LMFDB - Embedded newform 2312.4.a.q.1.3

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:	$18$
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			1.3
Root			$8.25076$ 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)

$f(q)$	$=$	$q-7.25076 q^{3} +0.749836 q^{5} -4.82634 q^{7} +25.5735 q^{9} +O(q^{10})$	$q-7.25076 q^{3} +0.749836 q^{5} -4.82634 q^{7} +25.5735 q^{9} -15.7900 q^{11} -47.6986 q^{13} -5.43688 q^{15} -41.6490 q^{19} +34.9947 q^{21} +91.0571 q^{23} -124.438 q^{25} +10.3431 q^{27} -37.6863 q^{29} -164.387 q^{31} +114.489 q^{33} -3.61897 q^{35} -107.681 q^{37} +345.851 q^{39} +61.2288 q^{41} -128.178 q^{43} +19.1759 q^{45} -286.294 q^{47} -319.706 q^{49} -496.225 q^{53} -11.8399 q^{55} +301.987 q^{57} +629.230 q^{59} +352.869 q^{61} -123.427 q^{63} -35.7661 q^{65} -252.213 q^{67} -660.233 q^{69} +270.531 q^{71} +311.320 q^{73} +902.268 q^{75} +76.2078 q^{77} -643.664 q^{79} -765.480 q^{81} +118.499 q^{83} +273.255 q^{87} -1371.14 q^{89} +230.210 q^{91} +1191.93 q^{93} -31.2299 q^{95} -1423.09 q^{97} -403.805 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.25076	−1.39541	−0.697705	−	0.716386i	$-0.745795\pi$
$3$	−7.25076	−1.39541	−0.697705	+	0.716386i	$0.745795\pi$
$4$	0	0
$5$	0.749836	0.0670674	0.0335337	−	0.999438i	$-0.489324\pi$
$5$	0.749836	0.0670674	0.0335337	+	0.999438i	$0.489324\pi$
$6$	0	0
$7$	−4.82634	−0.260598	−0.130299	−	0.991475i	$-0.541594\pi$
$7$	−4.82634	−0.260598	−0.130299	+	0.991475i	$0.541594\pi$
$8$	0	0
$9$	25.5735	0.947167
$10$	0	0
$11$	−15.7900	−0.432805	−0.216402	−	0.976304i	$-0.569432\pi$
$11$	−15.7900	−0.432805	−0.216402	+	0.976304i	$0.569432\pi$
$12$	0	0
$13$	−47.6986	−1.01763	−0.508816	−	0.860875i	$-0.669917\pi$
$13$	−47.6986	−1.01763	−0.508816	+	0.860875i	$0.669917\pi$
$14$	0	0
$15$	−5.43688	−0.0935864
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−41.6490	−0.502891	−0.251446	−	0.967871i	$-0.580906\pi$
$19$	−41.6490	−0.502891	−0.251446	+	0.967871i	$0.580906\pi$
$20$	0	0
$21$	34.9947	0.363641
$22$	0	0
$23$	91.0571	0.825510	0.412755	−	0.910842i	$-0.364567\pi$
$23$	91.0571	0.825510	0.412755	+	0.910842i	$0.364567\pi$
$24$	0	0
$25$	−124.438	−0.995502
$26$	0	0
$27$	10.3431	0.0737236
$28$	0	0
$29$	−37.6863	−0.241316	−0.120658	−	0.992694i	$-0.538501\pi$
$29$	−37.6863	−0.241316	−0.120658	+	0.992694i	$0.538501\pi$
$30$	0	0
$31$	−164.387	−0.952415	−0.476207	−	0.879333i	$-0.657989\pi$
$31$	−164.387	−0.952415	−0.476207	+	0.879333i	$0.657989\pi$
$32$	0	0
$33$	114.489	0.603940
$34$	0	0
$35$	−3.61897	−0.0174776
$36$	0	0
$37$	−107.681	−0.478450	−0.239225	−	0.970964i	$-0.576893\pi$
$37$	−107.681	−0.478450	−0.239225	+	0.970964i	$0.576893\pi$
$38$	0	0
$39$	345.851	1.42001
$40$	0	0
$41$	61.2288	0.233227	0.116614	−	0.993177i	$-0.462796\pi$
$41$	61.2288	0.233227	0.116614	+	0.993177i	$0.462796\pi$
$42$	0	0
$43$	−128.178	−0.454580	−0.227290	−	0.973827i	$-0.572986\pi$
$43$	−128.178	−0.454580	−0.227290	+	0.973827i	$0.572986\pi$
$44$	0	0
$45$	19.1759	0.0635240
$46$	0	0
$47$	−286.294	−0.888518	−0.444259	−	0.895898i	$-0.646533\pi$
$47$	−286.294	−0.888518	−0.444259	+	0.895898i	$0.646533\pi$
$48$	0	0
$49$	−319.706	−0.932089
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−496.225	−1.28607	−0.643035	−	0.765836i	$-0.722325\pi$
$53$	−496.225	−1.28607	−0.643035	+	0.765836i	$0.722325\pi$
$54$	0	0
$55$	−11.8399	−0.0290271
$56$	0	0
$57$	301.987	0.701739
$58$	0	0
$59$	629.230	1.38845	0.694227	−	0.719756i	$-0.255746\pi$
$59$	629.230	1.38845	0.694227	+	0.719756i	$0.255746\pi$
$60$	0	0
$61$	352.869	0.740660	0.370330	−	0.928900i	$-0.379245\pi$
$61$	352.869	0.740660	0.370330	+	0.928900i	$0.379245\pi$
$62$	0	0
$63$	−123.427	−0.246830
$64$	0	0
$65$	−35.7661	−0.0682499
$66$	0	0
$67$	−252.213	−0.459892	−0.229946	−	0.973203i	$-0.573855\pi$
$67$	−252.213	−0.459892	−0.229946	+	0.973203i	$0.573855\pi$
$68$	0	0
$69$	−660.233	−1.15192
$70$	0	0
$71$	270.531	0.452199	0.226099	−	0.974104i	$-0.427403\pi$
$71$	270.531	0.452199	0.226099	+	0.974104i	$0.427403\pi$
$72$	0	0
$73$	311.320	0.499140	0.249570	−	0.968357i	$-0.419711\pi$
$73$	311.320	0.499140	0.249570	+	0.968357i	$0.419711\pi$
$74$	0	0
$75$	902.268	1.38913
$76$	0	0
$77$	76.2078	0.112788
$78$	0	0
$79$	−643.664	−0.916682	−0.458341	−	0.888776i	$-0.651556\pi$
$79$	−643.664	−0.916682	−0.458341	+	0.888776i	$0.651556\pi$
$80$	0	0
$81$	−765.480	−1.05004
$82$	0	0
$83$	118.499	0.156711	0.0783553	−	0.996925i	$-0.475033\pi$
