LMFDB - Embedded newform 624.4.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.526440$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +19.3400 q^{5} -4.84136 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +19.3400 q^{5} -4.84136 q^{7} +9.00000 q^{9} +61.0728 q^{11} +13.0000 q^{13} -58.0199 q^{15} -41.7885 q^{17} +107.561 q^{19} +14.5241 q^{21} -28.5138 q^{23} +249.034 q^{25} -27.0000 q^{27} -89.8886 q^{29} -183.108 q^{31} -183.218 q^{33} -93.6318 q^{35} +418.029 q^{37} -39.0000 q^{39} -142.674 q^{41} +71.0935 q^{43} +174.060 q^{45} -323.711 q^{47} -319.561 q^{49} +125.365 q^{51} -25.1047 q^{53} +1181.15 q^{55} -322.683 q^{57} +684.508 q^{59} +308.125 q^{61} -43.5723 q^{63} +251.420 q^{65} -672.808 q^{67} +85.5413 q^{69} +326.837 q^{71} +24.3058 q^{73} -747.103 q^{75} -295.675 q^{77} -166.810 q^{79} +81.0000 q^{81} +201.093 q^{83} -808.188 q^{85} +269.666 q^{87} +108.834 q^{89} -62.9377 q^{91} +549.323 q^{93} +2080.23 q^{95} +1157.95 q^{97} +549.655 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 4 * q^5 - 30 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9} + 16 q^{11} + 39 q^{13} - 12 q^{15} - 146 q^{17} - 94 q^{19} + 90 q^{21} + 48 q^{23} + 145 q^{25} - 81 q^{27} - 2 q^{29} - 302 q^{31} - 48 q^{33} - 80 q^{35} + 374 q^{37} - 117 q^{39} + 480 q^{41} + 260 q^{43} + 36 q^{45} + 24 q^{47} + 447 q^{49} + 438 q^{51} - 678 q^{53} + 1552 q^{55} + 282 q^{57} + 1788 q^{59} + 230 q^{61} - 270 q^{63} + 52 q^{65} - 74 q^{67} - 144 q^{69} + 948 q^{71} - 222 q^{73} - 435 q^{75} + 112 q^{77} + 24 q^{79} + 243 q^{81} + 796 q^{83} - 248 q^{85} + 6 q^{87} + 1436 q^{89} - 390 q^{91} + 906 q^{93} + 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 4 * q^5 - 30 * q^7 + 27 * q^9 + 16 * q^11 + 39 * q^13 - 12 * q^15 - 146 * q^17 - 94 * q^19 + 90 * q^21 + 48 * q^23 + 145 * q^25 - 81 * q^27 - 2 * q^29 - 302 * q^31 - 48 * q^33 - 80 * q^35 + 374 * q^37 - 117 * q^39 + 480 * q^41 + 260 * q^43 + 36 * q^45 + 24 * q^47 + 447 * q^49 + 438 * q^51 - 678 * q^53 + 1552 * q^55 + 282 * q^57 + 1788 * q^59 + 230 * q^61 - 270 * q^63 + 52 * q^65 - 74 * q^67 - 144 * q^69 + 948 * q^71 - 222 * q^73 - 435 * q^75 + 112 * q^77 + 24 * q^79 + 243 * q^81 + 796 * q^83 - 248 * q^85 + 6 * q^87 + 1436 * q^89 - 390 * q^91 + 906 * q^93 + 4032 * q^95 + 3242 * q^97 + 144 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	19.3400	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$5$	19.3400	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$6$	0	0
$7$	−4.84136	−0.261409	−0.130704	−	0.991421i	$-0.541724\pi$
$7$	−4.84136	−0.261409	−0.130704	+	0.991421i	$0.541724\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	61.0728	1.67401	0.837006	−	0.547194i	$-0.184304\pi$
$11$	61.0728	1.67401	0.837006	+	0.547194i	$0.184304\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−58.0199	−0.998711
$16$	0	0
$17$	−41.7885	−0.596188	−0.298094	−	0.954537i	$-0.596351\pi$
$17$	−41.7885	−0.596188	−0.298094	+	0.954537i	$0.596351\pi$
$18$	0	0
$19$	107.561	1.29875	0.649374	−	0.760469i	$-0.275031\pi$
$19$	107.561	1.29875	0.649374	+	0.760469i	$0.275031\pi$
$20$	0	0
$21$	14.5241	0.150925
$22$	0	0
$23$	−28.5138	−0.258502	−0.129251	−	0.991612i	$-0.541257\pi$
$23$	−28.5138	−0.258502	−0.129251	+	0.991612i	$0.541257\pi$
$24$	0	0
$25$	249.034	1.99227
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−89.8886	−0.575583	−0.287791	−	0.957693i	$-0.592921\pi$
$29$	−89.8886	−0.575583	−0.287791	+	0.957693i	$0.592921\pi$
$30$	0	0
$31$	−183.108	−1.06087	−0.530437	−	0.847724i	$-0.677972\pi$
$31$	−183.108	−1.06087	−0.530437	+	0.847724i	$0.677972\pi$
$32$	0	0
$33$	−183.218	−0.966491
$34$	0	0
$35$	−93.6318	−0.452190
$36$	0	0
$37$	418.029	1.85739	0.928696	−	0.370843i	$-0.120931\pi$
$37$	418.029	1.85739	0.928696	+	0.370843i	$0.120931\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−142.674	−0.543460	−0.271730	−	0.962373i	$-0.587596\pi$
