LMFDB - Embedded newform 825.4.a.q.1.2

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:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
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			1.2
Root			$0.723686$ 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)

$f(q)$	$=$	\(q+0.723686 q^{2} -3.00000 q^{3} -7.47628 q^{4} -2.17106 q^{6} -1.13288 q^{7} -11.2000 q^{8} +9.00000 q^{9} +11.0000 q^{11} +22.4288 q^{12} +21.8566 q^{13} -0.819851 q^{14} +51.7050 q^{16} -6.18033 q^{17} +6.51317 q^{18} -92.8393 q^{19} +3.39865 q^{21} +7.96054 q^{22} +36.7317 q^{23} +33.5999 q^{24} +15.8173 q^{26} -27.0000 q^{27} +8.46975 q^{28} +71.1375 q^{29} +186.522 q^{31} +127.018 q^{32} -33.0000 q^{33} -4.47261 q^{34} -67.2865 q^{36} +356.581 q^{37} -67.1865 q^{38} -65.5697 q^{39} +271.940 q^{41} +2.45955 q^{42} -155.780 q^{43} -82.2391 q^{44} +26.5822 q^{46} -234.566 q^{47} -155.115 q^{48} -341.717 q^{49} +18.5410 q^{51} -163.406 q^{52} -195.018 q^{53} -19.5395 q^{54} +12.6882 q^{56} +278.518 q^{57} +51.4812 q^{58} -455.930 q^{59} -441.278 q^{61} +134.983 q^{62} -10.1959 q^{63} -321.719 q^{64} -23.8816 q^{66} -133.005 q^{67} +46.2058 q^{68} -110.195 q^{69} -1041.68 q^{71} -100.800 q^{72} +160.119 q^{73} +258.053 q^{74} +694.093 q^{76} -12.4617 q^{77} -47.4519 q^{78} -761.367 q^{79} +81.0000 q^{81} +196.799 q^{82} -51.7170 q^{83} -25.4092 q^{84} -112.736 q^{86} -213.412 q^{87} -123.200 q^{88} -1075.25 q^{89} -24.7609 q^{91} -274.616 q^{92} -559.565 q^{93} -169.752 q^{94} -381.054 q^{96} +703.238 q^{97} -247.295 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 9 q^{3} + 7 q^{4} - 3 q^{6} + 16 q^{7} - 3 q^{8} + 27 q^{9} + 33 q^{11} - 21 q^{12} + 45 q^{13} - 116 q^{14} - 85 q^{16} - 58 q^{17} + 9 q^{18} - 169 q^{19} - 48 q^{21} + 11 q^{22} + 155 q^{23}+ \cdots + 297 q^{99}+O(q^{100}) $ 3 * q + q^2 - 9 * q^3 + 7 * q^4 - 3 * q^6 + 16 * q^7 - 3 * q^8 + 27 * q^9 + 33 * q^11 - 21 * q^12 + 45 * q^13 - 116 * q^14 - 85 * q^16 - 58 * q^17 + 9 * q^18 - 169 * q^19 - 48 * q^21 + 11 * q^22 + 155 * q^23 + 9 * q^24 + 167 * q^26 - 81 * q^27 + 100 * q^28 - 277 * q^29 - 173 * q^31 + 97 * q^32 - 99 * q^33 - 146 * q^34 + 63 * q^36 + 60 * q^37 + 169 * q^38 - 135 * q^39 + 44 * q^41 + 348 * q^42 - 109 * q^43 + 77 * q^44 + 425 * q^46 + 270 * q^47 + 255 * q^48 - 427 * q^49 + 174 * q^51 + 45 * q^52 - 148 * q^53 - 27 * q^54 + 168 * q^56 + 507 * q^57 - 783 * q^58 - 684 * q^59 - 1038 * q^61 + 953 * q^62 + 144 * q^63 - 1129 * q^64 - 33 * q^66 + 314 * q^67 - 366 * q^68 - 465 * q^69 - 1459 * q^71 - 27 * q^72 - 1170 * q^73 - 1764 * q^74 + 211 * q^76 + 176 * q^77 - 501 * q^78 - 506 * q^79 + 243 * q^81 - 1040 * q^82 - 347 * q^83 - 300 * q^84 - 527 * q^86 + 831 * q^87 - 33 * q^88 - 607 * q^89 - 398 * q^91 + 687 * q^92 + 519 * q^93 - 1822 * q^94 - 291 * q^96 + 1263 * q^97 - 2273 * q^98 + 297 * 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 $

$2$	0.723686	0.255862	0.127931	−	0.991783i	$-0.459166\pi$
$2$	0.723686	0.255862	0.127931	+	0.991783i	$0.459166\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.47628	−0.934535
$5$	0	0
$5$	0	0
$6$	−2.17106	−0.147722
$7$	−1.13288	−0.0611699	−0.0305850	−	0.999532i	$-0.509737\pi$
$7$	−1.13288	−0.0611699	−0.0305850	+	0.999532i	$0.509737\pi$
$8$	−11.2000	−0.494973
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	22.4288	0.539554
$13$	21.8566	0.466302	0.233151	−	0.972441i	$-0.425096\pi$
$13$	21.8566	0.466302	0.233151	+	0.972441i	$0.425096\pi$
$14$	−0.819851	−0.0156510
$15$	0	0
$16$	51.7050	0.807890
$17$	−6.18033	−0.0881735	−0.0440867	−	0.999028i	$-0.514038\pi$
$17$	−6.18033	−0.0881735	−0.0440867	+	0.999028i	$0.514038\pi$
$18$	6.51317	0.0852872
$19$	−92.8393	−1.12099	−0.560495	−	0.828158i	$-0.689389\pi$
$19$	−92.8393	−1.12099	−0.560495	+	0.828158i	$0.689389\pi$
$20$	0	0
$21$	3.39865	0.0353165
$22$	7.96054	0.0771452
$23$	36.7317	0.333004	0.166502	−	0.986041i	$-0.446753\pi$
$23$	36.7317	0.333004	0.166502	+	0.986041i	$0.446753\pi$
$24$	33.5999	0.285773
$25$	0	0
$26$	15.8173	0.119309
$27$	−27.0000	−0.192450
$28$	8.46975	0.0571654
$29$	71.1375	0.455514	0.227757	−	0.973718i	$-0.426861\pi$
$29$	71.1375	0.455514	0.227757	+	0.973718i	$0.426861\pi$
$30$	0	0
$31$	186.522	1.08065	0.540327	−	0.841455i	$-0.318301\pi$
$31$	186.522	1.08065	0.540327	+	0.841455i	$0.318301\pi$
$32$	127.018	0.701681
$33$	−33.0000	−0.174078
$34$	−4.47261	−0.0225602
$35$	0	0
$36$	−67.2865	−0.311512
$37$	356.581	1.58437	0.792184	−	0.610282i	$-0.208944\pi$
$37$	356.581	1.58437	0.792184	+	0.610282i	$0.208944\pi$
$38$	−67.1865	−0.286818
$39$	−65.5697	−0.269219
$40$	0	0
$41$	271.940	1.03585	0.517925	−	0.855426i	$-0.326705\pi$
$41$	271.940	1.03585	0.517925	+	0.855426i	$0.326705\pi$
$42$	2.45955	0.00903613
$43$	−155.780	−0.552470	−0.276235	−	0.961090i	$-0.589087\pi$
$43$	−155.780	−0.552470	−0.276235	+	0.961090i	$0.589087\pi$
$44$	−82.2391	−0.281773
$45$	0	0
$46$	26.5822	0.0852028
$47$	−234.566	−0.727978	−0.363989	−	0.931403i	$-0.618585\pi$
$47$	−234.566	−0.727978	−0.363989	+	0.931403i	$0.618585\pi$
$48$	−155.115	−0.466436
$49$	−341.717	−0.996258
$50$	0	0
$51$	18.5410	0.0509070
$52$	−163.406	−0.435775
$53$	−195.018	−0.505429	−0.252715	−	0.967541i	$-0.581323\pi$
$53$	−195.018	−0.505429	−0.252715	+	0.967541i	$0.581323\pi$
$54$	−19.5395	−0.0492406
$55$	0	0
$56$	12.6882	0.0302775
$57$	278.518	0.647204
$58$	51.4812	0.116548
$59$	−455.930	−1.00605	−0.503025	−	0.864272i	$-0.667780\pi$
$59$	−455.930	−1.00605	−0.503025	+	0.864272i	$0.667780\pi$