$83$	118.499	0.156711	0.0783553	+	0.996925i	$0.475033\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	273.255	0.336735
$88$	0	0
$89$	−1371.14	−1.63304	−0.816520	−	0.577317i	$-0.804100\pi$
$89$	−1371.14	−1.63304	−0.816520	+	0.577317i	$0.804100\pi$
$90$	0	0
$91$	230.210	0.265193
$92$	0	0
$93$	1191.93	1.32901
$94$	0	0
$95$	−31.2299	−0.0337276
$96$	0	0
$97$	−1423.09	−1.48962	−0.744808	−	0.667279i	$-0.767459\pi$
$97$	−1423.09	−1.48962	−0.744808	+	0.667279i	$0.767459\pi$
$98$	0	0
$99$	−403.805	−0.409938
$100$	0	0
$101$	632.330	0.622962	0.311481	−	0.950252i	$-0.399175\pi$
$101$	632.330	0.622962	0.311481	+	0.950252i	$0.399175\pi$
$102$	0	0
$103$	−1563.36	−1.49556	−0.747781	−	0.663946i	$-0.768881\pi$
$103$	−1563.36	−1.49556	−0.747781	+	0.663946i	$0.768881\pi$
$104$	0	0
$105$	26.2403	0.0243884
$106$	0	0
$107$	1655.39	1.49563	0.747816	−	0.663906i	$-0.231102\pi$
$107$	1655.39	1.49563	0.747816	+	0.663906i	$0.231102\pi$
$108$	0	0
$109$	−989.106	−0.869166	−0.434583	−	0.900632i	$-0.643104\pi$
$109$	−989.106	−0.869166	−0.434583	+	0.900632i	$0.643104\pi$
$110$	0	0
$111$	780.769	0.667633
$112$	0	0
$113$	−356.633	−0.296896	−0.148448	−	0.988920i	$-0.547428\pi$
$113$	−356.633	−0.296896	−0.148448	+	0.988920i	$0.547428\pi$
$114$	0	0
$115$	68.2779	0.0553648
$116$	0	0
$117$	−1219.82	−0.963867
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1081.68	−0.812680
$122$	0	0
$123$	−443.955	−0.325448
$124$	0	0
$125$	−187.037	−0.133833
$126$	0	0
$127$	2156.46	1.50673	0.753366	−	0.657602i	$-0.228429\pi$
$127$	2156.46	1.50673	0.753366	+	0.657602i	$0.228429\pi$
$128$	0	0
$129$	929.386	0.634325
$130$	0	0
$131$	−1130.26	−0.753824	−0.376912	−	0.926249i	$-0.623014\pi$
$131$	−1130.26	−0.753824	−0.376912	+	0.926249i	$0.623014\pi$
$132$	0	0
$133$	201.012	0.131052
$134$	0	0
$135$	7.75566	0.00494445
$136$	0	0
$137$	−1429.70	−0.891590	−0.445795	−	0.895135i	$-0.647079\pi$
$137$	−1429.70	−0.891590	−0.445795	+	0.895135i	$0.647079\pi$
$138$	0	0
$139$	−981.499	−0.598918	−0.299459	−	0.954109i	$-0.596806\pi$
$139$	−981.499	−0.598918	−0.299459	+	0.954109i	$0.596806\pi$
$140$	0	0
$141$	2075.85	1.23985
$142$	0	0
$143$	753.159	0.440436
$144$	0	0
$145$	−28.2586	−0.0161845
$146$	0	0
$147$	2318.11	1.30065
$148$	0	0
$149$	−1030.25	−0.566452	−0.283226	−	0.959053i	$-0.591405\pi$
$149$	−1030.25	−0.566452	−0.283226	+	0.959053i	$0.591405\pi$
$150$	0	0
$151$	−2450.01	−1.32039	−0.660195	−	0.751095i	$-0.729526\pi$
$151$	−2450.01	−1.32039	−0.660195	+	0.751095i	$0.729526\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−123.264	−0.0638760
$156$	0	0
$157$	−3245.36	−1.64973	−0.824865	−	0.565330i	$-0.808749\pi$
$157$	−3245.36	−1.64973	−0.824865	+	0.565330i	$0.808749\pi$
$158$	0	0
$159$	3598.01	1.79460
$160$	0	0
$161$	−439.473	−0.215126
$162$	0	0
$163$	−1040.32	−0.499903	−0.249952	−	0.968258i	$-0.580415\pi$
$163$	−1040.32	−0.499903	−0.249952	+	0.968258i	$0.580415\pi$
$164$	0	0
$165$	85.8481	0.0405046
$166$	0	0
$167$	−3151.24	−1.46018	−0.730090	−	0.683351i	$-0.760522\pi$
$167$	−3151.24	−1.46018	−0.730090	+	0.683351i	$0.760522\pi$
$168$	0	0
$169$	78.1563	0.0355741
$170$	0	0
$171$	−1065.11	−0.476322
$172$	0	0
$173$	2214.52	0.973221	0.486610	−	0.873619i	$-0.338233\pi$
$173$	2214.52	0.973221	0.486610	+	0.873619i	$0.338233\pi$
$174$	0	0
$175$	600.579	0.259426
$176$	0	0
$177$	−4562.40	−1.93746
$178$	0	0
$179$	2845.75	1.18827	0.594137	−	0.804364i	$-0.297494\pi$
$179$	2845.75	1.18827	0.594137	+	0.804364i	$0.297494\pi$
$180$	0	0
$181$	−1917.31	−0.787361	−0.393681	−	0.919247i	$-0.628798\pi$
$181$	−1917.31	−0.787361	−0.393681	+	0.919247i	$0.628798\pi$
$182$	0	0
$183$	−2558.57	−1.03352
$184$	0	0
$185$	−80.7431	−0.0320884
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−49.9195	−0.0192122
$190$	0	0
$191$	3497.17	1.32485	0.662424	−	0.749129i	$-0.269528\pi$
$191$	3497.17	1.32485	0.662424	+	0.749129i	$0.269528\pi$
$192$	0	0
$193$	1637.05	0.610558	0.305279	−	0.952263i	$-0.401250\pi$
$193$	1637.05	0.610558	0.305279	+	0.952263i	$0.401250\pi$
$194$	0	0
$195$	259.332	0.0952365
$196$	0	0
$197$	2186.93	0.790925	0.395462	−	0.918482i	$-0.370584\pi$
$197$	2186.93	0.790925	0.395462	+	0.918482i	$0.370584\pi$
$198$	0	0
$199$	−62.8290	−0.0223811	−0.0111905	−	0.999937i	$-0.503562\pi$
$199$	−62.8290	−0.0223811	−0.0111905	+	0.999937i	$0.503562\pi$
$200$	0	0
$201$	1828.74	0.641737
$202$	0	0
$203$	181.887	0.0628866
$204$	0	0
$205$	45.9115	0.0156420
$206$	0	0
$207$	2328.65	0.781896
$208$	0	0
$209$	657.635	0.217654
$210$	0	0