$41$	−142.674	−0.543460	−0.271730	+	0.962373i	$0.587596\pi$
$42$	0	0
$43$	71.0935	0.252132	0.126066	−	0.992022i	$-0.459765\pi$
$43$	71.0935	0.252132	0.126066	+	0.992022i	$0.459765\pi$
$44$	0	0
$45$	174.060	0.576606
$46$	0	0
$47$	−323.711	−1.00464	−0.502321	−	0.864681i	$-0.667520\pi$
$47$	−323.711	−1.00464	−0.502321	+	0.864681i	$0.667520\pi$
$48$	0	0
$49$	−319.561	−0.931665
$50$	0	0
$51$	125.365	0.344209
$52$	0	0
$53$	−25.1047	−0.0650641	−0.0325321	−	0.999471i	$-0.510357\pi$
$53$	−25.1047	−0.0650641	−0.0325321	+	0.999471i	$0.510357\pi$
$54$	0	0
$55$	1181.15	2.89574
$56$	0	0
$57$	−322.683	−0.749832
$58$	0	0
$59$	684.508	1.51043	0.755215	−	0.655477i	$-0.227533\pi$
$59$	684.508	1.51043	0.755215	+	0.655477i	$0.227533\pi$
$60$	0	0
$61$	308.125	0.646744	0.323372	−	0.946272i	$-0.395184\pi$
$61$	308.125	0.646744	0.323372	+	0.946272i	$0.395184\pi$
$62$	0	0
$63$	−43.5723	−0.0871363
$64$	0	0
$65$	251.420	0.479765
$66$	0	0
$67$	−672.808	−1.22681	−0.613407	−	0.789767i	$-0.710202\pi$
$67$	−672.808	−1.22681	−0.613407	+	0.789767i	$0.710202\pi$
$68$	0	0
$69$	85.5413	0.149246
$70$	0	0
$71$	326.837	0.546315	0.273158	−	0.961969i	$-0.411932\pi$
$71$	326.837	0.546315	0.273158	+	0.961969i	$0.411932\pi$
$72$	0	0
$73$	24.3058	0.0389695	0.0194847	−	0.999810i	$-0.493797\pi$
$73$	24.3058	0.0389695	0.0194847	+	0.999810i	$0.493797\pi$
$74$	0	0
$75$	−747.103	−1.15024
$76$	0	0
$77$	−295.675	−0.437602
$78$	0	0
$79$	−166.810	−0.237565	−0.118783	−	0.992920i	$-0.537899\pi$
$79$	−166.810	−0.237565	−0.118783	+	0.992920i	$0.537899\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	201.093	0.265938	0.132969	−	0.991120i	$-0.457549\pi$
$83$	201.093	0.265938	0.132969	+	0.991120i	$0.457549\pi$
$84$	0	0
$85$	−808.188	−1.03130
$86$	0	0
$87$	269.666	0.332313
$88$	0	0
$89$	108.834	0.129622	0.0648109	−	0.997898i	$-0.479356\pi$
$89$	108.834	0.129622	0.0648109	+	0.997898i	$0.479356\pi$
$90$	0	0
$91$	−62.9377	−0.0725018
$92$	0	0
$93$	549.323	0.612496
$94$	0	0
$95$	2080.23	2.24660
$96$	0	0
$97$	1157.95	1.21208	0.606041	−	0.795434i	$-0.292757\pi$
$97$	1157.95	1.21208	0.606041	+	0.795434i	$0.292757\pi$
$98$	0	0
$99$	549.655	0.558004
$100$	0	0
$101$	1702.75	1.67752	0.838761	−	0.544500i	$-0.183281\pi$
$101$	1702.75	1.67752	0.838761	+	0.544500i	$0.183281\pi$
$102$	0	0
$103$	1455.14	1.39203	0.696015	−	0.718027i	$-0.254955\pi$
$103$	1455.14	1.39203	0.696015	+	0.718027i	$0.254955\pi$
$104$	0	0
$105$	280.895	0.261072
$106$	0	0
$107$	822.762	0.743359	0.371679	−	0.928361i	$-0.378782\pi$
$107$	822.762	0.743359	0.371679	+	0.928361i	$0.378782\pi$
$108$	0	0
$109$	457.264	0.401816	0.200908	−	0.979610i	$-0.435611\pi$
$109$	457.264	0.401816	0.200908	+	0.979610i	$0.435611\pi$
$110$	0	0
$111$	−1254.09	−1.07237
$112$	0	0
$113$	−381.693	−0.317758	−0.158879	−	0.987298i	$-0.550788\pi$
$113$	−381.693	−0.317758	−0.158879	+	0.987298i	$0.550788\pi$
$114$	0	0
$115$	−551.456	−0.447161
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	202.313	0.155849
$120$	0	0
$121$	2398.88	1.80232
$122$	0	0
$123$	428.021	0.313767
$124$	0	0
$125$	2398.82	1.71645
$126$	0	0
$127$	1129.09	0.788905	0.394452	−	0.918916i	$-0.370934\pi$
$127$	1129.09	0.788905	0.394452	+	0.918916i	$0.370934\pi$
$128$	0	0
$129$	−213.281	−0.145568
$130$	0	0
$131$	852.761	0.568749	0.284374	−	0.958713i	$-0.408214\pi$
$131$	852.761	0.568749	0.284374	+	0.958713i	$0.408214\pi$
$132$	0	0
$133$	−520.742	−0.339504
$134$	0	0
$135$	−522.179	−0.332904
$136$	0	0
$137$	−488.903	−0.304889	−0.152445	−	0.988312i	$-0.548715\pi$
$137$	−488.903	−0.304889	−0.152445	+	0.988312i	$0.548715\pi$
$138$	0	0
$139$	−407.123	−0.248430	−0.124215	−	0.992255i	$-0.539641\pi$
$139$	−407.123	−0.248430	−0.124215	+	0.992255i	$0.539641\pi$
$140$	0	0
$141$	971.134	0.580030
$142$	0	0
$143$	793.946	0.464287
$144$	0	0
$145$	−1738.44	−0.995654
$146$	0	0
$147$	958.684	0.537897
$148$	0	0
$149$	1717.63	0.944388	0.472194	−	0.881495i	$-0.343462\pi$
$149$	1717.63	0.944388	0.472194	+	0.881495i	$0.343462\pi$
