LMFDB - Embedded newform 825.4.a.x.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 21x^{3} + 17x^{2} + 78x - 30 $ x^5 - x^4 - 21x^3 + 17x^2 + 78*x - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.98707$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.98707 q^{2} +3.00000 q^{3} +7.89672 q^{4} +11.9612 q^{6} -12.5627 q^{7} -0.411800 q^{8} +9.00000 q^{9} -11.0000 q^{11} +23.6901 q^{12} -36.0726 q^{13} -50.0885 q^{14} -64.8156 q^{16} -39.7182 q^{17} +35.8836 q^{18} -148.094 q^{19} -37.6882 q^{21} -43.8578 q^{22} +35.0665 q^{23} -1.23540 q^{24} -143.824 q^{26} +27.0000 q^{27} -99.2043 q^{28} +88.2837 q^{29} -166.874 q^{31} -255.130 q^{32} -33.0000 q^{33} -158.359 q^{34} +71.0704 q^{36} -85.2847 q^{37} -590.460 q^{38} -108.218 q^{39} +329.944 q^{41} -150.265 q^{42} +278.695 q^{43} -86.8639 q^{44} +139.813 q^{46} +272.851 q^{47} -194.447 q^{48} -185.178 q^{49} -119.155 q^{51} -284.855 q^{52} -223.266 q^{53} +107.651 q^{54} +5.17334 q^{56} -444.282 q^{57} +351.993 q^{58} -467.463 q^{59} +752.249 q^{61} -665.338 q^{62} -113.065 q^{63} -498.695 q^{64} -131.573 q^{66} -733.563 q^{67} -313.643 q^{68} +105.200 q^{69} +537.226 q^{71} -3.70620 q^{72} -397.895 q^{73} -340.036 q^{74} -1169.46 q^{76} +138.190 q^{77} -431.471 q^{78} -1079.06 q^{79} +81.0000 q^{81} +1315.51 q^{82} -683.484 q^{83} -297.613 q^{84} +1111.18 q^{86} +264.851 q^{87} +4.52980 q^{88} -166.100 q^{89} +453.170 q^{91} +276.911 q^{92} -500.622 q^{93} +1087.87 q^{94} -765.390 q^{96} +694.031 q^{97} -738.317 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 15 q^{3} + 3 q^{4} - 3 q^{6} - 18 q^{7} + 3 q^{8} + 45 q^{9} - 55 q^{11} + 9 q^{12} - 31 q^{13} + 8 q^{14} - 125 q^{16} - 38 q^{17} - 9 q^{18} - 57 q^{19} - 54 q^{21} + 11 q^{22} - 161 q^{23}+ \cdots - 495 q^{99}+O(q^{100}) $ 5 * q - q^2 + 15 * q^3 + 3 * q^4 - 3 * q^6 - 18 * q^7 + 3 * q^8 + 45 * q^9 - 55 * q^11 + 9 * q^12 - 31 * q^13 + 8 * q^14 - 125 * q^16 - 38 * q^17 - 9 * q^18 - 57 * q^19 - 54 * q^21 + 11 * q^22 - 161 * q^23 + 9 * q^24 - 125 * q^26 + 135 * q^27 - 324 * q^28 - 107 * q^29 - 295 * q^31 + 23 * q^32 - 165 * q^33 + 34 * q^34 + 27 * q^36 + 260 * q^37 - 619 * q^38 - 93 * q^39 - 128 * q^41 + 24 * q^42 - 377 * q^43 - 33 * q^44 - 577 * q^46 + 114 * q^47 - 375 * q^48 - 415 * q^49 - 114 * q^51 + 395 * q^52 - 812 * q^53 - 27 * q^54 - 132 * q^56 - 171 * q^57 - 339 * q^58 - 1152 * q^59 - 344 * q^61 + 235 * q^62 - 162 * q^63 - 545 * q^64 + 33 * q^66 + 928 * q^67 - 654 * q^68 - 483 * q^69 - 707 * q^71 + 27 * q^72 - 322 * q^73 - 1176 * q^74 - 1699 * q^76 + 198 * q^77 - 375 * q^78 - 2494 * q^79 + 405 * q^81 + 2776 * q^82 - 1657 * q^83 - 972 * q^84 - 799 * q^86 - 321 * q^87 - 33 * q^88 - 2435 * q^89 - 804 * q^91 + 2775 * q^92 - 885 * q^93 - 502 * q^94 + 69 * q^96 + 1901 * q^97 - 343 * q^98 - 495 * 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 $

$2$	3.98707	1.40964	0.704821	−	0.709385i	$-0.251027\pi$
$2$	3.98707	1.40964	0.704821	+	0.709385i	$0.251027\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	7.89672	0.987090
$5$	0	0
$5$	0	0
$6$	11.9612	0.813857
$7$	−12.5627	−0.678324	−0.339162	−	0.940728i	$-0.610143\pi$
$7$	−12.5627	−0.678324	−0.339162	+	0.940728i	$0.610143\pi$
$8$	−0.411800	−0.0181992
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	23.6901	0.569896
$13$	−36.0726	−0.769594	−0.384797	−	0.923001i	$-0.625729\pi$
$13$	−36.0726	−0.769594	−0.384797	+	0.923001i	$0.625729\pi$
$14$	−50.0885	−0.956193
$15$	0	0
$16$	−64.8156	−1.01274
$17$	−39.7182	−0.566652	−0.283326	−	0.959024i	$-0.591438\pi$
$17$	−39.7182	−0.566652	−0.283326	+	0.959024i	$0.591438\pi$
$18$	35.8836	0.469881
$19$	−148.094	−1.78816	−0.894081	−	0.447906i	$-0.852170\pi$
$19$	−148.094	−1.78816	−0.894081	+	0.447906i	$0.852170\pi$
$20$	0	0
$21$	−37.6882	−0.391630
$22$	−43.8578	−0.425023
$23$	35.0665	0.317908	0.158954	−	0.987286i	$-0.449188\pi$
$23$	35.0665	0.317908	0.158954	+	0.987286i	$0.449188\pi$
$24$	−1.23540	−0.0105073
$25$	0	0
$26$	−143.824	−1.08485
$27$	27.0000	0.192450
$28$	−99.2043	−0.669566
$29$	88.2837	0.565306	0.282653	−	0.959222i	$-0.408785\pi$
$29$	88.2837	0.565306	0.282653	+	0.959222i	$0.408785\pi$
$30$	0	0
$31$	−166.874	−0.966820	−0.483410	−	0.875394i	$-0.660602\pi$
$31$	−166.874	−0.966820	−0.483410	+	0.875394i	$0.660602\pi$
$32$	−255.130	−1.40941
$33$	−33.0000	−0.174078
$34$	−158.359	−0.798776
$35$	0	0
$36$	71.0704	0.329030
$37$	−85.2847	−0.378939	−0.189469	−	0.981887i	$-0.560677\pi$
$37$	−85.2847	−0.378939	−0.189469	+	0.981887i	$0.560677\pi$
$38$	−590.460	−2.52067
$39$	−108.218	−0.444326
$40$	0	0
$41$	329.944	1.25679	0.628397	−	0.777893i	$-0.283711\pi$
$41$	329.944	1.25679	0.628397	+	0.777893i	$0.283711\pi$
$42$	−150.265	−0.552058
$43$	278.695	0.988386	0.494193	−	0.869352i	$-0.335464\pi$
$43$	278.695	0.988386	0.494193	+	0.869352i	$0.335464\pi$
$44$	−86.8639	−0.297619
$45$	0	0
$46$	139.813	0.448136
$47$	272.851	0.846795	0.423397	−	0.905944i	$-0.360837\pi$
$47$	272.851	0.846795	0.423397	+	0.905944i	$0.360837\pi$
$48$	−194.447	−0.584708
$49$	−185.178	−0.539877
$50$	0	0
$51$	−119.155	−0.327157
$52$	−284.855	−0.759659
$53$	−223.266	−0.578640	−0.289320	−	0.957232i	$-0.593429\pi$
$53$	−223.266	−0.578640	−0.289320	+	0.957232i	$0.593429\pi$
$54$	107.651	0.271286
$55$	0	0
$56$	5.17334	0.0123449
$57$	−444.282	−1.03240
$58$	351.993	0.796879
$59$	−467.463	−1.03150	−0.515750	−	0.856739i	$-0.672487\pi$
$59$	−467.463	−1.03150	−0.515750	+	0.856739i	$0.672487\pi$
