LMFDB - Embedded newform 2312.4.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 345 x^{16} - 182 x^{15} + 48165 x^{14} + 48078 x^{13} - 3485278 x^{12} - 4881882 x^{11} + \cdots - 119632152329 $ x^18 - 345x^16 - 182x^15 + 48165x^14 + 48078x^13 - 3485278x^12 - 4881882x^11 + 139131876x^10 + 239770606x^9 - 3013674039x^8 - 5867731896x^7 + 32610490343x^6 + 66112081788x^5 - 151840532421x^4 - 282437582582x^3 + 230444683878x^2 + 237530449410x - 119632152329
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.68427$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68427 q^{3} +3.61861 q^{5} -34.3516 q^{7} -19.7947 q^{9} +O(q^{10})$	$q-2.68427 q^{3} +3.61861 q^{5} -34.3516 q^{7} -19.7947 q^{9} +42.7402 q^{11} +52.0171 q^{13} -9.71334 q^{15} -105.635 q^{19} +92.2092 q^{21} +179.750 q^{23} -111.906 q^{25} +125.610 q^{27} +75.1233 q^{29} -213.036 q^{31} -114.726 q^{33} -124.305 q^{35} -280.857 q^{37} -139.628 q^{39} -382.779 q^{41} +519.334 q^{43} -71.6292 q^{45} +445.061 q^{47} +837.036 q^{49} +320.325 q^{53} +154.660 q^{55} +283.553 q^{57} -30.9564 q^{59} +411.797 q^{61} +679.980 q^{63} +188.230 q^{65} +300.784 q^{67} -482.498 q^{69} +170.421 q^{71} -420.355 q^{73} +300.385 q^{75} -1468.19 q^{77} +639.560 q^{79} +197.285 q^{81} +409.467 q^{83} -201.652 q^{87} -334.690 q^{89} -1786.87 q^{91} +571.847 q^{93} -382.252 q^{95} -764.915 q^{97} -846.027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9}+O(q^{10}) $ 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9	$ 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9} + 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} - 153 q^{23} + 594 q^{25} + 546 q^{27} - 447 q^{29} - 303 q^{31} + 153 q^{33} - 117 q^{35} - 939 q^{37} - 516 q^{39} - 1257 q^{41} + 306 q^{43} - 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} - 1494 q^{57} + 696 q^{59} - 1686 q^{61} - 1908 q^{63} - 855 q^{65} + 513 q^{67} - 1329 q^{69} - 324 q^{71} - 1863 q^{73} + 3054 q^{75} + 1833 q^{77} - 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} - 252 q^{91} - 1470 q^{93} - 2109 q^{95} - 4611 q^{97} + 3918 q^{99}+O(q^{100}) $ 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9 + 66 * q^11 - 30 * q^13 - 102 * q^15 - 168 * q^19 - 510 * q^21 - 153 * q^23 + 594 * q^25 + 546 * q^27 - 447 * q^29 - 303 * q^31 + 153 * q^33 - 117 * q^35 - 939 * q^37 - 516 * q^39 - 1257 * q^41 + 306 * q^43 - 672 * q^45 + 633 * q^47 + 1239 * q^49 - 489 * q^53 + 1089 * q^55 - 1494 * q^57 + 696 * q^59 - 1686 * q^61 - 1908 * q^63 - 855 * q^65 + 513 * q^67 - 1329 * q^69 - 324 * q^71 - 1863 * q^73 + 3054 * q^75 + 1833 * q^77 - 3699 * q^79 + 2622 * q^81 + 1188 * q^83 - 3927 * q^87 + 1713 * q^89 - 252 * q^91 - 1470 * q^93 - 2109 * q^95 - 4611 * q^97 + 3918 * 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
$3$	−2.68427	−0.516589	−0.258294	−	0.966066i	$-0.583160\pi$
$3$	−2.68427	−0.516589	−0.258294	+	0.966066i	$0.583160\pi$
$4$	0	0
$5$	3.61861	0.323658	0.161829	−	0.986819i	$-0.448261\pi$
$5$	3.61861	0.323658	0.161829	+	0.986819i	$0.448261\pi$
$6$	0	0
$7$	−34.3516	−1.85481	−0.927407	−	0.374053i	$-0.877968\pi$
$7$	−34.3516	−1.85481	−0.927407	+	0.374053i	$0.877968\pi$
$8$	0	0
$9$	−19.7947	−0.733136
$10$	0	0
$11$	42.7402	1.17151	0.585756	−	0.810487i	$-0.300798\pi$
$11$	42.7402	1.17151	0.585756	+	0.810487i	$0.300798\pi$
$12$	0	0
$13$	52.0171	1.10977	0.554883	−	0.831928i	$-0.312763\pi$
$13$	52.0171	1.10977	0.554883	+	0.831928i	$0.312763\pi$
$14$	0	0
$15$	−9.71334	−0.167198
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−105.635	−1.27549	−0.637745	−	0.770248i	$-0.720133\pi$
$19$	−105.635	−1.27549	−0.637745	+	0.770248i	$0.720133\pi$
$20$	0	0
$21$	92.2092	0.958176
$22$	0	0
$23$	179.750	1.62958	0.814792	−	0.579754i	$-0.196851\pi$
$23$	179.750	1.62958	0.814792	+	0.579754i	$0.196851\pi$
$24$	0	0
$25$	−111.906	−0.895245
$26$	0	0
$27$	125.610	0.895318
$28$	0	0
$29$	75.1233	0.481036	0.240518	−	0.970645i	$-0.422683\pi$
$29$	75.1233	0.481036	0.240518	+	0.970645i	$0.422683\pi$
$30$	0	0
$31$	−213.036	−1.23427	−0.617135	−	0.786857i	$-0.711707\pi$
$31$	−213.036	−1.23427	−0.617135	+	0.786857i	$0.711707\pi$
$32$	0	0
$33$	−114.726	−0.605190
$34$	0	0
$35$	−124.305	−0.600326
$36$	0	0
$37$	−280.857	−1.24791	−0.623953	−	0.781462i	$-0.714475\pi$
$37$	−280.857	−1.24791	−0.623953	+	0.781462i	$0.714475\pi$
$38$	0	0
$39$	−139.628	−0.573292
$40$	0	0
$41$	−382.779	−1.45805	−0.729024	−	0.684488i	$-0.760026\pi$
$41$	−382.779	−1.45805	−0.729024	+	0.684488i	$0.760026\pi$
$42$	0	0
$43$	519.334	1.84181	0.920903	−	0.389791i	$-0.127453\pi$
$43$	519.334	1.84181	0.920903	+	0.389791i	$0.127453\pi$
$44$	0	0
$45$	−71.6292	−0.237286
$46$	0	0
$47$	445.061	1.38125	0.690625	−	0.723213i	$-0.257335\pi$
$47$	445.061	1.38125	0.690625	+	0.723213i	$0.257335\pi$
$48$	0	0
$49$	837.036	2.44034
