LMFDB - Embedded newform 2312.4.a.n.1.5

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} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + \cdots - 176969301147 $ x^18 - 294x^16 - 14x^15 + 34371x^14 + 2670x^13 - 2054705x^12 - 160284x^11 + 67981059x^10 + 2824200x^9 - 1279285428x^8 + 12108606x^7 + 13538185208x^6 - 853720902x^5 - 75816227907x^4 + 7747530606x^3 + 195817930515x^2 - 12873873186x - 176969301147
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.86015$ 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)

$f(q)$	$=$	$q-4.86015 q^{3} +1.88035 q^{5} +7.34878 q^{7} -3.37898 q^{9} +O(q^{10})$	$q-4.86015 q^{3} +1.88035 q^{5} +7.34878 q^{7} -3.37898 q^{9} +66.6077 q^{11} +30.0692 q^{13} -9.13879 q^{15} -24.5842 q^{19} -35.7162 q^{21} -100.921 q^{23} -121.464 q^{25} +147.646 q^{27} -159.680 q^{29} -131.743 q^{31} -323.723 q^{33} +13.8183 q^{35} -5.27968 q^{37} -146.141 q^{39} +16.1052 q^{41} -136.331 q^{43} -6.35367 q^{45} +261.237 q^{47} -288.995 q^{49} +327.035 q^{53} +125.246 q^{55} +119.483 q^{57} -463.712 q^{59} +110.145 q^{61} -24.8314 q^{63} +56.5406 q^{65} +775.589 q^{67} +490.489 q^{69} -906.326 q^{71} +855.531 q^{73} +590.334 q^{75} +489.485 q^{77} -494.626 q^{79} -626.350 q^{81} +544.331 q^{83} +776.069 q^{87} -181.309 q^{89} +220.972 q^{91} +640.290 q^{93} -46.2270 q^{95} -495.027 q^{97} -225.066 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) $ 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9	$ 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) $ 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9 - 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 - 153 * q^23 + 306 * q^25 - 768 * q^27 - 51 * q^29 - 303 * q^31 + 525 * q^33 - 255 * q^35 - 717 * q^37 + 216 * q^39 + 393 * q^41 - 390 * q^43 - 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 - 810 * q^57 - 204 * q^59 - 534 * q^61 - 2556 * q^63 + 2127 * q^65 - 405 * q^67 + 2547 * q^69 + 426 * q^71 - 1149 * q^73 - 2226 * q^75 - 357 * q^77 - 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 - 6090 * q^91 + 606 * q^93 - 2109 * q^95 - 2349 * q^97 - 1428 * 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
$3$	−4.86015	−0.935336	−0.467668	−	0.883904i	$-0.654906\pi$
$3$	−4.86015	−0.935336	−0.467668	+	0.883904i	$0.654906\pi$
$4$	0	0
$5$	1.88035	0.168184	0.0840919	−	0.996458i	$-0.473201\pi$
$5$	1.88035	0.168184	0.0840919	+	0.996458i	$0.473201\pi$
$6$	0	0
$7$	7.34878	0.396797	0.198398	−	0.980121i	$-0.436426\pi$
$7$	7.34878	0.396797	0.198398	+	0.980121i	$0.436426\pi$
$8$	0	0
$9$	−3.37898	−0.125147
$10$	0	0
$11$	66.6077	1.82573	0.912863	−	0.408267i	$-0.133867\pi$
$11$	66.6077	1.82573	0.912863	+	0.408267i	$0.133867\pi$
$12$	0	0
$13$	30.0692	0.641514	0.320757	−	0.947161i	$-0.396063\pi$
$13$	30.0692	0.641514	0.320757	+	0.947161i	$0.396063\pi$
$14$	0	0
$15$	−9.13879	−0.157308
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−24.5842	−0.296842	−0.148421	−	0.988924i	$-0.547419\pi$
$19$	−24.5842	−0.296842	−0.148421	+	0.988924i	$0.547419\pi$
$20$	0	0
$21$	−35.7162	−0.371138
$22$	0	0
$23$	−100.921	−0.914931	−0.457466	−	0.889227i	$-0.651243\pi$
$23$	−100.921	−0.914931	−0.457466	+	0.889227i	$0.651243\pi$
$24$	0	0
$25$	−121.464	−0.971714
$26$	0	0
$27$	147.646	1.05239
$28$	0	0
$29$	−159.680	−1.02248	−0.511239	−	0.859438i	$-0.670813\pi$
$29$	−159.680	−1.02248	−0.511239	+	0.859438i	$0.670813\pi$
$30$	0	0
$31$	−131.743	−0.763282	−0.381641	−	0.924311i	$-0.624641\pi$
$31$	−131.743	−0.763282	−0.381641	+	0.924311i	$0.624641\pi$
$32$	0	0
$33$	−323.723	−1.70767
$34$	0	0
$35$	13.8183	0.0667348
$36$	0	0
$37$	−5.27968	−0.0234587	−0.0117294	−	0.999931i	$-0.503734\pi$
$37$	−5.27968	−0.0234587	−0.0117294	+	0.999931i	$0.503734\pi$
$38$	0	0
$39$	−146.141	−0.600031
$40$	0	0
$41$	16.1052	0.0613467	0.0306734	−	0.999529i	$-0.490235\pi$
$41$	16.1052	0.0613467	0.0306734	+	0.999529i	$0.490235\pi$
$42$	0	0
$43$	−136.331	−0.483494	−0.241747	−	0.970339i	$-0.577720\pi$
$43$	−136.331	−0.483494	−0.241747	+	0.970339i	$0.577720\pi$
$44$	0	0
$45$	−6.35367	−0.0210478
$46$	0	0
$47$	261.237	0.810753	0.405376	−	0.914150i	$-0.367141\pi$
$47$	261.237	0.810753	0.405376	+	0.914150i	$0.367141\pi$
$48$	0	0
$49$	−288.995	−0.842552
$50$	0	0
$51$	0	0
$52$	0	0
$53$	327.035	0.847579	0.423789	−	0.905761i	$-0.360700\pi$
$53$	327.035	0.847579	0.423789	+	0.905761i	$0.360700\pi$
$54$	0	0
$55$	125.246	0.307058
$56$	0	0
$57$	119.483	0.277647
$58$	0	0
$59$	−463.712	−1.02322	−0.511612	−	0.859217i	$-0.670951\pi$
$59$	−463.712	−1.02322	−0.511612	+	0.859217i	$0.670951\pi$
$60$	0	0
$61$	110.145	0.231192	0.115596	−	0.993296i	$-0.463122\pi$
$61$	110.145	0.231192	0.115596	+	0.993296i	$0.463122\pi$
