LMFDB - Embedded newform 592.4.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,4,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, 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("592.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	592.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:	$34.9291307234$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 172x^{6} - 325x^{5} + 7345x^{4} + 20706x^{3} - 73170x^{2} - 269101x - 182948 $ x^8 - 172x^6 - 325x^5 + 7345x^4 + 20706x^3 - 73170x^2 - 269101x - 182948
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.12820$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.88695 q^{3} +8.53113 q^{5} -34.6761 q^{7} -3.11767 q^{9} +O(q^{10})$	$q-4.88695 q^{3} +8.53113 q^{5} -34.6761 q^{7} -3.11767 q^{9} -39.6207 q^{11} -21.0951 q^{13} -41.6912 q^{15} -16.3404 q^{17} +65.9663 q^{19} +169.461 q^{21} -7.29485 q^{23} -52.2199 q^{25} +147.184 q^{27} +115.536 q^{29} -131.179 q^{31} +193.624 q^{33} -295.827 q^{35} -37.0000 q^{37} +103.091 q^{39} -214.780 q^{41} -181.840 q^{43} -26.5973 q^{45} +254.513 q^{47} +859.435 q^{49} +79.8547 q^{51} +143.485 q^{53} -338.009 q^{55} -322.375 q^{57} -234.441 q^{59} -74.9905 q^{61} +108.109 q^{63} -179.965 q^{65} +865.025 q^{67} +35.6496 q^{69} +382.418 q^{71} +851.842 q^{73} +255.196 q^{75} +1373.89 q^{77} -675.461 q^{79} -635.103 q^{81} -460.250 q^{83} -139.402 q^{85} -564.619 q^{87} -205.537 q^{89} +731.496 q^{91} +641.065 q^{93} +562.767 q^{95} +1012.06 q^{97} +123.524 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9}+O(q^{10}) $ 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9	$ 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9} - 33 q^{11} + 50 q^{13} - 22 q^{15} + 78 q^{17} - 158 q^{19} + 373 q^{21} - 76 q^{23} + 678 q^{25} - 469 q^{27} + 358 q^{29} - 390 q^{31} + 333 q^{33} - 546 q^{35} - 296 q^{37} - 442 q^{39} + 1141 q^{41} + 4 q^{43} + 1770 q^{45} + 735 q^{47} + 731 q^{49} - 132 q^{51} + 1349 q^{53} + 882 q^{55} + 2648 q^{57} - 1146 q^{59} + 1920 q^{61} + 534 q^{63} + 1748 q^{65} - 184 q^{67} + 1024 q^{69} + 881 q^{71} + 985 q^{73} + 1803 q^{75} + 1903 q^{77} + 658 q^{79} + 2296 q^{81} + 2773 q^{83} - 592 q^{85} + 3730 q^{87} + 234 q^{89} + 638 q^{91} - 904 q^{93} + 3660 q^{95} + 2174 q^{97} + 3582 q^{99}+O(q^{100}) $ 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9 - 33 * q^11 + 50 * q^13 - 22 * q^15 + 78 * q^17 - 158 * q^19 + 373 * q^21 - 76 * q^23 + 678 * q^25 - 469 * q^27 + 358 * q^29 - 390 * q^31 + 333 * q^33 - 546 * q^35 - 296 * q^37 - 442 * q^39 + 1141 * q^41 + 4 * q^43 + 1770 * q^45 + 735 * q^47 + 731 * q^49 - 132 * q^51 + 1349 * q^53 + 882 * q^55 + 2648 * q^57 - 1146 * q^59 + 1920 * q^61 + 534 * q^63 + 1748 * q^65 - 184 * q^67 + 1024 * q^69 + 881 * q^71 + 985 * q^73 + 1803 * q^75 + 1903 * q^77 + 658 * q^79 + 2296 * q^81 + 2773 * q^83 - 592 * q^85 + 3730 * q^87 + 234 * q^89 + 638 * q^91 - 904 * q^93 + 3660 * q^95 + 2174 * q^97 + 3582 * 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
$3$	−4.88695	−0.940495	−0.470247	−	0.882535i	$-0.655835\pi$
$3$	−4.88695	−0.940495	−0.470247	+	0.882535i	$0.655835\pi$
$4$	0	0
$5$	8.53113	0.763047	0.381524	−	0.924359i	$-0.375400\pi$
$5$	8.53113	0.763047	0.381524	+	0.924359i	$0.375400\pi$
$6$	0	0
$7$	−34.6761	−1.87234	−0.936168	−	0.351553i	$-0.885654\pi$
$7$	−34.6761	−1.87234	−0.936168	+	0.351553i	$0.885654\pi$
$8$	0	0
$9$	−3.11767	−0.115469
$10$	0	0
$11$	−39.6207	−1.08601	−0.543004	−	0.839730i	$-0.682713\pi$
$11$	−39.6207	−1.08601	−0.543004	+	0.839730i	$0.682713\pi$
$12$	0	0
$13$	−21.0951	−0.450056	−0.225028	−	0.974352i	$-0.572247\pi$
$13$	−21.0951	−0.450056	−0.225028	+	0.974352i	$0.572247\pi$
$14$	0	0
$15$	−41.6912	−0.717642
$16$	0	0
$17$	−16.3404	−0.233125	−0.116563	−	0.993183i	$-0.537188\pi$
$17$	−16.3404	−0.233125	−0.116563	+	0.993183i	$0.537188\pi$
$18$	0	0
$19$	65.9663	0.796511	0.398256	−	0.917274i	$-0.369616\pi$
$19$	65.9663	0.796511	0.398256	+	0.917274i	$0.369616\pi$
$20$	0	0
$21$	169.461	1.76092
$22$	0	0
$23$	−7.29485	−0.0661340	−0.0330670	−	0.999453i	$-0.510527\pi$
$23$	−7.29485	−0.0661340	−0.0330670	+	0.999453i	$0.510527\pi$
$24$	0	0
$25$	−52.2199	−0.417759
$26$	0	0
$27$	147.184	1.04909
$28$	0	0
$29$	115.536	0.739810	0.369905	−	0.929070i	$-0.379390\pi$
$29$	115.536	0.739810	0.369905	+	0.929070i	$0.379390\pi$
$30$	0	0
$31$	−131.179	−0.760013	−0.380007	−	0.924984i	$-0.624078\pi$
$31$	−131.179	−0.760013	−0.380007	+	0.924984i	$0.624078\pi$
$32$	0	0
$33$	193.624	1.02138
$34$	0	0
$35$	−295.827	−1.42868
$36$	0	0
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	0	0
$39$	103.091	0.423275
$40$	0	0
$41$	−214.780	−0.818122	−0.409061	−	0.912507i	$-0.634144\pi$
$41$	−214.780	−0.818122	−0.409061	+	0.912507i	$0.634144\pi$
$42$	0	0
$43$	−181.840	−0.644892	−0.322446	−	0.946588i	$-0.604505\pi$
$43$	−181.840	−0.644892	−0.322446	+	0.946588i	$0.604505\pi$
