LMFDB - Embedded newform 312.4.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.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:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.36248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 54x - 90 $ x^3 - x^2 - 54*x - 90
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.70213$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +18.8819 q^{5} -17.9266 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +18.8819 q^{5} -17.9266 q^{7} +9.00000 q^{9} +44.5723 q^{11} +13.0000 q^{13} -56.6456 q^{15} +34.0000 q^{17} -152.835 q^{19} +53.7799 q^{21} +107.234 q^{23} +231.525 q^{25} -27.0000 q^{27} +149.674 q^{29} +0.162885 q^{31} -133.717 q^{33} -338.489 q^{35} -256.342 q^{37} -39.0000 q^{39} +414.723 q^{41} +471.076 q^{43} +169.937 q^{45} +632.710 q^{47} -21.6354 q^{49} -102.000 q^{51} -236.489 q^{53} +841.608 q^{55} +458.505 q^{57} -108.018 q^{59} +888.292 q^{61} -161.340 q^{63} +245.464 q^{65} +637.149 q^{67} -321.702 q^{69} -362.597 q^{71} +723.177 q^{73} -694.576 q^{75} -799.031 q^{77} -964.684 q^{79} +81.0000 q^{81} -431.868 q^{83} +641.984 q^{85} -449.023 q^{87} +117.359 q^{89} -233.046 q^{91} -0.488655 q^{93} -2885.81 q^{95} -1153.38 q^{97} +401.151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9} - 20 q^{11} + 39 q^{13} - 48 q^{15} + 102 q^{17} - 38 q^{19} + 66 q^{21} + 32 q^{23} + 161 q^{25} - 81 q^{27} + 350 q^{29} + 50 q^{31} + 60 q^{33} + 232 q^{35} + 542 q^{37} - 117 q^{39} + 500 q^{41} + 420 q^{43} + 144 q^{45} + 324 q^{47} + 1119 q^{49} - 306 q^{51} + 538 q^{53} + 896 q^{55} + 114 q^{57} + 112 q^{59} + 1206 q^{61} - 198 q^{63} + 208 q^{65} + 950 q^{67} - 96 q^{69} - 1008 q^{71} + 1834 q^{73} - 483 q^{75} - 792 q^{77} - 272 q^{79} + 243 q^{81} - 1640 q^{83} + 544 q^{85} - 1050 q^{87} - 1576 q^{89} - 286 q^{91} - 150 q^{93} - 2760 q^{95} + 402 q^{97} - 180 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9 - 20 * q^11 + 39 * q^13 - 48 * q^15 + 102 * q^17 - 38 * q^19 + 66 * q^21 + 32 * q^23 + 161 * q^25 - 81 * q^27 + 350 * q^29 + 50 * q^31 + 60 * q^33 + 232 * q^35 + 542 * q^37 - 117 * q^39 + 500 * q^41 + 420 * q^43 + 144 * q^45 + 324 * q^47 + 1119 * q^49 - 306 * q^51 + 538 * q^53 + 896 * q^55 + 114 * q^57 + 112 * q^59 + 1206 * q^61 - 198 * q^63 + 208 * q^65 + 950 * q^67 - 96 * q^69 - 1008 * q^71 + 1834 * q^73 - 483 * q^75 - 792 * q^77 - 272 * q^79 + 243 * q^81 - 1640 * q^83 + 544 * q^85 - 1050 * q^87 - 1576 * q^89 - 286 * q^91 - 150 * q^93 - 2760 * q^95 + 402 * q^97 - 180 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	18.8819	1.68885	0.844423	−	0.535676i	$-0.179943\pi$
$5$	18.8819	1.68885	0.844423	+	0.535676i	$0.179943\pi$
$6$	0	0
$7$	−17.9266	−0.967948	−0.483974	−	0.875082i	$-0.660807\pi$
$7$	−17.9266	−0.967948	−0.483974	+	0.875082i	$0.660807\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	44.5723	1.22173	0.610866	−	0.791734i	$-0.290821\pi$
$11$	44.5723	1.22173	0.610866	+	0.791734i	$0.290821\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−56.6456	−0.975056
$16$	0	0
$17$	34.0000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	34.0000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−152.835	−1.84541	−0.922704	−	0.385510i	$-0.874026\pi$
$19$	−152.835	−1.84541	−0.922704	+	0.385510i	$0.874026\pi$
$20$	0	0
$21$	53.7799	0.558845
$22$	0	0
$23$	107.234	0.972168	0.486084	−	0.873912i	$-0.338425\pi$
$23$	107.234	0.972168	0.486084	+	0.873912i	$0.338425\pi$
$24$	0	0
$25$	231.525	1.85220
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	149.674	0.958407	0.479204	−	0.877704i	$-0.340926\pi$
$29$	149.674	0.958407	0.479204	+	0.877704i	$0.340926\pi$
$30$	0	0
$31$	0.162885	0.000943710 0	0.000471855	−	1.00000i	$-0.499850\pi$
$31$	0.162885	0.000943710 0	0.000471855		1.00000i	$0.499850\pi$
$32$	0	0
$33$	−133.717	−0.705367
$34$	0	0
$35$	−338.489	−1.63472
$36$	0	0
$37$	−256.342	−1.13898	−0.569491	−	0.821997i	$-0.692860\pi$
$37$	−256.342	−1.13898	−0.569491	+	0.821997i	$0.692860\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	414.723	1.57973	0.789865	−	0.613281i	$-0.210151\pi$
$41$	414.723	1.57973	0.789865	+	0.613281i	$0.210151\pi$
$42$	0	0
$43$	471.076	1.67066	0.835330	−	0.549749i	$-0.185277\pi$
$43$	471.076	1.67066	0.835330	+	0.549749i	$0.185277\pi$
$44$	0	0
$45$	169.937	0.562949
$46$	0	0
$47$	632.710	1.96362	0.981812	−	0.189857i	$-0.0608026\pi$
$47$	632.710	1.96362	0.981812	+	0.189857i	$0.0608026\pi$
$48$	0	0
$49$	−21.6354	−0.0630771
$50$	0	0
$51$	−102.000	−0.280056
$52$	0	0
$53$	−236.489	−0.612910	−0.306455	−	0.951885i	$-0.599143\pi$
$53$	−236.489	−0.612910	−0.306455	+	0.951885i	$0.599143\pi$
$54$	0	0
$55$	841.608	2.06332
$56$	0	0
$57$	458.505	1.06545
$58$	0	0
$59$	−108.018	−0.238351	−0.119176	−	0.992873i	$-0.538025\pi$