$62$	0	0
$63$	−24.8314	−0.0496581
$64$	0	0
$65$	56.5406	0.107892
$66$	0	0
$67$	775.589	1.41423	0.707114	−	0.707100i	$-0.249997\pi$
$67$	775.589	1.41423	0.707114	+	0.707100i	$0.249997\pi$
$68$	0	0
$69$	490.489	0.855768
$70$	0	0
$71$	−906.326	−1.51495	−0.757473	−	0.652867i	$-0.773566\pi$
$71$	−906.326	−1.51495	−0.757473	+	0.652867i	$0.773566\pi$
$72$	0	0
$73$	855.531	1.37168	0.685838	−	0.727754i	$-0.259436\pi$
$73$	855.531	1.37168	0.685838	+	0.727754i	$0.259436\pi$
$74$	0	0
$75$	590.334	0.908879
$76$	0	0
$77$	489.485	0.724442
$78$	0	0
$79$	−494.626	−0.704427	−0.352214	−	0.935920i	$-0.614571\pi$
$79$	−494.626	−0.704427	−0.352214	+	0.935920i	$0.614571\pi$
$80$	0	0
$81$	−626.350	−0.859191
$82$	0	0
$83$	544.331	0.719856	0.359928	−	0.932980i	$-0.382801\pi$
$83$	544.331	0.719856	0.359928	+	0.932980i	$0.382801\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	776.069	0.956361
$88$	0	0
$89$	−181.309	−0.215940	−0.107970	−	0.994154i	$-0.534435\pi$
$89$	−181.309	−0.215940	−0.107970	+	0.994154i	$0.534435\pi$
$90$	0	0
$91$	220.972	0.254551
$92$	0	0
$93$	640.290	0.713925
$94$	0	0
$95$	−46.2270	−0.0499241
$96$	0	0
$97$	−495.027	−0.518169	−0.259084	−	0.965855i	$-0.583421\pi$
$97$	−495.027	−0.518169	−0.259084	+	0.965855i	$0.583421\pi$
$98$	0	0
$99$	−225.066	−0.228485
$100$	0	0
$101$	341.955	0.336889	0.168445	−	0.985711i	$-0.446126\pi$
$101$	341.955	0.336889	0.168445	+	0.985711i	$0.446126\pi$
$102$	0	0
$103$	1022.89	0.978527	0.489264	−	0.872136i	$-0.337266\pi$
$103$	1022.89	0.978527	0.489264	+	0.872136i	$0.337266\pi$
$104$	0	0
$105$	−67.1590	−0.0624195
$106$	0	0
$107$	869.281	0.785389	0.392694	−	0.919669i	$-0.371543\pi$
$107$	869.281	0.785389	0.392694	+	0.919669i	$0.371543\pi$
$108$	0	0
$109$	143.872	0.126426	0.0632129	−	0.998000i	$-0.479865\pi$
$109$	143.872	0.126426	0.0632129	+	0.998000i	$0.479865\pi$
$110$	0	0
$111$	25.6600	0.0219418
$112$	0	0
$113$	−1995.79	−1.66149	−0.830744	−	0.556655i	$-0.812085\pi$
$113$	−1995.79	−1.66149	−0.830744	+	0.556655i	$0.812085\pi$
$114$	0	0
$115$	−189.767	−0.153877
$116$	0	0
$117$	−101.603	−0.0802838
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3105.59	2.33327
$122$	0	0
$123$	−78.2738	−0.0573798
$124$	0	0
$125$	−463.440	−0.331611
$126$	0	0
$127$	−204.651	−0.142991	−0.0714955	−	0.997441i	$-0.522777\pi$
$127$	−204.651	−0.142991	−0.0714955	+	0.997441i	$0.522777\pi$
$128$	0	0
$129$	662.588	0.452230
$130$	0	0
$131$	−2268.76	−1.51315	−0.756575	−	0.653907i	$-0.773129\pi$
$131$	−2268.76	−1.51315	−0.756575	+	0.653907i	$0.773129\pi$
$132$	0	0
$133$	−180.664	−0.117786
$134$	0	0
$135$	277.627	0.176995
$136$	0	0
$137$	−844.729	−0.526789	−0.263395	−	0.964688i	$-0.584842\pi$
$137$	−844.729	−0.526789	−0.263395	+	0.964688i	$0.584842\pi$
$138$	0	0
$139$	−1005.12	−0.613335	−0.306667	−	0.951817i	$-0.599214\pi$
$139$	−1005.12	−0.613335	−0.306667	+	0.951817i	$0.599214\pi$
$140$	0	0
$141$	−1269.65	−0.758326
$142$	0	0
$143$	2002.84	1.17123
$144$	0	0
$145$	−300.255	−0.171964
$146$	0	0
$147$	1404.56	0.788069
$148$	0	0
$149$	2896.88	1.59276	0.796380	−	0.604796i	$-0.206745\pi$
$149$	2896.88	1.59276	0.796380	+	0.604796i	$0.206745\pi$
$150$	0	0
$151$	−2010.95	−1.08377	−0.541884	−	0.840453i	$-0.682289\pi$
$151$	−2010.95	−1.08377	−0.541884	+	0.840453i	$0.682289\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−247.723	−0.128372
$156$	0	0
$157$	1610.82	0.818840	0.409420	−	0.912346i	$-0.365731\pi$
$157$	1610.82	0.818840	0.409420	+	0.912346i	$0.365731\pi$
$158$	0	0
$159$	−1589.44	−0.792770
$160$	0	0
$161$	−741.644	−0.363042
$162$	0	0
$163$	−1879.16	−0.902991	−0.451496	−	0.892273i	$-0.649109\pi$
$163$	−1879.16	−0.902991	−0.451496	+	0.892273i	$0.649109\pi$
$164$	0	0
$165$	−608.714	−0.287202
$166$	0	0
$167$	201.229	0.0932431	0.0466215	−	0.998913i	$-0.485155\pi$
$167$	201.229	0.0932431	0.0466215	+	0.998913i	$0.485155\pi$
$168$	0	0
$169$	−1292.85	−0.588460
$170$	0	0
$171$	83.0695	0.0371490
$172$	0	0
$173$	−3432.56	−1.50851	−0.754257	−	0.656579i	$-0.772003\pi$
$173$	−3432.56	−1.50851	−0.754257	+	0.656579i	$0.772003\pi$
$174$	0	0
$175$	−892.614	−0.385573
$176$	0	0
$177$	2253.71	0.957057
$178$	0	0
$179$	464.273	0.193863	0.0969313	−	0.995291i	$-0.469097\pi$
$179$	464.273	0.193863	0.0969313	+	0.995291i	$0.469097\pi$
$180$	0	0
$181$	1761.13	0.723225	0.361612	−	0.932329i	$-0.382226\pi$
$181$	1761.13	0.723225	0.361612	+	0.932329i	$0.382226\pi$
$182$	0	0
$183$	−535.323	−0.216242
$184$	0	0
$185$	−9.92766	−0.00394538
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1085.02	0.417585
$190$	0	0
$191$	−4421.70	−1.67509	−0.837547	−	0.546365i	$-0.816011\pi$
$191$	−4421.70	−1.67509	−0.837547	+	0.546365i	$0.816011\pi$