$211$	1645.33	0.536819	0.268410	−	0.963305i	$-0.413502\pi$
$211$	1645.33	0.536819	0.268410	+	0.963305i	$0.413502\pi$
$212$	0	0
$213$	−1961.55	−0.631002
$214$	0	0
$215$	−96.1124	−0.0304875
$216$	0	0
$217$	793.390	0.248197
$218$	0	0
$219$	−2257.30	−0.696504
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5883.42	1.76674	0.883370	−	0.468676i	$-0.155269\pi$
$223$	5883.42	1.76674	0.883370	+	0.468676i	$0.155269\pi$
$224$	0	0
$225$	−3182.31	−0.942907
$226$	0	0
$227$	6418.29	1.87664	0.938319	−	0.345772i	$-0.112383\pi$
$227$	6418.29	1.87664	0.938319	+	0.345772i	$0.112383\pi$
$228$	0	0
$229$	2565.51	0.740322	0.370161	−	0.928968i	$-0.379303\pi$
$229$	2565.51	0.740322	0.370161	+	0.928968i	$0.379303\pi$
$230$	0	0
$231$	−552.564	−0.157385
$232$	0	0
$233$	5381.66	1.51315	0.756576	−	0.653906i	$-0.226871\pi$
$233$	5381.66	1.51315	0.756576	+	0.653906i	$0.226871\pi$
$234$	0	0
$235$	−214.674	−0.0595906
$236$	0	0
$237$	4667.06	1.27915
$238$	0	0
$239$	1448.08	0.391919	0.195959	−	0.980612i	$-0.437218\pi$
$239$	1448.08	0.391919	0.195959	+	0.980612i	$0.437218\pi$
$240$	0	0
$241$	4480.61	1.19760	0.598800	−	0.800899i	$-0.295644\pi$
$241$	4480.61	1.19760	0.598800	+	0.800899i	$0.295644\pi$
$242$	0	0
$243$	5271.05	1.39151
$244$	0	0
$245$	−239.727	−0.0625127
$246$	0	0
$247$	1986.60	0.511758
$248$	0	0
$249$	−859.209	−0.218675
$250$	0	0
$251$	−2625.27	−0.660181	−0.330090	−	0.943949i	$-0.607079\pi$
$251$	−2625.27	−0.660181	−0.330090	+	0.943949i	$0.607079\pi$
$252$	0	0
$253$	−1437.79	−0.357284
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3208.86	0.778845	0.389422	−	0.921059i	$-0.372675\pi$
$257$	3208.86	0.778845	0.389422	+	0.921059i	$0.372675\pi$
$258$	0	0
$259$	519.705	0.124683
$260$	0	0
$261$	−963.772	−0.228567
$262$	0	0
$263$	−2402.24	−0.563226	−0.281613	−	0.959528i	$-0.590869\pi$
$263$	−2402.24	−0.563226	−0.281613	+	0.959528i	$0.590869\pi$
$264$	0	0
$265$	−372.088	−0.0862534
$266$	0	0
$267$	9941.80	2.27876
$268$	0	0
$269$	2719.47	0.616390	0.308195	−	0.951323i	$-0.400275\pi$
$269$	2719.47	0.616390	0.308195	+	0.951323i	$0.400275\pi$
$270$	0	0
$271$	−1580.07	−0.354179	−0.177090	−	0.984195i	$-0.556668\pi$
$271$	−1580.07	−0.354179	−0.177090	+	0.984195i	$0.556668\pi$
$272$	0	0
$273$	−1669.20	−0.370053
$274$	0	0
$275$	1964.87	0.430858
$276$	0	0
$277$	−4371.59	−0.948243	−0.474121	−	0.880459i	$-0.657234\pi$
$277$	−4371.59	−0.948243	−0.474121	+	0.880459i	$0.657234\pi$
$278$	0	0
$279$	−4203.96	−0.902096
$280$	0	0
$281$	−2297.86	−0.487824	−0.243912	−	0.969797i	$-0.578431\pi$
$281$	−2297.86	−0.487824	−0.243912	+	0.969797i	$0.578431\pi$
$282$	0	0
$283$	7004.56	1.47130	0.735650	−	0.677361i	$-0.236877\pi$
$283$	7004.56	1.47130	0.735650	+	0.677361i	$0.236877\pi$
$284$	0	0
$285$	226.441	0.0470638
$286$	0	0
$287$	−295.511	−0.0607786
$288$	0	0
$289$	0	0
$290$	0	0
$291$	10318.5	2.07862
$292$	0	0
$293$	−6789.83	−1.35381	−0.676905	−	0.736071i	$-0.736679\pi$
$293$	−6789.83	−1.35381	−0.676905	+	0.736071i	$0.736679\pi$
$294$	0	0
$295$	471.819	0.0931200
$296$	0	0
$297$	−163.318	−0.0319079
$298$	0	0
$299$	−4343.30	−0.840065
$300$	0	0
$301$	618.630	0.118463
$302$	0	0
$303$	−4584.87	−0.869287
$304$	0	0
$305$	264.594	0.0496741
$306$	0	0
$307$	6496.82	1.20779	0.603897	−	0.797062i	$-0.293614\pi$
$307$	6496.82	1.20779	0.603897	+	0.797062i	$0.293614\pi$
$308$	0	0
$309$	11335.6	2.08692
$310$	0	0
$311$	−2685.41	−0.489633	−0.244817	−	0.969569i	$-0.578728\pi$
$311$	−2685.41	−0.489633	−0.244817	+	0.969569i	$0.578728\pi$
$312$	0	0
$313$	−8605.55	−1.55404	−0.777020	−	0.629476i	$-0.783270\pi$
$313$	−8605.55	−1.55404	−0.777020	+	0.629476i	$0.783270\pi$
$314$	0	0
$315$	−92.5497	−0.0165542
$316$	0	0
$317$	3522.57	0.624125	0.312062	−	0.950062i	$-0.398980\pi$
$317$	3522.57	0.624125	0.312062	+	0.950062i	$0.398980\pi$
$318$	0	0
$319$	595.066	0.104443
$320$	0	0
$321$	−12002.8	−2.08702
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5935.51	1.01305
$326$	0	0
$327$	7171.77	1.21284
$328$	0	0
$329$	1381.76	0.231546
$330$	0	0
$331$	1471.27	0.244314	0.122157	−	0.992511i	$-0.461019\pi$
$331$	1471.27	0.244314	0.122157	+	0.992511i	$0.461019\pi$
$332$	0	0
$333$	−2753.78	−0.453172
$334$	0	0
$335$	−189.119	−0.0308437
$336$	0	0
$337$	11393.7	1.84170	0.920852	−	0.389913i	$-0.127495\pi$
$337$	11393.7	1.84170	0.920852	+	0.389913i	$0.127495\pi$
$338$	0	0
$339$	2585.86	0.414291
$340$	0	0
$341$	2595.67	0.412210
$342$	0	0
$343$	3198.45	0.503499
$344$	0	0
$345$	−495.067	−0.0772565
$346$	0	0
