LMFDB - Embedded newform 825.4.a.a.1.1

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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} +15.0000 q^{6} -3.00000 q^{7} -45.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} +15.0000 q^{6} -3.00000 q^{7} -45.0000 q^{8} +9.00000 q^{9} -11.0000 q^{11} -51.0000 q^{12} -32.0000 q^{13} +15.0000 q^{14} +89.0000 q^{16} -33.0000 q^{17} -45.0000 q^{18} +47.0000 q^{19} +9.00000 q^{21} +55.0000 q^{22} -113.000 q^{23} +135.000 q^{24} +160.000 q^{26} -27.0000 q^{27} -51.0000 q^{28} -54.0000 q^{29} +178.000 q^{31} -85.0000 q^{32} +33.0000 q^{33} +165.000 q^{34} +153.000 q^{36} -19.0000 q^{37} -235.000 q^{38} +96.0000 q^{39} +139.000 q^{41} -45.0000 q^{42} +308.000 q^{43} -187.000 q^{44} +565.000 q^{46} -195.000 q^{47} -267.000 q^{48} -334.000 q^{49} +99.0000 q^{51} -544.000 q^{52} -152.000 q^{53} +135.000 q^{54} +135.000 q^{56} -141.000 q^{57} +270.000 q^{58} -625.000 q^{59} +320.000 q^{61} -890.000 q^{62} -27.0000 q^{63} -287.000 q^{64} -165.000 q^{66} -200.000 q^{67} -561.000 q^{68} +339.000 q^{69} -947.000 q^{71} -405.000 q^{72} +448.000 q^{73} +95.0000 q^{74} +799.000 q^{76} +33.0000 q^{77} -480.000 q^{78} -721.000 q^{79} +81.0000 q^{81} -695.000 q^{82} -142.000 q^{83} +153.000 q^{84} -1540.00 q^{86} +162.000 q^{87} +495.000 q^{88} +404.000 q^{89} +96.0000 q^{91} -1921.00 q^{92} -534.000 q^{93} +975.000 q^{94} +255.000 q^{96} -79.0000 q^{97} +1670.00 q^{98} -99.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	17.0000	2.12500
$5$	0	0
$5$	0	0
$6$	15.0000	1.02062
$7$	−3.00000	−0.161985	−0.0809924	−	0.996715i	$-0.525809\pi$
$7$	−3.00000	−0.161985	−0.0809924	+	0.996715i	$0.525809\pi$
$8$	−45.0000	−1.98874
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−51.0000	−1.22687
$13$	−32.0000	−0.682708	−0.341354	−	0.939935i	$-0.610885\pi$
$13$	−32.0000	−0.682708	−0.341354	+	0.939935i	$0.610885\pi$
$14$	15.0000	0.286351
$15$	0	0
$16$	89.0000	1.39062
$17$	−33.0000	−0.470804	−0.235402	−	0.971898i	$-0.575641\pi$
$17$	−33.0000	−0.470804	−0.235402	+	0.971898i	$0.575641\pi$
$18$	−45.0000	−0.589256
$19$	47.0000	0.567502	0.283751	−	0.958898i	$-0.408421\pi$
$19$	47.0000	0.567502	0.283751	+	0.958898i	$0.408421\pi$
$20$	0	0
$21$	9.00000	0.0935220
$22$	55.0000	0.533002
$23$	−113.000	−1.02444	−0.512220	−	0.858854i	$-0.671177\pi$
$23$	−113.000	−1.02444	−0.512220	+	0.858854i	$0.671177\pi$
$24$	135.000	1.14820
$25$	0	0
$26$	160.000	1.20687
$27$	−27.0000	−0.192450
$28$	−51.0000	−0.344218
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	178.000	1.03128	0.515641	−	0.856805i	$-0.327554\pi$
$31$	178.000	1.03128	0.515641	+	0.856805i	$0.327554\pi$
$32$	−85.0000	−0.469563
$33$	33.0000	0.174078
$34$	165.000	0.832273
$35$	0	0
$36$	153.000	0.708333
$37$	−19.0000	−0.0844211	−0.0422106	−	0.999109i	$-0.513440\pi$
$37$	−19.0000	−0.0844211	−0.0422106	+	0.999109i	$0.513440\pi$
$38$	−235.000	−1.00321
$39$	96.0000	0.394162
$40$	0	0
$41$	139.000	0.529467	0.264734	−	0.964322i	$-0.414716\pi$
$41$	139.000	0.529467	0.264734	+	0.964322i	$0.414716\pi$
$42$	−45.0000	−0.165325
$43$	308.000	1.09232	0.546158	−	0.837682i	$-0.316090\pi$
$43$	308.000	1.09232	0.546158	+	0.837682i	$0.316090\pi$
$44$	−187.000	−0.640712
$45$	0	0
$46$	565.000	1.81097
$47$	−195.000	−0.605185	−0.302592	−	0.953120i	$-0.597852\pi$
$47$	−195.000	−0.605185	−0.302592	+	0.953120i	$0.597852\pi$
$48$	−267.000	−0.802878
$49$	−334.000	−0.973761
$50$	0	0
$51$	99.0000	0.271819
$52$	−544.000	−1.45075
$53$	−152.000	−0.393940	−0.196970	−	0.980410i	$-0.563110\pi$
$53$	−152.000	−0.393940	−0.196970	+	0.980410i	$0.563110\pi$
$54$	135.000	0.340207
$55$	0	0
$56$	135.000	0.322145
$57$	−141.000	−0.327647
$58$	270.000	0.611254
$59$	−625.000	−1.37912	−0.689560	−	0.724229i	$-0.742196\pi$
$59$	−625.000	−1.37912	−0.689560	+	0.724229i	$0.742196\pi$
$60$	0	0
$61$	320.000	0.671669	0.335834	−	0.941921i	$-0.390982\pi$
$61$	320.000	0.671669	0.335834	+	0.941921i	$0.390982\pi$
$62$	−890.000	−1.82307
$63$	−27.0000	−0.0539949
$64$	−287.000	−0.560547
$65$	0	0
$66$	−165.000	−0.307729
$67$	−200.000	−0.364685	−0.182342	−	0.983235i	$-0.558368\pi$
$67$	−200.000	−0.364685	−0.182342	+	0.983235i	$0.558368\pi$
$68$	−561.000	−1.00046
$69$	339.000	0.591461
$70$	0	0
$71$	−947.000	−1.58293	−0.791466	−	0.611213i	$-0.790682\pi$
$71$	−947.000	−1.58293	−0.791466	+	0.611213i	$0.790682\pi$
$72$	−405.000	−0.662913
$73$	448.000	0.718280	0.359140	−	0.933284i	$-0.383070\pi$
$73$	448.000	0.718280	0.359140	+	0.933284i	$0.383070\pi$
$74$	95.0000	0.149237
$75$	0	0
$76$	799.000	1.20594
$77$	33.0000	0.0488402
$78$	−480.000	−0.696786
$79$	−721.000	−1.02682	−0.513410	−	0.858143i	$-0.671618\pi$
$79$	−721.000	−1.02682	−0.513410	+	0.858143i	$0.671618\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−695.000	−0.935975
$83$	−142.000	−0.187789	−0.0938947	−	0.995582i	$-0.529932\pi$
$83$	−142.000	−0.187789	−0.0938947	+	0.995582i	$0.529932\pi$
