LMFDB - Embedded newform 2500.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2500,4,Mod(1,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2500.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2500 = 2^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2500.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:	$147.504775014$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 240 x^{12} + 242 x^{11} + 22134 x^{10} - 6820 x^{9} - 974680 x^{8} - 50130 x^{7} + \cdots - 43494224 $ x^14 - 2x^13 - 240x^12 + 242x^11 + 22134x^10 - 6820x^9 - 974680x^8 - 50130x^7 + 20874680x^6 + 929710x^5 - 195412686x^4 + 63905452x^3 + 546613815x^2 - 405237652*x - 43494224
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.60496$ of defining polynomial
Character	$\chi$	$=$	2500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.60496 q^{3} +17.7526 q^{7} +47.0453 q^{9} +O(q^{10})$	$q-8.60496 q^{3} +17.7526 q^{7} +47.0453 q^{9} +29.0132 q^{11} +54.6210 q^{13} +68.3775 q^{17} -53.7672 q^{19} -152.760 q^{21} -166.675 q^{23} -172.489 q^{27} -255.204 q^{29} +206.653 q^{31} -249.657 q^{33} -243.699 q^{37} -470.011 q^{39} -47.2594 q^{41} -35.8364 q^{43} +467.871 q^{47} -27.8456 q^{49} -588.385 q^{51} -673.560 q^{53} +462.665 q^{57} -330.452 q^{59} -396.058 q^{61} +835.175 q^{63} +356.450 q^{67} +1434.23 q^{69} -337.376 q^{71} +727.833 q^{73} +515.059 q^{77} -1271.45 q^{79} +214.034 q^{81} +64.3165 q^{83} +2196.02 q^{87} -698.706 q^{89} +969.664 q^{91} -1778.24 q^{93} +964.411 q^{97} +1364.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} + 8 q^{7} + 106 q^{9}+O(q^{10}) $ 14 * q - 2 * q^3 + 8 * q^7 + 106 * q^9	$ 14 q - 2 q^{3} + 8 q^{7} + 106 q^{9} - 30 q^{11} + 77 q^{13} + 111 q^{17} - 64 q^{19} - 156 q^{21} - 24 q^{23} - 614 q^{27} - 359 q^{29} - 156 q^{31} - 390 q^{33} - 307 q^{37} - 314 q^{39} - 791 q^{41} - 90 q^{43} + 276 q^{47} + 72 q^{49} - 832 q^{51} - 451 q^{53} + 192 q^{57} - 808 q^{59} - 1017 q^{61} + 2082 q^{63} + 1678 q^{67} - 1502 q^{69} - 148 q^{71} - 223 q^{73} - 1210 q^{77} - 92 q^{79} - 414 q^{81} - 474 q^{83} + 4262 q^{87} - 2111 q^{89} - 794 q^{91} - 2832 q^{93} + 353 q^{97} - 1050 q^{99}+O(q^{100}) $ 14 * q - 2 * q^3 + 8 * q^7 + 106 * q^9 - 30 * q^11 + 77 * q^13 + 111 * q^17 - 64 * q^19 - 156 * q^21 - 24 * q^23 - 614 * q^27 - 359 * q^29 - 156 * q^31 - 390 * q^33 - 307 * q^37 - 314 * q^39 - 791 * q^41 - 90 * q^43 + 276 * q^47 + 72 * q^49 - 832 * q^51 - 451 * q^53 + 192 * q^57 - 808 * q^59 - 1017 * q^61 + 2082 * q^63 + 1678 * q^67 - 1502 * q^69 - 148 * q^71 - 223 * q^73 - 1210 * q^77 - 92 * q^79 - 414 * q^81 - 474 * q^83 + 4262 * q^87 - 2111 * q^89 - 794 * q^91 - 2832 * q^93 + 353 * q^97 - 1050 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−8.60496	−1.65602	−0.828012	−	0.560710i	$-0.810528\pi$
$3$	−8.60496	−1.65602	−0.828012	+	0.560710i	$0.810528\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	17.7526	0.958550	0.479275	−	0.877665i	$-0.340900\pi$
$7$	17.7526	0.958550	0.479275	+	0.877665i	$0.340900\pi$
$8$	0	0
$9$	47.0453	1.74242
$10$	0	0
$11$	29.0132	0.795254	0.397627	−	0.917547i	$-0.369834\pi$
$11$	29.0132	0.795254	0.397627	+	0.917547i	$0.369834\pi$
$12$	0	0
$13$	54.6210	1.16532	0.582659	−	0.812717i	$-0.302012\pi$
$13$	54.6210	1.16532	0.582659	+	0.812717i	$0.302012\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	68.3775	0.975528	0.487764	−	0.872975i	$-0.337813\pi$
$17$	68.3775	0.975528	0.487764	+	0.872975i	$0.337813\pi$
$18$	0	0
$19$	−53.7672	−0.649213	−0.324607	−	0.945849i	$-0.605232\pi$
$19$	−53.7672	−0.649213	−0.324607	+	0.945849i	$0.605232\pi$
$20$	0	0
$21$	−152.760	−1.58738
$22$	0	0
$23$	−166.675	−1.51105	−0.755525	−	0.655119i	$-0.772618\pi$
$23$	−166.675	−1.51105	−0.755525	+	0.655119i	$0.772618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−172.489	−1.22946
$28$	0	0
$29$	−255.204	−1.63414	−0.817071	−	0.576537i	$-0.804404\pi$
$29$	−255.204	−1.63414	−0.817071	+	0.576537i	$0.804404\pi$
$30$	0	0
$31$	206.653	1.19729	0.598644	−	0.801015i	$-0.295706\pi$
$31$	206.653	1.19729	0.598644	+	0.801015i	$0.295706\pi$
$32$	0	0
$33$	−249.657	−1.31696
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−243.699	−1.08281	−0.541405	−	0.840762i	$-0.682107\pi$
$37$	−243.699	−1.08281	−0.541405	+	0.840762i	$0.682107\pi$
$38$	0	0
$39$	−470.011	−1.92979
$40$	0	0
$41$	−47.2594	−0.180017	−0.0900083	−	0.995941i	$-0.528689\pi$
$41$	−47.2594	−0.180017	−0.0900083	+	0.995941i	$0.528689\pi$
$42$	0	0
$43$	−35.8364	−0.127093	−0.0635464	−	0.997979i	$-0.520241\pi$
$43$	−35.8364	−0.127093	−0.0635464	+	0.997979i	$0.520241\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	467.871	1.45204	0.726022	−	0.687672i	$-0.241367\pi$
$47$	467.871	1.45204	0.726022	+	0.687672i	$0.241367\pi$
$48$	0	0
$49$	−27.8456	−0.0811824
$50$	0	0
$51$	−588.385	−1.61550
$52$	0	0
