LMFDB - Embedded newform 336.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(1,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.8246417619$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 14$ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	336.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +10.5498 q^{5} -7.00000 q^{7} +9.00000 q^{9} -34.7492 q^{11} -37.2990 q^{13} -31.6495 q^{15} -10.5498 q^{17} +58.5980 q^{19} +21.0000 q^{21} +125.347 q^{23} -13.7010 q^{25} -27.0000 q^{27} -35.4020 q^{29} -291.794 q^{31} +104.248 q^{33} -73.8488 q^{35} -259.897 q^{37} +111.897 q^{39} -338.248 q^{41} -6.80397 q^{43} +94.9485 q^{45} -250.694 q^{47} +49.0000 q^{49} +31.6495 q^{51} -536.900 q^{53} -366.598 q^{55} -175.794 q^{57} +35.8904 q^{59} +57.7940 q^{61} -63.0000 q^{63} -393.498 q^{65} -481.691 q^{67} -376.042 q^{69} -363.752 q^{71} +581.299 q^{73} +41.1030 q^{75} +243.244 q^{77} +693.691 q^{79} +81.0000 q^{81} -1334.39 q^{83} -111.299 q^{85} +106.206 q^{87} -353.038 q^{89} +261.093 q^{91} +875.382 q^{93} +618.199 q^{95} +1445.88 q^{97} -312.743 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{3} + 6 q^{5} - 14 q^{7} + 18 q^{9} + 6 q^{11} + 16 q^{13} - 18 q^{15} - 6 q^{17} - 64 q^{19} + 42 q^{21} - 6 q^{23} - 118 q^{25} - 54 q^{27} - 252 q^{29} - 40 q^{31} - 18 q^{33} - 42 q^{35} - 248 q^{37}+ \cdots + 54 q^{99}+O(q^{100})$ 2 * q - 6 * q^3 + 6 * q^5 - 14 * q^7 + 18 * q^9 + 6 * q^11 + 16 * q^13 - 18 * q^15 - 6 * q^17 - 64 * q^19 + 42 * q^21 - 6 * q^23 - 118 * q^25 - 54 * q^27 - 252 * q^29 - 40 * q^31 - 18 * q^33 - 42 * q^35 - 248 * q^37 - 48 * q^39 - 450 * q^41 - 376 * q^43 + 54 * q^45 + 12 * q^47 + 98 * q^49 + 18 * q^51 - 1104 * q^53 - 552 * q^55 + 192 * q^57 - 804 * q^59 - 428 * q^61 - 126 * q^63 - 636 * q^65 - 148 * q^67 + 18 * q^69 - 954 * q^71 + 1072 * q^73 + 354 * q^75 - 42 * q^77 + 572 * q^79 + 162 * q^81 - 1944 * q^83 - 132 * q^85 + 756 * q^87 + 366 * q^89 - 112 * q^91 + 120 * q^93 + 1176 * q^95 + 808 * q^97 + 54 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	10.5498	0.943606	0.471803	−	0.881704i	$-0.343603\pi$
$5$	10.5498	0.943606	0.471803	+	0.881704i	$0.343603\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−34.7492	−0.952479	−0.476240	−	0.879316i	$-0.658000\pi$
$11$	−34.7492	−0.952479	−0.476240	+	0.879316i	$0.658000\pi$
$12$	0	0
$13$	−37.2990	−0.795760	−0.397880	−	0.917437i	$-0.630254\pi$
$13$	−37.2990	−0.795760	−0.397880	+	0.917437i	$0.630254\pi$
$14$	0	0
$15$	−31.6495	−0.544791
$16$	0	0
$17$	−10.5498	−0.150512	−0.0752562	−	0.997164i	$-0.523977\pi$
$17$	−10.5498	−0.150512	−0.0752562	+	0.997164i	$0.523977\pi$
$18$	0	0
$19$	58.5980	0.707542	0.353771	−	0.935332i	$-0.384899\pi$
$19$	58.5980	0.707542	0.353771	+	0.935332i	$0.384899\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	125.347	1.13638	0.568189	−	0.822898i	$-0.307644\pi$
$23$	125.347	1.13638	0.568189	+	0.822898i	$0.307644\pi$
$24$	0	0
$25$	−13.7010	−0.109608
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−35.4020	−0.226689	−0.113345	−	0.993556i	$-0.536156\pi$
$29$	−35.4020	−0.226689	−0.113345	+	0.993556i	$0.536156\pi$
$30$	0	0
$31$	−291.794	−1.69057	−0.845286	−	0.534313i	$-0.820570\pi$
$31$	−291.794	−1.69057	−0.845286	+	0.534313i	$0.820570\pi$
$32$	0	0
$33$	104.248	0.549914
$34$	0	0
$35$	−73.8488	−0.356649
$36$	0	0
$37$	−259.897	−1.15478	−0.577389	−	0.816469i	$-0.695928\pi$
$37$	−259.897	−1.15478	−0.577389	+	0.816469i	$0.695928\pi$
$38$	0	0
$39$	111.897	0.459432
$40$	0	0
$41$	−338.248	−1.28842	−0.644212	−	0.764847i	$-0.722815\pi$
$41$	−338.248	−1.28842	−0.644212	+	0.764847i	$0.722815\pi$
$42$	0	0
$43$	−6.80397	−0.0241301	−0.0120651	−	0.999927i	$-0.503841\pi$
$43$	−6.80397	−0.0241301	−0.0120651	+	0.999927i	$0.503841\pi$
$44$	0	0
$45$	94.9485	0.314535
$46$	0	0
$47$	−250.694	−0.778033	−0.389016	−	0.921231i	$-0.627185\pi$
$47$	−250.694	−0.778033	−0.389016	+	0.921231i	$0.627185\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	31.6495	0.0868984
$52$	0	0
$53$	−536.900	−1.39149	−0.695745	−	0.718289i	$-0.744925\pi$
$53$	−536.900	−1.39149	−0.695745	+	0.718289i	$0.744925\pi$
$54$	0	0
$55$	−366.598	−0.898765
$56$	0	0
$57$	−175.794	−0.408500
$58$	0	0
$59$	35.8904	0.0791955	0.0395977	−	0.999216i	$-0.487392\pi$
$59$	35.8904	0.0791955	0.0395977	+	0.999216i	$0.487392\pi$
$60$	0	0
$61$	57.7940	0.121308	0.0606538	−	0.998159i	$-0.480681\pi$
$61$	57.7940	0.121308	0.0606538	+	0.998159i	$0.480681\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−393.498	−0.750884
$66$	0	0
$67$	−481.691	−0.878327	−0.439164	−	0.898407i	$-0.644725\pi$
