LMFDB - Embedded newform 312.4.a.c.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{113}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 28 $ x^2 - x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.81507$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -7.63015 q^{5} +5.63015 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -7.63015 q^{5} +5.63015 q^{7} +9.00000 q^{9} +34.5206 q^{11} +13.0000 q^{13} +22.8904 q^{15} +2.00000 q^{17} -88.1507 q^{19} -16.8904 q^{21} -64.0000 q^{23} -66.7809 q^{25} -27.0000 q^{27} +23.7809 q^{29} -284.452 q^{31} -103.562 q^{33} -42.9588 q^{35} +115.343 q^{37} -39.0000 q^{39} +1.41102 q^{41} -337.041 q^{43} -68.6713 q^{45} -198.219 q^{47} -311.301 q^{49} -6.00000 q^{51} +59.0412 q^{53} -263.397 q^{55} +264.452 q^{57} -188.301 q^{59} +336.987 q^{61} +50.6713 q^{63} -99.1919 q^{65} -531.411 q^{67} +192.000 q^{69} -510.247 q^{71} -164.219 q^{73} +200.343 q^{75} +194.356 q^{77} -29.3148 q^{79} +81.0000 q^{81} -117.507 q^{83} -15.2603 q^{85} -71.3426 q^{87} +508.671 q^{89} +73.1919 q^{91} +853.357 q^{93} +672.603 q^{95} -1020.88 q^{97} +310.685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9} - 16 q^{11} + 26 q^{13} - 18 q^{15} + 4 q^{17} - 70 q^{19} + 30 q^{21} - 128 q^{23} - 6 q^{25} - 54 q^{27} - 80 q^{29} - 250 q^{31} + 48 q^{33} - 256 q^{35} - 152 q^{37} - 78 q^{39} - 146 q^{41} - 504 q^{43} + 54 q^{45} - 524 q^{47} - 410 q^{49} - 12 q^{51} - 52 q^{53} - 952 q^{55} + 210 q^{57} - 164 q^{59} - 304 q^{61} - 90 q^{63} + 78 q^{65} - 914 q^{67} + 384 q^{69} - 456 q^{73} + 18 q^{75} + 984 q^{77} - 824 q^{79} + 162 q^{81} + 828 q^{83} + 12 q^{85} + 240 q^{87} + 826 q^{89} - 130 q^{91} + 750 q^{93} + 920 q^{95} + 552 q^{97} - 144 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9 - 16 * q^11 + 26 * q^13 - 18 * q^15 + 4 * q^17 - 70 * q^19 + 30 * q^21 - 128 * q^23 - 6 * q^25 - 54 * q^27 - 80 * q^29 - 250 * q^31 + 48 * q^33 - 256 * q^35 - 152 * q^37 - 78 * q^39 - 146 * q^41 - 504 * q^43 + 54 * q^45 - 524 * q^47 - 410 * q^49 - 12 * q^51 - 52 * q^53 - 952 * q^55 + 210 * q^57 - 164 * q^59 - 304 * q^61 - 90 * q^63 + 78 * q^65 - 914 * q^67 + 384 * q^69 - 456 * q^73 + 18 * q^75 + 984 * q^77 - 824 * q^79 + 162 * q^81 + 828 * q^83 + 12 * q^85 + 240 * q^87 + 826 * q^89 - 130 * q^91 + 750 * q^93 + 920 * q^95 + 552 * q^97 - 144 * 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
$5$	−7.63015	−0.682461	−0.341230	−	0.939980i	$-0.610844\pi$
$5$	−7.63015	−0.682461	−0.341230	+	0.939980i	$0.610844\pi$
$6$	0	0
$7$	5.63015	0.303999	0.152000	−	0.988381i	$-0.451429\pi$
$7$	5.63015	0.303999	0.152000	+	0.988381i	$0.451429\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	34.5206	0.946213	0.473107	−	0.881005i	$-0.343133\pi$
$11$	34.5206	0.946213	0.473107	+	0.881005i	$0.343133\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	22.8904	0.394019
$16$	0	0
$17$	2.00000	0.0285336	0.0142668	−	0.999898i	$-0.495459\pi$
$17$	2.00000	0.0285336	0.0142668	+	0.999898i	$0.495459\pi$
$18$	0	0
$19$	−88.1507	−1.06438	−0.532189	−	0.846626i	$-0.678630\pi$
$19$	−88.1507	−1.06438	−0.532189	+	0.846626i	$0.678630\pi$
$20$	0	0
$21$	−16.8904	−0.175514
$22$	0	0
$23$	−64.0000	−0.580214	−0.290107	−	0.956994i	$-0.593691\pi$
$23$	−64.0000	−0.580214	−0.290107	+	0.956994i	$0.593691\pi$
$24$	0	0
$25$	−66.7809	−0.534247
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	23.7809	0.152276	0.0761379	−	0.997097i	$-0.475741\pi$
$29$	23.7809	0.152276	0.0761379	+	0.997097i	$0.475741\pi$
$30$	0	0
$31$	−284.452	−1.64804	−0.824018	−	0.566564i	$-0.808273\pi$
$31$	−284.452	−1.64804	−0.824018	+	0.566564i	$0.808273\pi$
$32$	0	0
$33$	−103.562	−0.546297
$34$	0	0
$35$	−42.9588	−0.207468
$36$	0	0
$37$	115.343	0.512492	0.256246	−	0.966612i	$-0.417514\pi$
$37$	115.343	0.512492	0.256246	+	0.966612i	$0.417514\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	1.41102	0.00537474	0.00268737	−	0.999996i	$-0.499145\pi$
$41$	1.41102	0.00537474	0.00268737	+	0.999996i	$0.499145\pi$
$42$	0	0
$43$	−337.041	−1.19531	−0.597655	−	0.801754i	$-0.703901\pi$
$43$	−337.041	−1.19531	−0.597655	+	0.801754i	$0.703901\pi$
$44$	0	0
$45$	−68.6713	−0.227487
$46$	0	0
$47$	−198.219	−0.615175	−0.307588	−	0.951520i	$-0.599522\pi$
$47$	−198.219	−0.615175	−0.307588	+	0.951520i	$0.599522\pi$
$48$	0	0
$49$	−311.301	−0.907584
$50$	0	0
$51$	−6.00000	−0.0164739
$52$	0	0
$53$	59.0412	0.153018	0.0765088	−	0.997069i	$-0.475623\pi$
$53$	59.0412	0.153018	0.0765088	+	0.997069i	$0.475623\pi$
$54$	0	0
$55$	−263.397	−0.645754
$56$	0	0
$57$	264.452	0.614518
$58$	0	0
$59$	−188.301	−0.415504	−0.207752	−	0.978181i	$-0.566615\pi$
$59$	−188.301	−0.415504	−0.207752	+	0.978181i	$0.566615\pi$
$60$	0	0
$61$	336.987	0.707323	0.353662	−	0.935373i	$-0.384936\pi$
$61$	336.987	0.707323	0.353662	+	0.935373i	$0.384936\pi$
