LMFDB - Embedded newform 425.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,4,Mod(1,425)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("425.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$25.0758117524$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	\(q+0.267949 q^{2} -0.732051 q^{3} -7.92820 q^{4} -0.196152 q^{6} +14.5885 q^{7} -4.26795 q^{8} -26.4641 q^{9} +3.51666 q^{11} +5.80385 q^{12} +88.4974 q^{13} +3.90897 q^{14} +62.2820 q^{16} +17.0000 q^{17} -7.09103 q^{18} -79.6077 q^{19} -10.6795 q^{21} +0.942286 q^{22} -153.937 q^{23} +3.12436 q^{24} +23.7128 q^{26} +39.1384 q^{27} -115.660 q^{28} -28.3257 q^{29} -209.219 q^{31} +50.8320 q^{32} -2.57437 q^{33} +4.55514 q^{34} +209.813 q^{36} -359.597 q^{37} -21.3308 q^{38} -64.7846 q^{39} +417.951 q^{41} -2.86156 q^{42} -243.520 q^{43} -27.8808 q^{44} -41.2473 q^{46} +160.144 q^{47} -45.5936 q^{48} -130.177 q^{49} -12.4449 q^{51} -701.626 q^{52} -28.8513 q^{53} +10.4871 q^{54} -62.2628 q^{56} +58.2769 q^{57} -7.58984 q^{58} -832.315 q^{59} -502.190 q^{61} -56.0601 q^{62} -386.070 q^{63} -484.636 q^{64} -0.689801 q^{66} +555.472 q^{67} -134.779 q^{68} +112.690 q^{69} -961.114 q^{71} +112.947 q^{72} -512.823 q^{73} -96.3538 q^{74} +631.146 q^{76} +51.3027 q^{77} -17.3590 q^{78} +277.906 q^{79} +685.879 q^{81} +111.990 q^{82} +288.228 q^{83} +84.6692 q^{84} -65.2511 q^{86} +20.7358 q^{87} -15.0089 q^{88} -1387.18 q^{89} +1291.04 q^{91} +1220.44 q^{92} +153.159 q^{93} +42.9103 q^{94} -37.2116 q^{96} +249.174 q^{97} -34.8808 q^{98} -93.0653 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 10 q^{6} - 2 q^{7} - 12 q^{8} - 46 q^{9} - 38 q^{11} + 22 q^{12} + 80 q^{13} - 58 q^{14} - 14 q^{16} + 34 q^{17} - 80 q^{18} - 180 q^{19} - 56 q^{21} - 154 q^{22} + 42 q^{23}+ \cdots + 718 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^3 - 2 * q^4 + 10 * q^6 - 2 * q^7 - 12 * q^8 - 46 * q^9 - 38 * q^11 + 22 * q^12 + 80 * q^13 - 58 * q^14 - 14 * q^16 + 34 * q^17 - 80 * q^18 - 180 * q^19 - 56 * q^21 - 154 * q^22 + 42 * q^23 - 18 * q^24 - 8 * q^26 - 88 * q^27 - 214 * q^28 - 216 * q^29 + 70 * q^31 - 172 * q^32 - 116 * q^33 + 68 * q^34 + 94 * q^36 - 352 * q^37 - 396 * q^38 - 88 * q^39 + 344 * q^41 - 172 * q^42 + 88 * q^43 - 274 * q^44 + 690 * q^46 + 348 * q^47 - 254 * q^48 - 198 * q^49 + 34 * q^51 - 752 * q^52 + 164 * q^53 - 464 * q^54 + 66 * q^56 - 216 * q^57 - 708 * q^58 - 1436 * q^59 - 76 * q^61 + 986 * q^62 - 62 * q^63 - 706 * q^64 - 424 * q^66 + 44 * q^67 - 34 * q^68 + 648 * q^69 - 1510 * q^71 + 264 * q^72 - 1088 * q^73 - 68 * q^74 + 36 * q^76 + 740 * q^77 - 104 * q^78 - 958 * q^79 + 866 * q^81 - 164 * q^82 - 248 * q^83 - 184 * q^84 + 1172 * q^86 - 492 * q^87 + 306 * q^88 - 204 * q^89 + 1432 * q^91 + 2382 * q^92 + 916 * q^93 + 744 * q^94 - 646 * q^96 - 1012 * q^97 - 288 * q^98 + 718 * 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 $

$2$	0.267949	0.0947343	0.0473672	−	0.998878i	$-0.484917\pi$
$2$	0.267949	0.0947343	0.0473672	+	0.998878i	$0.484917\pi$
$3$	−0.732051	−0.140883	−0.0704416	−	0.997516i	$-0.522441\pi$
$3$	−0.732051	−0.140883	−0.0704416	+	0.997516i	$0.522441\pi$
$4$	−7.92820	−0.991025
$5$	0	0
$5$	0	0
$6$	−0.196152	−0.0133465
$7$	14.5885	0.787703	0.393851	−	0.919174i	$-0.371142\pi$
$7$	14.5885	0.787703	0.393851	+	0.919174i	$0.371142\pi$
$8$	−4.26795	−0.188618
$9$	−26.4641	−0.980152
$10$	0	0
$11$	3.51666	0.0963921	0.0481960	−	0.998838i	$-0.484653\pi$
$11$	3.51666	0.0963921	0.0481960	+	0.998838i	$0.484653\pi$
$12$	5.80385	0.139619
$13$	88.4974	1.88806	0.944030	−	0.329861i	$-0.107002\pi$
$13$	88.4974	1.88806	0.944030	+	0.329861i	$0.107002\pi$
$14$	3.90897	0.0746225
$15$	0	0
$16$	62.2820	0.973157
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−7.09103	−0.0928540
$19$	−79.6077	−0.961224	−0.480612	−	0.876933i	$-0.659585\pi$
$19$	−79.6077	−0.961224	−0.480612	+	0.876933i	$0.659585\pi$
$20$	0	0
$21$	−10.6795	−0.110974
$22$	0.942286	0.00913164
$23$	−153.937	−1.39557	−0.697785	−	0.716307i	$-0.745831\pi$
$23$	−153.937	−1.39557	−0.697785	+	0.716307i	$0.745831\pi$
$24$	3.12436	0.0265732
$25$	0	0
$26$	23.7128	0.178864
$27$	39.1384	0.278970
$28$	−115.660	−0.780633
$29$	−28.3257	−0.181377	−0.0906887	−	0.995879i	$-0.528907\pi$
$29$	−28.3257	−0.181377	−0.0906887	+	0.995879i	$0.528907\pi$
$30$	0	0
$31$	−209.219	−1.21216	−0.606079	−	0.795405i	$-0.707258\pi$
$31$	−209.219	−1.21216	−0.606079	+	0.795405i	$0.707258\pi$
$32$	50.8320	0.280810
$33$	−2.57437	−0.0135800
$34$	4.55514	0.0229765
$35$	0	0
$36$	209.813	0.971355
$37$	−359.597	−1.59777	−0.798884	−	0.601485i	$-0.794576\pi$
$37$	−359.597	−1.59777	−0.798884	+	0.601485i	$0.794576\pi$
$38$	−21.3308	−0.0910609
$39$	−64.7846	−0.265996
$40$	0	0
$41$	417.951	1.59202	0.796012	−	0.605280i	$-0.206939\pi$
$41$	417.951	1.59202	0.796012	+	0.605280i	$0.206939\pi$
$42$	−2.86156	−0.0105131
$43$	−243.520	−0.863640	−0.431820	−	0.901960i	$-0.642128\pi$
$43$	−243.520	−0.863640	−0.431820	+	0.901960i	$0.642128\pi$
$44$	−27.8808	−0.0955270
$45$	0	0
$46$	−41.2473	−0.132208
$47$	160.144	0.497007	0.248504	−	0.968631i	$-0.420061\pi$
$47$	160.144	0.497007	0.248504	+	0.968631i	$0.420061\pi$
$48$	−45.5936	−0.137101
$49$	−130.177	−0.379525
$50$	0	0
$51$	−12.4449	−0.0341692
$52$	−701.626	−1.87111
$53$	−28.8513	−0.0747740	−0.0373870	−	0.999301i	$-0.511903\pi$
$53$	−28.8513	−0.0747740	−0.0373870	+	0.999301i	$0.511903\pi$
$54$	10.4871	0.0264281
$55$	0	0
$56$	−62.2628	−0.148575
$57$	58.2769	0.135420
$58$	−7.58984	−0.0171827
