LMFDB - Embedded newform 425.2.a.i.1.5

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			1.5
Root			$1.18219$ 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)

$f(q)$	$=$	\(q+2.60242 q^{2} +1.18219 q^{3} +4.77260 q^{4} +3.07656 q^{6} -3.53650 q^{7} +7.21549 q^{8} -1.60242 q^{9} +2.94609 q^{11} +5.64213 q^{12} -4.01064 q^{13} -9.20348 q^{14} +9.23255 q^{16} +1.00000 q^{17} -4.17018 q^{18} -6.97745 q^{19} -4.18083 q^{21} +7.66698 q^{22} +6.12692 q^{23} +8.53009 q^{24} -10.4374 q^{26} -5.44095 q^{27} -16.8783 q^{28} +5.30040 q^{29} +6.49485 q^{31} +9.59601 q^{32} +3.48284 q^{33} +2.60242 q^{34} -7.64773 q^{36} -3.43224 q^{37} -18.1583 q^{38} -4.74135 q^{39} +4.61307 q^{41} -10.8803 q^{42} -10.2901 q^{43} +14.0605 q^{44} +15.9448 q^{46} -3.67705 q^{47} +10.9146 q^{48} +5.50686 q^{49} +1.18219 q^{51} -19.1412 q^{52} +6.77260 q^{53} -14.1596 q^{54} -25.5176 q^{56} -8.24868 q^{57} +13.7939 q^{58} +9.92573 q^{59} -2.36438 q^{61} +16.9024 q^{62} +5.66698 q^{63} +6.50778 q^{64} +9.06383 q^{66} +9.56650 q^{67} +4.77260 q^{68} +7.24319 q^{69} +5.51248 q^{71} -11.5623 q^{72} -2.00515 q^{73} -8.93214 q^{74} -33.3006 q^{76} -10.4189 q^{77} -12.3390 q^{78} +10.5803 q^{79} -1.62497 q^{81} +12.0052 q^{82} -9.07301 q^{83} -19.9534 q^{84} -26.7792 q^{86} +6.26609 q^{87} +21.2575 q^{88} +2.63321 q^{89} +14.1837 q^{91} +29.2414 q^{92} +7.67816 q^{93} -9.56923 q^{94} +11.3443 q^{96} -5.86816 q^{97} +14.3312 q^{98} -4.72088 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.60242	1.84019	0.920095	−	0.391694i	$-0.128111\pi$
$2$	2.60242	1.84019	0.920095	+	0.391694i	$0.128111\pi$
$3$	1.18219	0.682539	0.341269	−	0.939966i	$-0.389143\pi$
$3$	1.18219	0.682539	0.341269	+	0.939966i	$0.389143\pi$
$4$	4.77260	2.38630
$5$	0	0
$5$	0	0
$6$	3.07656	1.25600
$7$	−3.53650	−1.33667	−0.668337	−	0.743859i	$-0.732993\pi$
$7$	−3.53650	−1.33667	−0.668337	+	0.743859i	$0.732993\pi$
$8$	7.21549	2.55106
$9$	−1.60242	−0.534141
$10$	0	0
$11$	2.94609	0.888280	0.444140	−	0.895957i	$-0.353509\pi$
$11$	2.94609	0.888280	0.444140	+	0.895957i	$0.353509\pi$
$12$	5.64213	1.62874
$13$	−4.01064	−1.11235	−0.556176	−	0.831064i	$-0.687732\pi$
$13$	−4.01064	−1.11235	−0.556176	+	0.831064i	$0.687732\pi$
$14$	−9.20348	−2.45973
$15$	0	0
$16$	9.23255	2.30814
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−4.17018	−0.982921
$19$	−6.97745	−1.60074	−0.800368	−	0.599508i	$-0.795363\pi$
$19$	−6.97745	−1.60074	−0.800368	+	0.599508i	$0.795363\pi$
$20$	0	0
$21$	−4.18083	−0.912331
$22$	7.66698	1.63460
$23$	6.12692	1.27755	0.638775	−	0.769393i	$-0.279441\pi$
$23$	6.12692	1.27755	0.638775	+	0.769393i	$0.279441\pi$
$24$	8.53009	1.74120
$25$	0	0
$26$	−10.4374	−2.04694
$27$	−5.44095	−1.04711
$28$	−16.8783	−3.18971
$29$	5.30040	0.984260	0.492130	−	0.870522i	$-0.336218\pi$
$29$	5.30040	0.984260	0.492130	+	0.870522i	$0.336218\pi$
$30$	0	0
$31$	6.49485	1.16651	0.583255	−	0.812289i	$-0.301779\pi$
$31$	6.49485	1.16651	0.583255	+	0.812289i	$0.301779\pi$
$32$	9.59601	1.69635
$33$	3.48284	0.606285
$34$	2.60242	0.446312
$35$	0	0
$36$	−7.64773	−1.27462
$37$	−3.43224	−0.564257	−0.282128	−	0.959377i	$-0.591040\pi$
$37$	−3.43224	−0.564257	−0.282128	+	0.959377i	$0.591040\pi$
$38$	−18.1583	−2.94566
$39$	−4.74135	−0.759224
$40$	0	0
$41$	4.61307	0.720440	0.360220	−	0.932867i	$-0.382702\pi$
$41$	4.61307	0.720440	0.360220	+	0.932867i	$0.382702\pi$
$42$	−10.8803	−1.67886
$43$	−10.2901	−1.56923	−0.784614	−	0.619985i	$-0.787139\pi$
$43$	−10.2901	−1.56923	−0.784614	+	0.619985i	$0.787139\pi$
$44$	14.0605	2.11970
$45$	0	0
$46$	15.9448	2.35094
$47$	−3.67705	−0.536352	−0.268176	−	0.963370i	$-0.586421\pi$
$47$	−3.67705	−0.536352	−0.268176	+	0.963370i	$0.586421\pi$
$48$	10.9146	1.57539
$49$	5.50686	0.786695
$50$	0	0
$51$	1.18219	0.165540
$52$	−19.1412	−2.65441
$53$	6.77260	0.930289	0.465144	−	0.885235i	$-0.346002\pi$
$53$	6.77260	0.930289	0.465144	+	0.885235i	$0.346002\pi$
$54$	−14.1596	−1.92688
$55$	0	0
$56$	−25.5176	−3.40993
$57$	−8.24868	−1.09256
$58$	13.7939	1.81123
$59$	9.92573	1.29222	0.646110	−	0.763244i	$-0.276395\pi$
$59$	9.92573	1.29222	0.646110	+	0.763244i	$0.276395\pi$
$60$	0	0
$61$	−2.36438	−0.302728	−0.151364	−	0.988478i	$-0.548367\pi$
$61$	−2.36438	−0.302728	−0.151364	+	0.988478i	$0.548367\pi$
$62$	16.9024	2.14660
$63$	5.66698	0.713972
$64$	6.50778	0.813473
$65$	0	0
$66$	9.06383	1.11568
$67$	9.56650	1.16873	0.584367	−	0.811490i	$-0.301343\pi$
$67$	9.56650	1.16873	0.584367	+	0.811490i	$0.301343\pi$
$68$	4.77260	0.578763
$69$	7.24319	0.871978
$70$	0	0
$71$	5.51248	0.654212	0.327106	−	0.944988i	$-0.393927\pi$
$71$	5.51248	0.654212	0.327106	+	0.944988i	$0.393927\pi$
$72$	−11.5623	−1.36263
$73$	−2.00515	−0.234685	−0.117343	−	0.993091i	$-0.537438\pi$
$73$	−2.00515	−0.234685	−0.117343	+	0.993091i	$0.537438\pi$
$74$	−8.93214	−1.03834
$75$	0	0
$76$	−33.3006	−3.81984
$77$	−10.4189	−1.18734
$78$	−12.3390	−1.39712
$79$	10.5803	1.19038	0.595191	−	0.803584i	$-0.297076\pi$
$79$	10.5803	1.19038	0.595191	+	0.803584i	$0.297076\pi$
$80$	0	0
$81$	−1.62497	−0.180552
$82$	12.0052	1.32575
$83$	−9.07301	−0.995892	−0.497946	−	0.867208i	$-0.665912\pi$
