LMFDB - Embedded newform 308.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,4,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, 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("308.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	308.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:	$18.1725882818$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 43x^{2} - 11x + 222 $ x^4 - x^3 - 43x^2 - 11x + 222
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.79239$ of defining polynomial
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.79239 q^{3} -7.61779 q^{5} -7.00000 q^{7} -12.6178 q^{9} +11.0000 q^{11} -52.1907 q^{13} +28.8896 q^{15} +115.878 q^{17} -21.5449 q^{19} +26.5467 q^{21} -36.6714 q^{23} -66.9693 q^{25} +150.246 q^{27} +50.2172 q^{29} +153.232 q^{31} -41.7163 q^{33} +53.3245 q^{35} +303.168 q^{37} +197.927 q^{39} +301.043 q^{41} -273.623 q^{43} +96.1196 q^{45} +391.708 q^{47} +49.0000 q^{49} -439.453 q^{51} +119.005 q^{53} -83.7956 q^{55} +81.7067 q^{57} -243.783 q^{59} +824.563 q^{61} +88.3245 q^{63} +397.577 q^{65} +310.219 q^{67} +139.072 q^{69} -705.110 q^{71} +331.898 q^{73} +253.974 q^{75} -77.0000 q^{77} +812.935 q^{79} -229.111 q^{81} -1376.75 q^{83} -882.731 q^{85} -190.443 q^{87} -604.809 q^{89} +365.335 q^{91} -581.115 q^{93} +164.125 q^{95} +963.838 q^{97} -138.796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.79239	−0.729846	−0.364923	−	0.931038i	$-0.618905\pi$
$3$	−3.79239	−0.729846	−0.364923	+	0.931038i	$0.618905\pi$
$4$	0	0
$5$	−7.61779	−0.681355	−0.340678	−	0.940180i	$-0.610656\pi$
$5$	−7.61779	−0.681355	−0.340678	+	0.940180i	$0.610656\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−12.6178	−0.467325
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−52.1907	−1.11347	−0.556734	−	0.830691i	$-0.687946\pi$
$13$	−52.1907	−1.11347	−0.556734	+	0.830691i	$0.687946\pi$
$14$	0	0
$15$	28.8896	0.497284
$16$	0	0
$17$	115.878	1.65320	0.826602	−	0.562787i	$-0.190271\pi$
$17$	115.878	1.65320	0.826602	+	0.562787i	$0.190271\pi$
$18$	0	0
$19$	−21.5449	−0.260144	−0.130072	−	0.991505i	$-0.541521\pi$
$19$	−21.5449	−0.260144	−0.130072	+	0.991505i	$0.541521\pi$
$20$	0	0
$21$	26.5467	0.275856
$22$	0	0
$23$	−36.6714	−0.332457	−0.166228	−	0.986087i	$-0.553159\pi$
$23$	−36.6714	−0.332457	−0.166228	+	0.986087i	$0.553159\pi$
$24$	0	0
$25$	−66.9693	−0.535755
$26$	0	0
$27$	150.246	1.07092
$28$	0	0
$29$	50.2172	0.321555	0.160778	−	0.986991i	$-0.448600\pi$
$29$	50.2172	0.321555	0.160778	+	0.986991i	$0.448600\pi$
$30$	0	0
$31$	153.232	0.887783	0.443892	−	0.896081i	$-0.353598\pi$
$31$	153.232	0.887783	0.443892	+	0.896081i	$0.353598\pi$
$32$	0	0
$33$	−41.7163	−0.220057
$34$	0	0
$35$	53.3245	0.257528
$36$	0	0
$37$	303.168	1.34704	0.673520	−	0.739169i	$-0.264782\pi$
$37$	303.168	1.34704	0.673520	+	0.739169i	$0.264782\pi$
$38$	0	0
$39$	197.927	0.812660
$40$	0	0
$41$	301.043	1.14671	0.573353	−	0.819308i	$-0.305642\pi$
$41$	301.043	1.14671	0.573353	+	0.819308i	$0.305642\pi$
$42$	0	0
$43$	−273.623	−0.970398	−0.485199	−	0.874404i	$-0.661253\pi$
$43$	−273.623	−0.970398	−0.485199	+	0.874404i	$0.661253\pi$
$44$	0	0
$45$	96.1196	0.318415
$46$	0	0
$47$	391.708	1.21567	0.607835	−	0.794064i	$-0.292038\pi$
$47$	391.708	1.21567	0.607835	+	0.794064i	$0.292038\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−439.453	−1.20658
$52$	0	0
$53$	119.005	0.308426	0.154213	−	0.988038i	$-0.450716\pi$
$53$	119.005	0.308426	0.154213	+	0.988038i	$0.450716\pi$
$54$	0	0
$55$	−83.7956	−0.205436
$56$	0	0
$57$	81.7067	0.189865
$58$	0	0
$59$	−243.783	−0.537930	−0.268965	−	0.963150i	$-0.586682\pi$
$59$	−243.783	−0.537930	−0.268965	+	0.963150i	$0.586682\pi$
$60$	0	0
$61$	824.563	1.73073	0.865364	−	0.501143i	$-0.167087\pi$
$61$	824.563	1.73073	0.865364	+	0.501143i	$0.167087\pi$
$62$	0	0
$63$	88.3245	0.176632
$64$	0	0
$65$	397.577	0.758668
$66$	0	0
$67$	310.219	0.565661	0.282830	−	0.959170i	$-0.408727\pi$
$67$	310.219	0.565661	0.282830	+	0.959170i	$0.408727\pi$
$68$	0	0
$69$	139.072	0.242642
$70$	0	0
$71$	−705.110	−1.17861	−0.589304	−	0.807911i	$-0.700598\pi$
$71$	−705.110	−1.17861	−0.589304	+	0.807911i	$0.700598\pi$
$72$	0	0
$73$	331.898	0.532133	0.266066	−	0.963955i	$-0.414276\pi$
$73$	331.898	0.532133	0.266066	+	0.963955i	$0.414276\pi$
$74$	0	0
$75$	253.974	0.391018
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	812.935	1.15775	0.578875	−	0.815416i	$-0.303492\pi$
$79$	812.935	1.15775	0.578875	+	0.815416i	$0.303492\pi$
