LMFDB - Embedded newform 2475.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -7.00000 q^{4} +26.0000 q^{7} +15.0000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -7.00000 q^{4} +26.0000 q^{7} +15.0000 q^{8} -11.0000 q^{11} +32.0000 q^{13} -26.0000 q^{14} +41.0000 q^{16} +74.0000 q^{17} -60.0000 q^{19} +11.0000 q^{22} -182.000 q^{23} -32.0000 q^{26} -182.000 q^{28} +90.0000 q^{29} -8.00000 q^{31} -161.000 q^{32} -74.0000 q^{34} +66.0000 q^{37} +60.0000 q^{38} -422.000 q^{41} -408.000 q^{43} +77.0000 q^{44} +182.000 q^{46} -506.000 q^{47} +333.000 q^{49} -224.000 q^{52} +348.000 q^{53} +390.000 q^{56} -90.0000 q^{58} +200.000 q^{59} +132.000 q^{61} +8.00000 q^{62} -167.000 q^{64} +1036.00 q^{67} -518.000 q^{68} -762.000 q^{71} +542.000 q^{73} -66.0000 q^{74} +420.000 q^{76} -286.000 q^{77} -550.000 q^{79} +422.000 q^{82} -132.000 q^{83} +408.000 q^{86} -165.000 q^{88} -570.000 q^{89} +832.000 q^{91} +1274.00 q^{92} +506.000 q^{94} -14.0000 q^{97} -333.000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.0000	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$7$	26.0000	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$8$	15.0000	0.662913
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	32.0000	0.682708	0.341354	−	0.939935i	$-0.389115\pi$
$13$	32.0000	0.682708	0.341354	+	0.939935i	$0.389115\pi$
$14$	−26.0000	−0.496342
$15$	0	0
$16$	41.0000	0.640625
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	0	0
$19$	−60.0000	−0.724471	−0.362235	−	0.932087i	$-0.617986\pi$
$19$	−60.0000	−0.724471	−0.362235	+	0.932087i	$0.617986\pi$
$20$	0	0
$21$	0	0
$22$	11.0000	0.106600
$23$	−182.000	−1.64998	−0.824992	−	0.565145i	$-0.808820\pi$
$23$	−182.000	−1.64998	−0.824992	+	0.565145i	$0.808820\pi$
$24$	0	0
$25$	0	0
$26$	−32.0000	−0.241374
$27$	0	0
$28$	−182.000	−1.22838
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	−161.000	−0.889408
$33$	0	0
$34$	−74.0000	−0.373262
$35$	0	0
$36$	0	0
$37$	66.0000	0.293252	0.146626	−	0.989192i	$-0.453159\pi$
$37$	66.0000	0.293252	0.146626	+	0.989192i	$0.453159\pi$
$38$	60.0000	0.256139
$39$	0	0
$40$	0	0
$41$	−422.000	−1.60745	−0.803724	−	0.595003i	$-0.797151\pi$
$41$	−422.000	−1.60745	−0.803724	+	0.595003i	$0.797151\pi$
$42$	0	0
$43$	−408.000	−1.44696	−0.723482	−	0.690344i	$-0.757459\pi$
$43$	−408.000	−1.44696	−0.723482	+	0.690344i	$0.757459\pi$
$44$	77.0000	0.263822
$45$	0	0
$46$	182.000	0.583357
$47$	−506.000	−1.57038	−0.785188	−	0.619257i	$-0.787434\pi$
$47$	−506.000	−1.57038	−0.785188	+	0.619257i	$0.787434\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	0	0
$51$	0	0
$52$	−224.000	−0.597369
$53$	348.000	0.901915	0.450957	−	0.892546i	$-0.351083\pi$
$53$	348.000	0.901915	0.450957	+	0.892546i	$0.351083\pi$
$54$	0	0
$55$	0	0
$56$	390.000	0.930642
$57$	0	0
$58$	−90.0000	−0.203751
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	132.000	0.277063	0.138532	−	0.990358i	$-0.455762\pi$
$61$	132.000	0.277063	0.138532	+	0.990358i	$0.455762\pi$
$62$	8.00000	0.0163871
$63$	0	0
$64$	−167.000	−0.326172
$65$	0	0
$66$	0	0
$67$	1036.00	1.88907	0.944534	−	0.328414i	$-0.106514\pi$
$67$	1036.00	1.88907	0.944534	+	0.328414i	$0.106514\pi$
$68$	−518.000	−0.923775
$69$	0	0
$70$	0	0
$71$	−762.000	−1.27370	−0.636850	−	0.770987i	$-0.719763\pi$
$71$	−762.000	−1.27370	−0.636850	+	0.770987i	$0.719763\pi$
$72$	0	0
$73$	542.000	0.868990	0.434495	−	0.900674i	$-0.356927\pi$
$73$	542.000	0.868990	0.434495	+	0.900674i	$0.356927\pi$
$74$	−66.0000	−0.103680
$75$	0	0
$76$	420.000	0.633912
$77$	−286.000	−0.423282
$78$	0	0
$79$	−550.000	−0.783289	−0.391645	−	0.920117i	$-0.628094\pi$
$79$	−550.000	−0.783289	−0.391645	+	0.920117i	$0.628094\pi$
$80$	0	0
$81$	0	0
$82$	422.000	0.568318
$83$	−132.000	−0.174565	−0.0872824	−	0.996184i	$-0.527818\pi$
$83$	−132.000	−0.174565	−0.0872824	+	0.996184i	$0.527818\pi$
$84$	0	0
$85$	0	0
$86$	408.000	0.511579
$87$	0	0
$88$	−165.000	−0.199876
$89$	−570.000	−0.678875	−0.339438	−	0.940629i	$-0.610237\pi$
$89$	−570.000	−0.678875	−0.339438	+	0.940629i	$0.610237\pi$
$90$	0	0
$91$	832.000	0.958432
$92$	1274.00	1.44374
$93$	0	0
$94$	506.000	0.555212
$95$	0	0
$96$	0	0
$97$	−14.0000	−0.0146545	−0.00732724	−	0.999973i	$-0.502332\pi$
$97$	−14.0000	−0.0146545	−0.00732724	+	0.999973i	$0.502332\pi$
$98$	−333.000	−0.343246
$99$	0	0
$100$	0	0
$101$	−1702.00	−1.67679	−0.838393	−	0.545067i	$-0.816504\pi$