$62$	0	0
$63$	50.6713	0.101333
$64$	0	0
$65$	−99.1919	−0.189281
$66$	0	0
$67$	−531.411	−0.968988	−0.484494	−	0.874795i	$-0.660996\pi$
$67$	−531.411	−0.968988	−0.484494	+	0.874795i	$0.660996\pi$
$68$	0	0
$69$	192.000	0.334987
$70$	0	0
$71$	−510.247	−0.852890	−0.426445	−	0.904514i	$-0.640234\pi$
$71$	−510.247	−0.852890	−0.426445	+	0.904514i	$0.640234\pi$
$72$	0	0
$73$	−164.219	−0.263293	−0.131647	−	0.991297i	$-0.542026\pi$
$73$	−164.219	−0.263293	−0.131647	+	0.991297i	$0.542026\pi$
$74$	0	0
$75$	200.343	0.308448
$76$	0	0
$77$	194.356	0.287648
$78$	0	0
$79$	−29.3148	−0.0417490	−0.0208745	−	0.999782i	$-0.506645\pi$
$79$	−29.3148	−0.0417490	−0.0208745	+	0.999782i	$0.506645\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−117.507	−0.155399	−0.0776994	−	0.996977i	$-0.524757\pi$
$83$	−117.507	−0.155399	−0.0776994	+	0.996977i	$0.524757\pi$
$84$	0	0
$85$	−15.2603	−0.0194731
$86$	0	0
$87$	−71.3426	−0.0879165
$88$	0	0
$89$	508.671	0.605832	0.302916	−	0.953017i	$-0.402040\pi$
$89$	508.671	0.605832	0.302916	+	0.953017i	$0.402040\pi$
$90$	0	0
$91$	73.1919	0.0843142
$92$	0	0
$93$	853.357	0.951494
$94$	0	0
$95$	672.603	0.726396
$96$	0	0
$97$	−1020.88	−1.06860	−0.534301	−	0.845294i	$-0.679425\pi$
$97$	−1020.88	−1.06860	−0.534301	+	0.845294i	$0.679425\pi$
$98$	0	0
$99$	310.685	0.315404
$100$	0	0
$101$	−1028.44	−1.01320	−0.506602	−	0.862180i	$-0.669099\pi$
$101$	−1028.44	−1.01320	−0.506602	+	0.862180i	$0.669099\pi$
$102$	0	0
$103$	−1267.01	−1.21206	−0.606032	−	0.795440i	$-0.707240\pi$
$103$	−1267.01	−1.21206	−0.606032	+	0.795440i	$0.707240\pi$
$104$	0	0
$105$	128.877	0.119782
$106$	0	0
$107$	1934.74	1.74802	0.874012	−	0.485905i	$-0.161510\pi$
$107$	1934.74	1.74802	0.874012	+	0.485905i	$0.161510\pi$
$108$	0	0
$109$	2038.66	1.79145	0.895725	−	0.444608i	$-0.146657\pi$
$109$	2038.66	1.79145	0.895725	+	0.444608i	$0.146657\pi$
$110$	0	0
$111$	−346.028	−0.295887
$112$	0	0
$113$	−35.9455	−0.0299245	−0.0149623	−	0.999888i	$-0.504763\pi$
$113$	−35.9455	−0.0299245	−0.0149623	+	0.999888i	$0.504763\pi$
$114$	0	0
$115$	488.329	0.395973
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	11.2603	0.00867420
$120$	0	0
$121$	−139.329	−0.104680
$122$	0	0
$123$	−4.23306	−0.00310311
$124$	0	0
$125$	1463.32	1.04706
$126$	0	0
$127$	−1205.01	−0.841950	−0.420975	−	0.907072i	$-0.638312\pi$
$127$	−1205.01	−0.841950	−0.420975	+	0.907072i	$0.638312\pi$
$128$	0	0
$129$	1011.12	0.690112
$130$	0	0
$131$	1222.08	0.815067	0.407534	−	0.913190i	$-0.366389\pi$
$131$	1222.08	0.815067	0.407534	+	0.913190i	$0.366389\pi$
$132$	0	0
$133$	−496.301	−0.323570
$134$	0	0
$135$	206.014	0.131340
$136$	0	0
$137$	1466.40	0.914474	0.457237	−	0.889345i	$-0.348839\pi$
$137$	1466.40	0.914474	0.457237	+	0.889345i	$0.348839\pi$
$138$	0	0
$139$	−1225.40	−0.747748	−0.373874	−	0.927480i	$-0.621971\pi$
$139$	−1225.40	−0.747748	−0.373874	+	0.927480i	$0.621971\pi$
$140$	0	0
$141$	594.657	0.355172
$142$	0	0
$143$	448.768	0.262432
$144$	0	0
$145$	−181.452	−0.103922
$146$	0	0
$147$	933.904	0.523994
$148$	0	0
$149$	1601.88	0.880744	0.440372	−	0.897815i	$-0.354847\pi$
$149$	1601.88	0.880744	0.440372	+	0.897815i	$0.354847\pi$
$150$	0	0
$151$	588.234	0.317019	0.158509	−	0.987357i	$-0.449331\pi$
$151$	588.234	0.317019	0.158509	+	0.987357i	$0.449331\pi$
$152$	0	0
$153$	18.0000	0.00951120
$154$	0	0
$155$	2170.41	1.12472
$156$	0	0
$157$	−925.919	−0.470678	−0.235339	−	0.971913i	$-0.575620\pi$
$157$	−925.919	−0.470678	−0.235339	+	0.971913i	$0.575620\pi$
$158$	0	0
$159$	−177.123	−0.0883447
$160$	0	0
$161$	−360.329	−0.176385
$162$	0	0
$163$	−2947.52	−1.41637	−0.708183	−	0.706029i	$-0.750485\pi$
$163$	−2947.52	−1.41637	−0.708183	+	0.706029i	$0.750485\pi$
$164$	0	0
$165$	790.191	0.372826
$166$	0	0
$167$	−715.372	−0.331480	−0.165740	−	0.986169i	$-0.553001\pi$
$167$	−715.372	−0.331480	−0.165740	+	0.986169i	$0.553001\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−793.357	−0.354792
$172$	0	0
$173$	335.396	0.147397	0.0736985	−	0.997281i	$-0.476520\pi$
$173$	335.396	0.147397	0.0736985	+	0.997281i	$0.476520\pi$
$174$	0	0
$175$	−375.986	−0.162411
$176$	0	0
$177$	564.904	0.239892
$178$	0	0
$179$	−3549.04	−1.48194	−0.740972	−	0.671536i	$-0.765635\pi$
$179$	−3549.04	−1.48194	−0.740972	+	0.671536i	$0.765635\pi$
$180$	0	0
$181$	1169.26	0.480168	0.240084	−	0.970752i	$-0.422825\pi$
$181$	1169.26	0.480168	0.240084	+	0.970752i	$0.422825\pi$
$182$	0	0
$183$	−1010.96	−0.408373
$184$	0	0
$185$	−880.081	−0.349756
$186$	0	0
$187$	69.0412	0.0269989
$188$	0	0
$189$	−152.014	−0.0585047
$190$	0	0
$191$	4233.86	1.60394	0.801968	−	0.597367i	$-0.203787\pi$
$191$	4233.86	1.60394	0.801968	+	0.597367i	$0.203787\pi$
$192$	0	0
$193$	750.494	0.279905	0.139953	−	0.990158i	$-0.455305\pi$
$193$	750.494	0.279905	0.139953	+	0.990158i	$0.455305\pi$