$53$	−673.560	−1.74567	−0.872836	−	0.488013i	$-0.837722\pi$
$53$	−673.560	−1.74567	−0.872836	+	0.488013i	$0.837722\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	462.665	1.07511
$58$	0	0
$59$	−330.452	−0.729172	−0.364586	−	0.931170i	$-0.618790\pi$
$59$	−330.452	−0.729172	−0.364586	+	0.931170i	$0.618790\pi$
$60$	0	0
$61$	−396.058	−0.831312	−0.415656	−	0.909522i	$-0.636448\pi$
$61$	−396.058	−0.831312	−0.415656	+	0.909522i	$0.636448\pi$
$62$	0	0
$63$	835.175	1.67019
$64$	0	0
$65$	0	0
$66$	0	0
$67$	356.450	0.649960	0.324980	−	0.945721i	$-0.394642\pi$
$67$	356.450	0.649960	0.324980	+	0.945721i	$0.394642\pi$
$68$	0	0
$69$	1434.23	2.50234
$70$	0	0
$71$	−337.376	−0.563931	−0.281966	−	0.959424i	$-0.590986\pi$
$71$	−337.376	−0.563931	−0.281966	+	0.959424i	$0.590986\pi$
$72$	0	0
$73$	727.833	1.16694	0.583468	−	0.812136i	$-0.301695\pi$
$73$	727.833	1.16694	0.583468	+	0.812136i	$0.301695\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	515.059	0.762291
$78$	0	0
$79$	−1271.45	−1.81076	−0.905378	−	0.424606i	$-0.860413\pi$
$79$	−1271.45	−1.81076	−0.905378	+	0.424606i	$0.860413\pi$
$80$	0	0
$81$	214.034	0.293599
$82$	0	0
$83$	64.3165	0.0850561	0.0425281	−	0.999095i	$-0.486459\pi$
$83$	64.3165	0.0850561	0.0425281	+	0.999095i	$0.486459\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2196.02	2.70618
$88$	0	0
$89$	−698.706	−0.832165	−0.416083	−	0.909327i	$-0.636597\pi$
$89$	−698.706	−0.832165	−0.416083	+	0.909327i	$0.636597\pi$
$90$	0	0
$91$	969.664	1.11702
$92$	0	0
$93$	−1778.24	−1.98274
$94$	0	0
$95$	0	0
$96$	0	0
$97$	964.411	1.00950	0.504748	−	0.863267i	$-0.331585\pi$
$97$	964.411	1.00950	0.504748	+	0.863267i	$0.331585\pi$
$98$	0	0
$99$	1364.93	1.38566
$100$	0	0
$101$	−45.8510	−0.0451718	−0.0225859	−	0.999745i	$-0.507190\pi$
$101$	−45.8510	−0.0451718	−0.0225859	+	0.999745i	$0.507190\pi$
$102$	0	0
$103$	1138.01	1.08866	0.544328	−	0.838872i	$-0.316785\pi$
$103$	1138.01	1.08866	0.544328	+	0.838872i	$0.316785\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1828.84	1.65234	0.826169	−	0.563422i	$-0.190516\pi$
$107$	1828.84	1.65234	0.826169	+	0.563422i	$0.190516\pi$
$108$	0	0
$109$	2100.88	1.84613	0.923063	−	0.384648i	$-0.125677\pi$
$109$	2100.88	1.84613	0.923063	+	0.384648i	$0.125677\pi$
$110$	0	0
$111$	2097.02	1.79316
$112$	0	0
$113$	−1495.32	−1.24485	−0.622424	−	0.782680i	$-0.713852\pi$
$113$	−1495.32	−1.24485	−0.622424	+	0.782680i	$0.713852\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2569.66	2.03047
$118$	0	0
$119$	1213.88	0.935092
$120$	0	0
$121$	−489.237	−0.367571
$122$	0	0
$123$	406.665	0.298112
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2027.78	−1.41682	−0.708411	−	0.705800i	$-0.750588\pi$
$127$	−2027.78	−1.41682	−0.708411	+	0.705800i	$0.750588\pi$
$128$	0	0
$129$	308.370	0.210469
$130$	0	0
$131$	−346.097	−0.230830	−0.115415	−	0.993317i	$-0.536820\pi$
$131$	−346.097	−0.230830	−0.115415	+	0.993317i	$0.536820\pi$
$132$	0	0
$133$	−954.508	−0.622303
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−754.724	−0.470660	−0.235330	−	0.971916i	$-0.575617\pi$
$137$	−754.724	−0.470660	−0.235330	+	0.971916i	$0.575617\pi$
$138$	0	0
$139$	−2002.00	−1.22164	−0.610818	−	0.791771i	$-0.709159\pi$
$139$	−2002.00	−1.22164	−0.610818	+	0.791771i	$0.709159\pi$
$140$	0	0
$141$	−4026.01	−2.40462
$142$	0	0
$143$	1584.73	0.926724
$144$	0	0
$145$	0	0
$146$	0	0
$147$	239.610	0.134440
$148$	0	0
$149$	−1066.44	−0.586348	−0.293174	−	0.956059i	$-0.594711\pi$
$149$	−1066.44	−0.586348	−0.293174	+	0.956059i	$0.594711\pi$
$150$	0	0
$151$	763.019	0.411216	0.205608	−	0.978634i	$-0.434083\pi$
$151$	763.019	0.411216	0.205608	+	0.978634i	$0.434083\pi$
$152$	0	0
$153$	3216.84	1.69978
$154$	0	0
$155$	0	0
$156$	0	0
$157$	62.3483	0.0316939	0.0158469	−	0.999874i	$-0.494956\pi$
$157$	62.3483	0.0316939	0.0158469	+	0.999874i	$0.494956\pi$
$158$	0	0
$159$	5795.96	2.89088
$160$	0	0
$161$	−2958.91	−1.44842
$162$	0	0
$163$	−361.924	−0.173914	−0.0869572	−	0.996212i	$-0.527714\pi$
$163$	−361.924	−0.173914	−0.0869572	+	0.996212i	$0.527714\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1454.08	0.673772	0.336886	−	0.941545i	$-0.390626\pi$
$167$	1454.08	0.673772	0.336886	+	0.941545i	$0.390626\pi$
$168$	0	0
$169$	786.451	0.357966
$170$	0	0
$171$	−2529.49	−1.13120
$172$	0	0
$173$	−2022.97	−0.889039	−0.444519	−	0.895769i	$-0.646626\pi$
$173$	−2022.97	−0.889039	−0.444519	+	0.895769i	$0.646626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2843.52	1.20753
$178$	0	0
$179$	−1274.30	−0.532097	−0.266048	−	0.963960i	$-0.585718\pi$
$179$	−1274.30	−0.532097	−0.266048	+	0.963960i	$0.585718\pi$
$180$	0	0
$181$	−4308.75	−1.76943	−0.884716	−	0.466131i	$-0.845647\pi$
$181$	−4308.75	−1.76943	−0.884716	+	0.466131i	$0.845647\pi$
$182$	0	0
$183$	3408.06	1.37667
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1983.85	0.775793