$67$	−481.691	−0.878327	−0.439164	+	0.898407i	$0.644725\pi$
$68$	0	0
$69$	−376.042	−0.656088
$70$	0	0
$71$	−363.752	−0.608021	−0.304010	−	0.952669i	$-0.598326\pi$
$71$	−363.752	−0.608021	−0.304010	+	0.952669i	$0.598326\pi$
$72$	0	0
$73$	581.299	0.931999	0.465999	−	0.884785i	$-0.345695\pi$
$73$	581.299	0.931999	0.465999	+	0.884785i	$0.345695\pi$
$74$	0	0
$75$	41.1030	0.0632822
$76$	0	0
$77$	243.244	0.360003
$78$	0	0
$79$	693.691	0.987928	0.493964	−	0.869482i	$-0.335547\pi$
$79$	693.691	0.987928	0.493964	+	0.869482i	$0.335547\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1334.39	−1.76468	−0.882341	−	0.470611i	$-0.844033\pi$
$83$	−1334.39	−1.76468	−0.882341	+	0.470611i	$0.844033\pi$
$84$	0	0
$85$	−111.299	−0.142024
$86$	0	0
$87$	106.206	0.130879
$88$	0	0
$89$	−353.038	−0.420472	−0.210236	−	0.977651i	$-0.567423\pi$
$89$	−353.038	−0.420472	−0.210236	+	0.977651i	$0.567423\pi$
$90$	0	0
$91$	261.093	0.300769
$92$	0	0
$93$	875.382	0.976053
$94$	0	0
$95$	618.199	0.667641
$96$	0	0
$97$	1445.88	1.51347	0.756735	−	0.653722i	$-0.226793\pi$
$97$	1445.88	1.51347	0.756735	+	0.653722i	$0.226793\pi$
$98$	0	0
$99$	−312.743	−0.317493
$100$	0	0
$101$	474.852	0.467817	0.233909	−	0.972259i	$-0.424848\pi$
$101$	474.852	0.467817	0.233909	+	0.972259i	$0.424848\pi$
$102$	0	0
$103$	1999.59	1.91287	0.956433	−	0.291951i	$-0.0943044\pi$
$103$	1999.59	1.91287	0.956433	+	0.291951i	$0.0943044\pi$
$104$	0	0
$105$	221.547	0.205912
$106$	0	0
$107$	−1166.74	−1.05414	−0.527068	−	0.849823i	$-0.676709\pi$
$107$	−1166.74	−1.05414	−0.527068	+	0.849823i	$0.676709\pi$
$108$	0	0
$109$	−1337.18	−1.17503	−0.587515	−	0.809213i	$-0.699894\pi$
$109$	−1337.18	−1.17503	−0.587515	+	0.809213i	$0.699894\pi$
$110$	0	0
$111$	779.691	0.666712
$112$	0	0
$113$	906.578	0.754723	0.377361	−	0.926066i	$-0.376831\pi$
$113$	906.578	0.754723	0.377361	+	0.926066i	$0.376831\pi$
$114$	0	0
$115$	1322.39	1.07229
$116$	0	0
$117$	−335.691	−0.265253
$118$	0	0
$119$	73.8488	0.0568883
$120$	0	0
$121$	−123.495	−0.0927836
$122$	0	0
$123$	1014.74	0.743872
$124$	0	0
$125$	−1463.27	−1.04703
$126$	0	0
$127$	1714.89	1.19820	0.599101	−	0.800674i	$-0.295525\pi$
$127$	1714.89	1.19820	0.599101	+	0.800674i	$0.295525\pi$
$128$	0	0
$129$	20.4119	0.0139315
$130$	0	0
$131$	−470.611	−0.313874	−0.156937	−	0.987609i	$-0.550162\pi$
$131$	−470.611	−0.313874	−0.156937	+	0.987609i	$0.550162\pi$
$132$	0	0
$133$	−410.186	−0.267426
$134$	0	0
$135$	−284.846	−0.181597
$136$	0	0
$137$	−443.910	−0.276831	−0.138415	−	0.990374i	$-0.544201\pi$
$137$	−443.910	−0.276831	−0.138415	+	0.990374i	$0.544201\pi$
$138$	0	0
$139$	−1669.98	−1.01904	−0.509518	−	0.860460i	$-0.670176\pi$
$139$	−1669.98	−1.01904	−0.509518	+	0.860460i	$0.670176\pi$
$140$	0	0
$141$	752.083	0.449197
$142$	0	0
$143$	1296.11	0.757945
$144$	0	0
$145$	−373.485	−0.213905
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	743.871	0.408995	0.204497	−	0.978867i	$-0.434444\pi$
$149$	743.871	0.408995	0.204497	+	0.978867i	$0.434444\pi$
$150$	0	0
$151$	−606.764	−0.327005	−0.163503	−	0.986543i	$-0.552279\pi$
$151$	−606.764	−0.327005	−0.163503	+	0.986543i	$0.552279\pi$
$152$	0	0
$153$	−94.9485	−0.0501708
$154$	0	0
$155$	−3078.38	−1.59523
$156$	0	0
$157$	3114.78	1.58336	0.791678	−	0.610939i	$-0.209208\pi$
$157$	3114.78	1.58336	0.791678	+	0.610939i	$0.209208\pi$
$158$	0	0
$159$	1610.70	0.803377
$160$	0	0
$161$	−877.430	−0.429511
$162$	0	0
$163$	−2413.07	−1.15955	−0.579774	−	0.814777i	$-0.696859\pi$
$163$	−2413.07	−1.15955	−0.579774	+	0.814777i	$0.696859\pi$
$164$	0	0
$165$	1099.79	0.518902
$166$	0	0
$167$	610.475	0.282874	0.141437	−	0.989947i	$-0.454828\pi$
$167$	610.475	0.282874	0.141437	+	0.989947i	$0.454828\pi$
$168$	0	0
$169$	−805.784	−0.366766
$170$	0	0
$171$	527.382	0.235847
$172$	0	0
$173$	3793.81	1.66727	0.833636	−	0.552315i	$-0.186255\pi$
$173$	3793.81	1.66727	0.833636	+	0.552315i	$0.186255\pi$
$174$	0	0
$175$	95.9070	0.0414279
$176$	0	0
$177$	−107.671	−0.0457235
$178$	0	0
$179$	2804.68	1.17112	0.585562	−	0.810627i	$-0.300874\pi$
$179$	2804.68	1.17112	0.585562	+	0.810627i	$0.300874\pi$
$180$	0	0
$181$	3106.04	1.27553	0.637763	−	0.770232i	$-0.279860\pi$
$181$	3106.04	1.27553	0.637763	+	0.770232i	$0.279860\pi$
$182$	0	0
$183$	−173.382	−0.0700370
$184$	0	0
$185$	−2741.87	−1.08966
$186$	0	0
$187$	366.598	0.143360
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−261.952	−0.0992365	−0.0496182	−	0.998768i	$-0.515800\pi$
$191$	−261.952	−0.0992365	−0.0496182	+	0.998768i	$0.515800\pi$
$192$	0	0
$193$	4051.07	1.51089	0.755447	−	0.655210i	$-0.227420\pi$
$193$	4051.07	1.51089	0.755447	+	0.655210i	$0.227420\pi$
