LMFDB - Embedded newform 2312.4.a.p.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:	$0$
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} - 345 x^{16} - 182 x^{15} + 48165 x^{14} + 48078 x^{13} - 3485278 x^{12} - 4881882 x^{11} + \cdots - 119632152329 $ x^18 - 345x^16 - 182x^15 + 48165x^14 + 48078x^13 - 3485278x^12 - 4881882x^11 + 139131876x^10 + 239770606x^9 - 3013674039x^8 - 5867731896x^7 + 32610490343x^6 + 66112081788x^5 - 151840532421x^4 - 282437582582x^3 + 230444683878x^2 + 237530449410x - 119632152329
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.05066$ 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)

$f(q)$	$=$	$q-6.05066 q^{3} -8.16729 q^{5} -11.0156 q^{7} +9.61049 q^{9} +O(q^{10})$	$q-6.05066 q^{3} -8.16729 q^{5} -11.0156 q^{7} +9.61049 q^{9} +26.1221 q^{11} -34.2134 q^{13} +49.4175 q^{15} -130.200 q^{19} +66.6516 q^{21} -74.8766 q^{23} -58.2953 q^{25} +105.218 q^{27} +56.8343 q^{29} -90.6359 q^{31} -158.056 q^{33} +89.9676 q^{35} -206.726 q^{37} +207.013 q^{39} +171.097 q^{41} -169.648 q^{43} -78.4917 q^{45} +154.649 q^{47} -221.657 q^{49} -281.259 q^{53} -213.347 q^{55} +787.794 q^{57} -448.478 q^{59} -232.139 q^{61} -105.865 q^{63} +279.431 q^{65} -910.341 q^{67} +453.053 q^{69} -661.716 q^{71} -122.010 q^{73} +352.725 q^{75} -287.751 q^{77} +135.085 q^{79} -896.122 q^{81} -183.290 q^{83} -343.885 q^{87} -776.830 q^{89} +376.880 q^{91} +548.407 q^{93} +1063.38 q^{95} -1082.34 q^{97} +251.047 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 30 q^{5} + 33 q^{7} + 204 q^{9}+O(q^{10}) $ 18 * q + 30 * q^5 + 33 * q^7 + 204 * q^9	$ 18 q + 30 q^{5} + 33 q^{7} + 204 q^{9} - 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} + 153 q^{23} + 594 q^{25} - 546 q^{27} + 447 q^{29} + 303 q^{31} + 153 q^{33} - 117 q^{35} + 939 q^{37} + 516 q^{39} + 1257 q^{41} + 306 q^{43} + 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} + 1494 q^{57} + 696 q^{59} + 1686 q^{61} + 1908 q^{63} + 855 q^{65} + 513 q^{67} - 1329 q^{69} + 324 q^{71} + 1863 q^{73} - 3054 q^{75} + 1833 q^{77} + 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} + 252 q^{91} - 1470 q^{93} + 2109 q^{95} + 4611 q^{97} - 3918 q^{99}+O(q^{100}) $ 18 * q + 30 * q^5 + 33 * q^7 + 204 * q^9 - 66 * q^11 - 30 * q^13 - 102 * q^15 - 168 * q^19 - 510 * q^21 + 153 * q^23 + 594 * q^25 - 546 * q^27 + 447 * q^29 + 303 * q^31 + 153 * q^33 - 117 * q^35 + 939 * q^37 + 516 * q^39 + 1257 * q^41 + 306 * q^43 + 672 * q^45 + 633 * q^47 + 1239 * q^49 - 489 * q^53 + 1089 * q^55 + 1494 * q^57 + 696 * q^59 + 1686 * q^61 + 1908 * q^63 + 855 * q^65 + 513 * q^67 - 1329 * q^69 + 324 * q^71 + 1863 * q^73 - 3054 * q^75 + 1833 * q^77 + 3699 * q^79 + 2622 * q^81 + 1188 * q^83 - 3927 * q^87 + 1713 * q^89 + 252 * q^91 - 1470 * q^93 + 2109 * q^95 + 4611 * q^97 - 3918 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−6.05066	−1.16445	−0.582225	−	0.813028i	$-0.697818\pi$
$3$	−6.05066	−1.16445	−0.582225	+	0.813028i	$0.697818\pi$
$4$	0	0
$5$	−8.16729	−0.730505	−0.365252	−	0.930908i	$-0.619017\pi$
$5$	−8.16729	−0.730505	−0.365252	+	0.930908i	$0.619017\pi$
$6$	0	0
$7$	−11.0156	−0.594786	−0.297393	−	0.954755i	$-0.596117\pi$
$7$	−11.0156	−0.594786	−0.297393	+	0.954755i	$0.596117\pi$
$8$	0	0
$9$	9.61049	0.355944
$10$	0	0
$11$	26.1221	0.716011	0.358006	−	0.933719i	$-0.383457\pi$
$11$	26.1221	0.716011	0.358006	+	0.933719i	$0.383457\pi$
$12$	0	0
$13$	−34.2134	−0.729929	−0.364965	−	0.931021i	$-0.618919\pi$
$13$	−34.2134	−0.729929	−0.364965	+	0.931021i	$0.618919\pi$
$14$	0	0
$15$	49.4175	0.850637
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−130.200	−1.57210	−0.786049	−	0.618164i	$-0.787877\pi$
$19$	−130.200	−1.57210	−0.786049	+	0.618164i	$0.787877\pi$
$20$	0	0
$21$	66.6516	0.692598
$22$	0	0
$23$	−74.8766	−0.678820	−0.339410	−	0.940639i	$-0.610227\pi$
$23$	−74.8766	−0.678820	−0.339410	+	0.940639i	$0.610227\pi$
$24$	0	0
$25$	−58.2953	−0.466362
$26$	0	0
$27$	105.218	0.749971
$28$	0	0
$29$	56.8343	0.363926	0.181963	−	0.983305i	$-0.441755\pi$
$29$	56.8343	0.363926	0.181963	+	0.983305i	$0.441755\pi$
$30$	0	0
$31$	−90.6359	−0.525119	−0.262560	−	0.964916i	$-0.584567\pi$
$31$	−90.6359	−0.525119	−0.262560	+	0.964916i	$0.584567\pi$
$32$	0	0
$33$	−158.056	−0.833759
$34$	0	0
$35$	89.9676	0.434494
$36$	0	0
$37$	−206.726	−0.918529	−0.459265	−	0.888299i	$-0.651887\pi$
$37$	−206.726	−0.918529	−0.459265	+	0.888299i	$0.651887\pi$
$38$	0	0
$39$	207.013	0.849966
$40$	0	0
$41$	171.097	0.651727	0.325863	−	0.945417i	$-0.394345\pi$
$41$	171.097	0.651727	0.325863	+	0.945417i	$0.394345\pi$
$42$	0	0
$43$	−169.648	−0.601653	−0.300826	−	0.953679i	$-0.597263\pi$
$43$	−169.648	−0.601653	−0.300826	+	0.953679i	$0.597263\pi$
$44$	0	0
$45$	−78.4917	−0.260019
$46$	0	0
$47$	154.649	0.479954	0.239977	−	0.970779i	$-0.422860\pi$