$194$	0	0
$195$	297.576	0.109281
$196$	0	0
$197$	−745.986	−0.269793	−0.134897	−	0.990860i	$-0.543070\pi$
$197$	−745.986	−0.269793	−0.134897	+	0.990860i	$0.543070\pi$
$198$	0	0
$199$	2347.97	0.836400	0.418200	−	0.908355i	$-0.362661\pi$
$199$	2347.97	0.836400	0.418200	+	0.908355i	$0.362661\pi$
$200$	0	0
$201$	1594.23	0.559445
$202$	0	0
$203$	133.890	0.0462917
$204$	0	0
$205$	−10.7663	−0.00366805
$206$	0	0
$207$	−576.000	−0.193405
$208$	0	0
$209$	−3043.01	−1.00713
$210$	0	0
$211$	−684.056	−0.223186	−0.111593	−	0.993754i	$-0.535595\pi$
$211$	−684.056	−0.223186	−0.111593	+	0.993754i	$0.535595\pi$
$212$	0	0
$213$	1530.74	0.492416
$214$	0	0
$215$	2571.67	0.815752
$216$	0	0
$217$	−1601.51	−0.501002
$218$	0	0
$219$	492.657	0.152012
$220$	0	0
$221$	26.0000	0.00791380
$222$	0	0
$223$	−1583.90	−0.475632	−0.237816	−	0.971310i	$-0.576432\pi$
$223$	−1583.90	−0.475632	−0.237816	+	0.971310i	$0.576432\pi$
$224$	0	0
$225$	−601.028	−0.178082
$226$	0	0
$227$	3675.23	1.07460	0.537299	−	0.843392i	$-0.319445\pi$
$227$	3675.23	1.07460	0.537299	+	0.843392i	$0.319445\pi$
$228$	0	0
$229$	−2742.33	−0.791347	−0.395673	−	0.918391i	$-0.629489\pi$
$229$	−2742.33	−0.791347	−0.395673	+	0.918391i	$0.629489\pi$
$230$	0	0
$231$	−583.068	−0.166074
$232$	0	0
$233$	−2459.83	−0.691627	−0.345813	−	0.938303i	$-0.612397\pi$
$233$	−2459.83	−0.691627	−0.345813	+	0.938303i	$0.612397\pi$
$234$	0	0
$235$	1512.44	0.419833
$236$	0	0
$237$	87.9443	0.0241038
$238$	0	0
$239$	−334.688	−0.0905822	−0.0452911	−	0.998974i	$-0.514422\pi$
$239$	−334.688	−0.0905822	−0.0452911	+	0.998974i	$0.514422\pi$
$240$	0	0
$241$	−2697.81	−0.721083	−0.360542	−	0.932743i	$-0.617408\pi$
$241$	−2697.81	−0.721083	−0.360542	+	0.932743i	$0.617408\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	2375.28	0.619391
$246$	0	0
$247$	−1145.96	−0.295205
$248$	0	0
$249$	352.522	0.0897195
$250$	0	0
$251$	2665.64	0.670335	0.335167	−	0.942159i	$-0.391207\pi$
$251$	2665.64	0.670335	0.335167	+	0.942159i	$0.391207\pi$
$252$	0	0
$253$	−2209.32	−0.549006
$254$	0	0
$255$	45.7809	0.0112428
$256$	0	0
$257$	−6877.42	−1.66927	−0.834634	−	0.550805i	$-0.814321\pi$
$257$	−6877.42	−1.66927	−0.834634	+	0.550805i	$0.814321\pi$
$258$	0	0
$259$	649.396	0.155797
$260$	0	0
$261$	214.028	0.0507586
$262$	0	0
$263$	−55.5086	−0.0130145	−0.00650724	−	0.999979i	$-0.502071\pi$
$263$	−55.5086	−0.0130145	−0.00650724	+	0.999979i	$0.502071\pi$
$264$	0	0
$265$	−450.493	−0.104428
$266$	0	0
$267$	−1526.01	−0.349777
$268$	0	0
$269$	−3726.85	−0.844722	−0.422361	−	0.906428i	$-0.638798\pi$
$269$	−3726.85	−0.844722	−0.422361	+	0.906428i	$0.638798\pi$
$270$	0	0
$271$	1536.59	0.344432	0.172216	−	0.985059i	$-0.444907\pi$
$271$	1536.59	0.344432	0.172216	+	0.985059i	$0.444907\pi$
$272$	0	0
$273$	−219.576	−0.0486788
$274$	0	0
$275$	−2305.31	−0.505512
$276$	0	0
$277$	4917.23	1.06660	0.533300	−	0.845926i	$-0.320952\pi$
$277$	4917.23	1.06660	0.533300	+	0.845926i	$0.320952\pi$
$278$	0	0
$279$	−2560.07	−0.549345
$280$	0	0
$281$	−5281.66	−1.12127	−0.560636	−	0.828062i	$-0.689443\pi$
$281$	−5281.66	−1.12127	−0.560636	+	0.828062i	$0.689443\pi$
$282$	0	0
$283$	−5091.37	−1.06944	−0.534718	−	0.845030i	$-0.679582\pi$
$283$	−5091.37	−1.06944	−0.534718	+	0.845030i	$0.679582\pi$
$284$	0	0
$285$	−2017.81	−0.419385
$286$	0	0
$287$	7.94425	0.00163392
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	3062.63	0.616958
$292$	0	0
$293$	−4275.98	−0.852579	−0.426290	−	0.904587i	$-0.640180\pi$
$293$	−4275.98	−0.852579	−0.426290	+	0.904587i	$0.640180\pi$
$294$	0	0
$295$	1436.77	0.283566
$296$	0	0
$297$	−932.056	−0.182099
$298$	0	0
$299$	−832.000	−0.160922
$300$	0	0
$301$	−1897.59	−0.363373
$302$	0	0
$303$	3085.32	0.584973
$304$	0	0
$305$	−2571.26	−0.482721
$306$	0	0
$307$	306.312	0.0569450	0.0284725	−	0.999595i	$-0.490936\pi$
$307$	306.312	0.0569450	0.0284725	+	0.999595i	$0.490936\pi$
$308$	0	0
$309$	3801.04	0.699786
$310$	0	0
$311$	−3267.24	−0.595717	−0.297859	−	0.954610i	$-0.596272\pi$
$311$	−3267.24	−0.595717	−0.297859	+	0.954610i	$0.596272\pi$
$312$	0	0
$313$	−2456.19	−0.443553	−0.221776	−	0.975098i	$-0.571185\pi$
$313$	−2456.19	−0.443553	−0.221776	+	0.975098i	$0.571185\pi$
$314$	0	0
$315$	−386.630	−0.0691559
$316$	0	0
$317$	−1034.48	−0.183288	−0.0916441	−	0.995792i	$-0.529212\pi$
$317$	−1034.48	−0.183288	−0.0916441	+	0.995792i	$0.529212\pi$
$318$	0	0
$319$	820.930	0.144085
$320$	0	0
$321$	−5804.22	−1.00922
$322$	0	0
$323$	−176.301	−0.0303705
$324$	0	0
$325$	−868.151	−0.148173
$326$	0	0
$327$	−6115.98	−1.03429
$328$	0	0
$329$	−1116.00	−0.187013
$330$	0	0
$331$	4221.63	0.701033	0.350516	−	0.936557i	$-0.386006\pi$
$331$	4221.63	0.701033	0.350516	+	0.936557i	$0.386006\pi$
$332$	0	0
$333$	1038.08	0.170831
$334$	0	0
$335$	4054.74	0.661296
$336$	0	0