$60$	0	0
$61$	752.249	1.57894	0.789472	−	0.613786i	$-0.210354\pi$
$61$	752.249	1.57894	0.789472	+	0.613786i	$0.210354\pi$
$62$	−665.338	−1.36287
$63$	−113.065	−0.226108
$64$	−498.695	−0.974014
$65$	0	0
$66$	−131.573	−0.245387
$67$	−733.563	−1.33760	−0.668798	−	0.743444i	$-0.733191\pi$
$67$	−733.563	−1.33760	−0.668798	+	0.743444i	$0.733191\pi$
$68$	−313.643	−0.559336
$69$	105.200	0.183544
$70$	0	0
$71$	537.226	0.897985	0.448993	−	0.893535i	$-0.351783\pi$
$71$	537.226	0.897985	0.448993	+	0.893535i	$0.351783\pi$
$72$	−3.70620	−0.00606639
$73$	−397.895	−0.637947	−0.318974	−	0.947764i	$-0.603338\pi$
$73$	−397.895	−0.637947	−0.318974	+	0.947764i	$0.603338\pi$
$74$	−340.036	−0.534168
$75$	0	0
$76$	−1169.46	−1.76508
$77$	138.190	0.204522
$78$	−431.471	−0.626340
$79$	−1079.06	−1.53675	−0.768375	−	0.640000i	$-0.778934\pi$
$79$	−1079.06	−1.53675	−0.768375	+	0.640000i	$0.778934\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1315.51	1.77163
$83$	−683.484	−0.903881	−0.451941	−	0.892048i	$-0.649268\pi$
$83$	−683.484	−0.903881	−0.451941	+	0.892048i	$0.649268\pi$
$84$	−297.613	−0.386574
$85$	0	0
$86$	1111.18	1.39327
$87$	264.851	0.326380
$88$	4.52980	0.00548726
$89$	−166.100	−0.197826	−0.0989130	−	0.995096i	$-0.531537\pi$
$89$	−166.100	−0.197826	−0.0989130	+	0.995096i	$0.531537\pi$
$90$	0	0
$91$	453.170	0.522034
$92$	276.911	0.313803
$93$	−500.622	−0.558194
$94$	1087.87	1.19368
$95$	0	0
$96$	−765.390	−0.813721
$97$	694.031	0.726476	0.363238	−	0.931696i	$-0.381671\pi$
$97$	694.031	0.726476	0.363238	+	0.931696i	$0.381671\pi$
$98$	−738.317	−0.761033
$99$	−99.0000	−0.100504
$100$	0	0
$101$	447.846	0.441211	0.220605	−	0.975363i	$-0.429197\pi$
$101$	447.846	0.441211	0.220605	+	0.975363i	$0.429197\pi$
$102$	−475.078	−0.461174
$103$	1314.02	1.25703	0.628516	−	0.777797i	$-0.283663\pi$
$103$	1314.02	1.25703	0.628516	+	0.777797i	$0.283663\pi$
$104$	14.8547	0.0140060
$105$	0	0
$106$	−890.176	−0.815675
$107$	1048.81	0.947594	0.473797	−	0.880634i	$-0.342883\pi$
$107$	1048.81	0.947594	0.473797	+	0.880634i	$0.342883\pi$
$108$	213.211	0.189965
$109$	−1240.73	−1.09028	−0.545140	−	0.838345i	$-0.683523\pi$
$109$	−1240.73	−1.09028	−0.545140	+	0.838345i	$0.683523\pi$
$110$	0	0
$111$	−255.854	−0.218780
$112$	814.261	0.686968
$113$	−1331.27	−1.10828	−0.554138	−	0.832425i	$-0.686952\pi$
$113$	−1331.27	−1.10828	−0.554138	+	0.832425i	$0.686952\pi$
$114$	−1771.38	−1.45531
$115$	0	0
$116$	697.152	0.558008
$117$	−324.653	−0.256531
$118$	−1863.81	−1.45405
$119$	498.969	0.384373
$120$	0	0
$121$	121.000	0.0909091
$122$	2999.27	2.22575
$123$	989.832	0.725611
$124$	−1317.76	−0.954338
$125$	0	0
$126$	−450.796	−0.318731
$127$	−1250.59	−0.873791	−0.436896	−	0.899512i	$-0.643922\pi$
$127$	−1250.59	−0.873791	−0.436896	+	0.899512i	$0.643922\pi$
$128$	52.7060	0.0363953
$129$	836.085	0.570645
$130$	0	0
$131$	−168.067	−0.112092	−0.0560460	−	0.998428i	$-0.517849\pi$
$131$	−168.067	−0.112092	−0.0560460	+	0.998428i	$0.517849\pi$
$132$	−260.592	−0.171830
$133$	1860.46	1.21295
$134$	−2924.77	−1.88553
$135$	0	0
$136$	16.3560	0.0103126
$137$	1533.67	0.956428	0.478214	−	0.878243i	$-0.341284\pi$
$137$	1533.67	0.956428	0.478214	+	0.878243i	$0.341284\pi$
$138$	419.438	0.258731
$139$	−961.259	−0.586568	−0.293284	−	0.956025i	$-0.594748\pi$
$139$	−961.259	−0.586568	−0.293284	+	0.956025i	$0.594748\pi$
$140$	0	0
$141$	818.552	0.488897
$142$	2141.96	1.26584
$143$	396.798	0.232041
$144$	−583.340	−0.337581
$145$	0	0
$146$	−1586.44	−0.899277
$147$	−555.534	−0.311698
$148$	−673.469	−0.374046
$149$	−70.0037	−0.0384894	−0.0192447	−	0.999815i	$-0.506126\pi$
$149$	−70.0037	−0.0384894	−0.0192447	+	0.999815i	$0.506126\pi$
$150$	0	0
$151$	210.529	0.113461	0.0567306	−	0.998390i	$-0.481932\pi$
$151$	210.529	0.113461	0.0567306	+	0.998390i	$0.481932\pi$
$152$	60.9851	0.0325431
$153$	−357.464	−0.188884
$154$	550.973	0.288303
$155$	0	0
$156$	−854.564	−0.438589
$157$	2239.44	1.13839	0.569193	−	0.822204i	$-0.307256\pi$
$157$	2239.44	1.13839	0.569193	+	0.822204i	$0.307256\pi$
$158$	−4302.27	−2.16627
$159$	−669.798	−0.334078
$160$	0	0
$161$	−440.531	−0.215644
$162$	322.953	0.156627
$163$	1918.08	0.921692	0.460846	−	0.887480i	$-0.347546\pi$
$163$	1918.08	0.921692	0.460846	+	0.887480i	$0.347546\pi$
$164$	2605.47	1.24057
$165$	0	0
$166$	−2725.10	−1.27415
$167$	−3036.53	−1.40703	−0.703513	−	0.710682i	$-0.748386\pi$
$167$	−3036.53	−1.40703	−0.703513	+	0.710682i	$0.748386\pi$
$168$	15.5200	0.00712735
$169$	−895.771	−0.407724
$170$	0	0
$171$	−1332.84	−0.596054
$172$	2200.78	0.975625
$173$	1713.66	0.753104	0.376552	−	0.926396i	$-0.377110\pi$
$173$	1713.66	0.753104	0.376552	+	0.926396i	$0.377110\pi$
$174$	1055.98	0.460078
$175$	0	0
$176$	712.972	0.305354
$177$	−1402.39	−0.595537
$178$	−662.250	−0.278864
$179$	−901.687	−0.376510	−0.188255	−	0.982120i	$-0.560283\pi$
$179$	−901.687	−0.376510	−0.188255	+	0.982120i	$0.560283\pi$
$180$	0	0
$181$	−2970.07	−1.21969	−0.609844	−	0.792521i	$-0.708768\pi$
$181$	−2970.07	−1.21969	−0.609844	+	0.792521i	$0.708768\pi$
$182$	1806.82	0.735881
$183$	2256.75	0.911604
$184$	−14.4404	−0.00578566
$185$	0	0
$186$	−1996.01	−0.786854
$187$	436.900	0.170852
$188$	2154.62	0.835862
$189$	−339.194	−0.130543
$190$	0	0
$191$	611.910	0.231813	0.115906	−	0.993260i	$-0.463023\pi$
$191$	611.910	0.231813	0.115906	+	0.993260i	$0.463023\pi$
$192$	−1496.09	−0.562348
$193$	1902.58	0.709589	0.354795	−	0.934944i	$-0.384551\pi$