$194$	0	0
$195$	1180.50	0.433523
$196$	0	0
$197$	−2874.83	−1.03971	−0.519855	−	0.854254i	$-0.674014\pi$
$197$	−2874.83	−1.03971	−0.519855	+	0.854254i	$0.674014\pi$
$198$	0	0
$199$	3066.97	1.09252	0.546261	−	0.837615i	$-0.316051\pi$
$199$	3066.97	1.09252	0.546261	+	0.837615i	$0.316051\pi$
$200$	0	0
$201$	1445.07	0.507103
$202$	0	0
$203$	247.814	0.0856804
$204$	0	0
$205$	−3568.46	−1.21576
$206$	0	0
$207$	1128.12	0.378793
$208$	0	0
$209$	−2036.23	−0.673919
$210$	0	0
$211$	−595.422	−0.194268	−0.0971340	−	0.995271i	$-0.530968\pi$
$211$	−595.422	−0.194268	−0.0971340	+	0.995271i	$0.530968\pi$
$212$	0	0
$213$	1091.26	0.351041
$214$	0	0
$215$	−71.7808	−0.0227693
$216$	0	0
$217$	2042.56	0.638976
$218$	0	0
$219$	−1743.90	−0.538090
$220$	0	0
$221$	393.498	0.119772
$222$	0	0
$223$	3779.79	1.13504	0.567520	−	0.823360i	$-0.307903\pi$
$223$	3779.79	1.13504	0.567520	+	0.823360i	$0.307903\pi$
$224$	0	0
$225$	−123.309	−0.0365360
$226$	0	0
$227$	−1827.62	−0.534376	−0.267188	−	0.963644i	$-0.586094\pi$
$227$	−1827.62	−0.534376	−0.267188	+	0.963644i	$0.586094\pi$
$228$	0	0
$229$	−850.249	−0.245354	−0.122677	−	0.992447i	$-0.539148\pi$
$229$	−850.249	−0.245354	−0.122677	+	0.992447i	$0.539148\pi$
$230$	0	0
$231$	−729.733	−0.207848
$232$	0	0
$233$	−6591.10	−1.85321	−0.926604	−	0.376039i	$-0.877286\pi$
$233$	−6591.10	−1.85321	−0.926604	+	0.376039i	$0.877286\pi$
$234$	0	0
$235$	−2644.78	−0.734156
$236$	0	0
$237$	−2081.07	−0.570381
$238$	0	0
$239$	182.556	0.0494083	0.0247042	−	0.999695i	$-0.492136\pi$
$239$	182.556	0.0494083	0.0247042	+	0.999695i	$0.492136\pi$
$240$	0	0
$241$	1523.90	0.407315	0.203657	−	0.979042i	$-0.434717\pi$
$241$	1523.90	0.407315	0.203657	+	0.979042i	$0.434717\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	516.942	0.134801
$246$	0	0
$247$	−2185.65	−0.563034
$248$	0	0
$249$	4003.18	1.01884
$250$	0	0
$251$	−2357.73	−0.592903	−0.296451	−	0.955048i	$-0.595803\pi$
$251$	−2357.73	−0.592903	−0.296451	+	0.955048i	$0.595803\pi$
$252$	0	0
$253$	−4355.71	−1.08238
$254$	0	0
$255$	333.897	0.0819978
$256$	0	0
$257$	2782.55	0.675372	0.337686	−	0.941259i	$-0.390356\pi$
$257$	2782.55	0.675372	0.337686	+	0.941259i	$0.390356\pi$
$258$	0	0
$259$	1819.28	0.436465
$260$	0	0
$261$	−318.618	−0.0755630
$262$	0	0
$263$	−2043.78	−0.479183	−0.239591	−	0.970874i	$-0.577013\pi$
$263$	−2043.78	−0.479183	−0.239591	+	0.970874i	$0.577013\pi$
$264$	0	0
$265$	−5664.21	−1.31302
$266$	0	0
$267$	1059.11	0.242759
$268$	0	0
$269$	3452.84	0.782614	0.391307	−	0.920260i	$-0.372023\pi$
$269$	3452.84	0.782614	0.391307	+	0.920260i	$0.372023\pi$
$270$	0	0
$271$	−2644.29	−0.592728	−0.296364	−	0.955075i	$-0.595774\pi$
$271$	−2644.29	−0.592728	−0.296364	+	0.955075i	$0.595774\pi$
$272$	0	0
$273$	−783.279	−0.173649
$274$	0	0
$275$	476.098	0.104399
$276$	0	0
$277$	2679.49	0.581208	0.290604	−	0.956843i	$-0.406144\pi$
$277$	2679.49	0.581208	0.290604	+	0.956843i	$0.406144\pi$
$278$	0	0
$279$	−2626.15	−0.563524
$280$	0	0
$281$	−1019.69	−0.216476	−0.108238	−	0.994125i	$-0.534521\pi$
$281$	−1019.69	−0.216476	−0.108238	+	0.994125i	$0.534521\pi$
$282$	0	0
$283$	−432.206	−0.0907844	−0.0453922	−	0.998969i	$-0.514454\pi$
$283$	−432.206	−0.0907844	−0.0453922	+	0.998969i	$0.514454\pi$
$284$	0	0
$285$	−1854.60	−0.385463
$286$	0	0
$287$	2367.73	0.486979
$288$	0	0
$289$	−4801.70	−0.977346
$290$	0	0
$291$	−4337.63	−0.873802
$292$	0	0
$293$	−2245.92	−0.447809	−0.223904	−	0.974611i	$-0.571880\pi$
$293$	−2245.92	−0.447809	−0.223904	+	0.974611i	$0.571880\pi$
$294$	0	0
$295$	378.638	0.0747293
$296$	0	0
$297$	938.228	0.183305
$298$	0	0
$299$	−4675.33	−0.904284
$300$	0	0
$301$	47.6278	0.00912034
$302$	0	0
$303$	−1424.56	−0.270094
$304$	0	0
$305$	609.718	0.114467
$306$	0	0
$307$	3197.08	0.594354	0.297177	−	0.954822i	$-0.403955\pi$
$307$	3197.08	0.594354	0.297177	+	0.954822i	$0.403955\pi$
$308$	0	0
$309$	−5998.76	−1.10439
$310$	0	0
$311$	3355.60	0.611829	0.305915	−	0.952059i	$-0.401038\pi$
$311$	3355.60	0.611829	0.305915	+	0.952059i	$0.401038\pi$
$312$	0	0
$313$	−2256.39	−0.407472	−0.203736	−	0.979026i	$-0.565308\pi$
$313$	−2256.39	−0.407472	−0.203736	+	0.979026i	$0.565308\pi$
$314$	0	0
$315$	−664.640	−0.118883
$316$	0	0
$317$	−6139.19	−1.08773	−0.543866	−	0.839172i	$-0.683040\pi$
$317$	−6139.19	−1.08773	−0.543866	+	0.839172i	$0.683040\pi$
$318$	0	0
$319$	1230.19	0.215917
$320$	0	0
$321$	3500.21	0.608606
$322$	0	0
$323$	−618.199	−0.106494
$324$	0	0
$325$	511.033	0.0872216
$326$	0	0
$327$	4011.53	0.678404
$328$	0	0
$329$	1754.86	0.294069
$330$	0	0
$331$	−7029.81	−1.16735	−0.583676	−	0.811987i	$-0.698386\pi$