$44$	0	0
$45$	−26.5973	−0.0881086
$46$	0	0
$47$	254.513	0.789884	0.394942	−	0.918706i	$-0.370765\pi$
$47$	254.513	0.789884	0.394942	+	0.918706i	$0.370765\pi$
$48$	0	0
$49$	859.435	2.50564
$50$	0	0
$51$	79.8547	0.219253
$52$	0	0
$53$	143.485	0.371872	0.185936	−	0.982562i	$-0.440468\pi$
$53$	143.485	0.371872	0.185936	+	0.982562i	$0.440468\pi$
$54$	0	0
$55$	−338.009	−0.828675
$56$	0	0
$57$	−322.375	−0.749115
$58$	0	0
$59$	−234.441	−0.517317	−0.258658	−	0.965969i	$-0.583280\pi$
$59$	−234.441	−0.517317	−0.258658	+	0.965969i	$0.583280\pi$
$60$	0	0
$61$	−74.9905	−0.157403	−0.0787013	−	0.996898i	$-0.525077\pi$
$61$	−74.9905	−0.157403	−0.0787013	+	0.996898i	$0.525077\pi$
$62$	0	0
$63$	108.109	0.216198
$64$	0	0
$65$	−179.965	−0.343414
$66$	0	0
$67$	865.025	1.57731	0.788654	−	0.614838i	$-0.210778\pi$
$67$	865.025	1.57731	0.788654	+	0.614838i	$0.210778\pi$
$68$	0	0
$69$	35.6496	0.0621987
$70$	0	0
$71$	382.418	0.639220	0.319610	−	0.947549i	$-0.396448\pi$
$71$	382.418	0.639220	0.319610	+	0.947549i	$0.396448\pi$
$72$	0	0
$73$	851.842	1.36576	0.682881	−	0.730530i	$-0.260727\pi$
$73$	851.842	1.36576	0.682881	+	0.730530i	$0.260727\pi$
$74$	0	0
$75$	255.196	0.392900
$76$	0	0
$77$	1373.89	2.03337
$78$	0	0
$79$	−675.461	−0.961966	−0.480983	−	0.876730i	$-0.659720\pi$
$79$	−675.461	−0.961966	−0.480983	+	0.876730i	$0.659720\pi$
$80$	0	0
$81$	−635.103	−0.871197
$82$	0	0
$83$	−460.250	−0.608663	−0.304331	−	0.952566i	$-0.598433\pi$
$83$	−460.250	−0.608663	−0.304331	+	0.952566i	$0.598433\pi$
$84$	0	0
$85$	−139.402	−0.177885
$86$	0	0
$87$	−564.619	−0.695787
$88$	0	0
$89$	−205.537	−0.244796	−0.122398	−	0.992481i	$-0.539058\pi$
$89$	−205.537	−0.244796	−0.122398	+	0.992481i	$0.539058\pi$
$90$	0	0
$91$	731.496	0.842656
$92$	0	0
$93$	641.065	0.714788
$94$	0	0
$95$	562.767	0.607776
$96$	0	0
$97$	1012.06	1.05937	0.529684	−	0.848195i	$-0.322310\pi$
$97$	1012.06	1.05937	0.529684	+	0.848195i	$0.322310\pi$
$98$	0	0
$99$	123.524	0.125401
$100$	0	0
$101$	−768.499	−0.757114	−0.378557	−	0.925578i	$-0.623580\pi$
$101$	−768.499	−0.757114	−0.378557	+	0.925578i	$0.623580\pi$
$102$	0	0
$103$	1381.41	1.32149	0.660747	−	0.750609i	$-0.270240\pi$
$103$	1381.41	1.32149	0.660747	+	0.750609i	$0.270240\pi$
$104$	0	0
$105$	1445.69	1.34367
$106$	0	0
$107$	1857.00	1.67779	0.838894	−	0.544294i	$-0.183202\pi$
$107$	1857.00	1.67779	0.838894	+	0.544294i	$0.183202\pi$
$108$	0	0
$109$	928.847	0.816215	0.408108	−	0.912934i	$-0.366189\pi$
$109$	928.847	0.816215	0.408108	+	0.912934i	$0.366189\pi$
$110$	0	0
$111$	180.817	0.154616
$112$	0	0
$113$	104.630	0.0871040	0.0435520	−	0.999051i	$-0.486133\pi$
$113$	104.630	0.0871040	0.0435520	+	0.999051i	$0.486133\pi$
$114$	0	0
$115$	−62.2333	−0.0504633
$116$	0	0
$117$	65.7676	0.0519677
$118$	0	0
$119$	566.622	0.436488
$120$	0	0
$121$	238.797	0.179411
$122$	0	0
$123$	1049.62	0.769440
$124$	0	0
$125$	−1511.89	−1.08182
$126$	0	0
$127$	1871.74	1.30780	0.653899	−	0.756582i	$-0.273132\pi$
$127$	1871.74	1.30780	0.653899	+	0.756582i	$0.273132\pi$
$128$	0	0
$129$	888.644	0.606517
$130$	0	0
$131$	261.894	0.174670	0.0873350	−	0.996179i	$-0.472165\pi$
$131$	261.894	0.174670	0.0873350	+	0.996179i	$0.472165\pi$
$132$	0	0
$133$	−2287.46	−1.49134
$134$	0	0
$135$	1255.64	0.800507
$136$	0	0
$137$	−897.395	−0.559633	−0.279816	−	0.960054i	$-0.590274\pi$
$137$	−897.395	−0.559633	−0.279816	+	0.960054i	$0.590274\pi$
$138$	0	0
$139$	1722.54	1.05111	0.525554	−	0.850760i	$-0.323858\pi$
$139$	1722.54	1.05111	0.525554	+	0.850760i	$0.323858\pi$
$140$	0	0
$141$	−1243.79	−0.742882
$142$	0	0
$143$	835.802	0.488764
$144$	0	0
$145$	985.652	0.564510
$146$	0	0
$147$	−4200.02	−2.35654
$148$	0	0
$149$	−3084.97	−1.69618	−0.848088	−	0.529855i	$-0.822246\pi$
$149$	−3084.97	−1.69618	−0.848088	+	0.529855i	$0.822246\pi$
$150$	0	0
$151$	2783.85	1.50031	0.750155	−	0.661262i	$-0.229979\pi$
$151$	2783.85	1.50031	0.750155	+	0.661262i	$0.229979\pi$
$152$	0	0
$153$	50.9440	0.0269188
$154$	0	0
$155$	−1119.10	−0.579926
$156$	0	0
$157$	−665.985	−0.338544	−0.169272	−	0.985569i	$-0.554142\pi$
$157$	−665.985	−0.338544	−0.169272	+	0.985569i	$0.554142\pi$
$158$	0	0
$159$	−701.205	−0.349743
$160$	0	0
$161$	252.957	0.123825
$162$	0	0
$163$	−2602.10	−1.25038	−0.625190	−	0.780472i	$-0.714979\pi$
$163$	−2602.10	−1.25038	−0.625190	+	0.780472i	$0.714979\pi$
$164$	0	0
$165$	1651.83	0.779364
$166$	0	0
$167$	−4258.62	−1.97330	−0.986652	−	0.162845i	$-0.947933\pi$
$167$	−4258.62	−1.97330	−0.986652	+	0.162845i	$0.947933\pi$
$168$	0	0
$169$	−1752.00	−0.797450
$170$	0	0
$171$	−205.662	−0.0919727
$172$	0	0
$173$	−662.989	−0.291365	−0.145682	−	0.989331i	$-0.546538\pi$
$173$	−662.989	−0.291365	−0.145682	+	0.989331i	$0.546538\pi$
$174$	0	0
$175$	1810.78	0.782186
$176$	0	0
$177$	1145.70	0.486534
$178$	0	0
$179$	−2886.44	−1.20527	−0.602633	−	0.798019i	$-0.705882\pi$