$83$	−9.07301	−0.995892	−0.497946	+	0.867208i	$0.665912\pi$
$84$	−19.9534	−2.17710
$85$	0	0
$86$	−26.7792	−2.88768
$87$	6.26609	0.671796
$88$	21.2575	2.26606
$89$	2.63321	0.279119	0.139560	−	0.990214i	$-0.455431\pi$
$89$	2.63321	0.279119	0.139560	+	0.990214i	$0.455431\pi$
$90$	0	0
$91$	14.1837	1.48685
$92$	29.2414	3.04862
$93$	7.67816	0.796188
$94$	−9.56923	−0.986991
$95$	0	0
$96$	11.3443	1.15783
$97$	−5.86816	−0.595822	−0.297911	−	0.954594i	$-0.596290\pi$
$97$	−5.86816	−0.595822	−0.297911	+	0.954594i	$0.596290\pi$
$98$	14.3312	1.44767
$99$	−4.72088	−0.474467
$100$	0	0
$101$	−7.90283	−0.786361	−0.393180	−	0.919461i	$-0.628625\pi$
$101$	−7.90283	−0.786361	−0.393180	+	0.919461i	$0.628625\pi$
$102$	3.07656	0.304625
$103$	6.36826	0.627483	0.313742	−	0.949508i	$-0.398417\pi$
$103$	6.36826	0.627483	0.313742	+	0.949508i	$0.398417\pi$
$104$	−28.9388	−2.83768
$105$	0	0
$106$	17.6252	1.71191
$107$	−6.85432	−0.662632	−0.331316	−	0.943520i	$-0.607493\pi$
$107$	−6.85432	−0.662632	−0.331316	+	0.943520i	$0.607493\pi$
$108$	−25.9675	−2.49872
$109$	−14.6758	−1.40569	−0.702843	−	0.711345i	$-0.748086\pi$
$109$	−14.6758	−1.40569	−0.702843	+	0.711345i	$0.748086\pi$
$110$	0	0
$111$	−4.05757	−0.385127
$112$	−32.6509	−3.08522
$113$	−13.3994	−1.26051	−0.630255	−	0.776388i	$-0.717050\pi$
$113$	−13.3994	−1.26051	−0.630255	+	0.776388i	$0.717050\pi$
$114$	−21.4666	−2.01053
$115$	0	0
$116$	25.2967	2.34874
$117$	6.42675	0.594153
$118$	25.8309	2.37793
$119$	−3.53650	−0.324191
$120$	0	0
$121$	−2.32055	−0.210959
$122$	−6.15313	−0.557078
$123$	5.45353	0.491728
$124$	30.9974	2.78365
$125$	0	0
$126$	14.7479	1.31384
$127$	−4.63321	−0.411131	−0.205565	−	0.978643i	$-0.565903\pi$
$127$	−4.63321	−0.411131	−0.205565	+	0.978643i	$0.565903\pi$
$128$	−2.25602	−0.199405
$129$	−12.1649	−1.07106
$130$	0	0
$131$	−12.1496	−1.06151	−0.530757	−	0.847524i	$-0.678092\pi$
$131$	−12.1496	−1.06151	−0.530757	+	0.847524i	$0.678092\pi$
$132$	16.6222	1.44678
$133$	24.6758	2.13966
$134$	24.8961	2.15069
$135$	0	0
$136$	7.21549	0.618723
$137$	8.86852	0.757689	0.378844	−	0.925460i	$-0.376322\pi$
$137$	8.86852	0.757689	0.378844	+	0.925460i	$0.376322\pi$
$138$	18.8498	1.60461
$139$	7.32306	0.621134	0.310567	−	0.950552i	$-0.399481\pi$
$139$	7.32306	0.621134	0.310567	+	0.950552i	$0.399481\pi$
$140$	0	0
$141$	−4.34697	−0.366081
$142$	14.3458	1.20387
$143$	−11.8157	−0.988081
$144$	−14.7944	−1.23287
$145$	0	0
$146$	−5.21825	−0.431866
$147$	6.51017	0.536950
$148$	−16.3807	−1.34649
$149$	13.9059	1.13922	0.569608	−	0.821916i	$-0.307095\pi$
$149$	13.9059	1.13922	0.569608	+	0.821916i	$0.307095\pi$
$150$	0	0
$151$	14.3884	1.17091	0.585456	−	0.810704i	$-0.300916\pi$
$151$	14.3884	1.17091	0.585456	+	0.810704i	$0.300916\pi$
$152$	−50.3457	−4.08358
$153$	−1.60242	−0.129548
$154$	−27.1143	−2.18493
$155$	0	0
$156$	−22.6286	−1.81174
$157$	8.68608	0.693224	0.346612	−	0.938009i	$-0.387332\pi$
$157$	8.68608	0.693224	0.346612	+	0.938009i	$0.387332\pi$
$158$	27.5345	2.19053
$159$	8.00652	0.634958
$160$	0	0
$161$	−21.6679	−1.70767
$162$	−4.22887	−0.332251
$163$	−8.95868	−0.701698	−0.350849	−	0.936432i	$-0.614107\pi$
$163$	−8.95868	−0.701698	−0.350849	+	0.936432i	$0.614107\pi$
$164$	22.0163	1.71919
$165$	0	0
$166$	−23.6118	−1.83263
$167$	4.37318	0.338407	0.169203	−	0.985581i	$-0.445881\pi$
$167$	4.37318	0.338407	0.169203	+	0.985581i	$0.445881\pi$
$168$	−30.1667	−2.32741
$169$	3.08527	0.237328
$170$	0	0
$171$	11.1808	0.855019
$172$	−49.1106	−3.74465
$173$	8.82433	0.670901	0.335451	−	0.942058i	$-0.391111\pi$
$173$	8.82433	0.670901	0.335451	+	0.942058i	$0.391111\pi$
$174$	16.3070	1.23623
$175$	0	0
$176$	27.1999	2.05027
$177$	11.7341	0.881990
$178$	6.85272	0.513633
$179$	9.42951	0.704795	0.352397	−	0.935850i	$-0.385367\pi$
$179$	9.42951	0.704795	0.352397	+	0.935850i	$0.385367\pi$
$180$	0	0
$181$	−11.7939	−0.876633	−0.438317	−	0.898821i	$-0.644425\pi$
$181$	−11.7939	−0.876633	−0.438317	+	0.898821i	$0.644425\pi$
$182$	36.9119	2.73609
$183$	−2.79515	−0.206624
$184$	44.2087	3.25911
$185$	0	0
$186$	19.9818	1.46514
$187$	2.94609	0.215440
$188$	−17.5491	−1.27990
$189$	19.2419	1.39964
$190$	0	0
$191$	5.19969	0.376237	0.188118	−	0.982146i	$-0.439761\pi$
$191$	5.19969	0.376237	0.188118	+	0.982146i	$0.439761\pi$
$192$	7.69345	0.555227
$193$	−14.3936	−1.03607	−0.518035	−	0.855359i	$-0.673336\pi$
$193$	−14.3936	−1.03607	−0.518035	+	0.855359i	$0.673336\pi$
$194$	−15.2714	−1.09643
$195$	0	0
$196$	26.2821	1.87729
$197$	16.0840	1.14594	0.572969	−	0.819577i	$-0.305792\pi$
$197$	16.0840	1.14594	0.572969	+	0.819577i	$0.305792\pi$
$198$	−12.2857	−0.873109
$199$	18.1750	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$199$	18.1750	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$200$	0	0
$201$	11.3094	0.797706
$202$	−20.5665	−1.44705
$203$	−18.7449	−1.31563
$204$	5.64213	0.395028
$205$	0	0
$206$	16.5729	1.15469
$207$	−9.81791	−0.682392
$208$	−37.0285	−2.56746
$209$	−20.5562	−1.42190
$210$	0	0
$211$	−8.40614	−0.578702	−0.289351	−	0.957223i	$-0.593440\pi$
$211$	−8.40614	−0.578702	−0.289351	+	0.957223i	$0.593440\pi$
$212$	32.3230	2.21995