$50$	0	0
$51$	0	0
$52$	0	0
$53$	320.325	0.830188	0.415094	−	0.909778i	$-0.363749\pi$
$53$	320.325	0.830188	0.415094	+	0.909778i	$0.363749\pi$
$54$	0	0
$55$	154.660	0.379170
$56$	0	0
$57$	283.553	0.658904
$58$	0	0
$59$	−30.9564	−0.0683081	−0.0341541	−	0.999417i	$-0.510874\pi$
$59$	−30.9564	−0.0683081	−0.0341541	+	0.999417i	$0.510874\pi$
$60$	0	0
$61$	411.797	0.864348	0.432174	−	0.901790i	$-0.357747\pi$
$61$	411.797	0.864348	0.432174	+	0.901790i	$0.357747\pi$
$62$	0	0
$63$	679.980	1.35983
$64$	0	0
$65$	188.230	0.359185
$66$	0	0
$67$	300.784	0.548458	0.274229	−	0.961664i	$-0.411577\pi$
$67$	300.784	0.548458	0.274229	+	0.961664i	$0.411577\pi$
$68$	0	0
$69$	−482.498	−0.841824
$70$	0	0
$71$	170.421	0.284862	0.142431	−	0.989805i	$-0.454508\pi$
$71$	170.421	0.284862	0.142431	+	0.989805i	$0.454508\pi$
$72$	0	0
$73$	−420.355	−0.673957	−0.336979	−	0.941512i	$-0.609405\pi$
$73$	−420.355	−0.673957	−0.336979	+	0.941512i	$0.609405\pi$
$74$	0	0
$75$	300.385	0.462474
$76$	0	0
$77$	−1468.19	−2.17294
$78$	0	0
$79$	639.560	0.910836	0.455418	−	0.890278i	$-0.349490\pi$
$79$	639.560	0.910836	0.455418	+	0.890278i	$0.349490\pi$
$80$	0	0
$81$	197.285	0.270625
$82$	0	0
$83$	409.467	0.541504	0.270752	−	0.962649i	$-0.412728\pi$
$83$	409.467	0.541504	0.270752	+	0.962649i	$0.412728\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−201.652	−0.248498
$88$	0	0
$89$	−334.690	−0.398619	−0.199309	−	0.979937i	$-0.563870\pi$
$89$	−334.690	−0.398619	−0.199309	+	0.979937i	$0.563870\pi$
$90$	0	0
$91$	−1786.87	−2.05841
$92$	0	0
$93$	571.847	0.637610
$94$	0	0
$95$	−382.252	−0.412823
$96$	0	0
$97$	−764.915	−0.800674	−0.400337	−	0.916368i	$-0.631107\pi$
$97$	−764.915	−0.800674	−0.400337	+	0.916368i	$0.631107\pi$
$98$	0	0
$99$	−846.027	−0.858878
$100$	0	0
$101$	−1023.56	−1.00839	−0.504196	−	0.863589i	$-0.668211\pi$
$101$	−1023.56	−1.00839	−0.504196	+	0.863589i	$0.668211\pi$
$102$	0	0
$103$	1561.93	1.49419	0.747095	−	0.664717i	$-0.231448\pi$
$103$	1561.93	1.49419	0.747095	+	0.664717i	$0.231448\pi$
$104$	0	0
$105$	333.669	0.310122
$106$	0	0
$107$	−1360.47	−1.22918	−0.614588	−	0.788848i	$-0.710678\pi$
$107$	−1360.47	−1.22918	−0.614588	+	0.788848i	$0.710678\pi$
$108$	0	0
$109$	438.026	0.384911	0.192455	−	0.981306i	$-0.438355\pi$
$109$	438.026	0.384911	0.192455	+	0.981306i	$0.438355\pi$
$110$	0	0
$111$	753.896	0.644655
$112$	0	0
$113$	468.142	0.389726	0.194863	−	0.980830i	$-0.437574\pi$
$113$	468.142	0.389726	0.194863	+	0.980830i	$0.437574\pi$
$114$	0	0
$115$	650.445	0.527428
$116$	0	0
$117$	−1029.66	−0.813609
$118$	0	0
$119$	0	0
$120$	0	0
$121$	495.721	0.372442
$122$	0	0
$123$	1027.48	0.753212
$124$	0	0
$125$	−857.269	−0.613412
$126$	0	0
$127$	−1067.53	−0.745886	−0.372943	−	0.927854i	$-0.621651\pi$
$127$	−1067.53	−0.745886	−0.372943	+	0.927854i	$0.621651\pi$
$128$	0	0
$129$	−1394.03	−0.951456
$130$	0	0
$131$	−1293.47	−0.862677	−0.431338	−	0.902190i	$-0.641959\pi$
$131$	−1293.47	−0.862677	−0.431338	+	0.902190i	$0.641959\pi$
$132$	0	0
$133$	3628.73	2.36580
$134$	0	0
$135$	454.533	0.289777
$136$	0	0
$137$	1677.23	1.04595	0.522977	−	0.852347i	$-0.324821\pi$
$137$	1677.23	1.04595	0.522977	+	0.852347i	$0.324821\pi$
$138$	0	0
$139$	545.345	0.332774	0.166387	−	0.986061i	$-0.446790\pi$
$139$	545.345	0.332774	0.166387	+	0.986061i	$0.446790\pi$
$140$	0	0
$141$	−1194.66	−0.713538
$142$	0	0
$143$	2223.22	1.30010
$144$	0	0
$145$	271.842	0.155691
$146$	0	0
$147$	−2246.83	−1.26065
$148$	0	0
$149$	−2034.03	−1.11835	−0.559176	−	0.829049i	$-0.688882\pi$
$149$	−2034.03	−1.11835	−0.559176	+	0.829049i	$0.688882\pi$
$150$	0	0
$151$	998.913	0.538347	0.269173	−	0.963092i	$-0.413250\pi$
$151$	998.913	0.538347	0.269173	+	0.963092i	$0.413250\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−770.894	−0.399482
$156$	0	0
$157$	−173.428	−0.0881596	−0.0440798	−	0.999028i	$-0.514036\pi$
$157$	−173.428	−0.0881596	−0.0440798	+	0.999028i	$0.514036\pi$
$158$	0	0
$159$	−859.839	−0.428866
$160$	0	0
$161$	−6174.70	−3.02258
$162$	0	0
$163$	−2950.85	−1.41797	−0.708983	−	0.705226i	$-0.750846\pi$
$163$	−2950.85	−1.41797	−0.708983	+	0.705226i	$0.750846\pi$
$164$	0	0
$165$	−415.150	−0.195875
$166$	0	0
$167$	−998.939	−0.462876	−0.231438	−	0.972850i	$-0.574343\pi$
$167$	−998.939	−0.462876	−0.231438	+	0.972850i	$0.574343\pi$
$168$	0	0
$169$	508.782	0.231580
$170$	0	0
$171$	2091.01	0.935108
$172$	0	0
$173$	−3888.18	−1.70874	−0.854372	−	0.519662i	$-0.826058\pi$
$173$	−3888.18	−1.70874	−0.854372	+	0.519662i	$0.826058\pi$
$174$	0	0
$175$	3844.14	1.66051
$176$	0	0
$177$	83.0954	0.0352872
$178$	0	0
$179$	−2487.65	−1.03875	−0.519373	−	0.854548i	$-0.673834\pi$
$179$	−2487.65	−1.03875	−0.519373	+	0.854548i	$0.673834\pi$
$180$	0	0
$181$	−2130.60	−0.874953	−0.437477	−	0.899230i	$-0.644128\pi$
$181$	−2130.60	−0.874953	−0.437477	+	0.899230i	$0.644128\pi$