$47$	154.649	0.479954	0.239977	+	0.970779i	$0.422860\pi$
$48$	0	0
$49$	−221.657	−0.646230
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−281.259	−0.728941	−0.364470	−	0.931215i	$-0.618750\pi$
$53$	−281.259	−0.728941	−0.364470	+	0.931215i	$0.618750\pi$
$54$	0	0
$55$	−213.347	−0.523050
$56$	0	0
$57$	787.794	1.83063
$58$	0	0
$59$	−448.478	−0.989608	−0.494804	−	0.869005i	$-0.664760\pi$
$59$	−448.478	−0.989608	−0.494804	+	0.869005i	$0.664760\pi$
$60$	0	0
$61$	−232.139	−0.487252	−0.243626	−	0.969869i	$-0.578337\pi$
$61$	−232.139	−0.487252	−0.243626	+	0.969869i	$0.578337\pi$
$62$	0	0
$63$	−105.865	−0.211710
$64$	0	0
$65$	279.431	0.533217
$66$	0	0
$67$	−910.341	−1.65994	−0.829969	−	0.557810i	$-0.811642\pi$
$67$	−910.341	−1.65994	−0.829969	+	0.557810i	$0.811642\pi$
$68$	0	0
$69$	453.053	0.790452
$70$	0	0
$71$	−661.716	−1.10607	−0.553037	−	0.833157i	$-0.686531\pi$
$71$	−661.716	−1.10607	−0.553037	+	0.833157i	$0.686531\pi$
$72$	0	0
$73$	−122.010	−0.195619	−0.0978094	−	0.995205i	$-0.531184\pi$
$73$	−122.010	−0.195619	−0.0978094	+	0.995205i	$0.531184\pi$
$74$	0	0
$75$	352.725	0.543056
$76$	0	0
$77$	−287.751	−0.425873
$78$	0	0
$79$	135.085	0.192383	0.0961916	−	0.995363i	$-0.469334\pi$
$79$	135.085	0.192383	0.0961916	+	0.995363i	$0.469334\pi$
$80$	0	0
$81$	−896.122	−1.22925
$82$	0	0
$83$	−183.290	−0.242394	−0.121197	−	0.992628i	$-0.538673\pi$
$83$	−183.290	−0.242394	−0.121197	+	0.992628i	$0.538673\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−343.885	−0.423774
$88$	0	0
$89$	−776.830	−0.925212	−0.462606	−	0.886564i	$-0.653086\pi$
$89$	−776.830	−0.925212	−0.462606	+	0.886564i	$0.653086\pi$
$90$	0	0
$91$	376.880	0.434152
$92$	0	0
$93$	548.407	0.611475
$94$	0	0
$95$	1063.38	1.14843
$96$	0	0
$97$	−1082.34	−1.13294	−0.566471	−	0.824082i	$-0.691692\pi$
$97$	−1082.34	−1.13294	−0.566471	+	0.824082i	$0.691692\pi$
$98$	0	0
$99$	251.047	0.254860
$100$	0	0
$101$	−1508.36	−1.48601	−0.743005	−	0.669286i	$-0.766600\pi$
$101$	−1508.36	−1.48601	−0.743005	+	0.669286i	$0.766600\pi$
$102$	0	0
$103$	−525.326	−0.502543	−0.251271	−	0.967917i	$-0.580849\pi$
$103$	−525.326	−0.502543	−0.251271	+	0.967917i	$0.580849\pi$
$104$	0	0
$105$	−544.363	−0.505947
$106$	0	0
$107$	−333.202	−0.301046	−0.150523	−	0.988607i	$-0.548096\pi$
$107$	−333.202	−0.301046	−0.150523	+	0.988607i	$0.548096\pi$
$108$	0	0
$109$	−1736.00	−1.52549	−0.762744	−	0.646700i	$-0.776149\pi$
$109$	−1736.00	−1.52549	−0.762744	+	0.646700i	$0.776149\pi$
$110$	0	0
$111$	1250.83	1.06958
$112$	0	0
$113$	1841.67	1.53319	0.766593	−	0.642133i	$-0.221950\pi$
$113$	1841.67	1.53319	0.766593	+	0.642133i	$0.221950\pi$
$114$	0	0
$115$	611.540	0.495881
$116$	0	0
$117$	−328.807	−0.259814
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−648.633	−0.487328
$122$	0	0
$123$	−1035.25	−0.758903
$124$	0	0
$125$	1497.03	1.07119
$126$	0	0
$127$	−2445.75	−1.70886	−0.854431	−	0.519566i	$-0.826094\pi$
$127$	−2445.75	−1.70886	−0.854431	+	0.519566i	$0.826094\pi$
$128$	0	0
$129$	1026.48	0.700595
$130$	0	0
$131$	1833.58	1.22291	0.611454	−	0.791280i	$-0.290585\pi$
$131$	1833.58	1.22291	0.611454	+	0.791280i	$0.290585\pi$
$132$	0	0
$133$	1434.23	0.935062
$134$	0	0
$135$	−859.347	−0.547858
$136$	0	0
$137$	833.376	0.519709	0.259854	−	0.965648i	$-0.416325\pi$
$137$	833.376	0.519709	0.259854	+	0.965648i	$0.416325\pi$
$138$	0	0
$139$	−3068.90	−1.87267	−0.936335	−	0.351108i	$-0.885805\pi$
$139$	−3068.90	−1.87267	−0.936335	+	0.351108i	$0.885805\pi$
$140$	0	0
$141$	−935.726	−0.558882
$142$	0	0
$143$	−893.726	−0.522637
$144$	0	0
$145$	−464.182	−0.265850
$146$	0	0
$147$	1341.17	0.752502
$148$	0	0
$149$	2756.88	1.51579	0.757894	−	0.652378i	$-0.226229\pi$
$149$	2756.88	1.51579	0.757894	+	0.652378i	$0.226229\pi$
$150$	0	0
$151$	−1013.77	−0.546356	−0.273178	−	0.961964i	$-0.588075\pi$
$151$	−1013.77	−0.546356	−0.273178	+	0.961964i	$0.588075\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	740.250	0.383602
$156$	0	0
$157$	3360.41	1.70822	0.854108	−	0.520096i	$-0.174104\pi$
$157$	3360.41	1.70822	0.854108	+	0.520096i	$0.174104\pi$
$158$	0	0
$159$	1701.80	0.848815
$160$	0	0
$161$	824.810	0.403752
$162$	0	0
$163$	−1365.46	−0.656141	−0.328070	−	0.944653i	$-0.606398\pi$
$163$	−1365.46	−0.656141	−0.328070	+	0.944653i	$0.606398\pi$
$164$	0	0
$165$	1290.89	0.609065
$166$	0	0
$167$	−1587.31	−0.735509	−0.367754	−	0.929923i	$-0.619873\pi$
$167$	−1587.31	−0.735509	−0.367754	+	0.929923i	$0.619873\pi$
$168$	0	0
$169$	−1026.45	−0.467204
$170$	0	0
$171$	−1251.28	−0.559579
$172$	0	0
$173$	201.157	0.0884029	0.0442014	−	0.999023i	$-0.485926\pi$
$173$	201.157	0.0884029	0.0442014	+	0.999023i	$0.485926\pi$
$174$	0	0
$175$	642.157	0.277386
$176$	0	0
$177$	2713.59	1.15235
$178$	0	0
$179$	2805.71	1.17156	0.585779	−	0.810471i	$-0.300789\pi$
$179$	2805.71	1.17156	0.585779	+	0.810471i	$0.300789\pi$