$337$	5043.43	0.815232	0.407616	−	0.913154i	$-0.366360\pi$
$337$	5043.43	0.815232	0.407616	+	0.913154i	$0.366360\pi$
$338$	0	0
$339$	107.837	0.0172769
$340$	0	0
$341$	−9819.46	−1.55939
$342$	0	0
$343$	−3683.81	−0.579904
$344$	0	0
$345$	−1464.99	−0.228615
$346$	0	0
$347$	6689.76	1.03494	0.517471	−	0.855700i	$-0.326873\pi$
$347$	6689.76	1.03494	0.517471	+	0.855700i	$0.326873\pi$
$348$	0	0
$349$	12252.0	1.87918	0.939591	−	0.342299i	$-0.111206\pi$
$349$	12252.0	1.87918	0.939591	+	0.342299i	$0.111206\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−11099.6	−1.67358	−0.836790	−	0.547523i	$-0.815571\pi$
$353$	−11099.6	−1.67358	−0.836790	+	0.547523i	$0.815571\pi$
$354$	0	0
$355$	3893.26	0.582064
$356$	0	0
$357$	−33.7809	−0.00500805
$358$	0	0
$359$	9354.11	1.37518	0.687592	−	0.726097i	$-0.258668\pi$
$359$	9354.11	1.37518	0.687592	+	0.726097i	$0.258668\pi$
$360$	0	0
$361$	911.551	0.132899
$362$	0	0
$363$	417.988	0.0604371
$364$	0	0
$365$	1253.02	0.179687
$366$	0	0
$367$	1468.14	0.208818	0.104409	−	0.994534i	$-0.466705\pi$
$367$	1468.14	0.208818	0.104409	+	0.994534i	$0.466705\pi$
$368$	0	0
$369$	12.6992	0.00179158
$370$	0	0
$371$	332.410	0.0465172
$372$	0	0
$373$	6216.36	0.862925	0.431462	−	0.902131i	$-0.357998\pi$
$373$	6216.36	0.862925	0.431462	+	0.902131i	$0.357998\pi$
$374$	0	0
$375$	−4389.95	−0.604523
$376$	0	0
$377$	309.151	0.0422337
$378$	0	0
$379$	−1365.52	−0.185072	−0.0925358	−	0.995709i	$-0.529497\pi$
$379$	−1365.52	−0.185072	−0.0925358	+	0.995709i	$0.529497\pi$
$380$	0	0
$381$	3615.04	0.486100
$382$	0	0
$383$	8311.16	1.10883	0.554413	−	0.832242i	$-0.312943\pi$
$383$	8311.16	1.10883	0.554413	+	0.832242i	$0.312943\pi$
$384$	0	0
$385$	−1482.96	−0.196309
$386$	0	0
$387$	−3033.37	−0.398436
$388$	0	0
$389$	6160.09	0.802902	0.401451	−	0.915881i	$-0.368506\pi$
$389$	6160.09	0.802902	0.401451	+	0.915881i	$0.368506\pi$
$390$	0	0
$391$	−128.000	−0.0165556
$392$	0	0
$393$	−3666.25	−0.470579
$394$	0	0
$395$	223.676	0.0284920
$396$	0	0
$397$	12658.6	1.60029	0.800146	−	0.599805i	$-0.204755\pi$
$397$	12658.6	1.60029	0.800146	+	0.599805i	$0.204755\pi$
$398$	0	0
$399$	1488.90	0.186813
$400$	0	0
$401$	13609.8	1.69487	0.847434	−	0.530900i	$-0.178146\pi$
$401$	13609.8	1.69487	0.847434	+	0.530900i	$0.178146\pi$
$402$	0	0
$403$	−3697.88	−0.457083
$404$	0	0
$405$	−618.042	−0.0758290
$406$	0	0
$407$	3981.69	0.484927
$408$	0	0
$409$	5286.90	0.639170	0.319585	−	0.947558i	$-0.396457\pi$
$409$	5286.90	0.639170	0.319585	+	0.947558i	$0.396457\pi$
$410$	0	0
$411$	−4399.20	−0.527972
$412$	0	0
$413$	−1060.16	−0.126313
$414$	0	0
$415$	896.598	0.106054
$416$	0	0
$417$	3676.20	0.431712
$418$	0	0
$419$	−4058.86	−0.473241	−0.236621	−	0.971602i	$-0.576040\pi$
$419$	−4058.86	−0.473241	−0.236621	+	0.971602i	$0.576040\pi$
$420$	0	0
$421$	11383.0	1.31776	0.658878	−	0.752249i	$-0.271031\pi$
$421$	11383.0	1.31776	0.658878	+	0.752249i	$0.271031\pi$
$422$	0	0
$423$	−1783.97	−0.205058
$424$	0	0
$425$	−133.562	−0.0152440
$426$	0	0
$427$	1897.28	0.215026
$428$	0	0
$429$	−1346.30	−0.151515
$430$	0	0
$431$	−4896.19	−0.547195	−0.273598	−	0.961844i	$-0.588214\pi$
$431$	−4896.19	−0.547195	−0.273598	+	0.961844i	$0.588214\pi$
$432$	0	0
$433$	6775.12	0.751944	0.375972	−	0.926631i	$-0.377309\pi$
$433$	6775.12	0.751944	0.375972	+	0.926631i	$0.377309\pi$
$434$	0	0
$435$	544.355	0.0599996
$436$	0	0
$437$	5641.65	0.617566
$438$	0	0
$439$	4163.40	0.452639	0.226319	−	0.974053i	$-0.427331\pi$
$439$	4163.40	0.452639	0.226319	+	0.974053i	$0.427331\pi$
$440$	0	0
$441$	−2801.71	−0.302528
$442$	0	0
$443$	−11825.0	−1.26823	−0.634114	−	0.773240i	$-0.718635\pi$
$443$	−11825.0	−1.26823	−0.634114	+	0.773240i	$0.718635\pi$
$444$	0	0
$445$	−3881.24	−0.413457
$446$	0	0
$447$	−4805.63	−0.508498
$448$	0	0
$449$	−5455.22	−0.573381	−0.286690	−	0.958023i	$-0.592555\pi$
$449$	−5455.22	−0.573381	−0.286690	+	0.958023i	$0.592555\pi$
$450$	0	0
$451$	48.7093	0.00508565
$452$	0	0
$453$	−1764.70	−0.183031
$454$	0	0
$455$	−558.465	−0.0575412
$456$	0	0
$457$	−4052.72	−0.414832	−0.207416	−	0.978253i	$-0.566505\pi$
$457$	−4052.72	−0.414832	−0.207416	+	0.978253i	$0.566505\pi$
$458$	0	0
$459$	−54.0000	−0.00549129
$460$	0	0
$461$	17679.6	1.78616	0.893079	−	0.449900i	$-0.148540\pi$
$461$	17679.6	1.78616	0.893079	+	0.449900i	$0.148540\pi$
$462$	0	0
$463$	19202.9	1.92750	0.963752	−	0.266799i	$-0.0859661\pi$
$463$	19202.9	1.92750	0.963752	+	0.266799i	$0.0859661\pi$
$464$	0	0
$465$	−6511.23	−0.649358
$466$	0	0
$467$	−489.310	−0.0484851	−0.0242426	−	0.999706i	$-0.507717\pi$
$467$	−489.310	−0.0484851	−0.0242426	+	0.999706i	$0.507717\pi$
$468$	0	0
$469$	−2991.92	−0.294572
$470$	0	0
$471$	2777.76	0.271746
$472$	0	0
$473$	−11634.9	−1.13102
$474$	0	0
$475$	5886.78	0.568640
$476$	0	0
$477$	531.370	0.0510058
$478$	0	0