$182$	0	0
$183$	−1105.38	−0.446513
$184$	0	0
$185$	−1016.31	−0.403896
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4314.90	−1.66065
$190$	0	0
$191$	1837.76	0.696206	0.348103	−	0.937456i	$-0.386826\pi$
$191$	1837.76	0.696206	0.348103	+	0.937456i	$0.386826\pi$
$192$	0	0
$193$	−4784.05	−1.78427	−0.892133	−	0.451773i	$-0.850792\pi$
$193$	−4784.05	−1.78427	−0.892133	+	0.451773i	$0.850792\pi$
$194$	0	0
$195$	−505.260	−0.185551
$196$	0	0
$197$	863.953	0.312457	0.156229	−	0.987721i	$-0.450066\pi$
$197$	863.953	0.312457	0.156229	+	0.987721i	$0.450066\pi$
$198$	0	0
$199$	202.551	0.0721529	0.0360765	−	0.999349i	$-0.488514\pi$
$199$	202.551	0.0721529	0.0360765	+	0.999349i	$0.488514\pi$
$200$	0	0
$201$	−807.388	−0.283327
$202$	0	0
$203$	−2580.61	−0.892233
$204$	0	0
$205$	−1385.13	−0.471910
$206$	0	0
$207$	−3558.09	−1.19471
$208$	0	0
$209$	−4514.85	−1.49425
$210$	0	0
$211$	1586.59	0.517654	0.258827	−	0.965924i	$-0.416664\pi$
$211$	1586.59	0.517654	0.258827	+	0.965924i	$0.416664\pi$
$212$	0	0
$213$	−457.456	−0.147157
$214$	0	0
$215$	1879.27	0.596116
$216$	0	0
$217$	7318.13	2.28934
$218$	0	0
$219$	1128.35	0.348159
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−819.503	−0.246090	−0.123045	−	0.992401i	$-0.539266\pi$
$223$	−819.503	−0.246090	−0.123045	+	0.992401i	$0.539266\pi$
$224$	0	0
$225$	2215.14	0.656337
$226$	0	0
$227$	2270.28	0.663806	0.331903	−	0.943313i	$-0.392309\pi$
$227$	2270.28	0.663806	0.331903	+	0.943313i	$0.392309\pi$
$228$	0	0
$229$	3853.83	1.11209	0.556044	−	0.831153i	$-0.312319\pi$
$229$	3853.83	1.11209	0.556044	+	0.831153i	$0.312319\pi$
$230$	0	0
$231$	3941.04	1.12252
$232$	0	0
$233$	3400.20	0.956027	0.478014	−	0.878352i	$-0.341357\pi$
$233$	3400.20	0.956027	0.478014	+	0.878352i	$0.341357\pi$
$234$	0	0
$235$	1610.50	0.447053
$236$	0	0
$237$	−1716.75	−0.470528
$238$	0	0
$239$	−699.847	−0.189411	−0.0947057	−	0.995505i	$-0.530191\pi$
$239$	−699.847	−0.189411	−0.0947057	+	0.995505i	$0.530191\pi$
$240$	0	0
$241$	154.493	0.0412937	0.0206468	−	0.999787i	$-0.493427\pi$
$241$	154.493	0.0412937	0.0206468	+	0.999787i	$0.493427\pi$
$242$	0	0
$243$	−3921.03	−1.03512
$244$	0	0
$245$	3028.91	0.789836
$246$	0	0
$247$	−5494.82	−1.41550
$248$	0	0
$249$	−1099.12	−0.279735
$250$	0	0
$251$	−1378.18	−0.346573	−0.173286	−	0.984871i	$-0.555439\pi$
$251$	−1378.18	−0.346573	−0.173286	+	0.984871i	$0.555439\pi$
$252$	0	0
$253$	7682.53	1.90908
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6787.79	−1.64751	−0.823756	−	0.566945i	$-0.808125\pi$
$257$	−6787.79	−1.64751	−0.823756	+	0.566945i	$0.808125\pi$
$258$	0	0
$259$	9647.89	2.31464
$260$	0	0
$261$	−1487.04	−0.352665
$262$	0	0
$263$	2272.11	0.532716	0.266358	−	0.963874i	$-0.414180\pi$
$263$	2272.11	0.532716	0.266358	+	0.963874i	$0.414180\pi$
$264$	0	0
$265$	1159.13	0.268697
$266$	0	0
$267$	898.399	0.205922
$268$	0	0
$269$	−6525.02	−1.47895	−0.739475	−	0.673184i	$-0.764926\pi$
$269$	−6525.02	−1.47895	−0.739475	+	0.673184i	$0.764926\pi$
$270$	0	0
$271$	−452.308	−0.101387	−0.0506933	−	0.998714i	$-0.516143\pi$
$271$	−452.308	−0.101387	−0.0506933	+	0.998714i	$0.516143\pi$
$272$	0	0
$273$	4796.46	1.06335
$274$	0	0
$275$	−4782.86	−1.04879
$276$	0	0
$277$	3243.24	0.703493	0.351747	−	0.936095i	$-0.385588\pi$
$277$	3243.24	0.703493	0.351747	+	0.936095i	$0.385588\pi$
$278$	0	0
$279$	4216.98	0.904888
$280$	0	0
$281$	4900.08	1.04026	0.520132	−	0.854086i	$-0.325883\pi$
$281$	4900.08	1.04026	0.520132	+	0.854086i	$0.325883\pi$
$282$	0	0
$283$	−1524.43	−0.320205	−0.160102	−	0.987100i	$-0.551182\pi$
$283$	−1524.43	−0.320205	−0.160102	+	0.987100i	$0.551182\pi$
$284$	0	0
$285$	1026.07	0.213260
$286$	0	0
$287$	13149.1	2.70441
$288$	0	0
$289$	0	0
$290$	0	0
$291$	2053.24	0.413619
$292$	0	0
$293$	−6627.78	−1.32150	−0.660750	−	0.750606i	$-0.729761\pi$
$293$	−6627.78	−1.32150	−0.660750	+	0.750606i	$0.729761\pi$
$294$	0	0
$295$	−112.019	−0.0221085
$296$	0	0
$297$	5368.58	1.04888
$298$	0	0
$299$	9350.07	1.80846
$300$	0	0
$301$	−17840.0	−3.41621
$302$	0	0
$303$	2747.51	0.520924
$304$	0	0
$305$	1490.13	0.279754
$306$	0	0
$307$	−7804.25	−1.45085	−0.725427	−	0.688299i	$-0.758358\pi$
$307$	−7804.25	−1.45085	−0.725427	+	0.688299i	$0.758358\pi$
$308$	0	0
$309$	−4192.65	−0.771882
$310$	0	0
$311$	−9391.22	−1.71231	−0.856153	−	0.516722i	$-0.827152\pi$
$311$	−9391.22	−1.71231	−0.856153	+	0.516722i	$0.827152\pi$
$312$	0	0
$313$	−4979.40	−0.899208	−0.449604	−	0.893228i	$-0.648435\pi$
$313$	−4979.40	−0.899208	−0.449604	+	0.893228i	$0.648435\pi$
$314$	0	0
$315$	2460.58	0.440121
$316$	0	0
$317$	900.049	0.159469	0.0797347	−	0.996816i	$-0.474593\pi$
$317$	900.049	0.159469	0.0797347	+	0.996816i	$0.474593\pi$
$318$	0	0
$319$	3210.78	0.563540
$320$	0	0
$321$	3651.88	0.634978
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5821.01	−0.993513