$193$	1902.58	0.709589	0.354795	+	0.934944i	$0.384551\pi$
$194$	2767.15	1.02407
$195$	0	0
$196$	−1462.30	−0.532907
$197$	268.748	0.0971956	0.0485978	−	0.998818i	$-0.484525\pi$
$197$	268.748	0.0971956	0.0485978	+	0.998818i	$0.484525\pi$
$198$	−394.720	−0.141674
$199$	−229.583	−0.0817825	−0.0408912	−	0.999164i	$-0.513020\pi$
$199$	−229.583	−0.0817825	−0.0408912	+	0.999164i	$0.513020\pi$
$200$	0	0
$201$	−2200.69	−0.772262
$202$	1785.59	0.621949
$203$	−1109.08	−0.383460
$204$	−940.930	−0.322933
$205$	0	0
$206$	5239.09	1.77196
$207$	315.599	0.105969
$208$	2338.06	0.779402
$209$	1629.03	0.539151
$210$	0	0
$211$	−5111.79	−1.66782	−0.833910	−	0.551901i	$-0.813903\pi$
$211$	−5111.79	−1.66782	−0.833910	+	0.551901i	$0.813903\pi$
$212$	−1763.07	−0.571170
$213$	1611.68	0.518452
$214$	4181.69	1.33577
$215$	0	0
$216$	−11.1186	−0.00350243
$217$	2096.39	0.655817
$218$	−4946.88	−1.53690
$219$	−1193.69	−0.368319
$220$	0	0
$221$	1432.74	0.436092
$222$	−1020.11	−0.308402
$223$	1393.60	0.418486	0.209243	−	0.977864i	$-0.432900\pi$
$223$	1393.60	0.418486	0.209243	+	0.977864i	$0.432900\pi$
$224$	3205.13	0.956034
$225$	0	0
$226$	−5307.86	−1.56227
$227$	6244.88	1.82593	0.912967	−	0.408032i	$-0.133785\pi$
$227$	6244.88	1.82593	0.912967	+	0.408032i	$0.133785\pi$
$228$	−3508.37	−1.01907
$229$	3295.16	0.950873	0.475437	−	0.879750i	$-0.342290\pi$
$229$	3295.16	0.950873	0.475437	+	0.879750i	$0.342290\pi$
$230$	0	0
$231$	414.570	0.118081
$232$	−36.3553	−0.0102881
$233$	−961.850	−0.270442	−0.135221	−	0.990815i	$-0.543174\pi$
$233$	−961.850	−0.270442	−0.135221	+	0.990815i	$0.543174\pi$
$234$	−1294.41	−0.361617
$235$	0	0
$236$	−3691.42	−1.01818
$237$	−3237.17	−0.887242
$238$	1989.42	0.541829
$239$	−4738.03	−1.28233	−0.641166	−	0.767402i	$-0.721549\pi$
$239$	−4738.03	−1.28233	−0.641166	+	0.767402i	$0.721549\pi$
$240$	0	0
$241$	−3928.79	−1.05011	−0.525053	−	0.851070i	$-0.675954\pi$
$241$	−3928.79	−1.05011	−0.525053	+	0.851070i	$0.675954\pi$
$242$	482.435	0.128149
$243$	243.000	0.0641500
$244$	5940.30	1.55856
$245$	0	0
$246$	3946.53	1.02285
$247$	5342.12	1.37616
$248$	68.7187	0.0175953
$249$	−2050.45	−0.521856
$250$	0	0
$251$	7733.06	1.94465	0.972324	−	0.233638i	$-0.0750632\pi$
$251$	7733.06	1.94465	0.972324	+	0.233638i	$0.0750632\pi$
$252$	−892.839	−0.223189
$253$	−385.732	−0.0958528
$254$	−4986.17	−1.23173
$255$	0	0
$256$	4199.71	1.02532
$257$	1574.17	0.382079	0.191039	−	0.981582i	$-0.438814\pi$
$257$	1574.17	0.382079	0.191039	+	0.981582i	$0.438814\pi$
$258$	3333.53	0.804405
$259$	1071.41	0.257043
$260$	0	0
$261$	794.554	0.188435
$262$	−670.093	−0.158009
$263$	3172.64	0.743853	0.371927	−	0.928262i	$-0.378697\pi$
$263$	3172.64	0.743853	0.371927	+	0.928262i	$0.378697\pi$
$264$	13.5894	0.00316807
$265$	0	0
$266$	7417.79	1.70983
$267$	−498.299	−0.114215
$268$	−5792.74	−1.32033
$269$	−1238.83	−0.280791	−0.140396	−	0.990095i	$-0.544837\pi$
$269$	−1238.83	−0.280791	−0.140396	+	0.990095i	$0.544837\pi$
$270$	0	0
$271$	−2257.02	−0.505920	−0.252960	−	0.967477i	$-0.581404\pi$
$271$	−2257.02	−0.505920	−0.252960	+	0.967477i	$0.581404\pi$
$272$	2574.36	0.573873
$273$	1359.51	0.301396
$274$	6114.86	1.34822
$275$	0	0
$276$	830.732	0.181175
$277$	−8207.95	−1.78039	−0.890195	−	0.455580i	$-0.849432\pi$
$277$	−8207.95	−1.78039	−0.890195	+	0.455580i	$0.849432\pi$
$278$	−3832.60	−0.826850
$279$	−1501.87	−0.322273
$280$	0	0
$281$	−3499.20	−0.742864	−0.371432	−	0.928460i	$-0.621133\pi$
$281$	−3499.20	−0.742864	−0.371432	+	0.928460i	$0.621133\pi$
$282$	3263.62	0.689170
$283$	−8614.14	−1.80939	−0.904695	−	0.426059i	$-0.859901\pi$
$283$	−8614.14	−1.80939	−0.904695	+	0.426059i	$0.859901\pi$
$284$	4242.32	0.886392
$285$	0	0
$286$	1582.06	0.327095
$287$	−4145.00	−0.852513
$288$	−2296.17	−0.469802
$289$	−3335.46	−0.678906
$290$	0	0
$291$	2082.09	0.419431
$292$	−3142.07	−0.629711
$293$	1028.59	0.205088	0.102544	−	0.994728i	$-0.467302\pi$
$293$	1028.59	0.205088	0.102544	+	0.994728i	$0.467302\pi$
$294$	−2214.95	−0.439383
$295$	0	0
$296$	35.1203	0.00689637
$297$	−297.000	−0.0580259
$298$	−279.109	−0.0542563
$299$	−1264.94	−0.244660
$300$	0	0
$301$	−3501.17	−0.670446
$302$	839.395	0.159940
$303$	1343.54	0.254733
$304$	9598.79	1.81095
$305$	0	0
$306$	−1425.23	−0.266259
$307$	−2168.83	−0.403197	−0.201599	−	0.979468i	$-0.564614\pi$
$307$	−2168.83	−0.403197	−0.201599	+	0.979468i	$0.564614\pi$
$308$	1091.25	0.201882
$309$	3942.06	0.725748
$310$	0	0
$311$	4318.45	0.787386	0.393693	−	0.919242i	$-0.371197\pi$
$311$	4318.45	0.787386	0.393693	+	0.919242i	$0.371197\pi$
$312$	44.5641	0.00808636
$313$	4785.34	0.864164	0.432082	−	0.901834i	$-0.357779\pi$
$313$	4785.34	0.864164	0.432082	+	0.901834i	$0.357779\pi$
$314$	8928.79	1.60472
$315$	0	0
$316$	−8520.99	−1.51691
$317$	−6680.52	−1.18364	−0.591822	−	0.806069i	$-0.701591\pi$
$317$	−6680.52	−1.18364	−0.591822	+	0.806069i	$0.701591\pi$
$318$	−2670.53	−0.470930
$319$	−971.121	−0.170446
$320$	0	0
$321$	3146.44	0.547094
$322$	−1756.43	−0.303981
$323$	5882.02	1.01326
$324$	639.634	0.109677
$325$	0	0
$326$	7647.52	1.29926
$327$	−3722.19	−0.629473
$328$	−135.871	−0.0228726
$329$	−3427.75	−0.574401
$330$	0	0
$331$	3428.85	0.569386	0.284693	−	0.958619i	$-0.408108\pi$
$331$	3428.85	0.569386	0.284693	+	0.958619i	$0.408108\pi$
$332$	−5397.28	−0.892212
$333$	−767.563	−0.126313
$334$	−12106.8	−1.98340
$335$	0	0
$336$	2442.78	0.396621
$337$	7809.83	1.26240	0.631200	−	0.775620i	$-0.282563\pi$
