LMFDB - Embedded newform 7225.2.a.y.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,1,11,0,-3,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	7225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.19447 q^{2} -2.48887 q^{3} +2.81568 q^{4} -5.46174 q^{6} +3.05725 q^{7} +1.78998 q^{8} +3.19447 q^{9} +5.36180 q^{11} -7.00786 q^{12} +4.59895 q^{13} +6.70903 q^{14} -1.70331 q^{16} +7.01015 q^{18} +4.57325 q^{19} -7.60910 q^{21} +11.7663 q^{22} +1.24730 q^{23} -4.45503 q^{24} +10.0922 q^{26} -0.483999 q^{27} +8.60824 q^{28} +5.93018 q^{29} -9.84580 q^{31} -7.31781 q^{32} -13.3448 q^{33} +8.99459 q^{36} +4.20461 q^{37} +10.0358 q^{38} -11.4462 q^{39} -0.404485 q^{41} -16.6979 q^{42} +5.76142 q^{43} +15.0971 q^{44} +2.73715 q^{46} +3.35693 q^{47} +4.23931 q^{48} +2.34678 q^{49} +12.9492 q^{52} -4.81568 q^{53} -1.06212 q^{54} +5.47242 q^{56} -11.3822 q^{57} +13.0136 q^{58} +12.7392 q^{59} -4.97774 q^{61} -21.6063 q^{62} +9.76628 q^{63} -12.6521 q^{64} -29.2847 q^{66} -6.82926 q^{67} -3.10436 q^{69} -11.9408 q^{71} +5.71803 q^{72} +10.8876 q^{73} +9.22687 q^{74} +12.8768 q^{76} +16.3924 q^{77} -25.1183 q^{78} -16.7139 q^{79} -8.37879 q^{81} -0.887628 q^{82} -4.11450 q^{83} -21.4248 q^{84} +12.6432 q^{86} -14.7594 q^{87} +9.59752 q^{88} -10.4142 q^{89} +14.0601 q^{91} +3.51199 q^{92} +24.5049 q^{93} +7.36667 q^{94} +18.2131 q^{96} -2.27443 q^{97} +5.14994 q^{98} +17.1281 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + q^{3} + 11 q^{4} - 3 q^{6} + q^{7} + 9 q^{8} + 6 q^{9} - 4 q^{11} + 17 q^{12} + 3 q^{13} + 7 q^{14} + 27 q^{16} + 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} + 19 q^{24} - 5 q^{26}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q + q^2 + q^3 + 11 * q^4 - 3 * q^6 + q^7 + 9 * q^8 + 6 * q^9 - 4 * q^11 + 17 * q^12 + 3 * q^13 + 7 * q^14 + 27 * q^16 + 22 * q^18 + 6 * q^19 - 5 * q^21 + 18 * q^22 + 4 * q^23 + 19 * q^24 - 5 * q^26 - 5 * q^27 - 15 * q^28 - 2 * q^29 - 21 * q^31 + 9 * q^32 - 12 * q^33 + 22 * q^36 - 2 * q^37 + 34 * q^38 - 23 * q^39 + 8 * q^41 - 25 * q^42 + 4 * q^43 + 4 * q^44 + 6 * q^46 + 2 * q^47 + 51 * q^48 + 10 * q^49 + 13 * q^52 - 21 * q^53 + 25 * q^54 + 29 * q^56 - 20 * q^57 + 22 * q^58 + 12 * q^59 + 2 * q^61 + 3 * q^62 + 8 * q^63 + 43 * q^64 - 28 * q^66 + 12 * q^67 - 21 * q^71 + 58 * q^72 + 22 * q^73 + 50 * q^74 - 14 * q^76 + 32 * q^77 - 53 * q^78 - 41 * q^79 - 35 * q^81 + 28 * q^82 + 8 * q^83 - 71 * q^84 - 36 * q^86 + 4 * q^87 + 14 * q^88 - 10 * q^89 - 13 * q^91 + 56 * q^92 - 9 * q^93 - 10 * q^94 + 85 * q^96 - 20 * q^97 - 38 * q^98 + 14 * q^99

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	$0.217316\pi$
$3$	−2.48887	−1.43695	−0.718474	−	0.695553i	$-0.755159\pi$
$3$	−2.48887	−1.43695	−0.718474	+	0.695553i	$0.755159\pi$
$4$	2.81568	1.40784
$5$	0	0
$5$	0	0
$6$	−5.46174	−2.22974
$7$	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
$9$	3.19447	1.06482
$10$	0	0
$11$	5.36180	1.61664	0.808322	−	0.588741i	$-0.200376\pi$
$11$	5.36180	1.61664	0.808322	+	0.588741i	$0.200376\pi$
$12$	−7.00786	−2.02299
$13$	4.59895	1.27552	0.637760	−	0.770235i	$-0.279861\pi$
$13$	4.59895	1.27552	0.637760	+	0.770235i	$0.279861\pi$
$14$	6.70903	1.79306
$15$	0	0
$16$	−1.70331	−0.425827
$17$	0	0
$17$	0	0
$18$	7.01015	1.65231
$19$	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
$21$	−7.60910	−1.66044
$22$	11.7663	2.50858
$23$	1.24730	0.260079	0.130040	−	0.991509i	$-0.458490\pi$
$23$	1.24730	0.260079	0.130040	+	0.991509i	$0.458490\pi$
$24$	−4.45503	−0.909378
$25$	0	0
$26$	10.0922	1.97925
$27$	−0.483999	−0.0931457
$28$	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	$-0.845281\pi$
$31$	−9.84580	−1.76836	−0.884179	+	0.467149i	$0.845281\pi$
$32$	−7.31781	−1.29362
$33$	−13.3448	−2.32303
$34$	0	0
$35$	0	0
$36$	8.99459	1.49910
$37$	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
$39$	−11.4462	−1.83286
$40$	0	0
$41$	−0.404485	−0.0631699	−0.0315850	−	0.999501i	$-0.510055\pi$
$41$	−0.404485	−0.0631699	−0.0315850	+	0.999501i	$0.510055\pi$
$42$	−16.6979	−2.57654
$43$	5.76142	0.878608	0.439304	−	0.898339i	$-0.355225\pi$
$43$	5.76142	0.878608	0.439304	+	0.898339i	$0.355225\pi$
$44$	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	$0.421268\pi$
$48$	4.23931	0.611892
$49$	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	$0.607299\pi$
$54$	−1.06212	−0.144536
$55$	0	0
$56$	5.47242	0.731283
$57$	−11.3822	−1.50761