$326$	0	0
$327$	−1175.78	−0.198841
$328$	0	0
$329$	−15288.6	−2.56196
$330$	0	0
$331$	−6137.26	−1.01914	−0.509568	−	0.860430i	$-0.670195\pi$
$331$	−6137.26	−1.01914	−0.509568	+	0.860430i	$0.670195\pi$
$332$	0	0
$333$	5559.47	0.914886
$334$	0	0
$335$	1088.42	0.177513
$336$	0	0
$337$	3689.69	0.596410	0.298205	−	0.954502i	$-0.403612\pi$
$337$	3689.69	0.596410	0.298205	+	0.954502i	$0.403612\pi$
$338$	0	0
$339$	−1256.62	−0.201328
$340$	0	0
$341$	−9105.19	−1.44596
$342$	0	0
$343$	−16970.9	−2.67156
$344$	0	0
$345$	−1745.97	−0.272464
$346$	0	0
$347$	7511.00	1.16199	0.580997	−	0.813906i	$-0.302663\pi$
$347$	7511.00	1.16199	0.580997	+	0.813906i	$0.302663\pi$
$348$	0	0
$349$	−2621.99	−0.402155	−0.201077	−	0.979575i	$-0.564444\pi$
$349$	−2621.99	−0.402155	−0.201077	+	0.979575i	$0.564444\pi$
$350$	0	0
$351$	6533.86	0.993594
$352$	0	0
$353$	11579.9	1.74599	0.872997	−	0.487726i	$-0.162173\pi$
$353$	11579.9	1.74599	0.872997	+	0.487726i	$0.162173\pi$
$354$	0	0
$355$	616.686	0.0921980
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12165.6	−1.78852	−0.894259	−	0.447549i	$-0.852297\pi$
$359$	−12165.6	−1.78852	−0.894259	+	0.447549i	$0.852297\pi$
$360$	0	0
$361$	4299.73	0.626875
$362$	0	0
$363$	−1330.65	−0.192399
$364$	0	0
$365$	−1521.10	−0.218132
$366$	0	0
$367$	−8398.59	−1.19456	−0.597279	−	0.802033i	$-0.703752\pi$
$367$	−8398.59	−1.19456	−0.597279	+	0.802033i	$0.703752\pi$
$368$	0	0
$369$	7576.98	1.06895
$370$	0	0
$371$	−11003.7	−1.53985
$372$	0	0
$373$	5340.85	0.741391	0.370695	−	0.928755i	$-0.379119\pi$
$373$	5340.85	0.741391	0.370695	+	0.928755i	$0.379119\pi$
$374$	0	0
$375$	2301.15	0.316882
$376$	0	0
$377$	3907.70	0.533838
$378$	0	0
$379$	1059.64	0.143614	0.0718072	−	0.997419i	$-0.477123\pi$
$379$	1059.64	0.143614	0.0718072	+	0.997419i	$0.477123\pi$
$380$	0	0
$381$	2865.53	0.385316
$382$	0	0
$383$	−1277.76	−0.170471	−0.0852357	−	0.996361i	$-0.527164\pi$
$383$	−1277.76	−0.170471	−0.0852357	+	0.996361i	$0.527164\pi$
$384$	0	0
$385$	−5312.83	−0.703290
$386$	0	0
$387$	−10280.0	−1.35030
$388$	0	0
$389$	−1752.46	−0.228414	−0.114207	−	0.993457i	$-0.536433\pi$
$389$	−1752.46	−0.228414	−0.114207	+	0.993457i	$0.536433\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	3472.02	0.445649
$394$	0	0
$395$	2314.32	0.294800
$396$	0	0
$397$	−7611.24	−0.962209	−0.481105	−	0.876663i	$-0.659764\pi$
$397$	−7611.24	−0.962209	−0.481105	+	0.876663i	$0.659764\pi$
$398$	0	0
$399$	−9740.51	−1.22214
$400$	0	0
$401$	−15712.6	−1.95673	−0.978364	−	0.206892i	$-0.933665\pi$
$401$	−15712.6	−1.95673	−0.978364	+	0.206892i	$0.933665\pi$
$402$	0	0
$403$	−11081.5	−1.36975
$404$	0	0
$405$	713.899	0.0875900
$406$	0	0
$407$	−12003.9	−1.46194
$408$	0	0
$409$	7956.00	0.961856	0.480928	−	0.876760i	$-0.340300\pi$
$409$	7956.00	0.961856	0.480928	+	0.876760i	$0.340300\pi$
$410$	0	0
$411$	−4502.15	−0.540328
$412$	0	0
$413$	1063.40	0.126699
$414$	0	0
$415$	1481.70	0.175262
$416$	0	0
$417$	−1463.86	−0.171907
$418$	0	0
$419$	−3570.98	−0.416358	−0.208179	−	0.978091i	$-0.566754\pi$
$419$	−3570.98	−0.416358	−0.208179	+	0.978091i	$0.566754\pi$
$420$	0	0
$421$	−465.096	−0.0538418	−0.0269209	−	0.999638i	$-0.508570\pi$
$421$	−465.096	−0.0538418	−0.0269209	+	0.999638i	$0.508570\pi$
$422$	0	0
$423$	−8809.83	−1.01264
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14145.9	−1.60321
$428$	0	0
$429$	−5967.73	−0.671619
$430$	0	0
$431$	300.031	0.0335313	0.0167657	−	0.999859i	$-0.494663\pi$
$431$	300.031	0.0335313	0.0167657	+	0.999859i	$0.494663\pi$
$432$	0	0
$433$	−4129.93	−0.458364	−0.229182	−	0.973384i	$-0.573605\pi$
$433$	−4129.93	−0.458364	−0.229182	+	0.973384i	$0.573605\pi$
$434$	0	0
$435$	−729.699	−0.0804285
$436$	0	0
$437$	−18987.8	−2.07852
$438$	0	0
$439$	−3449.16	−0.374988	−0.187494	−	0.982266i	$-0.560036\pi$
$439$	−3449.16	−0.374988	−0.187494	+	0.982266i	$0.560036\pi$
$440$	0	0
$441$	−16568.8	−1.78910
$442$	0	0
$443$	11548.8	1.23860	0.619298	−	0.785156i	$-0.287417\pi$
$443$	11548.8	1.23860	0.619298	+	0.785156i	$0.287417\pi$
$444$	0	0
$445$	−1211.11	−0.129016
$446$	0	0
$447$	5459.90	0.577728
$448$	0	0
$449$	9447.67	0.993014	0.496507	−	0.868033i	$-0.334616\pi$
$449$	9447.67	0.993014	0.496507	+	0.868033i	$0.334616\pi$
$450$	0	0
$451$	−16360.0	−1.70812
$452$	0	0
$453$	−2681.36	−0.278104
$454$	0	0
$455$	−6466.00	−0.666222
$456$	0	0
$457$	12635.2	1.29333	0.646665	−	0.762774i	$-0.276163\pi$
$457$	12635.2	1.29333	0.646665	+	0.762774i	$0.276163\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12232.9	−1.23589	−0.617943	−	0.786223i	$-0.712034\pi$
$461$	−12232.9	−1.23589	−0.617943	+	0.786223i	$0.712034\pi$
$462$	0	0
$463$	11478.8	1.15219	0.576094	−	0.817384i	$-0.304576\pi$
$463$	11478.8	1.15219	0.576094	+	0.817384i	$0.304576\pi$
$464$	0	0
$465$	2069.29	0.206368
$466$	0	0