$192$	0	0
$193$	−4133.24	−1.54154	−0.770770	−	0.637114i	$-0.780128\pi$
$193$	−4133.24	−1.54154	−0.770770	+	0.637114i	$0.780128\pi$
$194$	0	0
$195$	−274.796	−0.100916
$196$	0	0
$197$	4372.25	1.58127	0.790634	−	0.612288i	$-0.209751\pi$
$197$	4372.25	1.58127	0.790634	+	0.612288i	$0.209751\pi$
$198$	0	0
$199$	−4998.72	−1.78065	−0.890327	−	0.455322i	$-0.849524\pi$
$199$	−4998.72	−1.78065	−0.890327	+	0.455322i	$0.849524\pi$
$200$	0	0
$201$	−3769.47	−1.32278
$202$	0	0
$203$	−1173.46	−0.405716
$204$	0	0
$205$	30.2835	0.0103175
$206$	0	0
$207$	341.009	0.114501
$208$	0	0
$209$	−1637.50	−0.541952
$210$	0	0
$211$	3042.16	0.992563	0.496281	−	0.868162i	$-0.334698\pi$
$211$	3042.16	0.992563	0.496281	+	0.868162i	$0.334698\pi$
$212$	0	0
$213$	4404.88	1.41698
$214$	0	0
$215$	−256.350	−0.0813160
$216$	0	0
$217$	−968.150	−0.302868
$218$	0	0
$219$	−4158.01	−1.28298
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−41.2331	−0.0123819	−0.00619097	−	0.999981i	$-0.501971\pi$
$223$	−41.2331	−0.0123819	−0.00619097	+	0.999981i	$0.501971\pi$
$224$	0	0
$225$	410.425	0.121607
$226$	0	0
$227$	−4091.37	−1.19627	−0.598136	−	0.801395i	$-0.704092\pi$
$227$	−4091.37	−1.19627	−0.598136	+	0.801395i	$0.704092\pi$
$228$	0	0
$229$	4241.60	1.22399	0.611993	−	0.790863i	$-0.290368\pi$
$229$	4241.60	1.22399	0.611993	+	0.790863i	$0.290368\pi$
$230$	0	0
$231$	−2378.97	−0.677596
$232$	0	0
$233$	5149.70	1.44793	0.723965	−	0.689836i	$-0.242318\pi$
$233$	5149.70	1.44793	0.723965	+	0.689836i	$0.242318\pi$
$234$	0	0
$235$	491.218	0.136356
$236$	0	0
$237$	2403.95	0.658876
$238$	0	0
$239$	−2260.63	−0.611831	−0.305916	−	0.952059i	$-0.598963\pi$
$239$	−2260.63	−0.611831	−0.305916	+	0.952059i	$0.598963\pi$
$240$	0	0
$241$	1489.18	0.398036	0.199018	−	0.979996i	$-0.436225\pi$
$241$	1489.18	0.398036	0.199018	+	0.979996i	$0.436225\pi$
$242$	0	0
$243$	−942.296	−0.248759
$244$	0	0
$245$	−543.413	−0.141704
$246$	0	0
$247$	−739.226	−0.190429
$248$	0	0
$249$	−2645.53	−0.673307
$250$	0	0
$251$	−5652.77	−1.42151	−0.710757	−	0.703438i	$-0.751647\pi$
$251$	−5652.77	−1.42151	−0.710757	+	0.703438i	$0.751647\pi$
$252$	0	0
$253$	−6722.10	−1.67041
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7791.40	1.89111	0.945553	−	0.325469i	$-0.105522\pi$
$257$	7791.40	1.89111	0.945553	+	0.325469i	$0.105522\pi$
$258$	0	0
$259$	−38.7992	−0.00930836
$260$	0	0
$261$	539.556	0.127960
$262$	0	0
$263$	−1282.76	−0.300755	−0.150377	−	0.988629i	$-0.548049\pi$
$263$	−1282.76	−0.300755	−0.150377	+	0.988629i	$0.548049\pi$
$264$	0	0
$265$	614.941	0.142549
$266$	0	0
$267$	881.187	0.201977
$268$	0	0
$269$	8114.80	1.83929	0.919643	−	0.392755i	$-0.128478\pi$
$269$	8114.80	1.83929	0.919643	+	0.392755i	$0.128478\pi$
$270$	0	0
$271$	7048.83	1.58002	0.790011	−	0.613093i	$-0.210075\pi$
$271$	7048.83	1.58002	0.790011	+	0.613093i	$0.210075\pi$
$272$	0	0
$273$	−1073.95	−0.238090
$274$	0	0
$275$	−8090.46	−1.77408
$276$	0	0
$277$	−7339.38	−1.59199	−0.795994	−	0.605304i	$-0.793051\pi$
$277$	−7339.38	−1.59199	−0.795994	+	0.605304i	$0.793051\pi$
$278$	0	0
$279$	445.156	0.0955227
$280$	0	0
$281$	−383.111	−0.0813327	−0.0406664	−	0.999173i	$-0.512948\pi$
$281$	−383.111	−0.0813327	−0.0406664	+	0.999173i	$0.512948\pi$
$282$	0	0
$283$	−3282.79	−0.689546	−0.344773	−	0.938686i	$-0.612044\pi$
$283$	−3282.79	−0.689546	−0.344773	+	0.938686i	$0.612044\pi$
$284$	0	0
$285$	224.670	0.0466958
$286$	0	0
$287$	118.354	0.0243422
$288$	0	0
$289$	0	0
$290$	0	0
$291$	2405.90	0.484662
$292$	0	0
$293$	−8764.12	−1.74746	−0.873729	−	0.486413i	$-0.838305\pi$
$293$	−8764.12	−1.74746	−0.873729	+	0.486413i	$0.838305\pi$
$294$	0	0
$295$	−871.943	−0.172090
$296$	0	0
$297$	9834.38	1.92138
$298$	0	0
$299$	−3034.60	−0.586941
$300$	0	0
$301$	−1001.87	−0.191849
$302$	0	0
$303$	−1661.95	−0.315104
$304$	0	0
$305$	207.112	0.0388827
$306$	0	0
$307$	−1580.88	−0.293895	−0.146947	−	0.989144i	$-0.546945\pi$
$307$	−1580.88	−0.293895	−0.146947	+	0.989144i	$0.546945\pi$
$308$	0	0
$309$	−4971.39	−0.915251
$310$	0	0
$311$	−4554.71	−0.830464	−0.415232	−	0.909716i	$-0.636300\pi$
$311$	−4554.71	−0.830464	−0.415232	+	0.909716i	$0.636300\pi$
$312$	0	0
$313$	1351.87	0.244128	0.122064	−	0.992522i	$-0.461049\pi$
$313$	1351.87	0.244128	0.122064	+	0.992522i	$0.461049\pi$
$314$	0	0
$315$	−46.6917	−0.00835168
$316$	0	0
$317$	−2966.66	−0.525628	−0.262814	−	0.964847i	$-0.584651\pi$
$317$	−2966.66	−0.525628	−0.262814	+	0.964847i	$0.584651\pi$
$318$	0	0
$319$	−10635.9	−1.86677
$320$	0	0
$321$	−4224.83	−0.734602
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3652.33	−0.623368
$326$	0	0
$327$	−699.238	−0.118251
$328$	0	0
$329$	1919.78	0.321704
$330$	0	0