$60$	0	0
$61$	−441.278	−0.926227	−0.463114	−	0.886299i	$-0.653268\pi$
$61$	−441.278	−0.926227	−0.463114	+	0.886299i	$0.653268\pi$
$62$	134.983	0.276498
$63$	−10.1959	−0.0203900
$64$	−321.719	−0.628357
$65$	0	0
$66$	−23.8816	−0.0445398
$67$	−133.005	−0.242524	−0.121262	−	0.992621i	$-0.538694\pi$
$67$	−133.005	−0.242524	−0.121262	+	0.992621i	$0.538694\pi$
$68$	46.2058	0.0824012
$69$	−110.195	−0.192260
$70$	0	0
$71$	−1041.68	−1.74119	−0.870596	−	0.491999i	$-0.836266\pi$
$71$	−1041.68	−1.74119	−0.870596	+	0.491999i	$0.836266\pi$
$72$	−100.800	−0.164991
$73$	160.119	0.256719	0.128360	−	0.991728i	$-0.459029\pi$
$73$	160.119	0.256719	0.128360	+	0.991728i	$0.459029\pi$
$74$	258.053	0.405379
$75$	0	0
$76$	694.093	1.04760
$77$	−12.4617	−0.0184434
$78$	−47.4519	−0.0688829
$79$	−761.367	−1.08431	−0.542155	−	0.840278i	$-0.682392\pi$
$79$	−761.367	−1.08431	−0.542155	+	0.840278i	$0.682392\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	196.799	0.265034
$83$	−51.7170	−0.0683938	−0.0341969	−	0.999415i	$-0.510887\pi$
$83$	−51.7170	−0.0683938	−0.0341969	+	0.999415i	$0.510887\pi$
$84$	−25.4092	−0.0330045
$85$	0	0
$86$	−112.736	−0.141356
$87$	−213.412	−0.262991
$88$	−123.200	−0.149240
$89$	−1075.25	−1.28064	−0.640318	−	0.768110i	$-0.721198\pi$
$89$	−1075.25	−1.28064	−0.640318	+	0.768110i	$0.721198\pi$
$90$	0	0
$91$	−24.7609	−0.0285236
$92$	−274.616	−0.311203
$93$	−559.565	−0.623916
$94$	−169.752	−0.186262
$95$	0	0
$96$	−381.054	−0.405116
$97$	703.238	0.736114	0.368057	−	0.929803i	$-0.380023\pi$
$97$	703.238	0.736114	0.368057	+	0.929803i	$0.380023\pi$
$98$	−247.295	−0.254904
$99$	99.0000	0.100504
$100$	0	0
$101$	−856.886	−0.844192	−0.422096	−	0.906551i	$-0.638705\pi$
$101$	−856.886	−0.844192	−0.422096	+	0.906551i	$0.638705\pi$
$102$	13.4178	0.0130251
$103$	805.989	0.771034	0.385517	−	0.922701i	$-0.374023\pi$
$103$	805.989	0.771034	0.385517	+	0.922701i	$0.374023\pi$
$104$	−244.793	−0.230807
$105$	0	0
$106$	−141.132	−0.129320
$107$	1608.55	1.45331	0.726654	−	0.687004i	$-0.241074\pi$
$107$	1608.55	1.45331	0.726654	+	0.687004i	$0.241074\pi$
$108$	201.860	0.179851
$109$	−925.724	−0.813471	−0.406735	−	0.913546i	$-0.633333\pi$
$109$	−925.724	−0.813471	−0.406735	+	0.913546i	$0.633333\pi$
$110$	0	0
$111$	−1069.74	−0.914735
$112$	−58.5757	−0.0494186
$113$	−1214.81	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$113$	−1214.81	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$114$	201.560	0.165595
$115$	0	0
$116$	−531.844	−0.425693
$117$	196.709	0.155434
$118$	−329.950	−0.257410
$119$	7.00159	0.00539357
$120$	0	0
$121$	121.000	0.0909091
$122$	−319.347	−0.236986
$123$	−815.819	−0.598048
$124$	−1394.49	−1.00991
$125$	0	0
$126$	−7.37866	−0.00521701
$127$	−1128.96	−0.788810	−0.394405	−	0.918937i	$-0.629049\pi$
$127$	−1128.96	−0.788810	−0.394405	+	0.918937i	$0.629049\pi$
$128$	−1248.97	−0.862454
$129$	467.340	0.318969
$130$	0	0
$131$	−1562.32	−1.04199	−0.520993	−	0.853561i	$-0.674438\pi$
$131$	−1562.32	−1.04199	−0.520993	+	0.853561i	$0.674438\pi$
$132$	246.717	0.162682
$133$	105.176	0.0685708
$134$	−96.2535	−0.0620525
$135$	0	0
$136$	69.2194	0.0436435
$137$	−44.9323	−0.0280206	−0.0140103	−	0.999902i	$-0.504460\pi$
$137$	−44.9323	−0.0280206	−0.0140103	+	0.999902i	$0.504460\pi$
$138$	−79.7466	−0.0491919
$139$	−415.574	−0.253587	−0.126793	−	0.991929i	$-0.540468\pi$
$139$	−415.574	−0.253587	−0.126793	+	0.991929i	$0.540468\pi$
$140$	0	0
$141$	703.698	0.420298
$142$	−753.849	−0.445504
$143$	240.422	0.140595
$144$	465.345	0.269297
$145$	0	0
$146$	115.876	0.0656846
$147$	1025.15	0.575190
$148$	−2665.90	−1.48065
$149$	678.402	0.372999	0.186499	−	0.982455i	$-0.440286\pi$
$149$	678.402	0.372999	0.186499	+	0.982455i	$0.440286\pi$
$150$	0	0
$151$	616.083	0.332027	0.166014	−	0.986123i	$-0.446910\pi$
$151$	616.083	0.332027	0.166014	+	0.986123i	$0.446910\pi$
$152$	1039.80	0.554860
$153$	−55.6229	−0.0293912
$154$	−9.01837	−0.00471896
$155$	0	0
$156$	490.217	0.251595
$157$	616.599	0.313439	0.156720	−	0.987643i	$-0.449908\pi$
$157$	616.599	0.313439	0.156720	+	0.987643i	$0.449908\pi$
$158$	−550.991	−0.277433
$159$	585.053	0.291810
$160$	0	0
$161$	−41.6127	−0.0203698
$162$	58.6186	0.0284291
$163$	−2031.15	−0.976026	−0.488013	−	0.872836i	$-0.662278\pi$
$163$	−2031.15	−0.976026	−0.488013	+	0.872836i	$0.662278\pi$
$164$	−2033.10	−0.968038
$165$	0	0
$166$	−37.4269	−0.0174993
$167$	2114.90	0.979973	0.489987	−	0.871730i	$-0.337002\pi$
$167$	2114.90	0.979973	0.489987	+	0.871730i	$0.337002\pi$
$168$	−38.0647	−0.0174807
$169$	−1719.29	−0.782563
$170$	0	0
$171$	−835.554	−0.373663
$172$	1164.65	0.516302
$173$	15.5096	0.00681602	0.00340801	−	0.999994i	$-0.498915\pi$
$173$	15.5096	0.00681602	0.00340801	+	0.999994i	$0.498915\pi$
$174$	−154.444	−0.0672893
$175$	0	0
$176$	568.755	0.243588
$177$	1367.79	0.580844
$178$	−778.146	−0.327666
$179$	487.991	0.203766	0.101883	−	0.994796i	$-0.467513\pi$
$179$	487.991	0.203766	0.101883	+	0.994796i	$0.467513\pi$
$180$	0	0
$181$	1248.71	0.512795	0.256397	−	0.966571i	$-0.417464\pi$
$181$	1248.71	0.512795	0.256397	+	0.966571i	$0.417464\pi$
$182$	−17.9191	−0.00729810
$183$	1323.83	0.534758
$184$	−411.393	−0.164828
$185$	0	0
$186$	−404.949	−0.159636
$187$	−67.9836	−0.0265853
$188$	1753.68	0.680321
$189$	30.5878	0.0117722
$190$	0	0
$191$	−2571.60	−0.974213	−0.487107	−	0.873342i	$-0.661948\pi$
$191$	−2571.60	−0.974213	−0.487107	+	0.873342i	$0.661948\pi$
$192$	965.156	0.362782
$193$	−1221.80	−0.455686	−0.227843	−	0.973698i	$-0.573167\pi$