$80$	0	0
$81$	−229.111	−0.314282
$82$	0	0
$83$	−1376.75	−1.82070	−0.910348	−	0.413844i	$-0.864186\pi$
$83$	−1376.75	−1.82070	−0.910348	+	0.413844i	$0.864186\pi$
$84$	0	0
$85$	−882.731	−1.12642
$86$	0	0
$87$	−190.443	−0.234686
$88$	0	0
$89$	−604.809	−0.720333	−0.360166	−	0.932888i	$-0.617280\pi$
$89$	−604.809	−0.720333	−0.360166	+	0.932888i	$0.617280\pi$
$90$	0	0
$91$	365.335	0.420851
$92$	0	0
$93$	−581.115	−0.647945
$94$	0	0
$95$	164.125	0.177251
$96$	0	0
$97$	963.838	1.00890	0.504448	−	0.863442i	$-0.331696\pi$
$97$	963.838	1.00890	0.504448	+	0.863442i	$0.331696\pi$
$98$	0	0
$99$	−138.796	−0.140904
$100$	0	0
$101$	−58.8600	−0.0579880	−0.0289940	−	0.999580i	$-0.509230\pi$
$101$	−58.8600	−0.0579880	−0.0289940	+	0.999580i	$0.509230\pi$
$102$	0	0
$103$	580.122	0.554962	0.277481	−	0.960731i	$-0.410500\pi$
$103$	580.122	0.554962	0.277481	+	0.960731i	$0.410500\pi$
$104$	0	0
$105$	−202.227	−0.187956
$106$	0	0
$107$	−924.890	−0.835631	−0.417815	−	0.908532i	$-0.637204\pi$
$107$	−924.890	−0.835631	−0.417815	+	0.908532i	$0.637204\pi$
$108$	0	0
$109$	−1136.51	−0.998695	−0.499348	−	0.866402i	$-0.666427\pi$
$109$	−1136.51	−0.998695	−0.499348	+	0.866402i	$0.666427\pi$
$110$	0	0
$111$	−1149.73	−0.983131
$112$	0	0
$113$	1157.61	0.963705	0.481852	−	0.876252i	$-0.339964\pi$
$113$	1157.61	0.963705	0.481852	+	0.876252i	$0.339964\pi$
$114$	0	0
$115$	279.355	0.226521
$116$	0	0
$117$	658.531	0.520352
$118$	0	0
$119$	−811.144	−0.624852
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1141.67	−0.836918
$124$	0	0
$125$	1462.38	1.04639
$126$	0	0
$127$	−1958.23	−1.36822	−0.684112	−	0.729377i	$-0.739810\pi$
$127$	−1958.23	−1.36822	−0.684112	+	0.729377i	$0.739810\pi$
$128$	0	0
$129$	1037.68	0.708240
$130$	0	0
$131$	−320.427	−0.213709	−0.106854	−	0.994275i	$-0.534078\pi$
$131$	−320.427	−0.213709	−0.106854	+	0.994275i	$0.534078\pi$
$132$	0	0
$133$	150.814	0.0983253
$134$	0	0
$135$	−1144.54	−0.729678
$136$	0	0
$137$	−1873.92	−1.16861	−0.584306	−	0.811533i	$-0.698633\pi$
$137$	−1873.92	−1.16861	−0.584306	+	0.811533i	$0.698633\pi$
$138$	0	0
$139$	−508.927	−0.310552	−0.155276	−	0.987871i	$-0.549627\pi$
$139$	−508.927	−0.310552	−0.155276	+	0.987871i	$0.549627\pi$
$140$	0	0
$141$	−1485.51	−0.887251
$142$	0	0
$143$	−574.097	−0.335723
$144$	0	0
$145$	−382.544	−0.219093
$146$	0	0
$147$	−185.827	−0.104264
$148$	0	0
$149$	1793.96	0.986358	0.493179	−	0.869928i	$-0.335835\pi$
$149$	1793.96	0.986358	0.493179	+	0.869928i	$0.335835\pi$
$150$	0	0
$151$	−1934.44	−1.04253	−0.521265	−	0.853395i	$-0.674540\pi$
$151$	−1934.44	−1.04253	−0.521265	+	0.853395i	$0.674540\pi$
$152$	0	0
$153$	−1462.12	−0.772584
$154$	0	0
$155$	−1167.29	−0.604896
$156$	0	0
$157$	3105.09	1.57843	0.789215	−	0.614117i	$-0.210488\pi$
$157$	3105.09	1.57843	0.789215	+	0.614117i	$0.210488\pi$
$158$	0	0
$159$	−451.313	−0.225103
$160$	0	0
$161$	256.700	0.125657
$162$	0	0
$163$	2738.18	1.31577	0.657887	−	0.753117i	$-0.271451\pi$
$163$	2738.18	1.31577	0.657887	+	0.753117i	$0.271451\pi$
$164$	0	0
$165$	317.786	0.149937
$166$	0	0
$167$	2794.07	1.29468	0.647340	−	0.762201i	$-0.275881\pi$
$167$	2794.07	1.29468	0.647340	+	0.762201i	$0.275881\pi$
$168$	0	0
$169$	526.865	0.239811
$170$	0	0
$171$	271.849	0.121572
$172$	0	0
$173$	−749.616	−0.329435	−0.164718	−	0.986341i	$-0.552671\pi$
$173$	−749.616	−0.329435	−0.164718	+	0.986341i	$0.552671\pi$
$174$	0	0
$175$	468.785	0.202496
$176$	0	0
$177$	924.521	0.392606
$178$	0	0
$179$	−2234.31	−0.932962	−0.466481	−	0.884531i	$-0.654478\pi$
$179$	−2234.31	−0.932962	−0.466481	+	0.884531i	$0.654478\pi$
$180$	0	0
$181$	−918.806	−0.377316	−0.188658	−	0.982043i	$-0.560414\pi$
$181$	−918.806	−0.377316	−0.188658	+	0.982043i	$0.560414\pi$
$182$	0	0
$183$	−3127.06	−1.26316
$184$	0	0
$185$	−2309.47	−0.917812
$186$	0	0
$187$	1274.65	0.498460
$188$	0	0
$189$	−1051.72	−0.404770
$190$	0	0
$191$	4208.11	1.59418	0.797089	−	0.603862i	$-0.206372\pi$
$191$	4208.11	1.59418	0.797089	+	0.603862i	$0.206372\pi$
$192$	0	0
$193$	4671.45	1.74227	0.871136	−	0.491042i	$-0.163384\pi$
$193$	4671.45	1.74227	0.871136	+	0.491042i	$0.163384\pi$
$194$	0	0
$195$	−1507.77	−0.553710
$196$	0	0
$197$	−2285.17	−0.826454	−0.413227	−	0.910628i	$-0.635598\pi$
$197$	−2285.17	−0.826454	−0.413227	+	0.910628i	$0.635598\pi$
$198$	0	0
$199$	−3198.19	−1.13926	−0.569632	−	0.821900i	$-0.692914\pi$
$199$	−3198.19	−1.13926	−0.569632	+	0.821900i	$0.692914\pi$
$200$	0	0
$201$	−1176.47	−0.412845
$202$	0	0
$203$	−351.520	−0.121536