$150$	0	0
$151$	−1341.79	−0.723133	−0.361567	−	0.932346i	$-0.617758\pi$
$151$	−1341.79	−0.723133	−0.361567	+	0.932346i	$0.617758\pi$
$152$	0	0
$153$	−376.096	−0.198729
$154$	0	0
$155$	−3541.30	−1.83512
$156$	0	0
$157$	−760.546	−0.386612	−0.193306	−	0.981138i	$-0.561921\pi$
$157$	−760.546	−0.386612	−0.193306	+	0.981138i	$0.561921\pi$
$158$	0	0
$159$	75.3142	0.0375648
$160$	0	0
$161$	138.046	0.0675746
$162$	0	0
$163$	−2712.09	−1.30323	−0.651616	−	0.758549i	$-0.725909\pi$
$163$	−2712.09	−1.30323	−0.651616	+	0.758549i	$0.725909\pi$
$164$	0	0
$165$	−3543.44	−1.67185
$166$	0	0
$167$	−1551.69	−0.719004	−0.359502	−	0.933144i	$-0.617053\pi$
$167$	−1551.69	−0.719004	−0.359502	+	0.933144i	$0.617053\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	968.050	0.432916
$172$	0	0
$173$	−3970.26	−1.74482	−0.872409	−	0.488777i	$-0.837443\pi$
$173$	−3970.26	−1.74482	−0.872409	+	0.488777i	$0.837443\pi$
$174$	0	0
$175$	−1205.66	−0.520798
$176$	0	0
$177$	−2053.52	−0.872047
$178$	0	0
$179$	2690.95	1.12364	0.561818	−	0.827261i	$-0.310102\pi$
$179$	2690.95	1.12364	0.561818	+	0.827261i	$0.310102\pi$
$180$	0	0
$181$	−4371.10	−1.79503	−0.897517	−	0.440980i	$-0.854631\pi$
$181$	−4371.10	−1.79503	−0.897517	+	0.440980i	$0.854631\pi$
$182$	0	0
$183$	−924.375	−0.373398
$184$	0	0
$185$	8084.66	3.21295
$186$	0	0
$187$	−2552.14	−0.998026
$188$	0	0
$189$	130.717	0.0503082
$190$	0	0
$191$	−1408.47	−0.533578	−0.266789	−	0.963755i	$-0.585963\pi$
$191$	−1408.47	−0.533578	−0.266789	+	0.963755i	$0.585963\pi$
$192$	0	0
$193$	−4131.69	−1.54096	−0.770481	−	0.637463i	$-0.779984\pi$
$193$	−4131.69	−1.54096	−0.770481	+	0.637463i	$0.779984\pi$
$194$	0	0
$195$	−754.259	−0.276993
$196$	0	0
$197$	−3401.23	−1.23009	−0.615045	−	0.788492i	$-0.710862\pi$
$197$	−3401.23	−1.23009	−0.615045	+	0.788492i	$0.710862\pi$
$198$	0	0
$199$	3520.74	1.25416	0.627081	−	0.778954i	$-0.284249\pi$
$199$	3520.74	1.25416	0.627081	+	0.778954i	$0.284249\pi$
$200$	0	0
$201$	2018.42	0.708302
$202$	0	0
$203$	435.183	0.150463
$204$	0	0
$205$	−2759.30	−0.940088
$206$	0	0
$207$	−256.624	−0.0861672
$208$	0	0
$209$	6569.05	2.17412
$210$	0	0
$211$	2245.22	0.732545	0.366272	−	0.930508i	$-0.380634\pi$
$211$	2245.22	0.732545	0.366272	+	0.930508i	$0.380634\pi$
$212$	0	0
$213$	−980.510	−0.315415
$214$	0	0
$215$	1374.95	0.436142
$216$	0	0
$217$	886.490	0.277322
$218$	0	0
$219$	−72.9173	−0.0224990
$220$	0	0
$221$	−543.250	−0.165353
$222$	0	0
$223$	−3431.26	−1.03038	−0.515188	−	0.857077i	$-0.672278\pi$
$223$	−3431.26	−1.03038	−0.515188	+	0.857077i	$0.672278\pi$
$224$	0	0
$225$	2241.31	0.664091
$226$	0	0
$227$	4757.91	1.39116	0.695581	−	0.718448i	$-0.255147\pi$
$227$	4757.91	1.39116	0.695581	+	0.718448i	$0.255147\pi$
$228$	0	0
$229$	−4368.93	−1.26073	−0.630364	−	0.776300i	$-0.717094\pi$
$229$	−4368.93	−1.26073	−0.630364	+	0.776300i	$0.717094\pi$
$230$	0	0
$231$	887.026	0.252649
$232$	0	0
$233$	−3642.00	−1.02401	−0.512007	−	0.858981i	$-0.671098\pi$
$233$	−3642.00	−1.02401	−0.512007	+	0.858981i	$0.671098\pi$
$234$	0	0
$235$	−6260.57	−1.73785
$236$	0	0
$237$	500.431	0.137158
$238$	0	0
$239$	−2236.17	−0.605213	−0.302606	−	0.953116i	$-0.597857\pi$
$239$	−2236.17	−0.605213	−0.302606	+	0.953116i	$0.597857\pi$
$240$	0	0
$241$	6538.78	1.74772	0.873858	−	0.486181i	$-0.161610\pi$
$241$	6538.78	1.74772	0.873858	+	0.486181i	$0.161610\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−6180.30	−1.61161
$246$	0	0
$247$	1398.29	0.360208
$248$	0	0
$249$	−603.280	−0.153539
$250$	0	0
$251$	−2507.12	−0.630470	−0.315235	−	0.949014i	$-0.602083\pi$
$251$	−2507.12	−0.630470	−0.315235	+	0.949014i	$0.602083\pi$
$252$	0	0
$253$	−1741.42	−0.432735
$254$	0	0
$255$	2424.56	0.595420
$256$	0	0
$257$	−808.131	−0.196147	−0.0980735	−	0.995179i	$-0.531268\pi$
$257$	−808.131	−0.196147	−0.0980735	+	0.995179i	$0.531268\pi$
$258$	0	0
$259$	−2023.83	−0.485539
$260$	0	0
$261$	−808.998	−0.191861
$262$	0	0
