LMFDB - Embedded newform 2312.4.a.j.1.2

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 92x^{4} + 123x^{3} + 2120x^{2} - 3573x - 261 $ x^6 - x^5 - 92x^4 + 123x^3 + 2120x^2 - 3573x - 261
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.84285$ 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)

$f(q)$	$=$	$q-5.84285 q^{3} +13.3245 q^{5} -23.1160 q^{7} +7.13888 q^{9} +O(q^{10})$	$q-5.84285 q^{3} +13.3245 q^{5} -23.1160 q^{7} +7.13888 q^{9} +43.8867 q^{11} -15.0943 q^{13} -77.8530 q^{15} +93.7217 q^{19} +135.064 q^{21} -69.4890 q^{23} +52.5420 q^{25} +116.046 q^{27} -129.193 q^{29} +93.0117 q^{31} -256.424 q^{33} -308.009 q^{35} +171.552 q^{37} +88.1937 q^{39} -416.025 q^{41} +195.374 q^{43} +95.1220 q^{45} +543.602 q^{47} +191.351 q^{49} -716.751 q^{53} +584.768 q^{55} -547.602 q^{57} -517.045 q^{59} +427.609 q^{61} -165.023 q^{63} -201.124 q^{65} -909.450 q^{67} +406.013 q^{69} +231.951 q^{71} +435.409 q^{73} -306.995 q^{75} -1014.49 q^{77} +279.357 q^{79} -870.786 q^{81} +1500.27 q^{83} +754.855 q^{87} +163.227 q^{89} +348.920 q^{91} -543.453 q^{93} +1248.79 q^{95} +1439.07 q^{97} +313.302 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9}+O(q^{10}) $ 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9	$ 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9} + 72 q^{11} + 15 q^{13} - 60 q^{15} + 83 q^{19} + 96 q^{21} + 141 q^{23} + 263 q^{25} + 94 q^{27} - 249 q^{29} + 106 q^{31} + 289 q^{33} - 267 q^{35} + 170 q^{37} + 329 q^{39} - 100 q^{41} - 90 q^{43} - 286 q^{45} + 372 q^{47} + 323 q^{49} - 23 q^{53} - 457 q^{55} - 193 q^{57} + 784 q^{59} + 92 q^{61} - 722 q^{63} + 1412 q^{65} + 238 q^{67} + 1178 q^{69} + 940 q^{71} + 692 q^{73} + 1814 q^{75} - 45 q^{77} + 84 q^{79} - 2182 q^{81} + 1393 q^{83} - 1247 q^{87} + 976 q^{89} - 384 q^{91} + 1437 q^{93} - 1022 q^{95} - 325 q^{97} + 3981 q^{99}+O(q^{100}) $ 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9 + 72 * q^11 + 15 * q^13 - 60 * q^15 + 83 * q^19 + 96 * q^21 + 141 * q^23 + 263 * q^25 + 94 * q^27 - 249 * q^29 + 106 * q^31 + 289 * q^33 - 267 * q^35 + 170 * q^37 + 329 * q^39 - 100 * q^41 - 90 * q^43 - 286 * q^45 + 372 * q^47 + 323 * q^49 - 23 * q^53 - 457 * q^55 - 193 * q^57 + 784 * q^59 + 92 * q^61 - 722 * q^63 + 1412 * q^65 + 238 * q^67 + 1178 * q^69 + 940 * q^71 + 692 * q^73 + 1814 * q^75 - 45 * q^77 + 84 * q^79 - 2182 * q^81 + 1393 * q^83 - 1247 * q^87 + 976 * q^89 - 384 * q^91 + 1437 * q^93 - 1022 * q^95 - 325 * q^97 + 3981 * 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
$3$	−5.84285	−1.12446	−0.562228	−	0.826982i	$-0.690056\pi$
$3$	−5.84285	−1.12446	−0.562228	+	0.826982i	$0.690056\pi$
$4$	0	0
$5$	13.3245	1.19178	0.595889	−	0.803067i	$-0.296800\pi$
$5$	13.3245	1.19178	0.595889	+	0.803067i	$0.296800\pi$
$6$	0	0
$7$	−23.1160	−1.24815	−0.624074	−	0.781365i	$-0.714524\pi$
$7$	−23.1160	−1.24815	−0.624074	+	0.781365i	$0.714524\pi$
$8$	0	0
$9$	7.13888	0.264403
$10$	0	0
$11$	43.8867	1.20294	0.601470	−	0.798895i	$-0.294582\pi$
$11$	43.8867	1.20294	0.601470	+	0.798895i	$0.294582\pi$
$12$	0	0
$13$	−15.0943	−0.322031	−0.161015	−	0.986952i	$-0.551477\pi$
$13$	−15.0943	−0.322031	−0.161015	+	0.986952i	$0.551477\pi$
$14$	0	0
$15$	−77.8530	−1.34010
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	93.7217	1.13164	0.565822	−	0.824528i	$-0.308559\pi$
$19$	93.7217	1.13164	0.565822	+	0.824528i	$0.308559\pi$
$20$	0	0
$21$	135.064	1.40349
$22$	0	0
$23$	−69.4890	−0.629976	−0.314988	−	0.949096i	$-0.602001\pi$
$23$	−69.4890	−0.629976	−0.314988	+	0.949096i	$0.602001\pi$
$24$	0	0
$25$	52.5420	0.420336
$26$	0	0
$27$	116.046	0.827147
$28$	0	0
$29$	−129.193	−0.827259	−0.413630	−	0.910445i	$-0.635739\pi$
$29$	−129.193	−0.827259	−0.413630	+	0.910445i	$0.635739\pi$
$30$	0	0
$31$	93.0117	0.538884	0.269442	−	0.963017i	$-0.413161\pi$
$31$	93.0117	0.538884	0.269442	+	0.963017i	$0.413161\pi$
$32$	0	0
$33$	−256.424	−1.35265
$34$	0	0
$35$	−308.009	−1.48752
$36$	0	0
$37$	171.552	0.762242	0.381121	−	0.924525i	$-0.375538\pi$
$37$	171.552	0.762242	0.381121	+	0.924525i	$0.375538\pi$
$38$	0	0
$39$	88.1937	0.362110
$40$	0	0
$41$	−416.025	−1.58469	−0.792344	−	0.610075i	$-0.791139\pi$
$41$	−416.025	−1.58469	−0.792344	+	0.610075i	$0.791139\pi$
$42$	0	0
$43$	195.374	0.692890	0.346445	−	0.938070i	$-0.387389\pi$
$43$	195.374	0.692890	0.346445	+	0.938070i	$0.387389\pi$
$44$	0	0
$45$	95.1220	0.315110
$46$	0	0
$47$	543.602	1.68707	0.843537	−	0.537071i	$-0.180469\pi$
$47$	543.602	1.68707	0.843537	+	0.537071i	$0.180469\pi$
$48$	0	0
$49$	191.351	0.557876
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−716.751	−1.85761	−0.928805	−	0.370568i	$-0.879163\pi$
$53$	−716.751	−1.85761	−0.928805	+	0.370568i	$0.879163\pi$
$54$	0	0
$55$	584.768	1.43364
$56$	0	0