$331$	−7029.81	−1.16735	−0.583676	+	0.811987i	$0.698386\pi$
$332$	0	0
$333$	−2339.07	−0.384926
$334$	0	0
$335$	−5081.76	−0.828795
$336$	0	0
$337$	10328.4	1.66951	0.834757	−	0.550619i	$-0.185608\pi$
$337$	10328.4	1.66951	0.834757	+	0.550619i	$0.185608\pi$
$338$	0	0
$339$	−2719.73	−0.435740
$340$	0	0
$341$	10139.6	1.61024
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3967.18	−0.619089
$346$	0	0
$347$	−1967.54	−0.304389	−0.152194	−	0.988351i	$-0.548634\pi$
$347$	−1967.54	−0.304389	−0.152194	+	0.988351i	$0.548634\pi$
$348$	0	0
$349$	−4365.46	−0.669564	−0.334782	−	0.942296i	$-0.608663\pi$
$349$	−4365.46	−0.669564	−0.334782	+	0.942296i	$0.608663\pi$
$350$	0	0
$351$	1007.07	0.153144
$352$	0	0
$353$	−6071.59	−0.915462	−0.457731	−	0.889091i	$-0.651338\pi$
$353$	−6071.59	−0.915462	−0.457731	+	0.889091i	$0.651338\pi$
$354$	0	0
$355$	−3837.53	−0.573732
$356$	0	0
$357$	−221.547	−0.0328445
$358$	0	0
$359$	−9638.04	−1.41693	−0.708463	−	0.705748i	$-0.750611\pi$
$359$	−9638.04	−1.41693	−0.708463	+	0.705748i	$0.750611\pi$
$360$	0	0
$361$	−3425.27	−0.499384
$362$	0	0
$363$	370.485	0.0535687
$364$	0	0
$365$	6132.61	0.879439
$366$	0	0
$367$	−522.725	−0.0743488	−0.0371744	−	0.999309i	$-0.511836\pi$
$367$	−522.725	−0.0743488	−0.0371744	+	0.999309i	$0.511836\pi$
$368$	0	0
$369$	−3044.23	−0.429475
$370$	0	0
$371$	3758.30	0.525934
$372$	0	0
$373$	3229.84	0.448351	0.224175	−	0.974549i	$-0.428031\pi$
$373$	3229.84	0.448351	0.224175	+	0.974549i	$0.428031\pi$
$374$	0	0
$375$	4389.82	0.604505
$376$	0	0
$377$	1320.46	0.180390
$378$	0	0
$379$	−6639.71	−0.899892	−0.449946	−	0.893056i	$-0.648557\pi$
$379$	−6639.71	−0.899892	−0.449946	+	0.893056i	$0.648557\pi$
$380$	0	0
$381$	−5144.66	−0.691782
$382$	0	0
$383$	14224.4	1.89774	0.948871	−	0.315664i	$-0.102227\pi$
$383$	14224.4	1.89774	0.948871	+	0.315664i	$0.102227\pi$
$384$	0	0
$385$	2566.19	0.339701
$386$	0	0
$387$	−61.2358	−0.00804338
$388$	0	0
$389$	2921.82	0.380828	0.190414	−	0.981704i	$-0.439017\pi$
$389$	2921.82	0.380828	0.190414	+	0.981704i	$0.439017\pi$
$390$	0	0
$391$	−1322.39	−0.171039
$392$	0	0
$393$	1411.83	0.181215
$394$	0	0
$395$	7318.33	0.932215
$396$	0	0
$397$	811.940	0.102645	0.0513226	−	0.998682i	$-0.483656\pi$
$397$	811.940	0.102645	0.0513226	+	0.998682i	$0.483656\pi$
$398$	0	0
$399$	1230.56	0.154398
$400$	0	0
$401$	2338.63	0.291237	0.145618	−	0.989341i	$-0.453483\pi$
$401$	2338.63	0.291237	0.145618	+	0.989341i	$0.453483\pi$
$402$	0	0
$403$	10883.6	1.34529
$404$	0	0
$405$	854.537	0.104845
$406$	0	0
$407$	9031.21	1.09990
$408$	0	0
$409$	−2727.57	−0.329755	−0.164877	−	0.986314i	$-0.552723\pi$
$409$	−2727.57	−0.329755	−0.164877	+	0.986314i	$0.552723\pi$
$410$	0	0
$411$	1331.73	0.159828
$412$	0	0
$413$	−251.233	−0.0299331
$414$	0	0
$415$	−14077.6	−1.66516
$416$	0	0
$417$	5009.94	0.588340
$418$	0	0
$419$	−13306.3	−1.55144	−0.775721	−	0.631076i	$-0.782614\pi$
$419$	−13306.3	−1.55144	−0.775721	+	0.631076i	$0.782614\pi$
$420$	0	0
$421$	−11007.5	−1.27428	−0.637138	−	0.770750i	$-0.719882\pi$
$421$	−11007.5	−1.27428	−0.637138	+	0.770750i	$0.719882\pi$
$422$	0	0
$423$	−2256.25	−0.259344
$424$	0	0
$425$	144.543	0.0164974
$426$	0	0
$427$	−404.558	−0.0458500
$428$	0	0
$429$	−3888.33	−0.437600
$430$	0	0
$431$	6525.62	0.729300	0.364650	−	0.931145i	$-0.381189\pi$
$431$	6525.62	0.729300	0.364650	+	0.931145i	$0.381189\pi$
$432$	0	0
$433$	−11716.3	−1.30034	−0.650171	−	0.759788i	$-0.725303\pi$
$433$	−11716.3	−1.30034	−0.650171	+	0.759788i	$0.725303\pi$
$434$	0	0
$435$	1120.46	0.123498
$436$	0	0
$437$	7345.10	0.804036
$438$	0	0
$439$	14611.4	1.58853	0.794264	−	0.607573i	$-0.207857\pi$
$439$	14611.4	1.58853	0.794264	+	0.607573i	$0.207857\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	15239.8	1.63446	0.817228	−	0.576314i	$-0.195510\pi$
$443$	15239.8	1.63446	0.817228	+	0.576314i	$0.195510\pi$
$444$	0	0
$445$	−3724.50	−0.396760
$446$	0	0
$447$	−2231.61	−0.236133
$448$	0	0
$449$	10678.8	1.12241	0.561206	−	0.827676i	$-0.310338\pi$
$449$	10678.8	1.12241	0.561206	+	0.827676i	$0.310338\pi$
$450$	0	0
$451$	11753.8	1.22720
$452$	0	0
$453$	1820.29	0.188796
$454$	0	0
$455$	2754.49	0.283807
$456$	0	0
$457$	4228.23	0.432797	0.216399	−	0.976305i	$-0.430569\pi$
$457$	4228.23	0.432797	0.216399	+	0.976305i	$0.430569\pi$
$458$	0	0
$459$	284.846	0.0289661
$460$	0	0
$461$	910.121	0.0919492	0.0459746	−	0.998943i	$-0.485361\pi$
$461$	910.121	0.0919492	0.0459746	+	0.998943i	$0.485361\pi$
$462$	0	0
$463$	−4456.16	−0.447290	−0.223645	−	0.974671i	$-0.571796\pi$
$463$	−4456.16	−0.447290	−0.223645	+	0.974671i	$0.571796\pi$
$464$	0	0
$465$	9235.14	0.921009
$466$	0	0