$84$	153.000	0.198734
$85$	0	0
$86$	−1540.00	−1.93096
$87$	162.000	0.199635
$88$	495.000	0.599627
$89$	404.000	0.481168	0.240584	−	0.970628i	$-0.422661\pi$
$89$	404.000	0.481168	0.240584	+	0.970628i	$0.422661\pi$
$90$	0	0
$91$	96.0000	0.110588
$92$	−1921.00	−2.17694
$93$	−534.000	−0.595411
$94$	975.000	1.06983
$95$	0	0
$96$	255.000	0.271102
$97$	−79.0000	−0.0826931	−0.0413466	−	0.999145i	$-0.513165\pi$
$97$	−79.0000	−0.0826931	−0.0413466	+	0.999145i	$0.513165\pi$
$98$	1670.00	1.72138
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−545.000	−0.536926	−0.268463	−	0.963290i	$-0.586516\pi$
$101$	−545.000	−0.536926	−0.268463	+	0.963290i	$0.586516\pi$
$102$	−495.000	−0.480513
$103$	1306.00	1.24936	0.624680	−	0.780881i	$-0.285230\pi$
$103$	1306.00	1.24936	0.624680	+	0.780881i	$0.285230\pi$
$104$	1440.00	1.35773
$105$	0	0
$106$	760.000	0.696394
$107$	−1938.00	−1.75097	−0.875484	−	0.483247i	$-0.839457\pi$
$107$	−1938.00	−1.75097	−0.875484	+	0.483247i	$0.839457\pi$
$108$	−459.000	−0.408956
$109$	−576.000	−0.506154	−0.253077	−	0.967446i	$-0.581443\pi$
$109$	−576.000	−0.506154	−0.253077	+	0.967446i	$0.581443\pi$
$110$	0	0
$111$	57.0000	0.0487405
$112$	−267.000	−0.225260
$113$	1104.00	0.919076	0.459538	−	0.888158i	$-0.348015\pi$
$113$	1104.00	0.919076	0.459538	+	0.888158i	$0.348015\pi$
$114$	705.000	0.579204
$115$	0	0
$116$	−918.000	−0.734777
$117$	−288.000	−0.227569
$118$	3125.00	2.43796
$119$	99.0000	0.0762632
$120$	0	0
$121$	121.000	0.0909091
$122$	−1600.00	−1.18735
$123$	−417.000	−0.305688
$124$	3026.00	2.19147
$125$	0	0
$126$	135.000	0.0954504
$127$	1739.00	1.21505	0.607525	−	0.794301i	$-0.292163\pi$
$127$	1739.00	1.21505	0.607525	+	0.794301i	$0.292163\pi$
$128$	2115.00	1.46048
$129$	−924.000	−0.630649
$130$	0	0
$131$	1818.00	1.21251	0.606257	−	0.795269i	$-0.292670\pi$
$131$	1818.00	1.21251	0.606257	+	0.795269i	$0.292670\pi$
$132$	561.000	0.369915
$133$	−141.000	−0.0919267
$134$	1000.00	0.644678
$135$	0	0
$136$	1485.00	0.936307
$137$	−870.000	−0.542548	−0.271274	−	0.962502i	$-0.587445\pi$
$137$	−870.000	−0.542548	−0.271274	+	0.962502i	$0.587445\pi$
$138$	−1695.00	−1.04557
$139$	−636.000	−0.388092	−0.194046	−	0.980992i	$-0.562161\pi$
$139$	−636.000	−0.388092	−0.194046	+	0.980992i	$0.562161\pi$
$140$	0	0
$141$	585.000	0.349403
$142$	4735.00	2.79826
$143$	352.000	0.205844
$144$	801.000	0.463542
$145$	0	0
$146$	−2240.00	−1.26975
$147$	1002.00	0.562201
$148$	−323.000	−0.179395
$149$	−239.000	−0.131407	−0.0657035	−	0.997839i	$-0.520929\pi$
$149$	−239.000	−0.131407	−0.0657035	+	0.997839i	$0.520929\pi$
$150$	0	0
$151$	1208.00	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	1208.00	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2115.00	−1.12861
$153$	−297.000	−0.156935
$154$	−165.000	−0.0863382
$155$	0	0
$156$	1632.00	0.837593
$157$	1874.00	0.952621	0.476310	−	0.879277i	$-0.341974\pi$
$157$	1874.00	0.952621	0.476310	+	0.879277i	$0.341974\pi$
$158$	3605.00	1.81518
$159$	456.000	0.227441
$160$	0	0
$161$	339.000	0.165944
$162$	−405.000	−0.196419
$163$	1904.00	0.914925	0.457463	−	0.889229i	$-0.348758\pi$
$163$	1904.00	0.914925	0.457463	+	0.889229i	$0.348758\pi$
$164$	2363.00	1.12512
$165$	0	0
$166$	710.000	0.331968
$167$	1180.00	0.546773	0.273387	−	0.961904i	$-0.411856\pi$
$167$	1180.00	0.546773	0.273387	+	0.961904i	$0.411856\pi$
$168$	−405.000	−0.185991
$169$	−1173.00	−0.533910
$170$	0	0
$171$	423.000	0.189167
$172$	5236.00	2.32117
$173$	3177.00	1.39620	0.698101	−	0.716000i	$-0.254029\pi$
$173$	3177.00	1.39620	0.698101	+	0.716000i	$0.254029\pi$
$174$	−810.000	−0.352908
$175$	0	0
$176$	−979.000	−0.419289
$177$	1875.00	0.796235
$178$	−2020.00	−0.850592
$179$	1787.00	0.746182	0.373091	−	0.927795i	$-0.378298\pi$
$179$	1787.00	0.746182	0.373091	+	0.927795i	$0.378298\pi$
$180$	0	0
$181$	−835.000	−0.342901	−0.171450	−	0.985193i	$-0.554845\pi$
$181$	−835.000	−0.342901	−0.171450	+	0.985193i	$0.554845\pi$
$182$	−480.000	−0.195494
$183$	−960.000	−0.387788
$184$	5085.00	2.03734
$185$	0	0
$186$	2670.00	1.05255
$187$	363.000	0.141953
$188$	−3315.00	−1.28602
$189$	81.0000	0.0311740
$190$	0	0
$191$	3613.00	1.36873	0.684365	−	0.729139i	$-0.260079\pi$
$191$	3613.00	1.36873	0.684365	+	0.729139i	$0.260079\pi$
$192$	861.000	0.323632
$193$	−4204.00	−1.56793	−0.783965	−	0.620805i	$-0.786806\pi$
$193$	−4204.00	−1.56793	−0.783965	+	0.620805i	$0.786806\pi$
$194$	395.000	0.146182
$195$	0	0
$196$	−5678.00	−2.06924
$197$	4517.00	1.63362	0.816809	−	0.576908i	$-0.195741\pi$
$197$	4517.00	1.63362	0.816809	+	0.576908i	$0.195741\pi$
$198$	495.000	0.177667
$199$	4164.00	1.48331	0.741654	−	0.670783i	$-0.234042\pi$
$199$	4164.00	1.48331	0.741654	+	0.670783i	$0.234042\pi$
$200$	0	0
$201$	600.000	0.210551
$202$	2725.00	0.949160
$203$	162.000	0.0560107
$204$	1683.00	0.577616
$205$	0	0
$206$	−6530.00	−2.20858
$207$	−1017.00	−0.341480
$208$	−2848.00	−0.949391
$209$	−517.000	−0.171108