$263$	−2940.70	−0.689472	−0.344736	−	0.938700i	$-0.612032\pi$
$263$	−2940.70	−0.689472	−0.344736	+	0.938700i	$0.612032\pi$
$264$	0	0
$265$	−485.525	−0.112549
$266$	0	0
$267$	−326.501	−0.0748371
$268$	0	0
$269$	7111.50	1.61188	0.805940	−	0.591997i	$-0.201660\pi$
$269$	7111.50	1.61188	0.805940	+	0.591997i	$0.201660\pi$
$270$	0	0
$271$	−2034.96	−0.456145	−0.228072	−	0.973644i	$-0.573242\pi$
$271$	−2034.96	−0.456145	−0.228072	+	0.973644i	$0.573242\pi$
$272$	0	0
$273$	188.813	0.0418589
$274$	0	0
$275$	15209.2	3.33509
$276$	0	0
$277$	2723.20	0.590689	0.295345	−	0.955391i	$-0.404566\pi$
$277$	2723.20	0.590689	0.295345	+	0.955391i	$0.404566\pi$
$278$	0	0
$279$	−1647.97	−0.353625
$280$	0	0
$281$	3265.56	0.693263	0.346632	−	0.938001i	$-0.387325\pi$
$281$	3265.56	0.693263	0.346632	+	0.938001i	$0.387325\pi$
$282$	0	0
$283$	−1144.02	−0.240299	−0.120150	−	0.992756i	$-0.538337\pi$
$283$	−1144.02	−0.240299	−0.120150	+	0.992756i	$0.538337\pi$
$284$	0	0
$285$	−6240.68	−1.29707
$286$	0	0
$287$	690.735	0.142065
$288$	0	0
$289$	−3166.72	−0.644560
$290$	0	0
$291$	−3473.85	−0.699796
$292$	0	0
$293$	−1677.35	−0.334444	−0.167222	−	0.985919i	$-0.553480\pi$
$293$	−1677.35	−0.334444	−0.167222	+	0.985919i	$0.553480\pi$
$294$	0	0
$295$	13238.4	2.61277
$296$	0	0
$297$	−1648.96	−0.322164
$298$	0	0
$299$	−370.679	−0.0716954
$300$	0	0
$301$	−344.190	−0.0659095
$302$	0	0
$303$	−5108.24	−0.968518
$304$	0	0
$305$	5959.13	1.11875
$306$	0	0
$307$	−7207.70	−1.33995	−0.669975	−	0.742383i	$-0.733695\pi$
$307$	−7207.70	−1.33995	−0.669975	+	0.742383i	$0.733695\pi$
$308$	0	0
$309$	−4365.42	−0.803689
$310$	0	0
$311$	−412.963	−0.0752958	−0.0376479	−	0.999291i	$-0.511987\pi$
$311$	−412.963	−0.0752958	−0.0376479	+	0.999291i	$0.511987\pi$
$312$	0	0
$313$	2936.39	0.530270	0.265135	−	0.964211i	$-0.414583\pi$
$313$	2936.39	0.530270	0.265135	+	0.964211i	$0.414583\pi$
$314$	0	0
$315$	−842.686	−0.150730
$316$	0	0
$317$	377.956	0.0669657	0.0334828	−	0.999439i	$-0.489340\pi$
$317$	377.956	0.0669657	0.0334828	+	0.999439i	$0.489340\pi$
$318$	0	0
$319$	−5489.75	−0.963533
$320$	0	0
$321$	−2468.28	−0.429178
$322$	0	0
$323$	−4494.81	−0.774298
$324$	0	0
$325$	3237.44	0.552557
$326$	0	0
$327$	−1371.79	−0.231989
$328$	0	0
$329$	1567.20	0.262622
$330$	0	0
$331$	4428.17	0.735330	0.367665	−	0.929958i	$-0.380157\pi$
$331$	4428.17	0.735330	0.367665	+	0.929958i	$0.380157\pi$
$332$	0	0
$333$	3762.26	0.619130
$334$	0	0
$335$	−13012.1	−2.12217
$336$	0	0
$337$	−1768.76	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$337$	−1768.76	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$338$	0	0
$339$	1145.08	0.183458
$340$	0	0
$341$	−11182.9	−1.77592
$342$	0	0
$343$	3207.70	0.504955
$344$	0	0
$345$	1654.37	0.258168
$346$	0	0
$347$	−2412.97	−0.373300	−0.186650	−	0.982426i	$-0.559763\pi$
$347$	−2412.97	−0.373300	−0.186650	+	0.982426i	$0.559763\pi$
$348$	0	0
$349$	−9967.45	−1.52878	−0.764392	−	0.644752i	$-0.776961\pi$
$349$	−9967.45	−1.52878	−0.764392	+	0.644752i	$0.776961\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	4516.30	0.680959	0.340479	−	0.940252i	$-0.389411\pi$
$353$	4516.30	0.680959	0.340479	+	0.940252i	$0.389411\pi$
$354$	0	0
$355$	6321.01	0.945027
$356$	0	0
$357$	−606.939	−0.0899794
$358$	0	0
$359$	−12159.8	−1.78767	−0.893833	−	0.448400i	$-0.851994\pi$
$359$	−12159.8	−1.78767	−0.893833	+	0.448400i	$0.851994\pi$
$360$	0	0
$361$	4710.38	0.686745
$362$	0	0
$363$	−7196.65	−1.04057
$364$	0	0
$365$	470.072	0.0674102
$366$	0	0
$367$	2674.25	0.380367	0.190183	−	0.981749i	$-0.439092\pi$
$367$	2674.25	0.380367	0.190183	+	0.981749i	$0.439092\pi$
$368$	0	0
$369$	−1284.06	−0.181153
$370$	0	0
$371$	121.541	0.0170083
$372$	0	0
$373$	9601.74	1.33287	0.666433	−	0.745564i	$-0.267820\pi$
$373$	9601.74	1.33287	0.666433	+	0.745564i	$0.267820\pi$
$374$	0	0
$375$	−7196.45	−0.990995
$376$	0	0
$377$	−1168.55	−0.159638
$378$	0	0
$379$	−9019.65	−1.22245	−0.611225	−	0.791457i	$-0.709323\pi$