$101$	−1702.00	−1.67679	−0.838393	+	0.545067i	$0.816504\pi$
$102$	0	0
$103$	1132.00	1.08291	0.541453	−	0.840731i	$-0.317874\pi$
$103$	1132.00	1.08291	0.541453	+	0.840731i	$0.317874\pi$
$104$	480.000	0.452576
$105$	0	0
$106$	−348.000	−0.318875
$107$	564.000	0.509570	0.254785	−	0.966998i	$-0.417995\pi$
$107$	564.000	0.509570	0.254785	+	0.966998i	$0.417995\pi$
$108$	0	0
$109$	−320.000	−0.281197	−0.140598	−	0.990067i	$-0.544903\pi$
$109$	−320.000	−0.281197	−0.140598	+	0.990067i	$0.544903\pi$
$110$	0	0
$111$	0	0
$112$	1066.00	0.899353
$113$	−2142.00	−1.78321	−0.891604	−	0.452817i	$-0.850419\pi$
$113$	−2142.00	−1.78321	−0.891604	+	0.452817i	$0.850419\pi$
$114$	0	0
$115$	0	0
$116$	−630.000	−0.504259
$117$	0	0
$118$	−200.000	−0.156030
$119$	1924.00	1.48212
$120$	0	0
$121$	121.000	0.0909091
$122$	−132.000	−0.0979567
$123$	0	0
$124$	56.0000	0.0405560
$125$	0	0
$126$	0	0
$127$	1606.00	1.12212	0.561061	−	0.827775i	$-0.310393\pi$
$127$	1606.00	1.12212	0.561061	+	0.827775i	$0.310393\pi$
$128$	1455.00	1.00473
$129$	0	0
$130$	0	0
$131$	1908.00	1.27254	0.636270	−	0.771466i	$-0.280476\pi$
$131$	1908.00	1.27254	0.636270	+	0.771466i	$0.280476\pi$
$132$	0	0
$133$	−1560.00	−1.01706
$134$	−1036.00	−0.667886
$135$	0	0
$136$	1110.00	0.699866
$137$	−2186.00	−1.36323	−0.681615	−	0.731711i	$-0.738722\pi$
$137$	−2186.00	−1.36323	−0.681615	+	0.731711i	$0.738722\pi$
$138$	0	0
$139$	2740.00	1.67197	0.835985	−	0.548753i	$-0.184897\pi$
$139$	2740.00	1.67197	0.835985	+	0.548753i	$0.184897\pi$
$140$	0	0
$141$	0	0
$142$	762.000	0.450321
$143$	−352.000	−0.205844
$144$	0	0
$145$	0	0
$146$	−542.000	−0.307235
$147$	0	0
$148$	−462.000	−0.256596
$149$	1310.00	0.720264	0.360132	−	0.932901i	$-0.382732\pi$
$149$	1310.00	0.720264	0.360132	+	0.932901i	$0.382732\pi$
$150$	0	0
$151$	−1198.00	−0.645641	−0.322821	−	0.946460i	$-0.604631\pi$
$151$	−1198.00	−0.645641	−0.322821	+	0.946460i	$0.604631\pi$
$152$	−900.000	−0.480261
$153$	0	0
$154$	286.000	0.149653
$155$	0	0
$156$	0	0
$157$	−2114.00	−1.07462	−0.537311	−	0.843384i	$-0.680560\pi$
$157$	−2114.00	−1.07462	−0.537311	+	0.843384i	$0.680560\pi$
$158$	550.000	0.276934
$159$	0	0
$160$	0	0
$161$	−4732.00	−2.31636
$162$	0	0
$163$	−3868.00	−1.85868	−0.929341	−	0.369223i	$-0.879624\pi$
$163$	−3868.00	−1.85868	−0.929341	+	0.369223i	$0.879624\pi$
$164$	2954.00	1.40652
$165$	0	0
$166$	132.000	0.0617180
$167$	2004.00	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	2004.00	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−1173.00	−0.533910
$170$	0	0
$171$	0	0
$172$	2856.00	1.26609
$173$	678.000	0.297962	0.148981	−	0.988840i	$-0.452401\pi$
$173$	678.000	0.297962	0.148981	+	0.988840i	$0.452401\pi$
$174$	0	0
$175$	0	0
$176$	−451.000	−0.193156
$177$	0	0
$178$	570.000	0.240019
$179$	1680.00	0.701503	0.350752	−	0.936469i	$-0.385926\pi$
$179$	1680.00	0.701503	0.350752	+	0.936469i	$0.385926\pi$
$180$	0	0
$181$	−4358.00	−1.78966	−0.894828	−	0.446412i	$-0.852702\pi$
$181$	−4358.00	−1.78966	−0.894828	+	0.446412i	$0.852702\pi$
$182$	−832.000	−0.338857
$183$	0	0
$184$	−2730.00	−1.09379
$185$	0	0
$186$	0	0
$187$	−814.000	−0.318319
$188$	3542.00	1.37408
$189$	0	0
$190$	0	0
$191$	1778.00	0.673568	0.336784	−	0.941582i	$-0.390661\pi$
$191$	1778.00	0.673568	0.336784	+	0.941582i	$0.390661\pi$
$192$	0	0
$193$	3962.00	1.47767	0.738837	−	0.673884i	$-0.235375\pi$
$193$	3962.00	1.47767	0.738837	+	0.673884i	$0.235375\pi$
$194$	14.0000	0.00518114
$195$	0	0
$196$	−2331.00	−0.849490
$197$	374.000	0.135261	0.0676304	−	0.997710i	$-0.478456\pi$
$197$	374.000	0.135261	0.0676304	+	0.997710i	$0.478456\pi$
$198$	0	0
$199$	2100.00	0.748066	0.374033	−	0.927415i	$-0.377975\pi$
$199$	2100.00	0.748066	0.374033	+	0.927415i	$0.377975\pi$
$200$	0	0
$201$	0	0
$202$	1702.00	0.592833
$203$	2340.00	0.809043
$204$	0	0
$205$	0	0
$206$	−1132.00	−0.382865
$207$	0	0
$208$	1312.00	0.437360
$209$	660.000	0.218436
$210$	0	0
$211$	2232.00	0.728233	0.364117	−	0.931353i	$-0.381371\pi$
$211$	2232.00	0.728233	0.364117	+	0.931353i	$0.381371\pi$
$212$	−2436.00	−0.789175
$213$	0	0
$214$	−564.000	−0.180160
$215$	0	0
$216$	0	0
$217$	−208.000	−0.0650689
$218$	320.000	0.0994180
$219$	0	0
$220$	0	0
$221$	2368.00	0.720764
$222$	0	0
$223$	−2128.00	−0.639020	−0.319510	−	0.947583i	$-0.603518\pi$
$223$	−2128.00	−0.639020	−0.319510	+	0.947583i	$0.603518\pi$
$224$	−4186.00	−1.24861
$225$	0	0
$226$	2142.00	0.630459
$227$	2964.00	0.866641	0.433321	−	0.901240i	$-0.357342\pi$