$59$	−832.315	−1.83658	−0.918290	−	0.395908i	$-0.870430\pi$
$59$	−832.315	−1.83658	−0.918290	+	0.395908i	$0.870430\pi$
$60$	0	0
$61$	−502.190	−1.05408	−0.527039	−	0.849841i	$-0.676698\pi$
$61$	−502.190	−1.05408	−0.527039	+	0.849841i	$0.676698\pi$
$62$	−56.0601	−0.114833
$63$	−386.070	−0.772068
$64$	−484.636	−0.946554
$65$	0	0
$66$	−0.689801	−0.00128650
$67$	555.472	1.01286	0.506430	−	0.862281i	$-0.330965\pi$
$67$	555.472	1.01286	0.506430	+	0.862281i	$0.330965\pi$
$68$	−134.779	−0.240359
$69$	112.690	0.196612
$70$	0	0
$71$	−961.114	−1.60652	−0.803262	−	0.595625i	$-0.796904\pi$
$71$	−961.114	−1.60652	−0.803262	+	0.595625i	$0.796904\pi$
$72$	112.947	0.184875
$73$	−512.823	−0.822211	−0.411105	−	0.911588i	$-0.634857\pi$
$73$	−512.823	−0.822211	−0.411105	+	0.911588i	$0.634857\pi$
$74$	−96.3538	−0.151364
$75$	0	0
$76$	631.146	0.952597
$77$	51.3027	0.0759283
$78$	−17.3590	−0.0251989
$79$	277.906	0.395783	0.197892	−	0.980224i	$-0.436591\pi$
$79$	277.906	0.395783	0.197892	+	0.980224i	$0.436591\pi$
$80$	0	0
$81$	685.879	0.940850
$82$	111.990	0.150819
$83$	288.228	0.381170	0.190585	−	0.981671i	$-0.438961\pi$
$83$	288.228	0.381170	0.190585	+	0.981671i	$0.438961\pi$
$84$	84.6692	0.109978
$85$	0	0
$86$	−65.2511	−0.0818164
$87$	20.7358	0.0255530
$88$	−15.0089	−0.0181813
$89$	−1387.18	−1.65215	−0.826073	−	0.563563i	$-0.809430\pi$
$89$	−1387.18	−1.65215	−0.826073	+	0.563563i	$0.809430\pi$
$90$	0	0
$91$	1291.04	1.48723
$92$	1220.44	1.38305
$93$	153.159	0.170773
$94$	42.9103	0.0470837
$95$	0	0
$96$	−37.2116	−0.0395614
$97$	249.174	0.260823	0.130411	−	0.991460i	$-0.458370\pi$
$97$	249.174	0.260823	0.130411	+	0.991460i	$0.458370\pi$
$98$	−34.8808	−0.0359540
$99$	−93.0653	−0.0944789
$100$	0	0
$101$	992.141	0.977443	0.488721	−	0.872440i	$-0.337463\pi$
$101$	992.141	0.977443	0.488721	+	0.872440i	$0.337463\pi$
$102$	−3.33459	−0.00323700
$103$	−592.144	−0.566463	−0.283231	−	0.959052i	$-0.591406\pi$
$103$	−592.144	−0.566463	−0.283231	+	0.959052i	$0.591406\pi$
$104$	−377.703	−0.356123
$105$	0	0
$106$	−7.73067	−0.00708367
$107$	−769.496	−0.695234	−0.347617	−	0.937637i	$-0.613009\pi$
$107$	−769.496	−0.695234	−0.347617	+	0.937637i	$0.613009\pi$
$108$	−310.297	−0.276467
$109$	381.159	0.334940	0.167470	−	0.985877i	$-0.446440\pi$
$109$	381.159	0.334940	0.167470	+	0.985877i	$0.446440\pi$
$110$	0	0
$111$	263.244	0.225099
$112$	908.599	0.766558
$113$	1081.41	0.900272	0.450136	−	0.892960i	$-0.351376\pi$
$113$	1081.41	0.900272	0.450136	+	0.892960i	$0.351376\pi$
$114$	15.6152	0.0128290
$115$	0	0
$116$	224.572	0.179750
$117$	−2342.00	−1.85058
$118$	−223.018	−0.173987
$119$	248.004	0.191046
$120$	0	0
$121$	−1318.63	−0.990709
$122$	−134.561	−0.0998574
$123$	−305.962	−0.224290
$124$	1658.73	1.20128
$125$	0	0
$126$	−103.447	−0.0731414
$127$	451.793	0.315670	0.157835	−	0.987465i	$-0.449549\pi$
$127$	451.793	0.315670	0.157835	+	0.987465i	$0.449549\pi$
$128$	−536.514	−0.370481
$129$	178.269	0.121672
$130$	0	0
$131$	−600.540	−0.400530	−0.200265	−	0.979742i	$-0.564180\pi$
$131$	−600.540	−0.400530	−0.200265	+	0.979742i	$0.564180\pi$
$132$	20.4102	0.0134582
$133$	−1161.35	−0.757159
$134$	148.838	0.0959527
$135$	0	0
$136$	−72.5551	−0.0457467
$137$	−1479.37	−0.922562	−0.461281	−	0.887254i	$-0.652610\pi$
$137$	−1479.37	−0.922562	−0.461281	+	0.887254i	$0.652610\pi$
$138$	30.1951	0.0186260
$139$	941.627	0.574588	0.287294	−	0.957842i	$-0.407244\pi$
$139$	941.627	0.574588	0.287294	+	0.957842i	$0.407244\pi$
$140$	0	0
$141$	−117.233	−0.0700200
$142$	−257.530	−0.152193
$143$	311.215	0.181994
$144$	−1648.24	−0.953841
$145$	0	0
$146$	−137.411	−0.0778916
$147$	95.2961	0.0534686
$148$	2850.96	1.58343
$149$	2427.95	1.33493	0.667467	−	0.744639i	$-0.267378\pi$
$149$	2427.95	1.33493	0.667467	+	0.744639i	$0.267378\pi$
$150$	0	0
$151$	−1927.95	−1.03903	−0.519517	−	0.854460i	$-0.673888\pi$
$151$	−1927.95	−1.03903	−0.519517	+	0.854460i	$0.673888\pi$
$152$	339.762	0.181305
$153$	−449.890	−0.237722
$154$	13.7465	0.00719302
$155$	0	0
$156$	513.626	0.263609
$157$	−3264.27	−1.65935	−0.829673	−	0.558250i	$-0.811473\pi$
$157$	−3264.27	−1.65935	−0.829673	+	0.558250i	$0.811473\pi$
$158$	74.4647	0.0374943
$159$	21.1206	0.0105344
$160$	0	0
$161$	−2245.71	−1.09929
$162$	183.781	0.0891308
$163$	3232.63	1.55337	0.776684	−	0.629891i	$-0.216900\pi$
$163$	3232.63	1.55337	0.776684	+	0.629891i	$0.216900\pi$
$164$	−3313.60	−1.57774
$165$	0	0
$166$	77.2305	0.0361099
$167$	3446.66	1.59707	0.798535	−	0.601948i	$-0.205609\pi$
$167$	3446.66	1.59707	0.798535	+	0.601948i	$0.205609\pi$
$168$	45.5795	0.0209318
$169$	5634.79	2.56477
$170$	0	0
$171$	2106.75	0.942146
$172$	1930.68	0.855889
$173$	2281.64	1.00272	0.501359	−	0.865239i	$-0.332834\pi$
$173$	2281.64	1.00272	0.501359	+	0.865239i	$0.332834\pi$
$174$	5.55615	0.00242075
$175$	0	0
$176$	219.025	0.0938046
$177$	609.297	0.258743
$178$	−371.694	−0.156515
$179$	202.290	0.0844685	0.0422342	−	0.999108i	$-0.486552\pi$
$179$	202.290	0.0844685	0.0422342	+	0.999108i	$0.486552\pi$
$180$	0	0
$181$	−3632.57	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$181$	−3632.57	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$182$	345.933	0.140892
$183$	367.628	0.148502
$184$	656.996	0.263230
$185$	0	0
$186$	41.0388	0.0161780
$187$	59.7832	0.0233785
$188$	−1269.65	−0.492547
$189$	570.969	0.219746
$190$	0	0
$191$	578.307	0.219083	0.109541	−	0.993982i	$-0.465062\pi$
$191$	578.307	0.219083	0.109541	+	0.993982i	$0.465062\pi$
$192$	354.778	0.133354