$204$	0	0
$205$	−2293.28	−0.781315
$206$	0	0
$207$	462.712	0.155366
$208$	0	0
$209$	−236.994	−0.0784365
$210$	0	0
$211$	−1762.25	−0.574968	−0.287484	−	0.957786i	$-0.592819\pi$
$211$	−1762.25	−0.574968	−0.287484	+	0.957786i	$0.592819\pi$
$212$	0	0
$213$	2674.05	0.860202
$214$	0	0
$215$	2084.40	0.661186
$216$	0	0
$217$	−1072.62	−0.335550
$218$	0	0
$219$	−1258.68	−0.388375
$220$	0	0
$221$	−6047.73	−1.84079
$222$	0	0
$223$	4092.23	1.22886	0.614430	−	0.788972i	$-0.289386\pi$
$223$	4092.23	1.22886	0.614430	+	0.788972i	$0.289386\pi$
$224$	0	0
$225$	845.005	0.250372
$226$	0	0
$227$	3892.12	1.13801	0.569007	−	0.822333i	$-0.307328\pi$
$227$	3892.12	1.13801	0.569007	+	0.822333i	$0.307328\pi$
$228$	0	0
$229$	2395.96	0.691395	0.345698	−	0.938346i	$-0.387642\pi$
$229$	2395.96	0.691395	0.345698	+	0.938346i	$0.387642\pi$
$230$	0	0
$231$	292.014	0.0831736
$232$	0	0
$233$	4291.37	1.20660	0.603298	−	0.797516i	$-0.293853\pi$
$233$	4291.37	1.20660	0.603298	+	0.797516i	$0.293853\pi$
$234$	0	0
$235$	−2983.95	−0.828303
$236$	0	0
$237$	−3082.96	−0.844979
$238$	0	0
$239$	3030.80	0.820277	0.410138	−	0.912023i	$-0.365480\pi$
$239$	3030.80	0.820277	0.410138	+	0.912023i	$0.365480\pi$
$240$	0	0
$241$	5922.81	1.58308	0.791539	−	0.611118i	$-0.209280\pi$
$241$	5922.81	1.58308	0.791539	+	0.611118i	$0.209280\pi$
$242$	0	0
$243$	−3187.76	−0.841544
$244$	0	0
$245$	−373.271	−0.0973365
$246$	0	0
$247$	1124.44	0.289662
$248$	0	0
$249$	5221.16	1.32883
$250$	0	0
$251$	−5819.20	−1.46336	−0.731682	−	0.681646i	$-0.761264\pi$
$251$	−5819.20	−1.46336	−0.731682	+	0.681646i	$0.761264\pi$
$252$	0	0
$253$	−403.385	−0.100240
$254$	0	0
$255$	3347.66	0.822112
$256$	0	0
$257$	−3259.53	−0.791143	−0.395572	−	0.918435i	$-0.629454\pi$
$257$	−3259.53	−0.791143	−0.395572	+	0.918435i	$0.629454\pi$
$258$	0	0
$259$	−2122.17	−0.509133
$260$	0	0
$261$	−633.630	−0.150271
$262$	0	0
$263$	1265.42	0.296688	0.148344	−	0.988936i	$-0.452606\pi$
$263$	1265.42	0.296688	0.148344	+	0.988936i	$0.452606\pi$
$264$	0	0
$265$	−906.554	−0.210148
$266$	0	0
$267$	2293.67	0.525732
$268$	0	0
$269$	5822.97	1.31982	0.659912	−	0.751343i	$-0.270593\pi$
$269$	5822.97	1.31982	0.659912	+	0.751343i	$0.270593\pi$
$270$	0	0
$271$	3043.07	0.682116	0.341058	−	0.940042i	$-0.389215\pi$
$271$	3043.07	0.682116	0.341058	+	0.940042i	$0.389215\pi$
$272$	0	0
$273$	−1385.49	−0.307157
$274$	0	0
$275$	−736.663	−0.161536
$276$	0	0
$277$	6368.24	1.38134	0.690669	−	0.723171i	$-0.257316\pi$
$277$	6368.24	1.38134	0.690669	+	0.723171i	$0.257316\pi$
$278$	0	0
$279$	−1933.45	−0.414884
$280$	0	0
$281$	5158.86	1.09520	0.547601	−	0.836740i	$-0.315541\pi$
$281$	5158.86	1.09520	0.547601	+	0.836740i	$0.315541\pi$
$282$	0	0
$283$	3364.31	0.706669	0.353335	−	0.935497i	$-0.385048\pi$
$283$	3364.31	0.706669	0.353335	+	0.935497i	$0.385048\pi$
$284$	0	0
$285$	−622.424	−0.129366
$286$	0	0
$287$	−2107.30	−0.433414
$288$	0	0
$289$	8514.64	1.73308
$290$	0	0
$291$	−3655.25	−0.736338
$292$	0	0
$293$	−2244.83	−0.447591	−0.223796	−	0.974636i	$-0.571845\pi$
$293$	−2244.83	−0.447591	−0.223796	+	0.974636i	$0.571845\pi$
$294$	0	0
$295$	1857.09	0.366522
$296$	0	0
$297$	1652.71	0.322895
$298$	0	0
$299$	1913.90	0.370180
$300$	0	0
$301$	1915.36	0.366776
$302$	0	0
$303$	223.220	0.0423223
$304$	0	0
$305$	−6281.34	−1.17924
$306$	0	0
$307$	−116.065	−0.0215771	−0.0107886	−	0.999942i	$-0.503434\pi$
$307$	−116.065	−0.0215771	−0.0107886	+	0.999942i	$0.503434\pi$
$308$	0	0
$309$	−2200.05	−0.405037
$310$	0	0
$311$	8639.49	1.57524	0.787621	−	0.616159i	$-0.211312\pi$
$311$	8639.49	1.57524	0.787621	+	0.616159i	$0.211312\pi$
$312$	0	0
$313$	4908.15	0.886341	0.443171	−	0.896437i	$-0.353854\pi$
$313$	4908.15	0.886341	0.443171	+	0.896437i	$0.353854\pi$
$314$	0	0
$315$	−672.837	−0.120349
$316$	0	0
$317$	1234.27	0.218686	0.109343	−	0.994004i	$-0.465125\pi$
$317$	1234.27	0.218686	0.109343	+	0.994004i	$0.465125\pi$
$318$	0	0
$319$	552.389	0.0969525
$320$	0	0
$321$	3507.54	0.609881
$322$	0	0
$323$	−2496.58	−0.430072
$324$	0	0
$325$	3495.17	0.596546
$326$	0	0
$327$	4310.08	0.728893
$328$	0	0
$329$	−2741.95	−0.459480
$330$	0	0
$331$	−374.394	−0.0621709	−0.0310855	−	0.999517i	$-0.509896\pi$
$331$	−374.394	−0.0621709	−0.0310855	+	0.999517i	$0.509896\pi$
$332$	0	0
$333$	−3825.30	−0.629506
$334$	0	0
$335$	−2363.18	−0.385416
$336$	0	0
$337$	3561.81	0.575740	0.287870	−	0.957670i	$-0.407053\pi$
$337$	3561.81	0.575740	0.287870	+	0.957670i	$0.407053\pi$
$338$	0	0
$339$	−4390.10	−0.703356
$340$	0	0