$188$	0	0
$189$	−3062.12	−1.17850
$190$	0	0
$191$	−311.274	−0.117921	−0.0589607	−	0.998260i	$-0.518779\pi$
$191$	−311.274	−0.117921	−0.0589607	+	0.998260i	$0.518779\pi$
$192$	0	0
$193$	2675.32	0.997790	0.498895	−	0.866662i	$-0.333739\pi$
$193$	2675.32	0.997790	0.498895	+	0.866662i	$0.333739\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	926.916	0.335229	0.167614	−	0.985853i	$-0.446394\pi$
$197$	926.916	0.335229	0.167614	+	0.985853i	$0.446394\pi$
$198$	0	0
$199$	−444.199	−0.158233	−0.0791166	−	0.996865i	$-0.525210\pi$
$199$	−444.199	−0.158233	−0.0791166	+	0.996865i	$0.525210\pi$
$200$	0	0
$201$	−3067.24	−1.07635
$202$	0	0
$203$	−4530.53	−1.56641
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7841.27	−2.63288
$208$	0	0
$209$	−1559.96	−0.516289
$210$	0	0
$211$	−970.577	−0.316670	−0.158335	−	0.987385i	$-0.550613\pi$
$211$	−970.577	−0.316670	−0.158335	+	0.987385i	$0.550613\pi$
$212$	0	0
$213$	2903.10	0.933884
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3668.62	1.14766
$218$	0	0
$219$	−6262.97	−1.93248
$220$	0	0
$221$	3734.85	1.13680
$222$	0	0
$223$	3780.83	1.13535	0.567675	−	0.823253i	$-0.307843\pi$
$223$	3780.83	1.13535	0.567675	+	0.823253i	$0.307843\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4528.45	−1.32407	−0.662035	−	0.749473i	$-0.730307\pi$
$227$	−4528.45	−1.32407	−0.662035	+	0.749473i	$0.730307\pi$
$228$	0	0
$229$	−4077.52	−1.17664	−0.588319	−	0.808629i	$-0.700210\pi$
$229$	−4077.52	−1.17664	−0.588319	+	0.808629i	$0.700210\pi$
$230$	0	0
$231$	−4432.06	−1.26237
$232$	0	0
$233$	−667.397	−0.187651	−0.0938254	−	0.995589i	$-0.529910\pi$
$233$	−667.397	−0.187651	−0.0938254	+	0.995589i	$0.529910\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10940.8	2.99866
$238$	0	0
$239$	3450.29	0.933810	0.466905	−	0.884308i	$-0.345369\pi$
$239$	3450.29	0.933810	0.466905	+	0.884308i	$0.345369\pi$
$240$	0	0
$241$	2231.68	0.596495	0.298247	−	0.954489i	$-0.403598\pi$
$241$	2231.68	0.596495	0.298247	+	0.954489i	$0.403598\pi$
$242$	0	0
$243$	2815.44	0.743252
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2936.82	−0.756540
$248$	0	0
$249$	−553.441	−0.140855
$250$	0	0
$251$	433.464	0.109004	0.0545021	−	0.998514i	$-0.482643\pi$
$251$	433.464	0.109004	0.0545021	+	0.998514i	$0.482643\pi$
$252$	0	0
$253$	−4835.77	−1.20167
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5696.10	−1.38254	−0.691270	−	0.722596i	$-0.742949\pi$
$257$	−5696.10	−1.38254	−0.691270	+	0.722596i	$0.742949\pi$
$258$	0	0
$259$	−4326.30	−1.03793
$260$	0	0
$261$	−12006.1	−2.84736
$262$	0	0
$263$	−3660.48	−0.858230	−0.429115	−	0.903250i	$-0.641175\pi$
$263$	−3660.48	−0.858230	−0.429115	+	0.903250i	$0.641175\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6012.33	1.37809
$268$	0	0
$269$	2546.02	0.577077	0.288539	−	0.957468i	$-0.406831\pi$
$269$	2546.02	0.577077	0.288539	+	0.957468i	$0.406831\pi$
$270$	0	0
$271$	4699.09	1.05332	0.526659	−	0.850076i	$-0.323444\pi$
$271$	4699.09	1.05332	0.526659	+	0.850076i	$0.323444\pi$
$272$	0	0
$273$	−8343.91	−1.84980
$274$	0	0
$275$	0	0
$276$	0	0
$277$	66.9402	0.0145200	0.00726001	−	0.999974i	$-0.497689\pi$
$277$	66.9402	0.0145200	0.00726001	+	0.999974i	$0.497689\pi$
$278$	0	0
$279$	9722.04	2.08618
$280$	0	0
$281$	4732.03	1.00459	0.502294	−	0.864697i	$-0.332490\pi$
$281$	4732.03	1.00459	0.502294	+	0.864697i	$0.332490\pi$
$282$	0	0
$283$	3747.57	0.787172	0.393586	−	0.919288i	$-0.371234\pi$
$283$	3747.57	0.787172	0.393586	+	0.919288i	$0.371234\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−838.977	−0.172555
$288$	0	0
$289$	−237.518	−0.0483447
$290$	0	0
$291$	−8298.71	−1.67175
$292$	0	0
$293$	−1309.24	−0.261046	−0.130523	−	0.991445i	$-0.541666\pi$
$293$	−1309.24	−0.261046	−0.130523	+	0.991445i	$0.541666\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5004.44	−0.977733
$298$	0	0
$299$	−9103.96	−1.76085
$300$	0	0
$301$	−636.188	−0.121825
$302$	0	0
$303$	394.546	0.0748055
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5371.12	0.998521	0.499260	−	0.866452i	$-0.333605\pi$
$307$	5371.12	0.998521	0.499260	+	0.866452i	$0.333605\pi$
$308$	0	0
$309$	−9792.54	−1.80284
$310$	0	0
$311$	3728.99	0.679909	0.339954	−	0.940442i	$-0.389588\pi$
$311$	3728.99	0.679909	0.339954	+	0.940442i	$0.389588\pi$
$312$	0	0
$313$	−2914.42	−0.526302	−0.263151	−	0.964755i	$-0.584762\pi$
$313$	−2914.42	−0.526302	−0.263151	+	0.964755i	$0.584762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3617.62	−0.640964	−0.320482	−	0.947255i	$-0.603845\pi$
$317$	−3617.62	−0.640964	−0.320482	+	0.947255i	$0.603845\pi$
$318$	0	0
$319$	−7404.26	−1.29956
$320$	0	0
$321$	−15737.0	−2.73631
$322$	0	0
$323$	−3676.47	−0.633326
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18078.0	−3.05723
$328$	0	0
$329$	8305.93	1.39186
$330$	0	0
$331$	−10486.5	−1.74136	−0.870680	−	0.491850i	$-0.836321\pi$