$180$	0	0
$181$	550.197	0.225944	0.112972	−	0.993598i	$-0.463963\pi$
$181$	550.197	0.225944	0.112972	+	0.993598i	$0.463963\pi$
$182$	0	0
$183$	1404.60	0.567381
$184$	0	0
$185$	1688.39	0.670990
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1159.04	−0.446072
$190$	0	0
$191$	−2439.44	−0.924145	−0.462073	−	0.886842i	$-0.652894\pi$
$191$	−2439.44	−0.924145	−0.462073	+	0.886842i	$0.652894\pi$
$192$	0	0
$193$	3690.38	1.37637	0.688184	−	0.725536i	$-0.258408\pi$
$193$	3690.38	1.37637	0.688184	+	0.725536i	$0.258408\pi$
$194$	0	0
$195$	−1690.74	−0.620904
$196$	0	0
$197$	−2057.48	−0.744107	−0.372054	−	0.928211i	$-0.621346\pi$
$197$	−2057.48	−0.744107	−0.372054	+	0.928211i	$0.621346\pi$
$198$	0	0
$199$	1667.29	0.593924	0.296962	−	0.954889i	$-0.404026\pi$
$199$	1667.29	0.593924	0.296962	+	0.954889i	$0.404026\pi$
$200$	0	0
$201$	5508.16	1.93291
$202$	0	0
$203$	−626.063	−0.216458
$204$	0	0
$205$	−1397.40	−0.476090
$206$	0	0
$207$	−719.601	−0.241622
$208$	0	0
$209$	−3401.10	−1.12564
$210$	0	0
$211$	659.954	0.215323	0.107661	−	0.994188i	$-0.465664\pi$
$211$	659.954	0.215323	0.107661	+	0.994188i	$0.465664\pi$
$212$	0	0
$213$	4003.82	1.28797
$214$	0	0
$215$	1385.57	0.439511
$216$	0	0
$217$	998.408	0.312333
$218$	0	0
$219$	738.240	0.227788
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3579.23	1.07481	0.537406	−	0.843323i	$-0.319404\pi$
$223$	3579.23	1.07481	0.537406	+	0.843323i	$0.319404\pi$
$224$	0	0
$225$	−560.246	−0.165999
$226$	0	0
$227$	−2160.44	−0.631690	−0.315845	−	0.948811i	$-0.602288\pi$
$227$	−2160.44	−0.631690	−0.315845	+	0.948811i	$0.602288\pi$
$228$	0	0
$229$	5620.77	1.62197	0.810985	−	0.585067i	$-0.198932\pi$
$229$	5620.77	1.62197	0.810985	+	0.585067i	$0.198932\pi$
$230$	0	0
$231$	1741.08	0.495908
$232$	0	0
$233$	6490.70	1.82498	0.912489	−	0.409102i	$-0.134158\pi$
$233$	6490.70	1.82498	0.912489	+	0.409102i	$0.134158\pi$
$234$	0	0
$235$	−1263.06	−0.350609
$236$	0	0
$237$	−817.355	−0.224021
$238$	0	0
$239$	−4988.35	−1.35008	−0.675041	−	0.737780i	$-0.735874\pi$
$239$	−4988.35	−1.35008	−0.675041	+	0.737780i	$0.735874\pi$
$240$	0	0
$241$	−4187.83	−1.11934	−0.559672	−	0.828714i	$-0.689073\pi$
$241$	−4187.83	−1.11934	−0.559672	+	0.828714i	$0.689073\pi$
$242$	0	0
$243$	2581.24	0.681427
$244$	0	0
$245$	1810.34	0.472074
$246$	0	0
$247$	4454.57	1.14752
$248$	0	0
$249$	1109.03	0.282256
$250$	0	0
$251$	−5296.31	−1.33187	−0.665936	−	0.746009i	$-0.731968\pi$
$251$	−5296.31	−1.33187	−0.665936	+	0.746009i	$0.731968\pi$
$252$	0	0
$253$	−1955.94	−0.486043
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2053.48	−0.498416	−0.249208	−	0.968450i	$-0.580170\pi$
$257$	−2053.48	−0.498416	−0.249208	+	0.968450i	$0.580170\pi$
$258$	0	0
$259$	2277.21	0.546328
$260$	0	0
$261$	546.205	0.129537
$262$	0	0
$263$	1810.46	0.424478	0.212239	−	0.977218i	$-0.431924\pi$
$263$	1810.46	0.424478	0.212239	+	0.977218i	$0.431924\pi$
$264$	0	0
$265$	2297.12	0.532495
$266$	0	0
$267$	4700.34	1.07736
$268$	0	0
$269$	5528.88	1.25317	0.626584	−	0.779354i	$-0.284453\pi$
$269$	5528.88	1.25317	0.626584	+	0.779354i	$0.284453\pi$
$270$	0	0
$271$	−1635.26	−0.366550	−0.183275	−	0.983062i	$-0.558670\pi$
$271$	−1635.26	−0.366550	−0.183275	+	0.983062i	$0.558670\pi$
$272$	0	0
$273$	−2280.37	−0.505548
$274$	0	0
$275$	−1522.80	−0.333921
$276$	0	0
$277$	−4962.54	−1.07643	−0.538214	−	0.842808i	$-0.680901\pi$
$277$	−4962.54	−1.07643	−0.538214	+	0.842808i	$0.680901\pi$
$278$	0	0
$279$	−871.055	−0.186913
$280$	0	0
$281$	1781.41	0.378186	0.189093	−	0.981959i	$-0.439445\pi$
$281$	1781.41	0.378186	0.189093	+	0.981959i	$0.439445\pi$
$282$	0	0
$283$	−6088.35	−1.27885	−0.639425	−	0.768853i	$-0.720828\pi$
$283$	−6088.35	−1.27885	−0.639425	+	0.768853i	$0.720828\pi$
$284$	0	0
$285$	−6434.15	−1.33728
$286$	0	0
$287$	−1884.73	−0.387638
$288$	0	0
$289$	0	0
$290$	0	0
$291$	6548.89	1.31925
$292$	0	0
$293$	7806.00	1.55642	0.778210	−	0.628004i	$-0.216128\pi$
$293$	7806.00	1.55642	0.778210	+	0.628004i	$0.216128\pi$
$294$	0	0
$295$	3662.85	0.722914
$296$	0	0
$297$	2748.52	0.536988
$298$	0	0
$299$	2561.78	0.495490
$300$	0	0
$301$	1868.77	0.357855
$302$	0	0
$303$	9126.55	1.73039
$304$	0	0
$305$	1895.95	0.355940
$306$	0	0
$307$	−10420.7	−1.93726	−0.968629	−	0.248513i	$-0.920058\pi$
$307$	−10420.7	−1.93726	−0.968629	+	0.248513i	$0.920058\pi$
$308$	0	0
$309$	3178.57	0.585186
$310$	0	0
$311$	3107.41	0.566576	0.283288	−	0.959035i	$-0.408575\pi$
$311$	3107.41	0.566576	0.283288	+	0.959035i	$0.408575\pi$
$312$	0	0
$313$	8444.23	1.52491	0.762454	−	0.647042i	$-0.223994\pi$
$313$	8444.23	1.52491	0.762454	+	0.647042i	$0.223994\pi$
$314$	0	0
$315$	864.632	0.154656
$316$	0	0
$317$	−3459.84	−0.613009	−0.306505	−	0.951869i	$-0.599160\pi$
$317$	−3459.84	−0.613009	−0.306505	+	0.951869i	$0.599160\pi$
$318$	0	0
$319$	1484.63	0.260575