$227$	2964.00	0.866641	0.433321	+	0.901240i	$0.357342\pi$
$228$	0	0
$229$	−2550.00	−0.735846	−0.367923	−	0.929856i	$-0.619931\pi$
$229$	−2550.00	−0.735846	−0.367923	+	0.929856i	$0.619931\pi$
$230$	0	0
$231$	0	0
$232$	1350.00	0.382034
$233$	−3042.00	−0.855314	−0.427657	−	0.903941i	$-0.640661\pi$
$233$	−3042.00	−0.855314	−0.427657	+	0.903941i	$0.640661\pi$
$234$	0	0
$235$	0	0
$236$	−1400.00	−0.386154
$237$	0	0
$238$	−1924.00	−0.524010
$239$	−2700.00	−0.730747	−0.365373	−	0.930861i	$-0.619059\pi$
$239$	−2700.00	−0.730747	−0.365373	+	0.930861i	$0.619059\pi$
$240$	0	0
$241$	−578.000	−0.154491	−0.0772453	−	0.997012i	$-0.524612\pi$
$241$	−578.000	−0.154491	−0.0772453	+	0.997012i	$0.524612\pi$
$242$	−121.000	−0.0321412
$243$	0	0
$244$	−924.000	−0.242430
$245$	0	0
$246$	0	0
$247$	−1920.00	−0.494602
$248$	−120.000	−0.0307258
$249$	0	0
$250$	0	0
$251$	−3752.00	−0.943522	−0.471761	−	0.881726i	$-0.656382\pi$
$251$	−3752.00	−0.943522	−0.471761	+	0.881726i	$0.656382\pi$
$252$	0	0
$253$	2002.00	0.497489
$254$	−1606.00	−0.396730
$255$	0	0
$256$	−119.000	−0.0290527
$257$	674.000	0.163591	0.0817957	−	0.996649i	$-0.473935\pi$
$257$	674.000	0.163591	0.0817957	+	0.996649i	$0.473935\pi$
$258$	0	0
$259$	1716.00	0.411687
$260$	0	0
$261$	0	0
$262$	−1908.00	−0.449911
$263$	−4352.00	−1.02036	−0.510182	−	0.860066i	$-0.670422\pi$
$263$	−4352.00	−1.02036	−0.510182	+	0.860066i	$0.670422\pi$
$264$	0	0
$265$	0	0
$266$	1560.00	0.359585
$267$	0	0
$268$	−7252.00	−1.65293
$269$	−500.000	−0.113329	−0.0566646	−	0.998393i	$-0.518047\pi$
$269$	−500.000	−0.113329	−0.0566646	+	0.998393i	$0.518047\pi$
$270$	0	0
$271$	−6538.00	−1.46552	−0.732759	−	0.680489i	$-0.761768\pi$
$271$	−6538.00	−1.46552	−0.732759	+	0.680489i	$0.761768\pi$
$272$	3034.00	0.676336
$273$	0	0
$274$	2186.00	0.481975
$275$	0	0
$276$	0	0
$277$	−124.000	−0.0268969	−0.0134484	−	0.999910i	$-0.504281\pi$
$277$	−124.000	−0.0268969	−0.0134484	+	0.999910i	$0.504281\pi$
$278$	−2740.00	−0.591131
$279$	0	0
$280$	0	0
$281$	−3642.00	−0.773180	−0.386590	−	0.922252i	$-0.626347\pi$
$281$	−3642.00	−0.773180	−0.386590	+	0.922252i	$0.626347\pi$
$282$	0	0
$283$	−4648.00	−0.976307	−0.488154	−	0.872758i	$-0.662329\pi$
$283$	−4648.00	−0.976307	−0.488154	+	0.872758i	$0.662329\pi$
$284$	5334.00	1.11449
$285$	0	0
$286$	352.000	0.0727769
$287$	−10972.0	−2.25664
$288$	0	0
$289$	563.000	0.114594
$290$	0	0
$291$	0	0
$292$	−3794.00	−0.760367
$293$	−3102.00	−0.618501	−0.309250	−	0.950981i	$-0.600078\pi$
$293$	−3102.00	−0.618501	−0.309250	+	0.950981i	$0.600078\pi$
$294$	0	0
$295$	0	0
$296$	990.000	0.194401
$297$	0	0
$298$	−1310.00	−0.254652
$299$	−5824.00	−1.12646
$300$	0	0
$301$	−10608.0	−2.03135
$302$	1198.00	0.228269
$303$	0	0
$304$	−2460.00	−0.464114
$305$	0	0
$306$	0	0
$307$	−1244.00	−0.231267	−0.115633	−	0.993292i	$-0.536890\pi$
$307$	−1244.00	−0.231267	−0.115633	+	0.993292i	$0.536890\pi$
$308$	2002.00	0.370372
$309$	0	0
$310$	0	0
$311$	−2082.00	−0.379612	−0.189806	−	0.981822i	$-0.560786\pi$
$311$	−2082.00	−0.379612	−0.189806	+	0.981822i	$0.560786\pi$
$312$	0	0
$313$	−2378.00	−0.429433	−0.214716	−	0.976676i	$-0.568883\pi$
$313$	−2378.00	−0.429433	−0.214716	+	0.976676i	$0.568883\pi$
$314$	2114.00	0.379936
$315$	0	0
$316$	3850.00	0.685378
$317$	−496.000	−0.0878806	−0.0439403	−	0.999034i	$-0.513991\pi$
$317$	−496.000	−0.0878806	−0.0439403	+	0.999034i	$0.513991\pi$
$318$	0	0
$319$	−990.000	−0.173760
$320$	0	0
$321$	0	0
$322$	4732.00	0.818957
$323$	−4440.00	−0.764855
$324$	0	0
$325$	0	0
$326$	3868.00	0.657143
$327$	0	0
$328$	−6330.00	−1.06560
$329$	−13156.0	−2.20460
$330$	0	0
$331$	−2708.00	−0.449683	−0.224842	−	0.974395i	$-0.572186\pi$
$331$	−2708.00	−0.449683	−0.224842	+	0.974395i	$0.572186\pi$
$332$	924.000	0.152744
$333$	0	0
$334$	−2004.00	−0.328305
$335$	0	0
$336$	0	0
$337$	−4034.00	−0.652065	−0.326033	−	0.945359i	$-0.605712\pi$
$337$	−4034.00	−0.652065	−0.326033	+	0.945359i	$0.605712\pi$
$338$	1173.00	0.188766
$339$	0	0
$340$	0	0
$341$	88.0000	0.0139750
$342$	0	0
$343$	−260.000	−0.0409291
$344$	−6120.00	−0.959210
$345$	0	0
$346$	−678.000	−0.105345
$347$	11084.0	1.71476	0.857378	−	0.514687i	$-0.172092\pi$
$347$	11084.0	1.71476	0.857378	+	0.514687i	$0.172092\pi$
$348$	0	0
$349$	−3120.00	−0.478538	−0.239269	−	0.970953i	$-0.576908\pi$
$349$	−3120.00	−0.478538	−0.239269	+	0.970953i	$0.576908\pi$
$350$	0	0
$351$	0	0
$352$	1771.00	0.268167