$59$	−108.018	−0.238351	−0.119176	+	0.992873i	$0.538025\pi$
$60$	0	0
$61$	888.292	1.86449	0.932247	−	0.361822i	$-0.117845\pi$
$61$	888.292	1.86449	0.932247	+	0.361822i	$0.117845\pi$
$62$	0	0
$63$	−161.340	−0.322649
$64$	0	0
$65$	245.464	0.468402
$66$	0	0
$67$	637.149	1.16179	0.580897	−	0.813977i	$-0.302702\pi$
$67$	637.149	1.16179	0.580897	+	0.813977i	$0.302702\pi$
$68$	0	0
$69$	−321.702	−0.561281
$70$	0	0
$71$	−362.597	−0.606090	−0.303045	−	0.952976i	$-0.598003\pi$
$71$	−362.597	−0.606090	−0.303045	+	0.952976i	$0.598003\pi$
$72$	0	0
$73$	723.177	1.15947	0.579736	−	0.814804i	$-0.303156\pi$
$73$	723.177	1.15947	0.579736	+	0.814804i	$0.303156\pi$
$74$	0	0
$75$	−694.576	−1.06937
$76$	0	0
$77$	−799.031	−1.18257
$78$	0	0
$79$	−964.684	−1.37387	−0.686933	−	0.726721i	$-0.741043\pi$
$79$	−964.684	−1.37387	−0.686933	+	0.726721i	$0.741043\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−431.868	−0.571128	−0.285564	−	0.958360i	$-0.592181\pi$
$83$	−431.868	−0.571128	−0.285564	+	0.958360i	$0.592181\pi$
$84$	0	0
$85$	641.984	0.819211
$86$	0	0
$87$	−449.023	−0.553337
$88$	0	0
$89$	117.359	0.139775	0.0698877	−	0.997555i	$-0.477736\pi$
$89$	117.359	0.139775	0.0698877	+	0.997555i	$0.477736\pi$
$90$	0	0
$91$	−233.046	−0.268460
$92$	0	0
$93$	−0.488655	−0.000544851 0
$94$	0	0
$95$	−2885.81	−3.11661
$96$	0	0
$97$	−1153.38	−1.20730	−0.603652	−	0.797248i	$-0.706288\pi$
$97$	−1153.38	−1.20730	−0.603652	+	0.797248i	$0.706288\pi$
$98$	0	0
$99$	401.151	0.407244
$100$	0	0
$101$	−848.183	−0.835618	−0.417809	−	0.908535i	$-0.637202\pi$
$101$	−848.183	−0.835618	−0.417809	+	0.908535i	$0.637202\pi$
$102$	0	0
$103$	−710.562	−0.679745	−0.339873	−	0.940471i	$-0.610384\pi$
$103$	−710.562	−0.679745	−0.339873	+	0.940471i	$0.610384\pi$
$104$	0	0
$105$	1015.47	0.943803
$106$	0	0
$107$	533.245	0.481782	0.240891	−	0.970552i	$-0.422560\pi$
$107$	533.245	0.481782	0.240891	+	0.970552i	$0.422560\pi$
$108$	0	0
$109$	−667.057	−0.586170	−0.293085	−	0.956086i	$-0.594682\pi$
$109$	−667.057	−0.586170	−0.293085	+	0.956086i	$0.594682\pi$
$110$	0	0
$111$	769.026	0.657592
$112$	0	0
$113$	302.916	0.252176	0.126088	−	0.992019i	$-0.459758\pi$
$113$	302.916	0.252176	0.126088	+	0.992019i	$0.459758\pi$
$114$	0	0
$115$	2024.78	1.64184
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−609.506	−0.469524
$120$	0	0
$121$	655.688	0.492628
$122$	0	0
$123$	−1244.17	−0.912057
$124$	0	0
$125$	2011.40	1.43924
$126$	0	0
$127$	−459.472	−0.321035	−0.160518	−	0.987033i	$-0.551316\pi$
$127$	−459.472	−0.321035	−0.160518	+	0.987033i	$0.551316\pi$
$128$	0	0
$129$	−1413.23	−0.964556
$130$	0	0
$131$	496.848	0.331373	0.165686	−	0.986179i	$-0.447016\pi$
$131$	496.848	0.331373	0.165686	+	0.986179i	$0.447016\pi$
$132$	0	0
$133$	2739.82	1.78626
$134$	0	0
$135$	−509.811	−0.325019
$136$	0	0
$137$	−2684.75	−1.67426	−0.837129	−	0.547006i	$-0.815768\pi$
$137$	−2684.75	−1.67426	−0.837129	+	0.547006i	$0.815768\pi$
$138$	0	0
$139$	−1728.99	−1.05504	−0.527522	−	0.849541i	$-0.676879\pi$
$139$	−1728.99	−1.05504	−0.527522	+	0.849541i	$0.676879\pi$
$140$	0	0
$141$	−1898.13	−1.13370
$142$	0	0
$143$	579.440	0.338847
$144$	0	0
$145$	2826.13	1.61860
$146$	0	0
$147$	64.9063	0.0364176
$148$	0	0
$149$	−1132.89	−0.622887	−0.311444	−	0.950265i	$-0.600812\pi$
$149$	−1132.89	−0.622887	−0.311444	+	0.950265i	$0.600812\pi$
$150$	0	0
$151$	−1258.52	−0.678259	−0.339130	−	0.940740i	$-0.610133\pi$
$151$	−1258.52	−0.678259	−0.339130	+	0.940740i	$0.610133\pi$
$152$	0	0
$153$	306.000	0.161690
$154$	0	0
$155$	3.07558	0.00159378
$156$	0	0
$157$	2469.89	1.25553	0.627766	−	0.778402i	$-0.283970\pi$
$157$	2469.89	1.25553	0.627766	+	0.778402i	$0.283970\pi$
$158$	0	0
$159$	709.466	0.353864
$160$	0	0
$161$	−1922.35	−0.941007
$162$	0	0
$163$	−733.892	−0.352656	−0.176328	−	0.984331i	$-0.556422\pi$
$163$	−733.892	−0.352656	−0.176328	+	0.984331i	$0.556422\pi$
$164$	0	0
$165$	−2524.83	−1.19126
$166$	0	0
$167$	2610.39	1.20957	0.604784	−	0.796389i	$-0.293259\pi$
$167$	2610.39	1.20957	0.604784	+	0.796389i	$0.293259\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1375.51	−0.615136
$172$	0	0
$173$	1164.04	0.511564	0.255782	−	0.966734i	$-0.417667\pi$
$173$	1164.04	0.511564	0.255782	+	0.966734i	$0.417667\pi$
$174$	0	0
$175$	−4150.47	−1.79284
$176$	0	0
$177$	324.053	0.137612
$178$	0	0
$179$	911.229	0.380494	0.190247	−	0.981736i	$-0.439071\pi$
$179$	911.229	0.380494	0.190247	+	0.981736i	$0.439071\pi$
$180$	0	0
$181$	−4817.82	−1.97848	−0.989242	−	0.146291i	$-0.953266\pi$
$181$	−4817.82	−1.97848	−0.989242	+	0.146291i	$0.953266\pi$
$182$	0	0
$183$	−2664.88	−1.07647
$184$	0	0
$185$	−4840.22	−1.92357
$186$	0	0
$187$	1515.46	0.592627
$188$	0	0
$189$	484.019	0.186282
$190$	0	0
$191$	674.312	0.255453	0.127726	−	0.991809i	$-0.459232\pi$
$191$	674.312	0.255453	0.127726	+	0.991809i	$0.459232\pi$
$192$	0	0