$320$	0	0
$321$	2016.09	0.350553
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1994.48	0.340412
$326$	0	0
$327$	10503.9	1.77636
$328$	0	0
$329$	−1703.55	−0.285470
$330$	0	0
$331$	−3692.74	−0.613206	−0.306603	−	0.951837i	$-0.599192\pi$
$331$	−3692.74	−0.613206	−0.306603	+	0.951837i	$0.599192\pi$
$332$	0	0
$333$	−1986.74	−0.326945
$334$	0	0
$335$	7435.02	1.21259
$336$	0	0
$337$	−3155.55	−0.510070	−0.255035	−	0.966932i	$-0.582087\pi$
$337$	−3155.55	−0.510070	−0.255035	+	0.966932i	$0.582087\pi$
$338$	0	0
$339$	−11143.3	−1.78532
$340$	0	0
$341$	−2367.60	−0.375991
$342$	0	0
$343$	6220.03	0.979154
$344$	0	0
$345$	−3700.22	−0.577429
$346$	0	0
$347$	−3541.97	−0.547962	−0.273981	−	0.961735i	$-0.588341\pi$
$347$	−3541.97	−0.547962	−0.273981	+	0.961735i	$0.588341\pi$
$348$	0	0
$349$	−2484.41	−0.381053	−0.190527	−	0.981682i	$-0.561020\pi$
$349$	−2484.41	−0.381053	−0.190527	+	0.981682i	$0.561020\pi$
$350$	0	0
$351$	−3599.86	−0.547426
$352$	0	0
$353$	7199.58	1.08554	0.542769	−	0.839882i	$-0.317376\pi$
$353$	7199.58	1.08554	0.542769	+	0.839882i	$0.317376\pi$
$354$	0	0
$355$	5404.43	0.807992
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11147.0	1.63877	0.819385	−	0.573243i	$-0.194315\pi$
$359$	11147.0	1.63877	0.819385	+	0.573243i	$0.194315\pi$
$360$	0	0
$361$	10093.0	1.47149
$362$	0	0
$363$	3924.66	0.567469
$364$	0	0
$365$	996.490	0.142901
$366$	0	0
$367$	5964.72	0.848381	0.424191	−	0.905573i	$-0.360559\pi$
$367$	5964.72	0.848381	0.424191	+	0.905573i	$0.360559\pi$
$368$	0	0
$369$	1644.32	0.231978
$370$	0	0
$371$	3098.23	0.433564
$372$	0	0
$373$	71.5986	0.00993897	0.00496948	−	0.999988i	$-0.498418\pi$
$373$	71.5986	0.00993897	0.00496948	+	0.999988i	$0.498418\pi$
$374$	0	0
$375$	−9058.00	−1.24734
$376$	0	0
$377$	−1944.49	−0.265640
$378$	0	0
$379$	−8149.73	−1.10455	−0.552274	−	0.833663i	$-0.686240\pi$
$379$	−8149.73	−1.10455	−0.552274	+	0.833663i	$0.686240\pi$
$380$	0	0
$381$	14798.4	1.98988
$382$	0	0
$383$	11615.3	1.54965	0.774823	−	0.632179i	$-0.217839\pi$
$383$	11615.3	1.54965	0.774823	+	0.632179i	$0.217839\pi$
$384$	0	0
$385$	2350.15	0.311103
$386$	0	0
$387$	−1630.40	−0.214155
$388$	0	0
$389$	−6557.69	−0.854725	−0.427362	−	0.904080i	$-0.640557\pi$
$389$	−6557.69	−0.854725	−0.427362	+	0.904080i	$0.640557\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−11094.4	−1.42401
$394$	0	0
$395$	−1103.28	−0.140537
$396$	0	0
$397$	3456.67	0.436990	0.218495	−	0.975838i	$-0.429885\pi$
$397$	3456.67	0.436990	0.218495	+	0.975838i	$0.429885\pi$
$398$	0	0
$399$	−8678.01	−1.08883
$400$	0	0
$401$	996.782	0.124132	0.0620660	−	0.998072i	$-0.480231\pi$
$401$	996.782	0.124132	0.0620660	+	0.998072i	$0.480231\pi$
$402$	0	0
$403$	3100.96	0.383300
$404$	0	0
$405$	7318.89	0.897972
$406$	0	0
$407$	−5400.13	−0.657677
$408$	0	0
$409$	−693.145	−0.0837990	−0.0418995	−	0.999122i	$-0.513341\pi$
$409$	−693.145	−0.0837990	−0.0418995	+	0.999122i	$0.513341\pi$
$410$	0	0
$411$	−5042.47	−0.605175
$412$	0	0
$413$	4940.25	0.588605
$414$	0	0
$415$	1496.99	0.177070
$416$	0	0
$417$	18568.9	2.18063
$418$	0	0
$419$	−15129.6	−1.76403	−0.882014	−	0.471224i	$-0.843812\pi$
$419$	−15129.6	−1.76403	−0.882014	+	0.471224i	$0.843812\pi$
$420$	0	0
$421$	−14703.2	−1.70212	−0.851058	−	0.525072i	$-0.824038\pi$
$421$	−14703.2	−1.70212	−0.851058	+	0.525072i	$0.824038\pi$
$422$	0	0
$423$	1486.25	0.170837
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2557.15	0.289811
$428$	0	0
$429$	5407.63	0.608585
$430$	0	0
$431$	−2596.52	−0.290185	−0.145093	−	0.989418i	$-0.546348\pi$
$431$	−2596.52	−0.290185	−0.145093	+	0.989418i	$0.546348\pi$
$432$	0	0
$433$	−2532.70	−0.281094	−0.140547	−	0.990074i	$-0.544886\pi$
$433$	−2532.70	−0.281094	−0.140547	+	0.990074i	$0.544886\pi$
$434$	0	0
$435$	2808.61	0.309569
$436$	0	0
$437$	9748.91	1.06717
$438$	0	0
$439$	−7464.46	−0.811524	−0.405762	−	0.913979i	$-0.632994\pi$
$439$	−7464.46	−0.811524	−0.405762	+	0.913979i	$0.632994\pi$
$440$	0	0
$441$	−2130.23	−0.230022
$442$	0	0
$443$	−2648.17	−0.284015	−0.142007	−	0.989866i	$-0.545356\pi$
$443$	−2648.17	−0.284015	−0.142007	+	0.989866i	$0.545356\pi$
$444$	0	0
$445$	6344.60	0.675872
$446$	0	0
$447$	−16680.9	−1.76506
$448$	0	0
$449$	10616.4	1.11585	0.557927	−	0.829890i	$-0.311597\pi$
$449$	10616.4	1.11585	0.557927	+	0.829890i	$0.311597\pi$
$450$	0	0
$451$	4469.41	0.466644
$452$	0	0
$453$	6134.00	0.636204
$454$	0	0
$455$	−3078.09	−0.317150
$456$	0	0
$457$	−3131.58	−0.320545	−0.160272	−	0.987073i	$-0.551237\pi$
$457$	−3131.58	−0.320545	−0.160272	+	0.987073i	$0.551237\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13970.4	1.41142	0.705712	−	0.708499i	$-0.250627\pi$
$461$	13970.4	1.41142	0.705712	+	0.708499i	$0.250627\pi$
$462$	0	0
$463$	3701.31	0.371522	0.185761	−	0.982595i	$-0.440525\pi$
$463$	3701.31	0.371522	0.185761	+	0.982595i	$0.440525\pi$