$353$	−5622.00	−0.847674	−0.423837	−	0.905739i	$-0.639317\pi$
$353$	−5622.00	−0.847674	−0.423837	+	0.905739i	$0.639317\pi$
$354$	0	0
$355$	0	0
$356$	3990.00	0.594016
$357$	0	0
$358$	−1680.00	−0.248019
$359$	8500.00	1.24962	0.624809	−	0.780778i	$-0.285177\pi$
$359$	8500.00	1.24962	0.624809	+	0.780778i	$0.285177\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	4358.00	0.632739
$363$	0	0
$364$	−5824.00	−0.838628
$365$	0	0
$366$	0	0
$367$	−7144.00	−1.01611	−0.508057	−	0.861324i	$-0.669636\pi$
$367$	−7144.00	−1.01611	−0.508057	+	0.861324i	$0.669636\pi$
$368$	−7462.00	−1.05702
$369$	0	0
$370$	0	0
$371$	9048.00	1.26617
$372$	0	0
$373$	632.000	0.0877312	0.0438656	−	0.999037i	$-0.486033\pi$
$373$	632.000	0.0877312	0.0438656	+	0.999037i	$0.486033\pi$
$374$	814.000	0.112543
$375$	0	0
$376$	−7590.00	−1.04102
$377$	2880.00	0.393442
$378$	0	0
$379$	−4220.00	−0.571944	−0.285972	−	0.958238i	$-0.592316\pi$
$379$	−4220.00	−0.571944	−0.285972	+	0.958238i	$0.592316\pi$
$380$	0	0
$381$	0	0
$382$	−1778.00	−0.238142
$383$	8458.00	1.12842	0.564208	−	0.825632i	$-0.309181\pi$
$383$	8458.00	1.12842	0.564208	+	0.825632i	$0.309181\pi$
$384$	0	0
$385$	0	0
$386$	−3962.00	−0.522437
$387$	0	0
$388$	98.0000	0.0128227
$389$	−1740.00	−0.226790	−0.113395	−	0.993550i	$-0.536173\pi$
$389$	−1740.00	−0.226790	−0.113395	+	0.993550i	$0.536173\pi$
$390$	0	0
$391$	−13468.0	−1.74196
$392$	4995.00	0.643586
$393$	0	0
$394$	−374.000	−0.0478219
$395$	0	0
$396$	0	0
$397$	5126.00	0.648027	0.324013	−	0.946053i	$-0.394968\pi$
$397$	5126.00	0.648027	0.324013	+	0.946053i	$0.394968\pi$
$398$	−2100.00	−0.264481
$399$	0	0
$400$	0	0
$401$	3098.00	0.385802	0.192901	−	0.981218i	$-0.438210\pi$
$401$	3098.00	0.385802	0.192901	+	0.981218i	$0.438210\pi$
$402$	0	0
$403$	−256.000	−0.0316433
$404$	11914.0	1.46719
$405$	0	0
$406$	−2340.00	−0.286040
$407$	−726.000	−0.0884189
$408$	0	0
$409$	6390.00	0.772531	0.386265	−	0.922388i	$-0.373765\pi$
$409$	6390.00	0.772531	0.386265	+	0.922388i	$0.373765\pi$
$410$	0	0
$411$	0	0
$412$	−7924.00	−0.947542
$413$	5200.00	0.619553
$414$	0	0
$415$	0	0
$416$	−5152.00	−0.607206
$417$	0	0
$418$	−660.000	−0.0772288
$419$	−9760.00	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$419$	−9760.00	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$420$	0	0
$421$	−5138.00	−0.594800	−0.297400	−	0.954753i	$-0.596119\pi$
$421$	−5138.00	−0.594800	−0.297400	+	0.954753i	$0.596119\pi$
$422$	−2232.00	−0.257469
$423$	0	0
$424$	5220.00	0.597891
$425$	0	0
$426$	0	0
$427$	3432.00	0.388960
$428$	−3948.00	−0.445873
$429$	0	0
$430$	0	0
$431$	7008.00	0.783210	0.391605	−	0.920133i	$-0.371920\pi$
$431$	7008.00	0.783210	0.391605	+	0.920133i	$0.371920\pi$
$432$	0	0
$433$	−5578.00	−0.619080	−0.309540	−	0.950886i	$-0.600175\pi$
$433$	−5578.00	−0.619080	−0.309540	+	0.950886i	$0.600175\pi$
$434$	208.000	0.0230053
$435$	0	0
$436$	2240.00	0.246047
$437$	10920.0	1.19536
$438$	0	0
$439$	−10430.0	−1.13393	−0.566967	−	0.823741i	$-0.691883\pi$
$439$	−10430.0	−1.13393	−0.566967	+	0.823741i	$0.691883\pi$
$440$	0	0
$441$	0	0
$442$	−2368.00	−0.254829
$443$	−4432.00	−0.475329	−0.237664	−	0.971347i	$-0.576382\pi$
$443$	−4432.00	−0.475329	−0.237664	+	0.971347i	$0.576382\pi$
$444$	0	0
$445$	0	0
$446$	2128.00	0.225928
$447$	0	0
$448$	−4342.00	−0.457902
$449$	6290.00	0.661121	0.330561	−	0.943785i	$-0.392762\pi$
$449$	6290.00	0.661121	0.330561	+	0.943785i	$0.392762\pi$
$450$	0	0
$451$	4642.00	0.484664
$452$	14994.0	1.56031
$453$	0	0
$454$	−2964.00	−0.306404
$455$	0	0
$456$	0	0
$457$	−3054.00	−0.312604	−0.156302	−	0.987709i	$-0.549957\pi$
$457$	−3054.00	−0.312604	−0.156302	+	0.987709i	$0.549957\pi$
$458$	2550.00	0.260161
$459$	0	0
$460$	0	0
$461$	−12882.0	−1.30146	−0.650732	−	0.759308i	$-0.725538\pi$
$461$	−12882.0	−1.30146	−0.650732	+	0.759308i	$0.725538\pi$
$462$	0	0
$463$	−6148.00	−0.617110	−0.308555	−	0.951207i	$-0.599845\pi$
$463$	−6148.00	−0.617110	−0.308555	+	0.951207i	$0.599845\pi$
$464$	3690.00	0.369190
$465$	0	0
$466$	3042.00	0.302399
$467$	5124.00	0.507731	0.253866	−	0.967240i	$-0.418298\pi$
$467$	5124.00	0.507731	0.253866	+	0.967240i	$0.418298\pi$
$468$	0	0
$469$	26936.0	2.65200
$470$	0	0
$471$	0	0
$472$	3000.00	0.292555
$473$	4488.00	0.436276
$474$	0	0
$475$	0	0
$476$	−13468.0	−1.29686
$477$	0	0
$478$	2700.00	0.258358
$479$	16520.0	1.57582	0.787910	−	0.615790i	$-0.211163\pi$
$479$	16520.0	1.57582	0.787910	+	0.615790i	$0.211163\pi$