$58$	13.0136	1.70877
$59$	12.7392	1.65850	0.829248	−	0.558881i	$-0.188769\pi$
$59$	12.7392	1.65850	0.829248	+	0.558881i	$0.188769\pi$
$60$	0	0
$61$	−4.97774	−0.637334	−0.318667	−	0.947867i	$-0.603235\pi$
$61$	−4.97774	−0.637334	−0.318667	+	0.947867i	$0.603235\pi$
$62$	−21.6063	−2.74400
$63$	9.76628	1.23044
$64$	−12.6521	−1.58151
$65$	0	0
$66$	−29.2847	−3.60470
$67$	−6.82926	−0.834327	−0.417163	−	0.908831i	$-0.636976\pi$
$67$	−6.82926	−0.834327	−0.417163	+	0.908831i	$0.636976\pi$
$68$	0	0
$69$	−3.10436	−0.373721
$70$	0	0
$71$	−11.9408	−1.41711	−0.708555	−	0.705656i	$-0.750652\pi$
$71$	−11.9408	−1.41711	−0.708555	+	0.705656i	$0.750652\pi$
$72$	5.71803	0.673877
$73$	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
$78$	−25.1183	−2.84408
$79$	−16.7139	−1.88046	−0.940230	−	0.340539i	$-0.889391\pi$
$79$	−16.7139	−1.88046	−0.940230	+	0.340539i	$0.889391\pi$
$80$	0	0
$81$	−8.37879	−0.930976
$82$	−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	$0.572504\pi$
$84$	−21.4248	−2.33763
$85$	0	0
$86$	12.6432	1.36335
$87$	−14.7594	−1.58238
$88$	9.59752	1.02310
$89$	−10.4142	−1.10391	−0.551953	−	0.833875i	$-0.686117\pi$
$89$	−10.4142	−1.10391	−0.551953	+	0.833875i	$0.686117\pi$
$90$	0	0
$91$	14.0601	1.47390
$92$	3.51199	0.366150
$93$	24.5049	2.54104
$94$	7.36667	0.759814
$95$	0	0
$96$	18.2131	1.85886
$97$	−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
$99$	17.1281	1.72144
$100$	0	0
$101$	8.12465	0.808433	0.404216	−	0.914663i	$-0.367544\pi$
$101$	8.12465	0.808433	0.404216	+	0.914663i	$0.367544\pi$
$102$	0	0
$103$	5.15706	0.508140	0.254070	−	0.967186i	$-0.418231\pi$
$103$	5.15706	0.508140	0.254070	+	0.967186i	$0.418231\pi$
$104$	8.23203	0.807217
$105$	0	0
$106$	−10.5678	−1.02644
$107$	5.61021	0.542360	0.271180	−	0.962529i	$-0.412586\pi$
$107$	5.61021	0.542360	0.271180	+	0.962529i	$0.412586\pi$
$108$	−1.36279	−0.131134
$109$	3.98158	0.381366	0.190683	−	0.981652i	$-0.438930\pi$
$109$	3.98158	0.381366	0.190683	+	0.981652i	$0.438930\pi$
$110$	0	0
$111$	−10.4647	−0.993268
$112$	−5.20744	−0.492057
$113$	−0.913734	−0.0859569	−0.0429785	−	0.999076i	$-0.513685\pi$
$113$	−0.913734	−0.0859569	−0.0429785	+	0.999076i	$0.513685\pi$
$114$	−24.9779	−2.33939
$115$	0	0
$116$	16.6975	1.55032
$117$	14.6912	1.35820
$118$	27.9556	2.57352
$119$	0	0
$120$	0	0
$121$	17.7489	1.61354
$122$	−10.9235	−0.988965
$123$	1.00671	0.0907720
$124$	−27.7226	−2.48956
$125$	0	0
$126$	21.4318	1.90929
$127$	−8.41422	−0.746642	−0.373321	−	0.927702i	$-0.621781\pi$
$127$	−8.41422	−0.746642	−0.373321	+	0.927702i	$0.621781\pi$
$128$	−13.1289	−1.16044
$129$	−14.3394	−1.26251
$130$	0	0
$131$	1.34723	0.117708	0.0588542	−	0.998267i	$-0.481255\pi$
$131$	1.34723	0.117708	0.0588542	+	0.998267i	$0.481255\pi$
$132$	−37.5747	−3.27046
$133$	13.9816	1.21236
$134$	−14.9866	−1.29464
$135$	0	0
$136$	0	0
$137$	−12.5650	−1.07350	−0.536749	−	0.843742i	$-0.680348\pi$
$137$	−12.5650	−1.07350	−0.536749	+	0.843742i	$0.680348\pi$
$138$	−6.81241	−0.579910
$139$	9.83024	0.833790	0.416895	−	0.908955i	$-0.363118\pi$
$139$	9.83024	0.833790	0.416895	+	0.908955i	$0.363118\pi$
$140$	0	0
$141$	−8.35496	−0.703614
$142$	−26.2036	−2.19896
$143$	24.6586	2.06206
$144$	−5.44116	−0.453430
$145$	0	0
$146$	23.8925	1.97736
$147$	−5.84084	−0.481744
$148$	11.8388	0.973146
$149$	−9.03003	−0.739769	−0.369884	−	0.929078i	$-0.620603\pi$
$149$	−9.03003	−0.739769	−0.369884	+	0.929078i	$0.620603\pi$
$150$	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
$156$	−32.2288	−2.58037
$157$	8.71002	0.695135	0.347568	−	0.937655i	$-0.387008\pi$
$157$	8.71002	0.695135	0.347568	+	0.937655i	$0.387008\pi$
$158$	−36.6781	−2.91795
$159$	11.9856	0.950519
$160$	0	0
$161$	3.81330	0.300530
$162$	−18.3870	−1.44462
$163$	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	$-0.483721\pi$
$167$	1.32122	0.102239	0.0511195	+	0.998693i	$0.483721\pi$
$168$	−13.6201	−1.05082
$169$	8.15035	0.626950
$170$	0	0
$171$	14.6091	1.11719
$172$	16.2223	1.23694
$173$	−7.49672	−0.569965	−0.284983	−	0.958533i	$-0.591988\pi$
$173$	−7.49672	−0.569965	−0.284983	+	0.958533i	$0.591988\pi$
$174$	−32.3891	−2.45541
$175$	0	0
$176$	−9.13279	−0.688410
$177$	−31.7061	−2.38317
$178$	−22.8537	−1.71295
$179$	15.9913	1.19525	0.597624	−	0.801777i	$-0.296112\pi$
$179$	15.9913	1.19525	0.597624	+	0.801777i	$0.296112\pi$
$180$	0	0
$181$	11.0136	0.818633	0.409317	−	0.912392i	$-0.365767\pi$