$210$	0	0
$211$	4660.00	1.52042	0.760208	−	0.649680i	$-0.225097\pi$
$211$	4660.00	1.52042	0.760208	+	0.649680i	$0.225097\pi$
$212$	−2584.00	−0.837122
$213$	2841.00	0.913907
$214$	9690.00	3.09530
$215$	0	0
$216$	1215.00	0.382733
$217$	−534.000	−0.167052
$218$	2880.00	0.894762
$219$	−1344.00	−0.414699
$220$	0	0
$221$	1056.00	0.321422
$222$	−285.000	−0.0861619
$223$	−3560.00	−1.06904	−0.534518	−	0.845157i	$-0.679507\pi$
$223$	−3560.00	−1.06904	−0.534518	+	0.845157i	$0.679507\pi$
$224$	255.000	0.0760621
$225$	0	0
$226$	−5520.00	−1.62471
$227$	4678.00	1.36780	0.683898	−	0.729577i	$-0.260283\pi$
$227$	4678.00	1.36780	0.683898	+	0.729577i	$0.260283\pi$
$228$	−2397.00	−0.696251
$229$	−4447.00	−1.28326	−0.641629	−	0.767015i	$-0.721741\pi$
$229$	−4447.00	−1.28326	−0.641629	+	0.767015i	$0.721741\pi$
$230$	0	0
$231$	−99.0000	−0.0281979
$232$	2430.00	0.687661
$233$	−411.000	−0.115560	−0.0577801	−	0.998329i	$-0.518402\pi$
$233$	−411.000	−0.115560	−0.0577801	+	0.998329i	$0.518402\pi$
$234$	1440.00	0.402290
$235$	0	0
$236$	−10625.0	−2.93063
$237$	2163.00	0.592835
$238$	−495.000	−0.134815
$239$	−6380.00	−1.72673	−0.863364	−	0.504582i	$-0.831647\pi$
$239$	−6380.00	−1.72673	−0.863364	+	0.504582i	$0.831647\pi$
$240$	0	0
$241$	7282.00	1.94637	0.973184	−	0.230027i	$-0.0738813\pi$
$241$	7282.00	1.94637	0.973184	+	0.230027i	$0.0738813\pi$
$242$	−605.000	−0.160706
$243$	−243.000	−0.0641500
$244$	5440.00	1.42730
$245$	0	0
$246$	2085.00	0.540385
$247$	−1504.00	−0.387438
$248$	−8010.00	−2.05095
$249$	426.000	0.108420
$250$	0	0
$251$	−4728.00	−1.18896	−0.594480	−	0.804111i	$-0.702642\pi$
$251$	−4728.00	−1.18896	−0.594480	+	0.804111i	$0.702642\pi$
$252$	−459.000	−0.114739
$253$	1243.00	0.308880
$254$	−8695.00	−2.14792
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−5418.00	−1.31504	−0.657521	−	0.753437i	$-0.728395\pi$
$257$	−5418.00	−1.31504	−0.657521	+	0.753437i	$0.728395\pi$
$258$	4620.00	1.11484
$259$	57.0000	0.0136749
$260$	0	0
$261$	−486.000	−0.115259
$262$	−9090.00	−2.14344
$263$	3354.00	0.786375	0.393187	−	0.919458i	$-0.371372\pi$
$263$	3354.00	0.786375	0.393187	+	0.919458i	$0.371372\pi$
$264$	−1485.00	−0.346195
$265$	0	0
$266$	705.000	0.162505
$267$	−1212.00	−0.277802
$268$	−3400.00	−0.774955
$269$	1062.00	0.240711	0.120356	−	0.992731i	$-0.461597\pi$
$269$	1062.00	0.240711	0.120356	+	0.992731i	$0.461597\pi$
$270$	0	0
$271$	−4821.00	−1.08065	−0.540323	−	0.841458i	$-0.681698\pi$
$271$	−4821.00	−1.08065	−0.540323	+	0.841458i	$0.681698\pi$
$272$	−2937.00	−0.654712
$273$	−288.000	−0.0638482
$274$	4350.00	0.959099
$275$	0	0
$276$	5763.00	1.25685
$277$	−4.00000	−0.000867642 0	−0.000433821	−	1.00000i	$-0.500138\pi$
$277$	−4.00000	−0.000867642 0	−0.000433821		1.00000i	$0.500138\pi$
$278$	3180.00	0.686057
$279$	1602.00	0.343761
$280$	0	0
$281$	4647.00	0.986537	0.493268	−	0.869877i	$-0.335802\pi$
$281$	4647.00	0.986537	0.493268	+	0.869877i	$0.335802\pi$
$282$	−2925.00	−0.617664
$283$	4283.00	0.899639	0.449820	−	0.893119i	$-0.351488\pi$
$283$	4283.00	0.899639	0.449820	+	0.893119i	$0.351488\pi$
$284$	−16099.0	−3.36373
$285$	0	0
$286$	−1760.00	−0.363885
$287$	−417.000	−0.0857656
$288$	−765.000	−0.156521
$289$	−3824.00	−0.778343
$290$	0	0
$291$	237.000	0.0477429
$292$	7616.00	1.52634
$293$	6811.00	1.35803	0.679015	−	0.734124i	$-0.262407\pi$
$293$	6811.00	1.35803	0.679015	+	0.734124i	$0.262407\pi$
$294$	−5010.00	−0.993841
$295$	0	0
$296$	855.000	0.167891
$297$	297.000	0.0580259
$298$	1195.00	0.232297
$299$	3616.00	0.699394
$300$	0	0
$301$	−924.000	−0.176938
$302$	−6040.00	−1.15087
$303$	1635.00	0.309994
$304$	4183.00	0.789183
$305$	0	0
$306$	1485.00	0.277424
$307$	460.000	0.0855166	0.0427583	−	0.999085i	$-0.486385\pi$
$307$	460.000	0.0855166	0.0427583	+	0.999085i	$0.486385\pi$
$308$	561.000	0.103786
$309$	−3918.00	−0.721318
$310$	0	0
$311$	8328.00	1.51845	0.759224	−	0.650829i	$-0.225579\pi$
$311$	8328.00	1.51845	0.759224	+	0.650829i	$0.225579\pi$
$312$	−4320.00	−0.783884
$313$	5929.00	1.07069	0.535346	−	0.844633i	$-0.320181\pi$
$313$	5929.00	1.07069	0.535346	+	0.844633i	$0.320181\pi$
$314$	−9370.00	−1.68401
$315$	0	0
$316$	−12257.0	−2.18199
$317$	−5040.00	−0.892980	−0.446490	−	0.894789i	$-0.647326\pi$
$317$	−5040.00	−0.892980	−0.446490	+	0.894789i	$0.647326\pi$
$318$	−2280.00	−0.402063
$319$	594.000	0.104256
$320$	0	0
$321$	5814.00	1.01092
$322$	−1695.00	−0.293350
$323$	−1551.00	−0.267183
$324$	1377.00	0.236111
$325$	0	0
$326$	−9520.00	−1.61737
$327$	1728.00	0.292228
$328$	−6255.00	−1.05297
$329$	585.000	0.0980307
$330$	0	0
$331$	10396.0	1.72633	0.863166	−	0.504920i	$-0.168478\pi$
$331$	10396.0	1.72633	0.863166	+	0.504920i	$0.168478\pi$
$332$	−2414.00	−0.399053
$333$	−171.000	−0.0281404
$334$	−5900.00	−0.966568
$335$	0	0
$336$	801.000	0.130054
$337$	7236.00	1.16964	0.584822	−	0.811162i	$-0.301164\pi$
$337$	7236.00	1.16964	0.584822	+	0.811162i	$0.301164\pi$