$480$	0	0
$481$	2112.00	0.200206
$482$	578.000	0.0546207
$483$	0	0
$484$	−847.000	−0.0795455
$485$	0	0
$486$	0	0
$487$	−524.000	−0.0487571	−0.0243785	−	0.999703i	$-0.507761\pi$
$487$	−524.000	−0.0487571	−0.0243785	+	0.999703i	$0.507761\pi$
$488$	1980.00	0.183669
$489$	0	0
$490$	0	0
$491$	15028.0	1.38127	0.690636	−	0.723203i	$-0.257331\pi$
$491$	15028.0	1.38127	0.690636	+	0.723203i	$0.257331\pi$
$492$	0	0
$493$	6660.00	0.608421
$494$	1920.00	0.174868
$495$	0	0
$496$	−328.000	−0.0296928
$497$	−19812.0	−1.78811
$498$	0	0
$499$	9020.00	0.809200	0.404600	−	0.914494i	$-0.367411\pi$
$499$	9020.00	0.809200	0.404600	+	0.914494i	$0.367411\pi$
$500$	0	0
$501$	0	0
$502$	3752.00	0.333586
$503$	−14812.0	−1.31299	−0.656495	−	0.754330i	$-0.727962\pi$
$503$	−14812.0	−1.31299	−0.656495	+	0.754330i	$0.727962\pi$
$504$	0	0
$505$	0	0
$506$	−2002.00	−0.175889
$507$	0	0
$508$	−11242.0	−0.981856
$509$	−12660.0	−1.10245	−0.551223	−	0.834358i	$-0.685839\pi$
$509$	−12660.0	−1.10245	−0.551223	+	0.834358i	$0.685839\pi$
$510$	0	0
$511$	14092.0	1.21995
$512$	−11521.0	−0.994455
$513$	0	0
$514$	−674.000	−0.0578383
$515$	0	0
$516$	0	0
$517$	5566.00	0.473486
$518$	−1716.00	−0.145553
$519$	0	0
$520$	0	0
$521$	3738.00	0.314328	0.157164	−	0.987573i	$-0.449765\pi$
$521$	3738.00	0.314328	0.157164	+	0.987573i	$0.449765\pi$
$522$	0	0
$523$	6352.00	0.531078	0.265539	−	0.964100i	$-0.414450\pi$
$523$	6352.00	0.531078	0.265539	+	0.964100i	$0.414450\pi$
$524$	−13356.0	−1.11347
$525$	0	0
$526$	4352.00	0.360753
$527$	−592.000	−0.0489334
$528$	0	0
$529$	20957.0	1.72245
$530$	0	0
$531$	0	0
$532$	10920.0	0.889929
$533$	−13504.0	−1.09742
$534$	0	0
$535$	0	0
$536$	15540.0	1.25229
$537$	0	0
$538$	500.000	0.0400679
$539$	−3663.00	−0.292721
$540$	0	0
$541$	−24728.0	−1.96514	−0.982569	−	0.185898i	$-0.940481\pi$
$541$	−24728.0	−1.96514	−0.982569	+	0.185898i	$0.940481\pi$
$542$	6538.00	0.518139
$543$	0	0
$544$	−11914.0	−0.938986
$545$	0	0
$546$	0	0
$547$	22756.0	1.77875	0.889375	−	0.457178i	$-0.151140\pi$
$547$	22756.0	1.77875	0.889375	+	0.457178i	$0.151140\pi$
$548$	15302.0	1.19283
$549$	0	0
$550$	0	0
$551$	−5400.00	−0.417509
$552$	0	0
$553$	−14300.0	−1.09963
$554$	124.000	0.00950949
$555$	0	0
$556$	−19180.0	−1.46297
$557$	−9526.00	−0.724649	−0.362325	−	0.932052i	$-0.618017\pi$
$557$	−9526.00	−0.724649	−0.362325	+	0.932052i	$0.618017\pi$
$558$	0	0
$559$	−13056.0	−0.987853
$560$	0	0
$561$	0	0
$562$	3642.00	0.273360
$563$	12068.0	0.903385	0.451692	−	0.892174i	$-0.350820\pi$
$563$	12068.0	0.903385	0.451692	+	0.892174i	$0.350820\pi$
$564$	0	0
$565$	0	0
$566$	4648.00	0.345177
$567$	0	0
$568$	−11430.0	−0.844352
$569$	−15090.0	−1.11179	−0.555893	−	0.831254i	$-0.687623\pi$
$569$	−15090.0	−1.11179	−0.555893	+	0.831254i	$0.687623\pi$
$570$	0	0
$571$	4412.00	0.323356	0.161678	−	0.986844i	$-0.448309\pi$
$571$	4412.00	0.323356	0.161678	+	0.986844i	$0.448309\pi$
$572$	2464.00	0.180114
$573$	0	0
$574$	10972.0	0.797844
$575$	0	0
$576$	0	0
$577$	3906.00	0.281818	0.140909	−	0.990023i	$-0.454998\pi$
$577$	3906.00	0.281818	0.140909	+	0.990023i	$0.454998\pi$
$578$	−563.000	−0.0405151
$579$	0	0
$580$	0	0
$581$	−3432.00	−0.245066
$582$	0	0
$583$	−3828.00	−0.271937
$584$	8130.00	0.576065
$585$	0	0
$586$	3102.00	0.218673
$587$	−12016.0	−0.844895	−0.422448	−	0.906387i	$-0.638829\pi$
$587$	−12016.0	−0.844895	−0.422448	+	0.906387i	$0.638829\pi$
$588$	0	0
$589$	480.000	0.0335790
$590$	0	0
$591$	0	0
$592$	2706.00	0.187865
$593$	−11342.0	−0.785430	−0.392715	−	0.919660i	$-0.628464\pi$
$593$	−11342.0	−0.785430	−0.392715	+	0.919660i	$0.628464\pi$
$594$	0	0
$595$	0	0
$596$	−9170.00	−0.630231
$597$	0	0
$598$	5824.00	0.398263
$599$	−20690.0	−1.41130	−0.705651	−	0.708559i	$-0.749346\pi$
$599$	−20690.0	−1.41130	−0.705651	+	0.708559i	$0.749346\pi$
$600$	0	0
$601$	−598.000	−0.0405872	−0.0202936	−	0.999794i	$-0.506460\pi$
$601$	−598.000	−0.0405872	−0.0202936	+	0.999794i	$0.506460\pi$
$602$	10608.0	0.718189
$603$	0	0
$604$	8386.00	0.564936
$605$	0	0
$606$	0	0
$607$	166.000	0.0111001	0.00555003	−	0.999985i	$-0.498233\pi$
$607$	166.000	0.0111001	0.00555003	+	0.999985i	$0.498233\pi$
$608$	9660.00	0.644350
$609$	0	0
$610$	0	0
$611$	−16192.0	−1.07211
$612$	0	0
$613$	−20108.0	−1.32488	−0.662442	−	0.749113i	$-0.730480\pi$
$613$	−20108.0	−1.32488	−0.662442	+	0.749113i	$0.730480\pi$
$614$	1244.00	0.0817651
$615$	0	0
$616$	−4290.00	−0.280599