$181$	11.0136	0.818633	0.409317	+	0.912392i	$0.365767\pi$
$182$	30.8545	2.28709
$183$	12.3889	0.915816
$184$	2.23264	0.164592
$185$	0	0
$186$	53.7752	3.94299
$187$	0	0
$188$	9.45204	0.689361
$189$	−1.47971	−0.107633
$190$	0	0
$191$	8.49870	0.614944	0.307472	−	0.951557i	$-0.400517\pi$
$191$	8.49870	0.614944	0.307472	+	0.951557i	$0.400517\pi$
$192$	31.4893	2.27255
$193$	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	$-0.374600\pi$
$197$	10.7750	0.767687	0.383843	+	0.923398i	$0.374600\pi$
$198$	37.5870	2.67119
$199$	12.7488	0.903735	0.451868	−	0.892085i	$-0.350758\pi$
$199$	12.7488	0.903735	0.451868	+	0.892085i	$0.350758\pi$
$200$	0	0
$201$	16.9971	1.19889
$202$	17.8293	1.25446
$203$	18.1301	1.27248
$204$	0	0
$205$	0	0
$206$	11.3170	0.788492
$207$	3.98445	0.276938
$208$	−7.83343	−0.543151
$209$	24.5209	1.69614
$210$	0	0
$211$	−26.3541	−1.81429	−0.907146	−	0.420817i	$-0.861743\pi$
$211$	−26.3541	−1.81429	−0.907146	+	0.420817i	$0.861743\pi$
$212$	−13.5594	−0.931264
$213$	29.7190	2.03631
$214$	12.3114	0.841591
$215$	0	0
$216$	−0.866350	−0.0589476
$217$	−30.1011	−2.04339
$218$	8.73744	0.591774
$219$	−27.0979	−1.83110
$220$	0	0
$221$	0	0
$222$	−22.9645	−1.54127
$223$	20.0300	1.34131	0.670655	−	0.741769i	$-0.266013\pi$
$223$	20.0300	1.34131	0.670655	+	0.741769i	$0.266013\pi$
$224$	−22.3724	−1.49482
$225$	0	0
$226$	−2.00516	−0.133381
$227$	13.2928	0.882277	0.441138	−	0.897439i	$-0.354575\pi$
$227$	13.2928	0.882277	0.441138	+	0.897439i	$0.354575\pi$
$228$	−32.0487	−2.12248
$229$	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
$231$	−40.7984	−2.68434
$232$	10.6149	0.696903
$233$	−5.12121	−0.335502	−0.167751	−	0.985829i	$-0.553650\pi$
$233$	−5.12121	−0.335502	−0.167751	+	0.985829i	$0.553650\pi$
$234$	32.2393	2.10755
$235$	0	0
$236$	35.8694	2.33490
$237$	41.5987	2.70213
$238$	0	0
$239$	−13.3667	−0.864618	−0.432309	−	0.901726i	$-0.642301\pi$
$239$	−13.3667	−0.864618	−0.432309	+	0.901726i	$0.642301\pi$
$240$	0	0
$241$	4.12595	0.265776	0.132888	−	0.991131i	$-0.457575\pi$
$241$	4.12595	0.265776	0.132888	+	0.991131i	$0.457575\pi$
$242$	38.9493	2.50376
$243$	22.3057	1.43091
$244$	−14.0157	−0.897264
$245$	0	0
$246$	2.20919	0.140853
$247$	21.0322	1.33824
$248$	−17.6238	−1.11911
$249$	10.2405	0.648962
$250$	0	0
$251$	−7.71502	−0.486968	−0.243484	−	0.969905i	$-0.578290\pi$
$251$	−7.71502	−0.486968	−0.243484	+	0.969905i	$0.578290\pi$
$252$	27.4987	1.73226
$253$	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	$0.304647\pi$
$258$	−31.4673	−1.95907
$259$	12.8546	0.798743
$260$	0	0
$261$	18.9438	1.17259
$262$	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	$0.303626\pi$
$264$	−23.8870	−1.47014
$265$	0	0
$266$	30.6821	1.88124
$267$	25.9196	1.58626
$268$	−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	$0.764143\pi$
$272$	0	0
$273$	−34.9939	−2.11792
$274$	−27.5734	−1.66577
$275$	0	0
$276$	−8.74088	−0.526139
$277$	−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
$279$	−31.4521	−1.88299
$280$	0	0
$281$	−0.651914	−0.0388899	−0.0194450	−	0.999811i	$-0.506190\pi$
$281$	−0.651914	−0.0388899	−0.0194450	+	0.999811i	$0.506190\pi$
$282$	−18.3347	−1.09181
$283$	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
$288$	−23.3765	−1.37747
$289$	0	0
$290$	0	0
$291$	5.66075	0.331839
$292$	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	$0.532242\pi$
$294$	−12.8175	−0.747533
$295$	0	0
$296$	7.52617	0.437450
$297$	−2.59511	−0.150583
$298$	−19.8161	−1.14792
$299$	5.73626	0.331736
$300$	0	0
$301$	17.6141	1.01526
$302$	−13.1136	−0.754603
$303$	−20.2212	−1.16168
$304$	−7.78966	−0.446767
$305$	0	0
$306$	0	0
$307$	19.7625	1.12790	0.563952	−	0.825808i	$-0.309280\pi$
$307$	19.7625	1.12790	0.563952	+	0.825808i	$0.309280\pi$
$308$	46.1557	2.62996
$309$	−12.8352	−0.730171
$310$	0	0
$311$	−26.2366	−1.48774	−0.743871	−	0.668324i	$-0.767012\pi$
$311$	−26.2366	−1.48774	−0.743871	+	0.668324i	$0.767012\pi$
$312$	−20.4884	−1.15993
$313$	−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	$-0.400582\pi$
$317$	10.9419	0.614558	0.307279	+	0.951619i	$0.400582\pi$
$318$	26.3020	1.47494
$319$	31.7964	1.78026
$320$	0	0
$321$	−13.9631	−0.779343
$322$	8.36815	0.466339
$323$	0	0
$324$	−23.5920	−1.31067
$325$	0	0
$326$	1.87080	0.103614
$327$	−9.90963	−0.548004
$328$	−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
$333$	13.4315	0.736041
$334$	2.89937	0.158646
$335$	0	0