$337$	7809.83	1.26240	0.631200	+	0.775620i	$0.282563\pi$
$338$	−3571.50	−0.574745
$339$	−3993.80	−0.639863
$340$	0	0
$341$	1835.61	0.291507
$342$	−5314.14	−0.840222
$343$	6635.35	1.04453
$344$	−114.767	−0.0179878
$345$	0	0
$346$	6832.47	1.06161
$347$	−5860.34	−0.906627	−0.453314	−	0.891351i	$-0.649758\pi$
$347$	−5860.34	−0.906627	−0.453314	+	0.891351i	$0.649758\pi$
$348$	2091.45	0.322166
$349$	5833.65	0.894750	0.447375	−	0.894346i	$-0.352359\pi$
$349$	5833.65	0.894750	0.447375	+	0.894346i	$0.352359\pi$
$350$	0	0
$351$	−973.959	−0.148109
$352$	2806.43	0.424952
$353$	4532.47	0.683397	0.341699	−	0.939810i	$-0.388998\pi$
$353$	4532.47	0.683397	0.341699	+	0.939810i	$0.388998\pi$
$354$	−5591.42	−0.839494
$355$	0	0
$356$	−1311.64	−0.195272
$357$	1496.91	0.221918
$358$	−3595.09	−0.530744
$359$	−8727.48	−1.28306	−0.641530	−	0.767098i	$-0.721700\pi$
$359$	−8727.48	−1.28306	−0.641530	+	0.767098i	$0.721700\pi$
$360$	0	0
$361$	15072.8	2.19752
$362$	−11841.9	−1.71932
$363$	363.000	0.0524864
$364$	3578.55	0.515294
$365$	0	0
$366$	8997.80	1.28503
$367$	−11397.8	−1.62114	−0.810572	−	0.585639i	$-0.800844\pi$
$367$	−11397.8	−1.62114	−0.810572	+	0.585639i	$0.800844\pi$
$368$	−2272.86	−0.321959
$369$	2969.50	0.418932
$370$	0	0
$371$	2804.83	0.392505
$372$	−3953.27	−0.550988
$373$	10698.2	1.48507	0.742535	−	0.669807i	$-0.233623\pi$
$373$	10698.2	1.48507	0.742535	+	0.669807i	$0.233623\pi$
$374$	1741.95	0.240840
$375$	0	0
$376$	−112.360	−0.0154110
$377$	−3184.62	−0.435056
$378$	−1352.39	−0.184019
$379$	6055.57	0.820723	0.410361	−	0.911923i	$-0.365403\pi$
$379$	6055.57	0.820723	0.410361	+	0.911923i	$0.365403\pi$
$380$	0	0
$381$	−3751.76	−0.504484
$382$	2439.73	0.326773
$383$	−13989.5	−1.86640	−0.933198	−	0.359363i	$-0.882994\pi$
$383$	−13989.5	−1.86640	−0.933198	+	0.359363i	$0.882994\pi$
$384$	158.118	0.0210128
$385$	0	0
$386$	7585.72	1.00027
$387$	2508.26	0.329462
$388$	5480.56	0.717097
$389$	4223.88	0.550538	0.275269	−	0.961367i	$-0.411233\pi$
$389$	4223.88	0.550538	0.275269	+	0.961367i	$0.411233\pi$
$390$	0	0
$391$	−1392.78	−0.180143
$392$	76.2563	0.00982532
$393$	−504.200	−0.0647163
$394$	1071.52	0.137011
$395$	0	0
$396$	−781.775	−0.0992062
$397$	3341.86	0.422476	0.211238	−	0.977435i	$-0.432250\pi$
$397$	3341.86	0.422476	0.211238	+	0.977435i	$0.432250\pi$
$398$	−915.364	−0.115284
$399$	5581.39	0.700298
$400$	0	0
$401$	−7167.91	−0.892639	−0.446320	−	0.894874i	$-0.647266\pi$
$401$	−7167.91	−0.892639	−0.446320	+	0.894874i	$0.647266\pi$
$402$	−8774.30	−1.08861
$403$	6019.57	0.744060
$404$	3536.51	0.435515
$405$	0	0
$406$	−4422.00	−0.540542
$407$	938.132	0.114254
$408$	49.0679	0.00595398
$409$	−10360.7	−1.25257	−0.626287	−	0.779592i	$-0.715426\pi$
$409$	−10360.7	−1.25257	−0.626287	+	0.779592i	$0.715426\pi$
$410$	0	0
$411$	4601.02	0.552194
$412$	10376.4	1.24080
$413$	5872.61	0.699691
$414$	1258.31	0.149379
$415$	0	0
$416$	9203.19	1.08467
$417$	−2883.78	−0.338655
$418$	6495.06	0.760009
$419$	−7640.60	−0.890853	−0.445427	−	0.895318i	$-0.646948\pi$
$419$	−7640.60	−0.890853	−0.445427	+	0.895318i	$0.646948\pi$
$420$	0	0
$421$	−1013.65	−0.117345	−0.0586724	−	0.998277i	$-0.518687\pi$
$421$	−1013.65	−0.117345	−0.0586724	+	0.998277i	$0.518687\pi$
$422$	−20381.0	−2.35103
$423$	2455.66	0.282265
$424$	91.9410	0.0105308
$425$	0	0
$426$	6425.87	0.730831
$427$	−9450.30	−1.07104
$428$	8282.18	0.935360
$429$	1190.39	0.133969
$430$	0	0
$431$	−13161.9	−1.47097	−0.735485	−	0.677541i	$-0.763046\pi$
$431$	−13161.9	−1.47097	−0.735485	+	0.677541i	$0.763046\pi$
$432$	−1750.02	−0.194903
$433$	3886.06	0.431298	0.215649	−	0.976471i	$-0.430813\pi$
$433$	3886.06	0.431298	0.215649	+	0.976471i	$0.430813\pi$
$434$	8358.46	0.924467
$435$	0	0
$436$	−9797.70	−1.07620
$437$	−5193.14	−0.568470
$438$	−4759.31	−0.519198
$439$	−384.942	−0.0418503	−0.0209252	−	0.999781i	$-0.506661\pi$
$439$	−384.942	−0.0418503	−0.0209252	+	0.999781i	$0.506661\pi$
$440$	0	0
$441$	−1666.60	−0.179959
$442$	5712.42	0.614734
$443$	13324.7	1.42907	0.714533	−	0.699601i	$-0.246639\pi$
$443$	13324.7	1.42907	0.714533	+	0.699601i	$0.246639\pi$
$444$	−2020.41	−0.215956
$445$	0	0
$446$	5556.38	0.589915
$447$	−210.011	−0.0222219
$448$	6264.97	0.660697
$449$	−3242.44	−0.340802	−0.170401	−	0.985375i	$-0.554506\pi$
$449$	−3242.44	−0.340802	−0.170401	+	0.985375i	$0.554506\pi$
$450$	0	0
$451$	−3629.38	−0.378938
$452$	−10512.6	−1.09397
$453$	631.588	0.0655068
$454$	24898.8	2.57391
$455$	0	0
$456$	182.955	0.0187887
$457$	3686.70	0.377366	0.188683	−	0.982038i	$-0.439578\pi$
$457$	3686.70	0.377366	0.188683	+	0.982038i	$0.439578\pi$
$458$	13138.0	1.34039
$459$	−1072.39	−0.109052
$460$	0	0
$461$	−19219.2	−1.94171	−0.970856	−	0.239664i	$-0.922963\pi$
$461$	−19219.2	−1.94171	−0.970856	+	0.239664i	$0.922963\pi$
$462$	1652.92	0.166452
$463$	1346.60	0.135166	0.0675832	−	0.997714i	$-0.478471\pi$
$463$	1346.60	0.135166	0.0675832	+	0.997714i	$0.478471\pi$
$464$	−5722.16	−0.572510
$465$	0	0
$466$	−3834.96	−0.381226
$467$	8913.27	0.883205	0.441603	−	0.897211i	$-0.354410\pi$
$467$	8913.27	0.883205	0.441603	+	0.897211i	$0.354410\pi$
$468$	−2563.69	−0.253220
$469$	9215.55	0.907323
$470$	0	0
$471$	6718.31	0.657247
$472$	192.501	0.0187725
$473$	−3065.65	−0.298010
$474$	−12906.8	−1.25069
$475$	0	0
$476$	3940.22	0.379411
$477$	−2009.39	−0.192880
$478$	−18890.8	−1.80763
$479$	−11726.8	−1.11860	−0.559300	−	0.828965i	$-0.688930\pi$
$479$	−11726.8	−1.11860	−0.559300	+	0.828965i	$0.688930\pi$