$179$	−2886.44	−1.20527	−0.602633	+	0.798019i	$0.705882\pi$
$180$	0	0
$181$	−1824.28	−0.749158	−0.374579	−	0.927195i	$-0.622213\pi$
$181$	−1824.28	−0.749158	−0.374579	+	0.927195i	$0.622213\pi$
$182$	0	0
$183$	366.475	0.148036
$184$	0	0
$185$	−315.652	−0.125444
$186$	0	0
$187$	647.417	0.253175
$188$	0	0
$189$	−5103.76	−1.96425
$190$	0	0
$191$	4759.56	1.80309	0.901544	−	0.432688i	$-0.142435\pi$
$191$	4759.56	1.80309	0.901544	+	0.432688i	$0.142435\pi$
$192$	0	0
$193$	3369.07	1.25653	0.628267	−	0.777998i	$-0.283764\pi$
$193$	3369.07	1.25653	0.628267	+	0.777998i	$0.283764\pi$
$194$	0	0
$195$	879.480	0.322979
$196$	0	0
$197$	1503.77	0.543853	0.271926	−	0.962318i	$-0.412339\pi$
$197$	1503.77	0.543853	0.271926	+	0.962318i	$0.412339\pi$
$198$	0	0
$199$	−1835.77	−0.653941	−0.326971	−	0.945035i	$-0.606028\pi$
$199$	−1835.77	−0.653941	−0.326971	+	0.945035i	$0.606028\pi$
$200$	0	0
$201$	−4227.34	−1.48345
$202$	0	0
$203$	−4006.34	−1.38517
$204$	0	0
$205$	−1832.32	−0.624266
$206$	0	0
$207$	22.7430	0.00763645
$208$	0	0
$209$	−2613.63	−0.865017
$210$	0	0
$211$	−5624.92	−1.83524	−0.917619	−	0.397461i	$-0.869891\pi$
$211$	−5624.92	−1.83524	−0.917619	+	0.397461i	$0.869891\pi$
$212$	0	0
$213$	−1868.86	−0.601183
$214$	0	0
$215$	−1551.30	−0.492083
$216$	0	0
$217$	4548.77	1.42300
$218$	0	0
$219$	−4162.91	−1.28449
$220$	0	0
$221$	344.702	0.104919
$222$	0	0
$223$	−1729.21	−0.519266	−0.259633	−	0.965707i	$-0.583602\pi$
$223$	−1729.21	−0.519266	−0.259633	+	0.965707i	$0.583602\pi$
$224$	0	0
$225$	162.805	0.0482384
$226$	0	0
$227$	5685.61	1.66241	0.831205	−	0.555966i	$-0.187652\pi$
$227$	5685.61	1.66241	0.831205	+	0.555966i	$0.187652\pi$
$228$	0	0
$229$	2617.98	0.755462	0.377731	−	0.925915i	$-0.376704\pi$
$229$	2617.98	0.755462	0.377731	+	0.925915i	$0.376704\pi$
$230$	0	0
$231$	−6714.15	−1.91237
$232$	0	0
$233$	4027.36	1.13236	0.566182	−	0.824280i	$-0.308420\pi$
$233$	4027.36	1.13236	0.566182	+	0.824280i	$0.308420\pi$
$234$	0	0
$235$	2171.28	0.602719
$236$	0	0
$237$	3300.95	0.904724
$238$	0	0
$239$	4815.63	1.30334	0.651668	−	0.758505i	$-0.274070\pi$
$239$	4815.63	1.30334	0.651668	+	0.758505i	$0.274070\pi$
$240$	0	0
$241$	4052.58	1.08319	0.541597	−	0.840638i	$-0.317820\pi$
$241$	4052.58	1.08319	0.541597	+	0.840638i	$0.317820\pi$
$242$	0	0
$243$	−870.241	−0.229737
$244$	0	0
$245$	7331.95	1.91192
$246$	0	0
$247$	−1391.57	−0.358475
$248$	0	0
$249$	2249.22	0.572444
$250$	0	0
$251$	−5247.62	−1.31963	−0.659814	−	0.751429i	$-0.729365\pi$
$251$	−5247.62	−1.31963	−0.659814	+	0.751429i	$0.729365\pi$
$252$	0	0
$253$	289.027	0.0718220
$254$	0	0
$255$	681.251	0.167300
$256$	0	0
$257$	−4966.96	−1.20557	−0.602783	−	0.797905i	$-0.705941\pi$
$257$	−4966.96	−1.20557	−0.602783	+	0.797905i	$0.705941\pi$
$258$	0	0
$259$	1283.02	0.307810
$260$	0	0
$261$	−360.204	−0.0854254
$262$	0	0
$263$	3380.69	0.792631	0.396316	−	0.918114i	$-0.370289\pi$
$263$	3380.69	0.792631	0.396316	+	0.918114i	$0.370289\pi$
$264$	0	0
$265$	1224.09	0.283756
$266$	0	0
$267$	1004.45	0.230229
$268$	0	0
$269$	6566.04	1.48825	0.744124	−	0.668042i	$-0.232867\pi$
$269$	6566.04	1.48825	0.744124	+	0.668042i	$0.232867\pi$
$270$	0	0
$271$	2771.49	0.621241	0.310620	−	0.950534i	$-0.399463\pi$
$271$	2771.49	0.621241	0.310620	+	0.950534i	$0.399463\pi$
$272$	0	0
$273$	−3574.79	−0.792513
$274$	0	0
$275$	2068.99	0.453689
$276$	0	0
$277$	1470.01	0.318861	0.159431	−	0.987209i	$-0.449034\pi$
$277$	1470.01	0.318861	0.159431	+	0.987209i	$0.449034\pi$
$278$	0	0
$279$	408.973	0.0877583
$280$	0	0
$281$	−6949.56	−1.47536	−0.737679	−	0.675151i	$-0.764078\pi$
$281$	−6949.56	−1.47536	−0.737679	+	0.675151i	$0.764078\pi$
$282$	0	0
$283$	−8998.73	−1.89017	−0.945086	−	0.326820i	$-0.894023\pi$
$283$	−8998.73	−1.89017	−0.945086	+	0.326820i	$0.894023\pi$
$284$	0	0
$285$	−2750.22	−0.571610
$286$	0	0
$287$	7447.74	1.53180
$288$	0	0
$289$	−4645.99	−0.945653
$290$	0	0
$291$	−4945.87	−0.996331
$292$	0	0
$293$	3018.98	0.601948	0.300974	−	0.953632i	$-0.402688\pi$
$293$	3018.98	0.601948	0.300974	+	0.953632i	$0.402688\pi$
$294$	0	0
$295$	−2000.05	−0.394737
$296$	0	0
$297$	−5831.52	−1.13932
$298$	0	0
$299$	153.886	0.0297640
$300$	0	0
$301$	6305.51	1.20745
$302$	0	0
$303$	3755.62	0.712061
$304$	0	0
$305$	−639.754	−0.120106
$306$	0	0
$307$	−6611.23	−1.22906	−0.614532	−	0.788892i	$-0.710655\pi$
$307$	−6611.23	−1.22906	−0.614532	+	0.788892i	$0.710655\pi$
$308$	0	0
$309$	−6750.86	−1.24286
$310$	0	0
$311$	5833.58	1.06364	0.531820	−	0.846857i	$-0.321508\pi$
$311$	5833.58	1.06364	0.531820	+	0.846857i	$0.321508\pi$
$312$	0	0
$313$	−7175.79	−1.29584	−0.647922	−	0.761706i	$-0.724362\pi$
$313$	−7175.79	−1.29584	−0.647922	+	0.761706i	$0.724362\pi$
$314$	0	0
$315$	922.291	0.164969
$316$	0	0
$317$	2847.99	0.504603	0.252302	−	0.967649i	$-0.418813\pi$
$317$	2847.99	0.504603	0.252302	+	0.967649i	$0.418813\pi$
$318$	0	0
$319$	−4577.61	−0.803439