$331$	−10486.5	−1.74136	−0.870680	+	0.491850i	$0.836321\pi$
$332$	0	0
$333$	−11464.9	−1.88670
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9342.60	−1.51016	−0.755080	−	0.655633i	$-0.772402\pi$
$337$	−9342.60	−1.51016	−0.755080	+	0.655633i	$0.772402\pi$
$338$	0	0
$339$	12867.2	2.06150
$340$	0	0
$341$	5995.65	0.952149
$342$	0	0
$343$	−6583.47	−1.03637
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1151.32	−0.178116	−0.0890579	−	0.996026i	$-0.528386\pi$
$347$	−1151.32	−0.178116	−0.0890579	+	0.996026i	$0.528386\pi$
$348$	0	0
$349$	8519.22	1.30666	0.653329	−	0.757074i	$-0.273372\pi$
$349$	8519.22	1.30666	0.653329	+	0.757074i	$0.273372\pi$
$350$	0	0
$351$	−9421.49	−1.43271
$352$	0	0
$353$	6377.84	0.961638	0.480819	−	0.876820i	$-0.340339\pi$
$353$	6377.84	0.961638	0.480819	+	0.876820i	$0.340339\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−10445.4	−1.54854
$358$	0	0
$359$	6066.25	0.891823	0.445912	−	0.895077i	$-0.352880\pi$
$359$	6066.25	0.891823	0.445912	+	0.895077i	$0.352880\pi$
$360$	0	0
$361$	−3968.09	−0.578522
$362$	0	0
$363$	4209.86	0.608707
$364$	0	0
$365$	0	0
$366$	0	0
$367$	310.224	0.0441242	0.0220621	−	0.999757i	$-0.492977\pi$
$367$	310.224	0.0441242	0.0220621	+	0.999757i	$0.492977\pi$
$368$	0	0
$369$	−2223.33	−0.313664
$370$	0	0
$371$	−11957.4	−1.67331
$372$	0	0
$373$	4388.68	0.609215	0.304607	−	0.952478i	$-0.401475\pi$
$373$	4388.68	0.609215	0.304607	+	0.952478i	$0.401475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13939.5	−1.90430
$378$	0	0
$379$	1612.66	0.218567	0.109283	−	0.994011i	$-0.465144\pi$
$379$	1612.66	0.218567	0.109283	+	0.994011i	$0.465144\pi$
$380$	0	0
$381$	17449.0	2.34629
$382$	0	0
$383$	5025.58	0.670483	0.335241	−	0.942132i	$-0.391182\pi$
$383$	5025.58	0.670483	0.335241	+	0.942132i	$0.391182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1685.93	−0.221449
$388$	0	0
$389$	11390.5	1.48463	0.742317	−	0.670049i	$-0.233727\pi$
$389$	11390.5	1.48463	0.742317	+	0.670049i	$0.233727\pi$
$390$	0	0
$391$	−11396.8	−1.47407
$392$	0	0
$393$	2978.15	0.382259
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7960.36	−1.00635	−0.503173	−	0.864186i	$-0.667834\pi$
$397$	−7960.36	−1.00635	−0.503173	+	0.864186i	$0.667834\pi$
$398$	0	0
$399$	8213.49	1.03055
$400$	0	0
$401$	−7624.87	−0.949546	−0.474773	−	0.880108i	$-0.657470\pi$
$401$	−7624.87	−0.949546	−0.474773	+	0.880108i	$0.657470\pi$
$402$	0	0
$403$	11287.6	1.39522
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7070.49	−0.861108
$408$	0	0
$409$	−10497.1	−1.26906	−0.634532	−	0.772897i	$-0.718807\pi$
$409$	−10497.1	−1.26906	−0.634532	+	0.772897i	$0.718807\pi$
$410$	0	0
$411$	6494.37	0.779425
$412$	0	0
$413$	−5866.38	−0.698948
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17227.1	2.02306
$418$	0	0
$419$	4135.74	0.482206	0.241103	−	0.970500i	$-0.422491\pi$
$419$	4135.74	0.482206	0.241103	+	0.970500i	$0.422491\pi$
$420$	0	0
$421$	6173.41	0.714664	0.357332	−	0.933978i	$-0.383687\pi$
$421$	6173.41	0.714664	0.357332	+	0.933978i	$0.383687\pi$
$422$	0	0
$423$	22011.1	2.53006
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7031.05	−0.796853
$428$	0	0
$429$	−13636.5	−1.53468
$430$	0	0
$431$	−11202.7	−1.25201	−0.626004	−	0.779820i	$-0.715310\pi$
$431$	−11202.7	−1.25201	−0.626004	+	0.779820i	$0.715310\pi$
$432$	0	0
$433$	−13542.0	−1.50297	−0.751486	−	0.659749i	$-0.770662\pi$
$433$	−13542.0	−1.50297	−0.751486	+	0.659749i	$0.770662\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8961.66	0.980994
$438$	0	0
$439$	−2167.31	−0.235626	−0.117813	−	0.993036i	$-0.537588\pi$
$439$	−2167.31	−0.235626	−0.117813	+	0.993036i	$0.537588\pi$
$440$	0	0
$441$	−1310.00	−0.141454
$442$	0	0
$443$	−13212.9	−1.41708	−0.708539	−	0.705672i	$-0.750645\pi$
$443$	−13212.9	−1.41708	−0.708539	+	0.705672i	$0.750645\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9176.63	0.971006
$448$	0	0
$449$	−1921.34	−0.201946	−0.100973	−	0.994889i	$-0.532195\pi$
$449$	−1921.34	−0.201946	−0.100973	+	0.994889i	$0.532195\pi$
$450$	0	0
$451$	−1371.14	−0.143159
$452$	0	0
$453$	−6565.74	−0.680983
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−117.313	−0.0120080	−0.00600399	−	0.999982i	$-0.501911\pi$
$457$	−117.313	−0.0120080	−0.00600399	+	0.999982i	$0.501911\pi$
$458$	0	0
$459$	−11794.3	−1.19937
$460$	0	0
$461$	−9636.20	−0.973541	−0.486771	−	0.873530i	$-0.661825\pi$
$461$	−9636.20	−0.973541	−0.486771	+	0.873530i	$0.661825\pi$
$462$	0	0
$463$	6344.08	0.636791	0.318396	−	0.947958i	$-0.396856\pi$
$463$	6344.08	0.636791	0.318396	+	0.947958i	$0.396856\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10606.5	−1.05099	−0.525494	−	0.850798i	$-0.676119\pi$
$467$	−10606.5	−1.05099	−0.525494	+	0.850798i	$0.676119\pi$
$468$	0	0
$469$	6327.92	0.623019
$470$	0	0
$471$	−536.504	−0.0524858
$472$	0	0