$193$	−1221.80	−0.455686	−0.227843	+	0.973698i	$0.573167\pi$
$194$	508.924	0.188343
$195$	0	0
$196$	2554.77	0.931038
$197$	1428.32	0.516568	0.258284	−	0.966069i	$-0.416843\pi$
$197$	1428.32	0.516568	0.258284	+	0.966069i	$0.416843\pi$
$198$	71.6449	0.0257151
$199$	−816.166	−0.290736	−0.145368	−	0.989378i	$-0.546437\pi$
$199$	−816.166	−0.290736	−0.145368	+	0.989378i	$0.546437\pi$
$200$	0	0
$201$	399.014	0.140021
$202$	−620.116	−0.215996
$203$	−80.5904	−0.0278637
$204$	−138.618	−0.0475743
$205$	0	0
$206$	583.283	0.197278
$207$	330.585	0.111001
$208$	1130.09	0.376721
$209$	−1021.23	−0.337991
$210$	0	0
$211$	−903.360	−0.294739	−0.147369	−	0.989082i	$-0.547081\pi$
$211$	−903.360	−0.294739	−0.147369	+	0.989082i	$0.547081\pi$
$212$	1458.01	0.472341
$213$	3125.04	1.00528
$214$	1164.08	0.371846
$215$	0	0
$216$	302.399	0.0952576
$217$	−211.307	−0.0661035
$218$	−669.934	−0.208136
$219$	−480.357	−0.148217
$220$	0	0
$221$	−135.081	−0.0411154
$222$	−774.159	−0.234046
$223$	−533.966	−0.160345	−0.0801727	−	0.996781i	$-0.525547\pi$
$223$	−533.966	−0.160345	−0.0801727	+	0.996781i	$0.525547\pi$
$224$	−143.896	−0.0429218
$225$	0	0
$226$	−879.142	−0.258760
$227$	−696.531	−0.203658	−0.101829	−	0.994802i	$-0.532469\pi$
$227$	−696.531	−0.203658	−0.101829	+	0.994802i	$0.532469\pi$
$228$	−2082.28	−0.604834
$229$	−796.794	−0.229929	−0.114964	−	0.993370i	$-0.536675\pi$
$229$	−796.794	−0.229929	−0.114964	+	0.993370i	$0.536675\pi$
$230$	0	0
$231$	37.3851	0.0106483
$232$	−796.737	−0.225467
$233$	5657.08	1.59059	0.795295	−	0.606223i	$-0.207316\pi$
$233$	5657.08	1.59059	0.795295	+	0.606223i	$0.207316\pi$
$234$	142.356	0.0397696
$235$	0	0
$236$	3408.66	0.940190
$237$	2284.10	0.626027
$238$	5.06695	0.00138001
$239$	−2023.06	−0.547534	−0.273767	−	0.961796i	$-0.588270\pi$
$239$	−2023.06	−0.547534	−0.273767	+	0.961796i	$0.588270\pi$
$240$	0	0
$241$	−3513.86	−0.939203	−0.469601	−	0.882879i	$-0.655602\pi$
$241$	−3513.86	−0.939203	−0.469601	+	0.882879i	$0.655602\pi$
$242$	87.5660	0.0232601
$243$	−243.000	−0.0641500
$244$	3299.12	0.865592
$245$	0	0
$246$	−590.397	−0.153018
$247$	−2029.15	−0.522719
$248$	−2089.03	−0.534895
$249$	155.151	0.0394872
$250$	0	0
$251$	1814.32	0.456250	0.228125	−	0.973632i	$-0.426740\pi$
$251$	1814.32	0.456250	0.228125	+	0.973632i	$0.426740\pi$
$252$	76.2277	0.0190551
$253$	404.048	0.100404
$254$	−817.011	−0.201826
$255$	0	0
$256$	1669.89	0.407688
$257$	−1445.46	−0.350839	−0.175419	−	0.984494i	$-0.556128\pi$
$257$	−1445.46	−0.350839	−0.175419	+	0.984494i	$0.556128\pi$
$258$	338.207	0.0816118
$259$	−403.965	−0.0969157
$260$	0	0
$261$	640.237	0.151838
$262$	−1130.63	−0.266604
$263$	4411.79	1.03438	0.517191	−	0.855870i	$-0.326978\pi$
$263$	4411.79	1.03438	0.517191	+	0.855870i	$0.326978\pi$
$264$	369.599	0.0861638
$265$	0	0
$266$	76.1144	0.0175446
$267$	3225.76	0.739376
$268$	994.379	0.226647
$269$	3955.14	0.896465	0.448232	−	0.893917i	$-0.352054\pi$
$269$	3955.14	0.896465	0.448232	+	0.893917i	$0.352054\pi$
$270$	0	0
$271$	6656.96	1.49218	0.746091	−	0.665844i	$-0.231928\pi$
$271$	6656.96	1.49218	0.746091	+	0.665844i	$0.231928\pi$
$272$	−319.554	−0.0712345
$273$	74.2828	0.0164681
$274$	−32.5169	−0.00716941
$275$	0	0
$276$	823.849	0.179673
$277$	−8523.79	−1.84890	−0.924450	−	0.381304i	$-0.875475\pi$
$277$	−8523.79	−1.84890	−0.924450	+	0.381304i	$0.875475\pi$
$278$	−300.745	−0.0648831
$279$	1678.69	0.360218
$280$	0	0
$281$	−4210.99	−0.893973	−0.446986	−	0.894541i	$-0.647503\pi$
$281$	−4210.99	−0.893973	−0.446986	+	0.894541i	$0.647503\pi$
$282$	509.256	0.107538
$283$	−3529.66	−0.741400	−0.370700	−	0.928753i	$-0.620882\pi$
$283$	−3529.66	−0.741400	−0.370700	+	0.928753i	$0.620882\pi$
$284$	7787.88	1.62720
$285$	0	0
$286$	173.990	0.0359729
$287$	−308.076	−0.0633629
$288$	1143.16	0.233894
$289$	−4874.80	−0.992225
$290$	0	0
$291$	−2109.71	−0.424995
$292$	−1197.09	−0.239913
$293$	7110.96	1.41784	0.708919	−	0.705290i	$-0.249183\pi$
$293$	7110.96	1.41784	0.708919	+	0.705290i	$0.249183\pi$
$294$	741.886	0.147169
$295$	0	0
$296$	−3993.70	−0.784220
$297$	−297.000	−0.0580259
$298$	490.950	0.0954361
$299$	802.828	0.155280
$300$	0	0
$301$	176.480	0.0337945
$302$	445.851	0.0849531
$303$	2570.66	0.487394
$304$	−4800.25	−0.905636
$305$	0	0
$306$	−40.2535	−0.00752007
$307$	−9474.72	−1.76140	−0.880702	−	0.473671i	$-0.842929\pi$
$307$	−9474.72	−1.76140	−0.880702	+	0.473671i	$0.842929\pi$
$308$	93.1672	0.0172360
$309$	−2417.97	−0.445156
$310$	0	0
$311$	−5210.73	−0.950075	−0.475038	−	0.879965i	$-0.657566\pi$
$311$	−5210.73	−0.950075	−0.475038	+	0.879965i	$0.657566\pi$
$312$	734.378	0.133256
$313$	−1944.27	−0.351108	−0.175554	−	0.984470i	$-0.556172\pi$
$313$	−1944.27	−0.351108	−0.175554	+	0.984470i	$0.556172\pi$
$314$	446.224	0.0801970
$315$	0	0
$316$	5692.20	1.01333
$317$	6852.73	1.21416	0.607079	−	0.794642i	$-0.292341\pi$
$317$	6852.73	1.21416	0.607079	+	0.794642i	$0.292341\pi$
$318$	423.395	0.0746629
$319$	782.512	0.137343
$320$	0	0
$321$	−4825.64	−0.839068
$322$	−30.1145	−0.00521185
$323$	573.777	0.0988415
$324$	−605.579	−0.103837
$325$	0	0
$326$	−1469.92	−0.249727
$327$	2777.17	0.469658
$328$	−3045.72	−0.512718
$329$	265.736	0.0445304
$330$	0	0
$331$	−2019.85	−0.335411	−0.167706	−	0.985837i	$-0.553636\pi$
$331$	−2019.85	−0.335411	−0.167706	+	0.985837i	$0.553636\pi$
$332$	386.651	0.0639163
$333$	3209.23	0.528123
$334$	1530.52	0.250737
$335$	0	0
$336$	175.727	0.0285318
$337$	−11738.7	−1.89748	−0.948739	−	0.316060i	$-0.897640\pi$