$193$	−1538.31	−0.573730	−0.286865	−	0.957971i	$-0.592613\pi$
$193$	−1538.31	−0.573730	−0.286865	+	0.957971i	$0.592613\pi$
$194$	66.7660	0.0247089
$195$	0	0
$196$	1032.07	0.376118
$197$	788.023	0.284997	0.142498	−	0.989795i	$-0.454486\pi$
$197$	788.023	0.284997	0.142498	+	0.989795i	$0.454486\pi$
$198$	−24.9368	−0.00895040
$199$	535.770	0.190853	0.0954265	−	0.995436i	$-0.469579\pi$
$199$	535.770	0.190853	0.0954265	+	0.995436i	$0.469579\pi$
$200$	0	0
$201$	−406.633	−0.142695
$202$	265.843	0.0925974
$203$	−413.228	−0.142871
$204$	98.6654	0.0338626
$205$	0	0
$206$	−158.664	−0.0536635
$207$	4073.81	1.36787
$208$	5511.80	1.83738
$209$	−279.953	−0.0926544
$210$	0	0
$211$	−3662.54	−1.19497	−0.597487	−	0.801879i	$-0.703834\pi$
$211$	−3662.54	−1.19497	−0.597487	+	0.801879i	$0.703834\pi$
$212$	228.739	0.0741030
$213$	703.584	0.226332
$214$	−206.186	−0.0658625
$215$	0	0
$216$	−167.041	−0.0526189
$217$	−3052.18	−0.954819
$218$	102.131	0.0317303
$219$	375.413	0.115836
$220$	0	0
$221$	1504.46	0.457922
$222$	70.5359	0.0213246
$223$	2869.99	0.861832	0.430916	−	0.902392i	$-0.358191\pi$
$223$	2869.99	0.861832	0.430916	+	0.902392i	$0.358191\pi$
$224$	741.561	0.221195
$225$	0	0
$226$	289.764	0.0852867
$227$	−4512.32	−1.31935	−0.659677	−	0.751549i	$-0.729307\pi$
$227$	−4512.32	−1.31935	−0.659677	+	0.751549i	$0.729307\pi$
$228$	−462.031	−0.134205
$229$	−422.427	−0.121899	−0.0609493	−	0.998141i	$-0.519413\pi$
$229$	−422.427	−0.121899	−0.0609493	+	0.998141i	$0.519413\pi$
$230$	0	0
$231$	−37.5561	−0.0106970
$232$	120.892	0.0342111
$233$	−1939.48	−0.545319	−0.272660	−	0.962111i	$-0.587903\pi$
$233$	−1939.48	−0.545319	−0.272660	+	0.962111i	$0.587903\pi$
$234$	−627.538	−0.175314
$235$	0	0
$236$	6598.77	1.82010
$237$	−203.441	−0.0557593
$238$	66.4524	0.0180986
$239$	3198.28	0.865605	0.432803	−	0.901489i	$-0.357525\pi$
$239$	3198.28	0.865605	0.432803	+	0.901489i	$0.357525\pi$
$240$	0	0
$241$	2855.26	0.763169	0.381584	−	0.924334i	$-0.375379\pi$
$241$	2855.26	0.763169	0.381584	+	0.924334i	$0.375379\pi$
$242$	−353.327	−0.0938541
$243$	−1558.84	−0.411520
$244$	3981.46	1.04462
$245$	0	0
$246$	−81.9821	−0.0212479
$247$	−7045.08	−1.81485
$248$	892.937	0.228635
$249$	−210.998	−0.0537005
$250$	0	0
$251$	−5912.18	−1.48675	−0.743373	−	0.668877i	$-0.766775\pi$
$251$	−5912.18	−1.48675	−0.743373	+	0.668877i	$0.766775\pi$
$252$	3060.84	0.765139
$253$	−541.345	−0.134522
$254$	121.057	0.0299048
$255$	0	0
$256$	3733.33	0.911457
$257$	2911.01	0.706553	0.353276	−	0.935519i	$-0.385067\pi$
$257$	2911.01	0.706553	0.353276	+	0.935519i	$0.385067\pi$
$258$	47.7671	0.0115266
$259$	−5245.97	−1.25857
$260$	0	0
$261$	749.613	0.177777
$262$	−160.914	−0.0379439
$263$	786.792	0.184470	0.0922351	−	0.995737i	$-0.470599\pi$
$263$	786.792	0.184470	0.0922351	+	0.995737i	$0.470599\pi$
$264$	10.9873	0.00256144
$265$	0	0
$266$	−311.184	−0.0717289
$267$	1015.49	0.232760
$268$	−4403.89	−1.00377
$269$	256.876	0.0582231	0.0291115	−	0.999576i	$-0.490732\pi$
$269$	256.876	0.0582231	0.0291115	+	0.999576i	$0.490732\pi$
$270$	0	0
$271$	−1617.40	−0.362546	−0.181273	−	0.983433i	$-0.558022\pi$
$271$	−1617.40	−0.362546	−0.181273	+	0.983433i	$0.558022\pi$
$272$	1058.79	0.236025
$273$	−945.108	−0.209526
$274$	−396.396	−0.0873983
$275$	0	0
$276$	−893.428	−0.194848
$277$	−1622.33	−0.351901	−0.175950	−	0.984399i	$-0.556300\pi$
$277$	−1622.33	−0.351901	−0.175950	+	0.984399i	$0.556300\pi$
$278$	252.308	0.0544332
$279$	5536.80	1.18810
$280$	0	0
$281$	−694.753	−0.147493	−0.0737464	−	0.997277i	$-0.523496\pi$
$281$	−694.753	−0.147493	−0.0737464	+	0.997277i	$0.523496\pi$
$282$	−31.4126	−0.00663330
$283$	3564.53	0.748725	0.374363	−	0.927282i	$-0.377862\pi$
$283$	3564.53	0.748725	0.374363	+	0.927282i	$0.377862\pi$
$284$	7619.91	1.59211
$285$	0	0
$286$	83.3899	0.0172411
$287$	6097.26	1.25404
$288$	−1345.22	−0.275236
$289$	289.000	0.0588235
$290$	0	0
$291$	−182.408	−0.0367455
$292$	4065.77	0.814832
$293$	−2382.23	−0.474988	−0.237494	−	0.971389i	$-0.576326\pi$
$293$	−2382.23	−0.474988	−0.237494	+	0.971389i	$0.576326\pi$
$294$	25.5345	0.00506532
$295$	0	0
$296$	1534.74	0.301369
$297$	137.637	0.0268905
$298$	650.567	0.126464
$299$	−13623.0	−2.63492
$300$	0	0
$301$	−3552.59	−0.680291
$302$	−516.591	−0.0984321
$303$	−726.297	−0.137705
$304$	−4958.13	−0.935422
$305$	0	0
$306$	−120.548	−0.0225204
$307$	−7795.16	−1.44916	−0.724582	−	0.689188i	$-0.757967\pi$
$307$	−7795.16	−1.44916	−0.724582	+	0.689188i	$0.757967\pi$
$308$	−406.738	−0.0752469
$309$	433.479	0.0798051
$310$	0	0
$311$	595.429	0.108565	0.0542824	−	0.998526i	$-0.482713\pi$
$311$	595.429	0.108565	0.0542824	+	0.998526i	$0.482713\pi$
$312$	276.497	0.0501717
$313$	−3297.09	−0.595408	−0.297704	−	0.954658i	$-0.596221\pi$
$313$	−3297.09	−0.595408	−0.297704	+	0.954658i	$0.596221\pi$
$314$	−874.659	−0.157197
$315$	0	0
$316$	−2203.30	−0.392231
$317$	−3437.55	−0.609060	−0.304530	−	0.952503i	$-0.598499\pi$
$317$	−3437.55	−0.609060	−0.304530	+	0.952503i	$0.598499\pi$
$318$	5.65924	0.000997970 0
$319$	−99.6117	−0.0174833
$320$	0	0
$321$	563.310	0.0979468
$322$	−601.735	−0.104141
$323$	−1353.33	−0.233131
$324$	−5437.79	−0.932406
$325$	0	0
$326$	866.180	0.147157
$327$	−279.028	−0.0471874
$328$	−1783.79	−0.300285
$329$	2336.25	0.391494
$330$	0	0
$331$	−7103.74	−1.17963	−0.589814	−	0.807539i	$-0.700799\pi$
$331$	−7103.74	−1.17963	−0.589814	+	0.807539i	$0.700799\pi$
$332$	−2285.13	−0.377750
$333$	9516.42	1.56606
$334$	923.530	0.151297
$335$	0	0
$336$	−665.140	−0.107995