$473$	−1039.73	−0.101071
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−31687.8	−3.04169
$478$	0	0
$479$	8317.33	0.793379	0.396690	−	0.917953i	$-0.370159\pi$
$479$	8317.33	0.793379	0.396690	+	0.917953i	$0.370159\pi$
$480$	0	0
$481$	−13311.1	−1.26182
$482$	0	0
$483$	25461.3	2.39861
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2232.13	−0.207695	−0.103848	−	0.994593i	$-0.533115\pi$
$487$	−2232.13	−0.207695	−0.103848	+	0.994593i	$0.533115\pi$
$488$	0	0
$489$	3114.34	0.288006
$490$	0	0
$491$	−7937.20	−0.729533	−0.364766	−	0.931099i	$-0.618851\pi$
$491$	−7937.20	−0.729533	−0.364766	+	0.931099i	$0.618851\pi$
$492$	0	0
$493$	−17450.2	−1.59415
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5989.29	−0.540556
$498$	0	0
$499$	16900.0	1.51612	0.758062	−	0.652182i	$-0.226146\pi$
$499$	16900.0	1.51612	0.758062	+	0.652182i	$0.226146\pi$
$500$	0	0
$501$	−12512.3	−1.11578
$502$	0	0
$503$	−261.773	−0.0232045	−0.0116023	−	0.999933i	$-0.503693\pi$
$503$	−261.773	−0.0232045	−0.0116023	+	0.999933i	$0.503693\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6767.37	−0.592800
$508$	0	0
$509$	−19481.9	−1.69650	−0.848252	−	0.529592i	$-0.822345\pi$
$509$	−19481.9	−1.69650	−0.848252	+	0.529592i	$0.822345\pi$
$510$	0	0
$511$	12920.9	1.11857
$512$	0	0
$513$	9274.23	0.798182
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13574.4	1.15474
$518$	0	0
$519$	17407.6	1.47227
$520$	0	0
$521$	−12256.2	−1.03062	−0.515311	−	0.857003i	$-0.672324\pi$
$521$	−12256.2	−1.03062	−0.515311	+	0.857003i	$0.672324\pi$
$522$	0	0
$523$	−11826.5	−0.988785	−0.494393	−	0.869239i	$-0.664610\pi$
$523$	−11826.5	−0.988785	−0.494393	+	0.869239i	$0.664610\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14130.4	1.16799
$528$	0	0
$529$	15613.6	1.28327
$530$	0	0
$531$	−15546.2	−1.27052
$532$	0	0
$533$	−2581.35	−0.209777
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10965.3	0.881165
$538$	0	0
$539$	−807.887	−0.0645606
$540$	0	0
$541$	834.380	0.0663083	0.0331542	−	0.999450i	$-0.489445\pi$
$541$	834.380	0.0663083	0.0331542	+	0.999450i	$0.489445\pi$
$542$	0	0
$543$	37076.6	2.93022
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12878.5	1.00667	0.503333	−	0.864093i	$-0.332107\pi$
$547$	12878.5	1.00667	0.503333	+	0.864093i	$0.332107\pi$
$548$	0	0
$549$	−18632.6	−1.44849
$550$	0	0
$551$	13721.6	1.06091
$552$	0	0
$553$	−22571.6	−1.73570
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10133.4	−0.770852	−0.385426	−	0.922739i	$-0.625945\pi$
$557$	−10133.4	−0.770852	−0.385426	+	0.922739i	$0.625945\pi$
$558$	0	0
$559$	−1957.42	−0.148104
$560$	0	0
$561$	−17070.9	−1.28473
$562$	0	0
$563$	19342.4	1.44793	0.723966	−	0.689836i	$-0.242317\pi$
$563$	19342.4	1.44793	0.723966	+	0.689836i	$0.242317\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3799.66	0.281430
$568$	0	0
$569$	−17040.8	−1.25551	−0.627757	−	0.778409i	$-0.716027\pi$
$569$	−17040.8	−1.25551	−0.627757	+	0.778409i	$0.716027\pi$
$570$	0	0
$571$	−19073.4	−1.39790	−0.698948	−	0.715172i	$-0.746348\pi$
$571$	−19073.4	−1.39790	−0.698948	+	0.715172i	$0.746348\pi$
$572$	0	0
$573$	2678.50	0.195281
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9637.82	0.695369	0.347684	−	0.937612i	$-0.386968\pi$
$577$	9637.82	0.695369	0.347684	+	0.937612i	$0.386968\pi$
$578$	0	0
$579$	−23021.0	−1.65237
$580$	0	0
$581$	1141.79	0.0815305
$582$	0	0
$583$	−19542.1	−1.38825
$584$	0	0
$585$	0	0
$586$	0	0
$587$	70.7132	0.00497214	0.00248607	−	0.999997i	$-0.499209\pi$
$587$	70.7132	0.00497214	0.00248607	+	0.999997i	$0.499209\pi$
$588$	0	0
$589$	−11111.2	−0.777296
$590$	0	0
$591$	−7976.07	−0.555147
$592$	0	0
$593$	10023.6	0.694134	0.347067	−	0.937840i	$-0.387178\pi$
$593$	10023.6	0.694134	0.347067	+	0.937840i	$0.387178\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3822.31	0.262038
$598$	0	0
$599$	−20888.2	−1.42483	−0.712413	−	0.701761i	$-0.752398\pi$
$599$	−20888.2	−1.42483	−0.712413	+	0.701761i	$0.752398\pi$
$600$	0	0
$601$	15619.9	1.06015	0.530073	−	0.847952i	$-0.322165\pi$
$601$	15619.9	1.06015	0.530073	+	0.847952i	$0.322165\pi$
$602$	0	0
$603$	16769.3	1.13250
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12370.5	0.827190	0.413595	−	0.910461i	$-0.364273\pi$
$607$	12370.5	0.827190	0.413595	+	0.910461i	$0.364273\pi$
$608$	0	0
$609$	38985.0	2.59401
$610$	0	0
$611$	25555.6	1.69209
$612$	0	0
$613$	2348.65	0.154749	0.0773744	−	0.997002i	$-0.475346\pi$
$613$	2348.65	0.154749	0.0773744	+	0.997002i	$0.475346\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4946.93	0.322781	0.161391	−	0.986891i	$-0.448402\pi$
$617$	4946.93	0.322781	0.161391	+	0.986891i	$0.448402\pi$
$618$	0	0
$619$	−12158.0	−0.789454	−0.394727	−	0.918798i	$-0.629161\pi$
$619$	−12158.0	−0.789454	−0.394727	+	0.918798i	$0.629161\pi$
$620$	0	0
$621$	28749.5	1.85778
$622$	0	0
$623$	−12403.8	−0.797672