$337$	−11738.7	−1.89748	−0.948739	+	0.316060i	$0.897640\pi$
$338$	−1244.23	−0.200228
$339$	3644.44	0.583890
$340$	0	0
$341$	2051.74	0.325829
$342$	−604.679	−0.0956061
$343$	775.704	0.122111
$344$	1744.73	0.273458
$345$	0	0
$346$	11.2241	0.00174396
$347$	−12692.2	−1.96355	−0.981777	−	0.190036i	$-0.939140\pi$
$347$	−12692.2	−1.96355	−0.981777	+	0.190036i	$0.939140\pi$
$348$	1595.53	0.245774
$349$	12026.8	1.84465	0.922324	−	0.386417i	$-0.126287\pi$
$349$	12026.8	1.84465	0.922324	+	0.386417i	$0.126287\pi$
$350$	0	0
$351$	−590.127	−0.0897398
$352$	1397.20	0.211565
$353$	8948.39	1.34922	0.674610	−	0.738174i	$-0.264312\pi$
$353$	8948.39	1.34922	0.674610	+	0.738174i	$0.264312\pi$
$354$	989.850	0.148616
$355$	0	0
$356$	8038.90	1.19680
$357$	−21.0048	−0.00311398
$358$	353.152	0.0521360
$359$	3906.52	0.574313	0.287156	−	0.957884i	$-0.407290\pi$
$359$	3906.52	0.574313	0.287156	+	0.957884i	$0.407290\pi$
$360$	0	0
$361$	1760.14	0.256617
$362$	903.673	0.131204
$363$	−363.000	−0.0524864
$364$	185.120	0.0266563
$365$	0	0
$366$	958.040	0.136824
$367$	−4225.04	−0.600941	−0.300471	−	0.953791i	$-0.597144\pi$
$367$	−4225.04	−0.600941	−0.300471	+	0.953791i	$0.597144\pi$
$368$	1899.21	0.269030
$369$	2447.46	0.345283
$370$	0	0
$371$	220.932	0.0309171
$372$	4183.46	0.583071
$373$	−4223.50	−0.586285	−0.293143	−	0.956069i	$-0.594701\pi$
$373$	−4223.50	−0.586285	−0.293143	+	0.956069i	$0.594701\pi$
$374$	−49.1988	−0.00680216
$375$	0	0
$376$	2627.13	0.360330
$377$	1554.82	0.212407
$378$	22.1360	0.00301204
$379$	−748.009	−0.101379	−0.0506895	−	0.998714i	$-0.516142\pi$
$379$	−748.009	−0.101379	−0.0506895	+	0.998714i	$0.516142\pi$
$380$	0	0
$381$	3386.87	0.455420
$382$	−1861.03	−0.249264
$383$	−4668.07	−0.622786	−0.311393	−	0.950281i	$-0.600796\pi$
$383$	−4668.07	−0.622786	−0.311393	+	0.950281i	$0.600796\pi$
$384$	3746.90	0.497938
$385$	0	0
$386$	−884.203	−0.116593
$387$	−1402.02	−0.184157
$388$	−5257.61	−0.687924
$389$	6544.12	0.852957	0.426478	−	0.904498i	$-0.359754\pi$
$389$	6544.12	0.852957	0.426478	+	0.904498i	$0.359754\pi$
$390$	0	0
$391$	−227.014	−0.0293621
$392$	3827.21	0.493121
$393$	4686.95	0.601591
$394$	1033.66	0.132170
$395$	0	0
$396$	−740.152	−0.0939243
$397$	−6439.84	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$397$	−6439.84	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$398$	−590.648	−0.0743882
$399$	−315.528	−0.0395894
$400$	0	0
$401$	7727.03	0.962268	0.481134	−	0.876647i	$-0.340225\pi$
$401$	7727.03	0.962268	0.481134	+	0.876647i	$0.340225\pi$
$402$	288.761	0.0358260
$403$	4076.72	0.503911
$404$	6406.32	0.788927
$405$	0	0
$406$	−58.3222	−0.00712926
$407$	3922.40	0.477705
$408$	−207.658	−0.0251976
$409$	−2539.97	−0.307074	−0.153537	−	0.988143i	$-0.549066\pi$
$409$	−2539.97	−0.307074	−0.153537	+	0.988143i	$0.549066\pi$
$410$	0	0
$411$	134.797	0.0161777
$412$	−6025.80	−0.720558
$413$	516.515	0.0615401
$414$	239.240	0.0284009
$415$	0	0
$416$	2776.17	0.327195
$417$	1246.72	0.146408
$418$	−739.052	−0.0864789
$419$	1170.17	0.136436	0.0682181	−	0.997670i	$-0.478269\pi$
$419$	1170.17	0.136436	0.0682181	+	0.997670i	$0.478269\pi$
$420$	0	0
$421$	4009.01	0.464103	0.232052	−	0.972703i	$-0.425456\pi$
$421$	4009.01	0.464103	0.232052	+	0.972703i	$0.425456\pi$
$422$	−653.749	−0.0754123
$423$	−2111.09	−0.242659
$424$	2184.19	0.250174
$425$	0	0
$426$	2261.55	0.257212
$427$	499.916	0.0566573
$428$	−12025.9	−1.35817
$429$	−721.267	−0.0811727
$430$	0	0
$431$	−2676.70	−0.299146	−0.149573	−	0.988751i	$-0.547790\pi$
$431$	−2676.70	−0.299146	−0.149573	+	0.988751i	$0.547790\pi$
$432$	−1396.03	−0.155479
$433$	−5655.83	−0.627718	−0.313859	−	0.949470i	$-0.601622\pi$
$433$	−5655.83	−0.627718	−0.313859	+	0.949470i	$0.601622\pi$
$434$	−152.920	−0.0169133
$435$	0	0
$436$	6920.97	0.760217
$437$	−3410.14	−0.373294
$438$	−347.628	−0.0379230
$439$	−6305.94	−0.685572	−0.342786	−	0.939414i	$-0.611371\pi$
$439$	−6305.94	−0.685572	−0.342786	+	0.939414i	$0.611371\pi$
$440$	0	0
$441$	−3075.45	−0.332086
$442$	−97.7560	−0.0105199
$443$	6772.22	0.726315	0.363158	−	0.931728i	$-0.381699\pi$
$443$	6772.22	0.726315	0.363158	+	0.931728i	$0.381699\pi$
$444$	7997.71	0.854852
$445$	0	0
$446$	−386.424	−0.0410262
$447$	−2035.20	−0.215351
$448$	364.470	0.0384365
$449$	−5049.16	−0.530701	−0.265351	−	0.964152i	$-0.585488\pi$
$449$	−5049.16	−0.530701	−0.265351	+	0.964152i	$0.585488\pi$
$450$	0	0
$451$	2991.34	0.312321
$452$	9082.27	0.945120
$453$	−1848.25	−0.191696
$454$	−504.070	−0.0521083
$455$	0	0
$456$	−3119.39	−0.320348
$457$	−3866.14	−0.395733	−0.197867	−	0.980229i	$-0.563401\pi$
$457$	−3866.14	−0.395733	−0.197867	+	0.980229i	$0.563401\pi$
$458$	−576.629	−0.0588299
$459$	166.869	0.0169690
$460$	0	0
$461$	−9108.85	−0.920263	−0.460132	−	0.887851i	$-0.652198\pi$
$461$	−9108.85	−0.920263	−0.460132	+	0.887851i	$0.652198\pi$
$462$	27.0551	0.00272450
$463$	17421.8	1.74873	0.874363	−	0.485273i	$-0.161280\pi$
$463$	17421.8	1.74873	0.874363	+	0.485273i	$0.161280\pi$
$464$	3678.16	0.368005
$465$	0	0
$466$	4093.95	0.406971
$467$	−2602.36	−0.257864	−0.128932	−	0.991653i	$-0.541155\pi$
$467$	−2602.36	−0.257864	−0.128932	+	0.991653i	$0.541155\pi$
$468$	−1470.65	−0.145258
$469$	150.679	0.0148352
$470$	0	0
$471$	−1849.80	−0.180964
$472$	5106.40	0.497968
$473$	−1713.58	−0.166576
$474$	1652.97	0.160176
$475$	0	0
$476$	−52.3458	−0.00504047
$477$	−1755.16	−0.168476
$478$	−1464.06	−0.140093
$479$	10751.8	1.02560	0.512801	−	0.858508i	$-0.328608\pi$
$479$	10751.8	1.02560	0.512801	+	0.858508i	$0.328608\pi$