$467$	−8629.77	−0.855114	−0.427557	−	0.903988i	$-0.640626\pi$
$467$	−8629.77	−0.855114	−0.427557	+	0.903988i	$0.640626\pi$
$468$	0	0
$469$	−10332.4	−1.01729
$470$	0	0
$471$	465.528	0.0455422
$472$	0	0
$473$	22196.4	2.15770
$474$	0	0
$475$	11821.1	1.14188
$476$	0	0
$477$	−6340.72	−0.608641
$478$	0	0
$479$	15366.0	1.46574	0.732871	−	0.680367i	$-0.238180\pi$
$479$	15366.0	1.46574	0.732871	+	0.680367i	$0.238180\pi$
$480$	0	0
$481$	−14609.4	−1.38488
$482$	0	0
$483$	16574.6	1.56143
$484$	0	0
$485$	−2767.93	−0.259145
$486$	0	0
$487$	−20187.0	−1.87836	−0.939181	−	0.343423i	$-0.888414\pi$
$487$	−20187.0	−1.87836	−0.939181	+	0.343423i	$0.888414\pi$
$488$	0	0
$489$	7920.89	0.732505
$490$	0	0
$491$	−1273.94	−0.117091	−0.0585457	−	0.998285i	$-0.518646\pi$
$491$	−1273.94	−0.117091	−0.0585457	+	0.998285i	$0.518646\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3061.44	−0.277983
$496$	0	0
$497$	−5854.23	−0.528366
$498$	0	0
$499$	4412.23	0.395828	0.197914	−	0.980219i	$-0.436583\pi$
$499$	4412.23	0.395828	0.197914	+	0.980219i	$0.436583\pi$
$500$	0	0
$501$	2681.43	0.239116
$502$	0	0
$503$	17102.6	1.51604	0.758018	−	0.652234i	$-0.226168\pi$
$503$	17102.6	1.51604	0.758018	+	0.652234i	$0.226168\pi$
$504$	0	0
$505$	−3703.85	−0.326375
$506$	0	0
$507$	−1365.71	−0.119632
$508$	0	0
$509$	−9812.45	−0.854478	−0.427239	−	0.904139i	$-0.640514\pi$
$509$	−9812.45	−0.854478	−0.427239	+	0.904139i	$0.640514\pi$
$510$	0	0
$511$	14439.9	1.25007
$512$	0	0
$513$	−13268.8	−1.14197
$514$	0	0
$515$	5652.02	0.483607
$516$	0	0
$517$	19022.0	1.61815
$518$	0	0
$519$	10436.9	0.882718
$520$	0	0
$521$	−3150.72	−0.264944	−0.132472	−	0.991187i	$-0.542291\pi$
$521$	−3150.72	−0.264944	−0.132472	+	0.991187i	$0.542291\pi$
$522$	0	0
$523$	−745.326	−0.0623151	−0.0311576	−	0.999514i	$-0.509919\pi$
$523$	−745.326	−0.0623151	−0.0311576	+	0.999514i	$0.509919\pi$
$524$	0	0
$525$	−10318.7	−0.857803
$526$	0	0
$527$	0	0
$528$	0	0
$529$	20143.0	1.65554
$530$	0	0
$531$	612.772	0.0500792
$532$	0	0
$533$	−19911.1	−1.61809
$534$	0	0
$535$	−4923.02	−0.397833
$536$	0	0
$537$	6677.53	0.536604
$538$	0	0
$539$	35775.0	2.85889
$540$	0	0
$541$	1846.55	0.146746	0.0733730	−	0.997305i	$-0.476624\pi$
$541$	1846.55	0.146746	0.0733730	+	0.997305i	$0.476624\pi$
$542$	0	0
$543$	5719.12	0.451991
$544$	0	0
$545$	1585.05	0.124580
$546$	0	0
$547$	9192.22	0.718521	0.359261	−	0.933237i	$-0.383029\pi$
$547$	9192.22	0.718521	0.359261	+	0.933237i	$0.383029\pi$
$548$	0	0
$549$	−8151.40	−0.633685
$550$	0	0
$551$	−7935.65	−0.613557
$552$	0	0
$553$	−21969.9	−1.68943
$554$	0	0
$555$	2728.06	0.208648
$556$	0	0
$557$	−19304.9	−1.46854	−0.734269	−	0.678859i	$-0.762475\pi$
$557$	−19304.9	−1.46854	−0.734269	+	0.678859i	$0.762475\pi$
$558$	0	0
$559$	27014.3	2.04397
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−238.639	−0.0178640	−0.00893200	−	0.999960i	$-0.502843\pi$
$563$	−238.639	−0.0178640	−0.00893200	+	0.999960i	$0.502843\pi$
$564$	0	0
$565$	1694.02	0.126138
$566$	0	0
$567$	−6777.08	−0.501959
$568$	0	0
$569$	−12470.9	−0.918815	−0.459408	−	0.888226i	$-0.651938\pi$
$569$	−12470.9	−0.918815	−0.459408	+	0.888226i	$0.651938\pi$
$570$	0	0
$571$	−9135.89	−0.669571	−0.334786	−	0.942294i	$-0.608664\pi$
$571$	−9135.89	−0.669571	−0.334786	+	0.942294i	$0.608664\pi$
$572$	0	0
$573$	−4933.04	−0.359652
$574$	0	0
$575$	−20115.0	−1.45888
$576$	0	0
$577$	2417.01	0.174387	0.0871936	−	0.996191i	$-0.472210\pi$
$577$	2417.01	0.174387	0.0871936	+	0.996191i	$0.472210\pi$
$578$	0	0
$579$	12841.7	0.921731
$580$	0	0
$581$	−14065.9	−1.00439
$582$	0	0
$583$	13690.7	0.972576
$584$	0	0
$585$	−3725.95	−0.263332
$586$	0	0
$587$	1638.89	0.115237	0.0576185	−	0.998339i	$-0.481649\pi$
$587$	1638.89	0.115237	0.0576185	+	0.998339i	$0.481649\pi$
$588$	0	0
$589$	22504.0	1.57430
$590$	0	0
$591$	−2319.09	−0.161412
$592$	0	0
$593$	−18659.0	−1.29213	−0.646067	−	0.763281i	$-0.723587\pi$
$593$	−18659.0	−1.29213	−0.646067	+	0.763281i	$0.723587\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−543.701	−0.0372734
$598$	0	0
$599$	18684.9	1.27453	0.637265	−	0.770645i	$-0.280066\pi$
$599$	18684.9	1.27453	0.637265	+	0.770645i	$0.280066\pi$
$600$	0	0
$601$	−216.698	−0.0147076	−0.00735381	−	0.999973i	$-0.502341\pi$
$601$	−216.698	−0.0147076	−0.00735381	+	0.999973i	$0.502341\pi$
$602$	0	0
$603$	−5953.93	−0.402094
$604$	0	0
$605$	1793.82	0.120544
$606$	0	0
$607$	13577.0	0.907867	0.453934	−	0.891036i	$-0.350020\pi$
$607$	13577.0	0.907867	0.453934	+	0.891036i	$0.350020\pi$
$608$	0	0
$609$	6927.06	0.460918
$610$	0	0
$611$	23150.8	1.53286
$612$	0	0
$613$	−19398.1	−1.27811	−0.639057	−	0.769159i	$-0.720675\pi$
$613$	−19398.1	−1.27811	−0.639057	+	0.769159i	$0.720675\pi$
$614$	0	0
$615$	3718.06	0.243783
$616$	0	0
$617$	−10815.2	−0.705681	−0.352841	−	0.935683i	$-0.614784\pi$
$617$	−10815.2	−0.705681	−0.352841	+	0.935683i	$0.614784\pi$