$617$	−2286.00	−0.149159	−0.0745793	−	0.997215i	$-0.523761\pi$
$617$	−2286.00	−0.149159	−0.0745793	+	0.997215i	$0.523761\pi$
$618$	0	0
$619$	−25660.0	−1.66618	−0.833088	−	0.553141i	$-0.813429\pi$
$619$	−25660.0	−1.66618	−0.833088	+	0.553141i	$0.813429\pi$
$620$	0	0
$621$	0	0
$622$	2082.00	0.134213
$623$	−14820.0	−0.953051
$624$	0	0
$625$	0	0
$626$	2378.00	0.151827
$627$	0	0
$628$	14798.0	0.940294
$629$	4884.00	0.309599
$630$	0	0
$631$	−11408.0	−0.719723	−0.359862	−	0.933006i	$-0.617176\pi$
$631$	−11408.0	−0.719723	−0.359862	+	0.933006i	$0.617176\pi$
$632$	−8250.00	−0.519252
$633$	0	0
$634$	496.000	0.0310705
$635$	0	0
$636$	0	0
$637$	10656.0	0.662804
$638$	990.000	0.0614333
$639$	0	0
$640$	0	0
$641$	3378.00	0.208148	0.104074	−	0.994570i	$-0.466812\pi$
$641$	3378.00	0.208148	0.104074	+	0.994570i	$0.466812\pi$
$642$	0	0
$643$	11212.0	0.687649	0.343824	−	0.939034i	$-0.388278\pi$
$643$	11212.0	0.687649	0.343824	+	0.939034i	$0.388278\pi$
$644$	33124.0	2.02681
$645$	0	0
$646$	4440.00	0.270417
$647$	−86.0000	−0.00522567	−0.00261284	−	0.999997i	$-0.500832\pi$
$647$	−86.0000	−0.00522567	−0.00261284	+	0.999997i	$0.500832\pi$
$648$	0	0
$649$	−2200.00	−0.133062
$650$	0	0
$651$	0	0
$652$	27076.0	1.62635
$653$	−4432.00	−0.265601	−0.132801	−	0.991143i	$-0.542397\pi$
$653$	−4432.00	−0.265601	−0.132801	+	0.991143i	$0.542397\pi$
$654$	0	0
$655$	0	0
$656$	−17302.0	−1.02977
$657$	0	0
$658$	13156.0	0.779444
$659$	−4580.00	−0.270731	−0.135365	−	0.990796i	$-0.543221\pi$
$659$	−4580.00	−0.270731	−0.135365	+	0.990796i	$0.543221\pi$
$660$	0	0
$661$	4282.00	0.251967	0.125984	−	0.992032i	$-0.459791\pi$
$661$	4282.00	0.251967	0.125984	+	0.992032i	$0.459791\pi$
$662$	2708.00	0.158987
$663$	0	0
$664$	−1980.00	−0.115721
$665$	0	0
$666$	0	0
$667$	−16380.0	−0.950879
$668$	−14028.0	−0.812514
$669$	0	0
$670$	0	0
$671$	−1452.00	−0.0835378
$672$	0	0
$673$	−8438.00	−0.483300	−0.241650	−	0.970363i	$-0.577689\pi$
$673$	−8438.00	−0.483300	−0.241650	+	0.970363i	$0.577689\pi$
$674$	4034.00	0.230540
$675$	0	0
$676$	8211.00	0.467171
$677$	34494.0	1.95822	0.979108	−	0.203341i	$-0.0651801\pi$
$677$	34494.0	1.95822	0.979108	+	0.203341i	$0.0651801\pi$
$678$	0	0
$679$	−364.000	−0.0205730
$680$	0	0
$681$	0	0
$682$	−88.0000	−0.00494090
$683$	−13712.0	−0.768192	−0.384096	−	0.923293i	$-0.625487\pi$
$683$	−13712.0	−0.768192	−0.384096	+	0.923293i	$0.625487\pi$
$684$	0	0
$685$	0	0
$686$	260.000	0.0144706
$687$	0	0
$688$	−16728.0	−0.926961
$689$	11136.0	0.615744
$690$	0	0
$691$	11372.0	0.626066	0.313033	−	0.949742i	$-0.398655\pi$
$691$	11372.0	0.626066	0.313033	+	0.949742i	$0.398655\pi$
$692$	−4746.00	−0.260717
$693$	0	0
$694$	−11084.0	−0.606258
$695$	0	0
$696$	0	0
$697$	−31228.0	−1.69705
$698$	3120.00	0.169189
$699$	0	0
$700$	0	0
$701$	6398.00	0.344721	0.172360	−	0.985034i	$-0.444861\pi$
$701$	6398.00	0.344721	0.172360	+	0.985034i	$0.444861\pi$
$702$	0	0
$703$	−3960.00	−0.212453
$704$	1837.00	0.0983445
$705$	0	0
$706$	5622.00	0.299698
$707$	−44252.0	−2.35399
$708$	0	0
$709$	−5830.00	−0.308816	−0.154408	−	0.988007i	$-0.549347\pi$
$709$	−5830.00	−0.308816	−0.154408	+	0.988007i	$0.549347\pi$
$710$	0	0
$711$	0	0
$712$	−8550.00	−0.450035
$713$	1456.00	0.0764763
$714$	0	0
$715$	0	0
$716$	−11760.0	−0.613815
$717$	0	0
$718$	−8500.00	−0.441807
$719$	−34530.0	−1.79103	−0.895516	−	0.445030i	$-0.853193\pi$
$719$	−34530.0	−1.79103	−0.895516	+	0.445030i	$0.853193\pi$
$720$	0	0
$721$	29432.0	1.52026
$722$	3259.00	0.167988
$723$	0	0
$724$	30506.0	1.56595
$725$	0	0
$726$	0	0
$727$	17316.0	0.883377	0.441688	−	0.897169i	$-0.354380\pi$
$727$	17316.0	0.883377	0.441688	+	0.897169i	$0.354380\pi$
$728$	12480.0	0.635357
$729$	0	0
$730$	0	0
$731$	−30192.0	−1.52762
$732$	0	0
$733$	27072.0	1.36416	0.682079	−	0.731279i	$-0.261076\pi$
$733$	27072.0	1.36416	0.682079	+	0.731279i	$0.261076\pi$
$734$	7144.00	0.359250
$735$	0	0
$736$	29302.0	1.46751
$737$	−11396.0	−0.569575
$738$	0	0
$739$	−17320.0	−0.862147	−0.431073	−	0.902317i	$-0.641865\pi$
$739$	−17320.0	−0.862147	−0.431073	+	0.902317i	$0.641865\pi$
$740$	0	0
$741$	0	0
$742$	−9048.00	−0.447658
$743$	14588.0	0.720299	0.360149	−	0.932895i	$-0.382726\pi$
$743$	14588.0	0.720299	0.360149	+	0.932895i	$0.382726\pi$
$744$	0	0
$745$	0	0
$746$	−632.000	−0.0310176
$747$	0	0
$748$	5698.00	0.278529
$749$	14664.0	0.715368
$750$	0	0
$751$	26152.0	1.27071	0.635353	−	0.772222i	$-0.280855\pi$
$751$	26152.0	1.27071	0.635353	+	0.772222i	$0.280855\pi$