$331$	7394.07	1.22784	0.613919	−	0.789369i	$-0.289592\pi$
$331$	7394.07	1.22784	0.613919	+	0.789369i	$0.289592\pi$
$332$	0	0
$333$	17.8399	0.00293580
$334$	0	0
$335$	1458.38	0.237850
$336$	0	0
$337$	−1740.27	−0.281302	−0.140651	−	0.990059i	$-0.544920\pi$
$337$	−1740.27	−0.281302	−0.140651	+	0.990059i	$0.544920\pi$
$338$	0	0
$339$	9699.83	1.55405
$340$	0	0
$341$	−8775.10	−1.39354
$342$	0	0
$343$	−4644.40	−0.731119
$344$	0	0
$345$	922.293	0.143926
$346$	0	0
$347$	1446.27	0.223746	0.111873	−	0.993723i	$-0.464315\pi$
$347$	1446.27	0.223746	0.111873	+	0.993723i	$0.464315\pi$
$348$	0	0
$349$	10823.4	1.66006	0.830032	−	0.557716i	$-0.188322\pi$
$349$	10823.4	1.66006	0.830032	+	0.557716i	$0.188322\pi$
$350$	0	0
$351$	4439.60	0.675123
$352$	0	0
$353$	−8195.32	−1.23567	−0.617837	−	0.786306i	$-0.711991\pi$
$353$	−8195.32	−1.23567	−0.617837	+	0.786306i	$0.711991\pi$
$354$	0	0
$355$	−1704.21	−0.254789
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−627.226	−0.0922109	−0.0461055	−	0.998937i	$-0.514681\pi$
$359$	−627.226	−0.0922109	−0.0461055	+	0.998937i	$0.514681\pi$
$360$	0	0
$361$	−6254.62	−0.911885
$362$	0	0
$363$	−15093.6	−2.18239
$364$	0	0
$365$	1608.70	0.230694
$366$	0	0
$367$	−4464.23	−0.634962	−0.317481	−	0.948265i	$-0.602837\pi$
$367$	−4464.23	−0.634962	−0.317481	+	0.948265i	$0.602837\pi$
$368$	0	0
$369$	−54.4192	−0.00767737
$370$	0	0
$371$	2403.31	0.336317
$372$	0	0
$373$	−13540.6	−1.87964	−0.939820	−	0.341670i	$-0.889008\pi$
$373$	−13540.6	−1.87964	−0.939820	+	0.341670i	$0.889008\pi$
$374$	0	0
$375$	2252.39	0.310167
$376$	0	0
$377$	−4801.45	−0.655935
$378$	0	0
$379$	−8762.89	−1.18765	−0.593825	−	0.804594i	$-0.702383\pi$
$379$	−8762.89	−1.18765	−0.593825	+	0.804594i	$0.702383\pi$
$380$	0	0
$381$	994.634	0.133745
$382$	0	0
$383$	−8698.39	−1.16049	−0.580244	−	0.814443i	$-0.697043\pi$
$383$	−8698.39	−1.16049	−0.580244	+	0.814443i	$0.697043\pi$
$384$	0	0
$385$	920.405	0.121839
$386$	0	0
$387$	460.659	0.0605080
$388$	0	0
$389$	−8977.99	−1.17019	−0.585093	−	0.810967i	$-0.698942\pi$
$389$	−8977.99	−1.17019	−0.585093	+	0.810967i	$0.698942\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	11026.5	1.41530
$394$	0	0
$395$	−930.071	−0.118473
$396$	0	0
$397$	−9307.15	−1.17661	−0.588303	−	0.808641i	$-0.700204\pi$
$397$	−9307.15	−1.17661	−0.588303	+	0.808641i	$0.700204\pi$
$398$	0	0
$399$	878.053	0.110170
$400$	0	0
$401$	1935.06	0.240978	0.120489	−	0.992715i	$-0.461554\pi$
$401$	1935.06	0.240978	0.120489	+	0.992715i	$0.461554\pi$
$402$	0	0
$403$	−3961.40	−0.489656
$404$	0	0
$405$	−1177.76	−0.144502
$406$	0	0
$407$	−351.667	−0.0428292
$408$	0	0
$409$	−5194.56	−0.628006	−0.314003	−	0.949422i	$-0.601670\pi$
$409$	−5194.56	−0.628006	−0.314003	+	0.949422i	$0.601670\pi$
$410$	0	0
$411$	4105.51	0.492725
$412$	0	0
$413$	−3407.72	−0.406012
$414$	0	0
$415$	1023.53	0.121068
$416$	0	0
$417$	4885.05	0.573674
$418$	0	0
$419$	3082.07	0.359353	0.179677	−	0.983726i	$-0.442495\pi$
$419$	3082.07	0.359353	0.179677	+	0.983726i	$0.442495\pi$
$420$	0	0
$421$	10602.0	1.22734	0.613669	−	0.789563i	$-0.289693\pi$
$421$	10602.0	1.22734	0.613669	+	0.789563i	$0.289693\pi$
$422$	0	0
$423$	−882.715	−0.101463
$424$	0	0
$425$	0	0
$426$	0	0
$427$	809.435	0.0917361
$428$	0	0
$429$	−9734.08	−1.09549
$430$	0	0
$431$	−16436.5	−1.83694	−0.918468	−	0.395495i	$-0.870573\pi$
$431$	−16436.5	−1.83694	−0.918468	+	0.395495i	$0.870573\pi$
$432$	0	0
$433$	3111.94	0.345382	0.172691	−	0.984976i	$-0.444754\pi$
$433$	3111.94	0.345382	0.172691	+	0.984976i	$0.444754\pi$
$434$	0	0
$435$	1459.28	0.160844
$436$	0	0
$437$	2481.06	0.271590
$438$	0	0
$439$	11911.3	1.29498	0.647491	−	0.762073i	$-0.275818\pi$
$439$	11911.3	1.29498	0.647491	+	0.762073i	$0.275818\pi$
$440$	0	0
$441$	976.509	0.105443
$442$	0	0
$443$	−13299.1	−1.42632	−0.713160	−	0.701001i	$-0.752737\pi$
$443$	−13299.1	−1.42632	−0.713160	+	0.701001i	$0.752737\pi$
$444$	0	0
$445$	−340.924	−0.0363177
$446$	0	0
$447$	−14079.2	−1.48977
$448$	0	0
$449$	8064.64	0.847648	0.423824	−	0.905745i	$-0.360688\pi$
$449$	8064.64	0.847648	0.423824	+	0.905745i	$0.360688\pi$
$450$	0	0
$451$	1072.73	0.112002
$452$	0	0
$453$	9773.52	1.01369
$454$	0	0
$455$	415.505	0.0428113
$456$	0	0
$457$	−10973.2	−1.12321	−0.561604	−	0.827406i	$-0.689816\pi$
$457$	−10973.2	−1.12321	−0.561604	+	0.827406i	$0.689816\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4613.10	−0.466060	−0.233030	−	0.972470i	$-0.574864\pi$
$461$	−4613.10	−0.466060	−0.233030	+	0.972470i	$0.574864\pi$
$462$	0	0
$463$	6514.21	0.653868	0.326934	−	0.945047i	$-0.393985\pi$
$463$	6514.21	0.653868	0.326934	+	0.945047i	$0.393985\pi$
$464$	0	0
$465$	1203.97	0.120071
$466$	0	0