$618$	0	0
$619$	−2001.27	−0.129948	−0.0649739	−	0.997887i	$-0.520696\pi$
$619$	−2001.27	−0.129948	−0.0649739	+	0.997887i	$0.520696\pi$
$620$	0	0
$621$	22578.3	1.45900
$622$	0	0
$623$	11497.2	0.739364
$624$	0	0
$625$	10886.1	0.696709
$626$	0	0
$627$	12119.1	0.771914
$628$	0	0
$629$	0	0
$630$	0	0
$631$	354.918	0.0223916	0.0111958	−	0.999937i	$-0.496436\pi$
$631$	354.918	0.0223916	0.0111958	+	0.999937i	$0.496436\pi$
$632$	0	0
$633$	−4258.83	−0.267414
$634$	0	0
$635$	−3862.96	−0.241412
$636$	0	0
$637$	43540.2	2.70820
$638$	0	0
$639$	−3373.42	−0.208843
$640$	0	0
$641$	−16857.5	−1.03874	−0.519368	−	0.854551i	$-0.673833\pi$
$641$	−16857.5	−1.03874	−0.519368	+	0.854551i	$0.673833\pi$
$642$	0	0
$643$	−3591.55	−0.220275	−0.110137	−	0.993916i	$-0.535129\pi$
$643$	−3591.55	−0.220275	−0.110137	+	0.993916i	$0.535129\pi$
$644$	0	0
$645$	−5044.47	−0.307947
$646$	0	0
$647$	−16226.1	−0.985957	−0.492978	−	0.870042i	$-0.664092\pi$
$647$	−16226.1	−0.985957	−0.492978	+	0.870042i	$0.664092\pi$
$648$	0	0
$649$	−1323.08	−0.0800238
$650$	0	0
$651$	−19643.9	−1.18265
$652$	0	0
$653$	−28790.3	−1.72534	−0.862672	−	0.505764i	$-0.831211\pi$
$653$	−28790.3	−1.72534	−0.862672	+	0.505764i	$0.831211\pi$
$654$	0	0
$655$	−4680.55	−0.279213
$656$	0	0
$657$	8320.80	0.494102
$658$	0	0
$659$	26851.5	1.58723	0.793617	−	0.608418i	$-0.208196\pi$
$659$	26851.5	1.58723	0.793617	+	0.608418i	$0.208196\pi$
$660$	0	0
$661$	12282.1	0.722721	0.361361	−	0.932426i	$-0.382312\pi$
$661$	12282.1	0.722721	0.361361	+	0.932426i	$0.382312\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13131.0	0.765710
$666$	0	0
$667$	13503.4	0.783889
$668$	0	0
$669$	2199.77	0.127127
$670$	0	0
$671$	17600.3	1.01260
$672$	0	0
$673$	−9507.86	−0.544578	−0.272289	−	0.962215i	$-0.587781\pi$
$673$	−9507.86	−0.544578	−0.272289	+	0.962215i	$0.587781\pi$
$674$	0	0
$675$	−14056.4	−0.801530
$676$	0	0
$677$	11675.6	0.662819	0.331409	−	0.943487i	$-0.392476\pi$
$677$	11675.6	0.662819	0.331409	+	0.943487i	$0.392476\pi$
$678$	0	0
$679$	26276.1	1.48510
$680$	0	0
$681$	−6094.06	−0.342915
$682$	0	0
$683$	−20077.2	−1.12479	−0.562395	−	0.826868i	$-0.690120\pi$
$683$	−20077.2	−1.12479	−0.562395	+	0.826868i	$0.690120\pi$
$684$	0	0
$685$	6069.26	0.338532
$686$	0	0
$687$	−10344.7	−0.574492
$688$	0	0
$689$	16662.4	0.921315
$690$	0	0
$691$	3531.05	0.194396	0.0971978	−	0.995265i	$-0.469012\pi$
$691$	3531.05	0.194396	0.0971978	+	0.995265i	$0.469012\pi$
$692$	0	0
$693$	29062.4	1.59306
$694$	0	0
$695$	1973.39	0.107705
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9127.06	−0.493873
$700$	0	0
$701$	1702.02	0.0917040	0.0458520	−	0.998948i	$-0.485400\pi$
$701$	1702.02	0.0917040	0.0458520	+	0.998948i	$0.485400\pi$
$702$	0	0
$703$	29668.3	1.59169
$704$	0	0
$705$	−4323.03	−0.230943
$706$	0	0
$707$	35160.8	1.87038
$708$	0	0
$709$	15533.0	0.822782	0.411391	−	0.911459i	$-0.365043\pi$
$709$	15533.0	0.822782	0.411391	+	0.911459i	$0.365043\pi$
$710$	0	0
$711$	−12659.9	−0.667767
$712$	0	0
$713$	−38293.2	−2.01135
$714$	0	0
$715$	8044.97	0.420790
$716$	0	0
$717$	1878.58	0.0978478
$718$	0	0
$719$	−23838.7	−1.23648	−0.618242	−	0.785988i	$-0.712155\pi$
$719$	−23838.7	−1.23648	−0.618242	+	0.785988i	$0.712155\pi$
$720$	0	0
$721$	−53654.9	−2.77145
$722$	0	0
$723$	−414.702	−0.0213318
$724$	0	0
$725$	−8406.73	−0.430645
$726$	0	0
$727$	−23473.5	−1.19750	−0.598751	−	0.800935i	$-0.704336\pi$
$727$	−23473.5	−1.19750	−0.598751	+	0.800935i	$0.704336\pi$
$728$	0	0
$729$	5198.41	0.264107
$730$	0	0
$731$	0	0
$732$	0	0
$733$	29236.5	1.47323	0.736613	−	0.676315i	$-0.236424\pi$
$733$	29236.5	1.47323	0.736613	+	0.676315i	$0.236424\pi$
$734$	0	0
$735$	−8130.41	−0.408020
$736$	0	0
$737$	12855.6	0.642525
$738$	0	0
$739$	−25147.9	−1.25180	−0.625901	−	0.779903i	$-0.715269\pi$
$739$	−25147.9	−1.25180	−0.625901	+	0.779903i	$0.715269\pi$
$740$	0	0
$741$	14749.6	0.731229
$742$	0	0
$743$	22738.2	1.12272	0.561361	−	0.827571i	$-0.310278\pi$
$743$	22738.2	1.12272	0.561361	+	0.827571i	$0.310278\pi$
$744$	0	0
$745$	−7360.37	−0.361964
$746$	0	0
$747$	−8105.26	−0.396996
$748$	0	0
$749$	46734.5	2.27989
$750$	0	0
$751$	15705.8	0.763132	0.381566	−	0.924342i	$-0.375385\pi$
$751$	15705.8	0.763132	0.381566	+	0.924342i	$0.375385\pi$
$752$	0	0
$753$	3699.41	0.179036
$754$	0	0
$755$	3614.68	0.174241
$756$	0	0
$757$	32813.2	1.57545	0.787726	−	0.616026i	$-0.211258\pi$
$757$	32813.2	1.57545	0.787726	+	0.616026i	$0.211258\pi$
$758$	0	0
$759$	−20622.0	−0.986208
$760$	0	0
$761$	10221.8	0.486910	0.243455	−	0.969912i	$-0.421719\pi$
$761$	10221.8	0.486910	0.243455	+	0.969912i	$0.421719\pi$
$762$	0	0
$763$	−15046.9	−0.713938
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1610.26	−0.0758060
$768$	0	0
$769$	−13974.9	−0.655331	−0.327665	−	0.944794i	$-0.606262\pi$
