LMFDB - Embedded newform 425.2.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.39364208590$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.48887$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.19447 q^{2} -2.48887 q^{3} +2.81568 q^{4} +5.46174 q^{6} +3.05725 q^{7} -1.78998 q^{8} +3.19447 q^{9} -5.36180 q^{11} -7.00786 q^{12} -4.59895 q^{13} -6.70903 q^{14} -1.70331 q^{16} +1.00000 q^{17} -7.01015 q^{18} +4.57325 q^{19} -7.60910 q^{21} +11.7663 q^{22} +1.24730 q^{23} +4.45503 q^{24} +10.0922 q^{26} -0.483999 q^{27} +8.60824 q^{28} -5.93018 q^{29} +9.84580 q^{31} +7.31781 q^{32} +13.3448 q^{33} -2.19447 q^{34} +8.99459 q^{36} +4.20461 q^{37} -10.0358 q^{38} +11.4462 q^{39} +0.404485 q^{41} +16.6979 q^{42} -5.76142 q^{43} -15.0971 q^{44} -2.73715 q^{46} -3.35693 q^{47} +4.23931 q^{48} +2.34678 q^{49} -2.48887 q^{51} -12.9492 q^{52} +4.81568 q^{53} +1.06212 q^{54} -5.47242 q^{56} -11.3822 q^{57} +13.0136 q^{58} +12.7392 q^{59} +4.97774 q^{61} -21.6063 q^{62} +9.76628 q^{63} -12.6521 q^{64} -29.2847 q^{66} +6.82926 q^{67} +2.81568 q^{68} -3.10436 q^{69} +11.9408 q^{71} -5.71803 q^{72} +10.8876 q^{73} -9.22687 q^{74} +12.8768 q^{76} -16.3924 q^{77} -25.1183 q^{78} +16.7139 q^{79} -8.37879 q^{81} -0.887628 q^{82} +4.11450 q^{83} -21.4248 q^{84} +12.6432 q^{86} +14.7594 q^{87} +9.59752 q^{88} -10.4142 q^{89} -14.0601 q^{91} +3.51199 q^{92} -24.5049 q^{93} +7.36667 q^{94} -18.2131 q^{96} -2.27443 q^{97} -5.14994 q^{98} -17.1281 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \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 + 5 * q^17 - 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 - q^34 + 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 + q^51 - 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 + 11 * q^68 + 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	$-0.782684\pi$
$2$	−2.19447	−1.55172	−0.775861	+	0.630904i	$0.782684\pi$
$3$	−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	$-0.799624\pi$
$11$	−5.36180	−1.61664	−0.808322	+	0.588741i	$0.799624\pi$
$12$	−7.00786	−2.02299
$13$	−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
$18$	−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	$-0.685602\pi$
$29$	−5.93018	−1.10121	−0.550604	+	0.834767i	$0.685602\pi$
$30$	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
$33$	13.3448	2.32303
$34$	−2.19447	−0.376348
$35$	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	$-0.489945\pi$
$41$	0.404485	0.0631699	0.0315850	+	0.999501i	$0.489945\pi$
$42$	16.6979	2.57654
$43$	−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	$-0.578732\pi$
$47$	−3.35693	−0.489659	−0.244829	+	0.969566i	$0.578732\pi$
$48$	4.23931	0.611892
$49$	2.34678	0.335255
$50$	0	0
$51$	−2.48887	−0.348511
$52$	−12.9492	−1.79573
$53$	4.81568	0.661484	0.330742	−	0.943721i	$-0.392701\pi$
$53$	4.81568	0.661484	0.330742	+	0.943721i	$0.392701\pi$
$54$	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	$-0.396765\pi$
$61$	4.97774	0.637334	0.318667	+	0.947867i	$0.396765\pi$
$62$	−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	$-0.363024\pi$
$67$	6.82926	0.834327	0.417163	+	0.908831i	$0.363024\pi$
$68$	2.81568	0.341451
$69$	−3.10436	−0.373721
$70$	0	0
$71$	11.9408	1.41711	0.708555	−	0.705656i	$-0.249348\pi$
$71$	11.9408	1.41711	0.708555	+	0.705656i	$0.249348\pi$
$72$	−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	$-0.110609\pi$
$79$	16.7139	1.88046	0.940230	+	0.340539i	$0.110609\pi$
$80$	0	0
$81$	−8.37879	−0.930976
$82$	−0.887628	−0.0980221
$83$	4.11450	0.451625	0.225813	−	0.974171i	$-0.427496\pi$
$83$	4.11450	0.451625	0.225813	+	0.974171i	$0.427496\pi$
$84$	−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	$0.367544\pi$