$479$	8629.64	0.823170	0.411585	−	0.911371i	$-0.364975\pi$
$479$	8629.64	0.823170	0.411585	+	0.911371i	$0.364975\pi$
$480$	0	0
$481$	1499.45	0.142140
$482$	0	0
$483$	1080.99	0.101836
$484$	0	0
$485$	7789.45	0.729279
$486$	0	0
$487$	−2511.57	−0.233697	−0.116848	−	0.993150i	$-0.537279\pi$
$487$	−2511.57	−0.233697	−0.116848	+	0.993150i	$0.537279\pi$
$488$	0	0
$489$	8842.57	0.817740
$490$	0	0
$491$	−19977.9	−1.83623	−0.918115	−	0.396314i	$-0.870289\pi$
$491$	−19977.9	−1.83623	−0.918115	+	0.396314i	$0.870289\pi$
$492$	0	0
$493$	47.5617	0.00434498
$494$	0	0
$495$	−2370.57	−0.215251
$496$	0	0
$497$	−2872.77	−0.259278
$498$	0	0
$499$	8946.90	0.802642	0.401321	−	0.915938i	$-0.368551\pi$
$499$	8946.90	0.802642	0.401321	+	0.915938i	$0.368551\pi$
$500$	0	0
$501$	2146.12	0.191380
$502$	0	0
$503$	−4061.20	−0.360000	−0.180000	−	0.983667i	$-0.557610\pi$
$503$	−4061.20	−0.360000	−0.180000	+	0.983667i	$0.557610\pi$
$504$	0	0
$505$	7847.14	0.691472
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	16439.6	1.43157	0.715787	−	0.698319i	$-0.246068\pi$
$509$	16439.6	1.43157	0.715787	+	0.698319i	$0.246068\pi$
$510$	0	0
$511$	−924.578	−0.0800409
$512$	0	0
$513$	2380.07	0.204839
$514$	0	0
$515$	9667.51	0.827187
$516$	0	0
$517$	−6842.64	−0.582087
$518$	0	0
$519$	−1006.19	−0.0850997
$520$	0	0
$521$	−19585.2	−1.64692	−0.823459	−	0.567376i	$-0.807958\pi$
$521$	−19585.2	−1.64692	−0.823459	+	0.567376i	$0.807958\pi$
$522$	0	0
$523$	1682.34	0.140657	0.0703283	−	0.997524i	$-0.477595\pi$
$523$	1682.34	0.140657	0.0703283	+	0.997524i	$0.477595\pi$
$524$	0	0
$525$	1127.96	0.0937679
$526$	0	0
$527$	−568.904	−0.0470244
$528$	0	0
$529$	−8071.00	−0.663352
$530$	0	0
$531$	−1694.71	−0.138501
$532$	0	0
$533$	18.3433	0.00149069
$534$	0	0
$535$	−14762.4	−1.19296
$536$	0	0
$537$	10647.1	0.855601
$538$	0	0
$539$	−10746.3	−0.858769
$540$	0	0
$541$	13519.3	1.07438	0.537189	−	0.843462i	$-0.319486\pi$
$541$	13519.3	1.07438	0.537189	+	0.843462i	$0.319486\pi$
$542$	0	0
$543$	−3507.78	−0.277225
$544$	0	0
$545$	−15555.3	−1.22259
$546$	0	0
$547$	−747.313	−0.0584147	−0.0292073	−	0.999573i	$-0.509298\pi$
$547$	−747.313	−0.0584147	−0.0292073	+	0.999573i	$0.509298\pi$
$548$	0	0
$549$	3032.88	0.235774
$550$	0	0
$551$	−2096.30	−0.162079
$552$	0	0
$553$	−165.046	−0.0126917
$554$	0	0
$555$	2640.24	0.201932
$556$	0	0
$557$	−779.939	−0.0593305	−0.0296653	−	0.999560i	$-0.509444\pi$
$557$	−779.939	−0.0593305	−0.0296653	+	0.999560i	$0.509444\pi$
$558$	0	0
$559$	−4381.54	−0.331519
$560$	0	0
$561$	−207.123	−0.0155878
$562$	0	0
$563$	−4041.76	−0.302558	−0.151279	−	0.988491i	$-0.548339\pi$
$563$	−4041.76	−0.302558	−0.151279	+	0.988491i	$0.548339\pi$
$564$	0	0
$565$	274.270	0.0204223
$566$	0	0
$567$	456.042	0.0337777
$568$	0	0
$569$	17848.8	1.31505	0.657524	−	0.753434i	$-0.271604\pi$
$569$	17848.8	1.31505	0.657524	+	0.753434i	$0.271604\pi$
$570$	0	0
$571$	10136.5	0.742908	0.371454	−	0.928451i	$-0.378859\pi$
$571$	10136.5	0.742908	0.371454	+	0.928451i	$0.378859\pi$
$572$	0	0
$573$	−12701.6	−0.926033
$574$	0	0
$575$	4273.98	0.309978
$576$	0	0
$577$	5111.16	0.368770	0.184385	−	0.982854i	$-0.440971\pi$
$577$	5111.16	0.368770	0.184385	+	0.982854i	$0.440971\pi$
$578$	0	0
$579$	−2251.48	−0.161603
$580$	0	0
$581$	−661.583	−0.0472411
$582$	0	0
$583$	2038.14	0.144787
$584$	0	0
$585$	−892.727	−0.0630935
$586$	0	0
$587$	−20129.9	−1.41542	−0.707708	−	0.706505i	$-0.750271\pi$
$587$	−20129.9	−1.41542	−0.707708	+	0.706505i	$0.750271\pi$
$588$	0	0
$589$	25074.7	1.75413
$590$	0	0
$591$	2237.96	0.155765
$592$	0	0
$593$	3531.09	0.244527	0.122264	−	0.992498i	$-0.460985\pi$
$593$	3531.09	0.244527	0.122264	+	0.992498i	$0.460985\pi$
$594$	0	0
$595$	−85.9177	−0.00591980
$596$	0	0
$597$	−7043.92	−0.482896
$598$	0	0
$599$	−14765.7	−1.00720	−0.503598	−	0.863938i	$-0.667991\pi$
$599$	−14765.7	−1.00720	−0.503598	+	0.863938i	$0.667991\pi$
$600$	0	0
$601$	−14004.6	−0.950513	−0.475257	−	0.879847i	$-0.657645\pi$
$601$	−14004.6	−0.950513	−0.475257	+	0.879847i	$0.657645\pi$
$602$	0	0
$603$	−4782.70	−0.322996
$604$	0	0
$605$	1063.10	0.0714401
$606$	0	0
$607$	−24883.8	−1.66393	−0.831963	−	0.554831i	$-0.812783\pi$
$607$	−24883.8	−1.66393	−0.831963	+	0.554831i	$0.812783\pi$
$608$	0	0
$609$	−401.669	−0.0267265
$610$	0	0
$611$	−2576.85	−0.170619
$612$	0	0
$613$	−9656.87	−0.636276	−0.318138	−	0.948044i	$-0.603058\pi$
$613$	−9656.87	−0.636276	−0.318138	+	0.948044i	$0.603058\pi$
$614$	0	0
$615$	32.2989	0.00211775
$616$	0	0
$617$	10779.4	0.703345	0.351672	−	0.936123i	$-0.385613\pi$
$617$	10779.4	0.703345	0.351672	+	0.936123i	$0.385613\pi$
$618$	0	0
$619$	11588.6	0.752481	0.376241	−	0.926522i	$-0.377217\pi$
$619$	11588.6	0.752481	0.376241	+	0.926522i	$0.377217\pi$
$620$	0	0
$621$	1728.00	0.111662
$622$	0	0
$623$	2863.89	0.184173
$624$	0	0
$625$	−2817.71	−0.180333