$769$	−13974.9	−0.655331	−0.327665	+	0.944794i	$0.606262\pi$
$770$	0	0
$771$	18220.3	0.851086
$772$	0	0
$773$	−627.655	−0.0292046	−0.0146023	−	0.999893i	$-0.504648\pi$
$773$	−627.655	−0.0292046	−0.0146023	+	0.999893i	$0.504648\pi$
$774$	0	0
$775$	23839.9	1.10497
$776$	0	0
$777$	−25897.6	−1.19571
$778$	0	0
$779$	40434.8	1.85973
$780$	0	0
$781$	7283.80	0.333720
$782$	0	0
$783$	9436.22	0.430681
$784$	0	0
$785$	−627.568	−0.0285336
$786$	0	0
$787$	33436.9	1.51448	0.757240	−	0.653136i	$-0.226547\pi$
$787$	33436.9	1.51448	0.757240	+	0.653136i	$0.226547\pi$
$788$	0	0
$789$	−6098.97	−0.275195
$790$	0	0
$791$	−16081.4	−0.722870
$792$	0	0
$793$	21420.5	0.959224
$794$	0	0
$795$	−3111.42	−0.138806
$796$	0	0
$797$	−30904.5	−1.37352	−0.686758	−	0.726886i	$-0.740967\pi$
$797$	−30904.5	−1.37352	−0.686758	+	0.726886i	$0.740967\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6625.08	0.292242
$802$	0	0
$803$	−17966.1	−0.789550
$804$	0	0
$805$	−22343.8	−0.978282
$806$	0	0
$807$	17514.9	0.764009
$808$	0	0
$809$	−42399.8	−1.84264	−0.921320	−	0.388805i	$-0.872888\pi$
$809$	−42399.8	−1.84264	−0.921320	+	0.388805i	$0.872888\pi$
$810$	0	0
$811$	36239.2	1.56909	0.784545	−	0.620072i	$-0.212897\pi$
$811$	36239.2	1.56909	0.784545	+	0.620072i	$0.212897\pi$
$812$	0	0
$813$	1214.12	0.0523751
$814$	0	0
$815$	−10678.0	−0.458937
$816$	0	0
$817$	−54859.8	−2.34921
$818$	0	0
$819$	35370.6	1.50909
$820$	0	0
$821$	14910.7	0.633844	0.316922	−	0.948452i	$-0.397351\pi$
$821$	14910.7	0.633844	0.316922	+	0.948452i	$0.397351\pi$
$822$	0	0
$823$	7343.78	0.311043	0.155521	−	0.987833i	$-0.450294\pi$
$823$	7343.78	0.311043	0.155521	+	0.987833i	$0.450294\pi$
$824$	0	0
$825$	12838.5	0.541794
$826$	0	0
$827$	22407.1	0.942164	0.471082	−	0.882089i	$-0.343864\pi$
$827$	22407.1	0.942164	0.471082	+	0.882089i	$0.343864\pi$
$828$	0	0
$829$	46433.6	1.94536	0.972682	−	0.232141i	$-0.0745730\pi$
$829$	46433.6	1.94536	0.972682	+	0.232141i	$0.0745730\pi$
$830$	0	0
$831$	−8705.75	−0.363417
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3614.77	−0.149814
$836$	0	0
$837$	−26759.4	−1.10506
$838$	0	0
$839$	13439.0	0.553000	0.276500	−	0.961014i	$-0.410825\pi$
$839$	13439.0	0.553000	0.276500	+	0.961014i	$0.410825\pi$
$840$	0	0
$841$	−18745.5	−0.768604
$842$	0	0
$843$	−13153.2	−0.537389
$844$	0	0
$845$	1841.08	0.0749529
$846$	0	0
$847$	−17028.8	−0.690811
$848$	0	0
$849$	4091.99	0.165414
$850$	0	0
$851$	−50483.9	−2.03357
$852$	0	0
$853$	35642.8	1.43070	0.715349	−	0.698767i	$-0.246268\pi$
$853$	35642.8	1.43070	0.715349	+	0.698767i	$0.246268\pi$
$854$	0	0
$855$	7566.55	0.302656
$856$	0	0
$857$	13653.3	0.544210	0.272105	−	0.962268i	$-0.412280\pi$
$857$	13653.3	0.544210	0.272105	+	0.962268i	$0.412280\pi$
$858$	0	0
$859$	9210.52	0.365843	0.182921	−	0.983128i	$-0.441445\pi$
$859$	9210.52	0.365843	0.182921	+	0.983128i	$0.441445\pi$
$860$	0	0
$861$	−35295.7	−1.39707
$862$	0	0
$863$	−23407.3	−0.923283	−0.461642	−	0.887066i	$-0.652739\pi$
$863$	−23407.3	−0.923283	−0.461642	+	0.887066i	$0.652739\pi$
$864$	0	0
$865$	−14069.8	−0.553050
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27334.9	1.06706
$870$	0	0
$871$	15645.9	0.608660
$872$	0	0
$873$	15141.2	0.587003
$874$	0	0
$875$	29448.6	1.13777
$876$	0	0
$877$	−45394.2	−1.74784	−0.873919	−	0.486072i	$-0.838429\pi$
$877$	−45394.2	−1.74784	−0.873919	+	0.486072i	$0.838429\pi$
$878$	0	0
$879$	17790.8	0.682671
$880$	0	0
$881$	17304.4	0.661747	0.330873	−	0.943675i	$-0.392657\pi$
$881$	17304.4	0.661747	0.330873	+	0.943675i	$0.392657\pi$
$882$	0	0
$883$	−12609.5	−0.480569	−0.240284	−	0.970703i	$-0.577241\pi$
$883$	−12609.5	−0.480569	−0.240284	+	0.970703i	$0.577241\pi$
$884$	0	0
$885$	300.690	0.0114210
$886$	0	0
$887$	−22698.1	−0.859220	−0.429610	−	0.903014i	$-0.641349\pi$
$887$	−22698.1	−0.859220	−0.429610	+	0.903014i	$0.641349\pi$
$888$	0	0
$889$	36671.2	1.38348
$890$	0	0
$891$	8432.01	0.317040
$892$	0	0
$893$	−47013.9	−1.76177
$894$	0	0
$895$	−9001.83	−0.336199
$896$	0	0
$897$	−25098.1	−0.934228
$898$	0	0
$899$	−16004.0	−0.593729
$900$	0	0
$901$	0	0
$902$	0	0
$903$	47887.4	1.76478
$904$	0	0
$905$	−7709.83	−0.283186
$906$	0	0
$907$	28834.9	1.05562	0.527811	−	0.849362i	$-0.323013\pi$
$907$	28834.9	1.05562	0.527811	+	0.849362i	$0.323013\pi$
$908$	0	0
$909$	20261.0	0.739289
$910$	0	0
$911$	18961.7	0.689605	0.344803	−	0.938675i	$-0.387946\pi$
$911$	18961.7	0.689605	0.344803	+	0.938675i	$0.387946\pi$
$912$	0	0
$913$	17500.7	0.634378
$914$	0	0
$915$	−3999.93	−0.144518
$916$	0	0
$917$	44432.7	1.60011
$918$	0	0
$919$	20722.0	0.743805	0.371902	−	0.928272i	$-0.378706\pi$
$919$	20722.0	0.743805	0.371902	+	0.928272i	$0.378706\pi$
$920$	0	0
$921$	20948.7	0.749494
$922$	0	0
$923$	8864.79	0.316130
$924$	0	0
$925$	31429.4	1.11718
$926$	0	0
$927$	−30917.9	−1.09544
$928$	0	0
$929$	−23634.9	−0.834701	−0.417350	−	0.908746i	$-0.637041\pi$