$213$	6.51681	0.446525
$214$	−17.8378	−1.21937
$215$	0	0
$216$	−39.2591	−2.67124
$217$	−22.9691	−1.55924
$218$	−38.1926	−2.58673
$219$	−2.37047	−0.160182
$220$	0	0
$221$	−4.01064	−0.269785
$222$	−10.5595	−0.708708
$223$	2.90591	0.194594	0.0972971	−	0.995255i	$-0.468980\pi$
$223$	2.90591	0.194594	0.0972971	+	0.995255i	$0.468980\pi$
$224$	−33.9363	−2.26747
$225$	0	0
$226$	−34.8709	−2.31958
$227$	15.8127	1.04952	0.524762	−	0.851249i	$-0.324154\pi$
$227$	15.8127	1.04952	0.524762	+	0.851249i	$0.324154\pi$
$228$	−39.3677	−2.60719
$229$	−23.1302	−1.52849	−0.764244	−	0.644927i	$-0.776888\pi$
$229$	−23.1302	−1.52849	−0.764244	+	0.644927i	$0.776888\pi$
$230$	0	0
$231$	−12.3171	−0.810405
$232$	38.2450	2.51091
$233$	14.5265	0.951665	0.475833	−	0.879536i	$-0.342147\pi$
$233$	14.5265	0.951665	0.475833	+	0.879536i	$0.342147\pi$
$234$	16.7251	1.09336
$235$	0	0
$236$	47.3716	3.08363
$237$	12.5080	0.812481
$238$	−9.20348	−0.596573
$239$	3.56923	0.230874	0.115437	−	0.993315i	$-0.463173\pi$
$239$	3.56923	0.230874	0.115437	+	0.993315i	$0.463173\pi$
$240$	0	0
$241$	−17.7990	−1.14654	−0.573269	−	0.819367i	$-0.694325\pi$
$241$	−17.7990	−1.14654	−0.573269	+	0.819367i	$0.694325\pi$
$242$	−6.03904	−0.388204
$243$	14.4018	0.923876
$244$	−11.2843	−0.722401
$245$	0	0
$246$	14.1924	0.904874
$247$	27.9841	1.78058
$248$	46.8636	2.97584
$249$	−10.7260	−0.679735
$250$	0	0
$251$	7.45480	0.470543	0.235271	−	0.971930i	$-0.424402\pi$
$251$	7.45480	0.470543	0.235271	+	0.971930i	$0.424402\pi$
$252$	27.0462	1.70375
$253$	18.0505	1.13482
$254$	−12.0576	−0.756559
$255$	0	0
$256$	−18.8867	−1.18042
$257$	−26.4740	−1.65140	−0.825702	−	0.564106i	$-0.809221\pi$
$257$	−26.4740	−1.65140	−0.825702	+	0.564106i	$0.809221\pi$
$258$	−31.6582	−1.97095
$259$	12.1381	0.754227
$260$	0	0
$261$	−8.49349	−0.525734
$262$	−31.6183	−1.95339
$263$	−12.4974	−0.770621	−0.385310	−	0.922787i	$-0.625906\pi$
$263$	−12.4974	−0.770621	−0.385310	+	0.922787i	$0.625906\pi$
$264$	25.1304	1.54667
$265$	0	0
$266$	64.2168	3.93739
$267$	3.11296	0.190510
$268$	45.6571	2.78895
$269$	2.83773	0.173020	0.0865098	−	0.996251i	$-0.472429\pi$
$269$	2.83773	0.173020	0.0865098	+	0.996251i	$0.472429\pi$
$270$	0	0
$271$	−7.60005	−0.461670	−0.230835	−	0.972993i	$-0.574146\pi$
$271$	−7.60005	−0.461670	−0.230835	+	0.972993i	$0.574146\pi$
$272$	9.23255	0.559805
$273$	16.7678	1.01483
$274$	23.0796	1.39429
$275$	0	0
$276$	34.5689	2.08080
$277$	30.4187	1.82768	0.913842	−	0.406070i	$-0.133101\pi$
$277$	30.4187	1.82768	0.913842	+	0.406070i	$0.133101\pi$
$278$	19.0577	1.14300
$279$	−10.4075	−0.623081
$280$	0	0
$281$	−20.5944	−1.22856	−0.614279	−	0.789089i	$-0.710553\pi$
$281$	−20.5944	−1.22856	−0.614279	+	0.789089i	$0.710553\pi$
$282$	−11.3127	−0.673659
$283$	4.14433	0.246355	0.123177	−	0.992385i	$-0.460692\pi$
$283$	4.14433	0.246355	0.123177	+	0.992385i	$0.460692\pi$
$284$	26.3089	1.56115
$285$	0	0
$286$	−30.7495	−1.81826
$287$	−16.3141	−0.962993
$288$	−15.3769	−0.906091
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.93729	−0.406671
$292$	−9.56980	−0.560030
$293$	7.85031	0.458620	0.229310	−	0.973353i	$-0.426353\pi$
$293$	7.85031	0.458620	0.229310	+	0.973353i	$0.426353\pi$
$294$	16.9422	0.988090
$295$	0	0
$296$	−24.7653	−1.43945
$297$	−16.0295	−0.930127
$298$	36.1891	2.09638
$299$	−24.5729	−1.42109
$300$	0	0
$301$	36.3910	2.09754
$302$	37.4447	2.15470
$303$	−9.34266	−0.536722
$304$	−64.4196	−3.69472
$305$	0	0
$306$	−4.17018	−0.238393
$307$	−0.473348	−0.0270154	−0.0135077	−	0.999909i	$-0.504300\pi$
$307$	−0.473348	−0.0270154	−0.0135077	+	0.999909i	$0.504300\pi$
$308$	−49.7251	−2.83335
$309$	7.52851	0.428282
$310$	0	0
$311$	18.3062	1.03805	0.519023	−	0.854760i	$-0.326296\pi$
$311$	18.3062	1.03805	0.519023	+	0.854760i	$0.326296\pi$
$312$	−34.2112	−1.93683
$313$	3.84193	0.217159	0.108579	−	0.994088i	$-0.465370\pi$
$313$	3.84193	0.217159	0.108579	+	0.994088i	$0.465370\pi$
$314$	22.6048	1.27567
$315$	0	0
$316$	50.4958	2.84061
$317$	−4.52266	−0.254018	−0.127009	−	0.991902i	$-0.540538\pi$
$317$	−4.52266	−0.254018	−0.127009	+	0.991902i	$0.540538\pi$
$318$	20.8363	1.16844
$319$	15.6155	0.874299
$320$	0	0
$321$	−8.10312	−0.452272
$322$	−56.3890	−3.14243
$323$	−6.97745	−0.388236
$324$	−7.75535	−0.430853
$325$	0	0
$326$	−23.3143	−1.29126
$327$	−17.3496	−0.959435
$328$	33.2855	1.83789
$329$	13.0039	0.716928
$330$	0	0
$331$	17.4347	0.958296	0.479148	−	0.877734i	$-0.340946\pi$
$331$	17.4347	0.958296	0.479148	+	0.877734i	$0.340946\pi$
$332$	−43.3019	−2.37650
$333$	5.49990	0.301393
$334$	11.3809	0.622733
$335$	0	0
$336$	−38.5997	−2.10578
$337$	−0.943903	−0.0514177	−0.0257088	−	0.999669i	$-0.508184\pi$
$337$	−0.943903	−0.0514177	−0.0257088	+	0.999669i	$0.508184\pi$
$338$	8.02917	0.436729
$339$	−15.8407	−0.860347
$340$	0	0
$341$	19.1344	1.03619
$342$	29.0972	1.57340
$343$	5.28048	0.285119
$344$	−74.2482	−4.00320
$345$	0	0
$346$	22.9646	1.23459
$347$	19.2108	1.03129	0.515645	−	0.856802i	$-0.327552\pi$
$347$	19.2108	1.03129	0.515645	+	0.856802i	$0.327552\pi$
$348$	29.9056	1.60311
$349$	−31.7831	−1.70131	−0.850655	−	0.525724i	$-0.823795\pi$