$320$	0	0
$321$	−9075.10	−1.57795
$322$	0	0
$323$	−1077.92	−0.185687
$324$	0	0
$325$	1101.58	0.188015
$326$	0	0
$327$	−4539.23	−0.767646
$328$	0	0
$329$	−8825.54	−1.47893
$330$	0	0
$331$	−8693.75	−1.44366	−0.721830	−	0.692070i	$-0.756699\pi$
$331$	−8693.75	−1.44366	−0.721830	+	0.692070i	$0.756699\pi$
$332$	0	0
$333$	115.354	0.0189831
$334$	0	0
$335$	7379.64	1.20356
$336$	0	0
$337$	8657.78	1.39946	0.699732	−	0.714405i	$-0.253303\pi$
$337$	8657.78	1.39946	0.699732	+	0.714405i	$0.253303\pi$
$338$	0	0
$339$	−511.322	−0.0819209
$340$	0	0
$341$	5197.39	0.825380
$342$	0	0
$343$	−17908.0	−2.81907
$344$	0	0
$345$	304.131	0.0474605
$346$	0	0
$347$	6541.09	1.01194	0.505971	−	0.862550i	$-0.331134\pi$
$347$	6541.09	1.01194	0.505971	+	0.862550i	$0.331134\pi$
$348$	0	0
$349$	2929.79	0.449364	0.224682	−	0.974432i	$-0.427866\pi$
$349$	2929.79	0.449364	0.224682	+	0.974432i	$0.427866\pi$
$350$	0	0
$351$	−3104.85	−0.472151
$352$	0	0
$353$	−4661.34	−0.702828	−0.351414	−	0.936220i	$-0.614299\pi$
$353$	−4661.34	−0.702828	−0.351414	+	0.936220i	$0.614299\pi$
$354$	0	0
$355$	3262.45	0.487755
$356$	0	0
$357$	−2769.05	−0.410515
$358$	0	0
$359$	795.892	0.117007	0.0585036	−	0.998287i	$-0.481367\pi$
$359$	795.892	0.117007	0.0585036	+	0.998287i	$0.481367\pi$
$360$	0	0
$361$	−2507.44	−0.365570
$362$	0	0
$363$	−1166.99	−0.168736
$364$	0	0
$365$	7267.17	1.04214
$366$	0	0
$367$	2119.84	0.301511	0.150755	−	0.988571i	$-0.451829\pi$
$367$	2119.84	0.301511	0.150755	+	0.988571i	$0.451829\pi$
$368$	0	0
$369$	669.614	0.0944681
$370$	0	0
$371$	−4975.51	−0.696269
$372$	0	0
$373$	7300.94	1.01348	0.506740	−	0.862099i	$-0.330850\pi$
$373$	7300.94	1.01348	0.506740	+	0.862099i	$0.330850\pi$
$374$	0	0
$375$	7388.51	1.01744
$376$	0	0
$377$	−2437.24	−0.332956
$378$	0	0
$379$	−7410.67	−1.00438	−0.502190	−	0.864757i	$-0.667472\pi$
$379$	−7410.67	−1.00438	−0.502190	+	0.864757i	$0.667472\pi$
$380$	0	0
$381$	−9147.12	−1.22998
$382$	0	0
$383$	2073.81	0.276676	0.138338	−	0.990385i	$-0.455824\pi$
$383$	2073.81	0.276676	0.138338	+	0.990385i	$0.455824\pi$
$384$	0	0
$385$	11720.8	1.55156
$386$	0	0
$387$	566.918	0.0744653
$388$	0	0
$389$	−7204.84	−0.939075	−0.469537	−	0.882913i	$-0.655579\pi$
$389$	−7204.84	−0.939075	−0.469537	+	0.882913i	$0.655579\pi$
$390$	0	0
$391$	119.201	0.0154175
$392$	0	0
$393$	−1279.86	−0.164276
$394$	0	0
$395$	−5762.44	−0.734025
$396$	0	0
$397$	−10953.8	−1.38478	−0.692389	−	0.721525i	$-0.743442\pi$
$397$	−10953.8	−1.38478	−0.692389	+	0.721525i	$0.743442\pi$
$398$	0	0
$399$	11178.7	1.40259
$400$	0	0
$401$	12632.3	1.57313	0.786567	−	0.617506i	$-0.211857\pi$
$401$	12632.3	1.57313	0.786567	+	0.617506i	$0.211857\pi$
$402$	0	0
$403$	2767.23	0.342048
$404$	0	0
$405$	−5418.14	−0.664765
$406$	0	0
$407$	1465.96	0.178538
$408$	0	0
$409$	−3729.10	−0.450837	−0.225418	−	0.974262i	$-0.572375\pi$
$409$	−3729.10	−0.450837	−0.225418	+	0.974262i	$0.572375\pi$
$410$	0	0
$411$	4385.53	0.526332
$412$	0	0
$413$	8129.52	0.968590
$414$	0	0
$415$	−3926.45	−0.464438
$416$	0	0
$417$	−8417.99	−0.988562
$418$	0	0
$419$	6325.64	0.737536	0.368768	−	0.929521i	$-0.379780\pi$
$419$	6325.64	0.737536	0.368768	+	0.929521i	$0.379780\pi$
$420$	0	0
$421$	10451.1	1.20987	0.604934	−	0.796276i	$-0.293199\pi$
$421$	10451.1	1.20987	0.604934	+	0.796276i	$0.293199\pi$
$422$	0	0
$423$	−793.489	−0.0912075
$424$	0	0
$425$	853.294	0.0973902
$426$	0	0
$427$	2600.38	0.294710
$428$	0	0
$429$	−4084.52	−0.459680
$430$	0	0
$431$	12370.6	1.38253	0.691267	−	0.722599i	$-0.257053\pi$
$431$	12370.6	1.38253	0.691267	+	0.722599i	$0.257053\pi$
$432$	0	0
$433$	5873.63	0.651891	0.325945	−	0.945389i	$-0.394317\pi$
$433$	5873.63	0.651891	0.325945	+	0.945389i	$0.394317\pi$
$434$	0	0
$435$	−4816.84	−0.530919
$436$	0	0
$437$	−481.215	−0.0526765
$438$	0	0
$439$	15763.3	1.71376	0.856882	−	0.515513i	$-0.172399\pi$
$439$	15763.3	1.71376	0.856882	+	0.515513i	$0.172399\pi$
$440$	0	0
$441$	−2679.44	−0.289325
$442$	0	0
$443$	10307.7	1.10549	0.552745	−	0.833350i	$-0.313580\pi$
$443$	10307.7	1.10549	0.552745	+	0.833350i	$0.313580\pi$
$444$	0	0
$445$	−1753.46	−0.186791
$446$	0	0
$447$	15076.1	1.59525
$448$	0	0
$449$	7698.62	0.809177	0.404589	−	0.914499i	$-0.367415\pi$
$449$	7698.62	0.809177	0.404589	+	0.914499i	$0.367415\pi$
$450$	0	0
$451$	8509.73	0.888486
$452$	0	0
$453$	−13604.6	−1.41103
$454$	0	0
$455$	6240.49	0.642986
$456$	0	0
$457$	−7302.83	−0.747510	−0.373755	−	0.927528i	$-0.621930\pi$
$457$	−7302.83	−0.747510	−0.373755	+	0.927528i	$0.621930\pi$
$458$	0	0
$459$	−2405.04	−0.244570
$460$	0	0
$461$	−9362.16	−0.945856	−0.472928	−	0.881101i	$-0.656803\pi$
$461$	−9362.16	−0.945856	−0.472928	+	0.881101i	$0.656803\pi$
$462$	0	0
$463$	251.785	0.0252731	0.0126365	−	0.999920i	$-0.495978\pi$
$463$	251.785	0.0252731	0.0126365	+	0.999920i	$0.495978\pi$
$464$	0	0
$465$	5469.00	0.545417
$466$	0	0
$467$	13453.4	1.33308	0.666539	−	0.745470i	$-0.267775\pi$