$193$	−963.634	−0.359398	−0.179699	−	0.983722i	$-0.557512\pi$
$193$	−963.634	−0.359398	−0.179699	+	0.983722i	$0.557512\pi$
$194$	0	0
$195$	−736.393	−0.270432
$196$	0	0
$197$	3995.21	1.44491	0.722454	−	0.691418i	$-0.243014\pi$
$197$	3995.21	1.44491	0.722454	+	0.691418i	$0.243014\pi$
$198$	0	0
$199$	−1619.07	−0.576749	−0.288374	−	0.957518i	$-0.593115\pi$
$199$	−1619.07	−0.576749	−0.288374	+	0.957518i	$0.593115\pi$
$200$	0	0
$201$	−1911.45	−0.670762
$202$	0	0
$203$	−2683.16	−0.927688
$204$	0	0
$205$	7830.76	2.66792
$206$	0	0
$207$	965.107	0.324056
$208$	0	0
$209$	−6812.20	−2.25459
$210$	0	0
$211$	543.991	0.177488	0.0887438	−	0.996054i	$-0.471715\pi$
$211$	543.991	0.177488	0.0887438	+	0.996054i	$0.471715\pi$
$212$	0	0
$213$	1087.79	0.349926
$214$	0	0
$215$	8894.79	2.82149
$216$	0	0
$217$	−2.91998	−0.000913462 0
$218$	0	0
$219$	−2169.53	−0.669421
$220$	0	0
$221$	442.000	0.134535
$222$	0	0
$223$	2447.45	0.734949	0.367474	−	0.930034i	$-0.380223\pi$
$223$	2447.45	0.734949	0.367474	+	0.930034i	$0.380223\pi$
$224$	0	0
$225$	2083.73	0.617401
$226$	0	0
$227$	467.798	0.136779	0.0683896	−	0.997659i	$-0.478214\pi$
$227$	467.798	0.136779	0.0683896	+	0.997659i	$0.478214\pi$
$228$	0	0
$229$	−361.180	−0.104225	−0.0521123	−	0.998641i	$-0.516595\pi$
$229$	−361.180	−0.104225	−0.0521123	+	0.998641i	$0.516595\pi$
$230$	0	0
$231$	2397.09	0.682759
$232$	0	0
$233$	−3068.38	−0.862732	−0.431366	−	0.902177i	$-0.641968\pi$
$233$	−3068.38	−0.862732	−0.431366	+	0.902177i	$0.641968\pi$
$234$	0	0
$235$	11946.8	3.31626
$236$	0	0
$237$	2894.05	0.793202
$238$	0	0
$239$	334.108	0.0904253	0.0452127	−	0.998977i	$-0.485603\pi$
$239$	334.108	0.0904253	0.0452127	+	0.998977i	$0.485603\pi$
$240$	0	0
$241$	514.961	0.137641	0.0688207	−	0.997629i	$-0.478076\pi$
$241$	514.961	0.137641	0.0688207	+	0.997629i	$0.478076\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−408.518	−0.106528
$246$	0	0
$247$	−1986.85	−0.511824
$248$	0	0
$249$	1295.60	0.329741
$250$	0	0
$251$	−7028.13	−1.76738	−0.883688	−	0.468076i	$-0.844947\pi$
$251$	−7028.13	−1.76738	−0.883688	+	0.468076i	$0.844947\pi$
$252$	0	0
$253$	4779.67	1.18773
$254$	0	0
$255$	−1925.95	−0.472972
$256$	0	0
$257$	−3572.23	−0.867040	−0.433520	−	0.901144i	$-0.642729\pi$
$257$	−3572.23	−0.867040	−0.433520	+	0.901144i	$0.642729\pi$
$258$	0	0
$259$	4595.35	1.10248
$260$	0	0
$261$	1347.07	0.319469
$262$	0	0
$263$	4276.74	1.00272	0.501359	−	0.865239i	$-0.332834\pi$
$263$	4276.74	1.00272	0.501359	+	0.865239i	$0.332834\pi$
$264$	0	0
$265$	−4465.35	−1.03511
$266$	0	0
$267$	−352.076	−0.0806993
$268$	0	0
$269$	2128.08	0.482347	0.241173	−	0.970482i	$-0.422468\pi$
$269$	2128.08	0.482347	0.241173	+	0.970482i	$0.422468\pi$
$270$	0	0
$271$	−4386.94	−0.983350	−0.491675	−	0.870779i	$-0.663615\pi$
$271$	−4386.94	−0.983350	−0.491675	+	0.870779i	$0.663615\pi$
$272$	0	0
$273$	699.139	0.154996
$274$	0	0
$275$	10319.6	2.26290
$276$	0	0
$277$	3411.86	0.740067	0.370034	−	0.929018i	$-0.379346\pi$
$277$	3411.86	0.740067	0.370034	+	0.929018i	$0.379346\pi$
$278$	0	0
$279$	1.46597	0.000314570 0
$280$	0	0
$281$	−4466.83	−0.948286	−0.474143	−	0.880448i	$-0.657242\pi$
$281$	−4466.83	−0.948286	−0.474143	+	0.880448i	$0.657242\pi$
$282$	0	0
$283$	−1879.30	−0.394746	−0.197373	−	0.980329i	$-0.563241\pi$
$283$	−1879.30	−0.394746	−0.197373	+	0.980329i	$0.563241\pi$
$284$	0	0
$285$	8657.43	1.79938
$286$	0	0
$287$	−7434.60	−1.52910
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	3460.15	0.697037
$292$	0	0
$293$	−207.996	−0.0414718	−0.0207359	−	0.999785i	$-0.506601\pi$
$293$	−207.996	−0.0414718	−0.0207359	+	0.999785i	$0.506601\pi$
$294$	0	0
$295$	−2039.58	−0.402539
$296$	0	0
$297$	−1203.45	−0.235122
$298$	0	0
$299$	1394.04	0.269631
$300$	0	0
$301$	−8444.80	−1.61711
$302$	0	0
$303$	2544.55	0.482444
$304$	0	0
$305$	16772.6	3.14885
$306$	0	0
$307$	−1743.07	−0.324046	−0.162023	−	0.986787i	$-0.551802\pi$
$307$	−1743.07	−0.324046	−0.162023	+	0.986787i	$0.551802\pi$
$308$	0	0
$309$	2131.69	0.392451
$310$	0	0
$311$	10027.9	1.82840	0.914198	−	0.405267i	$-0.132822\pi$
$311$	10027.9	1.82840	0.914198	+	0.405267i	$0.132822\pi$
$312$	0	0
$313$	−2092.80	−0.377929	−0.188965	−	0.981984i	$-0.560513\pi$
$313$	−2092.80	−0.377929	−0.188965	+	0.981984i	$0.560513\pi$
$314$	0	0
$315$	−3046.40	−0.544905
$316$	0	0
$317$	−10481.8	−1.85715	−0.928573	−	0.371151i	$-0.878963\pi$
$317$	−10481.8	−1.85715	−0.928573	+	0.371151i	$0.878963\pi$
$318$	0	0
$319$	6671.32	1.17092
$320$	0	0
$321$	−1599.73	−0.278157
$322$	0	0
$323$	−5196.39	−0.895154
$324$	0	0
$325$	3009.83	0.513709
$326$	0	0
$327$	2001.17	0.338425
$328$	0	0
$329$	−11342.4	−1.90068
$330$	0	0
$331$	3954.25	0.656631	0.328316	−	0.944568i	$-0.393519\pi$
$331$	3954.25	0.656631	0.328316	+	0.944568i	$0.393519\pi$
$332$	0	0
$333$	−2307.08	−0.379661
$334$	0	0