$57$	−547.602	−1.27248
$58$	0	0
$59$	−517.045	−1.14091	−0.570454	−	0.821330i	$-0.693233\pi$
$59$	−517.045	−1.14091	−0.570454	+	0.821330i	$0.693233\pi$
$60$	0	0
$61$	427.609	0.897537	0.448769	−	0.893648i	$-0.351863\pi$
$61$	427.609	0.897537	0.448769	+	0.893648i	$0.351863\pi$
$62$	0	0
$63$	−165.023	−0.330014
$64$	0	0
$65$	−201.124	−0.383790
$66$	0	0
$67$	−909.450	−1.65831	−0.829157	−	0.559016i	$-0.811179\pi$
$67$	−909.450	−1.65831	−0.829157	+	0.559016i	$0.811179\pi$
$68$	0	0
$69$	406.013	0.708381
$70$	0	0
$71$	231.951	0.387711	0.193856	−	0.981030i	$-0.437901\pi$
$71$	231.951	0.387711	0.193856	+	0.981030i	$0.437901\pi$
$72$	0	0
$73$	435.409	0.698093	0.349047	−	0.937105i	$-0.386505\pi$
$73$	435.409	0.698093	0.349047	+	0.937105i	$0.386505\pi$
$74$	0	0
$75$	−306.995	−0.472650
$76$	0	0
$77$	−1014.49	−1.50145
$78$	0	0
$79$	279.357	0.397850	0.198925	−	0.980015i	$-0.436255\pi$
$79$	279.357	0.397850	0.198925	+	0.980015i	$0.436255\pi$
$80$	0	0
$81$	−870.786	−1.19449
$82$	0	0
$83$	1500.27	1.98405	0.992024	−	0.126046i	$-0.0402288\pi$
$83$	1500.27	1.98405	0.992024	+	0.126046i	$0.0402288\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	754.855	0.930217
$88$	0	0
$89$	163.227	0.194405	0.0972024	−	0.995265i	$-0.469011\pi$
$89$	163.227	0.194405	0.0972024	+	0.995265i	$0.469011\pi$
$90$	0	0
$91$	348.920	0.401943
$92$	0	0
$93$	−543.453	−0.605951
$94$	0	0
$95$	1248.79	1.34867
$96$	0	0
$97$	1439.07	1.50634	0.753172	−	0.657823i	$-0.228523\pi$
$97$	1439.07	1.50634	0.753172	+	0.657823i	$0.228523\pi$
$98$	0	0
$99$	313.302	0.318061
$100$	0	0
$101$	−1897.79	−1.86967	−0.934836	−	0.355080i	$-0.884454\pi$
$101$	−1897.79	−1.86967	−0.934836	+	0.355080i	$0.884454\pi$
$102$	0	0
$103$	−1234.04	−1.18052	−0.590261	−	0.807213i	$-0.700975\pi$
$103$	−1234.04	−1.18052	−0.590261	+	0.807213i	$0.700975\pi$
$104$	0	0
$105$	1799.65	1.67265
$106$	0	0
$107$	−71.4326	−0.0645388	−0.0322694	−	0.999479i	$-0.510273\pi$
$107$	−71.4326	−0.0645388	−0.0322694	+	0.999479i	$0.510273\pi$
$108$	0	0
$109$	1228.39	1.07943	0.539716	−	0.841847i	$-0.318532\pi$
$109$	1228.39	1.07943	0.539716	+	0.841847i	$0.318532\pi$
$110$	0	0
$111$	−1002.35	−0.857109
$112$	0	0
$113$	325.073	0.270622	0.135311	−	0.990803i	$-0.456797\pi$
$113$	325.073	0.270622	0.135311	+	0.990803i	$0.456797\pi$
$114$	0	0
$115$	−925.905	−0.750792
$116$	0	0
$117$	−107.756	−0.0851460
$118$	0	0
$119$	0	0
$120$	0	0
$121$	595.045	0.447066
$122$	0	0
$123$	2430.77	1.78191
$124$	0	0
$125$	−965.465	−0.690831
$126$	0	0
$127$	980.295	0.684938	0.342469	−	0.939529i	$-0.388737\pi$
$127$	980.295	0.684938	0.342469	+	0.939529i	$0.388737\pi$
$128$	0	0
$129$	−1141.54	−0.779125
$130$	0	0
$131$	−1720.65	−1.14759	−0.573794	−	0.818999i	$-0.694529\pi$
$131$	−1720.65	−1.14759	−0.573794	+	0.818999i	$0.694529\pi$
$132$	0	0
$133$	−2166.47	−1.41246
$134$	0	0
$135$	1546.25	0.985776
$136$	0	0
$137$	−753.795	−0.470081	−0.235040	−	0.971986i	$-0.575522\pi$
$137$	−753.795	−0.470081	−0.235040	+	0.971986i	$0.575522\pi$
$138$	0	0
$139$	−716.662	−0.437313	−0.218656	−	0.975802i	$-0.570167\pi$
$139$	−716.662	−0.437313	−0.218656	+	0.975802i	$0.570167\pi$
$140$	0	0
$141$	−3176.18	−1.89704
$142$	0	0
$143$	−662.439	−0.387384
$144$	0	0
$145$	−1721.43	−0.985910
$146$	0	0
$147$	−1118.04	−0.627307
$148$	0	0
$149$	−2647.28	−1.45553	−0.727763	−	0.685829i	$-0.759440\pi$
$149$	−2647.28	−1.45553	−0.727763	+	0.685829i	$0.759440\pi$
$150$	0	0
$151$	1786.11	0.962594	0.481297	−	0.876558i	$-0.340166\pi$
$151$	1786.11	0.962594	0.481297	+	0.876558i	$0.340166\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1239.33	0.642230
$156$	0	0
$157$	−1400.78	−0.712068	−0.356034	−	0.934473i	$-0.615871\pi$
$157$	−1400.78	−0.712068	−0.356034	+	0.934473i	$0.615871\pi$
$158$	0	0
$159$	4187.87	2.08880
$160$	0	0
$161$	1606.31	0.786304
$162$	0	0
$163$	3762.54	1.80800	0.904002	−	0.427529i	$-0.140616\pi$
$163$	3762.54	1.80800	0.904002	+	0.427529i	$0.140616\pi$
$164$	0	0
$165$	−3416.71	−1.61206
$166$	0	0
$167$	−701.568	−0.325084	−0.162542	−	0.986702i	$-0.551969\pi$
$167$	−701.568	−0.325084	−0.162542	+	0.986702i	$0.551969\pi$
$168$	0	0
$169$	−1969.16	−0.896296
$170$	0	0
$171$	669.068	0.299210
$172$	0	0
$173$	2463.44	1.08261	0.541305	−	0.840826i	$-0.317930\pi$
$173$	2463.44	1.08261	0.541305	+	0.840826i	$0.317930\pi$
$174$	0	0
$175$	−1214.56	−0.524642
$176$	0	0
$177$	3021.02	1.28290
$178$	0	0
$179$	3001.58	1.25335	0.626673	−	0.779283i	$-0.284416\pi$
$179$	3001.58	1.25335	0.626673	+	0.779283i	$0.284416\pi$
$180$	0	0
$181$	−2903.99	−1.19255	−0.596275	−	0.802780i	$-0.703353\pi$
$181$	−2903.99	−1.19255	−0.596275	+	0.802780i	$0.703353\pi$
$182$	0	0
$183$	−2498.46	−1.00924
$184$	0	0
$185$	2285.84	0.908424
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2682.51	−1.03240
$190$	0	0