$467$	4429.42	0.438907	0.219453	−	0.975623i	$-0.429573\pi$
$467$	4429.42	0.438907	0.219453	+	0.975623i	$0.429573\pi$
$468$	0	0
$469$	3371.84	0.331977
$470$	0	0
$471$	−9344.35	−0.914151
$472$	0	0
$473$	236.432	0.0229835
$474$	0	0
$475$	−802.851	−0.0775523
$476$	0	0
$477$	−4832.10	−0.463830
$478$	0	0
$479$	−2752.85	−0.262591	−0.131296	−	0.991343i	$-0.541914\pi$
$479$	−2752.85	−0.262591	−0.131296	+	0.991343i	$0.541914\pi$
$480$	0	0
$481$	9693.90	0.918927
$482$	0	0
$483$	2632.29	0.247978
$484$	0	0
$485$	15253.8	1.42812
$486$	0	0
$487$	670.598	0.0623977	0.0311989	−	0.999513i	$-0.490067\pi$
$487$	670.598	0.0623977	0.0311989	+	0.999513i	$0.490067\pi$
$488$	0	0
$489$	7239.22	0.669466
$490$	0	0
$491$	8244.70	0.757797	0.378898	−	0.925438i	$-0.376303\pi$
$491$	8244.70	0.757797	0.378898	+	0.925438i	$0.376303\pi$
$492$	0	0
$493$	373.485	0.0341195
$494$	0	0
$495$	−3299.38	−0.299588
$496$	0	0
$497$	2546.27	0.229810
$498$	0	0
$499$	−8164.91	−0.732488	−0.366244	−	0.930519i	$-0.619356\pi$
$499$	−8164.91	−0.732488	−0.366244	+	0.930519i	$0.619356\pi$
$500$	0	0
$501$	−1831.43	−0.163317
$502$	0	0
$503$	−8175.59	−0.724715	−0.362357	−	0.932039i	$-0.618028\pi$
$503$	−8175.59	−0.724715	−0.362357	+	0.932039i	$0.618028\pi$
$504$	0	0
$505$	5009.61	0.441435
$506$	0	0
$507$	2417.35	0.211752
$508$	0	0
$509$	−878.448	−0.0764961	−0.0382480	−	0.999268i	$-0.512178\pi$
$509$	−878.448	−0.0764961	−0.0382480	+	0.999268i	$0.512178\pi$
$510$	0	0
$511$	−4069.09	−0.352262
$512$	0	0
$513$	−1582.15	−0.136167
$514$	0	0
$515$	21095.3	1.80499
$516$	0	0
$517$	8711.42	0.741060
$518$	0	0
$519$	−11381.4	−0.962600
$520$	0	0
$521$	11712.6	0.984910	0.492455	−	0.870338i	$-0.336100\pi$
$521$	11712.6	0.984910	0.492455	+	0.870338i	$0.336100\pi$
$522$	0	0
$523$	7341.82	0.613834	0.306917	−	0.951736i	$-0.400703\pi$
$523$	7341.82	0.613834	0.306917	+	0.951736i	$0.400703\pi$
$524$	0	0
$525$	−287.721	−0.0239184
$526$	0	0
$527$	3078.38	0.254452
$528$	0	0
$529$	3544.92	0.291355
$530$	0	0
$531$	323.014	0.0263985
$532$	0	0
$533$	12616.3	1.02528
$534$	0	0
$535$	−12308.9	−0.994690
$536$	0	0
$537$	−8414.03	−0.676149
$538$	0	0
$539$	−1702.71	−0.136068
$540$	0	0
$541$	−15868.7	−1.26109	−0.630545	−	0.776153i	$-0.717169\pi$
$541$	−15868.7	−1.26109	−0.630545	+	0.776153i	$0.717169\pi$
$542$	0	0
$543$	−9318.13	−0.736426
$544$	0	0
$545$	−14107.0	−1.10877
$546$	0	0
$547$	−2315.26	−0.180975	−0.0904875	−	0.995898i	$-0.528843\pi$
$547$	−2315.26	−0.180975	−0.0904875	+	0.995898i	$0.528843\pi$
$548$	0	0
$549$	520.146	0.0404359
$550$	0	0
$551$	−2074.49	−0.160392
$552$	0	0
$553$	−4855.84	−0.373402
$554$	0	0
$555$	8225.61	0.629113
$556$	0	0
$557$	−4819.05	−0.366588	−0.183294	−	0.983058i	$-0.558676\pi$
$557$	−4819.05	−0.366588	−0.183294	+	0.983058i	$0.558676\pi$
$558$	0	0
$559$	253.781	0.0192018
$560$	0	0
$561$	−1099.79	−0.0827689
$562$	0	0
$563$	−2540.86	−0.190203	−0.0951017	−	0.995468i	$-0.530318\pi$
$563$	−2540.86	−0.190203	−0.0951017	+	0.995468i	$0.530318\pi$
$564$	0	0
$565$	9564.25	0.712161
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−24220.0	−1.78445	−0.892227	−	0.451587i	$-0.850858\pi$
$569$	−24220.0	−1.78445	−0.892227	+	0.451587i	$0.850858\pi$
$570$	0	0
$571$	11772.1	0.862778	0.431389	−	0.902166i	$-0.358024\pi$
$571$	11772.1	0.862778	0.431389	+	0.902166i	$0.358024\pi$
$572$	0	0
$573$	785.855	0.0572942
$574$	0	0
$575$	−1717.38	−0.124556
$576$	0	0
$577$	10584.3	0.763655	0.381827	−	0.924234i	$-0.375295\pi$
$577$	10584.3	0.763655	0.381827	+	0.924234i	$0.375295\pi$
$578$	0	0
$579$	−12153.2	−0.872315
$580$	0	0
$581$	9340.74	0.666987
$582$	0	0
$583$	18656.8	1.32536
$584$	0	0
$585$	−3541.49	−0.250295
$586$	0	0
$587$	8712.63	0.612621	0.306311	−	0.951932i	$-0.400905\pi$
$587$	8712.63	0.612621	0.306311	+	0.951932i	$0.400905\pi$
$588$	0	0
$589$	−17098.6	−1.19615
$590$	0	0
$591$	8624.48	0.600277
$592$	0	0
$593$	−15362.9	−1.06387	−0.531937	−	0.846784i	$-0.678536\pi$
$593$	−15362.9	−1.06387	−0.531937	+	0.846784i	$0.678536\pi$
$594$	0	0
$595$	779.093	0.0536802
$596$	0	0
$597$	−9200.91	−0.630768
$598$	0	0
$599$	−26003.8	−1.77377	−0.886883	−	0.461994i	$-0.847134\pi$
$599$	−26003.8	−1.77377	−0.886883	+	0.461994i	$0.847134\pi$
$600$	0	0
$601$	20567.7	1.39596	0.697982	−	0.716115i	$-0.254082\pi$
$601$	20567.7	1.39596	0.697982	+	0.716115i	$0.254082\pi$
$602$	0	0
$603$	−4335.22	−0.292776
$604$	0	0
$605$	−1302.85	−0.0875512
$606$	0	0
$607$	−19642.1	−1.31342	−0.656711	−	0.754142i	$-0.728053\pi$
$607$	−19642.1	−1.31342	−0.656711	+	0.754142i	$0.728053\pi$
$608$	0	0
$609$	−743.442	−0.0494676
$610$	0	0
$611$	9350.65	0.619127
$612$	0	0