$335$	12030.6	1.96209
$336$	0	0
$337$	−7823.04	−1.26454	−0.632268	−	0.774750i	$-0.717876\pi$
$337$	−7823.04	−1.26454	−0.632268	+	0.774750i	$0.717876\pi$
$338$	0	0
$339$	−908.748	−0.145594
$340$	0	0
$341$	7.26016	0.00115296
$342$	0	0
$343$	6536.69	1.02900
$344$	0	0
$345$	−6074.34	−0.947918
$346$	0	0
$347$	4248.15	0.657213	0.328606	−	0.944467i	$-0.393421\pi$
$347$	4248.15	0.657213	0.328606	+	0.944467i	$0.393421\pi$
$348$	0	0
$349$	−3202.79	−0.491236	−0.245618	−	0.969367i	$-0.578991\pi$
$349$	−3202.79	−0.491236	−0.245618	+	0.969367i	$0.578991\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−2667.60	−0.402216	−0.201108	−	0.979569i	$-0.564454\pi$
$353$	−2667.60	−0.402216	−0.201108	+	0.979569i	$0.564454\pi$
$354$	0	0
$355$	−6846.52	−1.02359
$356$	0	0
$357$	1828.52	0.271080
$358$	0	0
$359$	9062.64	1.33233	0.666167	−	0.745803i	$-0.267934\pi$
$359$	9062.64	1.33233	0.666167	+	0.745803i	$0.267934\pi$
$360$	0	0
$361$	16499.5	2.40553
$362$	0	0
$363$	−1967.06	−0.284419
$364$	0	0
$365$	13654.9	1.95817
$366$	0	0
$367$	−4141.93	−0.589120	−0.294560	−	0.955633i	$-0.595173\pi$
$367$	−4141.93	−0.589120	−0.294560	+	0.955633i	$0.595173\pi$
$368$	0	0
$369$	3732.51	0.526576
$370$	0	0
$371$	4239.45	0.593265
$372$	0	0
$373$	−5350.96	−0.742794	−0.371397	−	0.928474i	$-0.621121\pi$
$373$	−5350.96	−0.742794	−0.371397	+	0.928474i	$0.621121\pi$
$374$	0	0
$375$	−6034.20	−0.830946
$376$	0	0
$377$	1945.76	0.265814
$378$	0	0
$379$	−9360.54	−1.26865	−0.634325	−	0.773066i	$-0.718722\pi$
$379$	−9360.54	−1.26865	−0.634325	+	0.773066i	$0.718722\pi$
$380$	0	0
$381$	1378.41	0.185350
$382$	0	0
$383$	7241.39	0.966104	0.483052	−	0.875592i	$-0.339528\pi$
$383$	7241.39	0.966104	0.483052	+	0.875592i	$0.339528\pi$
$384$	0	0
$385$	−15087.2	−1.99718
$386$	0	0
$387$	4239.68	0.556886
$388$	0	0
$389$	6258.22	0.815692	0.407846	−	0.913051i	$-0.366280\pi$
$389$	6258.22	0.815692	0.407846	+	0.913051i	$0.366280\pi$
$390$	0	0
$391$	3645.96	0.471571
$392$	0	0
$393$	−1490.54	−0.191318
$394$	0	0
$395$	−18215.0	−2.32025
$396$	0	0
$397$	−10114.6	−1.27868	−0.639342	−	0.768923i	$-0.720793\pi$
$397$	−10114.6	−1.27868	−0.639342	+	0.768923i	$0.720793\pi$
$398$	0	0
$399$	−8219.45	−1.03130
$400$	0	0
$401$	−12414.5	−1.54601	−0.773006	−	0.634398i	$-0.781248\pi$
$401$	−12414.5	−1.54601	−0.773006	+	0.634398i	$0.781248\pi$
$402$	0	0
$403$	2.11751	0.000261738 0
$404$	0	0
$405$	1529.43	0.187650
$406$	0	0
$407$	−11425.7	−1.39153
$408$	0	0
$409$	−14052.1	−1.69885	−0.849425	−	0.527709i	$-0.823051\pi$
$409$	−14052.1	−1.69885	−0.849425	+	0.527709i	$0.823051\pi$
$410$	0	0
$411$	8054.24	0.966633
$412$	0	0
$413$	1936.40	0.230711
$414$	0	0
$415$	−8154.48	−0.964548
$416$	0	0
$417$	5186.98	0.609130
$418$	0	0
$419$	1003.68	0.117024	0.0585119	−	0.998287i	$-0.481364\pi$
$419$	1003.68	0.117024	0.0585119	+	0.998287i	$0.481364\pi$
$420$	0	0
$421$	8376.14	0.969663	0.484832	−	0.874608i	$-0.338881\pi$
$421$	8376.14	0.969663	0.484832	+	0.874608i	$0.338881\pi$
$422$	0	0
$423$	5694.39	0.654541
$424$	0	0
$425$	7871.86	0.898450
$426$	0	0
$427$	−15924.1	−1.80473
$428$	0	0
$429$	−1738.32	−0.195634
$430$	0	0
$431$	−12296.3	−1.37423	−0.687116	−	0.726548i	$-0.741123\pi$
$431$	−12296.3	−1.37423	−0.687116	+	0.726548i	$0.741123\pi$
$432$	0	0
$433$	1010.07	0.112104	0.0560518	−	0.998428i	$-0.482149\pi$
$433$	1010.07	0.112104	0.0560518	+	0.998428i	$0.482149\pi$
$434$	0	0
$435$	−8478.39	−0.934501
$436$	0	0
$437$	−16389.1	−1.79405
$438$	0	0
$439$	1845.70	0.200662	0.100331	−	0.994954i	$-0.468010\pi$
$439$	1845.70	0.200662	0.100331	+	0.994954i	$0.468010\pi$
$440$	0	0
$441$	−194.719	−0.0210257
$442$	0	0
$443$	7313.67	0.784386	0.392193	−	0.919883i	$-0.371717\pi$
$443$	7313.67	0.784386	0.392193	+	0.919883i	$0.371717\pi$
$444$	0	0
$445$	2215.95	0.236059
$446$	0	0
$447$	3398.68	0.359624
$448$	0	0
$449$	8443.62	0.887481	0.443740	−	0.896155i	$-0.353651\pi$
$449$	8443.62	0.887481	0.443740	+	0.896155i	$0.353651\pi$
$450$	0	0
$451$	18485.2	1.93001
$452$	0	0
$453$	3775.57	0.391593
$454$	0	0
$455$	−4400.35	−0.453388
$456$	0	0
$457$	14302.7	1.46401	0.732004	−	0.681301i	$-0.238585\pi$
$457$	14302.7	1.46401	0.732004	+	0.681301i	$0.238585\pi$
$458$	0	0
$459$	−918.000	−0.0933520
$460$	0	0
$461$	−3862.08	−0.390184	−0.195092	−	0.980785i	$-0.562501\pi$
$461$	−3862.08	−0.390184	−0.195092	+	0.980785i	$0.562501\pi$
$462$	0	0
$463$	−14364.6	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$463$	−14364.6	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$464$	0	0
$465$	−9.22673	−0.000920171 0
$466$	0	0
$467$	14181.4	1.40522	0.702610	−	0.711575i	$-0.252018\pi$
$467$	14181.4	1.40522	0.702610	+	0.711575i	$0.252018\pi$
$468$	0	0
$469$	−11421.9	−1.12455
$470$	0	0
$471$	−7409.66	−0.724881
$472$	0	0
$473$	20996.9	2.04110
$474$	0	0
$475$	−35385.2	−3.41807
$476$	0	0
$477$	−2128.40	−0.204303