$467$	−9353.00	−0.926778	−0.463389	−	0.886155i	$-0.653367\pi$
$467$	−9353.00	−0.926778	−0.463389	+	0.886155i	$0.653367\pi$
$468$	0	0
$469$	5699.63	0.561161
$470$	0	0
$471$	−7828.84	−0.765890
$472$	0	0
$473$	−9080.69	−0.882728
$474$	0	0
$475$	2986.10	0.288446
$476$	0	0
$477$	−1105.04	−0.106072
$478$	0	0
$479$	−6821.43	−0.650687	−0.325343	−	0.945596i	$-0.605480\pi$
$479$	−6821.43	−0.650687	−0.325343	+	0.945596i	$0.605480\pi$
$480$	0	0
$481$	−158.755	−0.0150491
$482$	0	0
$483$	3604.50	0.339566
$484$	0	0
$485$	−930.825	−0.0871476
$486$	0	0
$487$	−10265.1	−0.955146	−0.477573	−	0.878592i	$-0.658483\pi$
$487$	−10265.1	−0.955146	−0.477573	+	0.878592i	$0.658483\pi$
$488$	0	0
$489$	9133.02	0.844600
$490$	0	0
$491$	−2334.74	−0.214594	−0.107297	−	0.994227i	$-0.534219\pi$
$491$	−2334.74	−0.214594	−0.107297	+	0.994227i	$0.534219\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−423.203	−0.0384274
$496$	0	0
$497$	−6660.39	−0.601125
$498$	0	0
$499$	1911.97	0.171527	0.0857633	−	0.996316i	$-0.472667\pi$
$499$	1911.97	0.171527	0.0857633	+	0.996316i	$0.472667\pi$
$500$	0	0
$501$	−978.004	−0.0872136
$502$	0	0
$503$	776.502	0.0688320	0.0344160	−	0.999408i	$-0.489043\pi$
$503$	776.502	0.0688320	0.0344160	+	0.999408i	$0.489043\pi$
$504$	0	0
$505$	642.996	0.0566593
$506$	0	0
$507$	6283.42	0.550407
$508$	0	0
$509$	5299.55	0.461490	0.230745	−	0.973014i	$-0.425884\pi$
$509$	5299.55	0.461490	0.230745	+	0.973014i	$0.425884\pi$
$510$	0	0
$511$	6287.11	0.544277
$512$	0	0
$513$	−3629.77	−0.312394
$514$	0	0
$515$	1923.39	0.164572
$516$	0	0
$517$	17400.4	1.48021
$518$	0	0
$519$	16682.8	1.41097
$520$	0	0
$521$	−14675.0	−1.23402	−0.617010	−	0.786955i	$-0.711656\pi$
$521$	−14675.0	−1.23402	−0.617010	+	0.786955i	$0.711656\pi$
$522$	0	0
$523$	−14469.9	−1.20980	−0.604901	−	0.796301i	$-0.706787\pi$
$523$	−14469.9	−1.20980	−0.604901	+	0.796301i	$0.706787\pi$
$524$	0	0
$525$	4338.24	0.360640
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1982.01	−0.162900
$530$	0	0
$531$	1566.87	0.128054
$532$	0	0
$533$	484.271	0.0393548
$534$	0	0
$535$	1634.56	0.132090
$536$	0	0
$537$	−2256.44	−0.181327
$538$	0	0
$539$	−19249.3	−1.53827
$540$	0	0
$541$	−21139.1	−1.67993	−0.839964	−	0.542642i	$-0.817424\pi$
$541$	−21139.1	−1.67993	−0.839964	+	0.542642i	$0.817424\pi$
$542$	0	0
$543$	−8559.34	−0.676458
$544$	0	0
$545$	270.530	0.0212628
$546$	0	0
$547$	−6560.30	−0.512794	−0.256397	−	0.966572i	$-0.582535\pi$
$547$	−6560.30	−0.512794	−0.256397	+	0.966572i	$0.582535\pi$
$548$	0	0
$549$	−372.179	−0.0289330
$550$	0	0
$551$	3925.61	0.303515
$552$	0	0
$553$	−3634.90	−0.279515
$554$	0	0
$555$	48.2499	0.00369026
$556$	0	0
$557$	563.086	0.0428343	0.0214172	−	0.999771i	$-0.493182\pi$
$557$	563.086	0.0428343	0.0214172	+	0.999771i	$0.493182\pi$
$558$	0	0
$559$	−4099.35	−0.310169
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12479.5	0.934187	0.467094	−	0.884208i	$-0.345301\pi$
$563$	12479.5	0.934187	0.467094	+	0.884208i	$0.345301\pi$
$564$	0	0
$565$	−3752.79	−0.279435
$566$	0	0
$567$	−4602.91	−0.340924
$568$	0	0
$569$	14926.7	1.09975	0.549876	−	0.835246i	$-0.314675\pi$
$569$	14926.7	1.09975	0.549876	+	0.835246i	$0.314675\pi$
$570$	0	0
$571$	−13095.6	−0.959780	−0.479890	−	0.877329i	$-0.659323\pi$
$571$	−13095.6	−0.959780	−0.479890	+	0.877329i	$0.659323\pi$
$572$	0	0
$573$	21490.1	1.56678
$574$	0	0
$575$	12258.3	0.889052
$576$	0	0
$577$	11696.4	0.843892	0.421946	−	0.906621i	$-0.361347\pi$
$577$	11696.4	0.843892	0.421946	+	0.906621i	$0.361347\pi$
$578$	0	0
$579$	20088.1	1.44186
$580$	0	0
$581$	4000.17	0.285637
$582$	0	0
$583$	21783.0	1.54745
$584$	0	0
$585$	−191.049	−0.0135024
$586$	0	0
$587$	2786.43	0.195925	0.0979626	−	0.995190i	$-0.468767\pi$
$587$	2786.43	0.195925	0.0979626	+	0.995190i	$0.468767\pi$
$588$	0	0
$589$	3238.80	0.226574
$590$	0	0
$591$	−21249.8	−1.47902
$592$	0	0
$593$	7990.34	0.553328	0.276664	−	0.960967i	$-0.410771\pi$
$593$	7990.34	0.553328	0.276664	+	0.960967i	$0.410771\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24294.5	1.66551
$598$	0	0
$599$	20127.2	1.37291	0.686457	−	0.727171i	$-0.259165\pi$
$599$	20127.2	1.37291	0.686457	+	0.727171i	$0.259165\pi$
$600$	0	0
$601$	24438.0	1.65865	0.829324	−	0.558769i	$-0.188726\pi$
$601$	24438.0	1.65865	0.829324	+	0.558769i	$0.188726\pi$
$602$	0	0
$603$	−2620.70	−0.176987
$604$	0	0
$605$	5839.60	0.392419
$606$	0	0
$607$	−5523.27	−0.369329	−0.184664	−	0.982802i	$-0.559120\pi$
$607$	−5523.27	−0.369329	−0.184664	+	0.982802i	$0.559120\pi$
$608$	0	0
$609$	5703.16	0.379481
$610$	0	0
$611$	7855.18	0.520109
$612$	0	0
$613$	−12039.6	−0.793268	−0.396634	−	0.917977i	$-0.629822\pi$
$613$	−12039.6	−0.793268	−0.396634	+	0.917977i	$0.629822\pi$