$347$	−5524.14	−0.854615	−0.427307	−	0.904106i	$-0.640538\pi$
$347$	−5524.14	−0.854615	−0.427307	+	0.904106i	$0.640538\pi$
$348$	0	0
$349$	11105.2	1.70328	0.851642	−	0.524124i	$-0.175607\pi$
$349$	11105.2	1.70328	0.851642	+	0.524124i	$0.175607\pi$
$350$	0	0
$351$	−493.353	−0.0750235
$352$	0	0
$353$	7049.91	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$353$	7049.91	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$354$	0	0
$355$	202.854	0.0303278
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10211.5	1.50123	0.750614	−	0.660741i	$-0.229758\pi$
$359$	10211.5	1.50123	0.750614	+	0.660741i	$0.229758\pi$
$360$	0	0
$361$	−5124.36	−0.747101
$362$	0	0
$363$	7842.98	1.13402
$364$	0	0
$365$	233.439	0.0334760
$366$	0	0
$367$	−729.522	−0.103762	−0.0518811	−	0.998653i	$-0.516522\pi$
$367$	−729.522	−0.103762	−0.0518811	+	0.998653i	$0.516522\pi$
$368$	0	0
$369$	1565.83	0.220905
$370$	0	0
$371$	2394.95	0.335148
$372$	0	0
$373$	3828.73	0.531486	0.265743	−	0.964044i	$-0.414383\pi$
$373$	3828.73	0.531486	0.265743	+	0.964044i	$0.414383\pi$
$374$	0	0
$375$	1356.16	0.186752
$376$	0	0
$377$	1797.59	0.245571
$378$	0	0
$379$	602.457	0.0816520	0.0408260	−	0.999166i	$-0.487001\pi$
$379$	602.457	0.0816520	0.0408260	+	0.999166i	$0.487001\pi$
$380$	0	0
$381$	−15636.0	−2.10251
$382$	0	0
$383$	459.622	0.0613201	0.0306600	−	0.999530i	$-0.490239\pi$
$383$	459.622	0.0613201	0.0306600	+	0.999530i	$0.490239\pi$
$384$	0	0
$385$	57.1433	0.00756440
$386$	0	0
$387$	−3277.96	−0.430563
$388$	0	0
$389$	−1050.03	−0.136860	−0.0684300	−	0.997656i	$-0.521799\pi$
$389$	−1050.03	−0.136860	−0.0684300	+	0.997656i	$0.521799\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	8195.22	1.05189
$394$	0	0
$395$	−482.643	−0.0614795
$396$	0	0
$397$	−5069.11	−0.640835	−0.320417	−	0.947276i	$-0.603823\pi$
$397$	−5069.11	−0.640835	−0.320417	+	0.947276i	$0.603823\pi$
$398$	0	0
$399$	−1457.49	−0.182872
$400$	0	0
$401$	−5546.31	−0.690697	−0.345348	−	0.938475i	$-0.612239\pi$
$401$	−5546.31	−0.690697	−0.345348	+	0.938475i	$0.612239\pi$
$402$	0	0
$403$	7841.05	0.969208
$404$	0	0
$405$	−573.985	−0.0704235
$406$	0	0
$407$	1700.28	0.207075
$408$	0	0
$409$	4819.31	0.582639	0.291319	−	0.956626i	$-0.405906\pi$
$409$	4819.31	0.582639	0.291319	+	0.956626i	$0.405906\pi$
$410$	0	0
$411$	10366.4	1.24413
$412$	0	0
$413$	−3036.88	−0.361828
$414$	0	0
$415$	88.8550	0.0105102
$416$	0	0
$417$	7116.61	0.835736
$418$	0	0
$419$	422.850	0.0493020	0.0246510	−	0.999696i	$-0.492153\pi$
$419$	422.850	0.0493020	0.0246510	+	0.999696i	$0.492153\pi$
$420$	0	0
$421$	2184.13	0.252845	0.126423	−	0.991976i	$-0.459650\pi$
$421$	2184.13	0.252845	0.126423	+	0.991976i	$0.459650\pi$
$422$	0	0
$423$	−7321.56	−0.841575
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1703.07	−0.193015
$428$	0	0
$429$	−5460.97	−0.614588
$430$	0	0
$431$	11365.4	1.27019	0.635095	−	0.772434i	$-0.280961\pi$
$431$	11365.4	1.27019	0.635095	+	0.772434i	$0.280961\pi$
$432$	0	0
$433$	−7122.15	−0.790459	−0.395230	−	0.918582i	$-0.629335\pi$
$433$	−7122.15	−0.790459	−0.395230	+	0.918582i	$0.629335\pi$
$434$	0	0
$435$	204.896	0.0225840
$436$	0	0
$437$	−3792.44	−0.415141
$438$	0	0
$439$	−875.552	−0.0951886	−0.0475943	−	0.998867i	$-0.515155\pi$
$439$	−875.552	−0.0951886	−0.0475943	+	0.998867i	$0.515155\pi$
$440$	0	0
$441$	−8176.01	−0.882844
$442$	0	0
$443$	−901.219	−0.0966551	−0.0483276	−	0.998832i	$-0.515389\pi$
$443$	−901.219	−0.0966551	−0.0483276	+	0.998832i	$0.515389\pi$
$444$	0	0
$445$	−1028.13	−0.109524
$446$	0	0
$447$	7470.10	0.790433
$448$	0	0
$449$	−6069.50	−0.637945	−0.318972	−	0.947764i	$-0.603338\pi$
$449$	−6069.50	−0.637945	−0.318972	+	0.947764i	$0.603338\pi$
$450$	0	0
$451$	−966.799	−0.100942
$452$	0	0
$453$	17764.4	1.84248
$454$	0	0
$455$	172.620	0.0177858
$456$	0	0
$457$	−4447.14	−0.455204	−0.227602	−	0.973754i	$-0.573089\pi$
$457$	−4447.14	−0.455204	−0.227602	+	0.973754i	$0.573089\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14639.2	1.47899	0.739495	−	0.673162i	$-0.235064\pi$
$461$	14639.2	1.47899	0.739495	+	0.673162i	$0.235064\pi$
$462$	0	0
$463$	−4429.45	−0.444609	−0.222305	−	0.974977i	$-0.571358\pi$
$463$	−4429.45	−0.444609	−0.222305	+	0.974977i	$0.571358\pi$
$464$	0	0
$465$	893.755	0.0891331
$466$	0	0
$467$	−2433.40	−0.241123	−0.120562	−	0.992706i	$-0.538470\pi$
$467$	−2433.40	−0.241123	−0.120562	+	0.992706i	$0.538470\pi$
$468$	0	0
$469$	1217.27	0.119847
$470$	0	0
$471$	23531.3	2.30205
$472$	0	0
$473$	2023.92	0.196744