$336$	12.9606	0.707060
$337$	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
$339$	2.27416	0.123516
$340$	0	0
$341$	−52.7912	−2.85880
$342$	32.0592	1.73356
$343$	−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	$-0.248161\pi$
$347$	26.4956	1.42236	0.711179	+	0.703011i	$0.248161\pi$
$348$	−41.5579	−2.22774
$349$	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
$351$	−2.22589	−0.118809
$352$	−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	$0.598103\pi$
$354$	−69.5779	−3.69802
$355$	0	0
$356$	−29.3231	−1.55412
$357$	0	0
$358$	35.0924	1.85469
$359$	12.2135	0.644601	0.322301	−	0.946637i	$-0.395544\pi$
$359$	12.2135	0.644601	0.322301	+	0.946637i	$0.395544\pi$
$360$	0	0
$361$	1.91463	0.100770
$362$	24.1689	1.27029
$363$	−44.1746	−2.31857
$364$	39.5889	2.07502
$365$	0	0
$366$	27.1871	1.42109
$367$	−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
$369$	−1.29211	−0.0672647
$370$	0	0
$371$	−14.7227	−0.764367
$372$	68.9979	3.57738
$373$	27.3487	1.41606	0.708030	−	0.706183i	$-0.249584\pi$
$373$	27.3487	1.41606	0.708030	+	0.706183i	$0.249584\pi$
$374$	0	0
$375$	0	0
$376$	6.00884	0.309882
$377$	27.2726	1.40461
$378$	−3.24717	−0.167016
$379$	−14.9563	−0.768255	−0.384127	−	0.923280i	$-0.625498\pi$
$379$	−14.9563	−0.768255	−0.384127	+	0.923280i	$0.625498\pi$
$380$	0	0
$381$	20.9419	1.07289
$382$	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	$0.758542\pi$
$384$	32.6761	1.66750
$385$	0	0
$386$	41.3951	2.10695
$387$	18.4046	0.935561
$388$	−6.40406	−0.325117
$389$	−22.2240	−1.12680	−0.563401	−	0.826184i	$-0.690507\pi$
$389$	−22.2240	−1.12680	−0.563401	+	0.826184i	$0.690507\pi$
$390$	0	0
$391$	0	0
$392$	4.20070	0.212167
$393$	−3.35309	−0.169141
$394$	23.6454	1.19124
$395$	0	0
$396$	48.2272	2.42351
$397$	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
$399$	−34.7983	−1.74209
$400$	0	0
$401$	4.79423	0.239412	0.119706	−	0.992809i	$-0.461805\pi$
$401$	4.79423	0.239412	0.119706	+	0.992809i	$0.461805\pi$
$402$	37.2996	1.86034
$403$	−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
$411$	31.2726	1.54256
$412$	14.5206	0.715379
$413$	38.9468	1.91645
$414$	8.74373	0.429731
$415$	0	0
$416$	−33.6543	−1.65004
$417$	−24.4662	−1.19811
$418$	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
$423$	10.7236	0.521399
$424$	−8.61997	−0.418623
$425$	0	0
$426$	65.2174	3.15979
$427$	−15.2182	−0.736460
$428$	15.7966	0.763556
$429$	−61.3721	−2.96307
$430$	0	0
$431$	17.5277	0.844282	0.422141	−	0.906530i	$-0.361279\pi$
$431$	17.5277	0.844282	0.422141	+	0.906530i	$0.361279\pi$
$432$	0.824400	0.0396640
$433$	11.1755	0.537059	0.268530	−	0.963271i	$-0.413462\pi$
$433$	11.1755	0.537059	0.268530	+	0.963271i	$0.413462\pi$
$434$	−66.0558	−3.17078
$435$	0	0
$436$	11.2109	0.536902
$437$	5.70420	0.272869
$438$	−59.4654	−2.84136
$439$	12.0419	0.574727	0.287363	−	0.957822i	$-0.407221\pi$
$439$	12.0419	0.574727	0.287363	+	0.957822i	$0.407221\pi$
$440$	0	0
$441$	7.49672	0.356987
$442$	0	0
$443$	−39.5984	−1.88138	−0.940689	−	0.339269i	$-0.889820\pi$
$443$	−39.5984	−1.88138	−0.940689	+	0.339269i	$0.889820\pi$
$444$	−29.4653	−1.39836
$445$	0	0
$446$	43.9552	2.08134
$447$	22.4746	1.06301
$448$	−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
$453$	14.8729	0.698789
$454$	29.1707	1.36905
$455$	0	0
$456$	−20.3740	−0.954098
$457$	−20.2148	−0.945606	−0.472803	−	0.881168i	$-0.656758\pi$
$457$	−20.2148	−0.945606	−0.472803	+	0.881168i	$0.656758\pi$
$458$	21.9973	1.02787
$459$	0	0
$460$	0	0
$461$	2.77786	0.129378	0.0646890	−	0.997905i	$-0.479394\pi$
$461$	2.77786	0.129378	0.0646890	+	0.997905i	$0.479394\pi$
$462$	−89.5308	−4.16535
$463$	−33.8186	−1.57168	−0.785842	−	0.618427i	$-0.787770\pi$
$463$	−33.8186	−1.57168	−0.785842	+	0.618427i	$0.787770\pi$
$464$	−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	$0.900877\pi$
$468$	41.3657	1.91213
$469$	−20.8788	−0.964092
$470$	0	0
$471$	−21.6781	−0.998874
$472$	22.8028	1.04959
$473$	30.8916	1.42039
$474$	91.2869	4.19295
$475$	0	0
$476$	0	0
$477$	−15.3835	−0.704363
$478$	−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
$483$	−9.49080	−0.431846
$484$	49.9752	2.27160
$485$	0	0
$486$	48.9491	2.22038
$487$	−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
$489$	−2.12178	−0.0959502
$490$	0	0
$491$	−28.6442	−1.29269	−0.646347	−	0.763043i	$-0.723704\pi$
$491$	−28.6442	−1.29269	−0.646347	+	0.763043i	$0.723704\pi$