$613$	8454.59	0.557060	0.278530	−	0.960428i	$-0.410153\pi$
$613$	8454.59	0.557060	0.278530	+	0.960428i	$0.410153\pi$
$614$	0	0
$615$	10705.4	0.701922
$616$	0	0
$617$	−24168.4	−1.57696	−0.788479	−	0.615061i	$-0.789131\pi$
$617$	−24168.4	−1.57696	−0.788479	+	0.615061i	$0.789131\pi$
$618$	0	0
$619$	2037.56	0.132305	0.0661523	−	0.997810i	$-0.478928\pi$
$619$	2037.56	0.132305	0.0661523	+	0.997810i	$0.478928\pi$
$620$	0	0
$621$	−3384.37	−0.218696
$622$	0	0
$623$	2471.27	0.158923
$624$	0	0
$625$	−13724.7	−0.878378
$626$	0	0
$627$	6108.70	0.389088
$628$	0	0
$629$	2741.87	0.173808
$630$	0	0
$631$	−12339.5	−0.778489	−0.389244	−	0.921135i	$-0.627264\pi$
$631$	−12339.5	−0.778489	−0.389244	+	0.921135i	$0.627264\pi$
$632$	0	0
$633$	1786.27	0.112161
$634$	0	0
$635$	18091.8	1.13063
$636$	0	0
$637$	−1827.65	−0.113680
$638$	0	0
$639$	−3273.77	−0.202674
$640$	0	0
$641$	−10222.6	−0.629906	−0.314953	−	0.949107i	$-0.601989\pi$
$641$	−10222.6	−0.629906	−0.314953	+	0.949107i	$0.601989\pi$
$642$	0	0
$643$	1211.75	0.0743187	0.0371594	−	0.999309i	$-0.488169\pi$
$643$	1211.75	0.0743187	0.0371594	+	0.999309i	$0.488169\pi$
$644$	0	0
$645$	215.342	0.0131459
$646$	0	0
$647$	2817.22	0.171184	0.0855922	−	0.996330i	$-0.472722\pi$
$647$	2817.22	0.171184	0.0855922	+	0.996330i	$0.472722\pi$
$648$	0	0
$649$	−1247.16	−0.0754320
$650$	0	0
$651$	−6127.67	−0.368913
$652$	0	0
$653$	20986.2	1.25766	0.628831	−	0.777542i	$-0.283534\pi$
$653$	20986.2	1.25766	0.628831	+	0.777542i	$0.283534\pi$
$654$	0	0
$655$	−4964.87	−0.296173
$656$	0	0
$657$	5231.69	0.310666
$658$	0	0
$659$	2384.09	0.140927	0.0704635	−	0.997514i	$-0.477552\pi$
$659$	2384.09	0.140927	0.0704635	+	0.997514i	$0.477552\pi$
$660$	0	0
$661$	−7577.10	−0.445862	−0.222931	−	0.974834i	$-0.571562\pi$
$661$	−7577.10	−0.445862	−0.222931	+	0.974834i	$0.571562\pi$
$662$	0	0
$663$	−1180.50	−0.0691503
$664$	0	0
$665$	−4327.40	−0.252345
$666$	0	0
$667$	−4437.54	−0.257605
$668$	0	0
$669$	−11339.4	−0.655315
$670$	0	0
$671$	−2008.30	−0.115543
$672$	0	0
$673$	11724.6	0.671547	0.335774	−	0.941943i	$-0.391002\pi$
$673$	11724.6	0.671547	0.335774	+	0.941943i	$0.391002\pi$
$674$	0	0
$675$	369.927	0.0210941
$676$	0	0
$677$	32304.3	1.83390	0.916952	−	0.398997i	$-0.130642\pi$
$677$	32304.3	1.83390	0.916952	+	0.398997i	$0.130642\pi$
$678$	0	0
$679$	−10121.1	−0.572038
$680$	0	0
$681$	5482.85	0.308522
$682$	0	0
$683$	−33367.1	−1.86934	−0.934669	−	0.355519i	$-0.884304\pi$
$683$	−33367.1	−1.86934	−0.934669	+	0.355519i	$0.884304\pi$
$684$	0	0
$685$	−4683.18	−0.261219
$686$	0	0
$687$	2550.75	0.141655
$688$	0	0
$689$	20025.8	1.10729
$690$	0	0
$691$	1043.67	0.0574577	0.0287288	−	0.999587i	$-0.490854\pi$
$691$	1043.67	0.0574577	0.0287288	+	0.999587i	$0.490854\pi$
$692$	0	0
$693$	2189.20	0.120001
$694$	0	0
$695$	−17618.0	−0.961567
$696$	0	0
$697$	3568.46	0.193924
$698$	0	0
$699$	19773.3	1.06995
$700$	0	0
$701$	−11305.7	−0.609143	−0.304572	−	0.952489i	$-0.598513\pi$
$701$	−11305.7	−0.609143	−0.304572	+	0.952489i	$0.598513\pi$
$702$	0	0
$703$	−15229.4	−0.817055
$704$	0	0
$705$	7934.35	0.423865
$706$	0	0
$707$	−3323.97	−0.176818
$708$	0	0
$709$	−13306.8	−0.704860	−0.352430	−	0.935838i	$-0.614645\pi$
$709$	−13306.8	−0.704860	−0.352430	+	0.935838i	$0.614645\pi$
$710$	0	0
$711$	6243.22	0.329309
$712$	0	0
$713$	−36575.6	−1.92113
$714$	0	0
$715$	13673.7	0.715201
$716$	0	0
$717$	−547.669	−0.0285259
$718$	0	0
$719$	−10701.2	−0.555062	−0.277531	−	0.960717i	$-0.589516\pi$
$719$	−10701.2	−0.555062	−0.277531	+	0.960717i	$0.589516\pi$
$720$	0	0
$721$	−13997.1	−0.722996
$722$	0	0
$723$	−4571.69	−0.235163
$724$	0	0
$725$	485.042	0.0248469
$726$	0	0
$727$	2121.14	0.108210	0.0541051	−	0.998535i	$-0.482769\pi$
$727$	2121.14	0.108210	0.0541051	+	0.998535i	$0.482769\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	71.7808	0.00363189
$732$	0	0
$733$	−21584.0	−1.08762	−0.543809	−	0.839209i	$-0.683019\pi$
$733$	−21584.0	−1.08762	−0.543809	+	0.839209i	$0.683019\pi$
$734$	0	0
$735$	−1550.83	−0.0778273
$736$	0	0
$737$	16738.4	0.836588
$738$	0	0
$739$	9945.21	0.495048	0.247524	−	0.968882i	$-0.420383\pi$
$739$	9945.21	0.495048	0.247524	+	0.968882i	$0.420383\pi$
$740$	0	0
$741$	6556.94	0.325068
$742$	0	0
$743$	−2867.01	−0.141562	−0.0707808	−	0.997492i	$-0.522549\pi$
$743$	−2867.01	−0.141562	−0.0707808	+	0.997492i	$0.522549\pi$
$744$	0	0
$745$	7847.71	0.385930
$746$	0	0
$747$	−12009.5	−0.588227
$748$	0	0
$749$	8167.15	0.398426
$750$	0	0
$751$	10824.1	0.525934	0.262967	−	0.964805i	$-0.415299\pi$
$751$	10824.1	0.525934	0.262967	+	0.964805i	$0.415299\pi$
$752$	0	0
$753$	7073.19	0.342313
$754$	0	0