$349$	−31.7831	−1.70131	−0.850655	+	0.525724i	$0.823795\pi$
$350$	0	0
$351$	21.8217	1.16476
$352$	28.2707	1.50683
$353$	−15.4511	−0.822380	−0.411190	−	0.911550i	$-0.634887\pi$
$353$	−15.4511	−0.822380	−0.411190	+	0.911550i	$0.634887\pi$
$354$	30.5371	1.62303
$355$	0	0
$356$	12.5673	0.666064
$357$	−4.18083	−0.221273
$358$	24.5396	1.29696
$359$	−27.9639	−1.47588	−0.737940	−	0.674866i	$-0.764201\pi$
$359$	−27.9639	−1.47588	−0.737940	+	0.674866i	$0.764201\pi$
$360$	0	0
$361$	29.6848	1.56236
$362$	−30.6927	−1.61317
$363$	−2.74333	−0.143987
$364$	67.6930	3.54808
$365$	0	0
$366$	−7.27417	−0.380227
$367$	22.7225	1.18610	0.593051	−	0.805165i	$-0.297923\pi$
$367$	22.7225	1.18610	0.593051	+	0.805165i	$0.297923\pi$
$368$	56.5671	2.94876
$369$	−7.39208	−0.384817
$370$	0	0
$371$	−23.9513	−1.24349
$372$	36.6448	1.89995
$373$	−35.5230	−1.83931	−0.919656	−	0.392725i	$-0.871532\pi$
$373$	−35.5230	−1.83931	−0.919656	+	0.392725i	$0.871532\pi$
$374$	7.66698	0.396450
$375$	0	0
$376$	−26.5317	−1.36827
$377$	−21.2580	−1.09484
$378$	50.0756	2.57561
$379$	−30.4727	−1.56528	−0.782640	−	0.622475i	$-0.786127\pi$
$379$	−30.4727	−1.56528	−0.782640	+	0.622475i	$0.786127\pi$
$380$	0	0
$381$	−5.47734	−0.280613
$382$	13.5318	0.692347
$383$	−25.9667	−1.32683	−0.663417	−	0.748250i	$-0.730895\pi$
$383$	−25.9667	−1.32683	−0.663417	+	0.748250i	$0.730895\pi$
$384$	−2.66704	−0.136102
$385$	0	0
$386$	−37.4581	−1.90657
$387$	16.4891	0.838189
$388$	−28.0064	−1.42181
$389$	6.37729	0.323342	0.161671	−	0.986845i	$-0.448312\pi$
$389$	6.37729	0.323342	0.161671	+	0.986845i	$0.448312\pi$
$390$	0	0
$391$	6.12692	0.309852
$392$	39.7347	2.00691
$393$	−14.3631	−0.724524
$394$	41.8574	2.10874
$395$	0	0
$396$	−22.5309	−1.13222
$397$	−13.1874	−0.661858	−0.330929	−	0.943656i	$-0.607362\pi$
$397$	−13.1874	−0.661858	−0.330929	+	0.943656i	$0.607362\pi$
$398$	47.2989	2.37088
$399$	29.1715	1.46040
$400$	0	0
$401$	−28.2411	−1.41030	−0.705148	−	0.709061i	$-0.749119\pi$
$401$	−28.2411	−1.41030	−0.705148	+	0.709061i	$0.749119\pi$
$402$	29.4319	1.46793
$403$	−26.0486	−1.29757
$404$	−37.7171	−1.87649
$405$	0	0
$406$	−48.7822	−2.42102
$407$	−10.1117	−0.501218
$408$	8.53009	0.422302
$409$	−21.0374	−1.04023	−0.520117	−	0.854095i	$-0.674112\pi$
$409$	−21.0374	−1.04023	−0.520117	+	0.854095i	$0.674112\pi$
$410$	0	0
$411$	10.4843	0.517152
$412$	30.3932	1.49737
$413$	−35.1024	−1.72728
$414$	−25.5504	−1.25573
$415$	0	0
$416$	−38.4862	−1.88694
$417$	8.65726	0.423948
$418$	−53.4959	−2.61657
$419$	−28.1482	−1.37513	−0.687565	−	0.726123i	$-0.741320\pi$
$419$	−28.1482	−1.37513	−0.687565	+	0.726123i	$0.741320\pi$
$420$	0	0
$421$	−16.6639	−0.812147	−0.406074	−	0.913840i	$-0.633102\pi$
$421$	−16.6639	−0.812147	−0.406074	+	0.913840i	$0.633102\pi$
$422$	−21.8763	−1.06492
$423$	5.89218	0.286488
$424$	48.8677	2.37322
$425$	0	0
$426$	16.9595	0.821691
$427$	8.36165	0.404649
$428$	−32.7130	−1.58124
$429$	−13.9685	−0.674403
$430$	0	0
$431$	−11.1833	−0.538682	−0.269341	−	0.963045i	$-0.586806\pi$
$431$	−11.1833	−0.538682	−0.269341	+	0.963045i	$0.586806\pi$
$432$	−50.2338	−2.41687
$433$	11.0440	0.530743	0.265372	−	0.964146i	$-0.414505\pi$
$433$	11.0440	0.530743	0.265372	+	0.964146i	$0.414505\pi$
$434$	−59.7753	−2.86930
$435$	0	0
$436$	−70.0417	−3.35439
$437$	−42.7503	−2.04502
$438$	−6.16897	−0.294765
$439$	−5.34654	−0.255176	−0.127588	−	0.991827i	$-0.540724\pi$
$439$	−5.34654	−0.255176	−0.127588	+	0.991827i	$0.540724\pi$
$440$	0	0
$441$	−8.82433	−0.420206
$442$	−10.4374	−0.496456
$443$	16.5863	0.788040	0.394020	−	0.919102i	$-0.371084\pi$
$443$	16.5863	0.788040	0.394020	+	0.919102i	$0.371084\pi$
$444$	−19.3652	−0.919030
$445$	0	0
$446$	7.56241	0.358091
$447$	16.4395	0.777559
$448$	−23.0148	−1.08735
$449$	35.2901	1.66544	0.832722	−	0.553691i	$-0.186781\pi$
$449$	35.2901	1.66544	0.832722	+	0.553691i	$0.186781\pi$
$450$	0	0
$451$	13.5905	0.639952
$452$	−63.9500	−3.00796
$453$	17.0098	0.799192
$454$	41.1513	1.93132
$455$	0	0
$456$	−59.5183	−2.78720
$457$	9.73794	0.455522	0.227761	−	0.973717i	$-0.426860\pi$
$457$	9.73794	0.455522	0.227761	+	0.973717i	$0.426860\pi$
$458$	−60.1947	−2.81271
$459$	−5.44095	−0.253962
$460$	0	0
$461$	−16.4097	−0.764276	−0.382138	−	0.924105i	$-0.624812\pi$
$461$	−16.4097	−0.764276	−0.382138	+	0.924105i	$0.624812\pi$
$462$	−32.0543	−1.49130
$463$	16.9720	0.788755	0.394378	−	0.918948i	$-0.370960\pi$
$463$	16.9720	0.788755	0.394378	+	0.918948i	$0.370960\pi$
$464$	48.9362	2.27181
$465$	0	0
$466$	37.8042	1.75125
$467$	−23.2884	−1.07766	−0.538828	−	0.842416i	$-0.681133\pi$
$467$	−23.2884	−1.07766	−0.538828	+	0.842416i	$0.681133\pi$
$468$	30.6723	1.41783
$469$	−33.8320	−1.56221
$470$	0	0
$471$	10.2686	0.473152
$472$	71.6190	3.29653
$473$	−30.3156	−1.39391
$474$	32.5511	1.49512
$475$	0	0
$476$	−16.8783	−0.773617
$477$	−10.8526	−0.496905
$478$	9.28864	0.424853
$479$	2.30406	0.105275	0.0526376	−	0.998614i	$-0.483237\pi$
$479$	2.30406	0.105275	0.0526376	+	0.998614i	$0.483237\pi$
$480$	0	0
$481$	13.7655	0.627653
$482$	−46.3206	−2.10985
$483$	−25.6156	−1.16555
$484$	−11.0750	−0.503411
$485$	0	0
$486$	37.4796	1.70011