$191$	714.901	0.270829	0.135415	−	0.990789i	$-0.456763\pi$
$191$	714.901	0.270829	0.135415	+	0.990789i	$0.456763\pi$
$192$	0	0
$193$	4559.82	1.70064	0.850319	−	0.526267i	$-0.176409\pi$
$193$	4559.82	1.70064	0.850319	+	0.526267i	$0.176409\pi$
$194$	0	0
$195$	1175.14	0.431555
$196$	0	0
$197$	2325.24	0.840948	0.420474	−	0.907305i	$-0.361864\pi$
$197$	2325.24	0.840948	0.420474	+	0.907305i	$0.361864\pi$
$198$	0	0
$199$	−1998.76	−0.712002	−0.356001	−	0.934486i	$-0.615860\pi$
$199$	−1998.76	−0.712002	−0.356001	+	0.934486i	$0.615860\pi$
$200$	0	0
$201$	5313.78	1.86470
$202$	0	0
$203$	2986.43	1.03254
$204$	0	0
$205$	−5543.32	−1.88860
$206$	0	0
$207$	−496.074	−0.166568
$208$	0	0
$209$	4113.14	1.36130
$210$	0	0
$211$	2095.01	0.683536	0.341768	−	0.939784i	$-0.388974\pi$
$211$	2095.01	0.683536	0.341768	+	0.939784i	$0.388974\pi$
$212$	0	0
$213$	−1355.25	−0.435964
$214$	0	0
$215$	2603.26	0.825772
$216$	0	0
$217$	−2150.06	−0.672607
$218$	0	0
$219$	−2544.03	−0.784976
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−619.487	−0.186026	−0.0930132	−	0.995665i	$-0.529650\pi$
$223$	−619.487	−0.186026	−0.0930132	+	0.995665i	$0.529650\pi$
$224$	0	0
$225$	375.091	0.111138
$226$	0	0
$227$	−840.815	−0.245845	−0.122923	−	0.992416i	$-0.539227\pi$
$227$	−840.815	−0.245845	−0.122923	+	0.992416i	$0.539227\pi$
$228$	0	0
$229$	2230.85	0.643750	0.321875	−	0.946782i	$-0.395687\pi$
$229$	2230.85	0.643750	0.321875	+	0.946782i	$0.395687\pi$
$230$	0	0
$231$	5927.50	1.68831
$232$	0	0
$233$	4798.66	1.34923	0.674615	−	0.738169i	$-0.264310\pi$
$233$	4798.66	1.34923	0.674615	+	0.738169i	$0.264310\pi$
$234$	0	0
$235$	7243.22	2.01062
$236$	0	0
$237$	−1632.24	−0.447365
$238$	0	0
$239$	2590.36	0.701072	0.350536	−	0.936549i	$-0.385999\pi$
$239$	2590.36	0.701072	0.350536	+	0.936549i	$0.385999\pi$
$240$	0	0
$241$	−3433.65	−0.917763	−0.458881	−	0.888497i	$-0.651750\pi$
$241$	−3433.65	−0.917763	−0.458881	+	0.888497i	$0.651750\pi$
$242$	0	0
$243$	1954.64	0.516010
$244$	0	0
$245$	2549.66	0.664864
$246$	0	0
$247$	−1414.66	−0.364424
$248$	0	0
$249$	−8765.85	−2.23098
$250$	0	0
$251$	−2633.50	−0.662251	−0.331126	−	0.943587i	$-0.607428\pi$
$251$	−2633.50	−0.662251	−0.331126	+	0.943587i	$0.607428\pi$
$252$	0	0
$253$	−3049.64	−0.757824
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2569.97	−0.623776	−0.311888	−	0.950119i	$-0.600961\pi$
$257$	−2569.97	−0.623776	−0.311888	+	0.950119i	$0.600961\pi$
$258$	0	0
$259$	−3965.60	−0.951392
$260$	0	0
$261$	−922.293	−0.218730
$262$	0	0
$263$	3399.64	0.797075	0.398538	−	0.917152i	$-0.369518\pi$
$263$	3399.64	0.797075	0.398538	+	0.917152i	$0.369518\pi$
$264$	0	0
$265$	−9550.35	−2.21386
$266$	0	0
$267$	−953.710	−0.218600
$268$	0	0
$269$	1577.18	0.357482	0.178741	−	0.983896i	$-0.442798\pi$
$269$	1577.18	0.357482	0.178741	+	0.983896i	$0.442798\pi$
$270$	0	0
$271$	2494.06	0.559052	0.279526	−	0.960138i	$-0.409823\pi$
$271$	2494.06	0.559052	0.279526	+	0.960138i	$0.409823\pi$
$272$	0	0
$273$	−2038.69	−0.451967
$274$	0	0
$275$	2305.90	0.505639
$276$	0	0
$277$	5199.09	1.12774	0.563869	−	0.825864i	$-0.309312\pi$
$277$	5199.09	1.12774	0.563869	+	0.825864i	$0.309312\pi$
$278$	0	0
$279$	664.000	0.142483
$280$	0	0
$281$	3223.01	0.684229	0.342115	−	0.939658i	$-0.388857\pi$
$281$	3223.01	0.684229	0.342115	+	0.939658i	$0.388857\pi$
$282$	0	0
$283$	8350.37	1.75399	0.876993	−	0.480503i	$-0.159546\pi$
$283$	8350.37	1.75399	0.876993	+	0.480503i	$0.159546\pi$
$284$	0	0
$285$	−7296.51	−1.51652
$286$	0	0
$287$	9616.85	1.97793
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−8408.27	−1.69382
$292$	0	0
$293$	8845.41	1.76367	0.881834	−	0.471560i	$-0.156309\pi$
$293$	8845.41	1.76367	0.881834	+	0.471560i	$0.156309\pi$
$294$	0	0
$295$	−6889.36	−1.35971
$296$	0	0
$297$	5092.86	0.995009
$298$	0	0
$299$	1048.89	0.202872
$300$	0	0
$301$	−4516.28	−0.864830
$302$	0	0
$303$	11088.5	2.10237
$304$	0	0
$305$	5697.68	1.06967
$306$	0	0
$307$	−6935.82	−1.28941	−0.644704	−	0.764432i	$-0.723019\pi$
$307$	−6935.82	−1.28941	−0.644704	+	0.764432i	$0.723019\pi$
$308$	0	0
$309$	7210.32	1.32745
$310$	0	0
$311$	−600.422	−0.109475	−0.0547377	−	0.998501i	$-0.517432\pi$
$311$	−600.422	−0.109475	−0.0547377	+	0.998501i	$0.517432\pi$
$312$	0	0
$313$	7621.53	1.37634	0.688170	−	0.725549i	$-0.258414\pi$
$313$	7621.53	1.37634	0.688170	+	0.725549i	$0.258414\pi$
$314$	0	0
$315$	−2198.84	−0.393304
$316$	0	0
$317$	−752.603	−0.133345	−0.0666726	−	0.997775i	$-0.521238\pi$
$317$	−752.603	−0.133345	−0.0666726	+	0.997775i	$0.521238\pi$
$318$	0	0
$319$	−5669.85	−0.995144
$320$	0	0
$321$	417.370	0.0725711
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−793.085	−0.135361
$326$	0	0
$327$	−7177.27	−1.21377
$328$	0	0
$329$	−12565.9	−2.10572
$330$	0	0
$331$	346.072	0.0574678	0.0287339	−	0.999587i	$-0.490852\pi$