$341$	1685.55	0.267677
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1059.42	−0.165326
$346$	0	0
$347$	2927.98	0.452975	0.226487	−	0.974014i	$-0.427276\pi$
$347$	2927.98	0.452975	0.226487	+	0.974014i	$0.427276\pi$
$348$	0	0
$349$	−913.271	−0.140075	−0.0700377	−	0.997544i	$-0.522312\pi$
$349$	−913.271	−0.140075	−0.0700377	+	0.997544i	$0.522312\pi$
$350$	0	0
$351$	−7841.44	−1.19244
$352$	0	0
$353$	8366.50	1.26148	0.630742	−	0.775993i	$-0.282751\pi$
$353$	8366.50	1.26148	0.630742	+	0.775993i	$0.282751\pi$
$354$	0	0
$355$	5371.38	0.803051
$356$	0	0
$357$	3076.17	0.456046
$358$	0	0
$359$	−1766.25	−0.259664	−0.129832	−	0.991536i	$-0.541444\pi$
$359$	−1766.25	−0.259664	−0.129832	+	0.991536i	$0.541444\pi$
$360$	0	0
$361$	−6394.82	−0.932325
$362$	0	0
$363$	−458.879	−0.0663496
$364$	0	0
$365$	−2528.32	−0.362571
$366$	0	0
$367$	4044.02	0.575194	0.287597	−	0.957751i	$-0.407144\pi$
$367$	4044.02	0.575194	0.287597	+	0.957751i	$0.407144\pi$
$368$	0	0
$369$	−3798.49	−0.535885
$370$	0	0
$371$	−833.034	−0.116574
$372$	0	0
$373$	−11145.2	−1.54712	−0.773562	−	0.633721i	$-0.781527\pi$
$373$	−11145.2	−1.54712	−0.773562	+	0.633721i	$0.781527\pi$
$374$	0	0
$375$	−5545.92	−0.763707
$376$	0	0
$377$	−2620.87	−0.358041
$378$	0	0
$379$	−14564.2	−1.97391	−0.986954	−	0.161005i	$-0.948526\pi$
$379$	−14564.2	−1.97391	−0.986954	+	0.161005i	$0.948526\pi$
$380$	0	0
$381$	7426.35	0.998592
$382$	0	0
$383$	−9754.74	−1.30142	−0.650710	−	0.759326i	$-0.725529\pi$
$383$	−9754.74	−1.30142	−0.650710	+	0.759326i	$0.725529\pi$
$384$	0	0
$385$	586.569	0.0776477
$386$	0	0
$387$	3452.51	0.453491
$388$	0	0
$389$	−10260.5	−1.33735	−0.668673	−	0.743557i	$-0.733137\pi$
$389$	−10260.5	−1.33735	−0.668673	+	0.743557i	$0.733137\pi$
$390$	0	0
$391$	−4249.39	−0.549619
$392$	0	0
$393$	1215.18	0.155974
$394$	0	0
$395$	−6192.76	−0.788840
$396$	0	0
$397$	−7308.66	−0.923957	−0.461979	−	0.886891i	$-0.652860\pi$
$397$	−7308.66	−0.923957	−0.461979	+	0.886891i	$0.652860\pi$
$398$	0	0
$399$	−571.947	−0.0717623
$400$	0	0
$401$	−13824.1	−1.72155	−0.860776	−	0.508984i	$-0.830021\pi$
$401$	−13824.1	−1.72155	−0.860776	+	0.508984i	$0.830021\pi$
$402$	0	0
$403$	−7997.28	−0.988518
$404$	0	0
$405$	1745.32	0.214137
$406$	0	0
$407$	3334.84	0.406148
$408$	0	0
$409$	5894.74	0.712656	0.356328	−	0.934361i	$-0.384029\pi$
$409$	5894.74	0.712656	0.356328	+	0.934361i	$0.384029\pi$
$410$	0	0
$411$	7106.64	0.852907
$412$	0	0
$413$	1706.48	0.203319
$414$	0	0
$415$	10487.8	1.24054
$416$	0	0
$417$	1930.05	0.226655
$418$	0	0
$419$	−1436.00	−0.167431	−0.0837153	−	0.996490i	$-0.526679\pi$
$419$	−1436.00	−0.167431	−0.0837153	+	0.996490i	$0.526679\pi$
$420$	0	0
$421$	5986.38	0.693013	0.346506	−	0.938048i	$-0.387368\pi$
$421$	5986.38	0.693013	0.346506	+	0.938048i	$0.387368\pi$
$422$	0	0
$423$	−4942.49	−0.568113
$424$	0	0
$425$	−7760.25	−0.885712
$426$	0	0
$427$	−5771.94	−0.654154
$428$	0	0
$429$	2177.20	0.245026
$430$	0	0
$431$	3482.47	0.389199	0.194599	−	0.980883i	$-0.437659\pi$
$431$	3482.47	0.389199	0.194599	+	0.980883i	$0.437659\pi$
$432$	0	0
$433$	−4646.70	−0.515719	−0.257860	−	0.966182i	$-0.583017\pi$
$433$	−4646.70	−0.515719	−0.257860	+	0.966182i	$0.583017\pi$
$434$	0	0
$435$	1450.75	0.159904
$436$	0	0
$437$	790.082	0.0864868
$438$	0	0
$439$	2252.14	0.244849	0.122425	−	0.992478i	$-0.460933\pi$
$439$	2252.14	0.244849	0.122425	+	0.992478i	$0.460933\pi$
$440$	0	0
$441$	−618.271	−0.0667608
$442$	0	0
$443$	−8872.75	−0.951596	−0.475798	−	0.879555i	$-0.657841\pi$
$443$	−8872.75	−0.951596	−0.475798	+	0.879555i	$0.657841\pi$
$444$	0	0
$445$	4607.30	0.490803
$446$	0	0
$447$	−6803.41	−0.719889
$448$	0	0
$449$	−15399.0	−1.61854	−0.809268	−	0.587440i	$-0.800136\pi$
$449$	−15399.0	−1.61854	−0.809268	+	0.587440i	$0.800136\pi$
$450$	0	0
$451$	3311.47	0.345745
$452$	0	0
$453$	7336.13	0.760886
$454$	0	0
$455$	−2783.04	−0.286749
$456$	0	0
$457$	15703.8	1.60742	0.803711	−	0.595020i	$-0.202856\pi$
$457$	15703.8	1.60742	0.803711	+	0.595020i	$0.202856\pi$
$458$	0	0
$459$	17410.2	1.77045
$460$	0	0
$461$	−8011.59	−0.809407	−0.404704	−	0.914448i	$-0.632625\pi$
$461$	−8011.59	−0.809407	−0.404704	+	0.914448i	$0.632625\pi$
$462$	0	0
$463$	−1708.52	−0.171494	−0.0857470	−	0.996317i	$-0.527328\pi$
$463$	−1708.52	−0.171494	−0.0857470	+	0.996317i	$0.527328\pi$
$464$	0	0
$465$	4426.81	0.441481
$466$	0	0
$467$	8807.46	0.872721	0.436361	−	0.899772i	$-0.356267\pi$
$467$	8807.46	0.872721	0.436361	+	0.899772i	$0.356267\pi$
$468$	0	0
$469$	−2171.53	−0.213800