$752$	−20746.0	−1.00602
$753$	0	0
$754$	−2880.00	−0.139103
$755$	0	0
$756$	0	0
$757$	1066.00	0.0511815	0.0255908	−	0.999673i	$-0.491853\pi$
$757$	1066.00	0.0511815	0.0255908	+	0.999673i	$0.491853\pi$
$758$	4220.00	0.202213
$759$	0	0
$760$	0	0
$761$	37518.0	1.78716	0.893578	−	0.448907i	$-0.148187\pi$
$761$	37518.0	1.78716	0.893578	+	0.448907i	$0.148187\pi$
$762$	0	0
$763$	−8320.00	−0.394763
$764$	−12446.0	−0.589372
$765$	0	0
$766$	−8458.00	−0.398956
$767$	6400.00	0.301292
$768$	0	0
$769$	−17290.0	−0.810785	−0.405392	−	0.914143i	$-0.632865\pi$
$769$	−17290.0	−0.810785	−0.405392	+	0.914143i	$0.632865\pi$
$770$	0	0
$771$	0	0
$772$	−27734.0	−1.29296
$773$	−17172.0	−0.799009	−0.399504	−	0.916731i	$-0.630818\pi$
$773$	−17172.0	−0.799009	−0.399504	+	0.916731i	$0.630818\pi$
$774$	0	0
$775$	0	0
$776$	−210.000	−0.00971464
$777$	0	0
$778$	1740.00	0.0801825
$779$	25320.0	1.16455
$780$	0	0
$781$	8382.00	0.384035
$782$	13468.0	0.615876
$783$	0	0
$784$	13653.0	0.621948
$785$	0	0
$786$	0	0
$787$	9536.00	0.431921	0.215960	−	0.976402i	$-0.430712\pi$
$787$	9536.00	0.431921	0.215960	+	0.976402i	$0.430712\pi$
$788$	−2618.00	−0.118353
$789$	0	0
$790$	0	0
$791$	−55692.0	−2.50339
$792$	0	0
$793$	4224.00	0.189153
$794$	−5126.00	−0.229112
$795$	0	0
$796$	−14700.0	−0.654557
$797$	−20516.0	−0.911812	−0.455906	−	0.890028i	$-0.650685\pi$
$797$	−20516.0	−0.911812	−0.455906	+	0.890028i	$0.650685\pi$
$798$	0	0
$799$	−37444.0	−1.65791
$800$	0	0
$801$	0	0
$802$	−3098.00	−0.136402
$803$	−5962.00	−0.262010
$804$	0	0
$805$	0	0
$806$	256.000	0.0111876
$807$	0	0
$808$	−25530.0	−1.11156
$809$	−22470.0	−0.976518	−0.488259	−	0.872699i	$-0.662368\pi$
$809$	−22470.0	−0.976518	−0.488259	+	0.872699i	$0.662368\pi$
$810$	0	0
$811$	−3368.00	−0.145828	−0.0729140	−	0.997338i	$-0.523230\pi$
$811$	−3368.00	−0.145828	−0.0729140	+	0.997338i	$0.523230\pi$
$812$	−16380.0	−0.707913
$813$	0	0
$814$	726.000	0.0312608
$815$	0	0
$816$	0	0
$817$	24480.0	1.04828
$818$	−6390.00	−0.273131
$819$	0	0
$820$	0	0
$821$	10738.0	0.456466	0.228233	−	0.973607i	$-0.426705\pi$
$821$	10738.0	0.456466	0.228233	+	0.973607i	$0.426705\pi$
$822$	0	0
$823$	15912.0	0.673946	0.336973	−	0.941514i	$-0.390597\pi$
$823$	15912.0	0.673946	0.336973	+	0.941514i	$0.390597\pi$
$824$	16980.0	0.717872
$825$	0	0
$826$	−5200.00	−0.219045
$827$	22924.0	0.963900	0.481950	−	0.876199i	$-0.339929\pi$
$827$	22924.0	0.963900	0.481950	+	0.876199i	$0.339929\pi$
$828$	0	0
$829$	−41690.0	−1.74663	−0.873313	−	0.487159i	$-0.838033\pi$
$829$	−41690.0	−1.74663	−0.873313	+	0.487159i	$0.838033\pi$
$830$	0	0
$831$	0	0
$832$	−5344.00	−0.222680
$833$	24642.0	1.02496
$834$	0	0
$835$	0	0
$836$	−4620.00	−0.191132
$837$	0	0
$838$	9760.00	0.402331
$839$	16450.0	0.676898	0.338449	−	0.940985i	$-0.390098\pi$
$839$	16450.0	0.676898	0.338449	+	0.940985i	$0.390098\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	5138.00	0.210294
$843$	0	0
$844$	−15624.0	−0.637204
$845$	0	0
$846$	0	0
$847$	3146.00	0.127624
$848$	14268.0	0.577789
$849$	0	0
$850$	0	0
$851$	−12012.0	−0.483861
$852$	0	0
$853$	30892.0	1.24000	0.620001	−	0.784601i	$-0.287132\pi$
$853$	30892.0	1.24000	0.620001	+	0.784601i	$0.287132\pi$
$854$	−3432.00	−0.137518
$855$	0	0
$856$	8460.00	0.337800
$857$	−38906.0	−1.55076	−0.775381	−	0.631493i	$-0.782442\pi$
$857$	−38906.0	−1.55076	−0.775381	+	0.631493i	$0.782442\pi$
$858$	0	0
$859$	−1020.00	−0.0405145	−0.0202572	−	0.999795i	$-0.506449\pi$
$859$	−1020.00	−0.0405145	−0.0202572	+	0.999795i	$0.506449\pi$
$860$	0	0
$861$	0	0
$862$	−7008.00	−0.276907
$863$	15078.0	0.594741	0.297370	−	0.954762i	$-0.403890\pi$
$863$	15078.0	0.594741	0.297370	+	0.954762i	$0.403890\pi$
$864$	0	0
$865$	0	0
$866$	5578.00	0.218878
$867$	0	0
$868$	1456.00	0.0569353
$869$	6050.00	0.236171
$870$	0	0
$871$	33152.0	1.28968
$872$	−4800.00	−0.186409
$873$	0	0
$874$	−10920.0	−0.422625
$875$	0	0
$876$	0	0
$877$	−22704.0	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$877$	−22704.0	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$878$	10430.0	0.400906
$879$	0	0
$880$	0	0
$881$	19358.0	0.740281	0.370141	−	0.928976i	$-0.379310\pi$
$881$	19358.0	0.740281	0.370141	+	0.928976i	$0.379310\pi$
$882$	0	0
$883$	11252.0	0.428833	0.214417	−	0.976742i	$-0.431215\pi$
$883$	11252.0	0.428833	0.214417	+	0.976742i	$0.431215\pi$
$884$	−16576.0	−0.630669
$885$	0	0
$886$	4432.00	0.168054