$474$	0	0
$475$	5182.70	0.500629
$476$	0	0
$477$	−12690.2	−1.21812
$478$	0	0
$479$	1654.13	0.157785	0.0788924	−	0.996883i	$-0.474862\pi$
$479$	1654.13	0.157785	0.0788924	+	0.996883i	$0.474862\pi$
$480$	0	0
$481$	5136.23	0.486886
$482$	0	0
$483$	3186.51	0.300189
$484$	0	0
$485$	−1067.08	−0.0999046
$486$	0	0
$487$	−7372.37	−0.685984	−0.342992	−	0.939338i	$-0.611440\pi$
$487$	−7372.37	−0.685984	−0.342992	+	0.939338i	$0.611440\pi$
$488$	0	0
$489$	7543.12	0.697570
$490$	0	0
$491$	843.794	0.0775558	0.0387779	−	0.999248i	$-0.487654\pi$
$491$	843.794	0.0775558	0.0387779	+	0.999248i	$0.487654\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−302.787	−0.0274935
$496$	0	0
$497$	−1305.67	−0.117842
$498$	0	0
$499$	13625.2	1.22234	0.611169	−	0.791500i	$-0.290700\pi$
$499$	13625.2	1.22234	0.611169	+	0.791500i	$0.290700\pi$
$500$	0	0
$501$	22848.9	2.03755
$502$	0	0
$503$	16315.6	1.44627	0.723136	−	0.690706i	$-0.242700\pi$
$503$	16315.6	1.44627	0.723136	+	0.690706i	$0.242700\pi$
$504$	0	0
$505$	474.144	0.0417804
$506$	0	0
$507$	−566.693	−0.0496405
$508$	0	0
$509$	3828.25	0.333368	0.166684	−	0.986010i	$-0.446694\pi$
$509$	3828.25	0.333368	0.166684	+	0.986010i	$0.446694\pi$
$510$	0	0
$511$	−1502.54	−0.130075
$512$	0	0
$513$	−430.781	−0.0370749
$514$	0	0
$515$	−1172.27	−0.100303
$516$	0	0
$517$	4520.58	0.384555
$518$	0	0
$519$	−16057.0	−1.35804
$520$	0	0
$521$	−20450.3	−1.71966	−0.859831	−	0.510579i	$-0.829431\pi$
$521$	−20450.3	−1.71966	−0.859831	+	0.510579i	$0.829431\pi$
$522$	0	0
$523$	−14235.2	−1.19017	−0.595087	−	0.803661i	$-0.702882\pi$
$523$	−14235.2	−1.19017	−0.595087	+	0.803661i	$0.702882\pi$
$524$	0	0
$525$	−4354.66	−0.362005
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3875.60	−0.318534
$530$	0	0
$531$	16091.6	1.31510
$532$	0	0
$533$	−2920.53	−0.237340
$534$	0	0
$535$	1241.27	0.100308
$536$	0	0
$537$	−20633.8	−1.65813
$538$	0	0
$539$	5048.15	0.403412
$540$	0	0
$541$	−8614.73	−0.684614	−0.342307	−	0.939588i	$-0.611208\pi$
$541$	−8614.73	−0.684614	−0.342307	+	0.939588i	$0.611208\pi$
$542$	0	0
$543$	13901.9	1.09869
$544$	0	0
$545$	−741.667	−0.0582927
$546$	0	0
$547$	10915.6	0.853229	0.426614	−	0.904434i	$-0.359706\pi$
$547$	10915.6	0.853229	0.426614	+	0.904434i	$0.359706\pi$
$548$	0	0
$549$	9024.10	0.701529
$550$	0	0
$551$	1569.60	0.121356
$552$	0	0
$553$	3106.55	0.238886
$554$	0	0
$555$	585.449	0.0447764
$556$	0	0
$557$	−15077.2	−1.14693	−0.573467	−	0.819229i	$-0.694402\pi$
$557$	−15077.2	−1.14693	−0.573467	+	0.819229i	$0.694402\pi$
$558$	0	0
$559$	6113.90	0.462595
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13204.0	−0.988427	−0.494213	−	0.869341i	$-0.664544\pi$
$563$	−13204.0	−0.988427	−0.494213	+	0.869341i	$0.664544\pi$
$564$	0	0
$565$	−267.416	−0.0199120
$566$	0	0
$567$	3694.47	0.273639
$568$	0	0
$569$	8946.08	0.659120	0.329560	−	0.944135i	$-0.393100\pi$
$569$	8946.08	0.659120	0.329560	+	0.944135i	$0.393100\pi$
$570$	0	0
$571$	23991.2	1.75832	0.879159	−	0.476529i	$-0.158105\pi$
$571$	23991.2	1.75832	0.879159	+	0.476529i	$0.158105\pi$
$572$	0	0
$573$	−25357.1	−1.84871
$574$	0	0
$575$	−11330.9	−0.821797
$576$	0	0
$577$	−23006.6	−1.65993	−0.829964	−	0.557817i	$-0.811639\pi$
$577$	−23006.6	−1.65993	−0.829964	+	0.557817i	$0.811639\pi$
$578$	0	0
$579$	−11869.9	−0.851979
$580$	0	0
$581$	−571.918	−0.0408385
$582$	0	0
$583$	7835.37	0.556617
$584$	0	0
$585$	−914.666	−0.0646440
$586$	0	0
$587$	10496.0	0.738019	0.369009	−	0.929426i	$-0.379697\pi$
$587$	10496.0	0.738019	0.369009	+	0.929426i	$0.379697\pi$
$588$	0	0
$589$	6846.57	0.478961
$590$	0	0
$591$	−15856.9	−1.10366
$592$	0	0
$593$	2016.15	0.139618	0.0698089	−	0.997560i	$-0.477761\pi$
$593$	2016.15	0.139618	0.0698089	+	0.997560i	$0.477761\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	455.558	0.0312308
$598$	0	0
$599$	−17365.7	−1.18455	−0.592273	−	0.805738i	$-0.701769\pi$
$599$	−17365.7	−1.18455	−0.592273	+	0.805738i	$0.701769\pi$
$600$	0	0
$601$	23059.4	1.56508	0.782539	−	0.622601i	$-0.213924\pi$
$601$	23059.4	1.56508	0.782539	+	0.622601i	$0.213924\pi$
$602$	0	0
$603$	−6449.98	−0.435594
$604$	0	0
$605$	−811.081	−0.0545043
$606$	0	0
$607$	−688.381	−0.0460305	−0.0230153	−	0.999735i	$-0.507327\pi$
$607$	−688.381	−0.0460305	−0.0230153	+	0.999735i	$0.507327\pi$
$608$	0	0
$609$	−1318.82	−0.0877526
$610$	0	0
$611$	13655.8	0.904184
$612$	0	0
$613$	4520.32	0.297837	0.148919	−	0.988849i	$-0.452421\pi$
$613$	4520.32	0.297837	0.148919	+	0.988849i	$0.452421\pi$