$331$	346.072	0.0574678	0.0287339	+	0.999587i	$0.490852\pi$
$332$	0	0
$333$	1224.69	0.201539
$334$	0	0
$335$	−12118.0	−1.97634
$336$	0	0
$337$	11341.5	1.83327	0.916633	−	0.399730i	$-0.130896\pi$
$337$	11341.5	1.83327	0.916633	+	0.399730i	$0.130896\pi$
$338$	0	0
$339$	−1899.35	−0.304303
$340$	0	0
$341$	4081.98	0.648245
$342$	0	0
$343$	3505.52	0.551837
$344$	0	0
$345$	5409.92	0.844233
$346$	0	0
$347$	−3171.28	−0.490615	−0.245307	−	0.969445i	$-0.578889\pi$
$347$	−3171.28	−0.490615	−0.245307	+	0.969445i	$0.578889\pi$
$348$	0	0
$349$	8476.24	1.30007	0.650033	−	0.759906i	$-0.274755\pi$
$349$	8476.24	1.30007	0.650033	+	0.759906i	$0.274755\pi$
$350$	0	0
$351$	−1751.62	−0.266367
$352$	0	0
$353$	4902.38	0.739171	0.369585	−	0.929197i	$-0.379500\pi$
$353$	4902.38	0.739171	0.369585	+	0.929197i	$0.379500\pi$
$354$	0	0
$355$	3090.63	0.462066
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7700.08	1.13202	0.566009	−	0.824399i	$-0.308487\pi$
$359$	7700.08	1.13202	0.566009	+	0.824399i	$0.308487\pi$
$360$	0	0
$361$	1924.75	0.280617
$362$	0	0
$363$	−3476.76	−0.502706
$364$	0	0
$365$	5801.61	0.831973
$366$	0	0
$367$	−12476.9	−1.77463	−0.887316	−	0.461162i	$-0.847433\pi$
$367$	−12476.9	−1.77463	−0.887316	+	0.461162i	$0.847433\pi$
$368$	0	0
$369$	−2969.95	−0.418996
$370$	0	0
$371$	16568.5	2.31857
$372$	0	0
$373$	−9711.07	−1.34804	−0.674022	−	0.738711i	$-0.735435\pi$
$373$	−9711.07	−1.34804	−0.674022	+	0.738711i	$0.735435\pi$
$374$	0	0
$375$	5641.07	0.776809
$376$	0	0
$377$	1950.07	0.266403
$378$	0	0
$379$	9417.61	1.27639	0.638193	−	0.769877i	$-0.279682\pi$
$379$	9417.61	1.27639	0.638193	+	0.769877i	$0.279682\pi$
$380$	0	0
$381$	−5727.71	−0.770183
$382$	0	0
$383$	−8432.22	−1.12498	−0.562489	−	0.826805i	$-0.690156\pi$
$383$	−8432.22	−1.12498	−0.562489	+	0.826805i	$0.690156\pi$
$384$	0	0
$385$	−13517.5	−1.78939
$386$	0	0
$387$	1394.75	0.183202
$388$	0	0
$389$	8121.97	1.05861	0.529306	−	0.848431i	$-0.322452\pi$
$389$	8121.97	1.05861	0.529306	+	0.848431i	$0.322452\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	10053.5	1.29041
$394$	0	0
$395$	3722.29	0.474149
$396$	0	0
$397$	8136.10	1.02856	0.514281	−	0.857622i	$-0.328059\pi$
$397$	8136.10	1.02856	0.514281	+	0.857622i	$0.328059\pi$
$398$	0	0
$399$	12658.4	1.58825
$400$	0	0
$401$	−255.211	−0.0317821	−0.0158911	−	0.999874i	$-0.505058\pi$
$401$	−255.211	−0.0317821	−0.0158911	+	0.999874i	$0.505058\pi$
$402$	0	0
$403$	−1403.95	−0.173537
$404$	0	0
$405$	−11602.8	−1.42357
$406$	0	0
$407$	7528.85	0.916932
$408$	0	0
$409$	4802.43	0.580598	0.290299	−	0.956936i	$-0.406245\pi$
$409$	4802.43	0.580598	0.290299	+	0.956936i	$0.406245\pi$
$410$	0	0
$411$	4404.31	0.528586
$412$	0	0
$413$	11952.0	1.42402
$414$	0	0
$415$	19990.3	2.36455
$416$	0	0
$417$	4187.35	0.491740
$418$	0	0
$419$	9288.73	1.08302	0.541508	−	0.840695i	$-0.317853\pi$
$419$	9288.73	1.08302	0.541508	+	0.840695i	$0.317853\pi$
$420$	0	0
$421$	6223.91	0.720510	0.360255	−	0.932854i	$-0.382690\pi$
$421$	6223.91	0.720510	0.360255	+	0.932854i	$0.382690\pi$
$422$	0	0
$423$	3880.71	0.446068
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9884.63	−1.12026
$428$	0	0
$429$	3870.53	0.435597
$430$	0	0
$431$	9316.16	1.04117	0.520584	−	0.853810i	$-0.325714\pi$
$431$	9316.16	1.04117	0.520584	+	0.853810i	$0.325714\pi$
$432$	0	0
$433$	10769.3	1.19524	0.597622	−	0.801778i	$-0.296113\pi$
$433$	10769.3	1.19524	0.597622	+	0.801778i	$0.296113\pi$
$434$	0	0
$435$	10058.1	1.10861
$436$	0	0
$437$	−6512.62	−0.712908
$438$	0	0
$439$	−9528.17	−1.03589	−0.517944	−	0.855415i	$-0.673302\pi$
$439$	−9528.17	−1.03589	−0.517944	+	0.855415i	$0.673302\pi$
$440$	0	0
$441$	1366.03	0.147504
$442$	0	0
$443$	7114.53	0.763029	0.381514	−	0.924363i	$-0.375403\pi$
$443$	7114.53	0.763029	0.381514	+	0.924363i	$0.375403\pi$
$444$	0	0
$445$	2174.92	0.231687
$446$	0	0
$447$	15467.6	1.63668
$448$	0	0
$449$	9654.45	1.01475	0.507374	−	0.861726i	$-0.330616\pi$
$449$	9654.45	1.01475	0.507374	+	0.861726i	$0.330616\pi$
$450$	0	0
$451$	−18258.0	−1.90629
$452$	0	0
$453$	−10436.0	−1.08240
$454$	0	0
$455$	4649.18	0.479027
$456$	0	0
$457$	−216.354	−0.0221457	−0.0110729	−	0.999939i	$-0.503525\pi$
$457$	−216.354	−0.0221457	−0.0110729	+	0.999939i	$0.503525\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13491.5	1.36304	0.681522	−	0.731797i	$-0.261318\pi$
$461$	13491.5	1.36304	0.681522	+	0.731797i	$0.261318\pi$
$462$	0	0
$463$	10159.3	1.01975	0.509875	−	0.860249i	$-0.329692\pi$
$463$	10159.3	1.01975	0.509875	+	0.860249i	$0.329692\pi$
$464$	0	0
$465$	−7241.24	−0.722160
$466$	0	0
$467$	6185.37	0.612901	0.306450	−	0.951887i	$-0.400859\pi$
$467$	6185.37	0.612901	0.306450	+	0.951887i	$0.400859\pi$
$468$	0	0
$469$	21022.9	2.06982
$470$	0	0
$471$	8184.57	0.800690
$472$	0	0