$467$	13453.4	1.33308	0.666539	+	0.745470i	$0.267775\pi$
$468$	0	0
$469$	−29995.7	−2.95325
$470$	0	0
$471$	3254.64	0.318399
$472$	0	0
$473$	7204.62	0.700357
$474$	0	0
$475$	−3444.76	−0.332750
$476$	0	0
$477$	−447.340	−0.0429398
$478$	0	0
$479$	−15016.3	−1.43238	−0.716191	−	0.697904i	$-0.754116\pi$
$479$	−15016.3	−1.43238	−0.716191	+	0.697904i	$0.754116\pi$
$480$	0	0
$481$	780.518	0.0739887
$482$	0	0
$483$	−1236.19	−0.116457
$484$	0	0
$485$	8633.98	0.808348
$486$	0	0
$487$	−15892.9	−1.47880	−0.739399	−	0.673268i	$-0.764890\pi$
$487$	−15892.9	−1.47880	−0.739399	+	0.673268i	$0.764890\pi$
$488$	0	0
$489$	12716.3	1.17598
$490$	0	0
$491$	−12824.5	−1.17874	−0.589368	−	0.807865i	$-0.700623\pi$
$491$	−12824.5	−1.17874	−0.589368	+	0.807865i	$0.700623\pi$
$492$	0	0
$493$	−1887.90	−0.172468
$494$	0	0
$495$	1053.80	0.0956866
$496$	0	0
$497$	−13260.8	−1.19683
$498$	0	0
$499$	3649.11	0.327368	0.163684	−	0.986513i	$-0.447662\pi$
$499$	3649.11	0.327368	0.163684	+	0.986513i	$0.447662\pi$
$500$	0	0
$501$	20811.7	1.85588
$502$	0	0
$503$	5721.32	0.507159	0.253579	−	0.967315i	$-0.418392\pi$
$503$	5721.32	0.507159	0.253579	+	0.967315i	$0.418392\pi$
$504$	0	0
$505$	−6556.16	−0.577713
$506$	0	0
$507$	8561.93	0.749997
$508$	0	0
$509$	19679.1	1.71368	0.856839	−	0.515583i	$-0.172425\pi$
$509$	19679.1	1.71368	0.856839	+	0.515583i	$0.172425\pi$
$510$	0	0
$511$	−29538.6	−2.55716
$512$	0	0
$513$	9709.17	0.835615
$514$	0	0
$515$	11784.9	1.00836
$516$	0	0
$517$	−10084.0	−0.857820
$518$	0	0
$519$	3239.99	0.274027
$520$	0	0
$521$	7616.99	0.640511	0.320256	−	0.947331i	$-0.396231\pi$
$521$	7616.99	0.640511	0.320256	+	0.947331i	$0.396231\pi$
$522$	0	0
$523$	−1064.29	−0.0889827	−0.0444914	−	0.999010i	$-0.514167\pi$
$523$	−1064.29	−0.0889827	−0.0444914	+	0.999010i	$0.514167\pi$
$524$	0	0
$525$	−8849.22	−0.735641
$526$	0	0
$527$	2143.51	0.177178
$528$	0	0
$529$	−12113.8	−0.995626
$530$	0	0
$531$	730.912	0.0597342
$532$	0	0
$533$	4530.81	0.368201
$534$	0	0
$535$	15842.3	1.28023
$536$	0	0
$537$	14105.9	1.13355
$538$	0	0
$539$	−34051.4	−2.72114
$540$	0	0
$541$	14970.6	1.18972	0.594859	−	0.803830i	$-0.297208\pi$
$541$	14970.6	1.18972	0.594859	+	0.803830i	$0.297208\pi$
$542$	0	0
$543$	8915.17	0.704579
$544$	0	0
$545$	7924.11	0.622810
$546$	0	0
$547$	4586.99	0.358548	0.179274	−	0.983799i	$-0.442625\pi$
$547$	4586.99	0.358548	0.179274	+	0.983799i	$0.442625\pi$
$548$	0	0
$549$	233.796	0.0181752
$550$	0	0
$551$	7621.48	0.589267
$552$	0	0
$553$	23422.4	1.80112
$554$	0	0
$555$	1542.58	0.117980
$556$	0	0
$557$	9941.29	0.756241	0.378120	−	0.925756i	$-0.376571\pi$
$557$	9941.29	0.756241	0.378120	+	0.925756i	$0.376571\pi$
$558$	0	0
$559$	3835.93	0.290237
$560$	0	0
$561$	−3163.90	−0.238110
$562$	0	0
$563$	22597.8	1.69162	0.845810	−	0.533484i	$-0.179118\pi$
$563$	22597.8	1.69162	0.845810	+	0.533484i	$0.179118\pi$
$564$	0	0
$565$	892.611	0.0664645
$566$	0	0
$567$	22022.9	1.63117
$568$	0	0
$569$	−15380.3	−1.13317	−0.566585	−	0.824003i	$-0.691736\pi$
$569$	−15380.3	−1.13317	−0.566585	+	0.824003i	$0.691736\pi$
$570$	0	0
$571$	−11541.7	−0.845895	−0.422947	−	0.906154i	$-0.639005\pi$
$571$	−11541.7	−0.845895	−0.422947	+	0.906154i	$0.639005\pi$
$572$	0	0
$573$	−23259.8	−1.69579
$574$	0	0
$575$	380.936	0.0276281
$576$	0	0
$577$	12144.8	0.876244	0.438122	−	0.898915i	$-0.355644\pi$
$577$	12144.8	0.876244	0.438122	+	0.898915i	$0.355644\pi$
$578$	0	0
$579$	−16464.5	−1.18176
$580$	0	0
$581$	15959.7	1.13962
$582$	0	0
$583$	−5684.98	−0.403855
$584$	0	0
$585$	561.072	0.0396538
$586$	0	0
$587$	−13094.7	−0.920741	−0.460370	−	0.887727i	$-0.652283\pi$
$587$	−13094.7	−0.920741	−0.460370	+	0.887727i	$0.652283\pi$
$588$	0	0
$589$	−8653.39	−0.605359
$590$	0	0
$591$	−7348.84	−0.511491
$592$	0	0
$593$	15246.4	1.05581	0.527903	−	0.849304i	$-0.322978\pi$
$593$	15246.4	1.05581	0.527903	+	0.849304i	$0.322978\pi$
$594$	0	0
$595$	4833.92	0.333061
$596$	0	0
$597$	8971.33	0.615028
$598$	0	0
$599$	1197.77	0.0817018	0.0408509	−	0.999165i	$-0.486993\pi$
$599$	1197.77	0.0817018	0.0408509	+	0.999165i	$0.486993\pi$
$600$	0	0
$601$	−3231.72	−0.219342	−0.109671	−	0.993968i	$-0.534980\pi$
$601$	−3231.72	−0.219342	−0.109671	+	0.993968i	$0.534980\pi$
$602$	0	0
$603$	−2696.87	−0.182131
$604$	0	0
$605$	2037.20	0.136899
$606$	0	0
$607$	10327.2	0.690557	0.345278	−	0.938500i	$-0.387784\pi$
$607$	10327.2	0.690557	0.345278	+	0.938500i	$0.387784\pi$
$608$	0	0
$609$	19578.8	1.30275
$610$	0	0
$611$	−5368.98	−0.355492
$612$	0	0
$613$	15969.5	1.05221	0.526103	−	0.850421i	$-0.323653\pi$
$613$	15969.5	1.05221	0.526103	+	0.850421i	$0.323653\pi$
$614$	0	0
$615$	8954.44	0.587119
$616$	0	0
$617$	−12473.6	−0.813884	−0.406942	−	0.913454i	$-0.633405\pi$
$617$	−12473.6	−0.813884	−0.406942	+	0.913454i	$0.633405\pi$
$618$	0	0
$619$	−9332.84	−0.606007	−0.303004	−	0.952989i	$-0.597989\pi$