$478$	0	0
$479$	1182.57	0.112804	0.0564019	−	0.998408i	$-0.482037\pi$
$479$	1182.57	0.112804	0.0564019	+	0.998408i	$0.482037\pi$
$480$	0	0
$481$	−3332.45	−0.315897
$482$	0	0
$483$	5767.04	0.543291
$484$	0	0
$485$	−21778.1	−2.03895
$486$	0	0
$487$	−16039.2	−1.49242	−0.746209	−	0.665712i	$-0.768128\pi$
$487$	−16039.2	−1.49242	−0.746209	+	0.665712i	$0.768128\pi$
$488$	0	0
$489$	2201.68	0.203606
$490$	0	0
$491$	−11867.2	−1.09075	−0.545376	−	0.838191i	$-0.683613\pi$
$491$	−11867.2	−1.09075	−0.545376	+	0.838191i	$0.683613\pi$
$492$	0	0
$493$	5088.92	0.464896
$494$	0	0
$495$	7574.48	0.687772
$496$	0	0
$497$	6500.15	0.586663
$498$	0	0
$499$	−6945.54	−0.623097	−0.311548	−	0.950230i	$-0.600848\pi$
$499$	−6945.54	−0.623097	−0.311548	+	0.950230i	$0.600848\pi$
$500$	0	0
$501$	−7831.17	−0.698345
$502$	0	0
$503$	5631.84	0.499227	0.249614	−	0.968346i	$-0.419696\pi$
$503$	5631.84	0.499227	0.249614	+	0.968346i	$0.419696\pi$
$504$	0	0
$505$	−16015.3	−1.41123
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	10395.3	0.905229	0.452614	−	0.891706i	$-0.350491\pi$
$509$	10395.3	0.905229	0.452614	+	0.891706i	$0.350491\pi$
$510$	0	0
$511$	−12964.1	−1.12231
$512$	0	0
$513$	4126.54	0.355149
$514$	0	0
$515$	−13416.7	−1.14799
$516$	0	0
$517$	28201.3	2.39902
$518$	0	0
$519$	−3492.13	−0.295352
$520$	0	0
$521$	7846.89	0.659844	0.329922	−	0.944008i	$-0.392978\pi$
$521$	7846.89	0.659844	0.329922	+	0.944008i	$0.392978\pi$
$522$	0	0
$523$	13670.9	1.14299	0.571497	−	0.820604i	$-0.306363\pi$
$523$	13670.9	1.14299	0.571497	+	0.820604i	$0.306363\pi$
$524$	0	0
$525$	12451.4	1.03509
$526$	0	0
$527$	5.53809	0.000457767 0
$528$	0	0
$529$	−667.849	−0.0548902
$530$	0	0
$531$	−972.160	−0.0794504
$532$	0	0
$533$	5391.40	0.438138
$534$	0	0
$535$	10068.7	0.813657
$536$	0	0
$537$	−2733.69	−0.219678
$538$	0	0
$539$	−964.341	−0.0770633
$540$	0	0
$541$	6194.38	0.492269	0.246134	−	0.969236i	$-0.420840\pi$
$541$	6194.38	0.492269	0.246134	+	0.969236i	$0.420840\pi$
$542$	0	0
$543$	14453.4	1.14228
$544$	0	0
$545$	−12595.3	−0.989951
$546$	0	0
$547$	−1856.88	−0.145145	−0.0725726	−	0.997363i	$-0.523121\pi$
$547$	−1856.88	−0.145145	−0.0725726	+	0.997363i	$0.523121\pi$
$548$	0	0
$549$	7994.63	0.621498
$550$	0	0
$551$	−22875.5	−1.76865
$552$	0	0
$553$	17293.5	1.32983
$554$	0	0
$555$	14520.7	1.11057
$556$	0	0
$557$	15017.2	1.14237	0.571183	−	0.820822i	$-0.306484\pi$
$557$	15017.2	1.14237	0.571183	+	0.820822i	$0.306484\pi$
$558$	0	0
$559$	6123.98	0.463358
$560$	0	0
$561$	−4546.37	−0.342153
$562$	0	0
$563$	13842.2	1.03620	0.518098	−	0.855321i	$-0.326640\pi$
$563$	13842.2	1.03620	0.518098	+	0.855321i	$0.326640\pi$
$564$	0	0
$565$	5719.62	0.425887
$566$	0	0
$567$	−1452.06	−0.107550
$568$	0	0
$569$	−17621.0	−1.29826	−0.649131	−	0.760677i	$-0.724867\pi$
$569$	−17621.0	−1.29826	−0.649131	+	0.760677i	$0.724867\pi$
$570$	0	0
$571$	16472.6	1.20728	0.603639	−	0.797258i	$-0.293717\pi$
$571$	16472.6	1.20728	0.603639	+	0.797258i	$0.293717\pi$
$572$	0	0
$573$	−2022.94	−0.147486
$574$	0	0
$575$	24827.4	1.80065
$576$	0	0
$577$	−19638.6	−1.41693	−0.708463	−	0.705748i	$-0.750611\pi$
$577$	−19638.6	−1.41693	−0.708463	+	0.705748i	$0.750611\pi$
$578$	0	0
$579$	2890.90	0.207499
$580$	0	0
$581$	7741.94	0.552822
$582$	0	0
$583$	−10540.8	−0.748811
$584$	0	0
$585$	2209.18	0.156134
$586$	0	0
$587$	−15965.5	−1.12260	−0.561300	−	0.827612i	$-0.689699\pi$
$587$	−15965.5	−1.12260	−0.561300	+	0.827612i	$0.689699\pi$
$588$	0	0
$589$	−24.8945	−0.00174153
$590$	0	0
$591$	−11985.6	−0.834219
$592$	0	0
$593$	−16128.6	−1.11690	−0.558452	−	0.829537i	$-0.688604\pi$
$593$	−16128.6	−1.11690	−0.558452	+	0.829537i	$0.688604\pi$
$594$	0	0
$595$	−11508.6	−0.792953
$596$	0	0
$597$	4857.22	0.332986
$598$	0	0
$599$	18322.6	1.24982	0.624910	−	0.780697i	$-0.285136\pi$
$599$	18322.6	1.24982	0.624910	+	0.780697i	$0.285136\pi$
$600$	0	0
$601$	10528.6	0.714594	0.357297	−	0.933991i	$-0.383698\pi$
$601$	10528.6	0.714594	0.357297	+	0.933991i	$0.383698\pi$
$602$	0	0
$603$	5734.34	0.387264
$604$	0	0
$605$	12380.6	0.831974
$606$	0	0
$607$	16560.5	1.10737	0.553683	−	0.832728i	$-0.313222\pi$
$607$	16560.5	1.10737	0.553683	+	0.832728i	$0.313222\pi$
$608$	0	0
$609$	8049.47	0.535601
$610$	0	0
$611$	8225.24	0.544611
$612$	0	0
$613$	23354.0	1.53876	0.769379	−	0.638793i	$-0.220566\pi$
$613$	23354.0	1.53876	0.769379	+	0.638793i	$0.220566\pi$
$614$	0	0
$615$	−23492.3	−1.54032
$616$	0	0
$617$	−23424.3	−1.52840	−0.764202	−	0.644977i	$-0.776867\pi$
$617$	−23424.3	−1.52840	−0.764202	+	0.644977i	$0.776867\pi$
$618$	0	0
$619$	−3275.77	−0.212705	−0.106352	−	0.994328i	$-0.533917\pi$
$619$	−3275.77	−0.212705	−0.106352	+	0.994328i	$0.533917\pi$
$620$	0	0
$621$	−2895.32	−0.187094
$622$	0	0
$623$	−2103.85	−0.135295
$624$	0	0
$625$	9038.33	0.578453
$626$	0	0
$627$	20436.6	1.30169