$379$	−9019.65	−1.22245	−0.611225	+	0.791457i	$0.709323\pi$
$380$	0	0
$381$	−3387.28	−0.455474
$382$	0	0
$383$	4015.34	0.535703	0.267852	−	0.963460i	$-0.413686\pi$
$383$	4015.34	0.535703	0.267852	+	0.963460i	$0.413686\pi$
$384$	0	0
$385$	−5718.35	−0.756972
$386$	0	0
$387$	639.842	0.0840439
$388$	0	0
$389$	−2725.35	−0.355221	−0.177610	−	0.984101i	$-0.556837\pi$
$389$	−2725.35	−0.355221	−0.177610	+	0.984101i	$0.556837\pi$
$390$	0	0
$391$	1191.55	0.154115
$392$	0	0
$393$	−2558.28	−0.328367
$394$	0	0
$395$	−3226.11	−0.410945
$396$	0	0
$397$	−4391.59	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$397$	−4391.59	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$398$	0	0
$399$	1562.23	0.196013
$400$	0	0
$401$	3762.48	0.468552	0.234276	−	0.972170i	$-0.424728\pi$
$401$	3762.48	0.468552	0.234276	+	0.972170i	$0.424728\pi$
$402$	0	0
$403$	−2380.40	−0.294234
$404$	0	0
$405$	1566.54	0.192202
$406$	0	0
$407$	25530.2	3.10930
$408$	0	0
$409$	−6797.81	−0.821833	−0.410917	−	0.911673i	$-0.634791\pi$
$409$	−6797.81	−0.821833	−0.410917	+	0.911673i	$0.634791\pi$
$410$	0	0
$411$	1466.71	0.176028
$412$	0	0
$413$	−3313.95	−0.394840
$414$	0	0
$415$	3889.14	0.460025
$416$	0	0
$417$	1221.37	0.143431
$418$	0	0
$419$	12594.2	1.46841	0.734207	−	0.678925i	$-0.237554\pi$
$419$	12594.2	1.46841	0.734207	+	0.678925i	$0.237554\pi$
$420$	0	0
$421$	6888.04	0.797393	0.398697	−	0.917083i	$-0.369463\pi$
$421$	6888.04	0.797393	0.398697	+	0.917083i	$0.369463\pi$
$422$	0	0
$423$	−2913.40	−0.334881
$424$	0	0
$425$	−10406.8	−1.18777
$426$	0	0
$427$	−1491.74	−0.169065
$428$	0	0
$429$	−2381.84	−0.268056
$430$	0	0
$431$	7384.53	0.825291	0.412645	−	0.910892i	$-0.364605\pi$
$431$	7384.53	0.825291	0.412645	+	0.910892i	$0.364605\pi$
$432$	0	0
$433$	9068.33	1.00646	0.503229	−	0.864153i	$-0.332145\pi$
$433$	9068.33	1.00646	0.503229	+	0.864153i	$0.332145\pi$
$434$	0	0
$435$	5215.33	0.574841
$436$	0	0
$437$	−3066.97	−0.335728
$438$	0	0
$439$	−16875.4	−1.83466	−0.917331	−	0.398125i	$-0.869661\pi$
$439$	−16875.4	−1.83466	−0.917331	+	0.398125i	$0.869661\pi$
$440$	0	0
$441$	−2876.05	−0.310555
$442$	0	0
$443$	6766.18	0.725668	0.362834	−	0.931854i	$-0.381809\pi$
$443$	6766.18	0.725668	0.362834	+	0.931854i	$0.381809\pi$
$444$	0	0
$445$	2104.84	0.224222
$446$	0	0
$447$	−5152.90	−0.545243
$448$	0	0
$449$	140.944	0.0148141	0.00740706	−	0.999973i	$-0.497642\pi$
$449$	140.944	0.0148141	0.00740706	+	0.999973i	$0.497642\pi$
$450$	0	0
$451$	−8713.47	−0.909759
$452$	0	0
$453$	4025.36	0.417501
$454$	0	0
$455$	−1217.21	−0.125415
$456$	0	0
$457$	−17733.1	−1.81514	−0.907571	−	0.419898i	$-0.862066\pi$
$457$	−17733.1	−1.81514	−0.907571	+	0.419898i	$0.862066\pi$
$458$	0	0
$459$	1128.29	0.114736
$460$	0	0
$461$	2293.37	0.231699	0.115849	−	0.993267i	$-0.463041\pi$
$461$	2293.37	0.231699	0.115849	+	0.993267i	$0.463041\pi$
$462$	0	0
$463$	−13770.9	−1.38226	−0.691129	−	0.722731i	$-0.742887\pi$
$463$	−13770.9	−1.38226	−0.691129	+	0.722731i	$0.742887\pi$
$464$	0	0
$465$	10623.9	1.05951
$466$	0	0
$467$	3477.37	0.344568	0.172284	−	0.985047i	$-0.444885\pi$
$467$	3477.37	0.344568	0.172284	+	0.985047i	$0.444885\pi$
$468$	0	0
$469$	3257.31	0.320700
$470$	0	0
$471$	2281.64	0.223211
$472$	0	0
$473$	4341.88	0.422072
$474$	0	0
$475$	26786.4	2.58746
$476$	0	0
$477$	−225.943	−0.0216880
$478$	0	0
$479$	3137.39	0.299271	0.149636	−	0.988741i	$-0.452190\pi$
$479$	3137.39	0.299271	0.149636	+	0.988741i	$0.452190\pi$
$480$	0	0
$481$	5434.37	0.515148
$482$	0	0
$483$	−414.137	−0.0390142
$484$	0	0
$485$	22394.7	2.09668
$486$	0	0
$487$	−5996.52	−0.557964	−0.278982	−	0.960296i	$-0.589997\pi$
$487$	−5996.52	−0.557964	−0.278982	+	0.960296i	$0.589997\pi$
$488$	0	0
$489$	8136.26	0.752422
$490$	0	0
$491$	9401.49	0.864121	0.432060	−	0.901845i	$-0.357787\pi$
$491$	9401.49	0.864121	0.432060	+	0.901845i	$0.357787\pi$
$492$	0	0
$493$	3756.31	0.343156
$494$	0	0
$495$	10630.3	0.965246
$496$	0	0
$497$	−1582.34	−0.142812
$498$	0	0