$755$	−6401.26	−0.308564
$756$	0	0
$757$	−14512.0	−0.696761	−0.348381	−	0.937353i	$-0.613268\pi$
$757$	−14512.0	−0.696761	−0.348381	+	0.937353i	$0.613268\pi$
$758$	0	0
$759$	13067.1	0.624910
$760$	0	0
$761$	−33075.8	−1.57556	−0.787778	−	0.615959i	$-0.788769\pi$
$761$	−33075.8	−1.57556	−0.787778	+	0.615959i	$0.788769\pi$
$762$	0	0
$763$	9360.23	0.444120
$764$	0	0
$765$	−1001.69	−0.0473415
$766$	0	0
$767$	−1338.68	−0.0630206
$768$	0	0
$769$	6728.44	0.315518	0.157759	−	0.987478i	$-0.449573\pi$
$769$	6728.44	0.315518	0.157759	+	0.987478i	$0.449573\pi$
$770$	0	0
$771$	−8347.65	−0.389926
$772$	0	0
$773$	−24233.3	−1.12757	−0.563784	−	0.825922i	$-0.690655\pi$
$773$	−24233.3	−1.12757	−0.563784	+	0.825922i	$0.690655\pi$
$774$	0	0
$775$	3997.87	0.185300
$776$	0	0
$777$	−5457.84	−0.251993
$778$	0	0
$779$	−19820.6	−0.911615
$780$	0	0
$781$	12640.1	0.579127
$782$	0	0
$783$	955.854	0.0436263
$784$	0	0
$785$	32860.5	1.49406
$786$	0	0
$787$	17200.4	0.779069	0.389535	−	0.921012i	$-0.372636\pi$
$787$	17200.4	0.779069	0.389535	+	0.921012i	$0.372636\pi$
$788$	0	0
$789$	6131.35	0.276656
$790$	0	0
$791$	−6346.05	−0.285258
$792$	0	0
$793$	−2155.66	−0.0965318
$794$	0	0
$795$	16992.6	0.758071
$796$	0	0
$797$	−4208.87	−0.187059	−0.0935295	−	0.995617i	$-0.529815\pi$
$797$	−4208.87	−0.187059	−0.0935295	+	0.995617i	$0.529815\pi$
$798$	0	0
$799$	2644.78	0.117104
$800$	0	0
$801$	−3177.34	−0.140157
$802$	0	0
$803$	−20199.7	−0.887709
$804$	0	0
$805$	−9256.74	−0.405289
$806$	0	0
$807$	−10358.5	−0.451842
$808$	0	0
$809$	−23632.1	−1.02702	−0.513511	−	0.858083i	$-0.671656\pi$
$809$	−23632.1	−1.02702	−0.513511	+	0.858083i	$0.671656\pi$
$810$	0	0
$811$	−28425.1	−1.23075	−0.615377	−	0.788233i	$-0.710996\pi$
$811$	−28425.1	−1.23075	−0.615377	+	0.788233i	$0.710996\pi$
$812$	0	0
$813$	7932.88	0.342212
$814$	0	0
$815$	−25457.5	−1.09416
$816$	0	0
$817$	−398.699	−0.0170731
$818$	0	0
$819$	2349.84	0.100256
$820$	0	0
$821$	39409.6	1.67528	0.837640	−	0.546223i	$-0.183935\pi$
$821$	39409.6	1.67528	0.837640	+	0.546223i	$0.183935\pi$
$822$	0	0
$823$	−16346.6	−0.692352	−0.346176	−	0.938170i	$-0.612520\pi$
$823$	−16346.6	−0.692352	−0.346176	+	0.938170i	$0.612520\pi$
$824$	0	0
$825$	−1428.29	−0.0602749
$826$	0	0
$827$	3738.87	0.157211	0.0786054	−	0.996906i	$-0.474953\pi$
$827$	3738.87	0.157211	0.0786054	+	0.996906i	$0.474953\pi$
$828$	0	0
$829$	−45196.2	−1.89352	−0.946761	−	0.321937i	$-0.895666\pi$
$829$	−45196.2	−1.89352	−0.946761	+	0.321937i	$0.895666\pi$
$830$	0	0
$831$	−8038.46	−0.335561
$832$	0	0
$833$	−516.942	−0.0215018
$834$	0	0
$835$	6440.41	0.266922
$836$	0	0
$837$	7878.44	0.325351
$838$	0	0
$839$	−15899.7	−0.654254	−0.327127	−	0.944980i	$-0.606080\pi$
$839$	−15899.7	−0.654254	−0.327127	+	0.944980i	$0.606080\pi$
$840$	0	0
$841$	−23135.7	−0.948612
$842$	0	0
$843$	3059.07	0.124982
$844$	0	0
$845$	−8500.89	−0.346082
$846$	0	0
$847$	864.465	0.0350689
$848$	0	0
$849$	1296.62	0.0524144
$850$	0	0
$851$	−32577.4	−1.31227
$852$	0	0
$853$	33926.7	1.36182	0.680908	−	0.732369i	$-0.261585\pi$
$853$	33926.7	1.36182	0.680908	+	0.732369i	$0.261585\pi$
$854$	0	0
$855$	5563.79	0.222547
$856$	0	0
$857$	−35432.4	−1.41231	−0.706154	−	0.708058i	$-0.749572\pi$
$857$	−35432.4	−1.41231	−0.706154	+	0.708058i	$0.749572\pi$
$858$	0	0
$859$	6780.17	0.269309	0.134655	−	0.990893i	$-0.457008\pi$
$859$	6780.17	0.269309	0.134655	+	0.990893i	$0.457008\pi$
$860$	0	0
$861$	−7103.20	−0.281157
$862$	0	0
$863$	30675.1	1.20995	0.604977	−	0.796243i	$-0.293182\pi$
$863$	30675.1	1.20995	0.604977	+	0.796243i	$0.293182\pi$
$864$	0	0
$865$	40024.1	1.57325
$866$	0	0
$867$	14405.1	0.564271
$868$	0	0
$869$	−24105.2	−0.940981
$870$	0	0
$871$	17966.6	0.698938
$872$	0	0
$873$	13012.9	0.504490
$874$	0	0
$875$	10242.9	0.395741
$876$	0	0
$877$	40861.3	1.57330	0.786652	−	0.617397i	$-0.211813\pi$
$877$	40861.3	1.57330	0.786652	+	0.617397i	$0.211813\pi$
$878$	0	0
$879$	6737.76	0.258543
$880$	0	0
$881$	−43839.0	−1.67647	−0.838236	−	0.545308i	$-0.816413\pi$
$881$	−43839.0	−1.67647	−0.838236	+	0.545308i	$0.816413\pi$
$882$	0	0
$883$	−44625.1	−1.70074	−0.850371	−	0.526183i	$-0.823623\pi$
$883$	−44625.1	−1.70074	−0.850371	+	0.526183i	$0.823623\pi$
$884$	0	0
$885$	−1135.91	−0.0431450
$886$	0	0
$887$	43967.5	1.66436	0.832178	−	0.554509i	$-0.187094\pi$
$887$	43967.5	1.66436	0.832178	+	0.554509i	$0.187094\pi$
$888$	0	0
$889$	−12004.2	−0.452878
$890$	0	0
$891$	−2814.68	−0.105831
$892$	0	0
$893$	−14690.2	−0.550491
$894$	0	0
$895$	29588.9	1.10508
$896$	0	0
$897$	14026.0	0.522089
$898$	0	0
$899$	10330.1	0.383234
$900$	0	0
$901$	5664.21	0.209436
$902$	0	0