$464$	0	0
$465$	−4479.00	−0.446685
$466$	0	0
$467$	7943.59	0.787121	0.393560	−	0.919299i	$-0.371243\pi$
$467$	7943.59	0.787121	0.393560	+	0.919299i	$0.371243\pi$
$468$	0	0
$469$	10027.9	0.987307
$470$	0	0
$471$	−20332.7	−1.98913
$472$	0	0
$473$	−4431.57	−0.430790
$474$	0	0
$475$	7590.03	0.733167
$476$	0	0
$477$	−2703.03	−0.259462
$478$	0	0
$479$	−10545.3	−1.00591	−0.502953	−	0.864314i	$-0.667753\pi$
$479$	−10545.3	−1.00591	−0.502953	+	0.864314i	$0.667753\pi$
$480$	0	0
$481$	7072.80	0.670461
$482$	0	0
$483$	−4990.65	−0.470150
$484$	0	0
$485$	8839.81	0.827619
$486$	0	0
$487$	7901.42	0.735210	0.367605	−	0.929982i	$-0.380178\pi$
$487$	7901.42	0.735210	0.367605	+	0.929982i	$0.380178\pi$
$488$	0	0
$489$	8261.92	0.764043
$490$	0	0
$491$	−4049.68	−0.372218	−0.186109	−	0.982529i	$-0.559588\pi$
$491$	−4049.68	−0.372218	−0.186109	+	0.982529i	$0.559588\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2050.37	−0.186176
$496$	0	0
$497$	7289.19	0.657877
$498$	0	0
$499$	18176.4	1.63063	0.815317	−	0.579015i	$-0.196563\pi$
$499$	18176.4	1.63063	0.815317	+	0.579015i	$0.196563\pi$
$500$	0	0
$501$	9604.29	0.856463
$502$	0	0
$503$	8247.64	0.731102	0.365551	−	0.930791i	$-0.380881\pi$
$503$	8247.64	0.731102	0.365551	+	0.930791i	$0.380881\pi$
$504$	0	0
$505$	12319.2	1.08554
$506$	0	0
$507$	6210.68	0.544035
$508$	0	0
$509$	−9848.53	−0.857620	−0.428810	−	0.903395i	$-0.641067\pi$
$509$	−9848.53	−0.857620	−0.428810	+	0.903395i	$0.641067\pi$
$510$	0	0
$511$	1344.01	0.116351
$512$	0	0
$513$	−13699.4	−1.17903
$514$	0	0
$515$	4290.49	0.367110
$516$	0	0
$517$	4039.76	0.343652
$518$	0	0
$519$	−1217.13	−0.102941
$520$	0	0
$521$	20161.0	1.69533	0.847666	−	0.530530i	$-0.178007\pi$
$521$	20161.0	1.69533	0.847666	+	0.530530i	$0.178007\pi$
$522$	0	0
$523$	1899.53	0.158816	0.0794078	−	0.996842i	$-0.474697\pi$
$523$	1899.53	0.158816	0.0794078	+	0.996842i	$0.474697\pi$
$524$	0	0
$525$	−3885.47	−0.323002
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−6560.49	−0.539204
$530$	0	0
$531$	−4310.09	−0.352245
$532$	0	0
$533$	−5853.79	−0.475714
$534$	0	0
$535$	2721.36	0.219915
$536$	0	0
$537$	−16976.4	−1.36422
$538$	0	0
$539$	−5790.15	−0.462708
$540$	0	0
$541$	23684.6	1.88222	0.941108	−	0.338106i	$-0.109786\pi$
$541$	23684.6	1.88222	0.941108	+	0.338106i	$0.109786\pi$
$542$	0	0
$543$	−3329.05	−0.263100
$544$	0	0
$545$	14178.4	1.11438
$546$	0	0
$547$	−488.623	−0.0381938	−0.0190969	−	0.999818i	$-0.506079\pi$
$547$	−488.623	−0.0381938	−0.0190969	+	0.999818i	$0.506079\pi$
$548$	0	0
$549$	−2230.97	−0.173435
$550$	0	0
$551$	−7399.80	−0.572128
$552$	0	0
$553$	−1488.04	−0.114427
$554$	0	0
$555$	−10215.9	−0.781335
$556$	0	0
$557$	5661.61	0.430683	0.215341	−	0.976539i	$-0.430914\pi$
$557$	5661.61	0.430683	0.215341	+	0.976539i	$0.430914\pi$
$558$	0	0
$559$	5804.23	0.439164
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1185.47	0.0887419	0.0443710	−	0.999015i	$-0.485872\pi$
$563$	1185.47	0.0887419	0.0443710	+	0.999015i	$0.485872\pi$
$564$	0	0
$565$	−15041.5	−1.12000
$566$	0	0
$567$	9871.31	0.731139
$568$	0	0
$569$	−19924.8	−1.46800	−0.733998	−	0.679152i	$-0.762348\pi$
$569$	−19924.8	−1.46800	−0.733998	+	0.679152i	$0.762348\pi$
$570$	0	0
$571$	−6795.71	−0.498059	−0.249029	−	0.968496i	$-0.580112\pi$
$571$	−6795.71	−0.498059	−0.249029	+	0.968496i	$0.580112\pi$
$572$	0	0
$573$	14760.2	1.07612
$574$	0	0
$575$	4364.96	0.316576
$576$	0	0
$577$	−9228.04	−0.665803	−0.332901	−	0.942962i	$-0.608028\pi$
$577$	−9228.04	−0.665803	−0.332901	+	0.942962i	$0.608028\pi$
$578$	0	0
$579$	−22329.2	−1.60271
$580$	0	0
$581$	2019.05	0.144173
$582$	0	0
$583$	−7347.08	−0.521930
$584$	0	0
$585$	2685.46	0.189795
$586$	0	0
$587$	−19925.6	−1.40105	−0.700527	−	0.713626i	$-0.747052\pi$
$587$	−19925.6	−1.40105	−0.700527	+	0.713626i	$0.747052\pi$
$588$	0	0
$589$	11800.8	0.825538
$590$	0	0
$591$	12449.1	0.866476
$592$	0	0
$593$	−11710.9	−0.810977	−0.405489	−	0.914100i	$-0.632899\pi$
$593$	−11710.9	−0.810977	−0.405489	+	0.914100i	$0.632899\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10088.2	−0.691595
$598$	0	0
$599$	−7693.75	−0.524805	−0.262403	−	0.964958i	$-0.584515\pi$
$599$	−7693.75	−0.524805	−0.262403	+	0.964958i	$0.584515\pi$
$600$	0	0
$601$	4209.18	0.285684	0.142842	−	0.989745i	$-0.454376\pi$
$601$	4209.18	0.285684	0.142842	+	0.989745i	$0.454376\pi$
$602$	0	0
$603$	−8748.82	−0.590845
$604$	0	0
$605$	5297.58	0.355995
$606$	0	0
$607$	4133.62	0.276406	0.138203	−	0.990404i	$-0.455867\pi$
$607$	4133.62	0.276406	0.138203	+	0.990404i	$0.455867\pi$
$608$	0	0
$609$	3788.09	0.252055
$610$	0	0
$611$	−5291.05	−0.350332
$612$	0	0
$613$	23145.7	1.52504	0.762518	−	0.646967i	$-0.223963\pi$
$613$	23145.7	1.52504	0.762518	+	0.646967i	$0.223963\pi$
$614$	0	0
$615$	8455.17	0.554383
$616$	0	0
$617$	19525.8	1.27403	0.637017	−	0.770849i	$-0.280168\pi$