$470$	0	0
$471$	−11775.7	−1.15201
$472$	0	0
$473$	−3009.85	−0.292586
$474$	0	0
$475$	1442.85	0.139374
$476$	0	0
$477$	−1501.58	−0.144135
$478$	0	0
$479$	2395.07	0.228462	0.114231	−	0.993454i	$-0.463560\pi$
$479$	2395.07	0.228462	0.114231	+	0.993454i	$0.463560\pi$
$480$	0	0
$481$	−15822.5	−1.49989
$482$	0	0
$483$	−973.505	−0.0917101
$484$	0	0
$485$	−7342.31	−0.687417
$486$	0	0
$487$	15541.8	1.44613	0.723065	−	0.690780i	$-0.242733\pi$
$487$	15541.8	1.44613	0.723065	+	0.690780i	$0.242733\pi$
$488$	0	0
$489$	−10384.3	−0.960312
$490$	0	0
$491$	−1905.13	−0.175107	−0.0875533	−	0.996160i	$-0.527905\pi$
$491$	−1905.13	−0.175107	−0.0875533	+	0.996160i	$0.527905\pi$
$492$	0	0
$493$	5819.05	0.531596
$494$	0	0
$495$	1057.32	0.0960056
$496$	0	0
$497$	4935.77	0.445472
$498$	0	0
$499$	6229.01	0.558816	0.279408	−	0.960173i	$-0.409862\pi$
$499$	6229.01	0.558816	0.279408	+	0.960173i	$0.409862\pi$
$500$	0	0
$501$	−10596.2	−0.944916
$502$	0	0
$503$	−2972.06	−0.263455	−0.131727	−	0.991286i	$-0.542052\pi$
$503$	−2972.06	−0.263455	−0.131727	+	0.991286i	$0.542052\pi$
$504$	0	0
$505$	448.383	0.0395104
$506$	0	0
$507$	−1998.08	−0.175025
$508$	0	0
$509$	−2988.49	−0.260241	−0.130120	−	0.991498i	$-0.541536\pi$
$509$	−2988.49	−0.260241	−0.130120	+	0.991498i	$0.541536\pi$
$510$	0	0
$511$	−2323.28	−0.201127
$512$	0	0
$513$	−3237.04	−0.278594
$514$	0	0
$515$	−4419.25	−0.378127
$516$	0	0
$517$	4308.79	0.366538
$518$	0	0
$519$	2842.84	0.240437
$520$	0	0
$521$	−11789.5	−0.991374	−0.495687	−	0.868501i	$-0.665084\pi$
$521$	−11789.5	−0.991374	−0.495687	+	0.868501i	$0.665084\pi$
$522$	0	0
$523$	13552.5	1.13310	0.566549	−	0.824028i	$-0.308278\pi$
$523$	13552.5	1.13310	0.566549	+	0.824028i	$0.308278\pi$
$524$	0	0
$525$	−1777.82	−0.147791
$526$	0	0
$527$	17756.2	1.46769
$528$	0	0
$529$	−10822.2	−0.889472
$530$	0	0
$531$	3076.01	0.251389
$532$	0	0
$533$	−15711.6	−1.27682
$534$	0	0
$535$	7045.61	0.569361
$536$	0	0
$537$	8473.38	0.680918
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	14450.2	1.14836	0.574180	−	0.818729i	$-0.305321\pi$
$541$	14450.2	1.14836	0.574180	+	0.818729i	$0.305321\pi$
$542$	0	0
$543$	3484.47	0.275383
$544$	0	0
$545$	8657.68	0.680466
$546$	0	0
$547$	10324.6	0.807038	0.403519	−	0.914971i	$-0.367787\pi$
$547$	10324.6	0.807038	0.403519	+	0.914971i	$0.367787\pi$
$548$	0	0
$549$	−10404.2	−0.808813
$550$	0	0
$551$	−1081.93	−0.0836507
$552$	0	0
$553$	−5690.54	−0.437589
$554$	0	0
$555$	8758.39	0.669861
$556$	0	0
$557$	−17839.1	−1.35703	−0.678516	−	0.734585i	$-0.737377\pi$
$557$	−17839.1	−1.35703	−0.678516	+	0.734585i	$0.737377\pi$
$558$	0	0
$559$	14280.6	1.08051
$560$	0	0
$561$	−4833.99	−0.363799
$562$	0	0
$563$	16178.5	1.21109	0.605543	−	0.795812i	$-0.292956\pi$
$563$	16178.5	1.21109	0.605543	+	0.795812i	$0.292956\pi$
$564$	0	0
$565$	−8818.41	−0.656626
$566$	0	0
$567$	1603.78	0.118787
$568$	0	0
$569$	8234.68	0.606706	0.303353	−	0.952878i	$-0.401894\pi$
$569$	8234.68	0.606706	0.303353	+	0.952878i	$0.401894\pi$
$570$	0	0
$571$	−9902.99	−0.725792	−0.362896	−	0.931830i	$-0.618212\pi$
$571$	−9902.99	−0.725792	−0.362896	+	0.931830i	$0.618212\pi$
$572$	0	0
$573$	−15958.8	−1.16350
$574$	0	0
$575$	2455.86	0.178115
$576$	0	0
$577$	−19446.7	−1.40308	−0.701538	−	0.712632i	$-0.747503\pi$
$577$	−19446.7	−1.40308	−0.701538	+	0.712632i	$0.747503\pi$
$578$	0	0
$579$	−17716.0	−1.27159
$580$	0	0
$581$	9637.24	0.688158
$582$	0	0
$583$	1309.05	0.0929939
$584$	0	0
$585$	−5016.54	−0.354545
$586$	0	0
$587$	−5284.52	−0.371577	−0.185788	−	0.982590i	$-0.559484\pi$
$587$	−5284.52	−0.371577	−0.185788	+	0.982590i	$0.559484\pi$
$588$	0	0
$589$	−3301.37	−0.230952
$590$	0	0
$591$	8666.24	0.603184
$592$	0	0
$593$	−5856.68	−0.405574	−0.202787	−	0.979223i	$-0.565000\pi$
$593$	−5856.68	−0.405574	−0.202787	+	0.979223i	$0.565000\pi$
$594$	0	0
$595$	6179.12	0.425747
$596$	0	0
$597$	12128.8	0.831487
$598$	0	0
$599$	636.088	0.0433888	0.0216944	−	0.999765i	$-0.493094\pi$
$599$	636.088	0.0433888	0.0216944	+	0.999765i	$0.493094\pi$
$600$	0	0
$601$	15468.9	1.04990	0.524951	−	0.851133i	$-0.324084\pi$
$601$	15468.9	1.04990	0.524951	+	0.851133i	$0.324084\pi$
$602$	0	0
$603$	−3914.28	−0.264348
$604$	0	0
$605$	−921.752	−0.0619414
$606$	0	0
$607$	26580.3	1.77737	0.888684	−	0.458521i	$-0.151620\pi$
$607$	26580.3	1.77737	0.888684	+	0.458521i	$0.151620\pi$
$608$	0	0
$609$	1333.10	0.0887028
$610$	0	0
$611$	−20443.5	−1.35361
$612$	0	0
$613$	24088.6	1.58716	0.793580	−	0.608466i	$-0.208215\pi$