$614$	0	0
$615$	−332.894	−0.0218269
$616$	0	0
$617$	11810.3	0.770607	0.385304	−	0.922790i	$-0.374097\pi$
$617$	11810.3	0.770607	0.385304	+	0.922790i	$0.374097\pi$
$618$	0	0
$619$	8636.93	0.560820	0.280410	−	0.959880i	$-0.409530\pi$
$619$	8636.93	0.560820	0.280410	+	0.959880i	$0.409530\pi$
$620$	0	0
$621$	941.816	0.0608596
$622$	0	0
$623$	6617.59	0.425567
$624$	0	0
$625$	15414.5	0.986526
$626$	0	0
$627$	−4768.36	−0.303716
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−16860.3	−1.06370	−0.531851	−	0.846838i	$-0.678504\pi$
$631$	−16860.3	−1.06370	−0.531851	+	0.846838i	$0.678504\pi$
$632$	0	0
$633$	−11929.9	−0.749083
$634$	0	0
$635$	1616.99	0.101053
$636$	0	0
$637$	15249.5	0.948523
$638$	0	0
$639$	6918.42	0.428308
$640$	0	0
$641$	−22833.6	−1.40698	−0.703489	−	0.710707i	$-0.748375\pi$
$641$	−22833.6	−1.40698	−0.703489	+	0.710707i	$0.748375\pi$
$642$	0	0
$643$	−8499.96	−0.521315	−0.260658	−	0.965431i	$-0.583939\pi$
$643$	−8499.96	−0.521315	−0.260658	+	0.965431i	$0.583939\pi$
$644$	0	0
$645$	696.888	0.0425425
$646$	0	0
$647$	13790.3	0.837945	0.418973	−	0.907999i	$-0.362390\pi$
$647$	13790.3	0.837945	0.418973	+	0.907999i	$0.362390\pi$
$648$	0	0
$649$	−9935.51	−0.600929
$650$	0	0
$651$	−5752.68	−0.346337
$652$	0	0
$653$	28598.1	1.71383	0.856914	−	0.515460i	$-0.172379\pi$
$653$	28598.1	1.71383	0.856914	+	0.515460i	$0.172379\pi$
$654$	0	0
$655$	−847.507	−0.0505570
$656$	0	0
$657$	7961.54	0.472769
$658$	0	0
$659$	−15819.3	−0.935103	−0.467552	−	0.883966i	$-0.654864\pi$
$659$	−15819.3	−0.935103	−0.467552	+	0.883966i	$0.654864\pi$
$660$	0	0
$661$	−6438.65	−0.378872	−0.189436	−	0.981893i	$-0.560666\pi$
$661$	−6438.65	−0.378872	−0.189436	+	0.981893i	$0.560666\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	150.726	0.00878934
$666$	0	0
$667$	−3431.61	−0.199209
$668$	0	0
$669$	−42659.3	−2.46533
$670$	0	0
$671$	−5571.79	−0.320561
$672$	0	0
$673$	27.7810	0.00159120	0.000795600	−	1.00000i	$-0.499747\pi$
$673$	27.7810	0.00159120	0.000795600		1.00000i	$0.499747\pi$
$674$	0	0
$675$	−1287.08	−0.0733920
$676$	0	0
$677$	15175.7	0.861522	0.430761	−	0.902466i	$-0.358245\pi$
$677$	15175.7	0.861522	0.430761	+	0.902466i	$0.358245\pi$
$678$	0	0
$679$	6868.31	0.388191
$680$	0	0
$681$	−46537.5	−2.61868
$682$	0	0
$683$	−23307.3	−1.30575	−0.652876	−	0.757465i	$-0.726438\pi$
$683$	−23307.3	−1.30575	−0.652876	+	0.757465i	$0.726438\pi$
$684$	0	0
$685$	−1072.04	−0.0597966
$686$	0	0
$687$	−18601.9	−1.03305
$688$	0	0
$689$	23669.2	1.30875
$690$	0	0
$691$	−27984.0	−1.54061	−0.770306	−	0.637674i	$-0.779897\pi$
$691$	−27984.0	−1.54061	−0.770306	+	0.637674i	$0.779897\pi$
$692$	0	0
$693$	1948.90	0.106829
$694$	0	0
$695$	−735.963	−0.0401679
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−39021.1	−2.11147
$700$	0	0
$701$	−1141.59	−0.0615082	−0.0307541	−	0.999527i	$-0.509791\pi$
$701$	−1141.59	−0.0615082	−0.0307541	+	0.999527i	$0.509791\pi$
$702$	0	0
$703$	4484.80	0.240608
$704$	0	0
$705$	1556.55	0.0831532
$706$	0	0
$707$	−3051.84	−0.162343
$708$	0	0
$709$	−28172.3	−1.49229	−0.746144	−	0.665785i	$-0.768097\pi$
$709$	−28172.3	−1.49229	−0.746144	+	0.665785i	$0.768097\pi$
$710$	0	0
$711$	−16460.8	−0.868251
$712$	0	0
$713$	−14968.7	−0.786228
$714$	0	0
$715$	564.746	0.0295389
$716$	0	0
$717$	−10499.7	−0.546887
$718$	0	0
$719$	27453.7	1.42399	0.711995	−	0.702185i	$-0.247792\pi$
$719$	27453.7	1.42399	0.711995	+	0.702185i	$0.247792\pi$
$720$	0	0
$721$	7545.33	0.389740
$722$	0	0
$723$	−32487.8	−1.67114
$724$	0	0
$725$	4689.60	0.240231
$726$	0	0
$727$	−23468.6	−1.19725	−0.598627	−	0.801028i	$-0.704287\pi$
$727$	−23468.6	−1.19725	−0.598627	+	0.801028i	$0.704287\pi$
$728$	0	0
$729$	−17551.1	−0.891690
$730$	0	0
$731$	0	0
$732$	0	0
$733$	24712.8	1.24528	0.622638	−	0.782510i	$-0.286061\pi$
$733$	24712.8	1.24528	0.622638	+	0.782510i	$0.286061\pi$
$734$	0	0
$735$	1738.21	0.0872309
$736$	0	0
$737$	3982.44	0.199043
$738$	0	0
$739$	−9896.16	−0.492606	−0.246303	−	0.969193i	$-0.579216\pi$
$739$	−9896.16	−0.492606	−0.246303	+	0.969193i	$0.579216\pi$
$740$	0	0
$741$	−14404.3	−0.714112
$742$	0	0
$743$	−15364.2	−0.758627	−0.379313	−	0.925268i	$-0.623840\pi$
$743$	−15364.2	−0.758627	−0.379313	+	0.925268i	$0.623840\pi$
$744$	0	0
$745$	−772.519	−0.0379905
$746$	0	0
$747$	3030.44	0.148431
$748$	0	0
$749$	−7989.49	−0.389759
$750$	0	0
$751$	−11646.2	−0.565882	−0.282941	−	0.959137i	$-0.591310\pi$
$751$	−11646.2	−0.565882	−0.282941	+	0.959137i	$0.591310\pi$
$752$	0	0
$753$	19035.2	0.921222