$487$	−14.0889	−0.638430	−0.319215	−	0.947682i	$-0.603419\pi$
$487$	−14.0889	−0.638430	−0.319215	+	0.947682i	$0.603419\pi$
$488$	−17.0602	−0.772278
$489$	−10.5909	−0.478936
$490$	0	0
$491$	21.0485	0.949905	0.474953	−	0.880011i	$-0.342465\pi$
$491$	21.0485	0.949905	0.474953	+	0.880011i	$0.342465\pi$
$492$	26.0275	1.17341
$493$	5.30040	0.238718
$494$	72.8264	3.27661
$495$	0	0
$496$	59.9640	2.69247
$497$	−19.4949	−0.874467
$498$	−27.9137	−1.25084
$499$	27.6747	1.23889	0.619446	−	0.785039i	$-0.287357\pi$
$499$	27.6747	1.23889	0.619446	+	0.785039i	$0.287357\pi$
$500$	0	0
$501$	5.16994	0.230976
$502$	19.4005	0.865889
$503$	31.0855	1.38603	0.693017	−	0.720921i	$-0.256281\pi$
$503$	31.0855	1.38603	0.693017	+	0.720921i	$0.256281\pi$
$504$	40.8900	1.82139
$505$	0	0
$506$	46.9749	2.08829
$507$	3.64738	0.161986
$508$	−22.1125	−0.981082
$509$	−20.5481	−0.910781	−0.455390	−	0.890292i	$-0.650500\pi$
$509$	−20.5481	−0.910781	−0.455390	+	0.890292i	$0.650500\pi$
$510$	0	0
$511$	7.09123	0.313697
$512$	−44.6391	−1.97279
$513$	37.9639	1.67615
$514$	−68.8966	−3.03890
$515$	0	0
$516$	−58.0582	−2.55587
$517$	−10.8329	−0.476431
$518$	31.5886	1.38792
$519$	10.4320	0.457916
$520$	0	0
$521$	−27.9505	−1.22453	−0.612267	−	0.790651i	$-0.709742\pi$
$521$	−27.9505	−1.22453	−0.612267	+	0.790651i	$0.709742\pi$
$522$	−22.1037	−0.967450
$523$	−0.0826499	−0.00361403	−0.00180701	−	0.999998i	$-0.500575\pi$
$523$	−0.0826499	−0.00361403	−0.00180701	+	0.999998i	$0.500575\pi$
$524$	−57.9851	−2.53309
$525$	0	0
$526$	−32.5234	−1.41809
$527$	6.49485	0.282920
$528$	32.1555	1.39939
$529$	14.5391	0.632136
$530$	0	0
$531$	−15.9052	−0.690228
$532$	117.768	5.10588
$533$	−18.5014	−0.801383
$534$	8.10123	0.350574
$535$	0	0
$536$	69.0270	2.98151
$537$	11.1475	0.481050
$538$	7.38498	0.318389
$539$	16.2237	0.698805
$540$	0	0
$541$	−10.7378	−0.461654	−0.230827	−	0.972995i	$-0.574143\pi$
$541$	−10.7378	−0.461654	−0.230827	+	0.972995i	$0.574143\pi$
$542$	−19.7785	−0.849561
$543$	−13.9426	−0.598336
$544$	9.59601	0.411426
$545$	0	0
$546$	43.6369	1.86749
$547$	36.8538	1.57576	0.787878	−	0.615832i	$-0.211180\pi$
$547$	36.8538	1.57576	0.787878	+	0.615832i	$0.211180\pi$
$548$	42.3259	1.80807
$549$	3.78874	0.161700
$550$	0	0
$551$	−36.9833	−1.57554
$552$	52.2632	2.22447
$553$	−37.4174	−1.59115
$554$	79.1624	3.36329
$555$	0	0
$556$	34.9501	1.48221
$557$	−2.41625	−0.102380	−0.0511899	−	0.998689i	$-0.516301\pi$
$557$	−2.41625	−0.102380	−0.0511899	+	0.998689i	$0.516301\pi$
$558$	−27.0847	−1.14659
$559$	41.2700	1.74553
$560$	0	0
$561$	3.48284	0.147046
$562$	−53.5953	−2.26078
$563$	−4.74857	−0.200129	−0.100064	−	0.994981i	$-0.531905\pi$
$563$	−4.74857	−0.200129	−0.100064	+	0.994981i	$0.531905\pi$
$564$	−20.7464	−0.873580
$565$	0	0
$566$	10.7853	0.453340
$567$	5.74672	0.241340
$568$	39.7753	1.66893
$569$	46.0041	1.92859	0.964296	−	0.264827i	$-0.0853150\pi$
$569$	46.0041	1.92859	0.964296	+	0.264827i	$0.0853150\pi$
$570$	0	0
$571$	9.95720	0.416696	0.208348	−	0.978055i	$-0.433191\pi$
$571$	9.95720	0.416696	0.208348	+	0.978055i	$0.433191\pi$
$572$	−56.3918	−2.35786
$573$	6.14704	0.256796
$574$	−42.4563	−1.77209
$575$	0	0
$576$	−10.4282	−0.434509
$577$	−18.8669	−0.785439	−0.392720	−	0.919658i	$-0.628466\pi$
$577$	−18.8669	−0.785439	−0.392720	+	0.919658i	$0.628466\pi$
$578$	2.60242	0.108247
$579$	−17.0159	−0.707158
$580$	0	0
$581$	32.0867	1.33118
$582$	−18.0538	−0.748353
$583$	19.9527	0.826357
$584$	−14.4682	−0.598696
$585$	0	0
$586$	20.4298	0.843949
$587$	23.8851	0.985843	0.492921	−	0.870074i	$-0.335929\pi$
$587$	23.8851	0.985843	0.492921	+	0.870074i	$0.335929\pi$
$588$	31.0705	1.28132
$589$	−45.3175	−1.86728
$590$	0	0
$591$	19.0144	0.782147
$592$	−31.6883	−1.30238
$593$	37.9019	1.55644	0.778222	−	0.627989i	$-0.216122\pi$
$593$	37.9019	1.55644	0.778222	+	0.627989i	$0.216122\pi$
$594$	−41.7156	−1.71161
$595$	0	0
$596$	66.3674	2.71852
$597$	21.4863	0.879375
$598$	−63.9490	−2.61507
$599$	17.8049	0.727488	0.363744	−	0.931499i	$-0.381498\pi$
$599$	17.8049	0.727488	0.363744	+	0.931499i	$0.381498\pi$
$600$	0	0
$601$	33.6532	1.37274	0.686372	−	0.727251i	$-0.259202\pi$
$601$	33.6532	1.37274	0.686372	+	0.727251i	$0.259202\pi$
$602$	94.7049	3.85988
$603$	−15.3296	−0.624269
$604$	68.6702	2.79415
$605$	0	0
$606$	−24.3135	−0.987670
$607$	−5.89699	−0.239351	−0.119676	−	0.992813i	$-0.538185\pi$
$607$	−5.89699	−0.239351	−0.119676	+	0.992813i	$0.538185\pi$
$608$	−66.9557	−2.71541
$609$	−22.1601	−0.897971
$610$	0	0
$611$	14.7473	0.596613
$612$	−7.64773	−0.309141
$613$	−7.04143	−0.284401	−0.142200	−	0.989838i	$-0.545418\pi$
$613$	−7.04143	−0.284401	−0.142200	+	0.989838i	$0.545418\pi$
$614$	−1.23185	−0.0497135
$615$	0	0
$616$	−75.1772	−3.02898
$617$	−6.04823	−0.243492	−0.121746	−	0.992561i	$-0.538849\pi$
$617$	−6.04823	−0.243492	−0.121746	+	0.992561i	$0.538849\pi$
$618$	19.5924	0.788120
$619$	−34.0992	−1.37056	−0.685282	−	0.728278i	$-0.740321\pi$
$619$	−34.0992	−1.37056	−0.685282	+	0.728278i	$0.740321\pi$
$620$	0	0
$621$	−33.3362	−1.33774
$622$	47.6404	1.91020
$623$	−9.31235	−0.373092
$624$	−43.7747	−1.75239
$625$	0	0
$626$	9.99833	0.399614