$617$	19525.8	1.27403	0.637017	+	0.770849i	$0.280168\pi$
$618$	0	0
$619$	−2147.87	−0.139467	−0.0697334	−	0.997566i	$-0.522215\pi$
$619$	−2147.87	−0.139467	−0.0697334	+	0.997566i	$0.522215\pi$
$620$	0	0
$621$	−7878.37	−0.509095
$622$	0	0
$623$	8557.25	0.550303
$624$	0	0
$625$	−4939.74	−0.316144
$626$	0	0
$627$	20578.9	1.31075
$628$	0	0
$629$	0	0
$630$	0	0
$631$	12501.0	0.788681	0.394341	−	0.918964i	$-0.370973\pi$
$631$	12501.0	0.788681	0.394341	+	0.918964i	$0.370973\pi$
$632$	0	0
$633$	−3993.16	−0.250733
$634$	0	0
$635$	19975.2	1.24833
$636$	0	0
$637$	7583.62	0.471702
$638$	0	0
$639$	−6359.41	−0.393700
$640$	0	0
$641$	11771.8	0.725362	0.362681	−	0.931913i	$-0.381861\pi$
$641$	11771.8	0.725362	0.362681	+	0.931913i	$0.381861\pi$
$642$	0	0
$643$	−8553.42	−0.524594	−0.262297	−	0.964987i	$-0.584480\pi$
$643$	−8553.42	−0.524594	−0.262297	+	0.964987i	$0.584480\pi$
$644$	0	0
$645$	−8383.58	−0.511788
$646$	0	0
$647$	−12054.0	−0.732446	−0.366223	−	0.930527i	$-0.619349\pi$
$647$	−12054.0	−0.732446	−0.366223	+	0.930527i	$0.619349\pi$
$648$	0	0
$649$	−11715.2	−0.708571
$650$	0	0
$651$	−6041.03	−0.363697
$652$	0	0
$653$	−30053.7	−1.80106	−0.900531	−	0.434793i	$-0.856822\pi$
$653$	−30053.7	−1.80106	−0.900531	+	0.434793i	$0.856822\pi$
$654$	0	0
$655$	−14975.4	−0.893340
$656$	0	0
$657$	−1172.57	−0.0696293
$658$	0	0
$659$	5430.34	0.320996	0.160498	−	0.987036i	$-0.448690\pi$
$659$	5430.34	0.320996	0.160498	+	0.987036i	$0.448690\pi$
$660$	0	0
$661$	−28422.2	−1.67246	−0.836231	−	0.548378i	$-0.815245\pi$
$661$	−28422.2	−1.67246	−0.836231	+	0.548378i	$0.815245\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11713.7	−0.683067
$666$	0	0
$667$	−4255.56	−0.247040
$668$	0	0
$669$	−21656.7	−1.25157
$670$	0	0
$671$	−6063.98	−0.348878
$672$	0	0
$673$	−8850.70	−0.506938	−0.253469	−	0.967343i	$-0.581572\pi$
$673$	−8850.70	−0.506938	−0.253469	+	0.967343i	$0.581572\pi$
$674$	0	0
$675$	−6133.72	−0.349758
$676$	0	0
$677$	−15147.9	−0.859941	−0.429971	−	0.902843i	$-0.641476\pi$
$677$	−15147.9	−0.859941	−0.429971	+	0.902843i	$0.641476\pi$
$678$	0	0
$679$	11922.6	0.673857
$680$	0	0
$681$	13072.1	0.735571
$682$	0	0
$683$	−27062.7	−1.51614	−0.758071	−	0.652172i	$-0.773858\pi$
$683$	−27062.7	−1.51614	−0.758071	+	0.652172i	$0.773858\pi$
$684$	0	0
$685$	−6806.43	−0.379650
$686$	0	0
$687$	−34009.4	−1.88870
$688$	0	0
$689$	9622.80	0.532075
$690$	0	0
$691$	3887.13	0.213999	0.107000	−	0.994259i	$-0.465876\pi$
$691$	3887.13	0.213999	0.107000	+	0.994259i	$0.465876\pi$
$692$	0	0
$693$	−2765.43	−0.151587
$694$	0	0
$695$	25064.6	1.36799
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−39273.0	−2.12510
$700$	0	0
$701$	−32911.4	−1.77325	−0.886624	−	0.462490i	$-0.846956\pi$
$701$	−32911.4	−1.77325	−0.886624	+	0.462490i	$0.846956\pi$
$702$	0	0
$703$	26915.7	1.44402
$704$	0	0
$705$	7642.35	0.408266
$706$	0	0
$707$	16615.4	0.883858
$708$	0	0
$709$	12063.7	0.639017	0.319509	−	0.947583i	$-0.396482\pi$
$709$	12063.7	0.639017	0.319509	+	0.947583i	$0.396482\pi$
$710$	0	0
$711$	1298.23	0.0684776
$712$	0	0
$713$	6786.51	0.356461
$714$	0	0
$715$	7299.33	0.381789
$716$	0	0
$717$	30182.8	1.57210
$718$	0	0
$719$	−7187.38	−0.372801	−0.186401	−	0.982474i	$-0.559682\pi$
$719$	−7187.38	−0.372801	−0.186401	+	0.982474i	$0.559682\pi$
$720$	0	0
$721$	5786.77	0.298905
$722$	0	0
$723$	25339.1	1.30342
$724$	0	0
$725$	−3313.17	−0.169722
$726$	0	0
$727$	19868.5	1.01359	0.506797	−	0.862065i	$-0.330829\pi$
$727$	19868.5	1.01359	0.506797	+	0.862065i	$0.330829\pi$
$728$	0	0
$729$	8577.08	0.435761
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−13813.1	−0.696041	−0.348020	−	0.937487i	$-0.613146\pi$
$733$	−13813.1	−0.696041	−0.348020	+	0.937487i	$0.613146\pi$
$734$	0	0
$735$	−10953.7	−0.549707
$736$	0	0
$737$	−23780.1	−1.18853
$738$	0	0
$739$	30918.0	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$739$	30918.0	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$740$	0	0
$741$	−26953.1	−1.33623
$742$	0	0
$743$	−31257.3	−1.54336	−0.771682	−	0.636009i	$-0.780584\pi$
$743$	−31257.3	−1.54336	−0.771682	+	0.636009i	$0.780584\pi$
$744$	0	0
$745$	−22516.2	−1.10729
$746$	0	0
$747$	−1761.51	−0.0862787
$748$	0	0
$749$	3670.42	0.179058
$750$	0	0
$751$	−37392.3	−1.81686	−0.908432	−	0.418032i	$-0.862720\pi$
$751$	−37392.3	−1.81686	−0.908432	+	0.418032i	$0.862720\pi$
$752$	0	0
$753$	32046.2	1.55090
$754$	0	0
$755$	8279.79	0.399116
$756$	0	0
$757$	−35247.9	−1.69235	−0.846173	−	0.532908i	$-0.821099\pi$
$757$	−35247.9	−1.69235	−0.846173	+	0.532908i	$0.821099\pi$
$758$	0	0
$759$	11834.7	0.565972
$760$	0	0
$761$	−6360.27	−0.302969	−0.151485	−	0.988460i	$-0.548405\pi$
$761$	−6360.27	−0.302969	−0.151485	+	0.988460i	$0.548405\pi$
$762$	0	0
$763$	19123.0	0.907339
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15343.9	0.722344