$480$	0	0
$481$	3076.44	0.291629
$482$	−15664.3	−1.48027
$483$	−1321.59	−0.124502
$484$	955.503	0.0897354
$485$	0	0
$486$	968.858	0.0904286
$487$	8115.91	0.755168	0.377584	−	0.925975i	$-0.376755\pi$
$487$	8115.91	0.755168	0.377584	+	0.925975i	$0.376755\pi$
$488$	−309.776	−0.0287355
$489$	5754.25	0.532139
$490$	0	0
$491$	−12161.5	−1.11780	−0.558901	−	0.829235i	$-0.688777\pi$
$491$	−12161.5	−1.11780	−0.558901	+	0.829235i	$0.688777\pi$
$492$	7816.42	0.716243
$493$	−3506.47	−0.320332
$494$	21299.4	1.93989
$495$	0	0
$496$	10816.0	0.979141
$497$	−6749.02	−0.609125
$498$	−8175.30	−0.735630
$499$	5696.54	0.511046	0.255523	−	0.966803i	$-0.417752\pi$
$499$	5696.54	0.511046	0.255523	+	0.966803i	$0.417752\pi$
$500$	0	0
$501$	−9109.58	−0.812347
$502$	30832.2	2.74126
$503$	9287.58	0.823286	0.411643	−	0.911345i	$-0.364955\pi$
$503$	9287.58	0.823286	0.411643	+	0.911345i	$0.364955\pi$
$504$	46.5600	0.00411498
$505$	0	0
$506$	−1537.94	−0.135118
$507$	−2687.31	−0.235400
$508$	−9875.52	−0.862510
$509$	−8316.30	−0.724192	−0.362096	−	0.932141i	$-0.617939\pi$
$509$	−8316.30	−0.724192	−0.362096	+	0.932141i	$0.617939\pi$
$510$	0	0
$511$	4998.65	0.432735
$512$	16322.9	1.40894
$513$	−3998.53	−0.344132
$514$	6276.34	0.538594
$515$	0	0
$516$	6602.33	0.563278
$517$	−3001.36	−0.255318
$518$	4271.78	0.362338
$519$	5140.97	0.434805
$520$	0	0
$521$	−4349.68	−0.365764	−0.182882	−	0.983135i	$-0.558543\pi$
$521$	−4349.68	−0.365764	−0.182882	+	0.983135i	$0.558543\pi$
$522$	3167.94	0.265626
$523$	6877.36	0.575002	0.287501	−	0.957780i	$-0.407176\pi$
$523$	6877.36	0.575002	0.287501	+	0.957780i	$0.407176\pi$
$524$	−1327.17	−0.110645
$525$	0	0
$526$	12649.5	1.04857
$527$	6627.93	0.547851
$528$	2138.91	0.176296
$529$	−10937.3	−0.898935
$530$	0	0
$531$	−4207.17	−0.343833
$532$	14691.5	1.19729
$533$	−11901.9	−0.967222
$534$	−1986.75	−0.161002
$535$	0	0
$536$	302.081	0.0243432
$537$	−2705.06	−0.217378
$538$	−4939.30	−0.395815
$539$	2036.96	0.162779
$540$	0	0
$541$	9908.33	0.787417	0.393708	−	0.919235i	$-0.371192\pi$
$541$	9908.33	0.787417	0.393708	+	0.919235i	$0.371192\pi$
$542$	−8998.90	−0.713166
$543$	−8910.21	−0.704187
$544$	10133.3	0.798643
$545$	0	0
$546$	5420.46	0.424861
$547$	8052.99	0.629472	0.314736	−	0.949179i	$-0.398084\pi$
$547$	8052.99	0.629472	0.314736	+	0.949179i	$0.398084\pi$
$548$	12111.0	0.944080
$549$	6770.24	0.526315
$550$	0	0
$551$	−13074.3	−1.01086
$552$	−43.3212	−0.00334035
$553$	13555.9	1.04241
$554$	−32725.7	−2.50971
$555$	0	0
$556$	−7590.79	−0.578995
$557$	22805.4	1.73482	0.867412	−	0.497591i	$-0.165782\pi$
$557$	22805.4	1.73482	0.867412	+	0.497591i	$0.165782\pi$
$558$	−5988.04	−0.454290
$559$	−10053.2	−0.760656
$560$	0	0
$561$	1310.70	0.0986414
$562$	−13951.5	−1.04717
$563$	−21527.6	−1.61151	−0.805754	−	0.592251i	$-0.798240\pi$
$563$	−21527.6	−1.61151	−0.805754	+	0.592251i	$0.798240\pi$
$564$	6463.87	0.482585
$565$	0	0
$566$	−34345.2	−2.55059
$567$	−1017.58	−0.0753693
$568$	−221.230	−0.0163426
$569$	−287.375	−0.0211729	−0.0105865	−	0.999944i	$-0.503370\pi$
$569$	−287.375	−0.0211729	−0.0105865	+	0.999944i	$0.503370\pi$
$570$	0	0
$571$	−14297.5	−1.04787	−0.523934	−	0.851759i	$-0.675536\pi$
$571$	−14297.5	−1.04787	−0.523934	+	0.851759i	$0.675536\pi$
$572$	3133.40	0.229046
$573$	1835.73	0.133837
$574$	−16526.4	−1.20174
$575$	0	0
$576$	−4488.26	−0.324671
$577$	16408.0	1.18383	0.591917	−	0.805999i	$-0.298371\pi$
$577$	16408.0	1.18383	0.591917	+	0.805999i	$0.298371\pi$
$578$	−13298.7	−0.957014
$579$	5707.74	0.409681
$580$	0	0
$581$	8586.43	0.613124
$582$	8301.45	0.591247
$583$	2455.92	0.174467
$584$	163.853	0.0116101
$585$	0	0
$586$	4101.06	0.289101
$587$	7700.65	0.541465	0.270732	−	0.962655i	$-0.412734\pi$
$587$	7700.65	0.541465	0.270732	+	0.962655i	$0.412734\pi$
$588$	−4386.89	−0.307674
$589$	24713.0	1.72883
$590$	0	0
$591$	806.245	0.0561159
$592$	5527.78	0.383768
$593$	−13161.1	−0.911402	−0.455701	−	0.890133i	$-0.650611\pi$
$593$	−13161.1	−0.911402	−0.455701	+	0.890133i	$0.650611\pi$
$594$	−1184.16	−0.0817957
$595$	0	0
$596$	−552.799	−0.0379925
$597$	−688.749	−0.0472171
$598$	−5043.40	−0.344883
$599$	11115.0	0.758175	0.379088	−	0.925361i	$-0.376238\pi$
$599$	11115.0	0.758175	0.379088	+	0.925361i	$0.376238\pi$
$600$	0	0
$601$	−12069.2	−0.819155	−0.409578	−	0.912275i	$-0.634324\pi$
$601$	−12069.2	−0.819155	−0.409578	+	0.912275i	$0.634324\pi$
$602$	−13959.4	−0.945088
$603$	−6602.07	−0.445866
$604$	1662.49	0.111996
$605$	0	0
$606$	5356.77	0.359083
$607$	−249.659	−0.0166941	−0.00834706	−	0.999965i	$-0.502657\pi$
$607$	−249.659	−0.0166941	−0.00834706	+	0.999965i	$0.502657\pi$
$608$	37783.2	2.52025
$609$	−3327.25	−0.221391
$610$	0	0
$611$	−9842.42	−0.651689
$612$	−2822.79	−0.186445
$613$	−19136.0	−1.26084	−0.630421	−	0.776253i	$-0.717118\pi$
$613$	−19136.0	−1.26084	−0.630421	+	0.776253i	$0.717118\pi$
$614$	−8647.26	−0.568363
$615$	0	0
$616$	−56.9067	−0.00372214
$617$	−22119.6	−1.44327	−0.721637	−	0.692272i	$-0.756610\pi$
$617$	−22119.6	−1.44327	−0.721637	+	0.692272i	$0.756610\pi$
$618$	15717.3	1.02304
$619$	19523.8	1.26773	0.633867	−	0.773442i	$-0.281467\pi$
$619$	19523.8	1.26773	0.633867	+	0.773442i	$0.281467\pi$
$620$	0	0
$621$	946.797	0.0611814
$622$	17218.0	1.10993
$623$	2086.66	0.134190
$624$	7014.19	0.449988
$625$	0	0
$626$	19079.5	1.21816
$627$	4887.10	0.311279
$628$	17684.2	1.12369
$629$	3387.36	0.214726
$630$	0	0
$631$	13656.2	0.861562	0.430781	−	0.902457i	$-0.358238\pi$