$337$	7973.12	1.28879	0.644397	−	0.764691i	$-0.277108\pi$
$337$	7973.12	1.28879	0.644397	+	0.764691i	$0.277108\pi$
$338$	1509.84	0.242972
$339$	−791.649	−0.126833
$340$	0	0
$341$	−735.753	−0.116842
$342$	564.501	0.0892536
$343$	−6902.92	−1.08666
$344$	1039.33	0.162898
$345$	0	0
$346$	611.365	0.0949918
$347$	−1456.46	−0.225322	−0.112661	−	0.993633i	$-0.535937\pi$
$347$	−1456.46	−0.225322	−0.112661	+	0.993633i	$0.535937\pi$
$348$	−164.398	−0.0253237
$349$	−3582.83	−0.549526	−0.274763	−	0.961512i	$-0.588599\pi$
$349$	−3582.83	−0.549526	−0.274763	+	0.961512i	$0.588599\pi$
$350$	0	0
$351$	3463.65	0.526712
$352$	178.759	0.0270679
$353$	173.891	0.0262190	0.0131095	−	0.999914i	$-0.495827\pi$
$353$	173.891	0.0262190	0.0131095	+	0.999914i	$0.495827\pi$
$354$	163.261	0.0245119
$355$	0	0
$356$	10997.9	1.63732
$357$	−181.551	−0.0269152
$358$	54.2034	0.00800207
$359$	4327.95	0.636269	0.318134	−	0.948046i	$-0.396944\pi$
$359$	4327.95	0.636269	0.318134	+	0.948046i	$0.396944\pi$
$360$	0	0
$361$	−521.615	−0.0760482
$362$	−973.343	−0.141320
$363$	965.306	0.139574
$364$	−10235.6	−1.47388
$365$	0	0
$366$	98.5057	0.0140682
$367$	−8700.38	−1.23748	−0.618741	−	0.785595i	$-0.712357\pi$
$367$	−8700.38	−1.23748	−0.618741	+	0.785595i	$0.712357\pi$
$368$	−9587.52	−1.35811
$369$	−11060.7	−1.56043
$370$	0	0
$371$	−420.895	−0.0588997
$372$	−1214.28	−0.169240
$373$	7208.52	1.00065	0.500326	−	0.865837i	$-0.333214\pi$
$373$	7208.52	1.00065	0.500326	+	0.865837i	$0.333214\pi$
$374$	16.0189	0.00221475
$375$	0	0
$376$	−683.485	−0.0937448
$377$	−2506.75	−0.342451
$378$	152.991	0.0208175
$379$	−8226.89	−1.11500	−0.557502	−	0.830175i	$-0.688240\pi$
$379$	−8226.89	−1.11500	−0.557502	+	0.830175i	$0.688240\pi$
$380$	0	0
$381$	−330.735	−0.0444726
$382$	154.957	0.0207547
$383$	−6579.76	−0.877833	−0.438916	−	0.898528i	$-0.644638\pi$
$383$	−6579.76	−0.877833	−0.438916	+	0.898528i	$0.644638\pi$
$384$	392.755	0.0521946
$385$	0	0
$386$	−412.189	−0.0543520
$387$	6444.55	0.846498
$388$	−1975.50	−0.258482
$389$	10161.7	1.32446	0.662232	−	0.749299i	$-0.269610\pi$
$389$	10161.7	1.32446	0.662232	+	0.749299i	$0.269610\pi$
$390$	0	0
$391$	−2616.93	−0.338475
$392$	555.588	0.0715853
$393$	439.626	0.0564279
$394$	211.150	0.0269990
$395$	0	0
$396$	737.840	0.0936310
$397$	5489.07	0.693925	0.346963	−	0.937879i	$-0.387213\pi$
$397$	5489.07	0.693925	0.346963	+	0.937879i	$0.387213\pi$
$398$	143.559	0.0180803
$399$	850.170	0.106671
$400$	0	0
$401$	5892.63	0.733825	0.366912	−	0.930255i	$-0.380415\pi$
$401$	5892.63	0.733825	0.366912	+	0.930255i	$0.380415\pi$
$402$	−108.957	−0.0135181
$403$	−18515.4	−2.28862
$404$	−7865.89	−0.968670
$405$	0	0
$406$	−110.724	−0.0135348
$407$	−1264.58	−0.154012
$408$	53.1140	0.00644494
$409$	4317.28	0.521946	0.260973	−	0.965346i	$-0.415957\pi$
$409$	4317.28	0.521946	0.260973	+	0.965346i	$0.415957\pi$
$410$	0	0
$411$	1082.97	0.129974
$412$	4694.63	0.561379
$413$	−12142.2	−1.44668
$414$	1091.57	0.129584
$415$	0	0
$416$	4498.50	0.530186
$417$	−689.318	−0.0809498
$418$	−75.0132	−0.00877755
$419$	−12364.2	−1.44160	−0.720799	−	0.693144i	$-0.756225\pi$
$419$	−12364.2	−1.44160	−0.720799	+	0.693144i	$0.756225\pi$
$420$	0	0
$421$	12077.2	1.39812	0.699059	−	0.715064i	$-0.253602\pi$
$421$	12077.2	1.39812	0.699059	+	0.715064i	$0.253602\pi$
$422$	−981.374	−0.113205
$423$	−4238.06	−0.487143
$424$	123.136	0.0141038
$425$	0	0
$426$	188.525	0.0214415
$427$	−7326.17	−0.830300
$428$	6100.72	0.688994
$429$	−227.825	−0.0256399
$430$	0	0
$431$	8966.16	1.00205	0.501026	−	0.865432i	$-0.332956\pi$
$431$	8966.16	1.00205	0.501026	+	0.865432i	$0.332956\pi$
$432$	2437.62	0.271482
$433$	294.478	0.0326829	0.0163414	−	0.999866i	$-0.494798\pi$
$433$	294.478	0.0326829	0.0163414	+	0.999866i	$0.494798\pi$
$434$	−817.830	−0.0904542
$435$	0	0
$436$	−3021.91	−0.331934
$437$	12254.6	1.34146
$438$	100.591	0.0109736
$439$	6088.60	0.661943	0.330971	−	0.943641i	$-0.392624\pi$
$439$	6088.60	0.661943	0.330971	+	0.943641i	$0.392624\pi$
$440$	0	0
$441$	3445.02	0.371992
$442$	403.118	0.0433809
$443$	11491.1	1.23241	0.616206	−	0.787585i	$-0.288669\pi$
$443$	11491.1	1.23241	0.616206	+	0.787585i	$0.288669\pi$
$444$	−2087.05	−0.223079
$445$	0	0
$446$	769.011	0.0816451
$447$	−1777.38	−0.188070
$448$	−7070.09	−0.745603
$449$	1292.46	0.135846	0.0679230	−	0.997691i	$-0.478363\pi$
$449$	1292.46	0.135846	0.0679230	+	0.997691i	$0.478363\pi$
$450$	0	0
$451$	1469.79	0.153459
$452$	−8573.66	−0.892192
$453$	1411.35	0.146382
$454$	−1209.07	−0.124988
$455$	0	0
$456$	−248.723	−0.0255428
$457$	−8532.20	−0.873346	−0.436673	−	0.899620i	$-0.643843\pi$
$457$	−8532.20	−0.873346	−0.436673	+	0.899620i	$0.643843\pi$
$458$	−113.189	−0.0115480
$459$	665.353	0.0676602
$460$	0	0
$461$	11874.1	1.19964	0.599818	−	0.800136i	$-0.295240\pi$
$461$	11874.1	1.19964	0.599818	+	0.800136i	$0.295240\pi$
$462$	−10.0631	−0.00101338
$463$	4687.65	0.470526	0.235263	−	0.971932i	$-0.424405\pi$
$463$	4687.65	0.470526	0.235263	+	0.971932i	$0.424405\pi$
$464$	−1764.18	−0.176509
$465$	0	0
$466$	−519.681	−0.0516605
$467$	9989.29	0.989827	0.494914	−	0.868942i	$-0.335200\pi$
$467$	9989.29	0.989827	0.494914	+	0.868942i	$0.335200\pi$
$468$	18567.9	1.83398
$469$	8103.47	0.797833
$470$	0	0
$471$	2389.61	0.233774
$472$	3552.28	0.346413
$473$	−856.379	−0.0832481
$474$	−54.5120	−0.00528232
$475$	0	0
$476$	−1966.22	−0.189331
$477$	763.522	0.0732899
$478$	856.977	0.0820026
$479$	5516.08	0.526171	0.263086	−	0.964772i	$-0.415260\pi$