$768$	0	0
$769$	−9497.21	−0.445355	−0.222678	−	0.974892i	$-0.571480\pi$
$769$	−9497.21	−0.445355	−0.222678	+	0.974892i	$0.571480\pi$
$770$	0	0
$771$	12424.9	0.580380
$772$	0	0
$773$	11638.8	0.541549	0.270774	−	0.962643i	$-0.412720\pi$
$773$	11638.8	0.541549	0.270774	+	0.962643i	$0.412720\pi$
$774$	0	0
$775$	5283.65	0.244896
$776$	0	0
$777$	−13778.6	−0.636172
$778$	0	0
$779$	−22276.7	−1.02458
$780$	0	0
$781$	−17285.4	−0.791961
$782$	0	0
$783$	5979.99	0.272934
$784$	0	0
$785$	−27445.5	−1.24786
$786$	0	0
$787$	10014.8	0.453609	0.226805	−	0.973940i	$-0.427172\pi$
$787$	10014.8	0.453609	0.226805	+	0.973940i	$0.427172\pi$
$788$	0	0
$789$	−10954.5	−0.494283
$790$	0	0
$791$	−20287.1	−0.911917
$792$	0	0
$793$	7942.27	0.355660
$794$	0	0
$795$	−13899.1	−0.620064
$796$	0	0
$797$	−42247.9	−1.87766	−0.938831	−	0.344378i	$-0.888090\pi$
$797$	−42247.9	−1.87766	−0.938831	+	0.344378i	$0.888090\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−7465.72	−0.329324
$802$	0	0
$803$	−3187.16	−0.140065
$804$	0	0
$805$	−6736.47	−0.294943
$806$	0	0
$807$	−33453.4	−1.45925
$808$	0	0
$809$	8305.80	0.360960	0.180480	−	0.983579i	$-0.442235\pi$
$809$	8305.80	0.360960	0.180480	+	0.983579i	$0.442235\pi$
$810$	0	0
$811$	−22044.8	−0.954496	−0.477248	−	0.878769i	$-0.658366\pi$
$811$	−22044.8	−0.954496	−0.477248	+	0.878769i	$0.658366\pi$
$812$	0	0
$813$	9894.42	0.426829
$814$	0	0
$815$	11152.1	0.479314
$816$	0	0
$817$	22088.1	0.945857
$818$	0	0
$819$	3622.00	0.154534
$820$	0	0
$821$	−23661.3	−1.00583	−0.502914	−	0.864337i	$-0.667739\pi$
$821$	−23661.3	−1.00583	−0.502914	+	0.864337i	$0.667739\pi$
$822$	0	0
$823$	−2378.42	−0.100737	−0.0503684	−	0.998731i	$-0.516040\pi$
$823$	−2378.42	−0.100737	−0.0503684	+	0.998731i	$0.516040\pi$
$824$	0	0
$825$	9213.94	0.388834
$826$	0	0
$827$	−34520.2	−1.45149	−0.725747	−	0.687962i	$-0.758506\pi$
$827$	−34520.2	−1.45149	−0.725747	+	0.687962i	$0.758506\pi$
$828$	0	0
$829$	23133.1	0.969176	0.484588	−	0.874743i	$-0.338970\pi$
$829$	23133.1	0.969176	0.484588	+	0.874743i	$0.338970\pi$
$830$	0	0
$831$	30026.7	1.25345
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12964.1	0.537293
$836$	0	0
$837$	−9536.53	−0.393824
$838$	0	0
$839$	−17405.3	−0.716206	−0.358103	−	0.933682i	$-0.616576\pi$
$839$	−17405.3	−0.716206	−0.358103	+	0.933682i	$0.616576\pi$
$840$	0	0
$841$	−21158.9	−0.867558
$842$	0	0
$843$	−10778.7	−0.440378
$844$	0	0
$845$	8383.29	0.341295
$846$	0	0
$847$	7145.08	0.289856
$848$	0	0
$849$	36838.5	1.48916
$850$	0	0
$851$	15479.0	0.623516
$852$	0	0
$853$	21595.6	0.866845	0.433422	−	0.901191i	$-0.357306\pi$
$853$	21595.6	0.866845	0.433422	+	0.901191i	$0.357306\pi$
$854$	0	0
$855$	10219.6	0.408775
$856$	0	0
$857$	40690.3	1.62188	0.810942	−	0.585127i	$-0.198955\pi$
$857$	40690.3	1.62188	0.810942	+	0.585127i	$0.198955\pi$
$858$	0	0
$859$	−16072.6	−0.638407	−0.319203	−	0.947686i	$-0.603415\pi$
$859$	−16072.6	−0.638407	−0.319203	+	0.947686i	$0.603415\pi$
$860$	0	0
$861$	11403.9	0.451385
$862$	0	0
$863$	−12221.3	−0.482061	−0.241030	−	0.970518i	$-0.577485\pi$
$863$	−12221.3	−0.482061	−0.241030	+	0.970518i	$0.577485\pi$
$864$	0	0
$865$	−1642.91	−0.0645787
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3528.72	0.137749
$870$	0	0
$871$	31145.8	1.21164
$872$	0	0
$873$	−10401.8	−0.403263
$874$	0	0
$875$	−16490.6	−0.637126
$876$	0	0
$877$	−16852.0	−0.648863	−0.324432	−	0.945909i	$-0.605173\pi$
$877$	−16852.0	−0.648863	−0.324432	+	0.945909i	$0.605173\pi$
$878$	0	0
$879$	−47231.4	−1.81237
$880$	0	0
$881$	−16901.3	−0.646331	−0.323166	−	0.946342i	$-0.604747\pi$
$881$	−16901.3	−0.646331	−0.323166	+	0.946342i	$0.604747\pi$
$882$	0	0
$883$	37696.3	1.43667	0.718336	−	0.695696i	$-0.244904\pi$
$883$	37696.3	1.43667	0.718336	+	0.695696i	$0.244904\pi$
$884$	0	0
$885$	−22162.7	−0.841797
$886$	0	0
$887$	40547.5	1.53489	0.767447	−	0.641112i	$-0.221527\pi$
$887$	40547.5	1.53489	0.767447	+	0.641112i	$0.221527\pi$
$888$	0	0
$889$	26941.4	1.01641
$890$	0	0
$891$	−23408.6	−0.880155
$892$	0	0
$893$	−20135.2	−0.754534
$894$	0	0
$895$	−22915.1	−0.855828
$896$	0	0
$897$	−15500.5	−0.576974
$898$	0	0
$899$	−5151.23	−0.191105
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−11307.3	−0.416704
$904$	0	0
$905$	−4493.62	−0.165053
$906$	0	0
$907$	−21162.0	−0.774722	−0.387361	−	0.921928i	$-0.626613\pi$
$907$	−21162.0	−0.774722	−0.387361	+	0.921928i	$0.626613\pi$
$908$	0	0
$909$	−14496.0	−0.528936
$910$	0	0
$911$	9149.80	0.332762	0.166381	−	0.986062i	$-0.446792\pi$
$911$	9149.80	0.332762	0.166381	+	0.986062i	$0.446792\pi$
$912$	0	0
$913$	−4787.93	−0.173557
$914$	0	0
$915$	−11471.7	−0.414475
$916$	0	0
$917$	−20198.0	−0.727368
$918$	0	0
$919$	24721.1	0.887349	0.443675	−	0.896188i	$-0.353675\pi$
$919$	24721.1	0.887349	0.443675	+	0.896188i	$0.353675\pi$
$920$	0	0
$921$	63051.8	2.25584