$631$	13656.2	0.861562	0.430781	+	0.902457i	$0.358238\pi$
$632$	444.355	0.0279676
$633$	−15335.4	−0.962916
$634$	−26635.7	−1.66851
$635$	0	0
$636$	−5289.20	−0.329765
$637$	6679.84	0.415486
$638$	−3871.93	−0.240268
$639$	4835.03	0.299328
$640$	0	0
$641$	−27823.7	−1.71446	−0.857230	−	0.514934i	$-0.827816\pi$
$641$	−27823.7	−1.71446	−0.857230	+	0.514934i	$0.827816\pi$
$642$	12545.1	0.771206
$643$	29045.0	1.78137	0.890687	−	0.454618i	$-0.150224\pi$
$643$	29045.0	1.78137	0.890687	+	0.454618i	$0.150224\pi$
$644$	−3478.75	−0.212860
$645$	0	0
$646$	23452.0	1.42834
$647$	−7596.84	−0.461612	−0.230806	−	0.973000i	$-0.574136\pi$
$647$	−7596.84	−0.461612	−0.230806	+	0.973000i	$0.574136\pi$
$648$	−33.3558	−0.00202213
$649$	5142.10	0.311009
$650$	0	0
$651$	6289.17	0.378636
$652$	15146.5	0.909792
$653$	−4744.20	−0.284310	−0.142155	−	0.989844i	$-0.545403\pi$
$653$	−4744.20	−0.284310	−0.142155	+	0.989844i	$0.545403\pi$
$654$	−14840.6	−0.887332
$655$	0	0
$656$	−21385.5	−1.27281
$657$	−3581.06	−0.212649
$658$	−13666.7	−0.809699
$659$	25736.9	1.52135	0.760674	−	0.649134i	$-0.224868\pi$
$659$	25736.9	1.52135	0.760674	+	0.649134i	$0.224868\pi$
$660$	0	0
$661$	14548.4	0.856080	0.428040	−	0.903760i	$-0.359204\pi$
$661$	14548.4	0.856080	0.428040	+	0.903760i	$0.359204\pi$
$662$	13671.1	0.802630
$663$	4298.21	0.251778
$664$	281.459	0.0164499
$665$	0	0
$666$	−3060.33	−0.178056
$667$	3095.81	0.179715
$668$	−23978.6	−1.38886
$669$	4180.80	0.241613
$670$	0	0
$671$	−8274.74	−0.476070
$672$	9615.38	0.551966
$673$	2228.16	0.127621	0.0638106	−	0.997962i	$-0.479675\pi$
$673$	2228.16	0.127621	0.0638106	+	0.997962i	$0.479675\pi$
$674$	31138.3	1.77953
$675$	0	0
$676$	−7073.65	−0.402461
$677$	−1975.02	−0.112121	−0.0560606	−	0.998427i	$-0.517854\pi$
$677$	−1975.02	−0.112121	−0.0560606	+	0.998427i	$0.517854\pi$
$678$	−15923.6	−0.901978
$679$	−8718.92	−0.492786
$680$	0	0
$681$	18734.6	1.05420
$682$	7318.71	0.410921
$683$	−14600.7	−0.817980	−0.408990	−	0.912539i	$-0.634119\pi$
$683$	−14600.7	−0.817980	−0.408990	+	0.912539i	$0.634119\pi$
$684$	−10525.1	−0.588358
$685$	0	0
$686$	26455.6	1.47242
$687$	9885.47	0.548987
$688$	−18063.8	−1.00098
$689$	8053.77	0.445318
$690$	0	0
$691$	−27053.0	−1.48936	−0.744678	−	0.667423i	$-0.767397\pi$
$691$	−27053.0	−1.48936	−0.744678	+	0.667423i	$0.767397\pi$
$692$	13532.3	0.743381
$693$	1243.71	0.0681741
$694$	−23365.6	−1.27802
$695$	0	0
$696$	−109.066	−0.00593984
$697$	−13104.8	−0.712165
$698$	23259.1	1.26128
$699$	−2885.55	−0.156140
$700$	0	0
$701$	−25541.0	−1.37613	−0.688067	−	0.725647i	$-0.741541\pi$
$701$	−25541.0	−1.37613	−0.688067	+	0.725647i	$0.741541\pi$
$702$	−3883.24	−0.208780
$703$	12630.1	0.677603
$704$	5485.65	0.293676
$705$	0	0
$706$	18071.3	0.963345
$707$	−5626.16	−0.299284
$708$	−11074.3	−0.587848
$709$	10650.6	0.564161	0.282081	−	0.959391i	$-0.408975\pi$
$709$	10650.6	0.564161	0.282081	+	0.959391i	$0.408975\pi$
$710$	0	0
$711$	−9711.50	−0.512250
$712$	68.3998	0.00360027
$713$	−5851.69	−0.307360
$714$	5968.27	0.312825
$715$	0	0
$716$	−7120.36	−0.371649
$717$	−14214.1	−0.740355
$718$	−34797.0	−1.80866
$719$	−26425.6	−1.37066	−0.685332	−	0.728231i	$-0.740343\pi$
$719$	−26425.6	−1.37066	−0.685332	+	0.728231i	$0.740343\pi$
$720$	0	0
$721$	−16507.7	−0.852675
$722$	60096.2	3.09772
$723$	−11786.4	−0.606279
$724$	−23453.8	−1.20394
$725$	0	0
$726$	1447.31	0.0739870
$727$	−5323.81	−0.271594	−0.135797	−	0.990737i	$-0.543360\pi$
$727$	−5323.81	−0.271594	−0.135797	+	0.990737i	$0.543360\pi$
$728$	−186.615	−0.00950059
$729$	729.000	0.0370370
$730$	0	0
$731$	−11069.3	−0.560071
$732$	17820.9	0.899835
$733$	−9265.08	−0.466867	−0.233434	−	0.972373i	$-0.574996\pi$
$733$	−9265.08	−0.466867	−0.233434	+	0.972373i	$0.574996\pi$
$734$	−45443.8	−2.28523
$735$	0	0
$736$	−8946.52	−0.448061
$737$	8069.19	0.403301
$738$	11839.6	0.590543
$739$	−31074.1	−1.54679	−0.773396	−	0.633923i	$-0.781443\pi$
$739$	−31074.1	−1.54679	−0.773396	+	0.633923i	$0.781443\pi$
$740$	0	0
$741$	16026.4	0.794526
$742$	11183.0	0.553292
$743$	−5376.26	−0.265459	−0.132729	−	0.991152i	$-0.542374\pi$
$743$	−5376.26	−0.265459	−0.132729	+	0.991152i	$0.542374\pi$
$744$	206.156	0.0101587
$745$	0	0
$746$	42654.4	2.09342
$747$	−6151.36	−0.301294
$748$	3450.08	0.168646
$749$	−13176.0	−0.642776
$750$	0	0
$751$	16857.3	0.819082	0.409541	−	0.912292i	$-0.365689\pi$
$751$	16857.3	0.819082	0.409541	+	0.912292i	$0.365689\pi$
$752$	−17685.0	−0.857586
$753$	23199.2	1.12274
$754$	−12697.3	−0.613274
$755$	0	0
$756$	−2678.52	−0.128858
$757$	−27458.7	−1.31837	−0.659183	−	0.751982i	$-0.729098\pi$
$757$	−27458.7	−1.31837	−0.659183	+	0.751982i	$0.729098\pi$
$758$	24144.0	1.15692
$759$	−1157.20	−0.0553406
$760$	0	0
$761$	−35829.9	−1.70675	−0.853373	−	0.521300i	$-0.825447\pi$
$761$	−35829.9	−1.70675	−0.853373	+	0.521300i	$0.825447\pi$
$762$	−14958.5	−0.711141
$763$	15587.0	0.739562
$764$	4832.08	0.228820
$765$	0	0
$766$	−55777.1	−2.63095
$767$	16862.6	0.793837
$768$	12599.1	0.591968
$769$	21200.9	0.994177	0.497089	−	0.867700i	$-0.334402\pi$
$769$	21200.9	0.994177	0.497089	+	0.867700i	$0.334402\pi$
$770$	0	0
$771$	4722.52	0.220593
$772$	15024.1	0.700428
$773$	29661.6	1.38015	0.690074	−	0.723739i	$-0.257578\pi$
$773$	29661.6	1.38015	0.690074	+	0.723739i	$0.257578\pi$
$774$	10000.6	0.464423
$775$	0	0
$776$	−285.802	−0.0132213
$777$	3214.23	0.148404
$778$	16840.9	0.776061
$779$	−48862.7	−2.24735
$780$	0	0
$781$	−5909.48	−0.270753