$102$	5.46174	0.540792
$103$	−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	$-0.412586\pi$
$107$	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	$-0.561070\pi$
$109$	−3.98158	−0.381366	−0.190683	+	0.981652i	$0.561070\pi$
$110$	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
$119$	3.05725	0.280258
$120$	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	$-0.378219\pi$
$127$	8.41422	0.746642	0.373321	+	0.927702i	$0.378219\pi$
$128$	13.1289	1.16044
$129$	14.3394	1.26251
$130$	0	0
$131$	−1.34723	−0.117708	−0.0588542	−	0.998267i	$-0.518745\pi$
$131$	−1.34723	−0.117708	−0.0588542	+	0.998267i	$0.518745\pi$
$132$	37.5747	3.27046
$133$	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	$-0.319652\pi$
$137$	12.5650	1.07350	0.536749	+	0.843742i	$0.319652\pi$
$138$	6.81241	0.579910
$139$	−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
$141$	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
$153$	3.19447	0.258257
$154$	35.9725	2.89875
$155$	0	0
$156$	32.2288	2.58037
$157$	−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
$159$	−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	$-0.634233\pi$
$181$	−11.0136	−0.818633	−0.409317	+	0.912392i	$0.634233\pi$
$182$	30.8545	2.28709
$183$	−12.3889	−0.915816
$184$	−2.23264	−0.164592
$185$	0	0
$186$	53.7752	3.94299
$187$	−5.36180	−0.392094
$188$	−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	$-0.649242\pi$
$199$	−12.7488	−0.903735	−0.451868	+	0.892085i	$0.649242\pi$
$200$	0	0
$201$	−16.9971	−1.19889
$202$	−17.8293	−1.25446
$203$	−18.1301	−1.27248
$204$	−7.00786	−0.490648
$205$	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	$-0.138257\pi$
$211$	26.3541	1.81429	0.907146	+	0.420817i	$0.138257\pi$
$212$	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
$221$	−4.59895	−0.309359
$222$	22.9645	1.54127
$223$	−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	$-0.354575\pi$
$227$	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
$238$	−6.70903	−0.434882
$239$	−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	$-0.542425\pi$
$241$	−4.12595	−0.265776	−0.132888	+	0.991131i	$0.542425\pi$
$242$	−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	$-0.695353\pi$
$257$	−18.4651	−1.15182	−0.575912	+	0.817512i	$0.695353\pi$
$258$	−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	$-0.696374\pi$
$263$	−18.7644	−1.15707	−0.578533	+	0.815659i	$0.696374\pi$
$264$	−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	$-0.351167\pi$
$269$	14.7847	0.901441	0.450721	+	0.892665i	$0.351167\pi$
$270$	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
$273$	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
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.66075	0.331839
$292$	30.6561	1.79401
$293$	3.46169	0.202234	0.101117	−	0.994875i	$-0.467758\pi$
$293$	3.46169	0.202234	0.101117	+	0.994875i	$0.467758\pi$
$294$	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
$306$	−7.01015	−0.400743
$307$	−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
$309$	12.8352	0.730171
$310$	0	0
$311$	26.2366	1.48774	0.743871	−	0.668324i	$-0.232988\pi$
$311$	26.2366	1.48774	0.743871	+	0.668324i	$0.232988\pi$
$312$	−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
$323$	4.57325	0.254463
$324$	−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	$-0.401897\pi$
$353$	11.3987	0.606690	0.303345	+	0.952881i	$0.401897\pi$
$354$	69.5779	3.69802
$355$	0	0
$356$	−29.3231	−1.55412
$357$	−7.60910	−0.402716
$358$	−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	$-0.750416\pi$
$373$	−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
$378$	3.24717	0.167016
$379$	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