$499$	5052.33	0.453253	0.226626	−	0.973982i	$-0.427230\pi$
$499$	5052.33	0.453253	0.226626	+	0.973982i	$0.427230\pi$
$500$	0	0
$501$	4655.08	0.415117
$502$	0	0
$503$	−8184.02	−0.725462	−0.362731	−	0.931894i	$-0.618156\pi$
$503$	−8184.02	−0.725462	−0.362731	+	0.931894i	$0.618156\pi$
$504$	0	0
$505$	32931.1	2.90181
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	6039.12	0.525892	0.262946	−	0.964811i	$-0.415306\pi$
$509$	6039.12	0.525892	0.262946	+	0.964811i	$0.415306\pi$
$510$	0	0
$511$	−117.673	−0.0101870
$512$	0	0
$513$	−2904.15	−0.249944
$514$	0	0
$515$	28142.3	2.40796
$516$	0	0
$517$	−19770.0	−1.68178
$518$	0	0
$519$	11910.8	1.00737
$520$	0	0
$521$	−14602.5	−1.22792	−0.613960	−	0.789337i	$-0.710424\pi$
$521$	−14602.5	−1.22792	−0.613960	+	0.789337i	$0.710424\pi$
$522$	0	0
$523$	−8910.70	−0.745005	−0.372502	−	0.928031i	$-0.621500\pi$
$523$	−8910.70	−0.745005	−0.372502	+	0.928031i	$0.621500\pi$
$524$	0	0
$525$	3616.99	0.300683
$526$	0	0
$527$	7651.79	0.632481
$528$	0	0
$529$	−11354.0	−0.933177
$530$	0	0
$531$	6160.57	0.503477
$532$	0	0
$533$	−1854.76	−0.150729
$534$	0	0
$535$	15912.2	1.28588
$536$	0	0
$537$	−8072.84	−0.648731
$538$	0	0
$539$	−19516.5	−1.55962
$540$	0	0
$541$	−13313.6	−1.05803	−0.529017	−	0.848611i	$-0.677439\pi$
$541$	−13313.6	−1.05803	−0.529017	+	0.848611i	$0.677439\pi$
$542$	0	0
$543$	13113.3	1.03636
$544$	0	0
$545$	8843.47	0.695069
$546$	0	0
$547$	4116.94	0.321806	0.160903	−	0.986970i	$-0.448559\pi$
$547$	4116.94	0.321806	0.160903	+	0.986970i	$0.448559\pi$
$548$	0	0
$549$	2773.12	0.215581
$550$	0	0
$551$	−9668.52	−0.747537
$552$	0	0
$553$	807.590	0.0621016
$554$	0	0
$555$	−24254.0	−1.85500
$556$	0	0
$557$	−6888.37	−0.524003	−0.262002	−	0.965067i	$-0.584383\pi$
$557$	−6888.37	−0.524003	−0.262002	+	0.965067i	$0.584383\pi$
$558$	0	0
$559$	924.216	0.0699288
$560$	0	0
$561$	7656.42	0.576211
$562$	0	0
$563$	−10537.1	−0.788782	−0.394391	−	0.918943i	$-0.629045\pi$
$563$	−10537.1	−0.788782	−0.394391	+	0.918943i	$0.629045\pi$
$564$	0	0
$565$	−7381.92	−0.549664
$566$	0	0
$567$	−392.150	−0.0290454
$568$	0	0
$569$	26930.1	1.98413	0.992065	−	0.125722i	$-0.0401248\pi$
$569$	26930.1	1.98413	0.992065	+	0.125722i	$0.0401248\pi$
$570$	0	0
$571$	3125.60	0.229076	0.114538	−	0.993419i	$-0.463461\pi$
$571$	3125.60	0.229076	0.114538	+	0.993419i	$0.463461\pi$
$572$	0	0
$573$	4225.42	0.308062
$574$	0	0
$575$	−7100.91	−0.515006
$576$	0	0
$577$	4787.13	0.345391	0.172696	−	0.984975i	$-0.444752\pi$
$577$	4787.13	0.345391	0.172696	+	0.984975i	$0.444752\pi$
$578$	0	0
$579$	12395.1	0.889675
$580$	0	0
$581$	−973.566	−0.0695186
$582$	0	0
$583$	−1533.22	−0.108918
$584$	0	0
$585$	2262.78	0.159922
$586$	0	0
$587$	−18380.8	−1.29243	−0.646214	−	0.763156i	$-0.723649\pi$
$587$	−18380.8	−1.29243	−0.646214	+	0.763156i	$0.723649\pi$
$588$	0	0
$589$	−19695.3	−1.37781
$590$	0	0
$591$	10203.7	0.710193
$592$	0	0
$593$	−13831.7	−0.957843	−0.478922	−	0.877858i	$-0.658972\pi$
$593$	−13831.7	−0.957843	−0.478922	+	0.877858i	$0.658972\pi$
$594$	0	0
$595$	3912.73	0.269590
$596$	0	0
$597$	−10562.2	−0.724091
$598$	0	0
$599$	−12248.6	−0.835502	−0.417751	−	0.908562i	$-0.637182\pi$
$599$	−12248.6	−0.835502	−0.417751	+	0.908562i	$0.637182\pi$
$600$	0	0
$601$	9719.56	0.659682	0.329841	−	0.944036i	$-0.393005\pi$
$601$	9719.56	0.659682	0.329841	+	0.944036i	$0.393005\pi$
$602$	0	0
$603$	−6055.27	−0.408938
$604$	0	0
$605$	46394.3	3.11768
$606$	0	0
$607$	−1607.83	−0.107512	−0.0537560	−	0.998554i	$-0.517119\pi$
$607$	−1607.83	−0.107512	−0.0537560	+	0.998554i	$0.517119\pi$
$608$	0	0
$609$	−1305.55	−0.0868696
$610$	0	0
$611$	−4208.25	−0.278638
$612$	0	0
$613$	14731.1	0.970610	0.485305	−	0.874345i	$-0.338709\pi$
$613$	14731.1	0.970610	0.485305	+	0.874345i	$0.338709\pi$
$614$	0	0
$615$	8277.91	0.542760
$616$	0	0
$617$	27951.8	1.82382	0.911909	−	0.410392i	$-0.134608\pi$
$617$	27951.8	1.82382	0.911909	+	0.410392i	$0.134608\pi$
$618$	0	0