$480$	0	0
$481$	7793.65	0.738794
$482$	−2542.93	−0.240306
$483$	124.838	0.0117605
$484$	−904.630	−0.0849577
$485$	0	0
$486$	−175.856	−0.0164135
$487$	2115.37	0.196831	0.0984154	−	0.995145i	$-0.468623\pi$
$487$	2115.37	0.196831	0.0984154	+	0.995145i	$0.468623\pi$
$488$	4942.30	0.458458
$489$	6093.46	0.563509
$490$	0	0
$491$	4634.30	0.425953	0.212977	−	0.977057i	$-0.431684\pi$
$491$	4634.30	0.425953	0.212977	+	0.977057i	$0.431684\pi$
$492$	6099.29	0.558897
$493$	−439.653	−0.0401642
$494$	−1468.47	−0.133744
$495$	0	0
$496$	9644.09	0.873049
$497$	1180.10	0.106509
$498$	112.281	0.0101032
$499$	2495.43	0.223870	0.111935	−	0.993716i	$-0.464295\pi$
$499$	2495.43	0.223870	0.111935	+	0.993716i	$0.464295\pi$
$500$	0	0
$501$	−6344.69	−0.565788
$502$	1313.00	0.116737
$503$	−18521.0	−1.64177	−0.820886	−	0.571092i	$-0.806520\pi$
$503$	−18521.0	−1.64177	−0.820886	+	0.571092i	$0.806520\pi$
$504$	114.194	0.0100925
$505$	0	0
$506$	292.404	0.0256896
$507$	5157.87	0.451813
$508$	8440.41	0.737170
$509$	−6102.40	−0.531403	−0.265702	−	0.964055i	$-0.585604\pi$
$509$	−6102.40	−0.531403	−0.265702	+	0.964055i	$0.585604\pi$
$510$	0	0
$511$	−181.396	−0.0157035
$512$	11200.2	0.966765
$513$	2506.66	0.215735
$514$	−1046.06	−0.0897661
$515$	0	0
$516$	−3493.96	−0.298087
$517$	−2580.23	−0.219494
$518$	−292.344	−0.0247970
$519$	−46.5288	−0.00393523
$520$	0	0
$521$	−2813.12	−0.236555	−0.118277	−	0.992981i	$-0.537737\pi$
$521$	−2813.12	−0.236555	−0.118277	+	0.992981i	$0.537737\pi$
$522$	463.331	0.0388495
$523$	−1563.27	−0.130702	−0.0653509	−	0.997862i	$-0.520817\pi$
$523$	−1563.27	−0.130702	−0.0653509	+	0.997862i	$0.520817\pi$
$524$	11680.3	0.973772
$525$	0	0
$526$	3192.75	0.264659
$527$	−1152.76	−0.0952850
$528$	−1706.26	−0.140636
$529$	−10817.8	−0.889109
$530$	0	0
$531$	−4103.37	−0.335350
$532$	−786.326	−0.0640818
$533$	5943.67	0.483019
$534$	2334.44	0.189178
$535$	0	0
$536$	1489.65	0.120043
$537$	−1463.97	−0.117645
$538$	2862.28	0.229371
$539$	−3758.88	−0.300383
$540$	0	0
$541$	−5959.61	−0.473611	−0.236806	−	0.971557i	$-0.576100\pi$
$541$	−5959.61	−0.473611	−0.236806	+	0.971557i	$0.576100\pi$
$542$	4817.55	0.381792
$543$	−3746.13	−0.296062
$544$	−785.012	−0.0618697
$545$	0	0
$546$	53.7574	0.00421356
$547$	−18084.2	−1.41357	−0.706787	−	0.707426i	$-0.749856\pi$
$547$	−18084.2	−1.41357	−0.706787	+	0.707426i	$0.749856\pi$
$548$	335.927	0.0261863
$549$	−3971.50	−0.308742
$550$	0	0
$551$	−6604.35	−0.510626
$552$	1234.18	0.0951634
$553$	862.540	0.0663272
$554$	−6168.55	−0.473062
$555$	0	0
$556$	3106.95	0.236985
$557$	−18129.5	−1.37912	−0.689561	−	0.724227i	$-0.742197\pi$
$557$	−18129.5	−1.37912	−0.689561	+	0.724227i	$0.742197\pi$
$558$	1214.85	0.0921659
$559$	−3404.81	−0.257618
$560$	0	0
$561$	203.951	0.0153490
$562$	−3047.43	−0.228733
$563$	17910.2	1.34072	0.670361	−	0.742035i	$-0.266139\pi$
$563$	17910.2	1.34072	0.670361	+	0.742035i	$0.266139\pi$
$564$	−5261.04	−0.392784
$565$	0	0
$566$	−2554.36	−0.189696
$567$	−91.7635	−0.00679666
$568$	11666.8	0.861843
$569$	24176.6	1.78126	0.890629	−	0.454731i	$-0.150265\pi$
$569$	24176.6	1.78126	0.890629	+	0.454731i	$0.150265\pi$
$570$	0	0
$571$	−13167.8	−0.965073	−0.482537	−	0.875876i	$-0.660285\pi$
$571$	−13167.8	−0.965073	−0.482537	+	0.875876i	$0.660285\pi$
$572$	−1797.46	−0.131391
$573$	7714.81	0.562462
$574$	−222.950	−0.0162121
$575$	0	0
$576$	−2895.47	−0.209452
$577$	722.513	0.0521293	0.0260646	−	0.999660i	$-0.491702\pi$
$577$	722.513	0.0521293	0.0260646	+	0.999660i	$0.491702\pi$
$578$	−3527.83	−0.253872
$579$	3665.41	0.263091
$580$	0	0
$581$	58.5893	0.00418364
$582$	−1526.77	−0.108740
$583$	−2145.19	−0.152393
$584$	−1793.33	−0.127069
$585$	0	0
$586$	5146.10	0.362770
$587$	17038.8	1.19807	0.599034	−	0.800723i	$-0.295551\pi$
$587$	17038.8	1.19807	0.599034	+	0.800723i	$0.295551\pi$
$588$	−7664.31	−0.537535
$589$	−17316.5	−1.21140
$590$	0	0
$591$	−4284.97	−0.298241
$592$	18437.0	1.28000
$593$	−6206.74	−0.429815	−0.214908	−	0.976634i	$-0.568945\pi$
$593$	−6206.74	−0.429815	−0.214908	+	0.976634i	$0.568945\pi$
$594$	−214.935	−0.0148466
$595$	0	0
$596$	−5071.92	−0.348580
$597$	2448.50	0.167856
$598$	580.996	0.0397302
$599$	−23050.7	−1.57233	−0.786164	−	0.618018i	$-0.787936\pi$
$599$	−23050.7	−1.57233	−0.786164	+	0.618018i	$0.787936\pi$
$600$	0	0
$601$	−6323.69	−0.429199	−0.214600	−	0.976702i	$-0.568845\pi$
$601$	−6323.69	−0.429199	−0.214600	+	0.976702i	$0.568845\pi$
$602$	127.716	0.00864673
$603$	−1197.04	−0.0808412
$604$	−4606.01	−0.310291
$605$	0	0
$606$	1860.35	0.124705
$607$	−4625.14	−0.309273	−0.154636	−	0.987971i	$-0.549421\pi$
$607$	−4625.14	−0.309273	−0.154636	+	0.987971i	$0.549421\pi$
$608$	−11792.3	−0.786577
$609$	241.771	0.0160871
$610$	0	0
$611$	−5126.81	−0.339457
$612$	415.853	0.0274671
$613$	28075.3	1.84984	0.924919	−	0.380164i	$-0.124132\pi$
$613$	28075.3	1.84984	0.924919	+	0.380164i	$0.124132\pi$
$614$	−6856.72	−0.450676
$615$	0	0
$616$	139.571	0.00912900
$617$	7264.85	0.474023	0.237011	−	0.971507i	$-0.423832\pi$
$617$	7264.85	0.474023	0.237011	+	0.971507i	$0.423832\pi$
$618$	−1749.85	−0.113898
$619$	−30583.2	−1.98585	−0.992927	−	0.118728i	$-0.962119\pi$
$619$	−30583.2	−1.98585	−0.992927	+	0.118728i	$0.962119\pi$
$620$	0	0
$621$	−991.755	−0.0640866
$622$	−3770.93	−0.243088
$623$	1218.14	0.0783365
$624$	−3390.28	−0.217500
$625$	0	0
$626$	−1407.04	−0.0898351
$627$	3063.70	0.195139
$628$	−4609.86	−0.292920
$629$	−2203.79	−0.139699
$630$	0	0