$613$	24088.6	1.58716	0.793580	+	0.608466i	$0.208215\pi$
$614$	0	0
$615$	8697.00	0.570239
$616$	0	0
$617$	375.545	0.0245038	0.0122519	−	0.999925i	$-0.496100\pi$
$617$	375.545	0.0245038	0.0122519	+	0.999925i	$0.496100\pi$
$618$	0	0
$619$	−14117.4	−0.916682	−0.458341	−	0.888776i	$-0.651556\pi$
$619$	−14117.4	−0.916682	−0.458341	+	0.888776i	$0.651556\pi$
$620$	0	0
$621$	−5509.73	−0.356035
$622$	0	0
$623$	4233.66	0.272260
$624$	0	0
$625$	−2768.94	−0.177212
$626$	0	0
$627$	898.774	0.0572465
$628$	0	0
$629$	35130.4	2.22693
$630$	0	0
$631$	366.133	0.0230991	0.0115496	−	0.999933i	$-0.496324\pi$
$631$	366.133	0.0230991	0.0115496	+	0.999933i	$0.496324\pi$
$632$	0	0
$633$	6683.13	0.419638
$634$	0	0
$635$	14917.3	0.932247
$636$	0	0
$637$	−2557.34	−0.159067
$638$	0	0
$639$	8896.93	0.550793
$640$	0	0
$641$	3605.83	0.222187	0.111093	−	0.993810i	$-0.464565\pi$
$641$	3605.83	0.222187	0.111093	+	0.993810i	$0.464565\pi$
$642$	0	0
$643$	10680.2	0.655035	0.327518	−	0.944845i	$-0.393788\pi$
$643$	10680.2	0.655035	0.327518	+	0.944845i	$0.393788\pi$
$644$	0	0
$645$	−7904.86	−0.482563
$646$	0	0
$647$	3282.78	0.199474	0.0997369	−	0.995014i	$-0.468200\pi$
$647$	3282.78	0.199474	0.0997369	+	0.995014i	$0.468200\pi$
$648$	0	0
$649$	−2681.62	−0.162192
$650$	0	0
$651$	4067.81	0.244900
$652$	0	0
$653$	10107.9	0.605745	0.302873	−	0.953031i	$-0.402054\pi$
$653$	10107.9	0.605745	0.302873	+	0.953031i	$0.402054\pi$
$654$	0	0
$655$	2440.94	0.145612
$656$	0	0
$657$	−4187.81	−0.248679
$658$	0	0
$659$	7414.35	0.438274	0.219137	−	0.975694i	$-0.429676\pi$
$659$	7414.35	0.438274	0.219137	+	0.975694i	$0.429676\pi$
$660$	0	0
$661$	215.146	0.0126599	0.00632997	−	0.999980i	$-0.497985\pi$
$661$	215.146	0.0126599	0.00632997	+	0.999980i	$0.497985\pi$
$662$	0	0
$663$	22935.4	1.34349
$664$	0	0
$665$	−1148.87	−0.0669945
$666$	0	0
$667$	−1841.53	−0.106903
$668$	0	0
$669$	−15519.3	−0.896878
$670$	0	0
$671$	9070.19	0.521834
$672$	0	0
$673$	1934.82	0.110820	0.0554100	−	0.998464i	$-0.482353\pi$
$673$	1934.82	0.110820	0.0554100	+	0.998464i	$0.482353\pi$
$674$	0	0
$675$	−10061.9	−0.573751
$676$	0	0
$677$	438.126	0.0248723	0.0124362	−	0.999923i	$-0.496041\pi$
$677$	438.126	0.0248723	0.0124362	+	0.999923i	$0.496041\pi$
$678$	0	0
$679$	−6746.86	−0.381327
$680$	0	0
$681$	−14760.4	−0.830574
$682$	0	0
$683$	−27875.2	−1.56166	−0.780831	−	0.624742i	$-0.785204\pi$
$683$	−27875.2	−1.56166	−0.780831	+	0.624742i	$0.785204\pi$
$684$	0	0
$685$	14275.1	0.796241
$686$	0	0
$687$	−9086.41	−0.504612
$688$	0	0
$689$	−6210.94	−0.343423
$690$	0	0
$691$	−21356.2	−1.17573	−0.587865	−	0.808959i	$-0.700031\pi$
$691$	−21356.2	−1.17573	−0.587865	+	0.808959i	$0.700031\pi$
$692$	0	0
$693$	971.569	0.0532567
$694$	0	0
$695$	3876.90	0.211596
$696$	0	0
$697$	34884.1	1.89574
$698$	0	0
$699$	−16274.5	−0.880629
$700$	0	0
$701$	−5192.45	−0.279766	−0.139883	−	0.990168i	$-0.544673\pi$
$701$	−5192.45	−0.279766	−0.139883	+	0.990168i	$0.544673\pi$
$702$	0	0
$703$	−6531.72	−0.350425
$704$	0	0
$705$	11316.3	0.604533
$706$	0	0
$707$	412.020	0.0219174
$708$	0	0
$709$	32712.8	1.73280	0.866400	−	0.499351i	$-0.166428\pi$
$709$	32712.8	1.73280	0.866400	+	0.499351i	$0.166428\pi$
$710$	0	0
$711$	−10257.4	−0.541046
$712$	0	0
$713$	−5619.23	−0.295150
$714$	0	0
$715$	4373.35	0.228747
$716$	0	0
$717$	−11494.0	−0.598675
$718$	0	0
$719$	25348.8	1.31482	0.657408	−	0.753535i	$-0.271653\pi$
$719$	25348.8	1.31482	0.657408	+	0.753535i	$0.271653\pi$
$720$	0	0
$721$	−4060.85	−0.209756
$722$	0	0
$723$	−22461.6	−1.15540
$724$	0	0
$725$	−3363.01	−0.172275
$726$	0	0
$727$	26700.6	1.36213	0.681067	−	0.732221i	$-0.261516\pi$
$727$	26700.6	1.36213	0.681067	+	0.732221i	$0.261516\pi$
$728$	0	0
$729$	18275.2	0.928479
$730$	0	0
$731$	−31706.8	−1.60427
$732$	0	0
$733$	−17254.0	−0.869430	−0.434715	−	0.900568i	$-0.643151\pi$
$733$	−17254.0	−0.869430	−0.434715	+	0.900568i	$0.643151\pi$
$734$	0	0
$735$	1415.59	0.0710406
$736$	0	0
$737$	3412.41	0.170553
$738$	0	0
$739$	16081.5	0.800495	0.400248	−	0.916407i	$-0.368924\pi$
$739$	16081.5	0.800495	0.400248	+	0.916407i	$0.368924\pi$
$740$	0	0
$741$	−4264.33	−0.211409
$742$	0	0
$743$	38812.0	1.91638	0.958192	−	0.286126i	$-0.0923677\pi$
$743$	38812.0	1.91638	0.958192	+	0.286126i	$0.0923677\pi$
$744$	0	0
$745$	−13666.0	−0.672060
$746$	0	0
$747$	17371.5	0.850857
$748$	0	0
$749$	6474.23	0.315839
$750$	0	0
$751$	−18200.2	−0.884336	−0.442168	−	0.896932i	$-0.645790\pi$
$751$	−18200.2	−0.884336	−0.442168	+	0.896932i	$0.645790\pi$
$752$	0	0