$627$	−24.3014	−0.970504
$628$	41.4552	1.65424
$629$	−3.43224	−0.136852
$630$	0	0
$631$	18.9841	0.755743	0.377872	−	0.925858i	$-0.376656\pi$
$631$	18.9841	0.755743	0.377872	+	0.925858i	$0.376656\pi$
$632$	76.3424	3.03674
$633$	−9.93767	−0.394987
$634$	−11.7699	−0.467441
$635$	0	0
$636$	38.2119	1.51520
$637$	−22.0861	−0.875082
$638$	40.6381	1.60888
$639$	−8.83333	−0.349441
$640$	0	0
$641$	−16.3869	−0.647245	−0.323623	−	0.946186i	$-0.604901\pi$
$641$	−16.3869	−0.647245	−0.323623	+	0.946186i	$0.604901\pi$
$642$	−21.0877	−0.832267
$643$	30.8332	1.21594	0.607972	−	0.793958i	$-0.291983\pi$
$643$	30.8332	1.21594	0.607972	+	0.793958i	$0.291983\pi$
$644$	−103.412	−4.07501
$645$	0	0
$646$	−18.1583	−0.714428
$647$	−15.8642	−0.623684	−0.311842	−	0.950134i	$-0.600946\pi$
$647$	−15.8642	−0.623684	−0.311842	+	0.950134i	$0.600946\pi$
$648$	−11.7250	−0.460600
$649$	29.2421	1.14785
$650$	0	0
$651$	−27.1539	−1.06424
$652$	−42.7562	−1.67446
$653$	−7.51669	−0.294151	−0.147075	−	0.989125i	$-0.546986\pi$
$653$	−7.51669	−0.294151	−0.147075	+	0.989125i	$0.546986\pi$
$654$	−45.1510	−1.76554
$655$	0	0
$656$	42.5904	1.66287
$657$	3.21310	0.125355
$658$	33.8416	1.31928
$659$	−3.89783	−0.151838	−0.0759190	−	0.997114i	$-0.524189\pi$
$659$	−3.89783	−0.151838	−0.0759190	+	0.997114i	$0.524189\pi$
$660$	0	0
$661$	−16.6761	−0.648625	−0.324313	−	0.945950i	$-0.605133\pi$
$661$	−16.6761	−0.648625	−0.324313	+	0.945950i	$0.605133\pi$
$662$	45.3724	1.76345
$663$	−4.74135	−0.184139
$664$	−65.4662	−2.54058
$665$	0	0
$666$	14.3131	0.554620
$667$	32.4751	1.25744
$668$	20.8715	0.807541
$669$	3.43534	0.132818
$670$	0	0
$671$	−6.96569	−0.268907
$672$	−40.1193	−1.54763
$673$	−39.2465	−1.51284	−0.756420	−	0.654086i	$-0.773054\pi$
$673$	−39.2465	−1.51284	−0.756420	+	0.654086i	$0.773054\pi$
$674$	−2.45644	−0.0946184
$675$	0	0
$676$	14.7248	0.566337
$677$	−15.3340	−0.589332	−0.294666	−	0.955600i	$-0.595208\pi$
$677$	−15.3340	−0.589332	−0.294666	+	0.955600i	$0.595208\pi$
$678$	−41.2241	−1.58320
$679$	20.7528	0.796419
$680$	0	0
$681$	18.6936	0.716341
$682$	49.7959	1.90678
$683$	15.0386	0.575435	0.287717	−	0.957715i	$-0.407104\pi$
$683$	15.0386	0.575435	0.287717	+	0.957715i	$0.407104\pi$
$684$	53.3617	2.04033
$685$	0	0
$686$	13.7420	0.524674
$687$	−27.3444	−1.04325
$688$	−95.0040	−3.62199
$689$	−27.1625	−1.03481
$690$	0	0
$691$	0.735679	0.0279866	0.0139933	−	0.999902i	$-0.495546\pi$
$691$	0.735679	0.0279866	0.0139933	+	0.999902i	$0.495546\pi$
$692$	42.1150	1.60097
$693$	16.6954	0.634207
$694$	49.9947	1.89777
$695$	0	0
$696$	45.2129	1.71379
$697$	4.61307	0.174732
$698$	−82.7131	−3.13074
$699$	17.1732	0.649548
$700$	0	0
$701$	−16.8115	−0.634962	−0.317481	−	0.948265i	$-0.602837\pi$
$701$	−16.8115	−0.634962	−0.317481	+	0.948265i	$0.602837\pi$
$702$	56.7893	2.14337
$703$	23.9483	0.903227
$704$	19.1725	0.722592
$705$	0	0
$706$	−40.2104	−1.51334
$707$	27.9484	1.05111
$708$	56.0023	2.10470
$709$	15.0110	0.563750	0.281875	−	0.959451i	$-0.409044\pi$
$709$	15.0110	0.563750	0.281875	+	0.959451i	$0.409044\pi$
$710$	0	0
$711$	−16.9542	−0.635832
$712$	18.9999	0.712051
$713$	39.7934	1.49028
$714$	−10.8803	−0.407184
$715$	0	0
$716$	45.0033	1.68185
$717$	4.21951	0.157581
$718$	−72.7740	−2.71590
$719$	37.4098	1.39515	0.697575	−	0.716512i	$-0.254262\pi$
$719$	37.4098	1.39515	0.697575	+	0.716512i	$0.254262\pi$
$720$	0	0
$721$	−22.5214	−0.838740
$722$	77.2524	2.87504
$723$	−21.0419	−0.782556
$724$	−56.2876	−2.09191
$725$	0	0
$726$	−7.13930	−0.264964
$727$	6.76798	0.251011	0.125505	−	0.992093i	$-0.459945\pi$
$727$	6.76798	0.251011	0.125505	+	0.992093i	$0.459945\pi$
$728$	102.342	3.79305
$729$	21.9006	0.811134
$730$	0	0
$731$	−10.2901	−0.380594
$732$	−13.3402	−0.493067
$733$	18.3230	0.676774	0.338387	−	0.941007i	$-0.390119\pi$
$733$	18.3230	0.676774	0.338387	+	0.941007i	$0.390119\pi$
$734$	59.1334	2.18266
$735$	0	0
$736$	58.7940	2.16717
$737$	28.1838	1.03816
$738$	−19.2373	−0.708136
$739$	−0.240801	−0.00885802	−0.00442901	−	0.999990i	$-0.501410\pi$
$739$	−0.240801	−0.00885802	−0.00442901	+	0.999990i	$0.501410\pi$
$740$	0	0
$741$	33.0825	1.21532
$742$	−62.3315	−2.28826
$743$	38.0128	1.39455	0.697277	−	0.716801i	$-0.254395\pi$
$743$	38.0128	1.39455	0.697277	+	0.716801i	$0.254395\pi$
$744$	55.4017	2.03113
$745$	0	0
$746$	−92.4459	−3.38468
$747$	14.5388	0.531947
$748$	14.0605	0.514104
$749$	24.2403	0.885722
$750$	0	0
$751$	51.5263	1.88022	0.940111	−	0.340868i	$-0.110721\pi$
$751$	51.5263	1.88022	0.940111	+	0.340868i	$0.110721\pi$
$752$	−33.9485	−1.23797
$753$	8.81301	0.321164
$754$	−55.3224	−2.01472
$755$	0	0
$756$	91.8341	3.33997
$757$	0.984558	0.0357844	0.0178922	−	0.999840i	$-0.494304\pi$
$757$	0.984558	0.0357844	0.0178922	+	0.999840i	$0.494304\pi$
$758$	−79.3029	−2.88041
$759$	21.3391	0.774560
$760$	0	0
$761$	−25.8638	−0.937563	−0.468781	−	0.883314i	$-0.655307\pi$
$761$	−25.8638	−0.937563	−0.468781	+	0.883314i	$0.655307\pi$
$762$	−14.2544	−0.516381
$763$	51.9010	1.87894
$764$	24.8161	0.897814
$765$	0	0
$766$	−67.5762	−2.44163
$767$	−39.8086	−1.43740
$768$	−22.3277	−0.805680
$769$	47.2099	1.70243	0.851216	−	0.524816i	$-0.175866\pi$