$782$	−5553.11	−0.253937
$783$	2383.66	0.108793
$784$	12002.4	0.546757
$785$	0	0
$786$	−2010.28	−0.0912268
$787$	340.379	0.0154170	0.00770852	−	0.999970i	$-0.497546\pi$
$787$	340.379	0.0154170	0.00770852	+	0.999970i	$0.497546\pi$
$788$	2122.23	0.0959407
$789$	9517.92	0.429464
$790$	0	0
$791$	16724.4	0.751769
$792$	40.7682	0.00182909
$793$	−27135.5	−1.21515
$794$	13324.2	0.595540
$795$	0	0
$796$	−1812.95	−0.0807266
$797$	−37442.3	−1.66408	−0.832041	−	0.554715i	$-0.812827\pi$
$797$	−37442.3	−1.66408	−0.832041	+	0.554715i	$0.812827\pi$
$798$	22253.4	0.987169
$799$	−10837.1	−0.479838
$800$	0	0
$801$	−1494.90	−0.0659420
$802$	−28578.9	−1.25830
$803$	4376.85	0.192348
$804$	−17378.2	−0.762292
$805$	0	0
$806$	24000.4	1.04886
$807$	−3716.49	−0.162115
$808$	−184.423	−0.00802967
$809$	−23629.8	−1.02692	−0.513461	−	0.858113i	$-0.671637\pi$
$809$	−23629.8	−1.02692	−0.513461	+	0.858113i	$0.671637\pi$
$810$	0	0
$811$	−40325.2	−1.74601	−0.873003	−	0.487715i	$-0.837830\pi$
$811$	−40325.2	−1.74601	−0.873003	+	0.487715i	$0.837830\pi$
$812$	−8758.13	−0.378510
$813$	−6771.06	−0.292093
$814$	3740.40	0.161058
$815$	0	0
$816$	7723.08	0.331326
$817$	−41273.0	−1.76739
$818$	−41308.8	−1.76568
$819$	4078.53	0.174011
$820$	0	0
$821$	20710.8	0.880403	0.440201	−	0.897899i	$-0.354907\pi$
$821$	20710.8	0.880403	0.440201	+	0.897899i	$0.354907\pi$
$822$	18344.6	0.778395
$823$	−14546.6	−0.616116	−0.308058	−	0.951368i	$-0.599679\pi$
$823$	−14546.6	−0.616116	−0.308058	+	0.951368i	$0.599679\pi$
$824$	−541.114	−0.0228769
$825$	0	0
$826$	23414.5	0.986314
$827$	−39477.0	−1.65992	−0.829958	−	0.557826i	$-0.811636\pi$
$827$	−39477.0	−1.65992	−0.829958	+	0.557826i	$0.811636\pi$
$828$	2492.19	0.104601
$829$	−37344.7	−1.56458	−0.782289	−	0.622915i	$-0.785948\pi$
$829$	−37344.7	−1.56458	−0.782289	+	0.622915i	$0.785948\pi$
$830$	0	0
$831$	−24623.9	−1.02791
$832$	17989.2	0.749596
$833$	7354.93	0.305922
$834$	−11497.8	−0.477382
$835$	0	0
$836$	12864.0	0.532190
$837$	−4505.60	−0.186065
$838$	−30463.6	−1.25578
$839$	−28412.4	−1.16914	−0.584568	−	0.811345i	$-0.698736\pi$
$839$	−28412.4	−1.16914	−0.584568	+	0.811345i	$0.698736\pi$
$840$	0	0
$841$	−16595.0	−0.680429
$842$	−4041.48	−0.165414
$843$	−10497.6	−0.428892
$844$	−40366.3	−1.64629
$845$	0	0
$846$	9790.87	0.397892
$847$	−1520.09	−0.0616658
$848$	14471.1	0.586014
$849$	−25842.4	−1.04465
$850$	0	0
$851$	−2990.64	−0.120468
$852$	12727.0	0.511759
$853$	−31070.6	−1.24717	−0.623585	−	0.781756i	$-0.714324\pi$
$853$	−31070.6	−1.24717	−0.623585	+	0.781756i	$0.714324\pi$
$854$	−37679.0	−1.50978
$855$	0	0
$856$	−431.902	−0.0172454
$857$	28889.1	1.15150	0.575749	−	0.817626i	$-0.304710\pi$
$857$	28889.1	1.15150	0.575749	+	0.817626i	$0.304710\pi$
$858$	4746.18	0.188849
$859$	17190.9	0.682825	0.341413	−	0.939913i	$-0.389095\pi$
$859$	17190.9	0.682825	0.341413	+	0.939913i	$0.389095\pi$
$860$	0	0
$861$	−12435.0	−0.492199
$862$	−52477.5	−2.07354
$863$	−39878.7	−1.57298	−0.786492	−	0.617600i	$-0.788105\pi$
$863$	−39878.7	−1.57298	−0.786492	+	0.617600i	$0.788105\pi$
$864$	−6888.51	−0.271240
$865$	0	0
$866$	15494.0	0.607976
$867$	−10006.4	−0.391966
$868$	16554.6	0.647350
$869$	11869.6	0.463347
$870$	0	0
$871$	26461.5	1.02941
$872$	510.933	0.0198422
$873$	6246.28	0.242159
$874$	−20705.4	−0.801339
$875$	0	0
$876$	−9426.20	−0.363564
$877$	−28622.5	−1.10207	−0.551034	−	0.834483i	$-0.685767\pi$
$877$	−28622.5	−1.10207	−0.551034	+	0.834483i	$0.685767\pi$
$878$	−1534.79	−0.0589939
$879$	3085.77	0.118408
$880$	0	0
$881$	34427.7	1.31657	0.658286	−	0.752768i	$-0.271282\pi$
$881$	34427.7	1.31657	0.658286	+	0.752768i	$0.271282\pi$
$882$	−6644.85	−0.253678
$883$	−14049.2	−0.535441	−0.267721	−	0.963497i	$-0.586270\pi$
$883$	−14049.2	−0.535441	−0.267721	+	0.963497i	$0.586270\pi$
$884$	11313.9	0.430462
$885$	0	0
$886$	53126.6	2.01447
$887$	29052.4	1.09976	0.549878	−	0.835245i	$-0.314674\pi$
$887$	29052.4	1.09976	0.549878	+	0.835245i	$0.314674\pi$
$888$	105.361	0.00398162
$889$	15710.8	0.592713
$890$	0	0
$891$	−891.000	−0.0335013
$892$	11004.9	0.413083
$893$	−40407.5	−1.51421
$894$	−837.328	−0.0313249
$895$	0	0
$896$	−662.132	−0.0246878
$897$	−3794.82	−0.141255
$898$	−12927.8	−0.480409
$899$	−14732.3	−0.546550
$900$	0	0
$901$	8867.72	0.327888
$902$	−14470.6	−0.534167
$903$	−10503.5	−0.387082
$904$	548.216	0.0201697
$905$	0	0
$906$	2518.18	0.0923411
$907$	39504.2	1.44621	0.723107	−	0.690736i	$-0.242713\pi$
$907$	39504.2	1.44621	0.723107	+	0.690736i	$0.242713\pi$
$908$	49314.0	1.80236
$909$	4030.61	0.147070
$910$	0	0
$911$	−45484.2	−1.65418	−0.827090	−	0.562070i	$-0.810005\pi$
$911$	−45484.2	−1.65418	−0.827090	+	0.562070i	$0.810005\pi$
$912$	28796.4	1.04555
$913$	7518.33	0.272530
$914$	14699.1	0.531951
$915$	0	0
$916$	26020.9	0.938597
$917$	2111.37	0.0760346
$918$	−4275.70	−0.153725
$919$	−14765.8	−0.530011	−0.265005	−	0.964247i	$-0.585374\pi$
$919$	−14765.8	−0.530011	−0.265005	+	0.964247i	$0.585374\pi$
$920$	0	0
$921$	−6506.48	−0.232786
$922$	−76628.4	−2.73712
$923$	−19379.1	−0.691084
$924$	3273.74	0.116556
$925$	0	0
$926$	5369.01	0.190536
$927$	11826.2	0.419011
$928$	−22523.8	−0.796746
$929$	31598.4	1.11594	0.557971	−	0.829861i	$-0.311580\pi$
$929$	31598.4	1.11594	0.557971	+	0.829861i	$0.311580\pi$
$930$	0	0
$931$	27423.7	0.965387
$932$	−7595.46	−0.266950
$933$	12955.4	0.454597
$934$	35537.8	1.24500
$935$	0	0
$936$	133.692	0.00466866
$937$	−14121.9	−0.492363	−0.246181	−	0.969224i	$-0.579176\pi$