$473$	8574.33	0.833506
$474$	0	0
$475$	4924.33	0.475671
$476$	0	0
$477$	−5116.80	−0.491158
$478$	0	0
$479$	7507.53	0.716133	0.358067	−	0.933696i	$-0.383436\pi$
$479$	7507.53	0.716133	0.358067	+	0.933696i	$0.383436\pi$
$480$	0	0
$481$	−2589.45	−0.245466
$482$	0	0
$483$	−9385.42	−0.884165
$484$	0	0
$485$	19174.9	1.79523
$486$	0	0
$487$	6242.72	0.580872	0.290436	−	0.956894i	$-0.406200\pi$
$487$	6242.72	0.580872	0.290436	+	0.956894i	$0.406200\pi$
$488$	0	0
$489$	−21983.9	−2.03302
$490$	0	0
$491$	11818.1	1.08624	0.543120	−	0.839655i	$-0.317243\pi$
$491$	11818.1	1.08624	0.543120	+	0.839655i	$0.317243\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4174.59	0.379058
$496$	0	0
$497$	−5361.78	−0.483921
$498$	0	0
$499$	−3641.42	−0.326678	−0.163339	−	0.986570i	$-0.552227\pi$
$499$	−3641.42	−0.326678	−0.163339	+	0.986570i	$0.552227\pi$
$500$	0	0
$501$	4099.16	0.365543
$502$	0	0
$503$	4964.92	0.440109	0.220055	−	0.975488i	$-0.429376\pi$
$503$	4964.92	0.440109	0.220055	+	0.975488i	$0.429376\pi$
$504$	0	0
$505$	−25287.0	−2.22824
$506$	0	0
$507$	11505.5	1.00785
$508$	0	0
$509$	−16091.2	−1.40124	−0.700618	−	0.713536i	$-0.747092\pi$
$509$	−16091.2	−1.40124	−0.700618	+	0.713536i	$0.747092\pi$
$510$	0	0
$511$	−10064.9	−0.871325
$512$	0	0
$513$	10876.0	0.936036
$514$	0	0
$515$	−16443.0	−1.40692
$516$	0	0
$517$	23856.9	2.02945
$518$	0	0
$519$	−14393.5	−1.21735
$520$	0	0
$521$	7377.79	0.620397	0.310198	−	0.950672i	$-0.399605\pi$
$521$	7377.79	0.620397	0.310198	+	0.950672i	$0.399605\pi$
$522$	0	0
$523$	18129.1	1.51573	0.757867	−	0.652409i	$-0.226242\pi$
$523$	18129.1	1.51573	0.757867	+	0.652409i	$0.226242\pi$
$524$	0	0
$525$	7096.51	0.589937
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7338.29	−0.603130
$530$	0	0
$531$	−3691.12	−0.301659
$532$	0	0
$533$	6279.60	0.510319
$534$	0	0
$535$	−951.803	−0.0769160
$536$	0	0
$537$	−17537.8	−1.40933
$538$	0	0
$539$	8397.78	0.671091
$540$	0	0
$541$	−9719.15	−0.772383	−0.386191	−	0.922419i	$-0.626210\pi$
$541$	−9719.15	−0.772383	−0.386191	+	0.922419i	$0.626210\pi$
$542$	0	0
$543$	16967.5	1.34097
$544$	0	0
$545$	16367.6	1.28644
$546$	0	0
$547$	10886.4	0.850947	0.425473	−	0.904971i	$-0.360108\pi$
$547$	10886.4	0.850947	0.425473	+	0.904971i	$0.360108\pi$
$548$	0	0
$549$	3052.65	0.237312
$550$	0	0
$551$	−12108.2	−0.936163
$552$	0	0
$553$	−6457.63	−0.496576
$554$	0	0
$555$	−13355.8	−1.02148
$556$	0	0
$557$	−18970.8	−1.44312	−0.721559	−	0.692353i	$-0.756574\pi$
$557$	−18970.8	−1.44312	−0.721559	+	0.692353i	$0.756574\pi$
$558$	0	0
$559$	−2949.03	−0.223132
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11723.6	−0.877601	−0.438801	−	0.898584i	$-0.644597\pi$
$563$	−11723.6	−0.877601	−0.438801	+	0.898584i	$0.644597\pi$
$564$	0	0
$565$	4331.44	0.322522
$566$	0	0
$567$	20129.1	1.49091
$568$	0	0
$569$	−23387.3	−1.72310	−0.861551	−	0.507672i	$-0.830506\pi$
$569$	−23387.3	−1.72310	−0.861551	+	0.507672i	$0.830506\pi$
$570$	0	0
$571$	−7828.92	−0.573783	−0.286891	−	0.957963i	$-0.592622\pi$
$571$	−7828.92	−0.573783	−0.286891	+	0.957963i	$0.592622\pi$
$572$	0	0
$573$	−4177.06	−0.304536
$574$	0	0
$575$	−3651.09	−0.264802
$576$	0	0
$577$	1341.87	0.0968159	0.0484079	−	0.998828i	$-0.484585\pi$
$577$	1341.87	0.0968159	0.0484079	+	0.998828i	$0.484585\pi$
$578$	0	0
$579$	−26642.4	−1.91229
$580$	0	0
$581$	−34680.3	−2.47639
$582$	0	0
$583$	−31455.9	−2.23460
$584$	0	0
$585$	−1435.80	−0.101475
$586$	0	0
$587$	7397.08	0.520120	0.260060	−	0.965593i	$-0.416258\pi$
$587$	7397.08	0.520120	0.260060	+	0.965593i	$0.416258\pi$
$588$	0	0
$589$	8717.21	0.609824
$590$	0	0
$591$	−13586.1	−0.945610
$592$	0	0
$593$	17436.9	1.20750	0.603750	−	0.797174i	$-0.293673\pi$
$593$	17436.9	1.20750	0.603750	+	0.797174i	$0.293673\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11678.5	0.800615
$598$	0	0
$599$	1739.00	0.118621	0.0593103	−	0.998240i	$-0.481110\pi$
$599$	1739.00	0.118621	0.0593103	+	0.998240i	$0.481110\pi$
$600$	0	0
$601$	−15636.4	−1.06127	−0.530635	−	0.847600i	$-0.678047\pi$
$601$	−15636.4	−1.06127	−0.530635	+	0.847600i	$0.678047\pi$
$602$	0	0
$603$	−6492.46	−0.438463
$604$	0	0
$605$	7928.67	0.532804
$606$	0	0
$607$	14022.4	0.937644	0.468822	−	0.883293i	$-0.344679\pi$
$607$	14022.4	0.937644	0.468822	+	0.883293i	$0.344679\pi$
$608$	0	0
$609$	−17449.2	−1.16105
$610$	0	0
$611$	−8205.28	−0.543290
$612$	0	0
$613$	−22907.9	−1.50936	−0.754682	−	0.656091i	$-0.772209\pi$
$613$	−22907.9	−1.50936	−0.754682	+	0.656091i	$0.772209\pi$
$614$	0	0
$615$	32388.8	2.12365
$616$	0	0
$617$	−10663.7	−0.695792	−0.347896	−	0.937533i	$-0.613104\pi$
$617$	−10663.7	−0.695792	−0.347896	+	0.937533i	$0.613104\pi$
$618$	0	0
$619$	−19281.4	−1.25200	−0.625999	−	0.779824i	$-0.715309\pi$
$619$	−19281.4	−1.25200	−0.625999	+	0.779824i	$0.715309\pi$
$620$	0	0