$631$	−16679.8	−1.05232	−0.526159	−	0.850386i	$-0.676368\pi$
$631$	−16679.8	−1.05232	−0.526159	+	0.850386i	$0.676368\pi$
$632$	8527.29	0.536705
$633$	2710.08	0.170167
$634$	4959.23	0.310656
$635$	0	0
$636$	−4374.02	−0.272706
$637$	−7468.75	−0.464557
$638$	566.293	0.0351407
$639$	−9375.11	−0.580397
$640$	0	0
$641$	4225.31	0.260358	0.130179	−	0.991490i	$-0.458445\pi$
$641$	4225.31	0.260358	0.130179	+	0.991490i	$0.458445\pi$
$642$	−3492.25	−0.214685
$643$	−23703.5	−1.45377	−0.726886	−	0.686758i	$-0.759033\pi$
$643$	−23703.5	−1.45377	−0.726886	+	0.686758i	$0.759033\pi$
$644$	311.108	0.0190363
$645$	0	0
$646$	415.234	0.0252898
$647$	10045.1	0.610374	0.305187	−	0.952292i	$-0.401281\pi$
$647$	10045.1	0.610374	0.305187	+	0.952292i	$0.401281\pi$
$648$	−907.197	−0.0549970
$649$	−5015.23	−0.303336
$650$	0	0
$651$	633.921	0.0381649
$652$	15185.5	0.912130
$653$	32314.2	1.93652	0.968262	−	0.249936i	$-0.0804096\pi$
$653$	32314.2	1.93652	0.968262	+	0.249936i	$0.0804096\pi$
$654$	2009.80	0.120167
$655$	0	0
$656$	14060.6	0.836853
$657$	1441.07	0.0855731
$658$	192.309	0.0113936
$659$	−12032.4	−0.711253	−0.355627	−	0.934628i	$-0.615733\pi$
$659$	−12032.4	−0.711253	−0.355627	+	0.934628i	$0.615733\pi$
$660$	0	0
$661$	12864.5	0.756988	0.378494	−	0.925604i	$-0.376442\pi$
$661$	12864.5	0.756988	0.378494	+	0.925604i	$0.376442\pi$
$662$	−1461.74	−0.0858189
$663$	405.242	0.0237380
$664$	579.229	0.0338531
$665$	0	0
$666$	2322.48	0.135126
$667$	2613.00	0.151688
$668$	−15811.5	−0.915819
$669$	1601.90	0.0925754
$670$	0	0
$671$	−4854.06	−0.279268
$672$	431.689	0.0247809
$673$	−12007.5	−0.687751	−0.343876	−	0.939015i	$-0.611740\pi$
$673$	−12007.5	−0.687751	−0.343876	+	0.939015i	$0.611740\pi$
$674$	−8495.16	−0.485492
$675$	0	0
$676$	12853.9	0.731332
$677$	10732.5	0.609284	0.304642	−	0.952467i	$-0.401463\pi$
$677$	10732.5	0.609284	0.304642	+	0.952467i	$0.401463\pi$
$678$	2637.43	0.149395
$679$	−796.687	−0.0450280
$680$	0	0
$681$	2089.59	0.117582
$682$	1484.81	0.0833672
$683$	−14126.0	−0.791383	−0.395692	−	0.918383i	$-0.629495\pi$
$683$	−14126.0	−0.791383	−0.395692	+	0.918383i	$0.629495\pi$
$684$	6246.83	0.349201
$685$	0	0
$686$	561.366	0.0312435
$687$	2390.38	0.132749
$688$	−8054.59	−0.446335
$689$	−4262.42	−0.235682
$690$	0	0
$691$	−18467.8	−1.01671	−0.508357	−	0.861147i	$-0.669747\pi$
$691$	−18467.8	−1.01671	−0.508357	+	0.861147i	$0.669747\pi$
$692$	−115.954	−0.00636981
$693$	−112.155	−0.00614781
$694$	−9185.17	−0.502398
$695$	0	0
$696$	2390.21	0.130173
$697$	−1680.68	−0.0913345
$698$	8703.66	0.471975
$699$	−16971.2	−0.918327
$700$	0	0
$701$	−24876.0	−1.34030	−0.670152	−	0.742224i	$-0.733771\pi$
$701$	−24876.0	−1.34030	−0.670152	+	0.742224i	$0.733771\pi$
$702$	−427.067	−0.0229610
$703$	−33104.8	−1.77606
$704$	−3538.91	−0.189457
$705$	0	0
$706$	6475.83	0.345214
$707$	970.752	0.0516391
$708$	−10226.0	−0.542819
$709$	21271.1	1.12673	0.563367	−	0.826207i	$-0.309506\pi$
$709$	21271.1	1.12673	0.563367	+	0.826207i	$0.309506\pi$
$710$	0	0
$711$	−6852.31	−0.361437
$712$	12042.8	0.633881
$713$	6851.25	0.359862
$714$	−15.2008	−0.000796747 0
$715$	0	0
$716$	−3648.36	−0.190427
$717$	6069.17	0.316119
$718$	2827.09	0.146945
$719$	19213.9	0.996603	0.498302	−	0.867004i	$-0.333957\pi$
$719$	19213.9	0.996603	0.498302	+	0.867004i	$0.333957\pi$
$720$	0	0
$721$	−913.091	−0.0471641
$722$	1273.79	0.0656585
$723$	10541.6	0.542249
$724$	−9335.70	−0.479224
$725$	0	0
$726$	−262.698	−0.0134293
$727$	27454.7	1.40060	0.700302	−	0.713846i	$-0.253049\pi$
$727$	27454.7	1.40060	0.700302	+	0.713846i	$0.253049\pi$
$728$	277.322	0.0141184
$729$	729.000	0.0370370
$730$	0	0
$731$	962.770	0.0487132
$732$	−9897.35	−0.499750
$733$	1182.16	0.0595691	0.0297846	−	0.999556i	$-0.490518\pi$
$733$	1182.16	0.0595691	0.0297846	+	0.999556i	$0.490518\pi$
$734$	−3057.61	−0.153758
$735$	0	0
$736$	4665.58	0.233662
$737$	−1463.05	−0.0731237
$738$	1771.19	0.0883447
$739$	25688.2	1.27869	0.639347	−	0.768918i	$-0.279205\pi$
$739$	25688.2	1.27869	0.639347	+	0.768918i	$0.279205\pi$
$740$	0	0
$741$	6087.45	0.301792
$742$	159.886	0.00791049
$743$	−32431.2	−1.60132	−0.800662	−	0.599116i	$-0.795519\pi$
$743$	−32431.2	−1.60132	−0.800662	+	0.599116i	$0.795519\pi$
$744$	6267.10	0.308822
$745$	0	0
$746$	−3056.49	−0.150008
$747$	−465.453	−0.0227979
$748$	508.264	0.0248449
$749$	−1822.29	−0.0888988
$750$	0	0
$751$	−32093.5	−1.55940	−0.779700	−	0.626154i	$-0.784628\pi$
$751$	−32093.5	−1.55940	−0.779700	+	0.626154i	$0.784628\pi$
$752$	−12128.2	−0.588127
$753$	−5442.96	−0.263416
$754$	1125.20	0.0543467
$755$	0	0
$756$	−228.683	−0.0110015
$757$	1293.36	0.0620977	0.0310488	−	0.999518i	$-0.490115\pi$
$757$	1293.36	0.0620977	0.0310488	+	0.999518i	$0.490115\pi$
$758$	−541.323	−0.0259390
$759$	−1212.15	−0.0579685
$760$	0	0
$761$	29013.1	1.38203	0.691014	−	0.722841i	$-0.257164\pi$
$761$	29013.1	1.38203	0.691014	+	0.722841i	$0.257164\pi$
$762$	2451.03	0.116524
$763$	1048.74	0.0497599
$764$	19226.0	0.910436
$765$	0	0
$766$	−3378.22	−0.159347
$767$	−9965.06	−0.469123
$768$	−5009.67	−0.235379
$769$	−31304.1	−1.46795	−0.733976	−	0.679176i	$-0.762337\pi$
$769$	−31304.1	−1.46795	−0.733976	+	0.679176i	$0.762337\pi$
$770$	0	0
$771$	4336.39	0.202557
$772$	9134.55	0.425855
$773$	−41913.5	−1.95023	−0.975113	−	0.221710i	$-0.928836\pi$
$773$	−41913.5	−1.95023	−0.975113	+	0.221710i	$0.928836\pi$
$774$	−1014.62	−0.0471186
$775$	0	0
$776$	−7876.24	−0.364357
$777$	1211.90	0.0559543
$778$	4735.89	0.218239
$779$	−25246.7	−1.16118
$780$	0	0
$781$	−11458.5	−0.524989