$338$	5865.00	0.943828
$339$	−3312.00	−0.530629
$340$	0	0
$341$	−1958.00	−0.310943
$342$	−2115.00	−0.334404
$343$	2031.00	0.319719
$344$	−13860.0	−2.17233
$345$	0	0
$346$	−15885.0	−2.46816
$347$	−1468.00	−0.227108	−0.113554	−	0.993532i	$-0.536223\pi$
$347$	−1468.00	−0.227108	−0.113554	+	0.993532i	$0.536223\pi$
$348$	2754.00	0.424224
$349$	5690.00	0.872718	0.436359	−	0.899773i	$-0.356268\pi$
$349$	5690.00	0.872718	0.436359	+	0.899773i	$0.356268\pi$
$350$	0	0
$351$	864.000	0.131387
$352$	935.000	0.141579
$353$	−5376.00	−0.810582	−0.405291	−	0.914188i	$-0.632830\pi$
$353$	−5376.00	−0.810582	−0.405291	+	0.914188i	$0.632830\pi$
$354$	−9375.00	−1.40756
$355$	0	0
$356$	6868.00	1.02248
$357$	−297.000	−0.0440306
$358$	−8935.00	−1.31908
$359$	3734.00	0.548950	0.274475	−	0.961594i	$-0.411496\pi$
$359$	3734.00	0.548950	0.274475	+	0.961594i	$0.411496\pi$
$360$	0	0
$361$	−4650.00	−0.677941
$362$	4175.00	0.606169
$363$	−363.000	−0.0524864
$364$	1632.00	0.235000
$365$	0	0
$366$	4800.00	0.685519
$367$	10274.0	1.46130	0.730652	−	0.682750i	$-0.239216\pi$
$367$	10274.0	1.46130	0.730652	+	0.682750i	$0.239216\pi$
$368$	−10057.0	−1.42461
$369$	1251.00	0.176489
$370$	0	0
$371$	456.000	0.0638122
$372$	−9078.00	−1.26525
$373$	−13662.0	−1.89649	−0.948246	−	0.317537i	$-0.897144\pi$
$373$	−13662.0	−1.89649	−0.948246	+	0.317537i	$0.897144\pi$
$374$	−1815.00	−0.250940
$375$	0	0
$376$	8775.00	1.20355
$377$	1728.00	0.236065
$378$	−405.000	−0.0551083
$379$	−7906.00	−1.07151	−0.535757	−	0.844372i	$-0.679974\pi$
$379$	−7906.00	−1.07151	−0.535757	+	0.844372i	$0.679974\pi$
$380$	0	0
$381$	−5217.00	−0.701509
$382$	−18065.0	−2.41960
$383$	3168.00	0.422656	0.211328	−	0.977415i	$-0.432221\pi$
$383$	3168.00	0.422656	0.211328	+	0.977415i	$0.432221\pi$
$384$	−6345.00	−0.843208
$385$	0	0
$386$	21020.0	2.77174
$387$	2772.00	0.364105
$388$	−1343.00	−0.175723
$389$	10770.0	1.40375	0.701877	−	0.712298i	$-0.252345\pi$
$389$	10770.0	1.40375	0.701877	+	0.712298i	$0.252345\pi$
$390$	0	0
$391$	3729.00	0.482311
$392$	15030.0	1.93656
$393$	−5454.00	−0.700046
$394$	−22585.0	−2.88786
$395$	0	0
$396$	−1683.00	−0.213571
$397$	−5670.00	−0.716799	−0.358399	−	0.933568i	$-0.616677\pi$
$397$	−5670.00	−0.716799	−0.358399	+	0.933568i	$0.616677\pi$
$398$	−20820.0	−2.62214
$399$	423.000	0.0530739
$400$	0	0
$401$	832.000	0.103611	0.0518056	−	0.998657i	$-0.483502\pi$
$401$	832.000	0.103611	0.0518056	+	0.998657i	$0.483502\pi$
$402$	−3000.00	−0.372205
$403$	−5696.00	−0.704064
$404$	−9265.00	−1.14097
$405$	0	0
$406$	−810.000	−0.0990139
$407$	209.000	0.0254539
$408$	−4455.00	−0.540577
$409$	−5712.00	−0.690563	−0.345281	−	0.938499i	$-0.612217\pi$
$409$	−5712.00	−0.690563	−0.345281	+	0.938499i	$0.612217\pi$
$410$	0	0
$411$	2610.00	0.313240
$412$	22202.0	2.65489
$413$	1875.00	0.223396
$414$	5085.00	0.603657
$415$	0	0
$416$	2720.00	0.320574
$417$	1908.00	0.224065
$418$	2585.00	0.302480
$419$	−4559.00	−0.531555	−0.265778	−	0.964034i	$-0.585629\pi$
$419$	−4559.00	−0.531555	−0.265778	+	0.964034i	$0.585629\pi$
$420$	0	0
$421$	6855.00	0.793568	0.396784	−	0.917912i	$-0.370126\pi$
$421$	6855.00	0.793568	0.396784	+	0.917912i	$0.370126\pi$
$422$	−23300.0	−2.68774
$423$	−1755.00	−0.201728
$424$	6840.00	0.783443
$425$	0	0
$426$	−14205.0	−1.61557
$427$	−960.000	−0.108800
$428$	−32946.0	−3.72081
$429$	−1056.00	−0.118844
$430$	0	0
$431$	10770.0	1.20365	0.601824	−	0.798628i	$-0.294441\pi$
$431$	10770.0	1.20365	0.601824	+	0.798628i	$0.294441\pi$
$432$	−2403.00	−0.267626
$433$	−8498.00	−0.943159	−0.471579	−	0.881824i	$-0.656316\pi$
$433$	−8498.00	−0.943159	−0.471579	+	0.881824i	$0.656316\pi$
$434$	2670.00	0.295309
$435$	0	0
$436$	−9792.00	−1.07558
$437$	−5311.00	−0.581372
$438$	6720.00	0.733091
$439$	9835.00	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$439$	9835.00	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$440$	0	0
$441$	−3006.00	−0.324587
$442$	−5280.00	−0.568199
$443$	10745.0	1.15239	0.576197	−	0.817311i	$-0.304536\pi$
$443$	10745.0	1.15239	0.576197	+	0.817311i	$0.304536\pi$
$444$	969.000	0.103574
$445$	0	0
$446$	17800.0	1.88981
$447$	717.000	0.0758679
$448$	861.000	0.0908001
$449$	8356.00	0.878272	0.439136	−	0.898421i	$-0.355285\pi$
$449$	8356.00	0.878272	0.439136	+	0.898421i	$0.355285\pi$
$450$	0	0
$451$	−1529.00	−0.159640
$452$	18768.0	1.95304
$453$	−3624.00	−0.375873
$454$	−23390.0	−2.41795
$455$	0	0
$456$	6345.00	0.651605
$457$	7058.00	0.722449	0.361225	−	0.932479i	$-0.382359\pi$
$457$	7058.00	0.722449	0.361225	+	0.932479i	$0.382359\pi$
$458$	22235.0	2.26850
$459$	891.000	0.0906064
$460$	0	0
$461$	646.000	0.0652651	0.0326326	−	0.999467i	$-0.489611\pi$
$461$	646.000	0.0652651	0.0326326	+	0.999467i	$0.489611\pi$
$462$	495.000	0.0498474
$463$	8982.00	0.901574	0.450787	−	0.892631i	$-0.351143\pi$
$463$	8982.00	0.901574	0.450787	+	0.892631i	$0.351143\pi$
$464$	−4806.00	−0.480847
$465$	0	0
$466$	2055.00	0.204283