$492$	2.83457	0.127792
$493$	0	0
$494$	46.1544	2.07658
$495$	0	0
$496$	16.7704	0.753014
$497$	−36.5060	−1.63752
$498$	22.4723	1.00701
$499$	10.5083	0.470418	0.235209	−	0.971945i	$-0.424423\pi$
$499$	10.5083	0.470418	0.235209	+	0.971945i	$0.424423\pi$
$500$	0	0
$501$	−3.28834	−0.146912
$502$	−16.9303	−0.755638
$503$	14.7844	0.659206	0.329603	−	0.944120i	$-0.393085\pi$
$503$	14.7844	0.659206	0.329603	+	0.944120i	$0.393085\pi$
$504$	17.4815	0.778686
$505$	0	0
$506$	14.6760	0.652430
$507$	−20.2851	−0.900895
$508$	−23.6918	−1.05115
$509$	32.5112	1.44103	0.720517	−	0.693438i	$-0.243905\pi$
$509$	32.5112	1.44103	0.720517	+	0.693438i	$0.243905\pi$
$510$	0	0
$511$	33.2862	1.47250
$512$	18.5623	0.820344
$513$	−2.21345	−0.0977263
$514$	40.5211	1.78731
$515$	0	0
$516$	−40.3752	−1.77742
$517$	17.9992	0.791603
$518$	28.2089	1.23943
$519$	18.6584	0.819011
$520$	0	0
$521$	−40.4949	−1.77411	−0.887057	−	0.461660i	$-0.847254\pi$
$521$	−40.4949	−1.77411	−0.887057	+	0.461660i	$0.847254\pi$
$522$	41.5714	1.81953
$523$	19.7050	0.861640	0.430820	−	0.902438i	$-0.358224\pi$
$523$	19.7050	0.861640	0.430820	+	0.902438i	$0.358224\pi$
$524$	3.79338	0.165715
$525$	0	0
$526$	41.1779	1.79544
$527$	0	0
$528$	22.7303	0.989210
$529$	−21.4443	−0.932359
$530$	0	0
$531$	40.6948	1.76600
$532$	39.3676	1.70680
$533$	−1.86021	−0.0805745
$534$	56.8797	2.46143
$535$	0	0
$536$	−12.2242	−0.528007
$537$	−39.8003	−1.71751
$538$	−32.4446	−1.39879
$539$	12.5830	0.541988
$540$	0	0
$541$	−21.0224	−0.903824	−0.451912	−	0.892062i	$-0.649258\pi$
$541$	−21.0224	−0.903824	−0.451912	+	0.892062i	$0.649258\pi$
$542$	−53.3080	−2.28977
$543$	−27.4114	−1.17633
$544$	0	0
$545$	0	0
$546$	−76.7928	−3.28643
$547$	−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
$549$	−15.9012	−0.678647
$550$	0	0
$551$	27.1202	1.15536
$552$	−5.55674	−0.236511
$553$	−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	$0.338401\pi$
$558$	−69.0205	−2.92187
$559$	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	$0.237006\pi$
$564$	−23.5249	−0.990576
$565$	0	0
$566$	10.1483	0.426567
$567$	−25.6161	−1.07577
$568$	−21.3738	−0.896823
$569$	6.87405	0.288175	0.144088	−	0.989565i	$-0.453975\pi$
$569$	6.87405	0.288175	0.144088	+	0.989565i	$0.453975\pi$
$570$	0	0
$571$	−24.7188	−1.03445	−0.517224	−	0.855850i	$-0.673035\pi$
$571$	−24.7188	−1.03445	−0.517224	+	0.855850i	$0.673035\pi$
$572$	69.4309	2.90305
$573$	−21.1521	−0.883643
$574$	−2.71370	−0.113268
$575$	0	0
$576$	−40.4166	−1.68403
$577$	4.89908	0.203951	0.101976	−	0.994787i	$-0.467484\pi$
$577$	4.89908	0.203951	0.101976	+	0.994787i	$0.467484\pi$
$578$	0	0
$579$	−46.9485	−1.95111
$580$	0	0
$581$	−12.5791	−0.521868
$582$	12.4223	0.514922
$583$	−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	$0.335997\pi$
$588$	−16.4459	−0.678219
$589$	−45.0273	−1.85532
$590$	0	0
$591$	−26.8175	−1.10313
$592$	−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	$0.783590\pi$
$594$	−5.69487	−0.233664
$595$	0	0
$596$	−25.4257	−1.04148
$597$	−31.7300	−1.29862
$598$	12.5880	0.514762
$599$	43.9867	1.79725	0.898625	−	0.438718i	$-0.144567\pi$
$599$	43.9867	1.79725	0.898625	+	0.438718i	$0.144567\pi$
$600$	0	0
$601$	−11.4083	−0.465355	−0.232678	−	0.972554i	$-0.574749\pi$
$601$	−11.4083	−0.465355	−0.232678	+	0.972554i	$0.574749\pi$
$602$	38.6535	1.57540
$603$	−21.8158	−0.888410
$604$	−16.8258	−0.684634
$605$	0	0
$606$	−44.3747	−1.80260
$607$	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
$609$	−45.1233	−1.82849
$610$	0	0
$611$	15.4384	0.624569
$612$	0	0
$613$	−0.620807	−0.0250741	−0.0125371	−	0.999921i	$-0.503991\pi$
$613$	−0.620807	−0.0250741	−0.0125371	+	0.999921i	$0.503991\pi$
$614$	43.3681	1.75019
$615$	0	0
$616$	29.3420	1.18222
$617$	−45.0359	−1.81308	−0.906539	−	0.422123i	$-0.861285\pi$
$617$	−45.0359	−1.81308	−0.906539	+	0.422123i	$0.861285\pi$
$618$	−28.1665	−1.13302
$619$	11.4780	0.461340	0.230670	−	0.973032i	$-0.425908\pi$
$619$	11.4780	0.461340	0.230670	+	0.973032i	$0.425908\pi$
$620$	0	0
$621$	−0.603691	−0.0242253
$622$	−57.5753	−2.30856
$623$	−31.8389	−1.27560
$624$	19.4964	0.780480
$625$	0	0
$626$	−50.4342	−2.01576
$627$	−61.0292	−2.43727
$628$	24.5246	0.978639
$629$	0	0
$630$	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
$633$	65.5919	2.60704
$634$	24.0116	0.953623
$635$	0	0
$636$	33.7476	1.33818
$637$	10.7927	0.427624
$638$	69.7762	2.76247
$639$	−38.1444	−1.50897
$640$	0	0