$903$	−142.883	−0.00526563
$904$	0	0
$905$	32768.2	1.20359
$906$	0	0
$907$	13584.3	0.497309	0.248654	−	0.968592i	$-0.420012\pi$
$907$	13584.3	0.497309	0.248654	+	0.968592i	$0.420012\pi$
$908$	0	0
$909$	4273.67	0.155939
$910$	0	0
$911$	16421.6	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$911$	16421.6	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$912$	0	0
$913$	46369.0	1.68082
$914$	0	0
$915$	−1829.15	−0.0660873
$916$	0	0
$917$	3294.28	0.118633
$918$	0	0
$919$	29487.3	1.05843	0.529214	−	0.848488i	$-0.322487\pi$
$919$	29487.3	1.05843	0.529214	+	0.848488i	$0.322487\pi$
$920$	0	0
$921$	−9591.23	−0.343151
$922$	0	0
$923$	13567.6	0.483839
$924$	0	0
$925$	3560.85	0.126573
$926$	0	0
$927$	17996.3	0.637622
$928$	0	0
$929$	3441.85	0.121554	0.0607769	−	0.998151i	$-0.480642\pi$
$929$	3441.85	0.121554	0.0607769	+	0.998151i	$0.480642\pi$
$930$	0	0
$931$	2871.30	0.101077
$932$	0	0
$933$	−10066.8	−0.353240
$934$	0	0
$935$	3867.55	0.135275
$936$	0	0
$937$	5646.60	0.196869	0.0984346	−	0.995144i	$-0.468616\pi$
$937$	5646.60	0.196869	0.0984346	+	0.995144i	$0.468616\pi$
$938$	0	0
$939$	6769.18	0.235254
$940$	0	0
$941$	−44680.1	−1.54785	−0.773927	−	0.633275i	$-0.781710\pi$
$941$	−44680.1	−1.54785	−0.773927	+	0.633275i	$0.781710\pi$
$942$	0	0
$943$	−42398.4	−1.46414
$944$	0	0
$945$	1993.92	0.0686372
$946$	0	0
$947$	48924.6	1.67881	0.839406	−	0.543505i	$-0.182903\pi$
$947$	48924.6	1.67881	0.839406	+	0.543505i	$0.182903\pi$
$948$	0	0
$949$	−21681.9	−0.741647
$950$	0	0
$951$	18417.6	0.628002
$952$	0	0
$953$	52014.3	1.76801	0.884003	−	0.467482i	$-0.154839\pi$
$953$	52014.3	1.76801	0.884003	+	0.467482i	$0.154839\pi$
$954$	0	0
$955$	−2763.55	−0.0936401
$956$	0	0
$957$	−3690.57	−0.124660
$958$	0	0
$959$	3107.37	0.104632
$960$	0	0
$961$	55352.8	1.85804
$962$	0	0
$963$	−10500.6	−0.351379
$964$	0	0
$965$	42738.2	1.42569
$966$	0	0
$967$	47117.7	1.56691	0.783456	−	0.621448i	$-0.213455\pi$
$967$	47117.7	1.56691	0.783456	+	0.621448i	$0.213455\pi$
$968$	0	0
$969$	1854.60	0.0614843
$970$	0	0
$971$	8195.04	0.270846	0.135423	−	0.990788i	$-0.456761\pi$
$971$	8195.04	0.270846	0.135423	+	0.990788i	$0.456761\pi$
$972$	0	0
$973$	11689.9	0.385159
$974$	0	0
$975$	−1533.10	−0.0503574
$976$	0	0
$977$	4643.51	0.152056	0.0760282	−	0.997106i	$-0.475776\pi$
$977$	4643.51	0.152056	0.0760282	+	0.997106i	$0.475776\pi$
$978$	0	0
$979$	12267.8	0.400490
$980$	0	0
$981$	−12034.6	−0.391677
$982$	0	0
$983$	−43986.5	−1.42721	−0.713607	−	0.700546i	$-0.752940\pi$
$983$	−43986.5	−1.42721	−0.713607	+	0.700546i	$0.752940\pi$
$984$	0	0
$985$	−30329.0	−0.981077
$986$	0	0
$987$	−5264.58	−0.169781
$988$	0	0
$989$	−852.859	−0.0274210
$990$	0	0
$991$	−1595.21	−0.0511337	−0.0255668	−	0.999673i	$-0.508139\pi$
$991$	−1595.21	−0.0511337	−0.0255668	+	0.999673i	$0.508139\pi$
$992$	0	0
$993$	21089.4	0.673971
$994$	0	0
$995$	32356.0	1.03091
$996$	0	0
$997$	21501.2	0.682998	0.341499	−	0.939882i	$-0.389065\pi$
$997$	21501.2	0.682998	0.341499	+	0.939882i	$0.389065\pi$
$998$	0	0
$999$	7017.22	0.222237

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.a.m.1.2		2
3.2	odd	2		1008.4.a.ba.1.1		2
4.3	odd	2		21.4.a.c.1.1	✓	2
7.6	odd	2		2352.4.a.bz.1.1		2
8.3	odd	2		1344.4.a.bg.1.1		2
8.5	even	2		1344.4.a.bo.1.1		2
12.11	even	2		63.4.a.e.1.2		2
20.3	even	4		525.4.d.g.274.4		4
20.7	even	4		525.4.d.g.274.1		4
20.19	odd	2		525.4.a.n.1.2		2
28.3	even	6		147.4.e.m.79.2		4
28.11	odd	6		147.4.e.l.79.2		4
28.19	even	6		147.4.e.m.67.2		4
28.23	odd	6		147.4.e.l.67.2		4
28.27	even	2		147.4.a.i.1.1		2
60.59	even	2		1575.4.a.p.1.1		2
84.11	even	6		441.4.e.q.226.1		4
84.23	even	6		441.4.e.q.361.1		4
84.47	odd	6		441.4.e.p.361.1		4
84.59	odd	6		441.4.e.p.226.1		4
84.83	odd	2		441.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.1	✓	2	4.3	odd	2
63.4.a.e.1.2		2	12.11	even	2
147.4.a.i.1.1		2	28.27	even	2
147.4.e.l.67.2		4	28.23	odd	6
147.4.e.l.79.2		4	28.11	odd	6
147.4.e.m.67.2		4	28.19	even	6
147.4.e.m.79.2		4	28.3	even	6
336.4.a.m.1.2		2	1.1	even	1	trivial
441.4.a.r.1.2		2	84.83	odd	2
441.4.e.p.226.1		4	84.59	odd	6
441.4.e.p.361.1		4	84.47	odd	6
441.4.e.q.226.1		4	84.11	even	6
441.4.e.q.361.1		4	84.23	even	6
525.4.a.n.1.2		2	20.19	odd	2
525.4.d.g.274.1		4	20.7	even	4
525.4.d.g.274.4		4	20.3	even	4
1008.4.a.ba.1.1		2	3.2	odd	2
1344.4.a.bg.1.1		2	8.3	odd	2
1344.4.a.bo.1.1		2	8.5	even	2
1575.4.a.p.1.1		2	60.59	even	2
2352.4.a.bz.1.1		2	7.6	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	336.4.a.m.1.2

Level	$336$
Weight	$4$
Character	336.1
Self dual	yes
Analytic conductor	$19.825$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$