$381$	−20.9419	−1.07289
$382$	−18.6501	−0.954222
$383$	28.4094	1.45165	0.725826	−	0.687879i	$-0.241458\pi$
$383$	28.4094	1.45165	0.725826	+	0.687879i	$0.241458\pi$
$384$	−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
$391$	1.24730	0.0630785
$392$	−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	$-0.538195\pi$
$401$	−4.79423	−0.239412	−0.119706	+	0.992809i	$0.538195\pi$
$402$	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
$408$	4.45503	0.220557
$409$	−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	$-0.703686\pi$
$419$	−24.4452	−1.19423	−0.597113	+	0.802157i	$0.703686\pi$
$420$	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	$-0.638721\pi$
$431$	−17.5277	−0.844282	−0.422141	+	0.906530i	$0.638721\pi$
$432$	0.824400	0.0396640
$433$	−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
$438$	59.4654	2.84136
$439$	−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
$441$	7.49672	0.356987
$442$	10.0922	0.480039
$443$	39.5984	1.88138	0.940689	−	0.339269i	$-0.110180\pi$
$443$	39.5984	1.88138	0.940689	+	0.339269i	$0.110180\pi$
$444$	−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	$-0.411139\pi$
$449$	11.6778	0.551107	0.275554	+	0.961286i	$0.411139\pi$
$450$	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	$-0.343242\pi$
$457$	20.2148	0.945606	0.472803	+	0.881168i	$0.343242\pi$
$458$	−21.9973	−1.02787
$459$	−0.483999	−0.0225912
$460$	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	$-0.212230\pi$
$463$	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	$-0.0991228\pi$
$467$	41.1417	1.90381	0.951904	+	0.306395i	$0.0991228\pi$
$468$	−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
$476$	8.60824	0.394558
$477$	15.3835	0.704363
$478$	29.3327	1.34165
$479$	−10.8273	−0.494710	−0.247355	−	0.968925i	$-0.579561\pi$
$479$	−10.8273	−0.494710	−0.247355	+	0.968925i	$0.579561\pi$
$480$	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
$493$	−5.93018	−0.267082
$494$	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	$-0.575577\pi$
$499$	−10.5083	−0.470418	−0.235209	+	0.971945i	$0.575577\pi$
$500$	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	$-0.152746\pi$
$521$	40.4949	1.77411	0.887057	+	0.461660i	$0.152746\pi$
$522$	41.5714	1.81953
$523$	−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
$528$	−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	$-0.350742\pi$
$541$	21.0224	0.903824	0.451912	+	0.892062i	$0.350742\pi$
$542$	53.3080	2.28977
$543$	27.4114	1.17633
$544$	7.31781	0.313749
$545$	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	$-0.661599\pi$
$557$	−22.9470	−0.972297	−0.486149	+	0.873876i	$0.661599\pi$
$558$	−69.0205	−2.92187
$559$	26.4965	1.12068
$560$	0	0
$561$	13.3448	0.563418
$562$	1.43060	0.0603463
$563$	−34.8974	−1.47075	−0.735374	−	0.677661i	$-0.762994\pi$
$563$	−34.8974	−1.47075	−0.735374	+	0.677661i	$0.762994\pi$
$564$	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	$-0.326965\pi$
$571$	24.7188	1.03445	0.517224	+	0.855850i	$0.326965\pi$
$572$	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	$-0.532516\pi$
$577$	−4.89908	−0.203951	−0.101976	+	0.994787i	$0.532516\pi$
$578$	−2.19447	−0.0912777
$579$	−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	$-0.664003\pi$
$587$	−23.8761	−0.985471	−0.492736	+	0.870179i	$0.664003\pi$
$588$	−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	$-0.216410\pi$
$593$	37.8742	1.55531	0.777654	+	0.628693i	$0.216410\pi$
$594$	−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	$-0.425251\pi$
$601$	11.4083	0.465355	0.232678	+	0.972554i	$0.425251\pi$
$602$	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
$612$	8.99459	0.363585
