
$839$	40.4717	1.39724	0.698618	−	0.715494i	$-0.253799\pi$
$839$	40.4717	1.39724	0.698618	+	0.715494i	$0.253799\pi$
$840$	0	0
$841$	6.16706	0.212657
$842$	−30.9221	−1.06565
$843$	0	0
$844$	74.2047	2.55423
$845$	0	0
$846$	0	0
$847$	54.2628	1.86449
$848$	8.20258	0.281678
$849$	0	0
$850$	0	0
$851$	−5.24440	−0.179776
$852$	0	0
$853$	7.32217	0.250706	0.125353	−	0.992112i	$-0.459994\pi$
$853$	7.32217	0.250706	0.125353	+	0.992112i	$0.459994\pi$
$854$	33.3958	1.14278
$855$	0	0
$856$	−10.0422	−0.343234
$857$	−4.74086	−0.161945	−0.0809724	−	0.996716i	$-0.525803\pi$
$857$	−4.74086	−0.161945	−0.0809724	+	0.996716i	$0.525803\pi$
$858$	0	0
$859$	8.14829	0.278016	0.139008	−	0.990291i	$-0.455609\pi$
$859$	8.14829	0.278016	0.139008	+	0.990291i	$0.455609\pi$
$860$	0	0
$861$	0	0
$862$	38.4640	1.31009
$863$	−15.8044	−0.537988	−0.268994	−	0.963142i	$-0.586691\pi$
$863$	−15.8044	−0.537988	−0.268994	+	0.963142i	$0.586691\pi$
$864$	0	0
$865$	0	0
$866$	−24.5242	−0.833366
$867$	0	0
$868$	84.7550	2.87677
$869$	89.6166	3.04003
$870$	0	0
$871$	−31.4074	−1.06420
$872$	−7.12695	−0.241349
$873$	0	0
$874$	−12.5177	−0.423417
$875$	0	0
$876$	0	0
$877$	−22.3063	−0.753231	−0.376616	−	0.926370i	$-0.622912\pi$
$877$	−22.3063	−0.753231	−0.376616	+	0.926370i	$0.622912\pi$
$878$	−26.4254	−0.891816
$879$	0	0
$880$	0	0
$881$	−9.67865	−0.326082	−0.163041	−	0.986619i	$-0.552130\pi$
$881$	−9.67865	−0.326082	−0.163041	+	0.986619i	$0.552130\pi$
$882$	0	0
$883$	21.7255	0.731120	0.365560	−	0.930788i	$-0.380877\pi$
$883$	21.7255	0.731120	0.365560	+	0.930788i	$0.380877\pi$
$884$	12.9492	0.435528
$885$	0	0
$886$	−86.8974	−2.91938
$887$	−2.17244	−0.0729434	−0.0364717	−	0.999335i	$-0.511612\pi$
$887$	−2.17244	−0.0729434	−0.0364717	+	0.999335i	$0.511612\pi$
$888$	0	0
$889$	25.7244	0.862768
$890$	0	0
$891$	0	0
$892$	−56.3981	−1.88835
$893$	15.3521	0.513738
$894$	0	0
$895$	0	0
$896$	−40.1384	−1.34093
$897$	0	0
$898$	−25.6264	−0.855165
$899$	58.3874	1.94733
$900$	0	0
$901$	4.81568	0.160434
$902$	−4.75928	−0.158467
$903$	0	0
$904$	1.63557	0.0543982
$905$	0	0
$906$	0	0
$907$	25.8704	0.859012	0.429506	−	0.903064i	$-0.358688\pi$
$907$	25.8704	0.859012	0.429506	+	0.903064i	$0.358688\pi$
$908$	−37.4284	−1.24210
$909$	0	0
$910$	0	0
$911$	36.1769	1.19859	0.599297	−	0.800527i	$-0.295447\pi$
$911$	36.1769	1.19859	0.599297	+	0.800527i	$0.295447\pi$
$912$	0	0
$913$	−22.0611	−0.730117
$914$	44.3606	1.46732
$915$	0	0
$916$	28.2243	0.932558
$917$	4.11883	0.136016
$918$	0	0
$919$	52.2475	1.72349	0.861743	−	0.507346i	$-0.169373\pi$
$919$	52.2475	1.72349	0.861743	+	0.507346i	$0.169373\pi$
$920$	0	0
$921$	0	0
$922$	−6.09592	−0.200759
$923$	54.9150	1.80755
$924$	0	0
$925$	0	0
$926$	74.2138	2.43882
$927$	0	0
$928$	−43.3960	−1.42454
$929$	−19.6884	−0.645955	−0.322978	−	0.946407i	$-0.604684\pi$
$929$	−19.6884	−0.645955	−0.322978	+	0.946407i	$0.604684\pi$
$930$	0	0
$931$	10.7324	0.351741
$932$	14.4197	0.472333
$933$	0	0
$934$	−90.2840	−2.95418
$935$	0	0
$936$	0	0
$937$	15.0805	0.492657	0.246328	−	0.969186i	$-0.420776\pi$
$937$	15.0805	0.492657	0.246328	+	0.969186i	$0.420776\pi$
$938$	45.8177	1.49600
$939$	0	0
$940$	0	0
$941$	12.2054	0.397886	0.198943	−	0.980011i	$-0.436249\pi$
$941$	12.2054	0.397886	0.198943	+	0.980011i	$0.436249\pi$
$942$	0	0
$943$	0.504513	0.0164292
$944$	21.6987	0.706232
$945$	0	0
$946$	−67.7904	−2.20406
$947$	23.4424	0.761776	0.380888	−	0.924621i	$-0.375618\pi$
$947$	23.4424	0.761776	0.380888	+	0.924621i	$0.375618\pi$
$948$	0	0
$949$	−50.0717	−1.62540
$950$	0	0
$951$	0	0