$624$	0	0
$625$	0	0
$626$	0	0
$627$	13423.4	0.854988
$628$	0	0
$629$	−16663.6	−1.05631
$630$	0	0
$631$	−1844.52	−0.116370	−0.0581849	−	0.998306i	$-0.518531\pi$
$631$	−1844.52	−0.116370	−0.0581849	+	0.998306i	$0.518531\pi$
$632$	0	0
$633$	8351.77	0.524413
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1520.95	−0.0946033
$638$	0	0
$639$	−15871.9	−0.982603
$640$	0	0
$641$	−13126.5	−0.808840	−0.404420	−	0.914573i	$-0.632527\pi$
$641$	−13126.5	−0.808840	−0.404420	+	0.914573i	$0.632527\pi$
$642$	0	0
$643$	−16752.9	−1.02748	−0.513740	−	0.857946i	$-0.671740\pi$
$643$	−16752.9	−1.02748	−0.513740	+	0.857946i	$0.671740\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7795.40	−0.473677	−0.236838	−	0.971549i	$-0.576111\pi$
$647$	−7795.40	−0.473677	−0.236838	+	0.971549i	$0.576111\pi$
$648$	0	0
$649$	−9587.45	−0.579877
$650$	0	0
$651$	−31568.3	−1.90055
$652$	0	0
$653$	−12057.1	−0.722559	−0.361279	−	0.932458i	$-0.617660\pi$
$653$	−12057.1	−0.722559	−0.361279	+	0.932458i	$0.617660\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	34241.1	2.03329
$658$	0	0
$659$	22121.7	1.30765	0.653823	−	0.756648i	$-0.273164\pi$
$659$	22121.7	1.30765	0.653823	+	0.756648i	$0.273164\pi$
$660$	0	0
$661$	−6977.45	−0.410577	−0.205289	−	0.978701i	$-0.565813\pi$
$661$	−6977.45	−0.410577	−0.205289	+	0.978701i	$0.565813\pi$
$662$	0	0
$663$	−32138.2	−1.88257
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42536.1	2.46927
$668$	0	0
$669$	−32533.8	−1.88017
$670$	0	0
$671$	−11490.9	−0.661104
$672$	0	0
$673$	−5466.88	−0.313125	−0.156562	−	0.987668i	$-0.550041\pi$
$673$	−5466.88	−0.313125	−0.156562	+	0.987668i	$0.550041\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1620.27	−0.0919824	−0.0459912	−	0.998942i	$-0.514645\pi$
$677$	−1620.27	−0.0919824	−0.0459912	+	0.998942i	$0.514645\pi$
$678$	0	0
$679$	17120.8	0.967652
$680$	0	0
$681$	38967.1	2.19269
$682$	0	0
$683$	2295.59	0.128607	0.0643033	−	0.997930i	$-0.479517\pi$
$683$	2295.59	0.128607	0.0643033	+	0.997930i	$0.479517\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	35086.9	1.94854
$688$	0	0
$689$	−36790.5	−2.03426
$690$	0	0
$691$	25221.2	1.38851	0.694254	−	0.719730i	$-0.255734\pi$
$691$	25221.2	1.38851	0.694254	+	0.719730i	$0.255734\pi$
$692$	0	0
$693$	24231.1	1.32823
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3231.48	−0.175611
$698$	0	0
$699$	5742.92	0.310754
$700$	0	0
$701$	9744.88	0.525049	0.262524	−	0.964925i	$-0.415445\pi$
$701$	9744.88	0.525049	0.262524	+	0.964925i	$0.415445\pi$
$702$	0	0
$703$	13103.0	0.702974
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−813.974	−0.0432994
$708$	0	0
$709$	32322.3	1.71212	0.856058	−	0.516879i	$-0.172906\pi$
$709$	32322.3	1.71212	0.856058	+	0.516879i	$0.172906\pi$
$710$	0	0
$711$	−59815.9	−3.15509
$712$	0	0
$713$	−34443.9	−1.80916
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−29689.6	−1.54641
$718$	0	0
$719$	37725.7	1.95679	0.978394	−	0.206749i	$-0.0662883\pi$
$719$	37725.7	1.95679	0.978394	+	0.206749i	$0.0662883\pi$
$720$	0	0
$721$	20202.7	1.04353
$722$	0	0
$723$	−19203.5	−0.987810
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17536.2	0.894612	0.447306	−	0.894381i	$-0.352384\pi$
$727$	17536.2	0.894612	0.447306	+	0.894381i	$0.352384\pi$
$728$	0	0
$729$	−30005.6	−1.52444
$730$	0	0
$731$	−2450.40	−0.123983
$732$	0	0
$733$	−12960.3	−0.653070	−0.326535	−	0.945185i	$-0.605881\pi$
$733$	−12960.3	−0.653070	−0.326535	+	0.945185i	$0.605881\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10341.7	0.516883
$738$	0	0
$739$	−18232.7	−0.907576	−0.453788	−	0.891110i	$-0.649928\pi$
$739$	−18232.7	−0.907576	−0.453788	+	0.891110i	$0.649928\pi$
$740$	0	0
$741$	25271.2	1.25285
$742$	0	0
$743$	11715.5	0.578464	0.289232	−	0.957259i	$-0.406600\pi$
$743$	11715.5	0.578464	0.289232	+	0.957259i	$0.406600\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3025.79	0.148203
$748$	0	0
$749$	32466.6	1.58385
$750$	0	0
$751$	37841.6	1.83870	0.919348	−	0.393444i	$-0.128717\pi$
$751$	37841.6	1.83870	0.919348	+	0.393444i	$0.128717\pi$
$752$	0	0
$753$	−3729.94	−0.180513
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19408.2	−0.931841	−0.465921	−	0.884826i	$-0.654277\pi$
$757$	−19408.2	−0.931841	−0.465921	+	0.884826i	$0.654277\pi$
$758$	0	0
$759$	41611.6	1.98999
$760$	0	0
$761$	20386.4	0.971098	0.485549	−	0.874209i	$-0.338620\pi$
$761$	20386.4	0.971098	0.485549	+	0.874209i	$0.338620\pi$
$762$	0	0
$763$	37296.1	1.76960
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18049.6	−0.849718
$768$	0	0
$769$	29602.5	1.38816	0.694078	−	0.719899i	$-0.255812\pi$
$769$	29602.5	1.38816	0.694078	+	0.719899i	$0.255812\pi$
$770$	0	0
$771$	49014.7	2.28952
$772$	0	0
$773$	−4483.92	−0.208636	−0.104318	−	0.994544i	$-0.533266\pi$
$773$	−4483.92	−0.208636	−0.104318	+	0.994544i	$0.533266\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	37227.6	1.71883