$613$	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	$-0.861285\pi$
$617$	−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	$-0.574092\pi$
$619$	−11.4780	−0.461340	−0.230670	+	0.973032i	$0.574092\pi$
$620$	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
$629$	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
$633$	−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	$-0.509505\pi$
$641$	−1.51186	−0.0597147	−0.0298574	+	0.999554i	$0.509505\pi$
$642$	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
$646$	−10.0358	−0.394855
$647$	−35.1787	−1.38302	−0.691508	−	0.722369i	$-0.743053\pi$
$647$	−35.1787	−1.38302	−0.691508	+	0.722369i	$0.743053\pi$
$648$	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
$663$	11.4462	0.444533
$664$	−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	$-0.135347\pi$
$691$	47.8923	1.82191	0.910955	+	0.412507i	$0.135347\pi$
$692$	−21.1084	−0.802420
$693$	−52.3649	−1.98918
$694$	−58.1437	−2.20710
$695$	0	0
$696$	−26.4191	−1.00141
$697$	0.404485	0.0153210
$698$	−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	$-0.358578\pi$
$709$	22.8895	0.859633	0.429817	+	0.902916i	$0.358578\pi$
$710$	0	0
$711$	53.3920	2.00236
$712$	18.6413	0.698611
$713$	12.2806	0.459913
$714$	16.6979	0.624903
$715$	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	$-0.495555\pi$
$719$	0.748823	0.0279264	0.0139632	+	0.999903i	$0.495555\pi$
$720$	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	$-0.910990\pi$
$727$	−51.8312	−1.92231	−0.961157	+	0.276004i	$0.910990\pi$
$728$	25.1674	0.932766
$729$	−30.3796	−1.12517
$730$	0	0
$731$	−5.76142	−0.213094
$732$	−34.8833	−1.28932
$733$	−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
$738$	−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
$748$	−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
$753$	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	$-0.377992\pi$
$757$	20.5792	0.747965	0.373982	+	0.927436i	$0.377992\pi$
$758$	−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	$-0.386536\pi$
$773$	19.4040	0.697912	0.348956	+	0.937139i	$0.386536\pi$
$774$	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
$782$	−2.73715	−0.0978803
$783$	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	$-0.474267\pi$
$797$	4.55967	0.161512	0.0807559	+	0.996734i	$0.474267\pi$
$798$	76.3637	2.70325
$799$	−3.35693	−0.118760
$800$	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	$-0.274553\pi$
$809$	37.0052	1.30103	0.650516	+	0.759493i	$0.274553\pi$
$810$	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
$813$	60.4596	2.12041
$814$	49.4726	1.73402
$815$	0	0
$816$	4.23931	0.148405
$817$	−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	$-0.486474\pi$
$821$	2.43436	0.0849596	0.0424798	+	0.999097i	$0.486474\pi$
$822$	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
$833$	2.34678	0.0813113
$834$	−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	$-0.746201\pi$
$839$	−40.4717	−1.39724	−0.698618	+	0.715494i	$0.746201\pi$
$840$	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	$-0.413309\pi$
$863$	15.8044	0.537988	0.268994	+	0.963142i	$0.413309\pi$
$864$	−3.54182	−0.120495
$865$	0	0
$866$	24.5242	0.833366
$867$	−2.48887	−0.0845264
$868$	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	$-0.447870\pi$
$881$	9.67865	0.326082	0.163041	+	0.986619i	$0.447870\pi$
$882$	−16.4513	−0.553944
$883$	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	$-0.488388\pi$
$887$	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
$901$	4.81568	0.160434
$902$	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	$-0.704553\pi$
$911$	−36.1769	−1.19859	−0.599297	+	0.800527i	$0.704553\pi$