$619$	−9332.84	−0.606007	−0.303004	+	0.952989i	$0.597989\pi$
$620$	0	0
$621$	−1073.68	−0.0693807
$622$	0	0
$623$	7127.22	0.458340
$624$	0	0
$625$	−6370.59	−0.407718
$626$	0	0
$627$	12772.7	0.813544
$628$	0	0
$629$	604.594	0.0383255
$630$	0	0
$631$	−11225.6	−0.708213	−0.354107	−	0.935205i	$-0.615215\pi$
$631$	−11225.6	−0.708213	−0.354107	+	0.935205i	$0.615215\pi$
$632$	0	0
$633$	27488.7	1.72603
$634$	0	0
$635$	15968.1	0.997911
$636$	0	0
$637$	−18129.9	−1.12768
$638$	0	0
$639$	−1192.25	−0.0738104
$640$	0	0
$641$	−2169.05	−0.133654	−0.0668269	−	0.997765i	$-0.521288\pi$
$641$	−2169.05	−0.133654	−0.0668269	+	0.997765i	$0.521288\pi$
$642$	0	0
$643$	13154.3	0.806776	0.403388	−	0.915029i	$-0.367833\pi$
$643$	13154.3	0.806776	0.403388	+	0.915029i	$0.367833\pi$
$644$	0	0
$645$	7581.13	0.462801
$646$	0	0
$647$	−11334.2	−0.688709	−0.344355	−	0.938840i	$-0.611902\pi$
$647$	−11334.2	−0.688709	−0.344355	+	0.938840i	$0.611902\pi$
$648$	0	0
$649$	9288.72	0.561809
$650$	0	0
$651$	−22229.7	−1.33832
$652$	0	0
$653$	16042.3	0.961384	0.480692	−	0.876889i	$-0.340385\pi$
$653$	16042.3	0.961384	0.480692	+	0.876889i	$0.340385\pi$
$654$	0	0
$655$	2234.25	0.133281
$656$	0	0
$657$	−2655.77	−0.157704
$658$	0	0
$659$	−16436.5	−0.971584	−0.485792	−	0.874075i	$-0.661469\pi$
$659$	−16436.5	−0.971584	−0.485792	+	0.874075i	$0.661469\pi$
$660$	0	0
$661$	27542.4	1.62069	0.810343	−	0.585955i	$-0.199281\pi$
$661$	27542.4	1.62069	0.810343	+	0.585955i	$0.199281\pi$
$662$	0	0
$663$	−1684.54	−0.0986761
$664$	0	0
$665$	−19514.6	−1.13796
$666$	0	0
$667$	−842.818	−0.0489266
$668$	0	0
$669$	8450.55	0.488367
$670$	0	0
$671$	2971.17	0.170940
$672$	0	0
$673$	25709.4	1.47255	0.736274	−	0.676684i	$-0.236584\pi$
$673$	25709.4	1.47255	0.736274	+	0.676684i	$0.236584\pi$
$674$	0	0
$675$	−7685.92	−0.438268
$676$	0	0
$677$	10268.4	0.582932	0.291466	−	0.956581i	$-0.405857\pi$
$677$	10268.4	0.582932	0.291466	+	0.956581i	$0.405857\pi$
$678$	0	0
$679$	−35094.2	−1.98349
$680$	0	0
$681$	−27785.3	−1.56349
$682$	0	0
$683$	9378.41	0.525410	0.262705	−	0.964876i	$-0.415385\pi$
$683$	9378.41	0.525410	0.262705	+	0.964876i	$0.415385\pi$
$684$	0	0
$685$	−7655.79	−0.427026
$686$	0	0
$687$	−12793.9	−0.710508
$688$	0	0
$689$	−3026.83	−0.167363
$690$	0	0
$691$	−9625.67	−0.529924	−0.264962	−	0.964259i	$-0.585359\pi$
$691$	−9625.67	−0.529924	−0.264962	+	0.964259i	$0.585359\pi$
$692$	0	0
$693$	−4283.35	−0.234792
$694$	0	0
$695$	14695.2	0.802045
$696$	0	0
$697$	3509.59	0.190725
$698$	0	0
$699$	−19681.5	−1.06498
$700$	0	0
$701$	−988.121	−0.0532394	−0.0266197	−	0.999646i	$-0.508474\pi$
$701$	−988.121	−0.0532394	−0.0266197	+	0.999646i	$0.508474\pi$
$702$	0	0
$703$	−2440.75	−0.130946
$704$	0	0
$705$	−10611.0	−0.566854
$706$	0	0
$707$	26648.6	1.41757
$708$	0	0
$709$	12441.9	0.659046	0.329523	−	0.944148i	$-0.393112\pi$
$709$	12441.9	0.659046	0.329523	+	0.944148i	$0.393112\pi$
$710$	0	0
$711$	2105.87	0.111078
$712$	0	0
$713$	956.930	0.0502627
$714$	0	0
$715$	7130.33	0.372950
$716$	0	0
$717$	−23533.8	−1.22578
$718$	0	0
$719$	−13807.2	−0.716162	−0.358081	−	0.933691i	$-0.616569\pi$
$719$	−13807.2	−0.716162	−0.358081	+	0.933691i	$0.616569\pi$
$720$	0	0
$721$	−47901.8	−2.47428
$722$	0	0
$723$	−19804.8	−1.01874
$724$	0	0
$725$	−6033.28	−0.309062
$726$	0	0
$727$	7675.95	0.391589	0.195794	−	0.980645i	$-0.437271\pi$
$727$	7675.95	0.391589	0.195794	+	0.980645i	$0.437271\pi$
$728$	0	0
$729$	21400.6	1.08726
$730$	0	0
$731$	2971.34	0.150340
$732$	0	0
$733$	21717.7	1.09435	0.547176	−	0.837017i	$-0.315703\pi$
$733$	21717.7	1.09435	0.547176	+	0.837017i	$0.315703\pi$
$734$	0	0
$735$	−35830.9	−1.79815
$736$	0	0
$737$	−34272.9	−1.71297
$738$	0	0
$739$	33592.6	1.67216	0.836079	−	0.548609i	$-0.184842\pi$
$739$	33592.6	1.67216	0.836079	+	0.548609i	$0.184842\pi$
$740$	0	0
$741$	6800.52	0.337144
$742$	0	0
$743$	14399.1	0.710972	0.355486	−	0.934682i	$-0.384315\pi$
$743$	14399.1	0.710972	0.355486	+	0.934682i	$0.384315\pi$
$744$	0	0
$745$	−26318.2	−1.29426
$746$	0	0
$747$	1434.91	0.0702819
$748$	0	0
$749$	−64393.8	−3.14138
$750$	0	0
$751$	21101.7	1.02532	0.512658	−	0.858593i	$-0.328661\pi$
$751$	21101.7	1.02532	0.512658	+	0.858593i	$0.328661\pi$
$752$	0	0
$753$	25644.9	1.24110
$754$	0	0
$755$	23749.4	1.14481
$756$	0	0
$757$	−9757.64	−0.468491	−0.234245	−	0.972178i	$-0.575262\pi$
$757$	−9757.64	−0.468491	−0.234245	+	0.972178i	$0.575262\pi$
$758$	0	0
$759$	−1412.46	−0.0675482
$760$	0	0
$761$	−26569.8	−1.26564	−0.632821	−	0.774298i	$-0.718103\pi$
$761$	−26569.8	−1.26564	−0.632821	+	0.774298i	$0.718103\pi$
$762$	0	0
$763$	−32208.8	−1.52823
$764$	0	0
$765$	434.610	0.0205403
$766$	0	0
$767$	4945.56	0.232821
$768$	0	0
$769$	23038.9	1.08037	0.540184	−	0.841547i	$-0.318354\pi$
$769$	23038.9	1.08037	0.540184	+	0.841547i	$0.318354\pi$
$770$	0	0
$771$	24273.3	1.13383
$772$	0	0
$773$	1084.39	0.0504562	0.0252281	−	0.999682i	$-0.491969\pi$