$467$	13476.0	1.33532	0.667661	−	0.744466i	$-0.267296\pi$
$467$	13476.0	1.33532	0.667661	+	0.744466i	$0.267296\pi$
$468$	−4896.00	−0.483585
$469$	600.000	0.0590734
$470$	0	0
$471$	−5622.00	−0.549996
$472$	28125.0	2.74271
$473$	−3388.00	−0.329345
$474$	−10815.0	−1.04799
$475$	0	0
$476$	1683.00	0.162059
$477$	−1368.00	−0.131313
$478$	31900.0	3.05245
$479$	12996.0	1.23967	0.619835	−	0.784732i	$-0.287199\pi$
$479$	12996.0	1.23967	0.619835	+	0.784732i	$0.287199\pi$
$480$	0	0
$481$	608.000	0.0576350
$482$	−36410.0	−3.44073
$483$	−1017.00	−0.0958077
$484$	2057.00	0.193182
$485$	0	0
$486$	1215.00	0.113402
$487$	6026.00	0.560707	0.280353	−	0.959897i	$-0.409548\pi$
$487$	6026.00	0.560707	0.280353	+	0.959897i	$0.409548\pi$
$488$	−14400.0	−1.33577
$489$	−5712.00	−0.528232
$490$	0	0
$491$	11698.0	1.07520	0.537600	−	0.843200i	$-0.319331\pi$
$491$	11698.0	1.07520	0.537600	+	0.843200i	$0.319331\pi$
$492$	−7089.00	−0.649587
$493$	1782.00	0.162794
$494$	7520.00	0.684900
$495$	0	0
$496$	15842.0	1.43413
$497$	2841.00	0.256411
$498$	−2130.00	−0.191662
$499$	−17052.0	−1.52976	−0.764882	−	0.644170i	$-0.777203\pi$
$499$	−17052.0	−1.52976	−0.764882	+	0.644170i	$0.777203\pi$
$500$	0	0
$501$	−3540.00	−0.315680
$502$	23640.0	2.10180
$503$	932.000	0.0826160	0.0413080	−	0.999146i	$-0.486848\pi$
$503$	932.000	0.0826160	0.0413080	+	0.999146i	$0.486848\pi$
$504$	1215.00	0.107382
$505$	0	0
$506$	−6215.00	−0.546029
$507$	3519.00	0.308253
$508$	29563.0	2.58198
$509$	4384.00	0.381763	0.190882	−	0.981613i	$-0.438865\pi$
$509$	4384.00	0.381763	0.190882	+	0.981613i	$0.438865\pi$
$510$	0	0
$511$	−1344.00	−0.116350
$512$	24475.0	2.11260
$513$	−1269.00	−0.109216
$514$	27090.0	2.32469
$515$	0	0
$516$	−15708.0	−1.34013
$517$	2145.00	0.182470
$518$	−285.000	−0.0241741
$519$	−9531.00	−0.806097
$520$	0	0
$521$	−2322.00	−0.195257	−0.0976283	−	0.995223i	$-0.531126\pi$
$521$	−2322.00	−0.195257	−0.0976283	+	0.995223i	$0.531126\pi$
$522$	2430.00	0.203751
$523$	9749.00	0.815094	0.407547	−	0.913184i	$-0.366384\pi$
$523$	9749.00	0.815094	0.407547	+	0.913184i	$0.366384\pi$
$524$	30906.0	2.57659
$525$	0	0
$526$	−16770.0	−1.39013
$527$	−5874.00	−0.485532
$528$	2937.00	0.242077
$529$	602.000	0.0494781
$530$	0	0
$531$	−5625.00	−0.459707
$532$	−2397.00	−0.195344
$533$	−4448.00	−0.361471
$534$	6060.00	0.491090
$535$	0	0
$536$	9000.00	0.725263
$537$	−5361.00	−0.430809
$538$	−5310.00	−0.425521
$539$	3674.00	0.293600
$540$	0	0
$541$	4208.00	0.334410	0.167205	−	0.985922i	$-0.446526\pi$
$541$	4208.00	0.334410	0.167205	+	0.985922i	$0.446526\pi$
$542$	24105.0	1.91033
$543$	2505.00	0.197974
$544$	2805.00	0.221072
$545$	0	0
$546$	1440.00	0.112869
$547$	−10179.0	−0.795654	−0.397827	−	0.917461i	$-0.630236\pi$
$547$	−10179.0	−0.795654	−0.397827	+	0.917461i	$0.630236\pi$
$548$	−14790.0	−1.15292
$549$	2880.00	0.223890
$550$	0	0
$551$	−2538.00	−0.196229
$552$	−15255.0	−1.17626
$553$	2163.00	0.166329
$554$	20.0000	0.00153379
$555$	0	0
$556$	−10812.0	−0.824696
$557$	2314.00	0.176028	0.0880138	−	0.996119i	$-0.471948\pi$
$557$	2314.00	0.176028	0.0880138	+	0.996119i	$0.471948\pi$
$558$	−8010.00	−0.607689
$559$	−9856.00	−0.745732
$560$	0	0
$561$	−1089.00	−0.0819565
$562$	−23235.0	−1.74397
$563$	−24330.0	−1.82129	−0.910646	−	0.413188i	$-0.864415\pi$
$563$	−24330.0	−1.82129	−0.910646	+	0.413188i	$0.864415\pi$
$564$	9945.00	0.742482
$565$	0	0
$566$	−21415.0	−1.59035
$567$	−243.000	−0.0179983
$568$	42615.0	3.14804
$569$	3445.00	0.253817	0.126909	−	0.991914i	$-0.459495\pi$
$569$	3445.00	0.253817	0.126909	+	0.991914i	$0.459495\pi$
$570$	0	0
$571$	−13056.0	−0.956877	−0.478438	−	0.878121i	$-0.658797\pi$
$571$	−13056.0	−0.956877	−0.478438	+	0.878121i	$0.658797\pi$
$572$	5984.00	0.437419
$573$	−10839.0	−0.790237
$574$	2085.00	0.151614
$575$	0	0
$576$	−2583.00	−0.186849
$577$	17347.0	1.25159	0.625793	−	0.779989i	$-0.284775\pi$
$577$	17347.0	1.25159	0.625793	+	0.779989i	$0.284775\pi$
$578$	19120.0	1.37593
$579$	12612.0	0.905245
$580$	0	0
$581$	426.000	0.0304190
$582$	−1185.00	−0.0843983
$583$	1672.00	0.118777
$584$	−20160.0	−1.42847
$585$	0	0
$586$	−34055.0	−2.40068
$587$	−8379.00	−0.589162	−0.294581	−	0.955626i	$-0.595180\pi$
$587$	−8379.00	−0.589162	−0.294581	+	0.955626i	$0.595180\pi$
$588$	17034.0	1.19468
$589$	8366.00	0.585255
$590$	0	0
$591$	−13551.0	−0.943170
$592$	−1691.00	−0.117398
$593$	−1958.00	−0.135591	−0.0677955	−	0.997699i	$-0.521597\pi$
$593$	−1958.00	−0.135591	−0.0677955	+	0.997699i	$0.521597\pi$
$594$	−1485.00	−0.102576
$595$	0	0
$596$	−4063.00	−0.279240
$597$	−12492.0	−0.856388
$598$	−18080.0	−1.23636
$599$	23583.0	1.60864	0.804320	−	0.594196i	$-0.202530\pi$
$599$	23583.0	1.60864	0.804320	+	0.594196i	$0.202530\pi$
$600$	0	0
$601$	−15328.0	−1.04034	−0.520168	−	0.854064i	$-0.674131\pi$
$601$	−15328.0	−1.04034	−0.520168	+	0.854064i	$0.674131\pi$
$602$	4620.00	0.312786
$603$	−1800.00	−0.121562
$604$	20536.0	1.38344
$605$	0	0