$754$	0	0
$755$	−1837.10	−0.0885551
$756$	0	0
$757$	4633.31	0.222458	0.111229	−	0.993795i	$-0.464521\pi$
$757$	4633.31	0.222458	0.111229	+	0.993795i	$0.464521\pi$
$758$	0	0
$759$	10425.1	0.498558
$760$	0	0
$761$	38552.5	1.83644	0.918218	−	0.396076i	$-0.129628\pi$
$761$	38552.5	1.83644	0.918218	+	0.396076i	$0.129628\pi$
$762$	0	0
$763$	4773.76	0.226503
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−30013.4	−1.41293
$768$	0	0
$769$	5342.15	0.250511	0.125255	−	0.992125i	$-0.460025\pi$
$769$	5342.15	0.250511	0.125255	+	0.992125i	$0.460025\pi$
$770$	0	0
$771$	−23266.7	−1.08681
$772$	0	0
$773$	40868.1	1.90158	0.950791	−	0.309834i	$-0.100273\pi$
$773$	40868.1	1.90158	0.950791	+	0.309834i	$0.100273\pi$
$774$	0	0
$775$	20456.0	0.948131
$776$	0	0
$777$	−3768.26	−0.173984
$778$	0	0
$779$	−2550.12	−0.117288
$780$	0	0
$781$	−4271.67	−0.195714
$782$	0	0
$783$	−389.795	−0.0177907
$784$	0	0
$785$	−2433.48	−0.110643
$786$	0	0
$787$	−16797.3	−0.760811	−0.380405	−	0.924820i	$-0.624216\pi$
$787$	−16797.3	−0.760811	−0.380405	+	0.924820i	$0.624216\pi$
$788$	0	0
$789$	17418.1	0.785931
$790$	0	0
$791$	1721.23	0.0773704
$792$	0	0
$793$	−16831.4	−0.753719
$794$	0	0
$795$	2697.92	0.120359
$796$	0	0
$797$	25038.7	1.11282	0.556410	−	0.830908i	$-0.312178\pi$
$797$	25038.7	1.11282	0.556410	+	0.830908i	$0.312178\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−35064.9	−1.54676
$802$	0	0
$803$	−4915.72	−0.216030
$804$	0	0
$805$	−329.533	−0.0144280
$806$	0	0
$807$	−19718.2	−0.860117
$808$	0	0
$809$	−42697.0	−1.85556	−0.927779	−	0.373131i	$-0.878284\pi$
$809$	−42697.0	−1.85556	−0.927779	+	0.373131i	$0.878284\pi$
$810$	0	0
$811$	−24530.1	−1.06211	−0.531054	−	0.847338i	$-0.678204\pi$
$811$	−24530.1	−1.06211	−0.531054	+	0.847338i	$0.678204\pi$
$812$	0	0
$813$	11456.7	0.494225
$814$	0	0
$815$	−780.070	−0.0335272
$816$	0	0
$817$	5338.47	0.228604
$818$	0	0
$819$	5887.27	0.251182
$820$	0	0
$821$	−42624.7	−1.81195	−0.905977	−	0.423328i	$-0.860862\pi$
$821$	−42624.7	−1.81195	−0.905977	+	0.423328i	$0.860862\pi$
$822$	0	0
$823$	17216.1	0.729180	0.364590	−	0.931168i	$-0.381209\pi$
$823$	17216.1	0.729180	0.364590	+	0.931168i	$0.381209\pi$
$824$	0	0
$825$	−14246.8	−0.601223
$826$	0	0
$827$	−27132.1	−1.14084	−0.570420	−	0.821353i	$-0.693219\pi$
$827$	−27132.1	−1.14084	−0.570420	+	0.821353i	$0.693219\pi$
$828$	0	0
$829$	−3051.44	−0.127842	−0.0639210	−	0.997955i	$-0.520361\pi$
$829$	−3051.44	−0.127842	−0.0639210	+	0.997955i	$0.520361\pi$
$830$	0	0
$831$	31697.3	1.32319
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2362.91	−0.0979305
$836$	0	0
$837$	−1700.28	−0.0702155
$838$	0	0
$839$	2354.61	0.0968895	0.0484447	−	0.998826i	$-0.484574\pi$
$839$	2354.61	0.0968895	0.0484447	+	0.998826i	$0.484574\pi$
$840$	0	0
$841$	−22968.7	−0.941766
$842$	0	0
$843$	16661.2	0.680714
$844$	0	0
$845$	58.6045	0.00238586
$846$	0	0
$847$	5220.55	0.211783
$848$	0	0
$849$	−50788.4	−2.05307
$850$	0	0
$851$	−9805.12	−0.394965
$852$	0	0
$853$	1410.82	0.0566300	0.0283150	−	0.999599i	$-0.490986\pi$
$853$	1410.82	0.0566300	0.0283150	+	0.999599i	$0.490986\pi$
$854$	0	0
$855$	−798.658	−0.0319457
$856$	0	0
$857$	10840.7	0.432103	0.216052	−	0.976382i	$-0.430682\pi$
$857$	10840.7	0.432103	0.216052	+	0.976382i	$0.430682\pi$
$858$	0	0
$859$	−3335.84	−0.132500	−0.0662499	−	0.997803i	$-0.521103\pi$
$859$	−3335.84	−0.132500	−0.0662499	+	0.997803i	$0.521103\pi$
$860$	0	0
$861$	2142.68	0.0848111
$862$	0	0
$863$	−4706.16	−0.185631	−0.0928154	−	0.995683i	$-0.529587\pi$
$863$	−4706.16	−0.185631	−0.0928154	+	0.995683i	$0.529587\pi$
$864$	0	0
$865$	1660.53	0.0652714
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10163.4	0.396744
$870$	0	0
$871$	12030.2	0.468001
$872$	0	0
$873$	−36393.4	−1.41092
$874$	0	0
$875$	902.707	0.0348766
$876$	0	0
$877$	10764.4	0.414466	0.207233	−	0.978292i	$-0.433554\pi$
$877$	10764.4	0.414466	0.207233	+	0.978292i	$0.433554\pi$
$878$	0	0
$879$	49231.4	1.88912
$880$	0	0
$881$	10346.4	0.395661	0.197831	−	0.980236i	$-0.436610\pi$
$881$	10346.4	0.395661	0.197831	+	0.980236i	$0.436610\pi$
$882$	0	0
$883$	20004.5	0.762406	0.381203	−	0.924491i	$-0.375510\pi$
$883$	20004.5	0.762406	0.381203	+	0.924491i	$0.375510\pi$
$884$	0	0
$885$	−3421.05	−0.129940
$886$	0	0
$887$	−13929.6	−0.527296	−0.263648	−	0.964619i	$-0.584926\pi$
$887$	−13929.6	−0.527296	−0.263648	+	0.964619i	$0.584926\pi$
$888$	0	0
$889$	−10407.8	−0.392651
$890$	0	0
$891$	12086.9	0.454463
$892$	0	0