$773$	1084.39	0.0504562	0.0252281	+	0.999682i	$0.491969\pi$
$774$	0	0
$775$	6850.14	0.317503
$776$	0	0
$777$	−6270.05	−0.289494
$778$	0	0
$779$	−14168.3	−0.651644
$780$	0	0
$781$	−15151.6	−0.694197
$782$	0	0
$783$	17005.0	0.776130
$784$	0	0
$785$	−5681.60	−0.258325
$786$	0	0
$787$	−37992.1	−1.72080	−0.860402	−	0.509616i	$-0.829787\pi$
$787$	−37992.1	−1.72080	−0.860402	+	0.509616i	$0.829787\pi$
$788$	0	0
$789$	−16521.3	−0.745465
$790$	0	0
$791$	−3628.16	−0.163088
$792$	0	0
$793$	1581.93	0.0708399
$794$	0	0
$795$	−5982.07	−0.266871
$796$	0	0
$797$	−34021.1	−1.51203	−0.756017	−	0.654552i	$-0.772857\pi$
$797$	−34021.1	−1.51203	−0.756017	+	0.654552i	$0.772857\pi$
$798$	0	0
$799$	−4158.84	−0.184142
$800$	0	0
$801$	640.797	0.0282665
$802$	0	0
$803$	−33750.5	−1.48323
$804$	0	0
$805$	2158.01	0.0944843
$806$	0	0
$807$	−32087.9	−1.39969
$808$	0	0
$809$	18907.2	0.821683	0.410841	−	0.911707i	$-0.365235\pi$
$809$	18907.2	0.821683	0.410841	+	0.911707i	$0.365235\pi$
$810$	0	0
$811$	28630.6	1.23965	0.619826	−	0.784740i	$-0.287203\pi$
$811$	28630.6	1.23965	0.619826	+	0.784740i	$0.287203\pi$
$812$	0	0
$813$	−13544.2	−0.584274
$814$	0	0
$815$	−22198.8	−0.954099
$816$	0	0
$817$	−11995.3	−0.513664
$818$	0	0
$819$	−2280.57	−0.0973010
$820$	0	0
$821$	−10403.5	−0.442247	−0.221124	−	0.975246i	$-0.570972\pi$
$821$	−10403.5	−0.442247	−0.221124	+	0.975246i	$0.570972\pi$
$822$	0	0
$823$	35016.9	1.48313	0.741563	−	0.670883i	$-0.234085\pi$
$823$	35016.9	1.48313	0.741563	+	0.670883i	$0.234085\pi$
$824$	0	0
$825$	−10111.0	−0.426693
$826$	0	0
$827$	−12350.9	−0.519328	−0.259664	−	0.965699i	$-0.583612\pi$
$827$	−12350.9	−0.519328	−0.259664	+	0.965699i	$0.583612\pi$
$828$	0	0
$829$	27827.4	1.16584	0.582922	−	0.812528i	$-0.301909\pi$
$829$	27827.4	1.16584	0.582922	+	0.812528i	$0.301909\pi$
$830$	0	0
$831$	−7183.90	−0.299888
$832$	0	0
$833$	−14043.5	−0.584128
$834$	0	0
$835$	−36330.8	−1.50572
$836$	0	0
$837$	−19307.4	−0.797325
$838$	0	0
$839$	−15986.4	−0.657821	−0.328911	−	0.944361i	$-0.606681\pi$
$839$	−15986.4	−0.657821	−0.328911	+	0.944361i	$0.606681\pi$
$840$	0	0
$841$	−11040.4	−0.452681
$842$	0	0
$843$	33962.2	1.38757
$844$	0	0
$845$	−14946.5	−0.608492
$846$	0	0
$847$	−8280.55	−0.335918
$848$	0	0
$849$	43976.4	1.77770
$850$	0	0
$851$	269.909	0.0108724
$852$	0	0
$853$	−3197.49	−0.128347	−0.0641735	−	0.997939i	$-0.520441\pi$
$853$	−3197.49	−0.128347	−0.0641735	+	0.997939i	$0.520441\pi$
$854$	0	0
$855$	−1754.52	−0.0701795
$856$	0	0
$857$	44590.4	1.77734	0.888670	−	0.458548i	$-0.151630\pi$
$857$	44590.4	1.77734	0.888670	+	0.458548i	$0.151630\pi$
$858$	0	0
$859$	13085.8	0.519771	0.259885	−	0.965639i	$-0.416315\pi$
$859$	13085.8	0.519771	0.259885	+	0.965639i	$0.416315\pi$
$860$	0	0
$861$	−36396.8	−1.44065
$862$	0	0
$863$	6189.46	0.244139	0.122069	−	0.992522i	$-0.461047\pi$
$863$	6189.46	0.244139	0.122069	+	0.992522i	$0.461047\pi$
$864$	0	0
$865$	−5656.04	−0.222325
$866$	0	0
$867$	22704.8	0.889381
$868$	0	0
$869$	26762.2	1.04470
$870$	0	0
$871$	−18247.8	−0.709876
$872$	0	0
$873$	−3155.26	−0.122325
$874$	0	0
$875$	52426.3	2.02552
$876$	0	0
$877$	−11906.2	−0.458431	−0.229216	−	0.973376i	$-0.573616\pi$
$877$	−11906.2	−0.458431	−0.229216	+	0.973376i	$0.573616\pi$
$878$	0	0
$879$	−14753.6	−0.566129
$880$	0	0
$881$	25601.8	0.979055	0.489527	−	0.871988i	$-0.337169\pi$
$881$	25601.8	0.979055	0.489527	+	0.871988i	$0.337169\pi$
$882$	0	0
$883$	2762.44	0.105281	0.0526407	−	0.998614i	$-0.483236\pi$
$883$	2762.44	0.105281	0.0526407	+	0.998614i	$0.483236\pi$
$884$	0	0
$885$	9774.15	0.371248
$886$	0	0
$887$	42963.8	1.62636	0.813181	−	0.582011i	$-0.197734\pi$
$887$	42963.8	1.62636	0.813181	+	0.582011i	$0.197734\pi$
$888$	0	0
$889$	−64904.8	−2.44864
$890$	0	0
$891$	25163.2	0.946127
$892$	0	0
$893$	16789.3	0.629152
$894$	0	0
$895$	−24624.6	−0.919674
$896$	0	0
$897$	−752.032	−0.0279929
$898$	0	0
$899$	−15155.9	−0.562265
$900$	0	0
$901$	−2344.60	−0.0866926
$902$	0	0
$903$	−30814.7	−1.13560
$904$	0	0
$905$	−15563.1	−0.571643
$906$	0	0
$907$	−10266.8	−0.375857	−0.187929	−	0.982183i	$-0.560177\pi$
$907$	−10266.8	−0.375857	−0.187929	+	0.982183i	$0.560177\pi$
$908$	0	0
$909$	2395.93	0.0874235
$910$	0	0
$911$	−17384.6	−0.632247	−0.316124	−	0.948718i	$-0.602381\pi$
$911$	−17384.6	−0.632247	−0.316124	+	0.948718i	$0.602381\pi$
$912$	0	0
$913$	18235.4	0.661012
$914$	0	0
$915$	3126.45	0.112959
$916$	0	0
$917$	−9081.46	−0.327041
$918$	0	0
$919$	−35855.7	−1.28702	−0.643510	−	0.765438i	$-0.722523\pi$
$919$	−35855.7	−1.28702	−0.643510	+	0.765438i	$0.722523\pi$
$920$	0	0
$921$	32308.8	1.15593
$922$	0	0
$923$	−8067.13	−0.287685
$924$	0	0
$925$	1932.14	0.0686792
$926$	0	0
$927$	−4306.77	−0.152592
$928$	0	0
$929$	16894.3	0.596646	0.298323	−	0.954465i	$-0.403573\pi$
$929$	16894.3	0.596646	0.298323	+	0.954465i	$0.403573\pi$
$930$	0	0
$931$	56693.8	1.99577
$932$	0	0