$887$	43684.0	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$887$	43684.0	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$888$	0	0
$889$	41756.0	1.57531
$890$	0	0
$891$	0	0
$892$	14896.0	0.559142
$893$	30360.0	1.13769
$894$	0	0
$895$	0	0
$896$	37830.0	1.41050
$897$	0	0
$898$	−6290.00	−0.233742
$899$	−720.000	−0.0267112
$900$	0	0
$901$	25752.0	0.952190
$902$	−4642.00	−0.171354
$903$	0	0
$904$	−32130.0	−1.18211
$905$	0	0
$906$	0	0
$907$	−45804.0	−1.67684	−0.838422	−	0.545022i	$-0.816521\pi$
$907$	−45804.0	−1.67684	−0.838422	+	0.545022i	$0.816521\pi$
$908$	−20748.0	−0.758311
$909$	0	0
$910$	0	0
$911$	15318.0	0.557089	0.278544	−	0.960423i	$-0.410148\pi$
$911$	15318.0	0.557089	0.278544	+	0.960423i	$0.410148\pi$
$912$	0	0
$913$	1452.00	0.0526333
$914$	3054.00	0.110522
$915$	0	0
$916$	17850.0	0.643865
$917$	49608.0	1.78648
$918$	0	0
$919$	11350.0	0.407401	0.203701	−	0.979033i	$-0.434703\pi$
$919$	11350.0	0.407401	0.203701	+	0.979033i	$0.434703\pi$
$920$	0	0
$921$	0	0
$922$	12882.0	0.460137
$923$	−24384.0	−0.869566
$924$	0	0
$925$	0	0
$926$	6148.00	0.218181
$927$	0	0
$928$	−14490.0	−0.512562
$929$	−33030.0	−1.16650	−0.583250	−	0.812292i	$-0.698219\pi$
$929$	−33030.0	−1.16650	−0.583250	+	0.812292i	$0.698219\pi$
$930$	0	0
$931$	−19980.0	−0.703349
$932$	21294.0	0.748399
$933$	0	0
$934$	−5124.00	−0.179510
$935$	0	0
$936$	0	0
$937$	10006.0	0.348860	0.174430	−	0.984670i	$-0.444192\pi$
$937$	10006.0	0.348860	0.174430	+	0.984670i	$0.444192\pi$
$938$	−26936.0	−0.937624
$939$	0	0
$940$	0	0
$941$	−2622.00	−0.0908340	−0.0454170	−	0.998968i	$-0.514462\pi$
$941$	−2622.00	−0.0908340	−0.0454170	+	0.998968i	$0.514462\pi$
$942$	0	0
$943$	76804.0	2.65226
$944$	8200.00	0.282720
$945$	0	0
$946$	−4488.00	−0.154247
$947$	−39876.0	−1.36832	−0.684158	−	0.729334i	$-0.739830\pi$
$947$	−39876.0	−1.36832	−0.684158	+	0.729334i	$0.739830\pi$
$948$	0	0
$949$	17344.0	0.593267
$950$	0	0
$951$	0	0
$952$	28860.0	0.982519
$953$	38918.0	1.32285	0.661426	−	0.750011i	$-0.269952\pi$
$953$	38918.0	1.32285	0.661426	+	0.750011i	$0.269952\pi$
$954$	0	0
$955$	0	0
$956$	18900.0	0.639403
$957$	0	0
$958$	−16520.0	−0.557137
$959$	−56836.0	−1.91380
$960$	0	0
$961$	−29727.0	−0.997852
$962$	−2112.00	−0.0707834
$963$	0	0
$964$	4046.00	0.135179
$965$	0	0
$966$	0	0
$967$	−1114.00	−0.0370464	−0.0185232	−	0.999828i	$-0.505896\pi$
$967$	−1114.00	−0.0370464	−0.0185232	+	0.999828i	$0.505896\pi$
$968$	1815.00	0.0602648
$969$	0	0
$970$	0	0
$971$	1688.00	0.0557884	0.0278942	−	0.999611i	$-0.491120\pi$
$971$	1688.00	0.0557884	0.0278942	+	0.999611i	$0.491120\pi$
$972$	0	0
$973$	71240.0	2.34722
$974$	524.000	0.0172382
$975$	0	0
$976$	5412.00	0.177494
$977$	−41826.0	−1.36963	−0.684817	−	0.728715i	$-0.740118\pi$
$977$	−41826.0	−1.36963	−0.684817	+	0.728715i	$0.740118\pi$
$978$	0	0
$979$	6270.00	0.204689
$980$	0	0
$981$	0	0
$982$	−15028.0	−0.488353
$983$	978.000	0.0317328	0.0158664	−	0.999874i	$-0.494949\pi$
$983$	978.000	0.0317328	0.0158664	+	0.999874i	$0.494949\pi$
$984$	0	0
$985$	0	0
$986$	−6660.00	−0.215109
$987$	0	0
$988$	13440.0	0.432777
$989$	74256.0	2.38747
$990$	0	0
$991$	47272.0	1.51528	0.757641	−	0.652671i	$-0.226352\pi$
$991$	47272.0	1.51528	0.757641	+	0.652671i	$0.226352\pi$
$992$	1288.00	0.0412238
$993$	0	0
$994$	19812.0	0.632192
$995$	0	0
$996$	0	0
$997$	−51104.0	−1.62335	−0.811675	−	0.584109i	$-0.801444\pi$
$997$	−51104.0	−1.62335	−0.811675	+	0.584109i	$0.801444\pi$
$998$	−9020.00	−0.286095
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.e.1.1		1
3.2	odd	2		825.4.a.f.1.1		1
5.4	even	2		99.4.a.a.1.1		1
15.2	even	4		825.4.c.f.199.2		2
15.8	even	4		825.4.c.f.199.1		2
15.14	odd	2		33.4.a.b.1.1	✓	1
20.19	odd	2		1584.4.a.l.1.1		1
55.54	odd	2		1089.4.a.e.1.1		1
60.59	even	2		528.4.a.h.1.1		1
105.104	even	2		1617.4.a.d.1.1		1
120.29	odd	2		2112.4.a.u.1.1		1
120.59	even	2		2112.4.a.h.1.1		1
165.164	even	2		363.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.b.1.1	✓	1	15.14	odd	2
99.4.a.a.1.1		1	5.4	even	2
363.4.a.d.1.1		1	165.164	even	2
528.4.a.h.1.1		1	60.59	even	2
825.4.a.f.1.1		1	3.2	odd	2
825.4.c.f.199.1		2	15.8	even	4
825.4.c.f.199.2		2	15.2	even	4
1089.4.a.e.1.1		1	55.54	odd	2
1584.4.a.l.1.1		1	20.19	odd	2
1617.4.a.d.1.1		1	105.104	even	2
2112.4.a.h.1.1		1	120.59	even	2
2112.4.a.u.1.1		1	120.29	odd	2
2475.4.a.e.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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