$753$	22068.7	1.06803
$754$	0	0
$755$	14736.1	0.710334
$756$	0	0
$757$	4770.66	0.229052	0.114526	−	0.993420i	$-0.463465\pi$
$757$	4770.66	0.229052	0.114526	+	0.993420i	$0.463465\pi$
$758$	0	0
$759$	1529.79	0.0731594
$760$	0	0
$761$	−21753.2	−1.03621	−0.518104	−	0.855318i	$-0.673362\pi$
$761$	−21753.2	−1.03621	−0.518104	+	0.855318i	$0.673362\pi$
$762$	0	0
$763$	7955.56	0.377471
$764$	0	0
$765$	11138.1	0.526405
$766$	0	0
$767$	12723.2	0.598968
$768$	0	0
$769$	−34112.8	−1.59966	−0.799831	−	0.600225i	$-0.795078\pi$
$769$	−34112.8	−1.59966	−0.799831	+	0.600225i	$0.795078\pi$
$770$	0	0
$771$	12361.4	0.577413
$772$	0	0
$773$	17175.1	0.799155	0.399578	−	0.916699i	$-0.369157\pi$
$773$	17175.1	0.799155	0.399578	+	0.916699i	$0.369157\pi$
$774$	0	0
$775$	−10261.8	−0.475634
$776$	0	0
$777$	8048.11	0.371588
$778$	0	0
$779$	−6485.94	−0.298309
$780$	0	0
$781$	−7756.21	−0.355364
$782$	0	0
$783$	7544.93	0.344360
$784$	0	0
$785$	−23653.9	−1.07547
$786$	0	0
$787$	−1855.57	−0.0840458	−0.0420229	−	0.999117i	$-0.513380\pi$
$787$	−1855.57	−0.0840458	−0.0420229	+	0.999117i	$0.513380\pi$
$788$	0	0
$789$	−4798.95	−0.216536
$790$	0	0
$791$	−8103.26	−0.364246
$792$	0	0
$793$	−43034.5	−1.92711
$794$	0	0
$795$	3438.00	0.153375
$796$	0	0
$797$	3010.78	0.133811	0.0669053	−	0.997759i	$-0.478687\pi$
$797$	3010.78	0.133811	0.0669053	+	0.997759i	$0.478687\pi$
$798$	0	0
$799$	45390.2	2.00975
$800$	0	0
$801$	7631.35	0.336630
$802$	0	0
$803$	3650.87	0.160444
$804$	0	0
$805$	−1955.48	−0.0856170
$806$	0	0
$807$	−22083.0	−0.963268
$808$	0	0
$809$	33945.5	1.47523	0.737614	−	0.675223i	$-0.235953\pi$
$809$	33945.5	1.47523	0.737614	+	0.675223i	$0.235953\pi$
$810$	0	0
$811$	−30150.4	−1.30545	−0.652726	−	0.757594i	$-0.726375\pi$
$811$	−30150.4	−1.30545	−0.652726	+	0.757594i	$0.726375\pi$
$812$	0	0
$813$	−11540.5	−0.497839
$814$	0	0
$815$	−20858.9	−0.896510
$816$	0	0
$817$	5895.18	0.252443
$818$	0	0
$819$	−4609.71	−0.196675
$820$	0	0
$821$	22041.9	0.936987	0.468493	−	0.883467i	$-0.344797\pi$
$821$	22041.9	0.936987	0.468493	+	0.883467i	$0.344797\pi$
$822$	0	0
$823$	3078.09	0.130371	0.0651855	−	0.997873i	$-0.479236\pi$
$823$	3078.09	0.130371	0.0651855	+	0.997873i	$0.479236\pi$
$824$	0	0
$825$	2793.71	0.117896
$826$	0	0
$827$	−14942.4	−0.628293	−0.314147	−	0.949375i	$-0.601718\pi$
$827$	−14942.4	−0.628293	−0.314147	+	0.949375i	$0.601718\pi$
$828$	0	0
$829$	18817.5	0.788370	0.394185	−	0.919031i	$-0.371027\pi$
$829$	18817.5	0.788370	0.394185	+	0.919031i	$0.371027\pi$
$830$	0	0
$831$	−24150.8	−1.00816
$832$	0	0
$833$	5678.01	0.236172
$834$	0	0
$835$	−21284.6	−0.882137
$836$	0	0
$837$	23022.5	0.950745
$838$	0	0
$839$	−16084.8	−0.661870	−0.330935	−	0.943654i	$-0.607364\pi$
$839$	−16084.8	−0.661870	−0.330935	+	0.943654i	$0.607364\pi$
$840$	0	0
$841$	−21867.2	−0.896602
$842$	0	0
$843$	−19564.4	−0.799328
$844$	0	0
$845$	−4013.55	−0.163397
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−12758.8	−0.515760
$850$	0	0
$851$	−11117.6	−0.447833
$852$	0	0
$853$	11194.6	0.449349	0.224674	−	0.974434i	$-0.427868\pi$
$853$	11194.6	0.449349	0.224674	+	0.974434i	$0.427868\pi$
$854$	0	0
$855$	−2070.89	−0.0828338
$856$	0	0
$857$	−22452.5	−0.894937	−0.447469	−	0.894300i	$-0.647674\pi$
$857$	−22452.5	−0.894937	−0.447469	+	0.894300i	$0.647674\pi$
$858$	0	0
$859$	−20481.7	−0.813534	−0.406767	−	0.913532i	$-0.633344\pi$
$859$	−20481.7	−0.813534	−0.406767	+	0.913532i	$0.633344\pi$
$860$	0	0
$861$	7991.69	0.316325
$862$	0	0
$863$	33713.7	1.32981	0.664905	−	0.746928i	$-0.268472\pi$
$863$	33713.7	1.32981	0.664905	+	0.746928i	$0.268472\pi$
$864$	0	0
$865$	5710.42	0.224462
$866$	0	0
$867$	−32290.8	−1.26488
$868$	0	0
$869$	8942.28	0.349075
$870$	0	0
$871$	−16190.5	−0.629845
$872$	0	0
$873$	−12161.5	−0.471483
$874$	0	0
$875$	−10236.7	−0.395500
$876$	0	0
$877$	−26653.6	−1.02626	−0.513130	−	0.858311i	$-0.671514\pi$
$877$	−26653.6	−1.02626	−0.513130	+	0.858311i	$0.671514\pi$
$878$	0	0
$879$	8513.26	0.326672
$880$	0	0
$881$	14001.8	0.535453	0.267726	−	0.963495i	$-0.413728\pi$
$881$	14001.8	0.535453	0.267726	+	0.963495i	$0.413728\pi$
$882$	0	0
$883$	−7233.02	−0.275663	−0.137832	−	0.990456i	$-0.544013\pi$
$883$	−7233.02	−0.275663	−0.137832	+	0.990456i	$0.544013\pi$
$884$	0	0
$885$	−7042.81	−0.267504
$886$	0	0
$887$	−33228.2	−1.25783	−0.628914	−	0.777475i	$-0.716500\pi$
$887$	−33228.2	−1.25783	−0.628914	+	0.777475i	$0.716500\pi$
$888$	0	0
$889$	13707.6	0.517140
$890$	0	0
$891$	−2520.22	−0.0947595
$892$	0	0
$893$	−8439.31	−0.316250