$626$	0	0
$627$	9129.04	0.581466
$628$	0	0
$629$	230.685	0.0146232
$630$	0	0
$631$	−69.9543	−0.00441337	−0.00220669	−	0.999998i	$-0.500702\pi$
$631$	−69.9543	−0.00441337	−0.00220669	+	0.999998i	$0.500702\pi$
$632$	0	0
$633$	2052.17	0.128857
$634$	0	0
$635$	9194.43	0.574598
$636$	0	0
$637$	−4046.92	−0.251719
$638$	0	0
$639$	−4592.22	−0.284297
$640$	0	0
$641$	22360.1	1.37780	0.688900	−	0.724856i	$-0.258094\pi$
$641$	22360.1	1.37780	0.688900	+	0.724856i	$0.258094\pi$
$642$	0	0
$643$	−18216.6	−1.11725	−0.558626	−	0.829419i	$-0.688671\pi$
$643$	−18216.6	−1.11725	−0.558626	+	0.829419i	$0.688671\pi$
$644$	0	0
$645$	−7715.02	−0.470975
$646$	0	0
$647$	−16594.2	−1.00832	−0.504162	−	0.863609i	$-0.668198\pi$
$647$	−16594.2	−1.00832	−0.504162	+	0.863609i	$0.668198\pi$
$648$	0	0
$649$	−6500.28	−0.393156
$650$	0	0
$651$	4804.52	0.289254
$652$	0	0
$653$	−14687.0	−0.880161	−0.440081	−	0.897958i	$-0.645050\pi$
$653$	−14687.0	−0.880161	−0.440081	+	0.897958i	$0.645050\pi$
$654$	0	0
$655$	−9324.67	−0.556252
$656$	0	0
$657$	−1477.97	−0.0877644
$658$	0	0
$659$	12906.0	0.762891	0.381446	−	0.924391i	$-0.375426\pi$
$659$	12906.0	0.762891	0.381446	+	0.924391i	$0.375426\pi$
$660$	0	0
$661$	−6640.83	−0.390769	−0.195385	−	0.980727i	$-0.562596\pi$
$661$	−6640.83	−0.390769	−0.195385	+	0.980727i	$0.562596\pi$
$662$	0	0
$663$	−78.0000	−0.00456903
$664$	0	0
$665$	3786.85	0.220824
$666$	0	0
$667$	−1521.98	−0.0883525
$668$	0	0
$669$	4751.71	0.274606
$670$	0	0
$671$	11633.0	0.669279
$672$	0	0
$673$	−26318.2	−1.50742	−0.753709	−	0.657209i	$-0.771737\pi$
$673$	−26318.2	−1.50742	−0.753709	+	0.657209i	$0.771737\pi$
$674$	0	0
$675$	1803.08	0.102816
$676$	0	0
$677$	12702.6	0.721121	0.360561	−	0.932736i	$-0.382585\pi$
$677$	12702.6	0.721121	0.360561	+	0.932736i	$0.382585\pi$
$678$	0	0
$679$	−5747.69	−0.324854
$680$	0	0
$681$	−11025.7	−0.620420
$682$	0	0
$683$	−32848.6	−1.84029	−0.920144	−	0.391580i	$-0.871929\pi$
$683$	−32848.6	−1.84029	−0.920144	+	0.391580i	$0.871929\pi$
$684$	0	0
$685$	−11188.8	−0.624093
$686$	0	0
$687$	8227.00	0.456884
$688$	0	0
$689$	767.535	0.0424394
$690$	0	0
$691$	−7183.85	−0.395494	−0.197747	−	0.980253i	$-0.563362\pi$
$691$	−7183.85	−0.395494	−0.197747	+	0.980253i	$0.563362\pi$
$692$	0	0
$693$	1749.20	0.0958827
$694$	0	0
$695$	9349.97	0.510309
$696$	0	0
$697$	2.82204	0.000153361 0
$698$	0	0
$699$	7379.50	0.399311
$700$	0	0
$701$	−34532.1	−1.86057	−0.930285	−	0.366838i	$-0.880440\pi$
$701$	−34532.1	−1.86057	−0.930285	+	0.366838i	$0.880440\pi$
$702$	0	0
$703$	−10167.5	−0.545485
$704$	0	0
$705$	−4537.32	−0.242391
$706$	0	0
$707$	−5790.26	−0.308013
$708$	0	0
$709$	6369.28	0.337381	0.168691	−	0.985669i	$-0.446046\pi$
$709$	6369.28	0.337381	0.168691	+	0.985669i	$0.446046\pi$
$710$	0	0
$711$	−263.833	−0.0139163
$712$	0	0
$713$	18204.9	0.956214
$714$	0	0
$715$	−3424.16	−0.179100
$716$	0	0
$717$	1004.06	0.0522977
$718$	0	0
$719$	18776.2	0.973899	0.486949	−	0.873430i	$-0.338110\pi$
$719$	18776.2	0.973899	0.486949	+	0.873430i	$0.338110\pi$
$720$	0	0
$721$	−7133.48	−0.368467
$722$	0	0
$723$	8093.42	0.416318
$724$	0	0
$725$	−1588.11	−0.0813529
$726$	0	0
$727$	−25307.8	−1.29108	−0.645539	−	0.763728i	$-0.723367\pi$
$727$	−25307.8	−1.29108	−0.645539	+	0.763728i	$0.723367\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−674.082	−0.0341065
$732$	0	0
$733$	377.025	0.0189983	0.00949914	−	0.999955i	$-0.496976\pi$
$733$	377.025	0.0189983	0.00949914	+	0.999955i	$0.496976\pi$
$734$	0	0
$735$	−7125.83	−0.357606
$736$	0	0
$737$	−18344.6	−0.916869
$738$	0	0
$739$	35868.3	1.78544	0.892718	−	0.450616i	$-0.148796\pi$
$739$	35868.3	1.78544	0.892718	+	0.450616i	$0.148796\pi$
$740$	0	0
$741$	3437.88	0.170437
$742$	0	0
$743$	−10183.8	−0.502835	−0.251417	−	0.967879i	$-0.580897\pi$
$743$	−10183.8	−0.502835	−0.251417	+	0.967879i	$0.580897\pi$
$744$	0	0
$745$	−12222.6	−0.601074
$746$	0	0
$747$	−1057.57	−0.0517996
$748$	0	0
$749$	10892.9	0.531398
$750$	0	0
$751$	−13074.3	−0.635271	−0.317635	−	0.948213i	$-0.602889\pi$
$751$	−13074.3	−0.635271	−0.317635	+	0.948213i	$0.602889\pi$
$752$	0	0
$753$	−7996.93	−0.387018
$754$	0	0
$755$	−4488.31	−0.216353
$756$	0	0
$757$	−26803.7	−1.28692	−0.643458	−	0.765481i	$-0.722501\pi$
$757$	−26803.7	−1.28692	−0.643458	+	0.765481i	$0.722501\pi$
$758$	0	0
$759$	6627.95	0.316969
$760$	0	0
$761$	30625.8	1.45885	0.729424	−	0.684062i	$-0.239788\pi$
$761$	30625.8	1.45885	0.729424	+	0.684062i	$0.239788\pi$
$762$	0	0
$763$	11477.9	0.544600
$764$	0	0
$765$	−137.343	−0.00649102
$766$	0	0
$767$	−2447.92	−0.115240
$768$	0	0
$769$	16457.3	0.771737	0.385868	−	0.922554i	$-0.373902\pi$
$769$	16457.3	0.771737	0.385868	+	0.922554i	$0.373902\pi$
$770$	0	0
$771$	20632.3	0.963753
$772$	0	0
$773$	−22499.4	−1.04689	−0.523446	−	0.852059i	$-0.675354\pi$