$937$	−14121.9	−0.492363	−0.246181	+	0.969224i	$0.579176\pi$
$938$	36743.0	1.27900
$939$	14356.0	0.498925
$940$	0	0
$941$	39525.5	1.36928	0.684641	−	0.728880i	$-0.259959\pi$
$941$	39525.5	1.36928	0.684641	+	0.728880i	$0.259959\pi$
$942$	26786.4	0.926483
$943$	11570.0	0.399545
$944$	30298.9	1.04465
$945$	0	0
$946$	−12222.9	−0.420087
$947$	39575.6	1.35801	0.679004	−	0.734135i	$-0.262412\pi$
$947$	39575.6	1.35801	0.679004	+	0.734135i	$0.262412\pi$
$948$	−25563.0	−0.875788
$949$	14353.1	0.490961
$950$	0	0
$951$	−20041.5	−0.683377
$952$	−205.476	−0.00699528
$953$	18421.4	0.626158	0.313079	−	0.949727i	$-0.398640\pi$
$953$	18421.4	0.626158	0.313079	+	0.949727i	$0.398640\pi$
$954$	−8011.59	−0.271892
$955$	0	0
$956$	−37414.8	−1.26578
$957$	−2913.36	−0.0984072
$958$	−46755.4	−1.57682
$959$	−19267.1	−0.648767
$960$	0	0
$961$	−1944.10	−0.0652581
$962$	12266.0	0.411092
$963$	9439.32	0.315865
$964$	−31024.5	−1.03655
$965$	0	0
$966$	−5269.29	−0.175504
$967$	−50038.2	−1.66403	−0.832016	−	0.554752i	$-0.812813\pi$
$967$	−50038.2	−1.66403	−0.832016	+	0.554752i	$0.812813\pi$
$968$	−49.8278	−0.00165447
$969$	17646.1	0.585009
$970$	0	0
$971$	53444.3	1.76633	0.883166	−	0.469060i	$-0.155407\pi$
$971$	53444.3	1.76633	0.883166	+	0.469060i	$0.155407\pi$
$972$	1918.90	0.0633218
$973$	12076.0	0.397883
$974$	32358.7	1.06452
$975$	0	0
$976$	−48757.5	−1.59907
$977$	−20163.0	−0.660256	−0.330128	−	0.943936i	$-0.607092\pi$
$977$	−20163.0	−0.660256	−0.330128	+	0.943936i	$0.607092\pi$
$978$	22942.6	0.750125
$979$	1827.09	0.0596468
$980$	0	0
$981$	−11166.6	−0.363427
$982$	−48488.7	−1.57570
$983$	−40813.1	−1.32425	−0.662124	−	0.749395i	$-0.730345\pi$
$983$	−40813.1	−1.32425	−0.662124	+	0.749395i	$0.730345\pi$
$984$	−407.613	−0.0132055
$985$	0	0
$986$	−13980.5	−0.451553
$987$	−10283.2	−0.331631
$988$	42185.2	1.35839
$989$	9772.87	0.314216
$990$	0	0
$991$	36663.8	1.17524	0.587620	−	0.809137i	$-0.300065\pi$
$991$	36663.8	1.17524	0.587620	+	0.809137i	$0.300065\pi$
$992$	42574.5	1.36264
$993$	10286.6	0.328735
$994$	−26908.8	−0.858647
$995$	0	0
$996$	−16191.8	−0.515119
$997$	−1330.33	−0.0422587	−0.0211294	−	0.999777i	$-0.506726\pi$
$997$	−1330.33	−0.0422587	−0.0211294	+	0.999777i	$0.506726\pi$
$998$	22712.5	0.720392
$999$	−2302.69	−0.0729268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.x.1.5	✓	5
3.2	odd	2		2475.4.a.bj.1.1		5
5.2	odd	4		825.4.c.s.199.9		10
5.3	odd	4		825.4.c.s.199.2		10
5.4	even	2		825.4.a.y.1.1	yes	5
15.14	odd	2		2475.4.a.bi.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.x.1.5	✓	5	1.1	even	1	trivial
825.4.a.y.1.1	yes	5	5.4	even	2
825.4.c.s.199.2		10	5.3	odd	4
825.4.c.s.199.9		10	5.2	odd	4
2475.4.a.bi.1.5		5	15.14	odd	2
2475.4.a.bj.1.1		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+3.98707 q^{2} +3.00000 q^{3} +7.89672 q^{4} +11.9612 q^{6} -12.5627 q^{7} -0.411800 q^{8} +9.00000 q^{9} -11.0000 q^{11} +23.6901 q^{12} -36.0726 q^{13} -50.0885 q^{14} -64.8156 q^{16} -39.7182 q^{17} +35.8836 q^{18} -148.094 q^{19} -37.6882 q^{21} -43.8578 q^{22} +35.0665 q^{23} -1.23540 q^{24} -143.824 q^{26} +27.0000 q^{27} -99.2043 q^{28} +88.2837 q^{29} -166.874 q^{31} -255.130 q^{32} -33.0000 q^{33} -158.359 q^{34} +71.0704 q^{36} -85.2847 q^{37} -590.460 q^{38} -108.218 q^{39} +329.944 q^{41} -150.265 q^{42} +278.695 q^{43} -86.8639 q^{44} +139.813 q^{46} +272.851 q^{47} -194.447 q^{48} -185.178 q^{49} -119.155 q^{51} -284.855 q^{52} -223.266 q^{53} +107.651 q^{54} +5.17334 q^{56} -444.282 q^{57} +351.993 q^{58} -467.463 q^{59} +752.249 q^{61} -665.338 q^{62} -113.065 q^{63} -498.695 q^{64} -131.573 q^{66} -733.563 q^{67} -313.643 q^{68} +105.200 q^{69} +537.226 q^{71} -3.70620 q^{72} -397.895 q^{73} -340.036 q^{74} -1169.46 q^{76} +138.190 q^{77} -431.471 q^{78} -1079.06 q^{79} +81.0000 q^{81} +1315.51 q^{82} -683.484 q^{83} -297.613 q^{84} +1111.18 q^{86} +264.851 q^{87} +4.52980 q^{88} -166.100 q^{89} +453.170 q^{91} +276.911 q^{92} -500.622 q^{93} +1087.87 q^{94} -765.390 q^{96} +694.031 q^{97} -738.317 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 15 q^{3} + 3 q^{4} - 3 q^{6} - 18 q^{7} + 3 q^{8} + 45 q^{9} - 55 q^{11} + 9 q^{12} - 31 q^{13} + 8 q^{14} - 125 q^{16} - 38 q^{17} - 9 q^{18} - 57 q^{19} - 54 q^{21} + 11 q^{22} - 161 q^{23}+ \cdots - 495 q^{99}+O(q^{100}) \) 5 * q - q^2 + 15 * q^3 + 3 * q^4 - 3 * q^6 - 18 * q^7 + 3 * q^8 + 45 * q^9 - 55 * q^11 + 9 * q^12 - 31 * q^13 + 8 * q^14 - 125 * q^16 - 38 * q^17 - 9 * q^18 - 57 * q^19 - 54 * q^21 + 11 * q^22 - 161 * q^23 + 9 * q^24 - 125 * q^26 + 135 * q^27 - 324 * q^28 - 107 * q^29 - 295 * q^31 + 23 * q^32 - 165 * q^33 + 34 * q^34 + 27 * q^36 + 260 * q^37 - 619 * q^38 - 93 * q^39 - 128 * q^41 + 24 * q^42 - 377 * q^43 - 33 * q^44 - 577 * q^46 + 114 * q^47 - 375 * q^48 - 415 * q^49 - 114 * q^51 + 395 * q^52 - 812 * q^53 - 27 * q^54 - 132 * q^56 - 171 * q^57 - 339 * q^58 - 1152 * q^59 - 344 * q^61 + 235 * q^62 - 162 * q^63 - 545 * q^64 + 33 * q^66 + 928 * q^67 - 654 * q^68 - 483 * q^69 - 707 * q^71 + 27 * q^72 - 322 * q^73 - 1176 * q^74 - 1699 * q^76 + 198 * q^77 - 375 * q^78 - 2494 * q^79 + 405 * q^81 + 2776 * q^82 - 1657 * q^83 - 972 * q^84 - 799 * q^86 - 321 * q^87 - 33 * q^88 - 2435 * q^89 - 804 * q^91 + 2775 * q^92 - 885 * q^93 - 502 * q^94 + 69 * q^96 + 1901 * q^97 - 343 * q^98 - 495 * q^99

Introduction

L-functions

Modular forms

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