$933$	−28508.4	−1.00035
$934$	0	0
$935$	5523.20	0.193185
$936$	0	0
$937$	−38445.7	−1.34041	−0.670206	−	0.742176i	$-0.733794\pi$
$937$	−38445.7	−1.34041	−0.670206	+	0.742176i	$0.733794\pi$
$938$	0	0
$939$	35067.7	1.21874
$940$	0	0
$941$	27735.3	0.960835	0.480417	−	0.877040i	$-0.340485\pi$
$941$	27735.3	0.960835	0.480417	+	0.877040i	$0.340485\pi$
$942$	0	0
$943$	1566.79	0.0541057
$944$	0	0
$945$	−43540.8	−1.49882
$946$	0	0
$947$	−36897.6	−1.26612	−0.633058	−	0.774105i	$-0.718200\pi$
$947$	−36897.6	−1.26612	−0.633058	+	0.774105i	$0.718200\pi$
$948$	0	0
$949$	−17969.7	−0.614669
$950$	0	0
$951$	−13918.0	−0.474577
$952$	0	0
$953$	−16611.3	−0.564629	−0.282314	−	0.959322i	$-0.591102\pi$
$953$	−16611.3	−0.564629	−0.282314	+	0.959322i	$0.591102\pi$
$954$	0	0
$955$	40604.4	1.37584
$956$	0	0
$957$	22370.6	0.755630
$958$	0	0
$959$	31118.2	1.04782
$960$	0	0
$961$	−12583.1	−0.422380
$962$	0	0
$963$	−5789.54	−0.193733
$964$	0	0
$965$	28742.0	0.958795
$966$	0	0
$967$	−43340.2	−1.44129	−0.720645	−	0.693305i	$-0.756154\pi$
$967$	−43340.2	−1.44129	−0.720645	+	0.693305i	$0.756154\pi$
$968$	0	0
$969$	5267.73	0.174637
$970$	0	0
$971$	7059.26	0.233308	0.116654	−	0.993173i	$-0.462783\pi$
$971$	7059.26	0.233308	0.116654	+	0.993173i	$0.462783\pi$
$972$	0	0
$973$	−59731.1	−1.96803
$974$	0	0
$975$	−5383.39	−0.176827
$976$	0	0
$977$	54881.8	1.79716	0.898580	−	0.438809i	$-0.144600\pi$
$977$	54881.8	1.79716	0.898580	+	0.438809i	$0.144600\pi$
$978$	0	0
$979$	8143.50	0.265850
$980$	0	0
$981$	−2895.84	−0.0942479
$982$	0	0
$983$	48347.8	1.56872	0.784361	−	0.620304i	$-0.212991\pi$
$983$	48347.8	1.56872	0.784361	+	0.620304i	$0.212991\pi$
$984$	0	0
$985$	12828.8	0.414985
$986$	0	0
$987$	43130.0	1.39092
$988$	0	0
$989$	1326.50	0.0426493
$990$	0	0
$991$	34191.7	1.09600	0.548000	−	0.836478i	$-0.315389\pi$
$991$	34191.7	1.09600	0.548000	+	0.836478i	$0.315389\pi$
$992$	0	0
$993$	42485.9	1.35775
$994$	0	0
$995$	−15661.2	−0.498988
$996$	0	0
$997$	−18588.3	−0.590468	−0.295234	−	0.955425i	$-0.595398\pi$
$997$	−18588.3	−0.590468	−0.295234	+	0.955425i	$0.595398\pi$
$998$	0	0
$999$	−5445.80	−0.172470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.4.a.k.1.3		8
4.3	odd	2		296.4.a.d.1.6	✓	8
8.3	odd	2		2368.4.a.u.1.3		8
8.5	even	2		2368.4.a.x.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.4.a.d.1.6	✓	8	4.3	odd	2
592.4.a.k.1.3		8	1.1	even	1	trivial
2368.4.a.u.1.3		8	8.3	odd	2
2368.4.a.x.1.6		8	8.5	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-4.88695 q^{3} +8.53113 q^{5} -34.6761 q^{7} -3.11767 q^{9} +O(q^{10})\)	\(q-4.88695 q^{3} +8.53113 q^{5} -34.6761 q^{7} -3.11767 q^{9} -39.6207 q^{11} -21.0951 q^{13} -41.6912 q^{15} -16.3404 q^{17} +65.9663 q^{19} +169.461 q^{21} -7.29485 q^{23} -52.2199 q^{25} +147.184 q^{27} +115.536 q^{29} -131.179 q^{31} +193.624 q^{33} -295.827 q^{35} -37.0000 q^{37} +103.091 q^{39} -214.780 q^{41} -181.840 q^{43} -26.5973 q^{45} +254.513 q^{47} +859.435 q^{49} +79.8547 q^{51} +143.485 q^{53} -338.009 q^{55} -322.375 q^{57} -234.441 q^{59} -74.9905 q^{61} +108.109 q^{63} -179.965 q^{65} +865.025 q^{67} +35.6496 q^{69} +382.418 q^{71} +851.842 q^{73} +255.196 q^{75} +1373.89 q^{77} -675.461 q^{79} -635.103 q^{81} -460.250 q^{83} -139.402 q^{85} -564.619 q^{87} -205.537 q^{89} +731.496 q^{91} +641.065 q^{93} +562.767 q^{95} +1012.06 q^{97} +123.524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9}+O(q^{10}) \) 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9	\( 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9} - 33 q^{11} + 50 q^{13} - 22 q^{15} + 78 q^{17} - 158 q^{19} + 373 q^{21} - 76 q^{23} + 678 q^{25} - 469 q^{27} + 358 q^{29} - 390 q^{31} + 333 q^{33} - 546 q^{35} - 296 q^{37} - 442 q^{39} + 1141 q^{41} + 4 q^{43} + 1770 q^{45} + 735 q^{47} + 731 q^{49} - 132 q^{51} + 1349 q^{53} + 882 q^{55} + 2648 q^{57} - 1146 q^{59} + 1920 q^{61} + 534 q^{63} + 1748 q^{65} - 184 q^{67} + 1024 q^{69} + 881 q^{71} + 985 q^{73} + 1803 q^{75} + 1903 q^{77} + 658 q^{79} + 2296 q^{81} + 2773 q^{83} - 592 q^{85} + 3730 q^{87} + 234 q^{89} + 638 q^{91} - 904 q^{93} + 3660 q^{95} + 2174 q^{97} + 3582 q^{99}+O(q^{100}) \) 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9 - 33 * q^11 + 50 * q^13 - 22 * q^15 + 78 * q^17 - 158 * q^19 + 373 * q^21 - 76 * q^23 + 678 * q^25 - 469 * q^27 + 358 * q^29 - 390 * q^31 + 333 * q^33 - 546 * q^35 - 296 * q^37 - 442 * q^39 + 1141 * q^41 + 4 * q^43 + 1770 * q^45 + 735 * q^47 + 731 * q^49 - 132 * q^51 + 1349 * q^53 + 882 * q^55 + 2648 * q^57 - 1146 * q^59 + 1920 * q^61 + 534 * q^63 + 1748 * q^65 - 184 * q^67 + 1024 * q^69 + 881 * q^71 + 985 * q^73 + 1803 * q^75 + 1903 * q^77 + 658 * q^79 + 2296 * q^81 + 2773 * q^83 - 592 * q^85 + 3730 * q^87 + 234 * q^89 + 638 * q^91 - 904 * q^93 + 3660 * q^95 + 2174 * q^97 + 3582 * q^99

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