$952$	−5.47242	−0.177362
$953$	50.7165	1.64287	0.821435	−	0.570302i	$-0.193174\pi$
$953$	50.7165	1.64287	0.821435	+	0.570302i	$0.193174\pi$
$954$	0	0
$955$	0	0
$956$	37.6363	1.21724
$957$	0	0
$958$	23.7600	0.767652
$959$	−38.4143	−1.24046
$960$	0	0
$961$	65.9397	2.12709
$962$	−42.4339	−1.36813
$963$	0	0
$964$	−11.6174	−0.374170
$965$	0	0
$966$	0	0
$967$	41.6170	1.33831	0.669157	−	0.743121i	$-0.266656\pi$
$967$	41.6170	1.33831	0.669157	+	0.743121i	$0.266656\pi$
$968$	31.7702	1.02113
$969$	0	0
$970$	0	0
$971$	−19.2045	−0.616302	−0.308151	−	0.951337i	$-0.599710\pi$
$971$	−19.2045	−0.616302	−0.308151	+	0.951337i	$0.599710\pi$
$972$	0	0
$973$	−30.0535	−0.963472
$974$	−37.8617	−1.21317
$975$	0	0
$976$	−8.47862	−0.271394
$977$	21.0264	0.672696	0.336348	−	0.941738i	$-0.390808\pi$
$977$	21.0264	0.672696	0.336348	+	0.941738i	$0.390808\pi$
$978$	0	0
$979$	55.8390	1.78462
$980$	0	0
$981$	0	0
$982$	62.8587	2.00590
$983$	−3.77964	−0.120552	−0.0602758	−	0.998182i	$-0.519198\pi$
$983$	−3.77964	−0.120552	−0.0602758	+	0.998182i	$0.519198\pi$
$984$	0	0
$985$	0	0
$986$	−13.0136	−0.414437
$987$	0	0
$988$	−59.2198	−1.88403
$989$	7.18619	0.228508
$990$	0	0
$991$	35.7886	1.13686	0.568431	−	0.822731i	$-0.307551\pi$
$991$	35.7886	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$992$	−72.0497	−2.28758
$993$	0	0
$994$	−80.1111	−2.54097
$995$	0	0
$996$	0	0
$997$	4.92887	0.156099	0.0780494	−	0.996949i	$-0.475131\pi$
$997$	4.92887	0.156099	0.0780494	+	0.996949i	$0.475131\pi$
$998$	−23.0602	−0.729958
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.bq.1.4		5
3.2	odd	2		425.2.a.i.1.2	✓	5
5.4	even	2		3825.2.a.bl.1.2		5
12.11	even	2		6800.2.a.bz.1.5		5
15.2	even	4		425.2.b.f.324.3		10
15.8	even	4		425.2.b.f.324.8		10
15.14	odd	2		425.2.a.j.1.4	yes	5
51.50	odd	2		7225.2.a.x.1.2		5
60.59	even	2		6800.2.a.cd.1.1		5
255.254	odd	2		7225.2.a.y.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.2	✓	5	3.2	odd	2
425.2.a.j.1.4	yes	5	15.14	odd	2
425.2.b.f.324.3		10	15.2	even	4
425.2.b.f.324.8		10	15.8	even	4
3825.2.a.bl.1.2		5	5.4	even	2
3825.2.a.bq.1.4		5	1.1	even	1	trivial
6800.2.a.bz.1.5		5	12.11	even	2
6800.2.a.cd.1.1		5	60.59	even	2
7225.2.a.x.1.2		5	51.50	odd	2
7225.2.a.y.1.4		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)

\(f(q)\)	\(=\)	\(q+2.19447 q^{2} +2.81568 q^{4} +3.05725 q^{7} +1.78998 q^{8} +5.36180 q^{11} -4.59895 q^{13} +6.70903 q^{14} -1.70331 q^{16} -1.00000 q^{17} +4.57325 q^{19} +11.7663 q^{22} -1.24730 q^{23} -10.0922 q^{26} +8.60824 q^{28} +5.93018 q^{29} +9.84580 q^{31} -7.31781 q^{32} -2.19447 q^{34} +4.20461 q^{37} +10.0358 q^{38} -0.404485 q^{41} -5.76142 q^{43} +15.0971 q^{44} -2.73715 q^{46} +3.35693 q^{47} +2.34678 q^{49} -12.9492 q^{52} -4.81568 q^{53} +5.47242 q^{56} +13.0136 q^{58} -12.7392 q^{59} +4.97774 q^{61} +21.6063 q^{62} -12.6521 q^{64} +6.82926 q^{67} -2.81568 q^{68} -11.9408 q^{71} +10.8876 q^{73} +9.22687 q^{74} +12.8768 q^{76} +16.3924 q^{77} +16.7139 q^{79} -0.887628 q^{82} -4.11450 q^{83} -12.6432 q^{86} +9.59752 q^{88} +10.4142 q^{89} -14.0601 q^{91} -3.51199 q^{92} +7.36667 q^{94} -2.27443 q^{97} +5.14994 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 11 q^{4} + q^{7} + 9 q^{8} - 4 q^{11} - 3 q^{13} + 7 q^{14} + 27 q^{16} - 5 q^{17} + 6 q^{19} + 18 q^{22} - 4 q^{23} + 5 q^{26} - 15 q^{28} - 2 q^{29} + 21 q^{31} + 9 q^{32} - q^{34} - 2 q^{37}+ \cdots - 38 q^{98}+O(q^{100}) \) 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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