$479$	5516.08	0.526171	0.263086	+	0.964772i	$0.415260\pi$
$480$	0	0
$481$	−31823.4	−3.01668
$482$	765.066	0.0722983
$483$	1643.97	0.154872
$484$	10454.4	0.981817
$485$	0	0
$486$	−417.689	−0.0389851
$487$	−4386.22	−0.408129	−0.204064	−	0.978957i	$-0.565415\pi$
$487$	−4386.22	−0.408129	−0.204064	+	0.978957i	$0.565415\pi$
$488$	2143.32	0.198819
$489$	−2366.45	−0.218843
$490$	0	0
$491$	−13513.6	−1.24208	−0.621041	−	0.783778i	$-0.713290\pi$
$491$	−13513.6	−1.24208	−0.621041	+	0.783778i	$0.713290\pi$
$492$	2425.73	0.222277
$493$	−481.536	−0.0439905
$494$	−1887.72	−0.171928
$495$	0	0
$496$	−13030.6	−1.17962
$497$	−14021.2	−1.26546
$498$	−56.5366	−0.00508728
$499$	−9724.59	−0.872410	−0.436205	−	0.899847i	$-0.643678\pi$
$499$	−9724.59	−0.872410	−0.436205	+	0.899847i	$0.643678\pi$
$500$	0	0
$501$	−2523.13	−0.225000
$502$	−1584.16	−0.140846
$503$	−12490.0	−1.10716	−0.553582	−	0.832795i	$-0.686739\pi$
$503$	−12490.0	−1.10716	−0.553582	+	0.832795i	$0.686739\pi$
$504$	1647.73	0.145626
$505$	0	0
$506$	−145.053	−0.0127438
$507$	−4124.96	−0.361333
$508$	−3581.90	−0.312837
$509$	−10366.0	−0.902682	−0.451341	−	0.892351i	$-0.649054\pi$
$509$	−10366.0	−0.902682	−0.451341	+	0.892351i	$0.649054\pi$
$510$	0	0
$511$	−7481.30	−0.647658
$512$	5292.45	0.456827
$513$	−3115.72	−0.268153
$514$	780.004	0.0669348
$515$	0	0
$516$	−1413.36	−0.120580
$517$	563.171	0.0479076
$518$	−1405.65	−0.119229
$519$	−1670.28	−0.141266
$520$	0	0
$521$	−10787.8	−0.907144	−0.453572	−	0.891220i	$-0.649850\pi$
$521$	−10787.8	−0.907144	−0.453572	+	0.891220i	$0.649850\pi$
$522$	200.858	0.0168416
$523$	−19147.5	−1.60088	−0.800442	−	0.599410i	$-0.795402\pi$
$523$	−19147.5	−1.60088	−0.800442	+	0.599410i	$0.795402\pi$
$524$	4761.20	0.396935
$525$	0	0
$526$	210.820	0.0174757
$527$	−3556.73	−0.293991
$528$	−160.337	−0.0132155
$529$	11529.6	0.947616
$530$	0	0
$531$	22026.5	1.80013
$532$	9207.45	0.750364
$533$	36987.6	3.00584
$534$	272.099	0.0220503
$535$	0	0
$536$	−2370.72	−0.191044
$537$	−148.087	−0.0119002
$538$	68.8297	0.00551572
$539$	−457.788	−0.0365832
$540$	0	0
$541$	−5507.20	−0.437658	−0.218829	−	0.975763i	$-0.570224\pi$
$541$	−5507.20	−0.437658	−0.218829	+	0.975763i	$0.570224\pi$
$542$	−433.381	−0.0343456
$543$	2659.22	0.210162
$544$	864.144	0.0681064
$545$	0	0
$546$	−253.241	−0.0198493
$547$	−17254.5	−1.34872	−0.674359	−	0.738404i	$-0.735580\pi$
$547$	−17254.5	−1.34872	−0.674359	+	0.738404i	$0.735580\pi$
$548$	11728.7	0.914283
$549$	13290.0	1.03316
$550$	0	0
$551$	2254.94	0.174344
$552$	−480.954	−0.0370847
$553$	4054.22	0.311760
$554$	−434.703	−0.0333371
$555$	0	0
$556$	−7465.41	−0.569431
$557$	11671.0	0.887824	0.443912	−	0.896071i	$-0.353590\pi$
$557$	11671.0	0.887824	0.443912	+	0.896071i	$0.353590\pi$
$558$	1483.58	0.112554
$559$	−21550.9	−1.63060
$560$	0	0
$561$	−43.7644	−0.00329364
$562$	−186.158	−0.0139726
$563$	15361.6	1.14994	0.574968	−	0.818176i	$-0.305014\pi$
$563$	15361.6	1.14994	0.574968	+	0.818176i	$0.305014\pi$
$564$	929.449	0.0693916
$565$	0	0
$566$	955.112	0.0709300
$567$	10005.9	0.741110
$568$	4101.99	0.303020
$569$	11898.7	0.876659	0.438330	−	0.898814i	$-0.355570\pi$
$569$	11898.7	0.876659	0.438330	+	0.898814i	$0.355570\pi$
$570$	0	0
$571$	11971.8	0.877415	0.438707	−	0.898630i	$-0.355437\pi$
$571$	11971.8	0.877415	0.438707	+	0.898630i	$0.355437\pi$
$572$	−2467.38	−0.180361
$573$	−423.350	−0.0308651
$574$	1633.76	0.118801
$575$	0	0
$576$	12825.5	0.927767
$577$	1260.22	0.0909248	0.0454624	−	0.998966i	$-0.485524\pi$
$577$	1260.22	0.0909248	0.0454624	+	0.998966i	$0.485524\pi$
$578$	77.4373	0.00557261
$579$	1126.12	0.0808290
$580$	0	0
$581$	4204.80	0.300249
$582$	−48.8761	−0.00348107
$583$	−101.460	−0.00720763
$584$	2188.70	0.155084
$585$	0	0
$586$	−638.317	−0.0449977
$587$	26168.6	1.84002	0.920012	−	0.391890i	$-0.128179\pi$
$587$	26168.6	1.84002	0.920012	+	0.391890i	$0.128179\pi$
$588$	−755.527	−0.0529888
$589$	16655.5	1.16515
$590$	0	0
$591$	−576.873	−0.0401513
$592$	−22396.5	−1.55488
$593$	18065.1	1.25100	0.625502	−	0.780222i	$-0.284894\pi$
$593$	18065.1	1.25100	0.625502	+	0.780222i	$0.284894\pi$
$594$	36.8796	0.00254746
$595$	0	0
$596$	−19249.3	−1.32295
$597$	−392.211	−0.0268880
$598$	−3650.28	−0.249617
$599$	19324.5	1.31816	0.659079	−	0.752074i	$-0.270946\pi$
$599$	19324.5	1.31816	0.659079	+	0.752074i	$0.270946\pi$
$600$	0	0
$601$	468.279	0.0317829	0.0158914	−	0.999874i	$-0.494941\pi$
$601$	468.279	0.0317829	0.0158914	+	0.999874i	$0.494941\pi$
$602$	−951.913	−0.0644470
$603$	−14700.1	−0.992757
$604$	15285.1	1.02971
$605$	0	0
$606$	−194.611	−0.0130454
$607$	−14125.2	−0.944523	−0.472261	−	0.881459i	$-0.656562\pi$
$607$	−14125.2	−0.944523	−0.472261	+	0.881459i	$0.656562\pi$
$608$	−4046.62	−0.269921
$609$	302.504	0.0201282
$610$	0	0
$611$	14172.3	0.938379
$612$	3566.82	0.235588
$613$	11132.7	0.733515	0.366757	−	0.930317i	$-0.380468\pi$
$613$	11132.7	0.733515	0.366757	+	0.930317i	$0.380468\pi$
$614$	−2088.71	−0.137286
$615$	0	0
$616$	−218.957	−0.0143215
$617$	20287.0	1.32370	0.661852	−	0.749635i	$-0.269771\pi$
$617$	20287.0	1.32370	0.661852	+	0.749635i	$0.269771\pi$
$618$	116.150	0.00756028
$619$	7693.25	0.499544	0.249772	−	0.968305i	$-0.419644\pi$
$619$	7693.25	0.499544	0.249772	+	0.968305i	$0.419644\pi$
$620$	0	0
$621$	−6024.86	−0.389322
$622$	159.545	0.0102848
$623$	−20236.8	−1.30140
$624$	−4034.92	−0.258856
$625$	0	0
$626$	−883.454	−0.0564056
$627$	204.940	0.0130535
$628$	25879.8	1.64445
$629$	−6113.16	−0.387516
$630$	0	0
$631$	2944.36	0.185758	0.0928788	−	0.995677i	$-0.470393\pi$