$773$	−22499.4	−1.04689	−0.523446	+	0.852059i	$0.675354\pi$
$774$	0	0
$775$	18996.0	0.880458
$776$	0	0
$777$	−1948.19	−0.0899496
$778$	0	0
$779$	−124.383	−0.00572075
$780$	0	0
$781$	−17614.0	−0.807016
$782$	0	0
$783$	−642.084	−0.0293055
$784$	0	0
$785$	7064.90	0.321219
$786$	0	0
$787$	26500.7	1.20031	0.600157	−	0.799882i	$-0.295105\pi$
$787$	26500.7	1.20031	0.600157	+	0.799882i	$0.295105\pi$
$788$	0	0
$789$	166.526	0.00751391
$790$	0	0
$791$	−202.379	−0.00909704
$792$	0	0
$793$	4380.83	0.196176
$794$	0	0
$795$	1351.48	0.0602918
$796$	0	0
$797$	27751.0	1.23337	0.616683	−	0.787212i	$-0.288476\pi$
$797$	27751.0	1.23337	0.616683	+	0.787212i	$0.288476\pi$
$798$	0	0
$799$	−396.438	−0.0175532
$800$	0	0
$801$	4578.04	0.201944
$802$	0	0
$803$	−5668.94	−0.249131
$804$	0	0
$805$	2749.37	0.120376
$806$	0	0
$807$	11180.6	0.487700
$808$	0	0
$809$	25964.3	1.12838	0.564189	−	0.825646i	$-0.309189\pi$
$809$	25964.3	1.12838	0.564189	+	0.825646i	$0.309189\pi$
$810$	0	0
$811$	−23042.2	−0.997685	−0.498843	−	0.866693i	$-0.666241\pi$
$811$	−23042.2	−0.997685	−0.498843	+	0.866693i	$0.666241\pi$
$812$	0	0
$813$	−4609.77	−0.198858
$814$	0	0
$815$	22490.0	0.966615
$816$	0	0
$817$	29710.4	1.27226
$818$	0	0
$819$	658.727	0.0281047
$820$	0	0
$821$	5645.55	0.239989	0.119994	−	0.992775i	$-0.461712\pi$
$821$	5645.55	0.239989	0.119994	+	0.992775i	$0.461712\pi$
$822$	0	0
$823$	−38009.9	−1.60989	−0.804946	−	0.593348i	$-0.797806\pi$
$823$	−38009.9	−1.60989	−0.804946	+	0.593348i	$0.797806\pi$
$824$	0	0
$825$	6915.94	0.291857
$826$	0	0
$827$	7195.40	0.302550	0.151275	−	0.988492i	$-0.451662\pi$
$827$	7195.40	0.302550	0.151275	+	0.988492i	$0.451662\pi$
$828$	0	0
$829$	−40503.6	−1.69692	−0.848461	−	0.529259i	$-0.822470\pi$
$829$	−40503.6	−1.69692	−0.848461	+	0.529259i	$0.822470\pi$
$830$	0	0
$831$	−14751.7	−0.615801
$832$	0	0
$833$	−622.603	−0.0258967
$834$	0	0
$835$	5458.39	0.226222
$836$	0	0
$837$	7680.21	0.317165
$838$	0	0
$839$	5322.63	0.219020	0.109510	−	0.993986i	$-0.465072\pi$
$839$	5322.63	0.219020	0.109510	+	0.993986i	$0.465072\pi$
$840$	0	0
$841$	−23823.5	−0.976812
$842$	0	0
$843$	15845.0	0.647367
$844$	0	0
$845$	−1289.49	−0.0524970
$846$	0	0
$847$	−784.444	−0.0318227
$848$	0	0
$849$	15274.1	0.617439
$850$	0	0
$851$	−7381.93	−0.297355
$852$	0	0
$853$	−44680.0	−1.79345	−0.896726	−	0.442586i	$-0.854061\pi$
$853$	−44680.0	−1.79345	−0.896726	+	0.442586i	$0.854061\pi$
$854$	0	0
$855$	6053.43	0.242132
$856$	0	0
$857$	−9103.80	−0.362870	−0.181435	−	0.983403i	$-0.558074\pi$
$857$	−9103.80	−0.362870	−0.181435	+	0.983403i	$0.558074\pi$
$858$	0	0
$859$	−41297.8	−1.64035	−0.820177	−	0.572110i	$-0.806125\pi$
$859$	−41297.8	−1.64035	−0.820177	+	0.572110i	$0.806125\pi$
$860$	0	0
$861$	−23.8328	−0.000943343 0
$862$	0	0
$863$	28878.2	1.13908	0.569540	−	0.821964i	$-0.307121\pi$
$863$	28878.2	1.13908	0.569540	+	0.821964i	$0.307121\pi$
$864$	0	0
$865$	−2559.12	−0.100593
$866$	0	0
$867$	14727.0	0.576880
$868$	0	0
$869$	−1011.96	−0.0395034
$870$	0	0
$871$	−6908.34	−0.268749
$872$	0	0
$873$	−9187.90	−0.356201
$874$	0	0
$875$	8238.68	0.318307
$876$	0	0
$877$	36361.7	1.40005	0.700027	−	0.714117i	$-0.253171\pi$
$877$	36361.7	1.40005	0.700027	+	0.714117i	$0.253171\pi$
$878$	0	0
$879$	12828.0	0.492237
$880$	0	0
$881$	−14003.7	−0.535523	−0.267761	−	0.963485i	$-0.586284\pi$
$881$	−14003.7	−0.535523	−0.267761	+	0.963485i	$0.586284\pi$
$882$	0	0
$883$	45492.4	1.73379	0.866897	−	0.498488i	$-0.166111\pi$
$883$	45492.4	1.73379	0.866897	+	0.498488i	$0.166111\pi$
$884$	0	0
$885$	−4310.30	−0.163717
$886$	0	0
$887$	−1388.94	−0.0525773	−0.0262886	−	0.999654i	$-0.508369\pi$
$887$	−1388.94	−0.0525773	−0.0262886	+	0.999654i	$0.508369\pi$
$888$	0	0
$889$	−6784.40	−0.255952
$890$	0	0
$891$	2796.17	0.105135
$892$	0	0
$893$	17473.2	0.654778
$894$	0	0
$895$	27079.7	1.01137
$896$	0	0
$897$	2496.00	0.0929086
$898$	0	0
$899$	−6764.52	−0.250956
$900$	0	0
$901$	118.082	0.00436614
$902$	0	0
$903$	5692.77	0.209794
$904$	0	0
$905$	−8921.63	−0.327696
$906$	0	0
$907$	28974.2	1.06072	0.530361	−	0.847772i	$-0.322057\pi$
$907$	28974.2	1.06072	0.530361	+	0.847772i	$0.322057\pi$
$908$	0	0
$909$	−9255.96	−0.337735
$910$	0	0
$911$	41047.3	1.49282	0.746408	−	0.665488i	$-0.231777\pi$
$911$	41047.3	1.49282	0.746408	+	0.665488i	$0.231777\pi$
$912$	0	0
$913$	−4056.42	−0.147040
$914$	0	0
$915$	7713.77	0.278699
$916$	0	0
$917$	6880.50	0.247780
$918$	0	0
$919$	25401.8	0.911783	0.455891	−	0.890035i	$-0.349321\pi$
$919$	25401.8	0.911783	0.455891	+	0.890035i	$0.349321\pi$
$920$	0	0
$921$	−918.935	−0.0328772
$922$	0	0
$923$	−6633.21	−0.236549
$924$	0	0
$925$	−7702.68	−0.273797
$926$	0	0
$927$	−11403.1	−0.404022
$928$	0	0
$929$	46342.1	1.63664	0.818319	−	0.574765i	$-0.194907\pi$
$929$	46342.1	1.63664	0.818319	+	0.574765i	$0.194907\pi$
$930$	0	0