$628$	0	0
$629$	−8715.63	−0.552488
$630$	0	0
$631$	−24146.4	−1.52338	−0.761691	−	0.647941i	$-0.775631\pi$
$631$	−24146.4	−1.52338	−0.761691	+	0.647941i	$0.775631\pi$
$632$	0	0
$633$	−1631.97	−0.102473
$634$	0	0
$635$	−8675.69	−0.542180
$636$	0	0
$637$	−281.261	−0.0174944
$638$	0	0
$639$	−3263.37	−0.202030
$640$	0	0
$641$	−23506.8	−1.44846	−0.724230	−	0.689558i	$-0.757805\pi$
$641$	−23506.8	−1.44846	−0.724230	+	0.689558i	$0.757805\pi$
$642$	0	0
$643$	5273.79	0.323449	0.161725	−	0.986836i	$-0.448294\pi$
$643$	5273.79	0.323449	0.161725	+	0.986836i	$0.448294\pi$
$644$	0	0
$645$	−26684.4	−1.62899
$646$	0	0
$647$	−7106.47	−0.431815	−0.215907	−	0.976414i	$-0.569271\pi$
$647$	−7106.47	−0.431815	−0.215907	+	0.976414i	$0.569271\pi$
$648$	0	0
$649$	−4814.60	−0.291201
$650$	0	0
$651$	8.75995	0.000527388 0
$652$	0	0
$653$	−8474.11	−0.507837	−0.253918	−	0.967226i	$-0.581719\pi$
$653$	−8474.11	−0.507837	−0.253918	+	0.967226i	$0.581719\pi$
$654$	0	0
$655$	9381.43	0.559638
$656$	0	0
$657$	6508.59	0.386491
$658$	0	0
$659$	−18444.7	−1.09030	−0.545148	−	0.838340i	$-0.683526\pi$
$659$	−18444.7	−1.09030	−0.545148	+	0.838340i	$0.683526\pi$
$660$	0	0
$661$	29184.5	1.71732	0.858659	−	0.512547i	$-0.171298\pi$
$661$	29184.5	1.71732	0.858659	+	0.512547i	$0.171298\pi$
$662$	0	0
$663$	−1326.00	−0.0776736
$664$	0	0
$665$	51732.9	3.01672
$666$	0	0
$667$	16050.2	0.931732
$668$	0	0
$669$	−7342.36	−0.424323
$670$	0	0
$671$	39593.2	2.27791
$672$	0	0
$673$	−28477.5	−1.63110	−0.815548	−	0.578689i	$-0.803564\pi$
$673$	−28477.5	−1.63110	−0.815548	+	0.578689i	$0.803564\pi$
$674$	0	0
$675$	−6251.18	−0.356457
$676$	0	0
$677$	6422.80	0.364621	0.182310	−	0.983241i	$-0.441642\pi$
$677$	6422.80	0.364621	0.182310	+	0.983241i	$0.441642\pi$
$678$	0	0
$679$	20676.3	1.16861
$680$	0	0
$681$	−1403.40	−0.0789695
$682$	0	0
$683$	−54.3801	−0.00304655	−0.00152328	−	0.999999i	$-0.500485\pi$
$683$	−54.3801	−0.00304655	−0.00152328	+	0.999999i	$0.500485\pi$
$684$	0	0
$685$	−50693.1	−2.82756
$686$	0	0
$687$	1083.54	0.0601741
$688$	0	0
$689$	−3074.35	−0.169991
$690$	0	0
$691$	14382.3	0.791794	0.395897	−	0.918295i	$-0.370434\pi$
$691$	14382.3	0.791794	0.395897	+	0.918295i	$0.370434\pi$
$692$	0	0
$693$	−7191.28	−0.394191
$694$	0	0
$695$	−32646.6	−1.78181
$696$	0	0
$697$	14100.6	0.766281
$698$	0	0
$699$	9205.15	0.498098
$700$	0	0
$701$	32081.3	1.72852	0.864262	−	0.503042i	$-0.167786\pi$
$701$	32081.3	1.72852	0.864262	+	0.503042i	$0.167786\pi$
$702$	0	0
$703$	39178.0	2.10189
$704$	0	0
$705$	−35840.3	−1.91464
$706$	0	0
$707$	15205.1	0.808835
$708$	0	0
$709$	11549.8	0.611793	0.305896	−	0.952065i	$-0.401044\pi$
$709$	11549.8	0.611793	0.305896	+	0.952065i	$0.401044\pi$
$710$	0	0
$711$	−8682.16	−0.457955
$712$	0	0
$713$	17.4668	0.000917445 0
$714$	0	0
$715$	10940.9	0.572261
$716$	0	0
$717$	−1002.32	−0.0522071
$718$	0	0
$719$	−33235.7	−1.72390	−0.861949	−	0.506995i	$-0.830756\pi$
$719$	−33235.7	−1.72390	−0.861949	+	0.506995i	$0.830756\pi$
$720$	0	0
$721$	12738.0	0.657958
$722$	0	0
$723$	−1544.88	−0.0794672
$724$	0	0
$725$	34653.4	1.77516
$726$	0	0
$727$	13863.5	0.707249	0.353625	−	0.935387i	$-0.384949\pi$
$727$	13863.5	0.707249	0.353625	+	0.935387i	$0.384949\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	16016.6	0.810389
$732$	0	0
$733$	13448.6	0.677675	0.338838	−	0.940845i	$-0.389966\pi$
$733$	13448.6	0.677675	0.338838	+	0.940845i	$0.389966\pi$
$734$	0	0
$735$	1225.55	0.0615037
$736$	0	0
$737$	28399.2	1.41940
$738$	0	0
$739$	−36402.4	−1.81202	−0.906012	−	0.423252i	$-0.860888\pi$
$739$	−36402.4	−1.81202	−0.906012	+	0.423252i	$0.860888\pi$
$740$	0	0
$741$	5960.56	0.295502
$742$	0	0
$743$	23695.5	1.16999	0.584996	−	0.811036i	$-0.301096\pi$
$743$	23695.5	1.16999	0.584996	+	0.811036i	$0.301096\pi$
$744$	0	0
$745$	−21391.1	−1.05196
$746$	0	0
$747$	−3886.81	−0.190376
$748$	0	0
$749$	−9559.29	−0.466340
$750$	0	0
$751$	−4052.82	−0.196923	−0.0984617	−	0.995141i	$-0.531392\pi$
$751$	−4052.82	−0.196923	−0.0984617	+	0.995141i	$0.531392\pi$
$752$	0	0
$753$	21084.4	1.02040
$754$	0	0
$755$	−23763.3	−1.14548
$756$	0	0
$757$	15508.4	0.744603	0.372301	−	0.928112i	$-0.378569\pi$
$757$	15508.4	0.744603	0.372301	+	0.928112i	$0.378569\pi$
$758$	0	0
$759$	−14339.0	−0.685735
$760$	0	0
$761$	29272.1	1.39436	0.697182	−	0.716894i	$-0.254437\pi$
$761$	29272.1	1.39436	0.697182	+	0.716894i	$0.254437\pi$
$762$	0	0
$763$	11958.1	0.567382
$764$	0	0
$765$	5777.86	0.273070
$766$	0	0
$767$	−1404.23	−0.0661067
$768$	0	0
$769$	35088.4	1.64541	0.822706	−	0.568468i	$-0.192464\pi$
$769$	35088.4	1.64541	0.822706	+	0.568468i	$0.192464\pi$
$770$	0	0
$771$	10716.7	0.500586
$772$	0	0
$773$	−11475.6	−0.533956	−0.266978	−	0.963703i	$-0.586025\pi$
$773$	−11475.6	−0.533956	−0.266978	+	0.963703i	$0.586025\pi$
$774$	0	0
$775$	37.7120	0.00174794