$912$	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
$918$	1.06212	0.0350552
$919$	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	$-0.395316\pi$
$929$	19.6884	0.645955	0.322978	+	0.946407i	$0.395316\pi$
$930$	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	$-0.420776\pi$
$937$	15.0805	0.492657	0.246328	+	0.969186i	$0.420776\pi$
$938$	−45.8177	−1.49600
$939$	57.2004	1.86666
$940$	0	0
$941$	−12.2054	−0.397886	−0.198943	−	0.980011i	$-0.563751\pi$
$941$	−12.2054	−0.397886	−0.198943	+	0.980011i	$0.563751\pi$
$942$	−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
$952$	−5.47242	−0.177362
$953$	−50.7165	−1.64287	−0.821435	−	0.570302i	$-0.806826\pi$
$953$	−50.7165	−1.64287	−0.821435	+	0.570302i	$0.806826\pi$
$954$	−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	$-0.266656\pi$
$967$	41.6170	1.33831	0.669157	+	0.743121i	$0.266656\pi$
$968$	−31.7702	−1.02113
$969$	−11.3822	−0.365650
$970$	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	$-0.609192\pi$
$977$	−21.0264	−0.672696	−0.336348	+	0.941738i	$0.609192\pi$
$978$	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
$986$	13.0136	0.414437
$987$	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	$-0.307551\pi$
$991$	35.7886	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$992$	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	425.2.a.i.1.2	✓	5
3.2	odd	2		3825.2.a.bq.1.4		5
4.3	odd	2		6800.2.a.bz.1.5		5
5.2	odd	4		425.2.b.f.324.3		10
5.3	odd	4		425.2.b.f.324.8		10
5.4	even	2		425.2.a.j.1.4	yes	5
15.14	odd	2		3825.2.a.bl.1.2		5
17.16	even	2		7225.2.a.x.1.2		5
20.19	odd	2		6800.2.a.cd.1.1		5
85.84	even	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	1.1	even	1	trivial
425.2.a.j.1.4	yes	5	5.4	even	2
425.2.b.f.324.3		10	5.2	odd	4
425.2.b.f.324.8		10	5.3	odd	4
3825.2.a.bl.1.2		5	15.14	odd	2
3825.2.a.bq.1.4		5	3.2	odd	2
6800.2.a.bz.1.5		5	4.3	odd	2
6800.2.a.cd.1.1		5	20.19	odd	2
7225.2.a.x.1.2		5	17.16	even	2
7225.2.a.y.1.4		5	85.84	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q-2.19447 q^{2} -2.48887 q^{3} +2.81568 q^{4} +5.46174 q^{6} +3.05725 q^{7} -1.78998 q^{8} +3.19447 q^{9} -5.36180 q^{11} -7.00786 q^{12} -4.59895 q^{13} -6.70903 q^{14} -1.70331 q^{16} +1.00000 q^{17} -7.01015 q^{18} +4.57325 q^{19} -7.60910 q^{21} +11.7663 q^{22} +1.24730 q^{23} +4.45503 q^{24} +10.0922 q^{26} -0.483999 q^{27} +8.60824 q^{28} -5.93018 q^{29} +9.84580 q^{31} +7.31781 q^{32} +13.3448 q^{33} -2.19447 q^{34} +8.99459 q^{36} +4.20461 q^{37} -10.0358 q^{38} +11.4462 q^{39} +0.404485 q^{41} +16.6979 q^{42} -5.76142 q^{43} -15.0971 q^{44} -2.73715 q^{46} -3.35693 q^{47} +4.23931 q^{48} +2.34678 q^{49} -2.48887 q^{51} -12.9492 q^{52} +4.81568 q^{53} +1.06212 q^{54} -5.47242 q^{56} -11.3822 q^{57} +13.0136 q^{58} +12.7392 q^{59} +4.97774 q^{61} -21.6063 q^{62} +9.76628 q^{63} -12.6521 q^{64} -29.2847 q^{66} +6.82926 q^{67} +2.81568 q^{68} -3.10436 q^{69} +11.9408 q^{71} -5.71803 q^{72} +10.8876 q^{73} -9.22687 q^{74} +12.8768 q^{76} -16.3924 q^{77} -25.1183 q^{78} +16.7139 q^{79} -8.37879 q^{81} -0.887628 q^{82} +4.11450 q^{83} -21.4248 q^{84} +12.6432 q^{86} +14.7594 q^{87} +9.59752 q^{88} -10.4142 q^{89} -14.0601 q^{91} +3.51199 q^{92} -24.5049 q^{93} +7.36667 q^{94} -18.2131 q^{96} -2.27443 q^{97} -5.14994 q^{98} -17.1281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 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 + 5 * q^17 - 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 - q^34 + 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 + q^51 - 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 + 11 * q^68 + 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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