$614$	0	0
$615$	−147.182	−0.00965035
$616$	0	0
$617$	13533.5	0.883042	0.441521	−	0.897251i	$-0.354439\pi$
$617$	13533.5	0.883042	0.441521	+	0.897251i	$0.354439\pi$
$618$	0	0
$619$	−28258.5	−1.83490	−0.917452	−	0.397846i	$-0.869758\pi$
$619$	−28258.5	−1.83490	−0.917452	+	0.397846i	$0.869758\pi$
$620$	0	0
$621$	−14900.6	−0.962865
$622$	0	0
$623$	−1332.40	−0.0856844
$624$	0	0
$625$	14311.6	0.915943
$626$	0	0
$627$	7958.48	0.506907
$628$	0	0
$629$	0	0
$630$	0	0
$631$	26358.2	1.66292	0.831459	−	0.555586i	$-0.187506\pi$
$631$	26358.2	1.66292	0.831459	+	0.555586i	$0.187506\pi$
$632$	0	0
$633$	−14785.3	−0.928379
$634$	0	0
$635$	−384.816	−0.0240488
$636$	0	0
$637$	−8689.85	−0.540509
$638$	0	0
$639$	3062.45	0.189591
$640$	0	0
$641$	6020.38	0.370969	0.185484	−	0.982647i	$-0.440615\pi$
$641$	6020.38	0.370969	0.185484	+	0.982647i	$0.440615\pi$
$642$	0	0
$643$	2913.65	0.178698	0.0893492	−	0.996000i	$-0.471521\pi$
$643$	2913.65	0.178698	0.0893492	+	0.996000i	$0.471521\pi$
$644$	0	0
$645$	1245.90	0.0760577
$646$	0	0
$647$	21574.1	1.31092	0.655459	−	0.755231i	$-0.272475\pi$
$647$	21574.1	1.31092	0.655459	+	0.755231i	$0.272475\pi$
$648$	0	0
$649$	−30886.8	−1.86812
$650$	0	0
$651$	4705.35	0.283283
$652$	0	0
$653$	20170.5	1.20878	0.604389	−	0.796690i	$-0.293417\pi$
$653$	20170.5	1.20878	0.604389	+	0.796690i	$0.293417\pi$
$654$	0	0
$655$	−4266.07	−0.254487
$656$	0	0
$657$	−2890.82	−0.171662
$658$	0	0
$659$	21469.2	1.26908	0.634538	−	0.772892i	$-0.281190\pi$
$659$	21469.2	1.26908	0.634538	+	0.772892i	$0.281190\pi$
$660$	0	0
$661$	−5040.07	−0.296575	−0.148287	−	0.988944i	$-0.547376\pi$
$661$	−5040.07	−0.296575	−0.148287	+	0.988944i	$0.547376\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−339.712	−0.0198097
$666$	0	0
$667$	16115.0	0.935498
$668$	0	0
$669$	200.399	0.0115813
$670$	0	0
$671$	7336.54	0.422092
$672$	0	0
$673$	9377.37	0.537104	0.268552	−	0.963265i	$-0.413455\pi$
$673$	9377.37	0.537104	0.268552	+	0.963265i	$0.413455\pi$
$674$	0	0
$675$	−17933.7	−1.02262
$676$	0	0
$677$	−32292.8	−1.83325	−0.916627	−	0.399743i	$-0.869099\pi$
$677$	−32292.8	−1.83325	−0.916627	+	0.399743i	$0.869099\pi$
$678$	0	0
$679$	−3637.84	−0.205608
$680$	0	0
$681$	19884.7	1.11892
$682$	0	0
$683$	21759.6	1.21904	0.609522	−	0.792769i	$-0.291362\pi$
$683$	21759.6	1.21904	0.609522	+	0.792769i	$0.291362\pi$
$684$	0	0
$685$	−1588.39	−0.0885974
$686$	0	0
$687$	−20614.8	−1.14484
$688$	0	0
$689$	9833.66	0.543734
$690$	0	0
$691$	−30906.6	−1.70151	−0.850754	−	0.525564i	$-0.823854\pi$
$691$	−30906.6	−1.70151	−0.850754	+	0.525564i	$0.823854\pi$
$692$	0	0
$693$	−1653.96	−0.0906620
$694$	0	0
$695$	−1889.99	−0.103153
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−25028.3	−1.35430
$700$	0	0
$701$	3500.06	0.188581	0.0942907	−	0.995545i	$-0.469942\pi$
$701$	3500.06	0.188581	0.0942907	+	0.995545i	$0.469942\pi$
$702$	0	0
$703$	129.797	0.00696355
$704$	0	0
$705$	−2387.39	−0.127538
$706$	0	0
$707$	2512.95	0.133677
$708$	0	0
$709$	−5319.84	−0.281792	−0.140896	−	0.990024i	$-0.544998\pi$
$709$	−5319.84	−0.281792	−0.140896	+	0.990024i	$0.544998\pi$
$710$	0	0
$711$	1671.33	0.0881572
$712$	0	0
$713$	13295.6	0.698351
$714$	0	0
$715$	3766.04	0.196982
$716$	0	0
$717$	10987.0	0.572268
$718$	0	0
$719$	−21796.5	−1.13056	−0.565280	−	0.824899i	$-0.691232\pi$
$719$	−21796.5	−1.13056	−0.565280	+	0.824899i	$0.691232\pi$
$720$	0	0
$721$	7516.99	0.388276
$722$	0	0
$723$	−7237.65	−0.372297
$724$	0	0
$725$	19395.4	0.993557
$726$	0	0
$727$	1056.97	0.0539211	0.0269606	−	0.999636i	$-0.491417\pi$
$727$	1056.97	0.0539211	0.0269606	+	0.999636i	$0.491417\pi$
$728$	0	0
$729$	21491.2	1.09186
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−11223.1	−0.565533	−0.282767	−	0.959189i	$-0.591252\pi$
$733$	−11223.1	−0.565533	−0.282767	+	0.959189i	$0.591252\pi$
$734$	0	0
$735$	2641.07	0.132541
$736$	0	0
$737$	51660.2	2.58199
$738$	0	0
$739$	−21079.8	−1.04930	−0.524649	−	0.851318i	$-0.675804\pi$
$739$	−21079.8	−1.04930	−0.524649	+	0.851318i	$0.675804\pi$
$740$	0	0
$741$	3592.75	0.178115
$742$	0	0
$743$	−37943.9	−1.87352	−0.936761	−	0.349971i	$-0.886192\pi$
$743$	−37943.9	−1.87352	−0.936761	+	0.349971i	$0.886192\pi$
$744$	0	0
$745$	5447.15	0.267877
$746$	0	0
$747$	−1839.28	−0.0900881
$748$	0	0
$749$	6388.16	0.311640
$750$	0	0
$751$	−236.001	−0.0114671	−0.00573356	−	0.999984i	$-0.501825\pi$
$751$	−236.001	−0.0114671	−0.00573356	+	0.999984i	$0.501825\pi$
$752$	0	0
$753$	27473.3	1.32959
$754$	0	0
$755$	−3781.30	−0.182272
$756$	0	0
$757$	−20961.3	−1.00641	−0.503203	−	0.864168i	$-0.667845\pi$
$757$	−20961.3	−1.00641	−0.503203	+	0.864168i	$0.667845\pi$
$758$	0	0
$759$	32670.4	1.56240
$760$	0	0