$776$	0	0
$777$	−13786.1	−0.636515
$778$	0	0
$779$	−63384.2	−2.91524
$780$	0	0
$781$	−16161.8	−0.740479
$782$	0	0
$783$	−4041.20	−0.184446
$784$	0	0
$785$	46636.1	2.12040
$786$	0	0
$787$	−17215.7	−0.779761	−0.389881	−	0.920865i	$-0.627484\pi$
$787$	−17215.7	−0.779761	−0.389881	+	0.920865i	$0.627484\pi$
$788$	0	0
$789$	−12830.2	−0.578920
$790$	0	0
$791$	−5430.27	−0.244094
$792$	0	0
$793$	11547.8	0.517118
$794$	0	0
$795$	13396.1	0.597622
$796$	0	0
$797$	43951.6	1.95338	0.976692	−	0.214646i	$-0.0688599\pi$
$797$	43951.6	1.95338	0.976692	+	0.214646i	$0.0688599\pi$
$798$	0	0
$799$	21512.2	0.952497
$800$	0	0
$801$	1056.23	0.0465918
$802$	0	0
$803$	32233.6	1.41656
$804$	0	0
$805$	−36297.5	−1.58922
$806$	0	0
$807$	−6384.24	−0.278483
$808$	0	0
$809$	−27914.7	−1.21314	−0.606569	−	0.795031i	$-0.707455\pi$
$809$	−27914.7	−1.21314	−0.606569	+	0.795031i	$0.707455\pi$
$810$	0	0
$811$	38016.3	1.64603	0.823017	−	0.568016i	$-0.192289\pi$
$811$	38016.3	1.64603	0.823017	+	0.568016i	$0.192289\pi$
$812$	0	0
$813$	13160.8	0.567737
$814$	0	0
$815$	−13857.3	−0.595581
$816$	0	0
$817$	−71996.8	−3.08305
$818$	0	0
$819$	−2097.42	−0.0894868
$820$	0	0
$821$	−15367.7	−0.653273	−0.326637	−	0.945150i	$-0.605915\pi$
$821$	−15367.7	−0.653273	−0.326637	+	0.945150i	$0.605915\pi$
$822$	0	0
$823$	377.068	0.0159705	0.00798527	−	0.999968i	$-0.497458\pi$
$823$	377.068	0.0159705	0.00798527	+	0.999968i	$0.497458\pi$
$824$	0	0
$825$	−30958.8	−1.30648
$826$	0	0
$827$	−5466.15	−0.229839	−0.114919	−	0.993375i	$-0.536661\pi$
$827$	−5466.15	−0.229839	−0.114919	+	0.993375i	$0.536661\pi$
$828$	0	0
$829$	−18760.0	−0.785962	−0.392981	−	0.919547i	$-0.628556\pi$
$829$	−18760.0	−0.785962	−0.392981	+	0.919547i	$0.628556\pi$
$830$	0	0
$831$	−10235.6	−0.427278
$832$	0	0
$833$	−735.605	−0.0305969
$834$	0	0
$835$	49289.1	2.04278
$836$	0	0
$837$	−4.39790	−0.000181617 0
$838$	0	0
$839$	−46530.4	−1.91467	−0.957334	−	0.288983i	$-0.906683\pi$
$839$	−46530.4	−1.91467	−0.957334	+	0.288983i	$0.906683\pi$
$840$	0	0
$841$	−1986.62	−0.0814558
$842$	0	0
$843$	13400.5	0.547493
$844$	0	0
$845$	3191.04	0.129911
$846$	0	0
$847$	−11754.3	−0.476839
$848$	0	0
$849$	5637.91	0.227906
$850$	0	0
$851$	−27488.6	−1.10728
$852$	0	0
$853$	−11378.2	−0.456719	−0.228359	−	0.973577i	$-0.573336\pi$
$853$	−11378.2	−0.456719	−0.228359	+	0.973577i	$0.573336\pi$
$854$	0	0
$855$	−25972.3	−1.03887
$856$	0	0
$857$	16515.4	0.658289	0.329145	−	0.944280i	$-0.393240\pi$
$857$	16515.4	0.658289	0.329145	+	0.944280i	$0.393240\pi$
$858$	0	0
$859$	1477.37	0.0586811	0.0293406	−	0.999569i	$-0.490659\pi$
$859$	1477.37	0.0586811	0.0293406	+	0.999569i	$0.490659\pi$
$860$	0	0
$861$	22303.8	0.882824
$862$	0	0
$863$	14215.0	0.560702	0.280351	−	0.959898i	$-0.409549\pi$
$863$	14215.0	0.560702	0.280351	+	0.959898i	$0.409549\pi$
$864$	0	0
$865$	21979.3	0.863953
$866$	0	0
$867$	11271.0	0.441503
$868$	0	0
$869$	−42998.2	−1.67850
$870$	0	0
$871$	8282.94	0.322223
$872$	0	0
$873$	−10380.5	−0.402434
$874$	0	0
$875$	−36057.6	−1.39311
$876$	0	0
$877$	15634.9	0.602001	0.301000	−	0.953624i	$-0.402679\pi$
$877$	15634.9	0.602001	0.301000	+	0.953624i	$0.402679\pi$
$878$	0	0
$879$	623.987	0.0239438
$880$	0	0
$881$	−35793.2	−1.36879	−0.684395	−	0.729111i	$-0.739934\pi$
$881$	−35793.2	−1.36879	−0.684395	+	0.729111i	$0.739934\pi$
$882$	0	0
$883$	−18937.7	−0.721749	−0.360875	−	0.932614i	$-0.617522\pi$
$883$	−18937.7	−0.721749	−0.360875	+	0.932614i	$0.617522\pi$
$884$	0	0
$885$	6118.74	0.232406
$886$	0	0
$887$	−16591.7	−0.628066	−0.314033	−	0.949412i	$-0.601680\pi$
$887$	−16591.7	−0.628066	−0.314033	+	0.949412i	$0.601680\pi$
$888$	0	0
$889$	8236.78	0.310746
$890$	0	0
$891$	3610.35	0.135748
$892$	0	0
$893$	−96700.3	−3.62369
$894$	0	0
$895$	17205.7	0.642596
$896$	0	0
$897$	−4182.13	−0.155671
$898$	0	0
$899$	24.3797	0.000904459 0
$900$	0	0
$901$	−8040.62	−0.297305
$902$	0	0
$903$	25334.4	0.933639
$904$	0	0
$905$	−90969.4	−3.34135
$906$	0	0
$907$	28947.7	1.05975	0.529875	−	0.848076i	$-0.322239\pi$
$907$	28947.7	1.05975	0.529875	+	0.848076i	$0.322239\pi$
$908$	0	0
$909$	−7633.65	−0.278539
$910$	0	0
$911$	13179.1	0.479302	0.239651	−	0.970859i	$-0.422967\pi$
$911$	13179.1	0.479302	0.239651	+	0.970859i	$0.422967\pi$
$912$	0	0
$913$	−19249.3	−0.697766
$914$	0	0
$915$	−50317.9	−1.81799
$916$	0	0
$917$	−8906.82	−0.320751
$918$	0	0
$919$	29617.7	1.06311	0.531555	−	0.847024i	$-0.321608\pi$
$919$	29617.7	1.06311	0.531555	+	0.847024i	$0.321608\pi$
$920$	0	0
$921$	5229.21	0.187088
$922$	0	0
$923$	−4713.76	−0.168099
$924$	0	0
$925$	−59349.7	−2.10963
$926$	0	0
$927$	−6395.06	−0.226582
$928$	0	0
$929$	−36633.5	−1.29376	−0.646882	−	0.762590i	$-0.723927\pi$
$929$	−36633.5	−1.29376	−0.646882	+	0.762590i	$0.723927\pi$
$930$	0	0
$931$	3306.65	0.116403
$932$	0	0
$933$	−30083.8	−1.05563
$934$	0	0