$619$	−16200.2	−1.05192	−0.525961	−	0.850509i	$-0.676294\pi$
$619$	−16200.2	−1.05192	−0.525961	+	0.850509i	$0.676294\pi$
$620$	0	0
$621$	769.872	0.0497486
$622$	0	0
$623$	−526.903	−0.0338843
$624$	0	0
$625$	15263.8	0.976880
$626$	0	0
$627$	−19707.2	−1.25523
$628$	0	0
$629$	−17468.8	−1.10735
$630$	0	0
$631$	−12731.8	−0.803239	−0.401619	−	0.915807i	$-0.631553\pi$
$631$	−12731.8	−0.803239	−0.401619	+	0.915807i	$0.631553\pi$
$632$	0	0
$633$	−6735.65	−0.422935
$634$	0	0
$635$	21836.6	1.36466
$636$	0	0
$637$	−4154.30	−0.258397
$638$	0	0
$639$	2941.53	0.182105
$640$	0	0
$641$	−11556.9	−0.712119	−0.356059	−	0.934463i	$-0.615880\pi$
$641$	−11556.9	−0.712119	−0.356059	+	0.934463i	$0.615880\pi$
$642$	0	0
$643$	9181.25	0.563100	0.281550	−	0.959547i	$-0.409152\pi$
$643$	9181.25	0.563100	0.281550	+	0.959547i	$0.409152\pi$
$644$	0	0
$645$	−4124.84	−0.251807
$646$	0	0
$647$	−5244.11	−0.318651	−0.159326	−	0.987226i	$-0.550932\pi$
$647$	−5244.11	−0.318651	−0.159326	+	0.987226i	$0.550932\pi$
$648$	0	0
$649$	41804.8	2.52848
$650$	0	0
$651$	−2659.47	−0.160112
$652$	0	0
$653$	−16421.4	−0.984106	−0.492053	−	0.870565i	$-0.663753\pi$
$653$	−16421.4	−0.984106	−0.492053	+	0.870565i	$0.663753\pi$
$654$	0	0
$655$	16492.4	0.983832
$656$	0	0
$657$	218.752	0.0129898
$658$	0	0
$659$	−1838.11	−0.108653	−0.0543266	−	0.998523i	$-0.517301\pi$
$659$	−1838.11	−0.108653	−0.0543266	+	0.998523i	$0.517301\pi$
$660$	0	0
$661$	−5500.93	−0.323694	−0.161847	−	0.986816i	$-0.551745\pi$
$661$	−5500.93	−0.323694	−0.161847	+	0.986816i	$0.551745\pi$
$662$	0	0
$663$	1629.75	0.0954665
$664$	0	0
$665$	−10071.1	−0.587281
$666$	0	0
$667$	2563.07	0.148789
$668$	0	0
$669$	10293.8	0.594888
$670$	0	0
$671$	18818.0	1.08266
$672$	0	0
$673$	−25986.7	−1.48843	−0.744216	−	0.667939i	$-0.767177\pi$
$673$	−25986.7	−1.48843	−0.744216	+	0.667939i	$0.767177\pi$
$674$	0	0
$675$	−6723.92	−0.383413
$676$	0	0
$677$	−11691.3	−0.663714	−0.331857	−	0.943330i	$-0.607675\pi$
$677$	−11691.3	−0.663714	−0.331857	+	0.943330i	$0.607675\pi$
$678$	0	0
$679$	−5606.05	−0.316849
$680$	0	0
$681$	−14273.7	−0.803188
$682$	0	0
$683$	11111.5	0.622501	0.311251	−	0.950328i	$-0.399252\pi$
$683$	11111.5	0.622501	0.311251	+	0.950328i	$0.399252\pi$
$684$	0	0
$685$	−9455.37	−0.527403
$686$	0	0
$687$	13106.8	0.727882
$688$	0	0
$689$	−326.361	−0.0180455
$690$	0	0
$691$	7542.55	0.415242	0.207621	−	0.978209i	$-0.433428\pi$
$691$	7542.55	0.415242	0.207621	+	0.978209i	$0.433428\pi$
$692$	0	0
$693$	−2661.08	−0.145867
$694$	0	0
$695$	−7873.75	−0.429738
$696$	0	0
$697$	5962.11	0.324005
$698$	0	0
$699$	10926.0	0.591215
$700$	0	0
$701$	−8231.17	−0.443491	−0.221745	−	0.975105i	$-0.571175\pi$
$701$	−8231.17	−0.443491	−0.221745	+	0.975105i	$0.571175\pi$
$702$	0	0
$703$	44963.6	2.41228
$704$	0	0
$705$	18781.7	1.00335
$706$	0	0
$707$	−8243.61	−0.438519
$708$	0	0
$709$	28044.6	1.48553	0.742764	−	0.669554i	$-0.233515\pi$
$709$	28044.6	1.48553	0.742764	+	0.669554i	$0.233515\pi$
$710$	0	0
$711$	−1501.29	−0.0791884
$712$	0	0
$713$	5221.09	0.274238
$714$	0	0
$715$	15354.9	0.803133
$716$	0	0
$717$	6708.51	0.349420
$718$	0	0
$719$	−29686.4	−1.53980	−0.769901	−	0.638163i	$-0.779694\pi$
$719$	−29686.4	−1.53980	−0.769901	+	0.638163i	$0.779694\pi$
$720$	0	0
$721$	−7044.86	−0.363889
$722$	0	0
$723$	−19616.3	−1.00904
$724$	0	0
$725$	−22385.3	−1.14672
$726$	0	0
$727$	27654.5	1.41080	0.705398	−	0.708812i	$-0.250768\pi$
$727$	27654.5	1.41080	0.705398	+	0.708812i	$0.250768\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2970.89	−0.150318
$732$	0	0
$733$	−13077.5	−0.658975	−0.329488	−	0.944160i	$-0.606876\pi$
$733$	−13077.5	−0.658975	−0.329488	+	0.944160i	$0.606876\pi$
$734$	0	0
$735$	18540.9	0.930465
$736$	0	0
$737$	−41090.3	−2.05370
$738$	0	0
$739$	4218.33	0.209978	0.104989	−	0.994473i	$-0.466519\pi$
$739$	4218.33	0.209978	0.104989	+	0.994473i	$0.466519\pi$
$740$	0	0
$741$	−4194.88	−0.207966
$742$	0	0
$743$	−7725.54	−0.381457	−0.190728	−	0.981643i	$-0.561085\pi$