$621$	−8063.88	−0.521083
$622$	0	0
$623$	−3773.16	−0.242646
$624$	0	0
$625$	−19432.1	−1.24365
$626$	0	0
$627$	−24032.4	−1.53072
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−2288.93	−0.144407	−0.0722035	−	0.997390i	$-0.523003\pi$
$631$	−2288.93	−0.144407	−0.0722035	+	0.997390i	$0.523003\pi$
$632$	0	0
$633$	−12240.8	−0.768607
$634$	0	0
$635$	13061.9	0.816294
$636$	0	0
$637$	−2888.31	−0.179653
$638$	0	0
$639$	1655.87	0.102512
$640$	0	0
$641$	17097.9	1.05355	0.526776	−	0.850004i	$-0.323401\pi$
$641$	17097.9	1.05355	0.526776	+	0.850004i	$0.323401\pi$
$642$	0	0
$643$	25880.4	1.58728	0.793642	−	0.608385i	$-0.208182\pi$
$643$	25880.4	1.58728	0.793642	+	0.608385i	$0.208182\pi$
$644$	0	0
$645$	−15210.5	−0.928544
$646$	0	0
$647$	−1089.29	−0.0661893	−0.0330946	−	0.999452i	$-0.510536\pi$
$647$	−1089.29	−0.0661893	−0.0330946	+	0.999452i	$0.510536\pi$
$648$	0	0
$649$	−22691.4	−1.37244
$650$	0	0
$651$	12562.5	0.756318
$652$	0	0
$653$	−16998.5	−1.01869	−0.509343	−	0.860564i	$-0.670111\pi$
$653$	−16998.5	−1.01869	−0.509343	+	0.860564i	$0.670111\pi$
$654$	0	0
$655$	−22926.8	−1.36767
$656$	0	0
$657$	3108.34	0.184578
$658$	0	0
$659$	−27155.9	−1.60523	−0.802613	−	0.596500i	$-0.796558\pi$
$659$	−27155.9	−1.60523	−0.802613	+	0.596500i	$0.796558\pi$
$660$	0	0
$661$	−18949.8	−1.11507	−0.557534	−	0.830154i	$-0.688252\pi$
$661$	−18949.8	−1.11507	−0.557534	+	0.830154i	$0.688252\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−28867.2	−1.68334
$666$	0	0
$667$	8977.48	0.521153
$668$	0	0
$669$	3619.57	0.209179
$670$	0	0
$671$	18766.4	1.07968
$672$	0	0
$673$	7356.01	0.421328	0.210664	−	0.977559i	$-0.432437\pi$
$673$	7356.01	0.421328	0.210664	+	0.977559i	$0.432437\pi$
$674$	0	0
$675$	6097.27	0.347680
$676$	0	0
$677$	11004.4	0.624719	0.312359	−	0.949964i	$-0.398881\pi$
$677$	11004.4	0.624719	0.312359	+	0.949964i	$0.398881\pi$
$678$	0	0
$679$	−33265.6	−1.88014
$680$	0	0
$681$	4912.76	0.276442
$682$	0	0
$683$	−21004.1	−1.17672	−0.588361	−	0.808599i	$-0.700226\pi$
$683$	−21004.1	−1.17672	−0.588361	+	0.808599i	$0.700226\pi$
$684$	0	0
$685$	−10043.9	−0.560232
$686$	0	0
$687$	−13034.5	−0.723869
$688$	0	0
$689$	10818.9	0.598208
$690$	0	0
$691$	612.926	0.0337436	0.0168718	−	0.999858i	$-0.494629\pi$
$691$	612.926	0.0337436	0.0168718	+	0.999858i	$0.494629\pi$
$692$	0	0
$693$	−7242.31	−0.396988
$694$	0	0
$695$	−9549.16	−0.521180
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−28037.9	−1.51715
$700$	0	0
$701$	19725.5	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$701$	19725.5	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$702$	0	0
$703$	16078.1	0.862587
$704$	0	0
$705$	−42321.0	−2.26085
$706$	0	0
$707$	43869.3	2.33363
$708$	0	0
$709$	9635.29	0.510382	0.255191	−	0.966891i	$-0.417862\pi$
$709$	9635.29	0.510382	0.255191	+	0.966891i	$0.417862\pi$
$710$	0	0
$711$	1994.30	0.105193
$712$	0	0
$713$	−6463.29	−0.339484
$714$	0	0
$715$	−8826.66	−0.461676
$716$	0	0
$717$	−15135.1	−0.788326
$718$	0	0
$719$	−36229.2	−1.87917	−0.939584	−	0.342318i	$-0.888788\pi$
$719$	−36229.2	−1.87917	−0.939584	+	0.342318i	$0.888788\pi$
$720$	0	0
$721$	28526.2	1.47347
$722$	0	0
$723$	20062.3	1.03198
$724$	0	0
$725$	−6788.06	−0.347727
$726$	0	0
$727$	38341.1	1.95597	0.977986	−	0.208670i	$-0.0669134\pi$
$727$	38341.1	1.95597	0.977986	+	0.208670i	$0.0669134\pi$
$728$	0	0
$729$	12090.5	0.614263
$730$	0	0
$731$	0	0
$732$	0	0
$733$	17288.4	0.871160	0.435580	−	0.900150i	$-0.356543\pi$
$733$	17288.4	0.871160	0.435580	+	0.900150i	$0.356543\pi$
$734$	0	0
$735$	−14897.3	−0.747611
$736$	0	0
$737$	−39912.8	−1.99485
$738$	0	0
$739$	−1522.94	−0.0758084	−0.0379042	−	0.999281i	$-0.512068\pi$
$739$	−1522.94	−0.0758084	−0.0379042	+	0.999281i	$0.512068\pi$
$740$	0	0
$741$	8265.66	0.409779
$742$	0	0
$743$	−8540.54	−0.421698	−0.210849	−	0.977519i	$-0.567623\pi$
$743$	−8540.54	−0.421698	−0.210849	+	0.977519i	$0.567623\pi$
$744$	0	0
$745$	−35273.6	−1.73466
$746$	0	0
$747$	10710.3	0.524589
$748$	0	0
$749$	1651.24	0.0805541
$750$	0	0
$751$	7904.05	0.384052	0.192026	−	0.981390i	$-0.438494\pi$
$751$	7904.05	0.384052	0.192026	+	0.981390i	$0.438494\pi$
$752$	0	0
$753$	15387.1	0.744673
$754$	0	0
$755$	23799.0	1.14720
$756$	0	0
$757$	37299.8	1.79086	0.895432	−	0.445199i	$-0.146867\pi$
$757$	37299.8	1.79086	0.895432	+	0.445199i	$0.146867\pi$
$758$	0	0
$759$	17818.6	0.852140
$760$	0	0
$761$	−16795.4	−0.800042	−0.400021	−	0.916506i	$-0.630997\pi$
$761$	−16795.4	−0.800042	−0.400021	+	0.916506i	$0.630997\pi$
$762$	0	0
$763$	−28395.4	−1.34729
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7804.43	0.367408
$768$	0	0
$769$	−7845.61	−0.367906	−0.183953	−	0.982935i	$-0.558889\pi$
$769$	−7845.61	−0.367906	−0.183953	+	0.982935i	$0.558889\pi$
$770$	0	0
$771$	15016.0	0.701410
$772$	0	0