$929$	−23634.9	−0.834701	−0.417350	+	0.908746i	$0.637041\pi$
$930$	0	0
$931$	−88420.2	−3.11263
$932$	0	0
$933$	25208.6	0.884558
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51305.2	−1.78876	−0.894381	−	0.447307i	$-0.852383\pi$
$937$	−51305.2	−1.78876	−0.894381	+	0.447307i	$0.852383\pi$
$938$	0	0
$939$	13366.1	0.464521
$940$	0	0
$941$	−16849.8	−0.583727	−0.291863	−	0.956460i	$-0.594275\pi$
$941$	−16849.8	−0.583727	−0.291863	+	0.956460i	$0.594275\pi$
$942$	0	0
$943$	−68804.4	−2.37601
$944$	0	0
$945$	−15613.9	−0.537483
$946$	0	0
$947$	−9570.45	−0.328403	−0.164202	−	0.986427i	$-0.552505\pi$
$947$	−9570.45	−0.328403	−0.164202	+	0.986427i	$0.552505\pi$
$948$	0	0
$949$	−21865.7	−0.747935
$950$	0	0
$951$	−2415.98	−0.0823801
$952$	0	0
$953$	17977.4	0.611066	0.305533	−	0.952181i	$-0.401165\pi$
$953$	17977.4	0.611066	0.305533	+	0.952181i	$0.401165\pi$
$954$	0	0
$955$	6650.13	0.225333
$956$	0	0
$957$	−8618.62	−0.291119
$958$	0	0
$959$	−57615.7	−1.94005
$960$	0	0
$961$	15593.3	0.523423
$962$	0	0
$963$	26930.1	0.901153
$964$	0	0
$965$	−17311.6	−0.577493
$966$	0	0
$967$	−53748.3	−1.78741	−0.893706	−	0.448653i	$-0.851904\pi$
$967$	−53748.3	−1.78741	−0.893706	+	0.448653i	$0.851904\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49026.9	1.62034	0.810169	−	0.586196i	$-0.199375\pi$
$971$	49026.9	1.62034	0.810169	+	0.586196i	$0.199375\pi$
$972$	0	0
$973$	−18733.5	−0.617234
$974$	0	0
$975$	15625.2	0.513237
$976$	0	0
$977$	56864.1	1.86207	0.931036	−	0.364928i	$-0.118906\pi$
$977$	56864.1	1.86207	0.931036	+	0.364928i	$0.118906\pi$
$978$	0	0
$979$	−14304.7	−0.466987
$980$	0	0
$981$	−8670.58	−0.282192
$982$	0	0
$983$	−20388.1	−0.661525	−0.330762	−	0.943714i	$-0.607306\pi$
$983$	−20388.1	−0.661525	−0.330762	+	0.943714i	$0.607306\pi$
$984$	0	0
$985$	3126.31	0.101129
$986$	0	0
$987$	41038.7	1.32348
$988$	0	0
$989$	93350.2	3.00138
$990$	0	0
$991$	−14412.0	−0.461968	−0.230984	−	0.972957i	$-0.574195\pi$
$991$	−14412.0	−0.461968	−0.230984	+	0.972957i	$0.574195\pi$
$992$	0	0
$993$	16474.1	0.526474
$994$	0	0
$995$	732.952	0.0233529
$996$	0	0
$997$	16952.6	0.538511	0.269255	−	0.963069i	$-0.413222\pi$
$997$	16952.6	0.538511	0.269255	+	0.963069i	$0.413222\pi$
$998$	0	0
$999$	−35278.3	−1.11727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.o.1.8	✓	18
17.16	even	2		2312.4.a.p.1.11	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.o.1.8	✓	18	1.1	even	1	trivial
2312.4.a.p.1.11	yes	18	17.16	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68427 q^{3} +3.61861 q^{5} -34.3516 q^{7} -19.7947 q^{9} +O(q^{10})\)	\(q-2.68427 q^{3} +3.61861 q^{5} -34.3516 q^{7} -19.7947 q^{9} +42.7402 q^{11} +52.0171 q^{13} -9.71334 q^{15} -105.635 q^{19} +92.2092 q^{21} +179.750 q^{23} -111.906 q^{25} +125.610 q^{27} +75.1233 q^{29} -213.036 q^{31} -114.726 q^{33} -124.305 q^{35} -280.857 q^{37} -139.628 q^{39} -382.779 q^{41} +519.334 q^{43} -71.6292 q^{45} +445.061 q^{47} +837.036 q^{49} +320.325 q^{53} +154.660 q^{55} +283.553 q^{57} -30.9564 q^{59} +411.797 q^{61} +679.980 q^{63} +188.230 q^{65} +300.784 q^{67} -482.498 q^{69} +170.421 q^{71} -420.355 q^{73} +300.385 q^{75} -1468.19 q^{77} +639.560 q^{79} +197.285 q^{81} +409.467 q^{83} -201.652 q^{87} -334.690 q^{89} -1786.87 q^{91} +571.847 q^{93} -382.252 q^{95} -764.915 q^{97} -846.027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9}+O(q^{10}) \) 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9	\( 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9} + 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} - 153 q^{23} + 594 q^{25} + 546 q^{27} - 447 q^{29} - 303 q^{31} + 153 q^{33} - 117 q^{35} - 939 q^{37} - 516 q^{39} - 1257 q^{41} + 306 q^{43} - 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} - 1494 q^{57} + 696 q^{59} - 1686 q^{61} - 1908 q^{63} - 855 q^{65} + 513 q^{67} - 1329 q^{69} - 324 q^{71} - 1863 q^{73} + 3054 q^{75} + 1833 q^{77} - 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} - 252 q^{91} - 1470 q^{93} - 2109 q^{95} - 4611 q^{97} + 3918 q^{99}+O(q^{100}) \) 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9 + 66 * q^11 - 30 * q^13 - 102 * q^15 - 168 * q^19 - 510 * q^21 - 153 * q^23 + 594 * q^25 + 546 * q^27 - 447 * q^29 - 303 * q^31 + 153 * q^33 - 117 * q^35 - 939 * q^37 - 516 * q^39 - 1257 * q^41 + 306 * q^43 - 672 * q^45 + 633 * q^47 + 1239 * q^49 - 489 * q^53 + 1089 * q^55 - 1494 * q^57 + 696 * q^59 - 1686 * q^61 - 1908 * q^63 - 855 * q^65 + 513 * q^67 - 1329 * q^69 - 324 * q^71 - 1863 * q^73 + 3054 * q^75 + 1833 * q^77 - 3699 * q^79 + 2622 * q^81 + 1188 * q^83 - 3927 * q^87 + 1713 * q^89 - 252 * q^91 - 1470 * q^93 - 2109 * q^95 - 4611 * q^97 + 3918 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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