$606$	−8175.00	−0.547998
$607$	−160.000	−0.0106988	−0.00534942	−	0.999986i	$-0.501703\pi$
$607$	−160.000	−0.0106988	−0.00534942	+	0.999986i	$0.501703\pi$
$608$	−3995.00	−0.266478
$609$	−486.000	−0.0323378
$610$	0	0
$611$	6240.00	0.413164
$612$	−5049.00	−0.333486
$613$	5948.00	0.391904	0.195952	−	0.980613i	$-0.437220\pi$
$613$	5948.00	0.391904	0.195952	+	0.980613i	$0.437220\pi$
$614$	−2300.00	−0.151173
$615$	0	0
$616$	−1485.00	−0.0971304
$617$	−334.000	−0.0217931	−0.0108965	−	0.999941i	$-0.503469\pi$
$617$	−334.000	−0.0217931	−0.0108965	+	0.999941i	$0.503469\pi$
$618$	19590.0	1.27512
$619$	−7202.00	−0.467646	−0.233823	−	0.972279i	$-0.575124\pi$
$619$	−7202.00	−0.467646	−0.233823	+	0.972279i	$0.575124\pi$
$620$	0	0
$621$	3051.00	0.197154
$622$	−41640.0	−2.68426
$623$	−1212.00	−0.0779418
$624$	8544.00	0.548131
$625$	0	0
$626$	−29645.0	−1.89274
$627$	1551.00	0.0987894
$628$	31858.0	2.02432
$629$	627.000	0.0397458
$630$	0	0
$631$	10306.0	0.650199	0.325099	−	0.945680i	$-0.394602\pi$
$631$	10306.0	0.650199	0.325099	+	0.945680i	$0.394602\pi$
$632$	32445.0	2.04208
$633$	−13980.0	−0.877812
$634$	25200.0	1.57858
$635$	0	0
$636$	7752.00	0.483313
$637$	10688.0	0.664794
$638$	−2970.00	−0.184300
$639$	−8523.00	−0.527644
$640$	0	0
$641$	−1228.00	−0.0756678	−0.0378339	−	0.999284i	$-0.512046\pi$
$641$	−1228.00	−0.0756678	−0.0378339	+	0.999284i	$0.512046\pi$
$642$	−29070.0	−1.78707
$643$	18454.0	1.13181	0.565906	−	0.824470i	$-0.308527\pi$
$643$	18454.0	1.13181	0.565906	+	0.824470i	$0.308527\pi$
$644$	5763.00	0.352630
$645$	0	0
$646$	7755.00	0.472316
$647$	−17647.0	−1.07230	−0.536148	−	0.844124i	$-0.680121\pi$
$647$	−17647.0	−1.07230	−0.536148	+	0.844124i	$0.680121\pi$
$648$	−3645.00	−0.220971
$649$	6875.00	0.415820
$650$	0	0
$651$	1602.00	0.0964475
$652$	32368.0	1.94422
$653$	−25918.0	−1.55322	−0.776608	−	0.629984i	$-0.783061\pi$
$653$	−25918.0	−1.55322	−0.776608	+	0.629984i	$0.783061\pi$
$654$	−8640.00	−0.516591
$655$	0	0
$656$	12371.0	0.736290
$657$	4032.00	0.239427
$658$	−2925.00	−0.173295
$659$	12864.0	0.760410	0.380205	−	0.924902i	$-0.375853\pi$
$659$	12864.0	0.760410	0.380205	+	0.924902i	$0.375853\pi$
$660$	0	0
$661$	−11419.0	−0.671933	−0.335966	−	0.941874i	$-0.609063\pi$
$661$	−11419.0	−0.671933	−0.335966	+	0.941874i	$0.609063\pi$
$662$	−51980.0	−3.05175
$663$	−3168.00	−0.185573
$664$	6390.00	0.373464
$665$	0	0
$666$	855.000	0.0497456
$667$	6102.00	0.354228
$668$	20060.0	1.16189
$669$	10680.0	0.617209
$670$	0	0
$671$	−3520.00	−0.202516
$672$	−765.000	−0.0439145
$673$	−15784.0	−0.904054	−0.452027	−	0.892004i	$-0.649299\pi$
$673$	−15784.0	−0.904054	−0.452027	+	0.892004i	$0.649299\pi$
$674$	−36180.0	−2.06766
$675$	0	0
$676$	−19941.0	−1.13456
$677$	26050.0	1.47885	0.739426	−	0.673238i	$-0.235097\pi$
$677$	26050.0	1.47885	0.739426	+	0.673238i	$0.235097\pi$
$678$	16560.0	0.938028
$679$	237.000	0.0133950
$680$	0	0
$681$	−14034.0	−0.789698
$682$	9790.00	0.549675
$683$	15095.0	0.845672	0.422836	−	0.906206i	$-0.361035\pi$
$683$	15095.0	0.845672	0.422836	+	0.906206i	$0.361035\pi$
$684$	7191.00	0.401981
$685$	0	0
$686$	−10155.0	−0.565189
$687$	13341.0	0.740889
$688$	27412.0	1.51900
$689$	4864.00	0.268946
$690$	0	0
$691$	15896.0	0.875126	0.437563	−	0.899188i	$-0.355842\pi$
$691$	15896.0	0.875126	0.437563	+	0.899188i	$0.355842\pi$
$692$	54009.0	2.96693
$693$	297.000	0.0162801
$694$	7340.00	0.401473
$695$	0	0
$696$	−7290.00	−0.397021
$697$	−4587.00	−0.249275
$698$	−28450.0	−1.54276
$699$	1233.00	0.0667187
$700$	0	0
$701$	10529.0	0.567296	0.283648	−	0.958928i	$-0.408455\pi$
$701$	10529.0	0.567296	0.283648	+	0.958928i	$0.408455\pi$
$702$	−4320.00	−0.232262
$703$	−893.000	−0.0479092
$704$	3157.00	0.169011
$705$	0	0
$706$	26880.0	1.43292
$707$	1635.00	0.0869738
$708$	31875.0	1.69200
$709$	−16087.0	−0.852130	−0.426065	−	0.904693i	$-0.640100\pi$
$709$	−16087.0	−0.852130	−0.426065	+	0.904693i	$0.640100\pi$
$710$	0	0
$711$	−6489.00	−0.342274
$712$	−18180.0	−0.956916
$713$	−20114.0	−1.05649
$714$	1485.00	0.0778358
$715$	0	0
$716$	30379.0	1.58564
$717$	19140.0	0.996927
$718$	−18670.0	−0.970415
$719$	24336.0	1.26228	0.631140	−	0.775669i	$-0.282587\pi$
$719$	24336.0	1.26228	0.631140	+	0.775669i	$0.282587\pi$
$720$	0	0
$721$	−3918.00	−0.202377
$722$	23250.0	1.19844
$723$	−21846.0	−1.12374
$724$	−14195.0	−0.728664
$725$	0	0
$726$	1815.00	0.0927837
$727$	−13960.0	−0.712170	−0.356085	−	0.934454i	$-0.615889\pi$
$727$	−13960.0	−0.712170	−0.356085	+	0.934454i	$0.615889\pi$
$728$	−4320.00	−0.219931
$729$	729.000	0.0370370
$730$	0	0
$731$	−10164.0	−0.514267
$732$	−16320.0	−0.824050
$733$	−9252.00	−0.466208	−0.233104	−	0.972452i	$-0.574888\pi$
$733$	−9252.00	−0.466208	−0.233104	+	0.972452i	$0.574888\pi$
$734$	−51370.0	−2.58324
$735$	0	0
$736$	9605.00	0.481039
$737$	2200.00	0.109957
$738$	−6255.00	−0.311992
$739$	28453.0	1.41632	0.708160	−	0.706052i	$-0.249526\pi$