$773$	−31023.8	−1.44353	−0.721766	−	0.692137i	$-0.756669\pi$
$773$	−31023.8	−1.44353	−0.721766	+	0.692137i	$0.756669\pi$
$774$	0	0
$775$	4887.02	0.226512
$776$	0	0
$777$	23170.4	1.06980
$778$	0	0
$779$	−38990.6	−1.79330
$780$	0	0
$781$	10179.6	0.466393
$782$	0	0
$783$	−14992.3	−0.684265
$784$	0	0
$785$	−18664.7	−0.848628
$786$	0	0
$787$	23818.2	1.07881	0.539407	−	0.842045i	$-0.318649\pi$
$787$	23818.2	1.07881	0.539407	+	0.842045i	$0.318649\pi$
$788$	0	0
$789$	−19863.6	−0.896277
$790$	0	0
$791$	−7514.41	−0.337777
$792$	0	0
$793$	−6454.46	−0.289035
$794$	0	0
$795$	55801.2	2.48939
$796$	0	0
$797$	4264.27	0.189521	0.0947604	−	0.995500i	$-0.469792\pi$
$797$	4264.27	0.189521	0.0947604	+	0.995500i	$0.469792\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	1165.26	0.0514012
$802$	0	0
$803$	19108.7	0.839765
$804$	0	0
$805$	21403.3	0.937100
$806$	0	0
$807$	−9215.24	−0.401973
$808$	0	0
$809$	−1644.67	−0.0714752	−0.0357376	−	0.999361i	$-0.511378\pi$
$809$	−1644.67	−0.0714752	−0.0357376	+	0.999361i	$0.511378\pi$
$810$	0	0
$811$	7192.73	0.311431	0.155716	−	0.987802i	$-0.450232\pi$
$811$	7192.73	0.311431	0.155716	+	0.987802i	$0.450232\pi$
$812$	0	0
$813$	−14572.4	−0.628630
$814$	0	0
$815$	50133.9	2.15474
$816$	0	0
$817$	18310.8	0.784105
$818$	0	0
$819$	2490.90	0.106275
$820$	0	0
$821$	−28659.7	−1.21831	−0.609154	−	0.793052i	$-0.708491\pi$
$821$	−28659.7	−1.21831	−0.609154	+	0.793052i	$0.708491\pi$
$822$	0	0
$823$	−32575.6	−1.37973	−0.689863	−	0.723940i	$-0.742329\pi$
$823$	−32575.6	−1.37973	−0.689863	+	0.723940i	$0.742329\pi$
$824$	0	0
$825$	−13473.0	−0.568570
$826$	0	0
$827$	12012.3	0.505090	0.252545	−	0.967585i	$-0.418732\pi$
$827$	12012.3	0.505090	0.252545	+	0.967585i	$0.418732\pi$
$828$	0	0
$829$	13917.4	0.583078	0.291539	−	0.956559i	$-0.405833\pi$
$829$	13917.4	0.583078	0.291539	+	0.956559i	$0.405833\pi$
$830$	0	0
$831$	−30377.5	−1.26809
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−9348.04	−0.387428
$836$	0	0
$837$	10793.6	0.445736
$838$	0	0
$839$	20282.2	0.834590	0.417295	−	0.908771i	$-0.362978\pi$
$839$	20282.2	0.834590	0.417295	+	0.908771i	$0.362978\pi$
$840$	0	0
$841$	−7698.20	−0.315642
$842$	0	0
$843$	−18831.5	−0.769386
$844$	0	0
$845$	−26238.1	−1.06819
$846$	0	0
$847$	−13755.1	−0.558005
$848$	0	0
$849$	−48790.0	−1.97228
$850$	0	0
$851$	−11921.0	−0.480194
$852$	0	0
$853$	2153.16	0.0864279	0.0432139	−	0.999066i	$-0.486240\pi$
$853$	2153.16	0.0864279	0.0432139	+	0.999066i	$0.486240\pi$
$854$	0	0
$855$	8914.99	0.356592
$856$	0	0
$857$	−21734.2	−0.866307	−0.433153	−	0.901320i	$-0.642599\pi$
$857$	−21734.2	−0.866307	−0.433153	+	0.901320i	$0.642599\pi$
$858$	0	0
$859$	−27746.7	−1.10210	−0.551050	−	0.834472i	$-0.685773\pi$
$859$	−27746.7	−1.10210	−0.551050	+	0.834472i	$0.685773\pi$
$860$	0	0
$861$	−56189.8	−2.22409
$862$	0	0
$863$	16382.7	0.646205	0.323102	−	0.946364i	$-0.395274\pi$
$863$	16382.7	0.646205	0.323102	+	0.946364i	$0.395274\pi$
$864$	0	0
$865$	32824.0	1.29023
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12260.1	0.478590
$870$	0	0
$871$	13727.5	0.534028
$872$	0	0
$873$	10273.4	0.398282
$874$	0	0
$875$	22317.7	0.862260
$876$	0	0
$877$	−34200.5	−1.31684	−0.658420	−	0.752651i	$-0.728775\pi$
$877$	−34200.5	−1.31684	−0.658420	+	0.752651i	$0.728775\pi$
$878$	0	0
$879$	−51682.4	−1.98317
$880$	0	0
$881$	6730.30	0.257377	0.128689	−	0.991685i	$-0.458923\pi$
$881$	6730.30	0.257377	0.128689	+	0.991685i	$0.458923\pi$
$882$	0	0
$883$	−38107.5	−1.45234	−0.726171	−	0.687514i	$-0.758702\pi$
$883$	−38107.5	−1.45234	−0.726171	+	0.687514i	$0.758702\pi$
$884$	0	0
$885$	40253.5	1.52893
$886$	0	0
$887$	−8162.76	−0.308995	−0.154498	−	0.987993i	$-0.549376\pi$
$887$	−8162.76	−0.308995	−0.154498	+	0.987993i	$0.549376\pi$
$888$	0	0
$889$	−22660.5	−0.854904
$890$	0	0
$891$	−38216.0	−1.43691
$892$	0	0
$893$	50947.3	1.90917
$894$	0	0
$895$	39994.6	1.49371
$896$	0	0
$897$	−6128.48	−0.228121
$898$	0	0
$899$	−12016.5	−0.445797
$900$	0	0
$901$	0	0
$902$	0	0
$903$	26387.9	0.972464
$904$	0	0
$905$	−38694.1	−1.42126
$906$	0	0
$907$	−41183.9	−1.50771	−0.753853	−	0.657043i	$-0.771807\pi$
$907$	−41183.9	−1.50771	−0.753853	+	0.657043i	$0.771807\pi$
$908$	0	0
$909$	−13548.1	−0.494347
$910$	0	0
$911$	−25001.3	−0.909252	−0.454626	−	0.890682i	$-0.650227\pi$
$911$	−25001.3	−0.909252	−0.454626	+	0.890682i	$0.650227\pi$
$912$	0	0
$913$	65841.9	2.38669
$914$	0	0
$915$	−33290.7	−1.20279
$916$	0	0
$917$	39774.7	1.43236
$918$	0	0
$919$	9793.16	0.351520	0.175760	−	0.984433i	$-0.443762\pi$
$919$	9793.16	0.351520	0.175760	+	0.984433i	$0.443762\pi$
$920$	0	0
$921$	40525.0	1.44988
$922$	0	0
$923$	−3501.13	−0.124855
$924$	0	0
$925$	9013.69	0.320398
$926$	0	0
$927$	−8809.68	−0.312134
$928$	0	0
$929$	49359.2	1.74319	0.871595	−	0.490227i	$-0.163086\pi$