$778$	0	0
$779$	2541.01	0.116869
$780$	0	0
$781$	−9788.33	−0.448469
$782$	0	0
$783$	44019.7	2.00911
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9467.98	0.428840	0.214420	−	0.976742i	$-0.431214\pi$
$787$	9467.98	0.428840	0.214420	+	0.976742i	$0.431214\pi$
$788$	0	0
$789$	31498.2	1.42125
$790$	0	0
$791$	−26545.8	−1.19325
$792$	0	0
$793$	−21633.1	−0.968742
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1715.34	0.0762367	0.0381183	−	0.999273i	$-0.487864\pi$
$797$	1715.34	0.0762367	0.0381183	+	0.999273i	$0.487864\pi$
$798$	0	0
$799$	31991.9	1.41651
$800$	0	0
$801$	−32870.8	−1.44998
$802$	0	0
$803$	21116.7	0.928011
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21908.4	−0.955654
$808$	0	0
$809$	2412.65	0.104851	0.0524254	−	0.998625i	$-0.483305\pi$
$809$	2412.65	0.104851	0.0524254	+	0.998625i	$0.483305\pi$
$810$	0	0
$811$	−17998.9	−0.779317	−0.389659	−	0.920959i	$-0.627407\pi$
$811$	−17998.9	−0.779317	−0.389659	+	0.920959i	$0.627407\pi$
$812$	0	0
$813$	−40435.5	−1.74432
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1926.82	0.0825104
$818$	0	0
$819$	45618.1	1.94631
$820$	0	0
$821$	−25396.4	−1.07959	−0.539793	−	0.841798i	$-0.681497\pi$
$821$	−25396.4	−1.07959	−0.539793	+	0.841798i	$0.681497\pi$
$822$	0	0
$823$	34943.8	1.48003	0.740015	−	0.672591i	$-0.234819\pi$
$823$	34943.8	1.48003	0.740015	+	0.672591i	$0.234819\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3635.92	0.152882	0.0764409	−	0.997074i	$-0.475644\pi$
$827$	3635.92	0.152882	0.0764409	+	0.997074i	$0.475644\pi$
$828$	0	0
$829$	9094.17	0.381006	0.190503	−	0.981687i	$-0.438988\pi$
$829$	9094.17	0.381006	0.190503	+	0.981687i	$0.438988\pi$
$830$	0	0
$831$	−576.017	−0.0240455
$832$	0	0
$833$	−1904.01	−0.0791957
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−35645.2	−1.47202
$838$	0	0
$839$	−1848.44	−0.0760613	−0.0380306	−	0.999277i	$-0.512108\pi$
$839$	−1848.44	−0.0760613	−0.0380306	+	0.999277i	$0.512108\pi$
$840$	0	0
$841$	40739.9	1.67042
$842$	0	0
$843$	−40718.9	−1.66362
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8685.22	−0.352335
$848$	0	0
$849$	−32247.6	−1.30358
$850$	0	0
$851$	40618.6	1.63618
$852$	0	0
$853$	−24229.5	−0.972571	−0.486286	−	0.873800i	$-0.661649\pi$
$853$	−24229.5	−0.972571	−0.486286	+	0.873800i	$0.661649\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.5262	−0.00137619	−0.000688093	−	1.00000i	$-0.500219\pi$
$857$	−34.5262	−0.00137619	−0.000688093		1.00000i	$0.500219\pi$
$858$	0	0
$859$	−16395.8	−0.651244	−0.325622	−	0.945500i	$-0.605574\pi$
$859$	−16395.8	−0.651244	−0.325622	+	0.945500i	$0.605574\pi$
$860$	0	0
$861$	7219.36	0.285755
$862$	0	0
$863$	−37070.5	−1.46222	−0.731109	−	0.682261i	$-0.760997\pi$
$863$	−37070.5	−1.46222	−0.731109	+	0.682261i	$0.760997\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2043.83	0.0800600
$868$	0	0
$869$	−36888.9	−1.44001
$870$	0	0
$871$	19469.7	0.757410
$872$	0	0
$873$	45371.0	1.75896
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29050.5	1.11855	0.559273	−	0.828984i	$-0.311081\pi$
$877$	29050.5	1.11855	0.559273	+	0.828984i	$0.311081\pi$
$878$	0	0
$879$	11265.9	0.432299
$880$	0	0
$881$	−38324.4	−1.46559	−0.732793	−	0.680452i	$-0.761784\pi$
$881$	−38324.4	−1.46559	−0.732793	+	0.680452i	$0.761784\pi$
$882$	0	0
$883$	−8029.30	−0.306011	−0.153005	−	0.988225i	$-0.548895\pi$
$883$	−8029.30	−0.306011	−0.153005	+	0.988225i	$0.548895\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4056.11	0.153541	0.0767705	−	0.997049i	$-0.475539\pi$
$887$	4056.11	0.153541	0.0767705	+	0.997049i	$0.475539\pi$
$888$	0	0
$889$	−35998.3	−1.35809
$890$	0	0
$891$	6209.80	0.233486
$892$	0	0
$893$	−25156.1	−0.942686
$894$	0	0
$895$	0	0
$896$	0	0
$897$	78339.1	2.91602
$898$	0	0
$899$	−52738.6	−1.95654
$900$	0	0
$901$	−46056.4	−1.70295
$902$	0	0
$903$	5474.37	0.201745
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30499.8	1.11657	0.558285	−	0.829649i	$-0.311460\pi$
$907$	30499.8	1.11657	0.558285	+	0.829649i	$0.311460\pi$
$908$	0	0
$909$	−2157.07	−0.0787080
$910$	0	0
$911$	32508.0	1.18226	0.591130	−	0.806576i	$-0.298682\pi$
$911$	32508.0	1.18226	0.591130	+	0.806576i	$0.298682\pi$
$912$	0	0
$913$	1866.03	0.0676412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6144.13	−0.221262
$918$	0	0
$919$	−19412.4	−0.696798	−0.348399	−	0.937346i	$-0.613275\pi$
$919$	−19412.4	−0.696798	−0.348399	+	0.937346i	$0.613275\pi$
$920$	0	0
$921$	−46218.2	−1.65358
$922$	0	0
$923$	−18427.8	−0.657159
$924$	0	0
$925$	0	0
$926$	0	0
$927$	53538.0	1.89689
$928$	0	0
$929$	−25333.1	−0.894673	−0.447336	−	0.894366i	$-0.647627\pi$
$929$	−25333.1	−0.894673	−0.447336	+	0.894366i	$0.647627\pi$
$930$	0	0
$931$	1497.18	0.0527047
$932$	0	0
$933$	−32087.8	−1.12595
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5118.91	0.178471	0.0892356	−	0.996011i	$-0.471558\pi$