$782$	−164.287	−0.00751263
$783$	−1920.71	−0.0876637
$784$	−17668.4	−0.804867
$785$	0	0
$786$	3391.88	0.153924
$787$	6648.00	0.301113	0.150556	−	0.988601i	$-0.451894\pi$
$787$	6648.00	0.301113	0.150556	+	0.988601i	$0.451894\pi$
$788$	−10678.6	−0.482751
$789$	−13235.4	−0.597201
$790$	0	0
$791$	1376.24	0.0618628
$792$	−1108.80	−0.0497467
$793$	−9644.82	−0.431901
$794$	−4660.42	−0.208302
$795$	0	0
$796$	6101.88	0.271703
$797$	−14798.4	−0.657701	−0.328850	−	0.944382i	$-0.606661\pi$
$797$	−14798.4	−0.657701	−0.328850	+	0.944382i	$0.606661\pi$
$798$	−228.343	−0.0101294
$799$	1449.69	0.0641884
$800$	0	0
$801$	−9677.28	−0.426879
$802$	5591.94	0.246207
$803$	1761.31	0.0774038
$804$	−2983.14	−0.130855
$805$	0	0
$806$	2950.27	0.128931
$807$	−11865.4	−0.517574
$808$	9597.09	0.417852
$809$	4500.04	0.195566	0.0977831	−	0.995208i	$-0.468825\pi$
$809$	4500.04	0.195566	0.0977831	+	0.995208i	$0.468825\pi$
$810$	0	0
$811$	−1909.49	−0.0826773	−0.0413386	−	0.999145i	$-0.513162\pi$
$811$	−1909.49	−0.0826773	−0.0413386	+	0.999145i	$0.513162\pi$
$812$	602.517	0.0260396
$813$	−19970.9	−0.861512
$814$	2838.58	0.122226
$815$	0	0
$816$	958.661	0.0411273
$817$	14462.5	0.619313
$818$	−1838.14	−0.0785685
$819$	−222.848	−0.00950788
$820$	0	0
$821$	10928.6	0.464569	0.232285	−	0.972648i	$-0.425380\pi$
$821$	10928.6	0.464569	0.232285	+	0.972648i	$0.425380\pi$
$822$	97.5507	0.00413926
$823$	−29278.4	−1.24007	−0.620036	−	0.784573i	$-0.712882\pi$
$823$	−29278.4	−1.24007	−0.620036	+	0.784573i	$0.712882\pi$
$824$	−9027.05	−0.381641
$825$	0	0
$826$	373.795	0.0157457
$827$	878.905	0.0369559	0.0184779	−	0.999829i	$-0.494118\pi$
$827$	878.905	0.0369559	0.0184779	+	0.999829i	$0.494118\pi$
$828$	−2471.55	−0.103734
$829$	16130.1	0.675779	0.337890	−	0.941186i	$-0.390287\pi$
$829$	16130.1	0.675779	0.337890	+	0.941186i	$0.390287\pi$
$830$	0	0
$831$	25571.4	1.06746
$832$	−7031.67	−0.293004
$833$	2111.92	0.0878436
$834$	902.235	0.0374603
$835$	0	0
$836$	7635.02	0.315864
$837$	−5036.08	−0.207972
$838$	846.838	0.0349088
$839$	−39531.8	−1.62669	−0.813343	−	0.581784i	$-0.802355\pi$
$839$	−39531.8	−1.62669	−0.813343	+	0.581784i	$0.802355\pi$
$840$	0	0
$841$	−19328.5	−0.792507
$842$	2901.27	0.118746
$843$	12633.0	0.516135
$844$	6753.77	0.275443
$845$	0	0
$846$	−1527.77	−0.0620872
$847$	−137.079	−0.00556090
$848$	−10083.4	−0.408331
$849$	10589.0	0.428048
$850$	0	0
$851$	13097.8	0.527600
$852$	−23363.7	−0.939467
$853$	5373.68	0.215699	0.107849	−	0.994167i	$-0.465604\pi$
$853$	5373.68	0.215699	0.107849	+	0.994167i	$0.465604\pi$
$854$	361.782	0.0144964
$855$	0	0
$856$	−18015.7	−0.719349
$857$	45378.4	1.80875	0.904374	−	0.426740i	$-0.140338\pi$
$857$	45378.4	1.80875	0.904374	+	0.426740i	$0.140338\pi$
$858$	−521.971	−0.0207690
$859$	35336.8	1.40358	0.701791	−	0.712383i	$-0.252384\pi$
$859$	35336.8	1.40358	0.701791	+	0.712383i	$0.252384\pi$
$860$	0	0
$861$	924.228	0.0365826
$862$	−1937.09	−0.0765401
$863$	24595.3	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$863$	24595.3	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$864$	−3429.48	−0.135039
$865$	0	0
$866$	−4093.05	−0.160609
$867$	14624.4	0.572862
$868$	1579.79	0.0617760
$869$	−8375.04	−0.326932
$870$	0	0
$871$	−2907.02	−0.113089
$872$	10368.1	0.402646
$873$	6329.14	0.245371
$874$	−2467.87	−0.0955115
$875$	0	0
$876$	3591.28	0.138514
$877$	8251.06	0.317695	0.158848	−	0.987303i	$-0.449222\pi$
$877$	8251.06	0.317695	0.158848	+	0.987303i	$0.449222\pi$
$878$	−4563.52	−0.175412
$879$	−21332.9	−0.818589
$880$	0	0
$881$	−3528.36	−0.134930	−0.0674651	−	0.997722i	$-0.521491\pi$
$881$	−3528.36	−0.134930	−0.0674651	+	0.997722i	$0.521491\pi$
$882$	−2225.66	−0.0849681
$883$	−35459.1	−1.35141	−0.675703	−	0.737174i	$-0.736160\pi$
$883$	−35459.1	−1.35141	−0.675703	+	0.737174i	$0.736160\pi$
$884$	1009.90	0.0384238
$885$	0	0
$886$	4900.96	0.185836
$887$	−32544.8	−1.23196	−0.615979	−	0.787763i	$-0.711239\pi$
$887$	−32544.8	−1.23196	−0.615979	+	0.787763i	$0.711239\pi$
$888$	11981.1	0.452770
$889$	1278.98	0.0482514
$890$	0	0
$891$	891.000	0.0335013
$892$	3992.08	0.149848
$893$	21777.0	0.816056
$894$	−1472.85	−0.0551000
$895$	0	0
$896$	1414.93	0.0527562
$897$	−2408.49	−0.0896510
$898$	−3654.01	−0.135786
$899$	13268.7	0.492253
$900$	0	0
$901$	1205.27	0.0445654
$902$	2164.79	0.0799108
$903$	−529.441	−0.0195113
$904$	13605.9	0.500579
$905$	0	0
$906$	−1337.55	−0.0490477
$907$	−20383.4	−0.746219	−0.373110	−	0.927787i	$-0.621709\pi$
$907$	−20383.4	−0.746219	−0.373110	+	0.927787i	$0.621709\pi$
$908$	5207.46	0.190326
$909$	−7711.98	−0.281397
$910$	0	0
$911$	7783.29	0.283065	0.141532	−	0.989934i	$-0.454797\pi$
$911$	7783.29	0.283065	0.141532	+	0.989934i	$0.454797\pi$
$912$	14400.8	0.522869
$913$	−568.887	−0.0206215
$914$	−2797.87	−0.101253
$915$	0	0
$916$	5957.05	0.214876
$917$	1769.92	0.0637382
$918$	120.761	0.00434171
$919$	27294.9	0.979734	0.489867	−	0.871797i	$-0.337045\pi$
$919$	27294.9	0.979734	0.489867	+	0.871797i	$0.337045\pi$
$920$	0	0
$921$	28424.2	1.01695
$922$	−6591.95	−0.235460
$923$	−22767.5	−0.811920
$924$	−279.502	−0.00995122
$925$	0	0
$926$	12607.9	0.447432
$927$	7253.90	0.257011
$928$	9035.73	0.319625
$929$	−46468.7	−1.64111	−0.820554	−	0.571568i	$-0.806335\pi$
$929$	−46468.7	−1.64111	−0.820554	+	0.571568i	$0.806335\pi$
$930$	0	0
$931$	31724.7	1.11679
$932$	−42293.9	−1.48646
$933$	15632.2	0.548526
$934$	−1883.29	−0.0659776
$935$	0	0
$936$	−2203.14	−0.0769356
$937$	−6922.45	−0.241352	−0.120676	−	0.992692i	$-0.538506\pi$