$631$	2944.36	0.185758	0.0928788	+	0.995677i	$0.470393\pi$
$632$	−1186.09	−0.0746521
$633$	2681.16	0.168352
$634$	−921.089	−0.0576989
$635$	0	0
$636$	−167.448	−0.0104399
$637$	−11520.3	−0.716565
$638$	−26.6909	−0.00165627
$639$	25435.0	1.57464
$640$	0	0
$641$	3174.60	0.195615	0.0978075	−	0.995205i	$-0.468817\pi$
$641$	3174.60	0.195615	0.0978075	+	0.995205i	$0.468817\pi$
$642$	150.939	0.00927892
$643$	−13368.1	−0.819888	−0.409944	−	0.912111i	$-0.634452\pi$
$643$	−13368.1	−0.819888	−0.409944	+	0.912111i	$0.634452\pi$
$644$	17804.4	1.08943
$645$	0	0
$646$	−362.624	−0.0220855
$647$	27235.3	1.65492	0.827458	−	0.561527i	$-0.189786\pi$
$647$	27235.3	1.65492	0.827458	+	0.561527i	$0.189786\pi$
$648$	−2927.30	−0.177462
$649$	−2926.97	−0.177032
$650$	0	0
$651$	2234.35	0.134518
$652$	−25628.9	−1.53943
$653$	−23322.4	−1.39767	−0.698833	−	0.715285i	$-0.746297\pi$
$653$	−23322.4	−1.39767	−0.698833	+	0.715285i	$0.746297\pi$
$654$	−74.7653	−0.00447027
$655$	0	0
$656$	26030.9	1.54929
$657$	13571.4	0.805892
$658$	625.996	0.0370879
$659$	−3387.03	−0.200213	−0.100106	−	0.994977i	$-0.531918\pi$
$659$	−3387.03	−0.200213	−0.100106	+	0.994977i	$0.531918\pi$
$660$	0	0
$661$	3070.54	0.180681	0.0903406	−	0.995911i	$-0.471204\pi$
$661$	3070.54	0.180681	0.0903406	+	0.995911i	$0.471204\pi$
$662$	−1903.44	−0.111751
$663$	−1101.34	−0.0645135
$664$	−1230.14	−0.0718958
$665$	0	0
$666$	2549.92	0.148359
$667$	4360.37	0.253125
$668$	−27325.8	−1.58274
$669$	−2100.98	−0.121418
$670$	0	0
$671$	−1766.03	−0.101605
$672$	−542.860	−0.0311626
$673$	22497.5	1.28858	0.644290	−	0.764781i	$-0.277153\pi$
$673$	22497.5	1.28858	0.644290	+	0.764781i	$0.277153\pi$
$674$	2136.39	0.122093
$675$	0	0
$676$	−44673.8	−2.54175
$677$	7304.72	0.414687	0.207344	−	0.978268i	$-0.433518\pi$
$677$	7304.72	0.414687	0.207344	+	0.978268i	$0.433518\pi$
$678$	−212.122	−0.0120155
$679$	3635.07	0.205451
$680$	0	0
$681$	3303.25	0.185875
$682$	−197.144	−0.0110690
$683$	35085.9	1.96563	0.982814	−	0.184600i	$-0.0590989\pi$
$683$	35085.9	1.96563	0.982814	+	0.184600i	$0.0590989\pi$
$684$	−16702.7	−0.933690
$685$	0	0
$686$	−1849.63	−0.102944
$687$	309.238	0.0171735
$688$	−15166.9	−0.840457
$689$	−2553.26	−0.141178
$690$	0	0
$691$	10233.0	0.563362	0.281681	−	0.959508i	$-0.409108\pi$
$691$	10233.0	0.563362	0.281681	+	0.959508i	$0.409108\pi$
$692$	−18089.3	−0.993719
$693$	−1357.68	−0.0744213
$694$	−390.257	−0.0213458
$695$	0	0
$696$	−88.4994	−0.00481977
$697$	7105.17	0.386123
$698$	−960.017	−0.0520590
$699$	1419.80	0.0768263
$700$	0	0
$701$	−1521.72	−0.0819892	−0.0409946	−	0.999159i	$-0.513053\pi$
$701$	−1521.72	−0.0819892	−0.0409946	+	0.999159i	$0.513053\pi$
$702$	928.082	0.0498977
$703$	28626.7	1.53581
$704$	−1704.30	−0.0912404
$705$	0	0
$706$	46.5941	0.00248384
$707$	14473.8	0.769934
$708$	−4830.63	−0.256421
$709$	3525.10	0.186725	0.0933624	−	0.995632i	$-0.470238\pi$
$709$	3525.10	0.186725	0.0933624	+	0.995632i	$0.470238\pi$
$710$	0	0
$711$	−7354.54	−0.387928
$712$	5920.42	0.311625
$713$	32206.6	1.69165
$714$	−48.6465	−0.00254979
$715$	0	0
$716$	−1603.80	−0.0837104
$717$	−2341.31	−0.121949
$718$	1159.67	0.0602765
$719$	−19858.2	−1.03002	−0.515011	−	0.857184i	$-0.672212\pi$
$719$	−19858.2	−1.03002	−0.515011	+	0.857184i	$0.672212\pi$
$720$	0	0
$721$	−8638.46	−0.446204
$722$	−139.766	−0.00720438
$723$	−2090.20	−0.107518
$724$	28799.7	1.47836
$725$	0	0
$726$	258.653	0.0132225
$727$	−1726.71	−0.0880880	−0.0440440	−	0.999030i	$-0.514024\pi$
$727$	−1726.71	−0.0880880	−0.0440440	+	0.999030i	$0.514024\pi$
$728$	−5510.10	−0.280519
$729$	−17377.6	−0.882873
$730$	0	0
$731$	−4139.85	−0.209463
$732$	−2914.63	−0.147169
$733$	785.389	0.0395757	0.0197879	−	0.999804i	$-0.493701\pi$
$733$	785.389	0.0395757	0.0197879	+	0.999804i	$0.493701\pi$
$734$	−2331.26	−0.117232
$735$	0	0
$736$	−7824.93	−0.391890
$737$	1953.41	0.0976318
$738$	−2963.71	−0.147826
$739$	−16332.6	−0.812999	−0.406499	−	0.913651i	$-0.633251\pi$
$739$	−16332.6	−0.812999	−0.406499	+	0.913651i	$0.633251\pi$
$740$	0	0
$741$	5157.35	0.255682
$742$	−112.779	−0.00557983
$743$	−872.429	−0.0430772	−0.0215386	−	0.999768i	$-0.506856\pi$
$743$	−872.429	−0.0430772	−0.0215386	+	0.999768i	$0.506856\pi$
$744$	−653.675	−0.0322109
$745$	0	0
$746$	1931.52	0.0947960
$747$	−7627.70	−0.373605
$748$	−473.974	−0.0231687
$749$	−11225.8	−0.547637
$750$	0	0
$751$	38661.9	1.87855	0.939275	−	0.343165i	$-0.111499\pi$
$751$	38661.9	1.87855	0.939275	+	0.343165i	$0.111499\pi$
$752$	9974.07	0.483666
$753$	4328.02	0.209458
$754$	−671.681	−0.0324419
$755$	0	0
$756$	−4526.76	−0.217773
$757$	−8857.80	−0.425287	−0.212644	−	0.977130i	$-0.568207\pi$
$757$	−8857.80	−0.425287	−0.212644	+	0.977130i	$0.568207\pi$
$758$	−2204.39	−0.105629
$759$	396.292	0.0189519
$760$	0	0
$761$	−38965.2	−1.85609	−0.928047	−	0.372464i	$-0.878513\pi$
$761$	−38965.2	−1.85609	−0.928047	+	0.372464i	$0.878513\pi$
$762$	−88.6202	−0.00421309
$763$	5560.52	0.263833
$764$	−4584.94	−0.217117
$765$	0	0
$766$	−1763.04	−0.0831609
$767$	−73657.8	−3.46757
$768$	−2732.99	−0.128409
$769$	27086.5	1.27017	0.635086	−	0.772441i	$-0.280965\pi$
$769$	27086.5	1.27017	0.635086	+	0.772441i	$0.280965\pi$
$770$	0	0
$771$	−2131.01	−0.0995415
$772$	12196.0	0.568581
$773$	1285.94	0.0598346	0.0299173	−	0.999552i	$-0.490476\pi$
$773$	1285.94	0.0598346	0.0299173	+	0.999552i	$0.490476\pi$
$774$	1726.81	0.0801925
$775$	0	0
$776$	−1063.46	−0.0491960
$777$	3840.32	0.177311
$778$	2722.81	0.125472