$641$	1.51186	0.0597147	0.0298574	−	0.999554i	$-0.490495\pi$
$641$	1.51186	0.0597147	0.0298574	+	0.999554i	$0.490495\pi$
$642$	−30.6415	−1.20932
$643$	−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
$646$	0	0
$647$	35.1787	1.38302	0.691508	−	0.722369i	$-0.256947\pi$
$647$	35.1787	1.38302	0.691508	+	0.722369i	$0.256947\pi$
$648$	−14.9979	−0.589172
$649$	68.3048	2.68120
$650$	0	0
$651$	74.9176	2.93625
$652$	2.40039	0.0940065
$653$	9.99410	0.391099	0.195550	−	0.980694i	$-0.437351\pi$
$653$	9.99410	0.391099	0.195550	+	0.980694i	$0.437351\pi$
$654$	−21.7463	−0.850349
$655$	0	0
$656$	0.688962	0.0268995
$657$	34.7802	1.35690
$658$	22.5218	0.877989
$659$	−23.9165	−0.931655	−0.465827	−	0.884876i	$-0.654243\pi$
$659$	−23.9165	−0.931655	−0.465827	+	0.884876i	$0.654243\pi$
$660$	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
$666$	29.4749	1.14213
$667$	7.39670	0.286401
$668$	3.72013	0.143936
$669$	−49.8521	−1.92739
$670$	0	0
$671$	−26.6896	−1.03034
$672$	55.6819	2.14798
$673$	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	$-0.649366\pi$
$677$	−23.5326	−0.904430	−0.452215	+	0.891909i	$0.649366\pi$
$678$	4.99058	0.191662
$679$	−6.95350	−0.266851
$680$	0	0
$681$	−33.0841	−1.26779
$682$	−115.848	−4.43607
$683$	25.4677	0.974495	0.487247	−	0.873264i	$-0.338001\pi$
$683$	25.4677	0.974495	0.487247	+	0.873264i	$0.338001\pi$
$684$	41.1345	1.57282
$685$	0	0
$686$	−31.2186	−1.19193
$687$	−24.9484	−0.951840
$688$	−9.81346	−0.374135
$689$	−22.1471	−0.843736
$690$	0	0
$691$	−47.8923	−1.82191	−0.910955	−	0.412507i	$-0.864653\pi$
$691$	−47.8923	−1.82191	−0.910955	+	0.412507i	$0.864653\pi$
$692$	−21.1084	−0.802420
$693$	52.3649	1.98918
$694$	58.1437	2.20710
$695$	0	0
$696$	−26.4191	−1.00141
$697$	0	0
$698$	25.9442	0.982001
$699$	12.7460	0.482099
$700$	0	0
$701$	−5.52783	−0.208783	−0.104392	−	0.994536i	$-0.533290\pi$
$701$	−5.52783	−0.208783	−0.104392	+	0.994536i	$0.533290\pi$
$702$	−4.88464	−0.184359
$703$	19.2287	0.725226
$704$	−67.8379	−2.55674
$705$	0	0
$706$	−25.0140	−0.941414
$707$	24.8391	0.934170
$708$	−89.2741	−3.35513
$709$	−22.8895	−0.859633	−0.429817	−	0.902916i	$-0.641422\pi$
$709$	−22.8895	−0.859633	−0.429817	+	0.902916i	$0.641422\pi$
$710$	0	0
$711$	−53.3920	−2.00236
$712$	−18.6413	−0.698611
$713$	−12.2806	−0.459913
$714$	0	0
$715$	0	0
$716$	45.0264	1.68272
$717$	33.2679	1.24241
$718$	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
$723$	−10.2690	−0.381906
$724$	31.0107	1.15250
$725$	0	0
$726$	−96.9398	−3.59777
$727$	51.8312	1.92231	0.961157	−	0.276004i	$-0.0890102\pi$
$727$	51.8312	1.92231	0.961157	+	0.276004i	$0.0890102\pi$
$728$	25.1674	0.932766
$729$	−30.3796	−1.12517
$730$	0	0
$731$	0	0
$732$	34.8833	1.28932
$733$	0.440590	0.0162735	0.00813677	−	0.999967i	$-0.497410\pi$
$733$	0.440590	0.0162735	0.00813677	+	0.999967i	$0.497410\pi$
$734$	−15.1780	−0.560232
$735$	0	0
$736$	−9.12748	−0.336443
$737$	−36.6171	−1.34881
$738$	−2.83550	−0.104376
$739$	−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
$741$	−52.3463	−1.92299
$742$	−32.3086	−1.18608
$743$	15.1434	0.555557	0.277779	−	0.960645i	$-0.410402\pi$
$743$	15.1434	0.555557	0.277779	+	0.960645i	$0.410402\pi$
$744$	43.8633	1.60811
$745$	0	0
$746$	60.0157	2.19733
$747$	−13.1436	−0.480901
$748$	0	0
$749$	17.1518	0.626714
$750$	0	0
$751$	−34.1610	−1.24655	−0.623277	−	0.782001i	$-0.714199\pi$
$751$	−34.1610	−1.24655	−0.623277	+	0.782001i	$0.714199\pi$
$752$	−5.71789	−0.208510
$753$	19.2017	0.699748
$754$	59.8488	2.17957
$755$	0	0
$756$	−4.16638	−0.151530
$757$	−20.5792	−0.747965	−0.373982	−	0.927436i	$-0.622008\pi$
$757$	−20.5792	−0.747965	−0.373982	+	0.927436i	$0.622008\pi$
$758$	−32.8211	−1.19212
$759$	−16.6449	−0.604173
$760$	0	0
$761$	−18.8045	−0.681661	−0.340831	−	0.940125i	$-0.610708\pi$
$761$	−18.8045	−0.681661	−0.340831	+	0.940125i	$0.610708\pi$
$762$	45.9563	1.66482
$763$	12.1727	0.440681
$764$	23.9296	0.865743
$765$	0	0
$766$	−62.3434	−2.25256
$767$	58.5867	2.11544
$768$	8.72798	0.314944
$769$	−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
$771$	−45.9573	−1.65511
$772$	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	$0.613464\pi$
$774$	40.3884	1.45173
$775$	0	0
$776$	−4.07118	−0.146147
$777$	−31.9933	−1.14775
$778$	−48.7698	−1.74848
$779$	−1.84981	−0.0662764
$780$	0	0
$781$	−64.0240	−2.29096
$782$	0	0
$783$	−2.87020	−0.102573
$784$	−3.99730	−0.142761
$785$	0	0