$922$	0	0
$923$	22639.5	0.807355
$924$	0	0
$925$	12051.2	0.428368
$926$	0	0
$927$	−5048.64	−0.178877
$928$	0	0
$929$	43193.6	1.52544	0.762722	−	0.646726i	$-0.223862\pi$
$929$	43193.6	1.52544	0.762722	+	0.646726i	$0.223862\pi$
$930$	0	0
$931$	28859.6	1.01594
$932$	0	0
$933$	−18801.9	−0.659750
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25038.3	0.872961	0.436481	−	0.899714i	$-0.356225\pi$
$937$	25038.3	0.872961	0.436481	+	0.899714i	$0.356225\pi$
$938$	0	0
$939$	−51093.2	−1.77568
$940$	0	0
$941$	−5088.51	−0.176281	−0.0881406	−	0.996108i	$-0.528092\pi$
$941$	−5088.51	−0.176281	−0.0881406	+	0.996108i	$0.528092\pi$
$942$	0	0
$943$	−12811.1	−0.442405
$944$	0	0
$945$	9466.21	0.325858
$946$	0	0
$947$	−19993.0	−0.686044	−0.343022	−	0.939327i	$-0.611451\pi$
$947$	−19993.0	−0.686044	−0.343022	+	0.939327i	$0.611451\pi$
$948$	0	0
$949$	4174.37	0.142788
$950$	0	0
$951$	20934.3	0.713819
$952$	0	0
$953$	34716.1	1.18003	0.590013	−	0.807394i	$-0.299123\pi$
$953$	34716.1	1.18003	0.590013	+	0.807394i	$0.299123\pi$
$954$	0	0
$955$	19923.6	0.675093
$956$	0	0
$957$	−8983.01	−0.303427
$958$	0	0
$959$	−9180.13	−0.309115
$960$	0	0
$961$	−21576.1	−0.724250
$962$	0	0
$963$	−3202.24	−0.107155
$964$	0	0
$965$	−30140.4	−1.00544
$966$	0	0
$967$	1823.37	0.0606366	0.0303183	−	0.999540i	$-0.490348\pi$
$967$	1823.37	0.0606366	0.0303183	+	0.999540i	$0.490348\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21168.7	0.699624	0.349812	−	0.936820i	$-0.386246\pi$
$971$	21168.7	0.699624	0.349812	+	0.936820i	$0.386246\pi$
$972$	0	0
$973$	33805.8	1.11384
$974$	0	0
$975$	−12067.9	−0.396392
$976$	0	0
$977$	36209.7	1.18572	0.592861	−	0.805305i	$-0.297999\pi$
$977$	36209.7	1.18572	0.592861	+	0.805305i	$0.297999\pi$
$978$	0	0
$979$	−20292.5	−0.662462
$980$	0	0
$981$	−16683.8	−0.542988
$982$	0	0
$983$	−39058.4	−1.26731	−0.633656	−	0.773615i	$-0.718447\pi$
$983$	−39058.4	−1.26731	−0.633656	+	0.773615i	$0.718447\pi$
$984$	0	0
$985$	16804.0	0.543574
$986$	0	0
$987$	10307.6	0.332415
$988$	0	0
$989$	12702.7	0.408414
$990$	0	0
$991$	−4464.65	−0.143112	−0.0715561	−	0.997437i	$-0.522797\pi$
$991$	−4464.65	−0.143112	−0.0715561	+	0.997437i	$0.522797\pi$
$992$	0	0
$993$	22343.5	0.714048
$994$	0	0
$995$	−13617.2	−0.433865
$996$	0	0
$997$	−19525.3	−0.620233	−0.310117	−	0.950699i	$-0.600368\pi$
$997$	−19525.3	−0.620233	−0.310117	+	0.950699i	$0.600368\pi$
$998$	0	0
$999$	−21751.3	−0.688871

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.o.1.14	✓	18	17.16	even	2
2312.4.a.p.1.5	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)

\(f(q)\)	\(=\)	\(q-6.05066 q^{3} -8.16729 q^{5} -11.0156 q^{7} +9.61049 q^{9} +O(q^{10})\)	\(q-6.05066 q^{3} -8.16729 q^{5} -11.0156 q^{7} +9.61049 q^{9} +26.1221 q^{11} -34.2134 q^{13} +49.4175 q^{15} -130.200 q^{19} +66.6516 q^{21} -74.8766 q^{23} -58.2953 q^{25} +105.218 q^{27} +56.8343 q^{29} -90.6359 q^{31} -158.056 q^{33} +89.9676 q^{35} -206.726 q^{37} +207.013 q^{39} +171.097 q^{41} -169.648 q^{43} -78.4917 q^{45} +154.649 q^{47} -221.657 q^{49} -281.259 q^{53} -213.347 q^{55} +787.794 q^{57} -448.478 q^{59} -232.139 q^{61} -105.865 q^{63} +279.431 q^{65} -910.341 q^{67} +453.053 q^{69} -661.716 q^{71} -122.010 q^{73} +352.725 q^{75} -287.751 q^{77} +135.085 q^{79} -896.122 q^{81} -183.290 q^{83} -343.885 q^{87} -776.830 q^{89} +376.880 q^{91} +548.407 q^{93} +1063.38 q^{95} -1082.34 q^{97} +251.047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 30 q^{5} + 33 q^{7} + 204 q^{9}+O(q^{10}) \) 18 * q + 30 * q^5 + 33 * q^7 + 204 * q^9	\( 18 q + 30 q^{5} + 33 q^{7} + 204 q^{9} - 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} + 153 q^{23} + 594 q^{25} - 546 q^{27} + 447 q^{29} + 303 q^{31} + 153 q^{33} - 117 q^{35} + 939 q^{37} + 516 q^{39} + 1257 q^{41} + 306 q^{43} + 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} + 1494 q^{57} + 696 q^{59} + 1686 q^{61} + 1908 q^{63} + 855 q^{65} + 513 q^{67} - 1329 q^{69} + 324 q^{71} + 1863 q^{73} - 3054 q^{75} + 1833 q^{77} + 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} + 252 q^{91} - 1470 q^{93} + 2109 q^{95} + 4611 q^{97} - 3918 q^{99}+O(q^{100}) \) 18 * q + 30 * q^5 + 33 * q^7 + 204 * q^9 - 66 * q^11 - 30 * q^13 - 102 * q^15 - 168 * q^19 - 510 * q^21 + 153 * q^23 + 594 * q^25 - 546 * q^27 + 447 * q^29 + 303 * q^31 + 153 * q^33 - 117 * q^35 + 939 * q^37 + 516 * q^39 + 1257 * q^41 + 306 * q^43 + 672 * q^45 + 633 * q^47 + 1239 * q^49 - 489 * q^53 + 1089 * q^55 + 1494 * q^57 + 696 * q^59 + 1686 * q^61 + 1908 * q^63 + 855 * q^65 + 513 * q^67 - 1329 * q^69 + 324 * q^71 + 1863 * q^73 - 3054 * q^75 + 1833 * q^77 + 3699 * q^79 + 2622 * q^81 + 1188 * q^83 - 3927 * q^87 + 1713 * q^89 + 252 * q^91 - 1470 * q^93 + 2109 * q^95 + 4611 * q^97 - 3918 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more