$761$	6389.97	0.304384	0.152192	−	0.988351i	$-0.451367\pi$
$761$	6389.97	0.304384	0.152192	+	0.988351i	$0.451367\pi$
$762$	0	0
$763$	1057.28	0.0501654
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13943.4	−0.656412
$768$	0	0
$769$	−16802.1	−0.787906	−0.393953	−	0.919131i	$-0.628893\pi$
$769$	−16802.1	−0.787906	−0.393953	+	0.919131i	$0.628893\pi$
$770$	0	0
$771$	−37867.3	−1.76882
$772$	0	0
$773$	−23223.2	−1.08057	−0.540286	−	0.841481i	$-0.681684\pi$
$773$	−23223.2	−1.08057	−0.540286	+	0.841481i	$0.681684\pi$
$774$	0	0
$775$	16002.1	0.741692
$776$	0	0
$777$	188.570	0.00870644
$778$	0	0
$779$	−395.934	−0.0182103
$780$	0	0
$781$	−60368.3	−2.76587
$782$	0	0
$783$	−23576.2	−1.07605
$784$	0	0
$785$	3028.92	0.137716
$786$	0	0
$787$	18873.2	0.854838	0.427419	−	0.904054i	$-0.359423\pi$
$787$	18873.2	0.854838	0.427419	+	0.904054i	$0.359423\pi$
$788$	0	0
$789$	6234.41	0.281307
$790$	0	0
$791$	−14666.6	−0.659273
$792$	0	0
$793$	3311.98	0.148313
$794$	0	0
$795$	−2988.70	−0.133331
$796$	0	0
$797$	−25559.3	−1.13596	−0.567979	−	0.823043i	$-0.692274\pi$
$797$	−25559.3	−1.13596	−0.567979	+	0.823043i	$0.692274\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	612.638	0.0270243
$802$	0	0
$803$	56985.0	2.50430
$804$	0	0
$805$	−1394.55	−0.0610578
$806$	0	0
$807$	−39439.1	−1.72035
$808$	0	0
$809$	−8082.03	−0.351235	−0.175617	−	0.984458i	$-0.556192\pi$
$809$	−8082.03	−0.351235	−0.175617	+	0.984458i	$0.556192\pi$
$810$	0	0
$811$	32339.0	1.40022	0.700109	−	0.714036i	$-0.253135\pi$
$811$	32339.0	1.40022	0.700109	+	0.714036i	$0.253135\pi$
$812$	0	0
$813$	−34258.3	−1.47785
$814$	0	0
$815$	−3533.49	−0.151869
$816$	0	0
$817$	3351.59	0.143522
$818$	0	0
$819$	−746.658	−0.0318563
$820$	0	0
$821$	6477.35	0.275348	0.137674	−	0.990478i	$-0.456037\pi$
$821$	6477.35	0.275348	0.137674	+	0.990478i	$0.456037\pi$
$822$	0	0
$823$	15924.3	0.674468	0.337234	−	0.941421i	$-0.390509\pi$
$823$	15924.3	0.674468	0.337234	+	0.941421i	$0.390509\pi$
$824$	0	0
$825$	39320.8	1.65936
$826$	0	0
$827$	9919.77	0.417103	0.208551	−	0.978011i	$-0.433125\pi$
$827$	9919.77	0.417103	0.208551	+	0.978011i	$0.433125\pi$
$828$	0	0
$829$	39930.2	1.67290	0.836450	−	0.548043i	$-0.184627\pi$
$829$	39930.2	1.67290	0.836450	+	0.548043i	$0.184627\pi$
$830$	0	0
$831$	35670.5	1.48904
$832$	0	0
$833$	0	0
$834$	0	0
$835$	378.382	0.0156820
$836$	0	0
$837$	−19451.4	−0.803270
$838$	0	0
$839$	11055.6	0.454926	0.227463	−	0.973787i	$-0.426957\pi$
$839$	11055.6	0.454926	0.227463	+	0.973787i	$0.426957\pi$
$840$	0	0
$841$	1108.79	0.0454626
$842$	0	0
$843$	1861.98	0.0760734
$844$	0	0
$845$	−2431.01	−0.0989694
$846$	0	0
$847$	22822.3	0.925835
$848$	0	0
$849$	15954.8	0.644956
$850$	0	0
$851$	532.829	0.0214631
$852$	0	0
$853$	44703.2	1.79438	0.897192	−	0.441640i	$-0.145603\pi$
$853$	44703.2	1.79438	0.897192	+	0.441640i	$0.145603\pi$
$854$	0	0
$855$	156.200	0.00624786
$856$	0	0
$857$	−13065.5	−0.520782	−0.260391	−	0.965503i	$-0.583852\pi$
$857$	−13065.5	−0.520782	−0.260391	+	0.965503i	$0.583852\pi$
$858$	0	0
$859$	−7524.54	−0.298875	−0.149438	−	0.988771i	$-0.547746\pi$
$859$	−7524.54	−0.298875	−0.149438	+	0.988771i	$0.547746\pi$
$860$	0	0
$861$	−575.217	−0.0227681
$862$	0	0
$863$	−11000.3	−0.433899	−0.216950	−	0.976183i	$-0.569611\pi$
$863$	−11000.3	−0.433899	−0.216950	+	0.976183i	$0.569611\pi$
$864$	0	0
$865$	−6454.43	−0.253708
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−32945.9	−1.28609
$870$	0	0
$871$	23321.3	0.907247
$872$	0	0
$873$	1672.68	0.0648474
$874$	0	0
$875$	−3405.72	−0.131582
$876$	0	0
$877$	−4521.45	−0.174092	−0.0870458	−	0.996204i	$-0.527743\pi$
$877$	−4521.45	−0.174092	−0.0870458	+	0.996204i	$0.527743\pi$
$878$	0	0
$879$	42594.9	1.63446
$880$	0	0
$881$	14154.9	0.541306	0.270653	−	0.962677i	$-0.412760\pi$
$881$	14154.9	0.541306	0.270653	+	0.962677i	$0.412760\pi$
$882$	0	0
$883$	−10737.6	−0.409229	−0.204615	−	0.978843i	$-0.565594\pi$
$883$	−10737.6	−0.409229	−0.204615	+	0.978843i	$0.565594\pi$
$884$	0	0
$885$	4237.77	0.160962
$886$	0	0
$887$	29366.5	1.11165	0.555823	−	0.831301i	$-0.312403\pi$
$887$	29366.5	1.11165	0.555823	+	0.831301i	$0.312403\pi$
$888$	0	0
$889$	−1503.94	−0.0567384
$890$	0	0
$891$	−41719.7	−1.56865
$892$	0	0
$893$	−6422.31	−0.240666
$894$	0	0
$895$	872.998	0.0326046
$896$	0	0
$897$	14748.6	0.548987
$898$	0	0
$899$	21036.8	0.780439
$900$	0	0
$901$	0	0
$902$	0	0
$903$	4869.21	0.179443
$904$	0	0
$905$	3311.54	0.121635
$906$	0	0
$907$	−6477.17	−0.237123	−0.118562	−	0.992947i	$-0.537828\pi$
$907$	−6477.17	−0.237123	−0.118562	+	0.992947i	$0.537828\pi$
$908$	0	0
$909$	−1155.46	−0.0421608
$910$	0	0
$911$	−7714.04	−0.280546	−0.140273	−	0.990113i	$-0.544798\pi$