$786$	−7.35823	−0.262460
$787$	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
$789$	−46.7022	−1.66264
$790$	0	0
$791$	−2.79352	−0.0993260
$792$	30.6589	1.08942
$793$	−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	$0.525733\pi$
$798$	−76.3637	−2.70325
$799$	0	0
$800$	0	0
$801$	−33.2679	−1.17546
$802$	10.5208	0.371501
$803$	58.3773	2.06009
$804$	47.8585	1.68784
$805$	0	0
$806$	−99.3662	−3.50002
$807$	36.7973	1.29532
$808$	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	$-0.320607\pi$
$811$	30.4268	1.06843	0.534215	+	0.845349i	$0.320607\pi$
$812$	51.0484	1.79145
$813$	60.4596	2.12041
$814$	49.4726	1.73402
$815$	0	0
$816$	0	0
$817$	26.3484	0.921814
$818$	−79.6227	−2.78394
$819$	44.9147	1.56945
$820$	0	0
$821$	−2.43436	−0.0849596	−0.0424798	−	0.999097i	$-0.513526\pi$
$821$	−2.43436	−0.0849596	−0.0424798	+	0.999097i	$0.513526\pi$
$822$	68.6266	2.39363
$823$	−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	$-0.600123\pi$
$827$	−17.7944	−0.618771	−0.309385	+	0.950937i	$0.600123\pi$
$828$	11.2189	0.389885
$829$	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
$831$	14.5470	0.504629
$832$	−58.1863	−2.01725
$833$	0	0
$834$	−53.6902	−1.85914
$835$	0	0
$836$	69.0429	2.38790
$837$	4.76536	0.164715
$838$	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
$843$	1.62253	0.0558828
$844$	−74.2047	−2.55423
$845$	0	0
$846$	23.5326	0.809066
$847$	54.2628	1.86449
$848$	8.20258	0.281678
$849$	−11.5098	−0.395016
$850$	0	0
$851$	5.24440	0.179776
$852$	83.6792	2.86680
$853$	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	$-0.474197\pi$
$857$	4.74086	0.161945	0.0809724	+	0.996716i	$0.474197\pi$
$858$	−134.679	−4.59787
$859$	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
$861$	3.07776	0.104890
$862$	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	$0.586691\pi$
$864$	3.54182	0.120495
$865$	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
$873$	−7.26559	−0.245903
$874$	12.5177	0.423417
$875$	0	0
$876$	−76.2989	−2.57790
$877$	−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
$879$	8.61570	0.290600
$880$	0	0
$881$	−9.67865	−0.326082	−0.163041	−	0.986619i	$-0.552130\pi$
$881$	−9.67865	−0.326082	−0.163041	+	0.986619i	$0.552130\pi$
$882$	16.4513	0.553944
$883$	−21.7255	−0.731120	−0.365560	−	0.930788i	$-0.619123\pi$
$883$	−21.7255	−0.731120	−0.365560	+	0.930788i	$0.619123\pi$
$884$	0	0
$885$	0	0
$886$	−86.8974	−2.91938
$887$	2.17244	0.0729434	0.0364717	−	0.999335i	$-0.488388\pi$
$887$	2.17244	0.0729434	0.0364717	+	0.999335i	$0.488388\pi$
$888$	−18.7317	−0.628593
$889$	−25.7244	−0.862768
$890$	0	0
$891$	−44.9254	−1.50506
$892$	56.3981	1.88835
$893$	15.3521	0.513738
$894$	49.3196	1.64950
$895$	0	0
$896$	−40.1384	−1.34093
$897$	−14.2768	−0.476688
$898$	−25.6264	−0.855165
$899$	−58.3874	−1.94733
$900$	0	0
$901$	0	0
$902$	−4.75928	−0.158467
$903$	−43.8392	−1.45888
$904$	−1.63557	−0.0543982
$905$	0	0
$906$	32.6380	1.08433
$907$	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
$909$	25.9539	0.860837
$910$	0	0
$911$	36.1769	1.19859	0.599297	−	0.800527i	$-0.295447\pi$
$911$	36.1769	1.19859	0.599297	+	0.800527i	$0.295447\pi$
$912$	19.3874	0.641982
$913$	−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
$921$	−49.1862	−1.62074
$922$	6.09592	0.200759
$923$	−54.9150	−1.80755
$924$	−114.875	−3.77912
$925$	0	0
$926$	−74.2138	−2.43882
$927$	16.4740	0.541078
$928$	−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
$933$	65.2994	2.13781
$934$	−90.2840	−2.95418
$935$	0	0
$936$	26.2969	0.859543
$937$	−15.0805	−0.492657	−0.246328	−	0.969186i	$-0.579224\pi$
$937$	−15.0805	−0.492657	−0.246328	+	0.969186i	$0.579224\pi$
$938$	−45.8177	−1.49600
$939$	57.2004	1.86666
$940$	0	0
$941$	12.2054	0.397886	0.198943	−	0.980011i	$-0.436249\pi$
$941$	12.2054	0.397886	0.198943	+	0.980011i	$0.436249\pi$
$942$	−47.5718	−1.54997
$943$	−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	$-0.624382\pi$
$947$	−23.4424	−0.761776	−0.380888	+	0.924621i	$0.624382\pi$
$948$	117.129	3.80416
$949$	50.0717	1.62540
$950$	0	0
$951$	−27.2329	−0.883088
$952$	0	0
$953$	50.7165	1.64287	0.821435	−	0.570302i	$-0.193174\pi$
$953$	50.7165	1.64287	0.821435	+	0.570302i	$0.193174\pi$
$954$	−33.7586	−1.09298
$955$	0	0
$956$	−37.6363	−1.21724
$957$	−79.1372	−2.55814
$958$	23.7600	0.767652
$959$	−38.4143	−1.24046
$960$	0	0
$961$	65.9397	2.12709
$962$	42.4339	1.36813