$739$	28453.0	1.41632	0.708160	+	0.706052i	$0.249526\pi$
$740$	0	0
$741$	4512.00	0.223688
$742$	−2280.00	−0.112805
$743$	−512.000	−0.0252806	−0.0126403	−	0.999920i	$-0.504024\pi$
$743$	−512.000	−0.0252806	−0.0126403	+	0.999920i	$0.504024\pi$
$744$	24030.0	1.18412
$745$	0	0
$746$	68310.0	3.35256
$747$	−1278.00	−0.0625965
$748$	6171.00	0.301650
$749$	5814.00	0.283630
$750$	0	0
$751$	772.000	0.0375109	0.0187554	−	0.999824i	$-0.494030\pi$
$751$	772.000	0.0375109	0.0187554	+	0.999824i	$0.494030\pi$
$752$	−17355.0	−0.841585
$753$	14184.0	0.686446
$754$	−8640.00	−0.417308
$755$	0	0
$756$	1377.00	0.0662447
$757$	−8058.00	−0.386886	−0.193443	−	0.981111i	$-0.561966\pi$
$757$	−8058.00	−0.386886	−0.193443	+	0.981111i	$0.561966\pi$
$758$	39530.0	1.89419
$759$	−3729.00	−0.178332
$760$	0	0
$761$	18650.0	0.888386	0.444193	−	0.895931i	$-0.353490\pi$
$761$	18650.0	0.888386	0.444193	+	0.895931i	$0.353490\pi$
$762$	26085.0	1.24010
$763$	1728.00	0.0819893
$764$	61421.0	2.90855
$765$	0	0
$766$	−15840.0	−0.747157
$767$	20000.0	0.941536
$768$	24837.0	1.16696
$769$	−7144.00	−0.335005	−0.167503	−	0.985872i	$-0.553570\pi$
$769$	−7144.00	−0.335005	−0.167503	+	0.985872i	$0.553570\pi$
$770$	0	0
$771$	16254.0	0.759239
$772$	−71468.0	−3.33185
$773$	1904.00	0.0885927	0.0442963	−	0.999018i	$-0.485895\pi$
$773$	1904.00	0.0885927	0.0442963	+	0.999018i	$0.485895\pi$
$774$	−13860.0	−0.643653
$775$	0	0
$776$	3555.00	0.164455
$777$	−171.000	−0.00789523
$778$	−53850.0	−2.48151
$779$	6533.00	0.300474
$780$	0	0
$781$	10417.0	0.477272
$782$	−18645.0	−0.852614
$783$	1458.00	0.0665449
$784$	−29726.0	−1.35414
$785$	0	0
$786$	27270.0	1.23752
$787$	−7555.00	−0.342194	−0.171097	−	0.985254i	$-0.554731\pi$
$787$	−7555.00	−0.342194	−0.171097	+	0.985254i	$0.554731\pi$
$788$	76789.0	3.47144
$789$	−10062.0	−0.454014
$790$	0	0
$791$	−3312.00	−0.148876
$792$	4455.00	0.199876
$793$	−10240.0	−0.458554
$794$	28350.0	1.26713
$795$	0	0
$796$	70788.0	3.15203
$797$	−24950.0	−1.10888	−0.554438	−	0.832225i	$-0.687067\pi$
$797$	−24950.0	−1.10888	−0.554438	+	0.832225i	$0.687067\pi$
$798$	−2115.00	−0.0938223
$799$	6435.00	0.284924
$800$	0	0
$801$	3636.00	0.160389
$802$	−4160.00	−0.183160
$803$	−4928.00	−0.216570
$804$	10200.0	0.447421
$805$	0	0
$806$	28480.0	1.24462
$807$	−3186.00	−0.138975
$808$	24525.0	1.06781
$809$	19893.0	0.864525	0.432262	−	0.901748i	$-0.357715\pi$
$809$	19893.0	0.864525	0.432262	+	0.901748i	$0.357715\pi$
$810$	0	0
$811$	34503.0	1.49391	0.746957	−	0.664872i	$-0.231514\pi$
$811$	34503.0	1.49391	0.746957	+	0.664872i	$0.231514\pi$
$812$	2754.00	0.119023
$813$	14463.0	0.623911
$814$	−1045.00	−0.0449966
$815$	0	0
$816$	8811.00	0.377998
$817$	14476.0	0.619891
$818$	28560.0	1.22075
$819$	864.000	0.0368628
$820$	0	0
$821$	16890.0	0.717984	0.358992	−	0.933341i	$-0.383120\pi$
$821$	16890.0	0.717984	0.358992	+	0.933341i	$0.383120\pi$
$822$	−13050.0	−0.553736
$823$	−34692.0	−1.46936	−0.734682	−	0.678411i	$-0.762669\pi$
$823$	−34692.0	−1.46936	−0.734682	+	0.678411i	$0.762669\pi$
$824$	−58770.0	−2.48465
$825$	0	0
$826$	−9375.00	−0.394913
$827$	−41424.0	−1.74178	−0.870891	−	0.491476i	$-0.836457\pi$
$827$	−41424.0	−1.74178	−0.870891	+	0.491476i	$0.836457\pi$
$828$	−17289.0	−0.725645
$829$	−18494.0	−0.774817	−0.387408	−	0.921908i	$-0.626630\pi$
$829$	−18494.0	−0.774817	−0.387408	+	0.921908i	$0.626630\pi$
$830$	0	0
$831$	12.0000	0.000500933 0
$832$	9184.00	0.382690
$833$	11022.0	0.458451
$834$	−9540.00	−0.396095
$835$	0	0
$836$	−8789.00	−0.363605
$837$	−4806.00	−0.198470
$838$	22795.0	0.939666
$839$	6680.00	0.274874	0.137437	−	0.990511i	$-0.456114\pi$
$839$	6680.00	0.274874	0.137437	+	0.990511i	$0.456114\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	−34275.0	−1.40284
$843$	−13941.0	−0.569577
$844$	79220.0	3.23088
$845$	0	0
$846$	8775.00	0.356608
$847$	−363.000	−0.0147259
$848$	−13528.0	−0.547822
$849$	−12849.0	−0.519407
$850$	0	0
$851$	2147.00	0.0864844
$852$	48297.0	1.94205
$853$	−43358.0	−1.74039	−0.870193	−	0.492711i	$-0.836006\pi$
$853$	−43358.0	−1.74039	−0.870193	+	0.492711i	$0.836006\pi$
$854$	4800.00	0.192333
$855$	0	0
$856$	87210.0	3.48222
$857$	−15585.0	−0.621206	−0.310603	−	0.950540i	$-0.600531\pi$
$857$	−15585.0	−0.621206	−0.310603	+	0.950540i	$0.600531\pi$
$858$	5280.00	0.210089
$859$	−17036.0	−0.676672	−0.338336	−	0.941025i	$-0.609864\pi$
$859$	−17036.0	−0.676672	−0.338336	+	0.941025i	$0.609864\pi$
$860$	0	0
$861$	1251.00	0.0495168
$862$	−53850.0	−2.12777
$863$	−28064.0	−1.10696	−0.553482	−	0.832861i	$-0.686701\pi$
$863$	−28064.0	−1.10696	−0.553482	+	0.832861i	$0.686701\pi$
$864$	2295.00	0.0903675
$865$	0	0
$866$	42490.0	1.66729
$867$	11472.0	0.449377
$868$	−9078.00	−0.354985
$869$	7931.00	0.309598
$870$	0	0
$871$	6400.00	0.248973
$872$	25920.0	1.00661
$873$	−711.000	−0.0275644
$874$	26555.0	1.02773
$875$	0	0
$876$	−22848.0	−0.881236
$877$	22654.0	0.872259	0.436130	−	0.899884i	$-0.356349\pi$