$929$	49359.2	1.74319	0.871595	+	0.490227i	$0.163086\pi$
$930$	0	0
$931$	17933.8	0.631316
$932$	0	0
$933$	3508.18	0.123100
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42068.8	1.46673	0.733366	−	0.679834i	$-0.237948\pi$
$937$	42068.8	1.46673	0.733366	+	0.679834i	$0.237948\pi$
$938$	0	0
$939$	−44531.5	−1.54764
$940$	0	0
$941$	−35447.4	−1.22801	−0.614003	−	0.789304i	$-0.710442\pi$
$941$	−35447.4	−1.22801	−0.614003	+	0.789304i	$0.710442\pi$
$942$	0	0
$943$	28909.1	0.998315
$944$	0	0
$945$	−35743.1	−1.23040
$946$	0	0
$947$	9075.93	0.311434	0.155717	−	0.987802i	$-0.450231\pi$
$947$	9075.93	0.311434	0.155717	+	0.987802i	$0.450231\pi$
$948$	0	0
$949$	−6572.20	−0.224808
$950$	0	0
$951$	4397.35	0.149941
$952$	0	0
$953$	−6231.06	−0.211798	−0.105899	−	0.994377i	$-0.533772\pi$
$953$	−6231.06	−0.211798	−0.105899	+	0.994377i	$0.533772\pi$
$954$	0	0
$955$	9525.69	0.322769
$956$	0	0
$957$	33128.1	1.11900
$958$	0	0
$959$	17424.8	0.586731
$960$	0	0
$961$	−21139.8	−0.709604
$962$	0	0
$963$	−509.949	−0.0170643
$964$	0	0
$965$	60757.3	2.02678
$966$	0	0
$967$	−402.531	−0.0133863	−0.00669314	−	0.999978i	$-0.502131\pi$
$967$	−402.531	−0.0133863	−0.00669314	+	0.999978i	$0.502131\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10668.4	0.352591	0.176295	−	0.984337i	$-0.443589\pi$
$971$	10668.4	0.352591	0.176295	+	0.984337i	$0.443589\pi$
$972$	0	0
$973$	16566.4	0.545832
$974$	0	0
$975$	4633.87	0.152208
$976$	0	0
$977$	14528.9	0.475764	0.237882	−	0.971294i	$-0.423547\pi$
$977$	14528.9	0.475764	0.237882	+	0.971294i	$0.423547\pi$
$978$	0	0
$979$	7163.50	0.233857
$980$	0	0
$981$	8769.30	0.285405
$982$	0	0
$983$	−34056.4	−1.10502	−0.552508	−	0.833507i	$-0.686329\pi$
$983$	−34056.4	−1.10502	−0.552508	+	0.833507i	$0.686329\pi$
$984$	0	0
$985$	30982.7	1.00222
$986$	0	0
$987$	73420.8	2.36779
$988$	0	0
$989$	−13576.3	−0.436504
$990$	0	0
$991$	18300.0	0.586597	0.293299	−	0.956021i	$-0.405247\pi$
$991$	18300.0	0.586597	0.293299	+	0.956021i	$0.405247\pi$
$992$	0	0
$993$	−2022.05	−0.0646200
$994$	0	0
$995$	−26632.5	−0.848548
$996$	0	0
$997$	−2533.05	−0.0804640	−0.0402320	−	0.999190i	$-0.512810\pi$
$997$	−2533.05	−0.0804640	−0.0402320	+	0.999190i	$0.512810\pi$
$998$	0	0
$999$	19907.8	0.630486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.j.1.2	yes	6
17.16	even	2		2312.4.a.f.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.f.1.5	✓	6	17.16	even	2
2312.4.a.j.1.2	yes	6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-5.84285 q^{3} +13.3245 q^{5} -23.1160 q^{7} +7.13888 q^{9} +O(q^{10})\)	\(q-5.84285 q^{3} +13.3245 q^{5} -23.1160 q^{7} +7.13888 q^{9} +43.8867 q^{11} -15.0943 q^{13} -77.8530 q^{15} +93.7217 q^{19} +135.064 q^{21} -69.4890 q^{23} +52.5420 q^{25} +116.046 q^{27} -129.193 q^{29} +93.0117 q^{31} -256.424 q^{33} -308.009 q^{35} +171.552 q^{37} +88.1937 q^{39} -416.025 q^{41} +195.374 q^{43} +95.1220 q^{45} +543.602 q^{47} +191.351 q^{49} -716.751 q^{53} +584.768 q^{55} -547.602 q^{57} -517.045 q^{59} +427.609 q^{61} -165.023 q^{63} -201.124 q^{65} -909.450 q^{67} +406.013 q^{69} +231.951 q^{71} +435.409 q^{73} -306.995 q^{75} -1014.49 q^{77} +279.357 q^{79} -870.786 q^{81} +1500.27 q^{83} +754.855 q^{87} +163.227 q^{89} +348.920 q^{91} -543.453 q^{93} +1248.79 q^{95} +1439.07 q^{97} +313.302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9}+O(q^{10}) \) 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9	\( 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9} + 72 q^{11} + 15 q^{13} - 60 q^{15} + 83 q^{19} + 96 q^{21} + 141 q^{23} + 263 q^{25} + 94 q^{27} - 249 q^{29} + 106 q^{31} + 289 q^{33} - 267 q^{35} + 170 q^{37} + 329 q^{39} - 100 q^{41} - 90 q^{43} - 286 q^{45} + 372 q^{47} + 323 q^{49} - 23 q^{53} - 457 q^{55} - 193 q^{57} + 784 q^{59} + 92 q^{61} - 722 q^{63} + 1412 q^{65} + 238 q^{67} + 1178 q^{69} + 940 q^{71} + 692 q^{73} + 1814 q^{75} - 45 q^{77} + 84 q^{79} - 2182 q^{81} + 1393 q^{83} - 1247 q^{87} + 976 q^{89} - 384 q^{91} + 1437 q^{93} - 1022 q^{95} - 325 q^{97} + 3981 q^{99}+O(q^{100}) \) 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9 + 72 * q^11 + 15 * q^13 - 60 * q^15 + 83 * q^19 + 96 * q^21 + 141 * q^23 + 263 * q^25 + 94 * q^27 - 249 * q^29 + 106 * q^31 + 289 * q^33 - 267 * q^35 + 170 * q^37 + 329 * q^39 - 100 * q^41 - 90 * q^43 - 286 * q^45 + 372 * q^47 + 323 * q^49 - 23 * q^53 - 457 * q^55 - 193 * q^57 + 784 * q^59 + 92 * q^61 - 722 * q^63 + 1412 * q^65 + 238 * q^67 + 1178 * q^69 + 940 * q^71 + 692 * q^73 + 1814 * q^75 - 45 * q^77 + 84 * q^79 - 2182 * q^81 + 1393 * q^83 - 1247 * q^87 + 976 * q^89 - 384 * q^91 + 1437 * q^93 - 1022 * q^95 - 325 * q^97 + 3981 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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