$963$	17.9216	0.577517
$964$	11.6174	0.374170
$965$	0	0
$966$	−20.8272	−0.670105
$967$	−41.6170	−1.33831	−0.669157	−	0.743121i	$-0.733344\pi$
$967$	−41.6170	−1.33831	−0.669157	+	0.743121i	$0.733344\pi$
$968$	31.7702	1.02113
$969$	0	0
$970$	0	0
$971$	19.2045	0.616302	0.308151	−	0.951337i	$-0.400290\pi$
$971$	19.2045	0.616302	0.308151	+	0.951337i	$0.400290\pi$
$972$	62.8057	2.01449
$973$	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	$0.390808\pi$
$978$	−4.65618	−0.148888
$979$	−55.8390	−1.78462
$980$	0	0
$981$	12.7190	0.406087
$982$	−62.8587	−2.00590
$983$	3.77964	0.120552	0.0602758	−	0.998182i	$-0.480802\pi$
$983$	3.77964	0.120552	0.0602758	+	0.998182i	$0.480802\pi$
$984$	1.80199	0.0574454
$985$	0	0
$986$	0	0
$987$	−25.5432	−0.813049
$988$	59.2198	1.88403
$989$	7.18619	0.228508
$990$	0	0
$991$	−35.7886	−1.13686	−0.568431	−	0.822731i	$-0.692449\pi$
$991$	−35.7886	−1.13686	−0.568431	+	0.822731i	$0.692449\pi$
$992$	72.0497	2.28758
$993$	−27.6356	−0.876990
$994$	−80.1111	−2.54097
$995$	0	0
$996$	28.8338	0.913635
$997$	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
$999$	−2.03503	−0.0643855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.y.1.4		5
5.4	even	2		7225.2.a.x.1.2		5
17.16	even	2		425.2.a.j.1.4	yes	5
51.50	odd	2		3825.2.a.bl.1.2		5
68.67	odd	2		6800.2.a.cd.1.1		5
85.33	odd	4		425.2.b.f.324.3		10
85.67	odd	4		425.2.b.f.324.8		10
85.84	even	2		425.2.a.i.1.2	✓	5
255.254	odd	2		3825.2.a.bq.1.4		5
340.339	odd	2		6800.2.a.bz.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.2	✓	5	85.84	even	2
425.2.a.j.1.4	yes	5	17.16	even	2
425.2.b.f.324.3		10	85.33	odd	4
425.2.b.f.324.8		10	85.67	odd	4
3825.2.a.bl.1.2		5	51.50	odd	2
3825.2.a.bq.1.4		5	255.254	odd	2
6800.2.a.bz.1.5		5	340.339	odd	2
6800.2.a.cd.1.1		5	68.67	odd	2
7225.2.a.x.1.2		5	5.4	even	2
7225.2.a.y.1.4		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.19447 q^{2} -2.48887 q^{3} +2.81568 q^{4} -5.46174 q^{6} +3.05725 q^{7} +1.78998 q^{8} +3.19447 q^{9} +5.36180 q^{11} -7.00786 q^{12} +4.59895 q^{13} +6.70903 q^{14} -1.70331 q^{16} +7.01015 q^{18} +4.57325 q^{19} -7.60910 q^{21} +11.7663 q^{22} +1.24730 q^{23} -4.45503 q^{24} +10.0922 q^{26} -0.483999 q^{27} +8.60824 q^{28} +5.93018 q^{29} -9.84580 q^{31} -7.31781 q^{32} -13.3448 q^{33} +8.99459 q^{36} +4.20461 q^{37} +10.0358 q^{38} -11.4462 q^{39} -0.404485 q^{41} -16.6979 q^{42} +5.76142 q^{43} +15.0971 q^{44} +2.73715 q^{46} +3.35693 q^{47} +4.23931 q^{48} +2.34678 q^{49} +12.9492 q^{52} -4.81568 q^{53} -1.06212 q^{54} +5.47242 q^{56} -11.3822 q^{57} +13.0136 q^{58} +12.7392 q^{59} -4.97774 q^{61} -21.6063 q^{62} +9.76628 q^{63} -12.6521 q^{64} -29.2847 q^{66} -6.82926 q^{67} -3.10436 q^{69} -11.9408 q^{71} +5.71803 q^{72} +10.8876 q^{73} +9.22687 q^{74} +12.8768 q^{76} +16.3924 q^{77} -25.1183 q^{78} -16.7139 q^{79} -8.37879 q^{81} -0.887628 q^{82} -4.11450 q^{83} -21.4248 q^{84} +12.6432 q^{86} -14.7594 q^{87} +9.59752 q^{88} -10.4142 q^{89} +14.0601 q^{91} +3.51199 q^{92} +24.5049 q^{93} +7.36667 q^{94} +18.2131 q^{96} -2.27443 q^{97} +5.14994 q^{98} +17.1281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + q^{3} + 11 q^{4} - 3 q^{6} + q^{7} + 9 q^{8} + 6 q^{9} - 4 q^{11} + 17 q^{12} + 3 q^{13} + 7 q^{14} + 27 q^{16} + 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} + 19 q^{24} - 5 q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q + q^2 + q^3 + 11 * q^4 - 3 * q^6 + q^7 + 9 * q^8 + 6 * q^9 - 4 * q^11 + 17 * q^12 + 3 * q^13 + 7 * q^14 + 27 * q^16 + 22 * q^18 + 6 * q^19 - 5 * q^21 + 18 * q^22 + 4 * q^23 + 19 * q^24 - 5 * q^26 - 5 * q^27 - 15 * q^28 - 2 * q^29 - 21 * q^31 + 9 * q^32 - 12 * q^33 + 22 * q^36 - 2 * q^37 + 34 * q^38 - 23 * q^39 + 8 * q^41 - 25 * q^42 + 4 * q^43 + 4 * q^44 + 6 * q^46 + 2 * q^47 + 51 * q^48 + 10 * q^49 + 13 * q^52 - 21 * q^53 + 25 * q^54 + 29 * q^56 - 20 * q^57 + 22 * q^58 + 12 * q^59 + 2 * q^61 + 3 * q^62 + 8 * q^63 + 43 * q^64 - 28 * q^66 + 12 * q^67 - 21 * q^71 + 58 * q^72 + 22 * q^73 + 50 * q^74 - 14 * q^76 + 32 * q^77 - 53 * q^78 - 41 * q^79 - 35 * q^81 + 28 * q^82 + 8 * q^83 - 71 * q^84 - 36 * q^86 + 4 * q^87 + 14 * q^88 - 10 * q^89 - 13 * q^91 + 56 * q^92 - 9 * q^93 - 10 * q^94 + 85 * q^96 - 20 * q^97 - 38 * q^98 + 14 * q^99

Introduction

L-functions

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Database

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