$779$	−33272.1	−1.53029
$780$	0	0
$781$	−3379.91	−0.154856
$782$	−701.205	−0.0320653
$783$	−1108.62	−0.0505989
$784$	−8107.68	−0.369337
$785$	0	0
$786$	117.797	0.00534566
$787$	32037.1	1.45108	0.725541	−	0.688179i	$-0.241590\pi$
$787$	32037.1	1.45108	0.725541	+	0.688179i	$0.241590\pi$
$788$	−6247.61	−0.282439
$789$	−575.972	−0.0259888
$790$	0	0
$791$	15776.1	0.709147
$792$	397.198	0.0178205
$793$	−44442.5	−1.99016
$794$	1470.79	0.0657385
$795$	0	0
$796$	−4247.69	−0.189140
$797$	9820.04	0.436441	0.218221	−	0.975899i	$-0.429975\pi$
$797$	9820.04	0.436441	0.218221	+	0.975899i	$0.429975\pi$
$798$	227.802	0.0101054
$799$	2722.44	0.120542
$800$	0	0
$801$	36710.5	1.61935
$802$	1578.93	0.0695184
$803$	−1803.42	−0.0792546
$804$	3223.87	0.141414
$805$	0	0
$806$	−4961.17	−0.216811
$807$	−188.046	−0.00820265
$808$	−4234.41	−0.184364
$809$	−11298.4	−0.491013	−0.245506	−	0.969395i	$-0.578954\pi$
$809$	−11298.4	−0.491013	−0.245506	+	0.969395i	$0.578954\pi$
$810$	0	0
$811$	−33456.5	−1.44860	−0.724302	−	0.689483i	$-0.757838\pi$
$811$	−33456.5	−1.44860	−0.724302	+	0.689483i	$0.757838\pi$
$812$	3276.15	0.141589
$813$	1184.02	0.0510767
$814$	−338.844	−0.0145903
$815$	0	0
$816$	−775.091	−0.0332520
$817$	19386.1	0.830152
$818$	1156.81	0.0494462
$819$	−34166.2	−1.45771
$820$	0	0
$821$	−42691.4	−1.81479	−0.907394	−	0.420280i	$-0.861932\pi$
$821$	−42691.4	−1.81479	−0.907394	+	0.420280i	$0.861932\pi$
$822$	290.182	0.0123130
$823$	−19073.6	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$823$	−19073.6	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$824$	2527.24	0.106845
$825$	0	0
$826$	−3253.49	−0.137050
$827$	36029.9	1.51497	0.757486	−	0.652852i	$-0.226427\pi$
$827$	36029.9	1.51497	0.757486	+	0.652852i	$0.226427\pi$
$828$	−32298.0	−1.35559
$829$	15725.4	0.658827	0.329413	−	0.944186i	$-0.393149\pi$
$829$	15725.4	0.658827	0.329413	+	0.944186i	$0.393149\pi$
$830$	0	0
$831$	1187.63	0.0495769
$832$	−42889.0	−1.78715
$833$	−2213.01	−0.0920482
$834$	−184.702	−0.00766873
$835$	0	0
$836$	2219.53	0.0918229
$837$	−8188.51	−0.338156
$838$	−3312.97	−0.136569
$839$	−31204.9	−1.28404	−0.642021	−	0.766687i	$-0.721904\pi$
$839$	−31204.9	−1.28404	−0.642021	+	0.766687i	$0.721904\pi$
$840$	0	0
$841$	−23586.7	−0.967102
$842$	3236.08	0.132450
$843$	508.594	0.0207793
$844$	29037.3	1.18425
$845$	0	0
$846$	−1135.58	−0.0461491
$847$	−19236.8	−0.780384
$848$	−1796.91	−0.0727669
$849$	−2609.42	−0.105483
$850$	0	0
$851$	55355.4	2.22980
$852$	−5578.16	−0.224301
$853$	5921.82	0.237701	0.118851	−	0.992912i	$-0.462079\pi$
$853$	5921.82	0.237701	0.118851	+	0.992912i	$0.462079\pi$
$854$	−1963.04	−0.0786580
$855$	0	0
$856$	3284.17	0.131134
$857$	−41898.6	−1.67004	−0.835022	−	0.550216i	$-0.814545\pi$
$857$	−41898.6	−1.67004	−0.835022	+	0.550216i	$0.814545\pi$
$858$	−61.0457	−0.00242898
$859$	26802.0	1.06458	0.532289	−	0.846563i	$-0.321332\pi$
$859$	26802.0	1.06458	0.532289	+	0.846563i	$0.321332\pi$
$860$	0	0
$861$	−4463.51	−0.176674
$862$	2402.47	0.0949288
$863$	−30601.3	−1.20704	−0.603522	−	0.797346i	$-0.706236\pi$
$863$	−30601.3	−1.20704	−0.603522	+	0.797346i	$0.706236\pi$
$864$	1989.49	0.0783376
$865$	0	0
$866$	78.9050	0.00309619
$867$	−211.563	−0.00828725
$868$	24198.3	0.946250
$869$	977.302	0.0381504
$870$	0	0
$871$	49157.8	1.91234
$872$	−1626.77	−0.0631758
$873$	−6594.17	−0.255646
$874$	3283.60	0.127082
$875$	0	0
$876$	−2976.35	−0.114796
$877$	9324.64	0.359031	0.179516	−	0.983755i	$-0.442547\pi$
$877$	9324.64	0.359031	0.179516	+	0.983755i	$0.442547\pi$
$878$	1631.43	0.0627087
$879$	1743.91	0.0669178
$880$	0	0
$881$	41605.3	1.59106	0.795528	−	0.605917i	$-0.207194\pi$
$881$	41605.3	1.59106	0.795528	+	0.605917i	$0.207194\pi$
$882$	923.089	0.0352404
$883$	27653.1	1.05391	0.526953	−	0.849894i	$-0.323334\pi$
$883$	27653.1	1.05391	0.526953	+	0.849894i	$0.323334\pi$
$884$	−11927.6	−0.453812
$885$	0	0
$886$	3079.03	0.116752
$887$	−9467.39	−0.358381	−0.179191	−	0.983814i	$-0.557348\pi$
$887$	−9467.39	−0.358381	−0.179191	+	0.983814i	$0.557348\pi$
$888$	−1123.51	−0.0424578
$889$	6590.96	0.248654
$890$	0	0
$891$	2412.01	0.0906905
$892$	−22753.8	−0.854097
$893$	−12748.7	−0.477735
$894$	−476.248	−0.0178167
$895$	0	0
$896$	−7826.91	−0.291829
$897$	9972.76	0.371216
$898$	346.313	0.0128693
$899$	5926.27	0.219858
$900$	0	0
$901$	−490.471	−0.0181354
$902$	393.830	0.0145378
$903$	2600.67	0.0958417
$904$	−4615.41	−0.169808
$905$	0	0
$906$	378.171	0.0138674
$907$	−13047.1	−0.477642	−0.238821	−	0.971064i	$-0.576761\pi$
$907$	−13047.1	−0.477642	−0.238821	+	0.971064i	$0.576761\pi$
$908$	35774.6	1.30751
$909$	−26256.1	−0.958042
$910$	0	0
$911$	39715.3	1.44437	0.722187	−	0.691697i	$-0.243137\pi$
$911$	39715.3	1.44437	0.722187	+	0.691697i	$0.243137\pi$
$912$	3629.60	0.131785
$913$	1013.60	0.0367418
$914$	−2286.19	−0.0827359
$915$	0	0
$916$	3349.09	0.120805
$917$	−8760.95	−0.315498
$918$	178.281	0.00640975
$919$	−32719.0	−1.17443	−0.587215	−	0.809431i	$-0.699776\pi$
$919$	−32719.0	−1.17443	−0.587215	+	0.809431i	$0.699776\pi$
$920$	0	0
$921$	5706.45	0.204163
$922$	3181.66	0.113647
$923$	−85056.1	−3.03321
$924$	297.753	0.0106010
$925$	0	0
$926$	1256.05	0.0445750
$927$	15670.5	0.555219
$928$	−1439.85	−0.0509326
$929$	−7793.81	−0.275249	−0.137625	−	0.990484i	$-0.543947\pi$
$929$	−7793.81	−0.275249	−0.137625	+	0.990484i	$0.543947\pi$
$930$	0	0
$931$	10363.1	0.364808
$932$	15376.6	0.540425
$933$	−435.884	−0.0152950
$934$	2676.62	0.0937706
$935$	0	0
$936$	9995.56	0.349055