$769$	47.2099	1.70243	0.851216	+	0.524816i	$0.175866\pi$
$770$	0	0
$771$	−31.2974	−1.12715
$772$	−68.6947	−2.47238
$773$	1.18874	0.0427560	0.0213780	−	0.999771i	$-0.493195\pi$
$773$	1.18874	0.0427560	0.0213780	+	0.999771i	$0.493195\pi$
$774$	42.9116	1.54243
$775$	0	0
$776$	−42.3417	−1.51998
$777$	14.3496	0.514789
$778$	16.5964	0.595010
$779$	−32.1874	−1.15323
$780$	0	0
$781$	16.2403	0.581123
$782$	15.9448	0.570186
$783$	−28.8392	−1.03063
$784$	50.8424	1.81580
$785$	0	0
$786$	−37.3789	−1.33326
$787$	−38.6087	−1.37625	−0.688125	−	0.725592i	$-0.741566\pi$
$787$	−38.6087	−1.37625	−0.688125	+	0.725592i	$0.741566\pi$
$788$	76.7626	2.73455
$789$	−14.7743	−0.525978
$790$	0	0
$791$	47.3870	1.68489
$792$	−34.0635	−1.21039
$793$	9.48270	0.336741
$794$	−34.3193	−1.21795
$795$	0	0
$796$	86.7419	3.07448
$797$	−7.77134	−0.275275	−0.137638	−	0.990483i	$-0.543951\pi$
$797$	−7.77134	−0.275275	−0.137638	+	0.990483i	$0.543951\pi$
$798$	75.9166	2.68742
$799$	−3.67705	−0.130085
$800$	0	0
$801$	−4.21951	−0.149089
$802$	−73.4954	−2.59521
$803$	−5.90736	−0.208466
$804$	53.9755	1.90357
$805$	0	0
$806$	−67.7893	−2.38778
$807$	3.35474	0.118093
$808$	−57.0228	−2.00605
$809$	−52.6740	−1.85192	−0.925960	−	0.377621i	$-0.876742\pi$
$809$	−52.6740	−1.85192	−0.925960	+	0.377621i	$0.876742\pi$
$810$	0	0
$811$	−38.0502	−1.33612	−0.668062	−	0.744105i	$-0.732876\pi$
$811$	−38.0502	−1.33612	−0.668062	+	0.744105i	$0.732876\pi$
$812$	−89.4620	−3.13950
$813$	−8.98471	−0.315108
$814$	−26.3149	−0.922337
$815$	0	0
$816$	10.9146	0.382089
$817$	71.7988	2.51192
$818$	−54.7483	−1.91423
$819$	−22.7282	−0.794188
$820$	0	0
$821$	22.8093	0.796051	0.398026	−	0.917374i	$-0.369695\pi$
$821$	22.8093	0.796051	0.398026	+	0.917374i	$0.369695\pi$
$822$	27.2845	0.951658
$823$	−26.6089	−0.927530	−0.463765	−	0.885958i	$-0.653502\pi$
$823$	−26.6089	−0.927530	−0.463765	+	0.885958i	$0.653502\pi$
$824$	45.9501	1.60075
$825$	0	0
$826$	−91.3513	−3.17852
$827$	0.707585	0.0246052	0.0123026	−	0.999924i	$-0.496084\pi$
$827$	0.707585	0.0246052	0.0123026	+	0.999924i	$0.496084\pi$
$828$	−46.8570	−1.62839
$829$	39.7559	1.38078	0.690390	−	0.723437i	$-0.257439\pi$
$829$	39.7559	1.38078	0.690390	+	0.723437i	$0.257439\pi$
$830$	0	0
$831$	35.9608	1.24746
$832$	−26.1004	−0.904869
$833$	5.50686	0.190802
$834$	22.5298	0.780145
$835$	0	0
$836$	−98.1067	−3.39309
$837$	−35.3382	−1.22147
$838$	−73.2535	−2.53050
$839$	15.4254	0.532542	0.266271	−	0.963898i	$-0.414208\pi$
$839$	15.4254	0.532542	0.266271	+	0.963898i	$0.414208\pi$
$840$	0	0
$841$	−0.905714	−0.0312315
$842$	−43.3664	−1.49451
$843$	−24.3465	−0.838539
$844$	−40.1192	−1.38096
$845$	0	0
$846$	15.3340	0.527192
$847$	8.20662	0.281983
$848$	62.5284	2.14723
$849$	4.89939	0.168147
$850$	0	0
$851$	−21.0291	−0.720867
$852$	31.1022	1.06554
$853$	50.0605	1.71404	0.857020	−	0.515284i	$-0.172314\pi$
$853$	50.0605	1.71404	0.857020	+	0.515284i	$0.172314\pi$
$854$	21.7606	0.744631
$855$	0	0
$856$	−49.4573	−1.69041
$857$	46.0441	1.57284	0.786418	−	0.617695i	$-0.211933\pi$
$857$	46.0441	1.57284	0.786418	+	0.617695i	$0.211933\pi$
$858$	−36.3518	−1.24103
$859$	−37.1872	−1.26881	−0.634406	−	0.773000i	$-0.718755\pi$
$859$	−37.1872	−1.26881	−0.634406	+	0.773000i	$0.718755\pi$
$860$	0	0
$861$	−19.2864	−0.657280
$862$	−29.1038	−0.991279
$863$	−48.2016	−1.64080	−0.820401	−	0.571788i	$-0.806250\pi$
$863$	−48.2016	−1.64080	−0.820401	+	0.571788i	$0.806250\pi$
$864$	−52.2114	−1.77627
$865$	0	0
$866$	28.7413	0.976668
$867$	1.18219	0.0401493
$868$	−109.622	−3.72083
$869$	31.1707	1.05739
$870$	0	0
$871$	−38.3678	−1.30004
$872$	−105.893	−3.58599
$873$	9.40328	0.318253
$874$	−111.254	−3.76323
$875$	0	0
$876$	−11.3133	−0.382242
$877$	−7.97840	−0.269411	−0.134706	−	0.990886i	$-0.543009\pi$
$877$	−7.97840	−0.269411	−0.134706	+	0.990886i	$0.543009\pi$
$878$	−13.9139	−0.469573
$879$	9.28057	0.313026
$880$	0	0
$881$	48.5755	1.63655	0.818276	−	0.574826i	$-0.194930\pi$
$881$	48.5755	1.63655	0.818276	+	0.574826i	$0.194930\pi$
$882$	−22.9646	−0.773259
$883$	−42.6792	−1.43627	−0.718134	−	0.695905i	$-0.755004\pi$
$883$	−42.6792	−1.43627	−0.718134	+	0.695905i	$0.755004\pi$
$884$	−19.1412	−0.643789
$885$	0	0
$886$	43.1646	1.45014
$887$	53.1721	1.78535	0.892673	−	0.450706i	$-0.148828\pi$
$887$	53.1721	1.78535	0.892673	+	0.450706i	$0.148828\pi$
$888$	−29.2773	−0.982483
$889$	16.3854	0.549547
$890$	0	0
$891$	−4.78732	−0.160381
$892$	13.8688	0.464361
$893$	25.6564	0.858559
$894$	42.7824	1.43086
$895$	0	0
$896$	7.97841	0.266540
$897$	−29.0499	−0.969947
$898$	91.8398	3.06473
$899$	34.4254	1.14815
$900$	0	0
$901$	6.77260	0.225628
$902$	35.3683	1.17763
$903$	43.0212	1.43166
$904$	−96.6832	−3.21564
$905$	0	0
$906$	44.2668	1.47067
$907$	−42.0949	−1.39774	−0.698868	−	0.715250i	$-0.746313\pi$
$907$	−42.0949	−1.39774	−0.698868	+	0.715250i	$0.746313\pi$
$908$	75.4676	2.50448
$909$	12.6637	0.420027
$910$	0	0
$911$	−32.3227	−1.07090	−0.535450	−	0.844567i	$-0.679858\pi$
$911$	−32.3227	−1.07090	−0.535450	+	0.844567i	$0.679858\pi$
$912$	−76.1564	−2.52179
$913$	−26.7299	−0.884631
$914$	25.3422	0.838247
$915$	0	0
$916$	−110.391	−3.64744