$894$	0	0
$895$	17020.5	0.635679
$896$	0	0
$897$	−7258.27	−0.270174
$898$	0	0
$899$	7694.88	0.285471
$900$	0	0
$901$	13790.0	0.509891
$902$	0	0
$903$	−7263.79	−0.267690
$904$	0	0
$905$	6999.26	0.257087
$906$	0	0
$907$	−31926.4	−1.16880	−0.584399	−	0.811466i	$-0.698670\pi$
$907$	−31926.4	−1.16880	−0.584399	+	0.811466i	$0.698670\pi$
$908$	0	0
$909$	742.683	0.0270993
$910$	0	0
$911$	31486.0	1.14509	0.572545	−	0.819874i	$-0.305956\pi$
$911$	31486.0	1.14509	0.572545	+	0.819874i	$0.305956\pi$
$912$	0	0
$913$	−15144.2	−0.548960
$914$	0	0
$915$	23821.3	0.860664
$916$	0	0
$917$	2242.99	0.0807743
$918$	0	0
$919$	−18031.8	−0.647240	−0.323620	−	0.946187i	$-0.604900\pi$
$919$	−18031.8	−0.647240	−0.323620	+	0.946187i	$0.604900\pi$
$920$	0	0
$921$	440.163	0.0157480
$922$	0	0
$923$	36800.2	1.31234
$924$	0	0
$925$	−20302.9	−0.721683
$926$	0	0
$927$	−7319.86	−0.259348
$928$	0	0
$929$	9760.08	0.344691	0.172345	−	0.985037i	$-0.444865\pi$
$929$	9760.08	0.344691	0.172345	+	0.985037i	$0.444865\pi$
$930$	0	0
$931$	−1055.70	−0.0371635
$932$	0	0
$933$	−32764.3	−1.14968
$934$	0	0
$935$	−9710.05	−0.339628
$936$	0	0
$937$	−16441.1	−0.573221	−0.286611	−	0.958047i	$-0.592529\pi$
$937$	−16441.1	−0.573221	−0.286611	+	0.958047i	$0.592529\pi$
$938$	0	0
$939$	−18613.6	−0.646892
$940$	0	0
$941$	−38520.8	−1.33448	−0.667239	−	0.744844i	$-0.732524\pi$
$941$	−38520.8	−1.33448	−0.667239	+	0.744844i	$0.732524\pi$
$942$	0	0
$943$	−11039.6	−0.381230
$944$	0	0
$945$	8011.80	0.275792
$946$	0	0
$947$	−36712.7	−1.25977	−0.629885	−	0.776688i	$-0.716898\pi$
$947$	−36712.7	−1.25977	−0.629885	+	0.776688i	$0.716898\pi$
$948$	0	0
$949$	−17322.0	−0.592513
$950$	0	0
$951$	−4680.82	−0.159607
$952$	0	0
$953$	2875.31	0.0977340	0.0488670	−	0.998805i	$-0.484439\pi$
$953$	2875.31	0.0977340	0.0488670	+	0.998805i	$0.484439\pi$
$954$	0	0
$955$	−32056.4	−1.08620
$956$	0	0
$957$	−2094.87	−0.0707603
$958$	0	0
$959$	13117.5	0.441694
$960$	0	0
$961$	−6310.96	−0.211841
$962$	0	0
$963$	11670.1	0.390511
$964$	0	0
$965$	−35586.1	−1.18711
$966$	0	0
$967$	18435.7	0.613082	0.306541	−	0.951857i	$-0.400828\pi$
$967$	18435.7	0.613082	0.306541	+	0.951857i	$0.400828\pi$
$968$	0	0
$969$	9467.99	0.313886
$970$	0	0
$971$	−10382.8	−0.343152	−0.171576	−	0.985171i	$-0.554886\pi$
$971$	−10382.8	−0.343152	−0.171576	+	0.985171i	$0.554886\pi$
$972$	0	0
$973$	3562.49	0.117377
$974$	0	0
$975$	−13255.1	−0.435386
$976$	0	0
$977$	44121.6	1.44481	0.722403	−	0.691472i	$-0.243037\pi$
$977$	44121.6	1.44481	0.722403	+	0.691472i	$0.243037\pi$
$978$	0	0
$979$	−6652.90	−0.217188
$980$	0	0
$981$	14340.2	0.466716
$982$	0	0
$983$	−14664.0	−0.475798	−0.237899	−	0.971290i	$-0.576459\pi$
$983$	−14664.0	−0.475798	−0.237899	+	0.971290i	$0.576459\pi$
$984$	0	0
$985$	17407.9	0.563109
$986$	0	0
$987$	10398.6	0.335349
$988$	0	0
$989$	10034.1	0.322615
$990$	0	0
$991$	−39530.0	−1.26711	−0.633557	−	0.773696i	$-0.718406\pi$
$991$	−39530.0	−1.26711	−0.633557	+	0.773696i	$0.718406\pi$
$992$	0	0
$993$	1419.85	0.0453752
$994$	0	0
$995$	24363.1	0.776244
$996$	0	0
$997$	47679.5	1.51457	0.757284	−	0.653086i	$-0.226526\pi$
$997$	47679.5	1.51457	0.757284	+	0.653086i	$0.226526\pi$
$998$	0	0
$999$	45549.7	1.44257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.4.a.d.1.2	✓	4
4.3	odd	2		1232.4.a.v.1.3		4
7.6	odd	2		2156.4.a.f.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.d.1.2	✓	4	1.1	even	1	trivial
1232.4.a.v.1.3		4	4.3	odd	2
2156.4.a.f.1.3		4	7.6	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:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

\(f(q)\)	\(=\)	\(q-3.79239 q^{3} -7.61779 q^{5} -7.00000 q^{7} -12.6178 q^{9} +11.0000 q^{11} -52.1907 q^{13} +28.8896 q^{15} +115.878 q^{17} -21.5449 q^{19} +26.5467 q^{21} -36.6714 q^{23} -66.9693 q^{25} +150.246 q^{27} +50.2172 q^{29} +153.232 q^{31} -41.7163 q^{33} +53.3245 q^{35} +303.168 q^{37} +197.927 q^{39} +301.043 q^{41} -273.623 q^{43} +96.1196 q^{45} +391.708 q^{47} +49.0000 q^{49} -439.453 q^{51} +119.005 q^{53} -83.7956 q^{55} +81.7067 q^{57} -243.783 q^{59} +824.563 q^{61} +88.3245 q^{63} +397.577 q^{65} +310.219 q^{67} +139.072 q^{69} -705.110 q^{71} +331.898 q^{73} +253.974 q^{75} -77.0000 q^{77} +812.935 q^{79} -229.111 q^{81} -1376.75 q^{83} -882.731 q^{85} -190.443 q^{87} -604.809 q^{89} +365.335 q^{91} -581.115 q^{93} +164.125 q^{95} +963.838 q^{97} -138.796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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