$893$	11923.9	0.446828
$894$	0	0
$895$	2133.84	0.0796944
$896$	0	0
$897$	31492.2	1.17223
$898$	0	0
$899$	6195.16	0.229833
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−4485.54	−0.165304
$904$	0	0
$905$	−1437.67	−0.0528063
$906$	0	0
$907$	31913.7	1.16833	0.584166	−	0.811634i	$-0.301422\pi$
$907$	31913.7	1.16833	0.584166	+	0.811634i	$0.301422\pi$
$908$	0	0
$909$	16170.9	0.590049
$910$	0	0
$911$	24978.8	0.908436	0.454218	−	0.890891i	$-0.349919\pi$
$911$	24978.8	0.908436	0.454218	+	0.890891i	$0.349919\pi$
$912$	0	0
$913$	−1871.10	−0.0678251
$914$	0	0
$915$	−1918.51	−0.0693157
$916$	0	0
$917$	5455.01	0.196445
$918$	0	0
$919$	51879.7	1.86219	0.931096	−	0.364774i	$-0.118854\pi$
$919$	51879.7	1.86219	0.931096	+	0.364774i	$0.118854\pi$
$920$	0	0
$921$	−47106.9	−1.68537
$922$	0	0
$923$	−12903.9	−0.460172
$924$	0	0
$925$	13399.6	0.476298
$926$	0	0
$927$	−39980.7	−1.41655
$928$	0	0
$929$	−31639.1	−1.11738	−0.558690	−	0.829376i	$-0.688696\pi$
$929$	−31639.1	−1.11738	−0.558690	+	0.829376i	$0.688696\pi$
$930$	0	0
$931$	13315.4	0.468739
$932$	0	0
$933$	19471.3	0.683238
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24094.9	−0.840072	−0.420036	−	0.907507i	$-0.637983\pi$
$937$	−24094.9	−0.840072	−0.420036	+	0.907507i	$0.637983\pi$
$938$	0	0
$939$	62396.8	2.16852
$940$	0	0
$941$	−32589.5	−1.12900	−0.564499	−	0.825433i	$-0.690931\pi$
$941$	−32589.5	−1.12900	−0.564499	+	0.825433i	$0.690931\pi$
$942$	0	0
$943$	5575.31	0.192532
$944$	0	0
$945$	−37.4315	−0.00128851
$946$	0	0
$947$	25158.7	0.863304	0.431652	−	0.902040i	$-0.357931\pi$
$947$	25158.7	0.863304	0.431652	+	0.902040i	$0.357931\pi$
$948$	0	0
$949$	−14849.5	−0.507941
$950$	0	0
$951$	−25541.3	−0.870909
$952$	0	0
$953$	42902.2	1.45828	0.729138	−	0.684366i	$-0.239921\pi$
$953$	42902.2	1.45828	0.729138	+	0.684366i	$0.239921\pi$
$954$	0	0
$955$	2622.30	0.0888541
$956$	0	0
$957$	−4314.68	−0.145741
$958$	0	0
$959$	6900.24	0.232347
$960$	0	0
$961$	−2767.76	−0.0929058
$962$	0	0
$963$	42334.2	1.41661
$964$	0	0
$965$	1227.52	0.0409485
$966$	0	0
$967$	45649.9	1.51810	0.759050	−	0.651033i	$-0.225664\pi$
$967$	45649.9	1.51810	0.759050	+	0.651033i	$0.225664\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27834.0	0.919915	0.459957	−	0.887941i	$-0.347865\pi$
$971$	27834.0	0.919915	0.459957	+	0.887941i	$0.347865\pi$
$972$	0	0
$973$	4737.05	0.156077
$974$	0	0
$975$	−43036.9	−1.41363
$976$	0	0
$977$	−55483.4	−1.81686	−0.908430	−	0.418037i	$-0.862718\pi$
$977$	−55483.4	−1.81686	−0.908430	+	0.418037i	$0.862718\pi$
$978$	0	0
$979$	21650.2	0.706787
$980$	0	0
$981$	−25294.9	−0.823246
$982$	0	0
$983$	9555.91	0.310057	0.155029	−	0.987910i	$-0.450453\pi$
$983$	9555.91	0.310057	0.155029	+	0.987910i	$0.450453\pi$
$984$	0	0
$985$	1639.84	0.0530453
$986$	0	0
$987$	−10018.8	−0.323102
$988$	0	0
$989$	−11671.5	−0.375260
$990$	0	0
$991$	−21577.0	−0.691641	−0.345820	−	0.938301i	$-0.612399\pi$
$991$	−21577.0	−0.691641	−0.345820	+	0.938301i	$0.612399\pi$
$992$	0	0
$993$	−10667.8	−0.340919
$994$	0	0
$995$	−47.1115	−0.00150104
$996$	0	0
$997$	50807.0	1.61392	0.806958	−	0.590609i	$-0.201112\pi$
$997$	50807.0	1.61392	0.806958	+	0.590609i	$0.201112\pi$
$998$	0	0
$999$	−1113.76	−0.0352731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.q.1.3	yes	18
17.16	even	2		2312.4.a.n.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.n.1.16	✓	18	17.16	even	2
2312.4.a.q.1.3	yes	18	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)

\(f(q)\)	\(=\)	\(q-7.25076 q^{3} +0.749836 q^{5} -4.82634 q^{7} +25.5735 q^{9} +O(q^{10})\)	\(q-7.25076 q^{3} +0.749836 q^{5} -4.82634 q^{7} +25.5735 q^{9} -15.7900 q^{11} -47.6986 q^{13} -5.43688 q^{15} -41.6490 q^{19} +34.9947 q^{21} +91.0571 q^{23} -124.438 q^{25} +10.3431 q^{27} -37.6863 q^{29} -164.387 q^{31} +114.489 q^{33} -3.61897 q^{35} -107.681 q^{37} +345.851 q^{39} +61.2288 q^{41} -128.178 q^{43} +19.1759 q^{45} -286.294 q^{47} -319.706 q^{49} -496.225 q^{53} -11.8399 q^{55} +301.987 q^{57} +629.230 q^{59} +352.869 q^{61} -123.427 q^{63} -35.7661 q^{65} -252.213 q^{67} -660.233 q^{69} +270.531 q^{71} +311.320 q^{73} +902.268 q^{75} +76.2078 q^{77} -643.664 q^{79} -765.480 q^{81} +118.499 q^{83} +273.255 q^{87} -1371.14 q^{89} +230.210 q^{91} +1191.93 q^{93} -31.2299 q^{95} -1423.09 q^{97} -403.805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9}+O(q^{10}) \) 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

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