$911$	−7714.04	−0.280546	−0.140273	+	0.990113i	$0.544798\pi$
$912$	0	0
$913$	36256.6	1.31426
$914$	0	0
$915$	−1006.60	−0.0363684
$916$	0	0
$917$	−16672.6	−0.600413
$918$	0	0
$919$	36977.2	1.32727	0.663636	−	0.748055i	$-0.269012\pi$
$919$	36977.2	1.32727	0.663636	+	0.748055i	$0.269012\pi$
$920$	0	0
$921$	7683.32	0.274890
$922$	0	0
$923$	−27252.5	−0.971859
$924$	0	0
$925$	641.292	0.0227952
$926$	0	0
$927$	−3456.32	−0.122460
$928$	0	0
$929$	−4302.06	−0.151933	−0.0759666	−	0.997110i	$-0.524204\pi$
$929$	−4302.06	−0.151933	−0.0759666	+	0.997110i	$0.524204\pi$
$930$	0	0
$931$	7104.72	0.250105
$932$	0	0
$933$	22136.6	0.776762
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42979.7	−1.49849	−0.749245	−	0.662293i	$-0.769584\pi$
$937$	−42979.7	−1.49849	−0.749245	+	0.662293i	$0.769584\pi$
$938$	0	0
$939$	−6570.27	−0.228342
$940$	0	0
$941$	38762.6	1.34285	0.671427	−	0.741071i	$-0.265682\pi$
$941$	38762.6	1.34285	0.671427	+	0.741071i	$0.265682\pi$
$942$	0	0
$943$	−1625.35	−0.0561280
$944$	0	0
$945$	2040.22	0.0702311
$946$	0	0
$947$	−25527.9	−0.875971	−0.437986	−	0.898982i	$-0.644308\pi$
$947$	−25527.9	−0.875971	−0.437986	+	0.898982i	$0.644308\pi$
$948$	0	0
$949$	25725.1	0.879950
$950$	0	0
$951$	14418.4	0.491639
$952$	0	0
$953$	−8804.92	−0.299286	−0.149643	−	0.988740i	$-0.547812\pi$
$953$	−8804.92	−0.299286	−0.149643	+	0.988740i	$0.547812\pi$
$954$	0	0
$955$	−8314.36	−0.281724
$956$	0	0
$957$	51692.2	1.74605
$958$	0	0
$959$	−6207.73	−0.209028
$960$	0	0
$961$	−12434.8	−0.417401
$962$	0	0
$963$	−2937.28	−0.0982892
$964$	0	0
$965$	−7771.95	−0.259262
$966$	0	0
$967$	50613.0	1.68315	0.841574	−	0.540142i	$-0.181629\pi$
$967$	50613.0	1.68315	0.841574	+	0.540142i	$0.181629\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16260.6	−0.537412	−0.268706	−	0.963222i	$-0.586596\pi$
$971$	−16260.6	−0.537412	−0.268706	+	0.963222i	$0.586596\pi$
$972$	0	0
$973$	−7386.44	−0.243369
$974$	0	0
$975$	17750.9	0.583059
$976$	0	0
$977$	56489.7	1.84981	0.924906	−	0.380195i	$-0.124143\pi$
$977$	56489.7	1.84981	0.924906	+	0.380195i	$0.124143\pi$
$978$	0	0
$979$	−12076.6	−0.394248
$980$	0	0
$981$	−486.139	−0.0158218
$982$	0	0
$983$	−3279.27	−0.106401	−0.0532006	−	0.998584i	$-0.516942\pi$
$983$	−3279.27	−0.106401	−0.0532006	+	0.998584i	$0.516942\pi$
$984$	0	0
$985$	8221.37	0.265944
$986$	0	0
$987$	−9330.39	−0.300901
$988$	0	0
$989$	13758.6	0.442364
$990$	0	0
$991$	−6329.45	−0.202888	−0.101444	−	0.994841i	$-0.532346\pi$
$991$	−6329.45	−0.202888	−0.101444	+	0.994841i	$0.532346\pi$
$992$	0	0
$993$	−35936.2	−1.14844
$994$	0	0
$995$	−9399.36	−0.299477
$996$	0	0
$997$	−25839.5	−0.820809	−0.410404	−	0.911904i	$-0.634612\pi$
$997$	−25839.5	−0.820809	−0.410404	+	0.911904i	$0.634612\pi$
$998$	0	0
$999$	−779.525	−0.0246878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.n.1.5	✓	18
17.16	even	2		2312.4.a.q.1.14	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.n.1.5	✓	18	1.1	even	1	trivial
2312.4.a.q.1.14	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)

\(f(q)\)	\(=\)	\(q-4.86015 q^{3} +1.88035 q^{5} +7.34878 q^{7} -3.37898 q^{9} +O(q^{10})\)	\(q-4.86015 q^{3} +1.88035 q^{5} +7.34878 q^{7} -3.37898 q^{9} +66.6077 q^{11} +30.0692 q^{13} -9.13879 q^{15} -24.5842 q^{19} -35.7162 q^{21} -100.921 q^{23} -121.464 q^{25} +147.646 q^{27} -159.680 q^{29} -131.743 q^{31} -323.723 q^{33} +13.8183 q^{35} -5.27968 q^{37} -146.141 q^{39} +16.1052 q^{41} -136.331 q^{43} -6.35367 q^{45} +261.237 q^{47} -288.995 q^{49} +327.035 q^{53} +125.246 q^{55} +119.483 q^{57} -463.712 q^{59} +110.145 q^{61} -24.8314 q^{63} +56.5406 q^{65} +775.589 q^{67} +490.489 q^{69} -906.326 q^{71} +855.531 q^{73} +590.334 q^{75} +489.485 q^{77} -494.626 q^{79} -626.350 q^{81} +544.331 q^{83} +776.069 q^{87} -181.309 q^{89} +220.972 q^{91} +640.290 q^{93} -46.2270 q^{95} -495.027 q^{97} -225.066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) \) 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) \) 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9 - 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 - 153 * q^23 + 306 * q^25 - 768 * q^27 - 51 * q^29 - 303 * q^31 + 525 * q^33 - 255 * q^35 - 717 * q^37 + 216 * q^39 + 393 * q^41 - 390 * q^43 - 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 - 810 * q^57 - 204 * q^59 - 534 * q^61 - 2556 * q^63 + 2127 * q^65 - 405 * q^67 + 2547 * q^69 + 426 * q^71 - 1149 * q^73 - 2226 * q^75 - 357 * q^77 - 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 - 6090 * q^91 + 606 * q^93 - 2109 * q^95 - 2349 * q^97 - 1428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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