$931$	27441.5	0.966012
$932$	0	0
$933$	9801.71	0.343937
$934$	0	0
$935$	−526.794	−0.0184257
$936$	0	0
$937$	8386.36	0.292391	0.146196	−	0.989256i	$-0.453297\pi$
$937$	8386.36	0.292391	0.146196	+	0.989256i	$0.453297\pi$
$938$	0	0
$939$	7368.57	0.256085
$940$	0	0
$941$	−7119.76	−0.246650	−0.123325	−	0.992366i	$-0.539356\pi$
$941$	−7119.76	−0.246650	−0.123325	+	0.992366i	$0.539356\pi$
$942$	0	0
$943$	−90.3053	−0.00311850
$944$	0	0
$945$	1159.89	0.0399272
$946$	0	0
$947$	−19592.2	−0.672293	−0.336147	−	0.941810i	$-0.609124\pi$
$947$	−19592.2	−0.672293	−0.336147	+	0.941810i	$0.609124\pi$
$948$	0	0
$949$	−2134.85	−0.0730244
$950$	0	0
$951$	3103.45	0.105821
$952$	0	0
$953$	48924.0	1.66296	0.831482	−	0.555552i	$-0.187493\pi$
$953$	48924.0	1.66296	0.831482	+	0.555552i	$0.187493\pi$
$954$	0	0
$955$	−32305.0	−1.09462
$956$	0	0
$957$	−2462.79	−0.0831877
$958$	0	0
$959$	8256.04	0.277999
$960$	0	0
$961$	51122.0	1.71602
$962$	0	0
$963$	17412.7	0.582674
$964$	0	0
$965$	−5726.38	−0.191025
$966$	0	0
$967$	29095.8	0.967588	0.483794	−	0.875182i	$-0.339258\pi$
$967$	29095.8	0.967588	0.483794	+	0.875182i	$0.339258\pi$
$968$	0	0
$969$	528.904	0.0175344
$970$	0	0
$971$	−41888.1	−1.38440	−0.692200	−	0.721706i	$-0.743358\pi$
$971$	−41888.1	−1.38440	−0.692200	+	0.721706i	$0.743358\pi$
$972$	0	0
$973$	−6899.17	−0.227315
$974$	0	0
$975$	2604.45	0.0855480
$976$	0	0
$977$	−37197.0	−1.21805	−0.609027	−	0.793150i	$-0.708440\pi$
$977$	−37197.0	−1.21805	−0.609027	+	0.793150i	$0.708440\pi$
$978$	0	0
$979$	17559.6	0.573246
$980$	0	0
$981$	18347.9	0.597150
$982$	0	0
$983$	44554.4	1.44564	0.722820	−	0.691037i	$-0.242846\pi$
$983$	44554.4	1.44564	0.722820	+	0.691037i	$0.242846\pi$
$984$	0	0
$985$	5691.98	0.184123
$986$	0	0
$987$	3348.01	0.107972
$988$	0	0
$989$	21570.6	0.693535
$990$	0	0
$991$	−23053.2	−0.738961	−0.369480	−	0.929238i	$-0.620464\pi$
$991$	−23053.2	−0.738961	−0.369480	+	0.929238i	$0.620464\pi$
$992$	0	0
$993$	−12664.9	−0.404742
$994$	0	0
$995$	−17915.4	−0.570810
$996$	0	0
$997$	−36408.5	−1.15654	−0.578269	−	0.815846i	$-0.696271\pi$
$997$	−36408.5	−1.15654	−0.578269	+	0.815846i	$0.696271\pi$
$998$	0	0
$999$	−3114.25	−0.0986292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.c.1.1	✓	2
3.2	odd	2		936.4.a.d.1.2		2
4.3	odd	2		624.4.a.q.1.1		2
8.3	odd	2		2496.4.a.v.1.2		2
8.5	even	2		2496.4.a.be.1.2		2
12.11	even	2		1872.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.c.1.1	✓	2	1.1	even	1	trivial
624.4.a.q.1.1		2	4.3	odd	2
936.4.a.d.1.2		2	3.2	odd	2
1872.4.a.x.1.2		2	12.11	even	2
2496.4.a.v.1.2		2	8.3	odd	2
2496.4.a.be.1.2		2	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -7.63015 q^{5} +5.63015 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -7.63015 q^{5} +5.63015 q^{7} +9.00000 q^{9} +34.5206 q^{11} +13.0000 q^{13} +22.8904 q^{15} +2.00000 q^{17} -88.1507 q^{19} -16.8904 q^{21} -64.0000 q^{23} -66.7809 q^{25} -27.0000 q^{27} +23.7809 q^{29} -284.452 q^{31} -103.562 q^{33} -42.9588 q^{35} +115.343 q^{37} -39.0000 q^{39} +1.41102 q^{41} -337.041 q^{43} -68.6713 q^{45} -198.219 q^{47} -311.301 q^{49} -6.00000 q^{51} +59.0412 q^{53} -263.397 q^{55} +264.452 q^{57} -188.301 q^{59} +336.987 q^{61} +50.6713 q^{63} -99.1919 q^{65} -531.411 q^{67} +192.000 q^{69} -510.247 q^{71} -164.219 q^{73} +200.343 q^{75} +194.356 q^{77} -29.3148 q^{79} +81.0000 q^{81} -117.507 q^{83} -15.2603 q^{85} -71.3426 q^{87} +508.671 q^{89} +73.1919 q^{91} +853.357 q^{93} +672.603 q^{95} -1020.88 q^{97} +310.685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9} - 16 q^{11} + 26 q^{13} - 18 q^{15} + 4 q^{17} - 70 q^{19} + 30 q^{21} - 128 q^{23} - 6 q^{25} - 54 q^{27} - 80 q^{29} - 250 q^{31} + 48 q^{33} - 256 q^{35} - 152 q^{37} - 78 q^{39} - 146 q^{41} - 504 q^{43} + 54 q^{45} - 524 q^{47} - 410 q^{49} - 12 q^{51} - 52 q^{53} - 952 q^{55} + 210 q^{57} - 164 q^{59} - 304 q^{61} - 90 q^{63} + 78 q^{65} - 914 q^{67} + 384 q^{69} - 456 q^{73} + 18 q^{75} + 984 q^{77} - 824 q^{79} + 162 q^{81} + 828 q^{83} + 12 q^{85} + 240 q^{87} + 826 q^{89} - 130 q^{91} + 750 q^{93} + 920 q^{95} + 552 q^{97} - 144 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9 - 16 * q^11 + 26 * q^13 - 18 * q^15 + 4 * q^17 - 70 * q^19 + 30 * q^21 - 128 * q^23 - 6 * q^25 - 54 * q^27 - 80 * q^29 - 250 * q^31 + 48 * q^33 - 256 * q^35 - 152 * q^37 - 78 * q^39 - 146 * q^41 - 504 * q^43 + 54 * q^45 - 524 * q^47 - 410 * q^49 - 12 * q^51 - 52 * q^53 - 952 * q^55 + 210 * q^57 - 164 * q^59 - 304 * q^61 - 90 * q^63 + 78 * q^65 - 914 * q^67 + 384 * q^69 - 456 * q^73 + 18 * q^75 + 984 * q^77 - 824 * q^79 + 162 * q^81 + 828 * q^83 + 12 * q^85 + 240 * q^87 + 826 * q^89 - 130 * q^91 + 750 * q^93 + 920 * q^95 + 552 * q^97 - 144 * q^99

Introduction

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