$937$	−6922.45	−0.241352	−0.120676	+	0.992692i	$0.538506\pi$
$938$	109.044	0.00379575
$939$	5832.82	0.202712
$940$	0	0
$941$	−23772.1	−0.823536	−0.411768	−	0.911289i	$-0.635089\pi$
$941$	−23772.1	−0.823536	−0.411768	+	0.911289i	$0.635089\pi$
$942$	−1338.67	−0.0463018
$943$	9988.80	0.344942
$944$	−23573.8	−0.812779
$945$	0	0
$946$	−1240.09	−0.0426204
$947$	−7177.76	−0.246300	−0.123150	−	0.992388i	$-0.539300\pi$
$947$	−7177.76	−0.246300	−0.123150	+	0.992388i	$0.539300\pi$
$948$	−17076.6	−0.585044
$949$	3499.65	0.119709
$950$	0	0
$951$	−20558.2	−0.700994
$952$	−78.4175	−0.00266967
$953$	27530.2	0.935772	0.467886	−	0.883789i	$-0.345016\pi$
$953$	27530.2	0.935772	0.467886	+	0.883789i	$0.345016\pi$
$954$	−1270.18	−0.0431066
$955$	0	0
$956$	15124.9	0.511690
$957$	−2347.54	−0.0792948
$958$	7780.94	0.262412
$959$	50.9031	0.00171402
$960$	0	0
$961$	4999.29	0.167812
$962$	5640.15	0.189029
$963$	14476.9	0.484436
$964$	26270.6	0.877718
$965$	0	0
$966$	90.3435	0.00300906
$967$	55407.1	1.84258	0.921288	−	0.388881i	$-0.127138\pi$
$967$	55407.1	1.84258	0.921288	+	0.388881i	$0.127138\pi$
$968$	−1355.20	−0.0449976
$969$	−1721.33	−0.0570662
$970$	0	0
$971$	−55100.7	−1.82108	−0.910539	−	0.413423i	$-0.864333\pi$
$971$	−55100.7	−1.82108	−0.910539	+	0.413423i	$0.864333\pi$
$972$	1816.74	0.0599504
$973$	470.797	0.0155119
$974$	1530.86	0.0503615
$975$	0	0
$976$	−22816.3	−0.748290
$977$	−44473.9	−1.45634	−0.728170	−	0.685396i	$-0.759629\pi$
$977$	−44473.9	−1.45634	−0.728170	+	0.685396i	$0.759629\pi$
$978$	4409.75	0.144180
$979$	−11827.8	−0.386127
$980$	0	0
$981$	−8331.52	−0.271157
$982$	3353.78	0.108985
$983$	28003.0	0.908605	0.454302	−	0.890848i	$-0.349889\pi$
$983$	28003.0	0.908605	0.454302	+	0.890848i	$0.349889\pi$
$984$	9137.15	0.296018
$985$	0	0
$986$	−318.170	−0.0102765
$987$	−797.208	−0.0257096
$988$	15170.5	0.488499
$989$	−5722.06	−0.183974
$990$	0	0
$991$	−40719.5	−1.30524	−0.652622	−	0.757684i	$-0.726331\pi$
$991$	−40719.5	−1.30524	−0.652622	+	0.757684i	$0.726331\pi$
$992$	23691.6	0.758274
$993$	6059.56	0.193650
$994$	854.022	0.0272514
$995$	0	0
$996$	−1159.95	−0.0369021
$997$	−18161.1	−0.576898	−0.288449	−	0.957495i	$-0.593140\pi$
$997$	−18161.1	−0.576898	−0.288449	+	0.957495i	$0.593140\pi$
$998$	1805.91	0.0572796
$999$	−9627.70	−0.304912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.q.1.2	yes	3
3.2	odd	2		2475.4.a.v.1.2		3
5.2	odd	4		825.4.c.n.199.4		6
5.3	odd	4		825.4.c.n.199.3		6
5.4	even	2		825.4.a.o.1.2	✓	3
15.14	odd	2		2475.4.a.y.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.2	✓	3	5.4	even	2
825.4.a.q.1.2	yes	3	1.1	even	1	trivial
825.4.c.n.199.3		6	5.3	odd	4
825.4.c.n.199.4		6	5.2	odd	4
2475.4.a.v.1.2		3	3.2	odd	2
2475.4.a.y.1.2		3	15.14	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)

\(f(q)\)	\(=\)	\(q+0.723686 q^{2} -3.00000 q^{3} -7.47628 q^{4} -2.17106 q^{6} -1.13288 q^{7} -11.2000 q^{8} +9.00000 q^{9} +11.0000 q^{11} +22.4288 q^{12} +21.8566 q^{13} -0.819851 q^{14} +51.7050 q^{16} -6.18033 q^{17} +6.51317 q^{18} -92.8393 q^{19} +3.39865 q^{21} +7.96054 q^{22} +36.7317 q^{23} +33.5999 q^{24} +15.8173 q^{26} -27.0000 q^{27} +8.46975 q^{28} +71.1375 q^{29} +186.522 q^{31} +127.018 q^{32} -33.0000 q^{33} -4.47261 q^{34} -67.2865 q^{36} +356.581 q^{37} -67.1865 q^{38} -65.5697 q^{39} +271.940 q^{41} +2.45955 q^{42} -155.780 q^{43} -82.2391 q^{44} +26.5822 q^{46} -234.566 q^{47} -155.115 q^{48} -341.717 q^{49} +18.5410 q^{51} -163.406 q^{52} -195.018 q^{53} -19.5395 q^{54} +12.6882 q^{56} +278.518 q^{57} +51.4812 q^{58} -455.930 q^{59} -441.278 q^{61} +134.983 q^{62} -10.1959 q^{63} -321.719 q^{64} -23.8816 q^{66} -133.005 q^{67} +46.2058 q^{68} -110.195 q^{69} -1041.68 q^{71} -100.800 q^{72} +160.119 q^{73} +258.053 q^{74} +694.093 q^{76} -12.4617 q^{77} -47.4519 q^{78} -761.367 q^{79} +81.0000 q^{81} +196.799 q^{82} -51.7170 q^{83} -25.4092 q^{84} -112.736 q^{86} -213.412 q^{87} -123.200 q^{88} -1075.25 q^{89} -24.7609 q^{91} -274.616 q^{92} -559.565 q^{93} -169.752 q^{94} -381.054 q^{96} +703.238 q^{97} -247.295 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 9 q^{3} + 7 q^{4} - 3 q^{6} + 16 q^{7} - 3 q^{8} + 27 q^{9} + 33 q^{11} - 21 q^{12} + 45 q^{13} - 116 q^{14} - 85 q^{16} - 58 q^{17} + 9 q^{18} - 169 q^{19} - 48 q^{21} + 11 q^{22} + 155 q^{23}+ \cdots + 297 q^{99}+O(q^{100}) \) 3 * q + q^2 - 9 * q^3 + 7 * q^4 - 3 * q^6 + 16 * q^7 - 3 * q^8 + 27 * q^9 + 33 * q^11 - 21 * q^12 + 45 * q^13 - 116 * q^14 - 85 * q^16 - 58 * q^17 + 9 * q^18 - 169 * q^19 - 48 * q^21 + 11 * q^22 + 155 * q^23 + 9 * q^24 + 167 * q^26 - 81 * q^27 + 100 * q^28 - 277 * q^29 - 173 * q^31 + 97 * q^32 - 99 * q^33 - 146 * q^34 + 63 * q^36 + 60 * q^37 + 169 * q^38 - 135 * q^39 + 44 * q^41 + 348 * q^42 - 109 * q^43 + 77 * q^44 + 425 * q^46 + 270 * q^47 + 255 * q^48 - 427 * q^49 + 174 * q^51 + 45 * q^52 - 148 * q^53 - 27 * q^54 + 168 * q^56 + 507 * q^57 - 783 * q^58 - 684 * q^59 - 1038 * q^61 + 953 * q^62 + 144 * q^63 - 1129 * q^64 - 33 * q^66 + 314 * q^67 - 366 * q^68 - 465 * q^69 - 1459 * q^71 - 27 * q^72 - 1170 * q^73 - 1764 * q^74 + 211 * q^76 + 176 * q^77 - 501 * q^78 - 506 * q^79 + 243 * q^81 - 1040 * q^82 - 347 * q^83 - 300 * q^84 - 527 * q^86 + 831 * q^87 - 33 * q^88 - 607 * q^89 - 398 * q^91 + 687 * q^92 + 519 * q^93 - 1822 * q^94 - 291 * q^96 + 1263 * q^97 - 2273 * q^98 + 297 * q^99

Introduction

L-functions

Modular forms

Varieties

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