$877$	22654.0	0.872259	0.436130	+	0.899884i	$0.356349\pi$
$878$	−49175.0	−1.89018
$879$	−20433.0	−0.784059
$880$	0	0
$881$	−22380.0	−0.855847	−0.427924	−	0.903815i	$-0.640755\pi$
$881$	−22380.0	−0.855847	−0.427924	+	0.903815i	$0.640755\pi$
$882$	15030.0	0.573794
$883$	−35174.0	−1.34054	−0.670271	−	0.742116i	$-0.733822\pi$
$883$	−35174.0	−1.34054	−0.670271	+	0.742116i	$0.733822\pi$
$884$	17952.0	0.683022
$885$	0	0
$886$	−53725.0	−2.03716
$887$	−30868.0	−1.16848	−0.584242	−	0.811579i	$-0.698608\pi$
$887$	−30868.0	−1.16848	−0.584242	+	0.811579i	$0.698608\pi$
$888$	−2565.00	−0.0969322
$889$	−5217.00	−0.196820
$890$	0	0
$891$	−891.000	−0.0335013
$892$	−60520.0	−2.27170
$893$	−9165.00	−0.343443
$894$	−3585.00	−0.134117
$895$	0	0
$896$	−6345.00	−0.236575
$897$	−10848.0	−0.403795
$898$	−41780.0	−1.55258
$899$	−9612.00	−0.356594
$900$	0	0
$901$	5016.00	0.185469
$902$	7645.00	0.282207
$903$	2772.00	0.102155
$904$	−49680.0	−1.82780
$905$	0	0
$906$	18120.0	0.664455
$907$	−10070.0	−0.368654	−0.184327	−	0.982865i	$-0.559010\pi$
$907$	−10070.0	−0.368654	−0.184327	+	0.982865i	$0.559010\pi$
$908$	79526.0	2.90657
$909$	−4905.00	−0.178975
$910$	0	0
$911$	1885.00	0.0685542	0.0342771	−	0.999412i	$-0.489087\pi$
$911$	1885.00	0.0685542	0.0342771	+	0.999412i	$0.489087\pi$
$912$	−12549.0	−0.455635
$913$	1562.00	0.0566207
$914$	−35290.0	−1.27712
$915$	0	0
$916$	−75599.0	−2.72692
$917$	−5454.00	−0.196409
$918$	−4455.00	−0.160171
$919$	23703.0	0.850805	0.425403	−	0.905004i	$-0.360133\pi$
$919$	23703.0	0.850805	0.425403	+	0.905004i	$0.360133\pi$
$920$	0	0
$921$	−1380.00	−0.0493730
$922$	−3230.00	−0.115374
$923$	30304.0	1.08068
$924$	−1683.00	−0.0599206
$925$	0	0
$926$	−44910.0	−1.59377
$927$	11754.0	0.416453
$928$	4590.00	0.162364
$929$	53804.0	1.90016	0.950082	−	0.312001i	$-0.100999\pi$
$929$	53804.0	1.90016	0.950082	+	0.312001i	$0.100999\pi$
$930$	0	0
$931$	−15698.0	−0.552611
$932$	−6987.00	−0.245565
$933$	−24984.0	−0.876677
$934$	−67380.0	−2.36054
$935$	0	0
$936$	12960.0	0.452576
$937$	−1326.00	−0.0462311	−0.0231155	−	0.999733i	$-0.507359\pi$
$937$	−1326.00	−0.0462311	−0.0231155	+	0.999733i	$0.507359\pi$
$938$	−3000.00	−0.104428
$939$	−17787.0	−0.618165
$940$	0	0
$941$	−27109.0	−0.939137	−0.469569	−	0.882896i	$-0.655591\pi$
$941$	−27109.0	−0.939137	−0.469569	+	0.882896i	$0.655591\pi$
$942$	28110.0	0.972265
$943$	−15707.0	−0.542408
$944$	−55625.0	−1.91784
$945$	0	0
$946$	16940.0	0.582206
$947$	−31143.0	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$947$	−31143.0	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$948$	36771.0	1.25977
$949$	−14336.0	−0.490375
$950$	0	0
$951$	15120.0	0.515562
$952$	−4455.00	−0.151667
$953$	879.000	0.0298779	0.0149389	−	0.999888i	$-0.495245\pi$
$953$	879.000	0.0298779	0.0149389	+	0.999888i	$0.495245\pi$
$954$	6840.00	0.232131
$955$	0	0
$956$	−108460.	−3.66930
$957$	−1782.00	−0.0601921
$958$	−64980.0	−2.19145
$959$	2610.00	0.0878846
$960$	0	0
$961$	1893.00	0.0635427
$962$	−3040.00	−0.101885
$963$	−17442.0	−0.583656
$964$	123794.	4.13603
$965$	0	0
$966$	5085.00	0.169366
$967$	14824.0	0.492976	0.246488	−	0.969146i	$-0.420723\pi$
$967$	14824.0	0.492976	0.246488	+	0.969146i	$0.420723\pi$
$968$	−5445.00	−0.180794
$969$	4653.00	0.154258
$970$	0	0
$971$	34089.0	1.12664	0.563320	−	0.826239i	$-0.309524\pi$
$971$	34089.0	1.12664	0.563320	+	0.826239i	$0.309524\pi$
$972$	−4131.00	−0.136319
$973$	1908.00	0.0628650
$974$	−30130.0	−0.991199
$975$	0	0
$976$	28480.0	0.934040
$977$	−33446.0	−1.09522	−0.547611	−	0.836733i	$-0.684463\pi$
$977$	−33446.0	−1.09522	−0.547611	+	0.836733i	$0.684463\pi$
$978$	28560.0	0.933792
$979$	−4444.00	−0.145077
$980$	0	0
$981$	−5184.00	−0.168718
$982$	−58490.0	−1.90070
$983$	52025.0	1.68804	0.844018	−	0.536315i	$-0.180184\pi$
$983$	52025.0	1.68804	0.844018	+	0.536315i	$0.180184\pi$
$984$	18765.0	0.607933
$985$	0	0
$986$	−8910.00	−0.287781
$987$	−1755.00	−0.0565980
$988$	−25568.0	−0.823306
$989$	−34804.0	−1.11901
$990$	0	0
$991$	−41260.0	−1.32257	−0.661285	−	0.750135i	$-0.729989\pi$
$991$	−41260.0	−1.32257	−0.661285	+	0.750135i	$0.729989\pi$
$992$	−15130.0	−0.484252
$993$	−31188.0	−0.996698
$994$	−14205.0	−0.453275
$995$	0	0
$996$	7242.00	0.230393
$997$	190.000	0.00603547	0.00301773	−	0.999995i	$-0.499039\pi$
$997$	190.000	0.00603547	0.00301773	+	0.999995i	$0.499039\pi$
$998$	85260.0	2.70427
$999$	513.000	0.0162468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.a.1.1	✓	1
3.2	odd	2		2475.4.a.k.1.1		1
5.2	odd	4		825.4.c.b.199.1		2
5.3	odd	4		825.4.c.b.199.2		2
5.4	even	2		825.4.a.j.1.1	yes	1
15.14	odd	2		2475.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.a.1.1	✓	1	1.1	even	1	trivial
825.4.a.j.1.1	yes	1	5.4	even	2
825.4.c.b.199.1		2	5.2	odd	4
825.4.c.b.199.2		2	5.3	odd	4
2475.4.a.a.1.1		1	15.14	odd	2
2475.4.a.k.1.1		1	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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