$917$	42.9670	1.41890
$918$	−14.1596	−0.467338
$919$	20.8713	0.688480	0.344240	−	0.938882i	$-0.388137\pi$
$919$	20.8713	0.688480	0.344240	+	0.938882i	$0.388137\pi$
$920$	0	0
$921$	−0.559588	−0.0184391
$922$	−42.7050	−1.40641
$923$	−22.1086	−0.727714
$924$	−58.7846	−1.93387
$925$	0	0
$926$	44.1683	1.45146
$927$	−10.2046	−0.335165
$928$	50.8627	1.66965
$929$	41.3293	1.35597	0.677985	−	0.735076i	$-0.262854\pi$
$929$	41.3293	1.35597	0.677985	+	0.735076i	$0.262854\pi$
$930$	0	0
$931$	−38.4239	−1.25929
$932$	69.3294	2.27096
$933$	21.6414	0.708507
$934$	−60.6061	−1.98309
$935$	0	0
$936$	46.3721	1.51572
$937$	24.3395	0.795138	0.397569	−	0.917572i	$-0.369854\pi$
$937$	24.3395	0.795138	0.397569	+	0.917572i	$0.369854\pi$
$938$	−88.0451	−2.87477
$939$	4.54190	0.148219
$940$	0	0
$941$	−33.4007	−1.08883	−0.544415	−	0.838816i	$-0.683248\pi$
$941$	−33.4007	−1.08883	−0.544415	+	0.838816i	$0.683248\pi$
$942$	26.7233	0.870691
$943$	28.2639	0.920399
$944$	91.6398	2.98262
$945$	0	0
$946$	−78.8940	−2.56507
$947$	14.7087	0.477969	0.238985	−	0.971023i	$-0.423185\pi$
$947$	14.7087	0.477969	0.238985	+	0.971023i	$0.423185\pi$
$948$	59.6957	1.93883
$949$	8.04195	0.261053
$950$	0	0
$951$	−5.34665	−0.173377
$952$	−25.5176	−0.827031
$953$	20.9652	0.679128	0.339564	−	0.940583i	$-0.389720\pi$
$953$	20.9652	0.679128	0.339564	+	0.940583i	$0.389720\pi$
$954$	−28.2430	−0.914401
$955$	0	0
$956$	17.0345	0.550936
$957$	18.4605	0.596743
$958$	5.99614	0.193727
$959$	−31.3635	−1.01278
$960$	0	0
$961$	11.1831	0.360746
$962$	35.8236	1.15500
$963$	10.9835	0.353939
$964$	−84.9478	−2.73598
$965$	0	0
$966$	−66.6626	−2.14483
$967$	−31.1916	−1.00306	−0.501528	−	0.865141i	$-0.667229\pi$
$967$	−31.1916	−1.00306	−0.501528	+	0.865141i	$0.667229\pi$
$968$	−16.7439	−0.538168
$969$	−8.24868	−0.264986
$970$	0	0
$971$	3.55989	0.114242	0.0571211	−	0.998367i	$-0.481808\pi$
$971$	3.55989	0.114242	0.0571211	+	0.998367i	$0.481808\pi$
$972$	68.7341	2.20465
$973$	−25.8980	−0.830253
$974$	−36.6653	−1.17483
$975$	0	0
$976$	−21.8293	−0.698738
$977$	51.0404	1.63293	0.816463	−	0.577397i	$-0.195932\pi$
$977$	51.0404	1.63293	0.816463	+	0.577397i	$0.195932\pi$
$978$	−27.5619	−0.881333
$979$	7.75767	0.247936
$980$	0	0
$981$	23.5168	0.750834
$982$	54.7771	1.74801
$983$	43.4903	1.38712	0.693562	−	0.720397i	$-0.256040\pi$
$983$	43.4903	1.38712	0.693562	+	0.720397i	$0.256040\pi$
$984$	39.3499	1.25443
$985$	0	0
$986$	13.7939	0.439287
$987$	15.3731	0.489331
$988$	133.557	4.24901
$989$	−63.0467	−2.00477
$990$	0	0
$991$	−20.5455	−0.652650	−0.326325	−	0.945258i	$-0.605810\pi$
$991$	−20.5455	−0.652650	−0.326325	+	0.945258i	$0.605810\pi$
$992$	62.3247	1.97881
$993$	20.6111	0.654074
$994$	−50.7340	−1.60919
$995$	0	0
$996$	−51.1911	−1.62205
$997$	48.6107	1.53952	0.769758	−	0.638336i	$-0.220377\pi$
$997$	48.6107	1.53952	0.769758	+	0.638336i	$0.220377\pi$
$998$	72.0214	2.27980
$999$	18.6746	0.590839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.i.1.5	✓	5
3.2	odd	2		3825.2.a.bq.1.1		5
4.3	odd	2		6800.2.a.bz.1.3		5
5.2	odd	4		425.2.b.f.324.9		10
5.3	odd	4		425.2.b.f.324.2		10
5.4	even	2		425.2.a.j.1.1	yes	5
15.14	odd	2		3825.2.a.bl.1.5		5
17.16	even	2		7225.2.a.x.1.5		5
20.19	odd	2		6800.2.a.cd.1.3		5
85.84	even	2		7225.2.a.y.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.5	✓	5	1.1	even	1	trivial
425.2.a.j.1.1	yes	5	5.4	even	2
425.2.b.f.324.2		10	5.3	odd	4
425.2.b.f.324.9		10	5.2	odd	4
3825.2.a.bl.1.5		5	15.14	odd	2
3825.2.a.bq.1.1		5	3.2	odd	2
6800.2.a.bz.1.3		5	4.3	odd	2
6800.2.a.cd.1.3		5	20.19	odd	2
7225.2.a.x.1.5		5	17.16	even	2
7225.2.a.y.1.1		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)

\(f(q)\)	\(=\)	\(q+2.60242 q^{2} +1.18219 q^{3} +4.77260 q^{4} +3.07656 q^{6} -3.53650 q^{7} +7.21549 q^{8} -1.60242 q^{9} +2.94609 q^{11} +5.64213 q^{12} -4.01064 q^{13} -9.20348 q^{14} +9.23255 q^{16} +1.00000 q^{17} -4.17018 q^{18} -6.97745 q^{19} -4.18083 q^{21} +7.66698 q^{22} +6.12692 q^{23} +8.53009 q^{24} -10.4374 q^{26} -5.44095 q^{27} -16.8783 q^{28} +5.30040 q^{29} +6.49485 q^{31} +9.59601 q^{32} +3.48284 q^{33} +2.60242 q^{34} -7.64773 q^{36} -3.43224 q^{37} -18.1583 q^{38} -4.74135 q^{39} +4.61307 q^{41} -10.8803 q^{42} -10.2901 q^{43} +14.0605 q^{44} +15.9448 q^{46} -3.67705 q^{47} +10.9146 q^{48} +5.50686 q^{49} +1.18219 q^{51} -19.1412 q^{52} +6.77260 q^{53} -14.1596 q^{54} -25.5176 q^{56} -8.24868 q^{57} +13.7939 q^{58} +9.92573 q^{59} -2.36438 q^{61} +16.9024 q^{62} +5.66698 q^{63} +6.50778 q^{64} +9.06383 q^{66} +9.56650 q^{67} +4.77260 q^{68} +7.24319 q^{69} +5.51248 q^{71} -11.5623 q^{72} -2.00515 q^{73} -8.93214 q^{74} -33.3006 q^{76} -10.4189 q^{77} -12.3390 q^{78} +10.5803 q^{79} -1.62497 q^{81} +12.0052 q^{82} -9.07301 q^{83} -19.9534 q^{84} -26.7792 q^{86} +6.26609 q^{87} +21.2575 q^{88} +2.63321 q^{89} +14.1837 q^{91} +29.2414 q^{92} +7.67816 q^{93} -9.56923 q^{94} +11.3443 q^{96} -5.86816 q^{97} +14.3312 q^{98} -4.72088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 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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