$937$	5118.91	0.178471	0.0892356	+	0.996011i	$0.471558\pi$
$938$	0	0
$939$	25078.4	0.871569
$940$	0	0
$941$	−24570.3	−0.851189	−0.425595	−	0.904914i	$-0.639935\pi$
$941$	−24570.3	−0.851189	−0.425595	+	0.904914i	$0.639935\pi$
$942$	0	0
$943$	7876.97	0.272014
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7233.27	0.248204	0.124102	−	0.992269i	$-0.460395\pi$
$947$	7233.27	0.248204	0.124102	+	0.992269i	$0.460395\pi$
$948$	0	0
$949$	39754.9	1.35985
$950$	0	0
$951$	31129.4	1.06145
$952$	0	0
$953$	24004.9	0.815944	0.407972	−	0.912994i	$-0.366236\pi$
$953$	24004.9	0.815944	0.407972	+	0.912994i	$0.366236\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	63713.4	2.15210
$958$	0	0
$959$	−13398.3	−0.451151
$960$	0	0
$961$	12914.4	0.433501
$962$	0	0
$963$	86038.0	2.87906
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15780.9	−0.524796	−0.262398	−	0.964960i	$-0.584513\pi$
$967$	−15780.9	−0.524796	−0.262398	+	0.964960i	$0.584513\pi$
$968$	0	0
$969$	31635.8	1.04880
$970$	0	0
$971$	18395.0	0.607955	0.303978	−	0.952679i	$-0.401685\pi$
$971$	18395.0	0.607955	0.303978	+	0.952679i	$0.401685\pi$
$972$	0	0
$973$	−35540.7	−1.17100
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−744.780	−0.0243886	−0.0121943	−	0.999926i	$-0.503882\pi$
$977$	−744.780	−0.0243886	−0.0121943	+	0.999926i	$0.503882\pi$
$978$	0	0
$979$	−20271.7	−0.661783
$980$	0	0
$981$	98836.4	3.21672
$982$	0	0
$983$	−13427.1	−0.435664	−0.217832	−	0.975986i	$-0.569898\pi$
$983$	−13427.1	−0.435664	−0.217832	+	0.975986i	$0.569898\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−71472.1	−2.30495
$988$	0	0
$989$	5973.03	0.192044
$990$	0	0
$991$	−13106.8	−0.420133	−0.210067	−	0.977687i	$-0.567368\pi$
$991$	−13106.8	−0.420133	−0.210067	+	0.977687i	$0.567368\pi$
$992$	0	0
$993$	90235.9	2.88373
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−27333.6	−0.868270	−0.434135	−	0.900848i	$-0.642946\pi$
$997$	−27333.6	−0.868270	−0.434135	+	0.900848i	$0.642946\pi$
$998$	0	0
$999$	42035.3	1.33127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2500.4.a.c.1.2		14
5.4	even	2		2500.4.a.d.1.13		14
25.3	odd	20		500.4.i.b.49.14		56
25.4	even	10		500.4.g.a.201.7		28
25.6	even	5		100.4.g.a.61.1	yes	28
25.8	odd	20		500.4.i.b.449.1		56
25.17	odd	20		500.4.i.b.449.14		56
25.19	even	10		500.4.g.a.301.7		28
25.21	even	5		100.4.g.a.41.1	✓	28
25.22	odd	20		500.4.i.b.49.1		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.g.a.41.1	✓	28	25.21	even	5
100.4.g.a.61.1	yes	28	25.6	even	5
500.4.g.a.201.7		28	25.4	even	10
500.4.g.a.301.7		28	25.19	even	10
500.4.i.b.49.1		56	25.22	odd	20
500.4.i.b.49.14		56	25.3	odd	20
500.4.i.b.449.1		56	25.8	odd	20
500.4.i.b.449.14		56	25.17	odd	20
2500.4.a.c.1.2		14	1.1	even	1	trivial
2500.4.a.d.1.13		14	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2500.a (trivial)

\(f(q)\)	\(=\)	\(q-8.60496 q^{3} +17.7526 q^{7} +47.0453 q^{9} +O(q^{10})\)	\(q-8.60496 q^{3} +17.7526 q^{7} +47.0453 q^{9} +29.0132 q^{11} +54.6210 q^{13} +68.3775 q^{17} -53.7672 q^{19} -152.760 q^{21} -166.675 q^{23} -172.489 q^{27} -255.204 q^{29} +206.653 q^{31} -249.657 q^{33} -243.699 q^{37} -470.011 q^{39} -47.2594 q^{41} -35.8364 q^{43} +467.871 q^{47} -27.8456 q^{49} -588.385 q^{51} -673.560 q^{53} +462.665 q^{57} -330.452 q^{59} -396.058 q^{61} +835.175 q^{63} +356.450 q^{67} +1434.23 q^{69} -337.376 q^{71} +727.833 q^{73} +515.059 q^{77} -1271.45 q^{79} +214.034 q^{81} +64.3165 q^{83} +2196.02 q^{87} -698.706 q^{89} +969.664 q^{91} -1778.24 q^{93} +964.411 q^{97} +1364.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} + 8 q^{7} + 106 q^{9}+O(q^{10}) \) 14 * q - 2 * q^3 + 8 * q^7 + 106 * q^9	\( 14 q - 2 q^{3} + 8 q^{7} + 106 q^{9} - 30 q^{11} + 77 q^{13} + 111 q^{17} - 64 q^{19} - 156 q^{21} - 24 q^{23} - 614 q^{27} - 359 q^{29} - 156 q^{31} - 390 q^{33} - 307 q^{37} - 314 q^{39} - 791 q^{41} - 90 q^{43} + 276 q^{47} + 72 q^{49} - 832 q^{51} - 451 q^{53} + 192 q^{57} - 808 q^{59} - 1017 q^{61} + 2082 q^{63} + 1678 q^{67} - 1502 q^{69} - 148 q^{71} - 223 q^{73} - 1210 q^{77} - 92 q^{79} - 414 q^{81} - 474 q^{83} + 4262 q^{87} - 2111 q^{89} - 794 q^{91} - 2832 q^{93} + 353 q^{97} - 1050 q^{99}+O(q^{100}) \) 14 * q - 2 * q^3 + 8 * q^7 + 106 * q^9 - 30 * q^11 + 77 * q^13 + 111 * q^17 - 64 * q^19 - 156 * q^21 - 24 * q^23 - 614 * q^27 - 359 * q^29 - 156 * q^31 - 390 * q^33 - 307 * q^37 - 314 * q^39 - 791 * q^41 - 90 * q^43 + 276 * q^47 + 72 * q^49 - 832 * q^51 - 451 * q^53 + 192 * q^57 - 808 * q^59 - 1017 * q^61 + 2082 * q^63 + 1678 * q^67 - 1502 * q^69 - 148 * q^71 - 223 * q^73 - 1210 * q^77 - 92 * q^79 - 414 * q^81 - 474 * q^83 + 4262 * q^87 - 2111 * q^89 - 794 * q^91 - 2832 * q^93 + 353 * q^97 - 1050 * q^99

Introduction

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