$937$	36379.6	1.26838	0.634189	−	0.773178i	$-0.281334\pi$
$937$	36379.6	1.26838	0.634189	+	0.773178i	$0.281334\pi$
$938$	2171.32	0.0755822
$939$	2413.64	0.0838830
$940$	0	0
$941$	40703.7	1.41010	0.705049	−	0.709159i	$-0.250925\pi$
$941$	40703.7	1.41010	0.705049	+	0.709159i	$0.250925\pi$
$942$	640.295	0.0221464
$943$	−64338.2	−2.22178
$944$	−51838.3	−1.78728
$945$	0	0
$946$	−229.466	−0.00788645
$947$	37956.0	1.30243	0.651216	−	0.758893i	$-0.274259\pi$
$947$	37956.0	1.30243	0.651216	+	0.758893i	$0.274259\pi$
$948$	1612.93	0.0552588
$949$	−45383.5	−1.55238
$950$	0	0
$951$	2516.46	0.0858064
$952$	−1058.47	−0.0360348
$953$	6618.25	0.224959	0.112480	−	0.993654i	$-0.464121\pi$
$953$	6618.25	0.224959	0.112480	+	0.993654i	$0.464121\pi$
$954$	204.585	0.00694307
$955$	0	0
$956$	−25356.6	−0.857837
$957$	72.9209	0.00246311
$958$	1478.03	0.0498465
$959$	−21581.7	−0.726705
$960$	0	0
$961$	13981.7	0.469325
$962$	−8527.07	−0.285783
$963$	20364.0	0.681434
$964$	−22637.1	−0.756320
$965$	0	0
$966$	440.501	0.0146717
$967$	−23783.2	−0.790916	−0.395458	−	0.918484i	$-0.629414\pi$
$967$	−23783.2	−0.790916	−0.395458	+	0.918484i	$0.629414\pi$
$968$	5627.86	0.186866
$969$	990.707	0.0328443
$970$	0	0
$971$	−17994.7	−0.594723	−0.297362	−	0.954765i	$-0.596107\pi$
$971$	−17994.7	−0.594723	−0.297362	+	0.954765i	$0.596107\pi$
$972$	12358.8	0.407827
$973$	13736.9	0.452604
$974$	−1175.28	−0.0386638
$975$	0	0
$976$	−31277.4	−1.02578
$977$	38188.8	1.25053	0.625266	−	0.780412i	$-0.284991\pi$
$977$	38188.8	1.25053	0.625266	+	0.780412i	$0.284991\pi$
$978$	−634.088	−0.0207320
$979$	−4878.25	−0.159254
$980$	0	0
$981$	−10087.0	−0.328292
$982$	−3620.97	−0.117668
$983$	−38392.9	−1.24572	−0.622860	−	0.782333i	$-0.714030\pi$
$983$	−38392.9	−1.24572	−0.622860	+	0.782333i	$0.714030\pi$
$984$	1305.83	0.0423052
$985$	0	0
$986$	−129.027	−0.00416741
$987$	−1710.25	−0.0551549
$988$	55854.8	1.79856
$989$	37486.8	1.20527
$990$	0	0
$991$	−51686.4	−1.65678	−0.828392	−	0.560148i	$-0.810744\pi$
$991$	−51686.4	−1.65678	−0.828392	+	0.560148i	$0.810744\pi$
$992$	−10635.0	−0.340386
$993$	5200.30	0.166190
$994$	−3756.96	−0.119883
$995$	0	0
$996$	1672.83	0.0532186
$997$	−4420.86	−0.140431	−0.0702156	−	0.997532i	$-0.522369\pi$
$997$	−4420.86	−0.140431	−0.0702156	+	0.997532i	$0.522369\pi$
$998$	−2605.70	−0.0826472
$999$	−14074.1	−0.445730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.a.e.1.1		2
5.2	odd	4		425.4.b.e.324.3		4
5.3	odd	4		425.4.b.e.324.2		4
5.4	even	2		85.4.a.d.1.2	✓	2
15.14	odd	2		765.4.a.i.1.1		2
20.19	odd	2		1360.4.a.m.1.1		2
85.84	even	2		1445.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.d.1.2	✓	2	5.4	even	2
425.4.a.e.1.1		2	1.1	even	1	trivial
425.4.b.e.324.2		4	5.3	odd	4
425.4.b.e.324.3		4	5.2	odd	4
765.4.a.i.1.1		2	15.14	odd	2
1360.4.a.m.1.1		2	20.19	odd	2
1445.4.a.i.1.2		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+0.267949 q^{2} -0.732051 q^{3} -7.92820 q^{4} -0.196152 q^{6} +14.5885 q^{7} -4.26795 q^{8} -26.4641 q^{9} +3.51666 q^{11} +5.80385 q^{12} +88.4974 q^{13} +3.90897 q^{14} +62.2820 q^{16} +17.0000 q^{17} -7.09103 q^{18} -79.6077 q^{19} -10.6795 q^{21} +0.942286 q^{22} -153.937 q^{23} +3.12436 q^{24} +23.7128 q^{26} +39.1384 q^{27} -115.660 q^{28} -28.3257 q^{29} -209.219 q^{31} +50.8320 q^{32} -2.57437 q^{33} +4.55514 q^{34} +209.813 q^{36} -359.597 q^{37} -21.3308 q^{38} -64.7846 q^{39} +417.951 q^{41} -2.86156 q^{42} -243.520 q^{43} -27.8808 q^{44} -41.2473 q^{46} +160.144 q^{47} -45.5936 q^{48} -130.177 q^{49} -12.4449 q^{51} -701.626 q^{52} -28.8513 q^{53} +10.4871 q^{54} -62.2628 q^{56} +58.2769 q^{57} -7.58984 q^{58} -832.315 q^{59} -502.190 q^{61} -56.0601 q^{62} -386.070 q^{63} -484.636 q^{64} -0.689801 q^{66} +555.472 q^{67} -134.779 q^{68} +112.690 q^{69} -961.114 q^{71} +112.947 q^{72} -512.823 q^{73} -96.3538 q^{74} +631.146 q^{76} +51.3027 q^{77} -17.3590 q^{78} +277.906 q^{79} +685.879 q^{81} +111.990 q^{82} +288.228 q^{83} +84.6692 q^{84} -65.2511 q^{86} +20.7358 q^{87} -15.0089 q^{88} -1387.18 q^{89} +1291.04 q^{91} +1220.44 q^{92} +153.159 q^{93} +42.9103 q^{94} -37.2116 q^{96} +249.174 q^{97} -34.8808 q^{98} -93.0653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 10 q^{6} - 2 q^{7} - 12 q^{8} - 46 q^{9} - 38 q^{11} + 22 q^{12} + 80 q^{13} - 58 q^{14} - 14 q^{16} + 34 q^{17} - 80 q^{18} - 180 q^{19} - 56 q^{21} - 154 q^{22} + 42 q^{23}+ \cdots + 718 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^3 - 2 * q^4 + 10 * q^6 - 2 * q^7 - 12 * q^8 - 46 * q^9 - 38 * q^11 + 22 * q^12 + 80 * q^13 - 58 * q^14 - 14 * q^16 + 34 * q^17 - 80 * q^18 - 180 * q^19 - 56 * q^21 - 154 * q^22 + 42 * q^23 - 18 * q^24 - 8 * q^26 - 88 * q^27 - 214 * q^28 - 216 * q^29 + 70 * q^31 - 172 * q^32 - 116 * q^33 + 68 * q^34 + 94 * q^36 - 352 * q^37 - 396 * q^38 - 88 * q^39 + 344 * q^41 - 172 * q^42 + 88 * q^43 - 274 * q^44 + 690 * q^46 + 348 * q^47 - 254 * q^48 - 198 * q^49 + 34 * q^51 - 752 * q^52 + 164 * q^53 - 464 * q^54 + 66 * q^56 - 216 * q^57 - 708 * q^58 - 1436 * q^59 - 76 * q^61 + 986 * q^62 - 62 * q^63 - 706 * q^64 - 424 * q^66 + 44 * q^67 - 34 * q^68 + 648 * q^69 - 1510 * q^71 + 264 * q^72 - 1088 * q^73 - 68 * q^74 + 36 * q^76 + 740 * q^77 - 104 * q^78 - 958 * q^79 + 866 * q^81 - 164 * q^82 - 248 * q^83 - 184 * q^84 + 1172 * q^86 - 492 * q^87 + 306 * q^88 - 204 * q^89 + 1432 * q^91 + 2382 * q^92 + 916 * q^93 + 744 * q^94 - 646 * q^96 - 1012 * q^97 - 288 * q^98 + 718 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more