$935$	28614.7	1.00086
$936$	0	0
$937$	3252.46	0.113397	0.0566986	−	0.998391i	$-0.481943\pi$
$937$	3252.46	0.113397	0.0566986	+	0.998391i	$0.481943\pi$
$938$	0	0
$939$	6278.39	0.218198
$940$	0	0
$941$	−20638.9	−0.714995	−0.357497	−	0.933914i	$-0.616370\pi$
$941$	−20638.9	−0.714995	−0.357497	+	0.933914i	$0.616370\pi$
$942$	0	0
$943$	44472.5	1.53576
$944$	0	0
$945$	9139.20	0.314601
$946$	0	0
$947$	−21790.8	−0.747737	−0.373869	−	0.927482i	$-0.621969\pi$
$947$	−21790.8	−0.747737	−0.373869	+	0.927482i	$0.621969\pi$
$948$	0	0
$949$	9401.30	0.321580
$950$	0	0
$951$	31445.3	1.07222
$952$	0	0
$953$	31780.0	1.08023	0.540114	−	0.841592i	$-0.318381\pi$
$953$	31780.0	1.08023	0.540114	+	0.841592i	$0.318381\pi$
$954$	0	0
$955$	12732.3	0.431421
$956$	0	0
$957$	−20014.0	−0.676029
$958$	0	0
$959$	48128.5	1.62059
$960$	0	0
$961$	−29791.0	−0.999999
$962$	0	0
$963$	4799.20	0.160594
$964$	0	0
$965$	−18195.2	−0.606969
$966$	0	0
$967$	−38528.7	−1.28128	−0.640640	−	0.767841i	$-0.721331\pi$
$967$	−38528.7	−1.28128	−0.640640	+	0.767841i	$0.721331\pi$
$968$	0	0
$969$	15589.2	0.516818
$970$	0	0
$971$	22579.5	0.746252	0.373126	−	0.927781i	$-0.378286\pi$
$971$	22579.5	0.746252	0.373126	+	0.927781i	$0.378286\pi$
$972$	0	0
$973$	30995.0	1.02123
$974$	0	0
$975$	−9029.49	−0.296590
$976$	0	0
$977$	30254.4	0.990712	0.495356	−	0.868690i	$-0.335038\pi$
$977$	30254.4	0.990712	0.495356	+	0.868690i	$0.335038\pi$
$978$	0	0
$979$	5230.95	0.170768
$980$	0	0
$981$	−6003.51	−0.195390
$982$	0	0
$983$	−7071.61	−0.229450	−0.114725	−	0.993397i	$-0.536599\pi$
$983$	−7071.61	−0.229450	−0.114725	+	0.993397i	$0.536599\pi$
$984$	0	0
$985$	75437.1	2.44023
$986$	0	0
$987$	34027.1	1.09736
$988$	0	0
$989$	50515.4	1.62416
$990$	0	0
$991$	2808.28	0.0900180	0.0450090	−	0.998987i	$-0.485668\pi$
$991$	2808.28	0.0900180	0.0450090	+	0.998987i	$0.485668\pi$
$992$	0	0
$993$	−11862.7	−0.379106
$994$	0	0
$995$	−30571.1	−0.974040
$996$	0	0
$997$	−15225.5	−0.483648	−0.241824	−	0.970320i	$-0.577746\pi$
$997$	−15225.5	−0.483648	−0.241824	+	0.970320i	$0.577746\pi$
$998$	0	0
$999$	6921.23	0.219197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.g.1.3	✓	3
3.2	odd	2		936.4.a.k.1.1		3
4.3	odd	2		624.4.a.u.1.3		3
8.3	odd	2		2496.4.a.bk.1.1		3
8.5	even	2		2496.4.a.bo.1.1		3
12.11	even	2		1872.4.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.g.1.3	✓	3	1.1	even	1	trivial
624.4.a.u.1.3		3	4.3	odd	2
936.4.a.k.1.1		3	3.2	odd	2
1872.4.a.bj.1.1		3	12.11	even	2
2496.4.a.bk.1.1		3	8.3	odd	2
2496.4.a.bo.1.1		3	8.5	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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +18.8819 q^{5} -17.9266 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +18.8819 q^{5} -17.9266 q^{7} +9.00000 q^{9} +44.5723 q^{11} +13.0000 q^{13} -56.6456 q^{15} +34.0000 q^{17} -152.835 q^{19} +53.7799 q^{21} +107.234 q^{23} +231.525 q^{25} -27.0000 q^{27} +149.674 q^{29} +0.162885 q^{31} -133.717 q^{33} -338.489 q^{35} -256.342 q^{37} -39.0000 q^{39} +414.723 q^{41} +471.076 q^{43} +169.937 q^{45} +632.710 q^{47} -21.6354 q^{49} -102.000 q^{51} -236.489 q^{53} +841.608 q^{55} +458.505 q^{57} -108.018 q^{59} +888.292 q^{61} -161.340 q^{63} +245.464 q^{65} +637.149 q^{67} -321.702 q^{69} -362.597 q^{71} +723.177 q^{73} -694.576 q^{75} -799.031 q^{77} -964.684 q^{79} +81.0000 q^{81} -431.868 q^{83} +641.984 q^{85} -449.023 q^{87} +117.359 q^{89} -233.046 q^{91} -0.488655 q^{93} -2885.81 q^{95} -1153.38 q^{97} +401.151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9} - 20 q^{11} + 39 q^{13} - 48 q^{15} + 102 q^{17} - 38 q^{19} + 66 q^{21} + 32 q^{23} + 161 q^{25} - 81 q^{27} + 350 q^{29} + 50 q^{31} + 60 q^{33} + 232 q^{35} + 542 q^{37} - 117 q^{39} + 500 q^{41} + 420 q^{43} + 144 q^{45} + 324 q^{47} + 1119 q^{49} - 306 q^{51} + 538 q^{53} + 896 q^{55} + 114 q^{57} + 112 q^{59} + 1206 q^{61} - 198 q^{63} + 208 q^{65} + 950 q^{67} - 96 q^{69} - 1008 q^{71} + 1834 q^{73} - 483 q^{75} - 792 q^{77} - 272 q^{79} + 243 q^{81} - 1640 q^{83} + 544 q^{85} - 1050 q^{87} - 1576 q^{89} - 286 q^{91} - 150 q^{93} - 2760 q^{95} + 402 q^{97} - 180 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9 - 20 * q^11 + 39 * q^13 - 48 * q^15 + 102 * q^17 - 38 * q^19 + 66 * q^21 + 32 * q^23 + 161 * q^25 - 81 * q^27 + 350 * q^29 + 50 * q^31 + 60 * q^33 + 232 * q^35 + 542 * q^37 - 117 * q^39 + 500 * q^41 + 420 * q^43 + 144 * q^45 + 324 * q^47 + 1119 * q^49 - 306 * q^51 + 538 * q^53 + 896 * q^55 + 114 * q^57 + 112 * q^59 + 1206 * q^61 - 198 * q^63 + 208 * q^65 + 950 * q^67 - 96 * q^69 - 1008 * q^71 + 1834 * q^73 - 483 * q^75 - 792 * q^77 - 272 * q^79 + 243 * q^81 - 1640 * q^83 + 544 * q^85 - 1050 * q^87 - 1576 * q^89 - 286 * q^91 - 150 * q^93 - 2760 * q^95 + 402 * q^97 - 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more