$743$	−7725.54	−0.381457	−0.190728	+	0.981643i	$0.561085\pi$
$744$	0	0
$745$	33218.9	1.63362
$746$	0	0
$747$	1809.84	0.0886460
$748$	0	0
$749$	−3983.29	−0.194321
$750$	0	0
$751$	−7506.12	−0.364717	−0.182358	−	0.983232i	$-0.558373\pi$
$751$	−7506.12	−0.364717	−0.182358	+	0.983232i	$0.558373\pi$
$752$	0	0
$753$	7521.35	0.364002
$754$	0	0
$755$	−25950.1	−1.25089
$756$	0	0
$757$	−2741.62	−0.131632	−0.0658162	−	0.997832i	$-0.520965\pi$
$757$	−2741.62	−0.131632	−0.0658162	+	0.997832i	$0.520965\pi$
$758$	0	0
$759$	5224.25	0.249839
$760$	0	0
$761$	29740.0	1.41666	0.708328	−	0.705883i	$-0.249450\pi$
$761$	29740.0	1.41666	0.708328	+	0.705883i	$0.249450\pi$
$762$	0	0
$763$	−2213.78	−0.105038
$764$	0	0
$765$	−7273.69	−0.343766
$766$	0	0
$767$	8898.60	0.418918
$768$	0	0
$769$	−19896.3	−0.933004	−0.466502	−	0.884520i	$-0.654486\pi$
$769$	−19896.3	−0.933004	−0.466502	+	0.884520i	$0.654486\pi$
$770$	0	0
$771$	2424.39	0.113246
$772$	0	0
$773$	−13601.3	−0.632866	−0.316433	−	0.948615i	$-0.602485\pi$
$773$	−13601.3	−0.632866	−0.316433	+	0.948615i	$0.602485\pi$
$774$	0	0
$775$	−45600.1	−2.11355
$776$	0	0
$777$	6071.48	0.280326
$778$	0	0
$779$	−15346.1	−0.705818
$780$	0	0
$781$	19960.8	0.914539
$782$	0	0
$783$	2426.99	0.110771
$784$	0	0
$785$	−14708.9	−0.668770
$786$	0	0
$787$	1498.29	0.0678631	0.0339315	−	0.999424i	$-0.489197\pi$
$787$	1498.29	0.0678631	0.0339315	+	0.999424i	$0.489197\pi$
$788$	0	0
$789$	8822.09	0.398067
$790$	0	0
$791$	1847.91	0.0830647
$792$	0	0
$793$	4005.62	0.179374
$794$	0	0
$795$	1456.57	0.0649803
$796$	0	0
$797$	−3713.30	−0.165034	−0.0825168	−	0.996590i	$-0.526296\pi$
$797$	−3713.30	−0.165034	−0.0825168	+	0.996590i	$0.526296\pi$
$798$	0	0
$799$	13527.4	0.598955
$800$	0	0
$801$	979.502	0.0432072
$802$	0	0
$803$	1484.42	0.0652354
$804$	0	0
$805$	2669.80	0.116892
$806$	0	0
$807$	−21334.5	−0.930620
$808$	0	0
$809$	−34527.6	−1.50053	−0.750263	−	0.661139i	$-0.770073\pi$
$809$	−34527.6	−1.50053	−0.750263	+	0.661139i	$0.770073\pi$
$810$	0	0
$811$	37279.2	1.61412	0.807059	−	0.590471i	$-0.201058\pi$
$811$	37279.2	1.61412	0.807059	+	0.590471i	$0.201058\pi$
$812$	0	0
$813$	6104.89	0.263355
$814$	0	0
$815$	−52451.6	−2.25436
$816$	0	0
$817$	7646.90	0.327455
$818$	0	0
$819$	−566.439	−0.0241673
$820$	0	0
$821$	13877.9	0.589943	0.294972	−	0.955506i	$-0.404690\pi$
$821$	13877.9	0.589943	0.294972	+	0.955506i	$0.404690\pi$
$822$	0	0
$823$	−18945.1	−0.802410	−0.401205	−	0.915988i	$-0.631408\pi$
$823$	−18945.1	−0.802410	−0.401205	+	0.915988i	$0.631408\pi$
$824$	0	0
$825$	−45627.6	−1.92552
$826$	0	0
$827$	7804.75	0.328171	0.164086	−	0.986446i	$-0.447533\pi$
$827$	7804.75	0.328171	0.164086	+	0.986446i	$0.447533\pi$
$828$	0	0
$829$	5784.85	0.242360	0.121180	−	0.992631i	$-0.461332\pi$
$829$	5784.85	0.242360	0.121180	+	0.992631i	$0.461332\pi$
$830$	0	0
$831$	−8169.59	−0.341035
$832$	0	0
$833$	13354.0	0.555448
$834$	0	0
$835$	−30009.7	−1.24375
$836$	0	0
$837$	4943.91	0.204165
$838$	0	0
$839$	5011.42	0.206214	0.103107	−	0.994670i	$-0.467122\pi$
$839$	5011.42	0.206214	0.103107	+	0.994670i	$0.467122\pi$
$840$	0	0
$841$	−16309.0	−0.668704
$842$	0	0
$843$	−9796.68	−0.400256
$844$	0	0
$845$	3268.45	0.133063
$846$	0	0
$847$	−11613.9	−0.471142
$848$	0	0
$849$	3432.05	0.138737
$850$	0	0
$851$	−11919.6	−0.480138
$852$	0	0
$853$	−22059.0	−0.885446	−0.442723	−	0.896659i	$-0.645987\pi$
$853$	−22059.0	−0.885446	−0.442723	+	0.896659i	$0.645987\pi$
$854$	0	0
$855$	18722.0	0.748866
$856$	0	0
$857$	13956.2	0.556283	0.278141	−	0.960540i	$-0.410282\pi$
$857$	13956.2	0.556283	0.278141	+	0.960540i	$0.410282\pi$
$858$	0	0
$859$	−12498.5	−0.496442	−0.248221	−	0.968703i	$-0.579846\pi$
$859$	−12498.5	−0.496442	−0.248221	+	0.968703i	$0.579846\pi$
$860$	0	0
$861$	−2072.20	−0.0820215
$862$	0	0
$863$	38631.2	1.52378	0.761890	−	0.647707i	$-0.224272\pi$
$863$	38631.2	1.52378	0.761890	+	0.647707i	$0.224272\pi$
$864$	0	0
$865$	−76784.7	−3.01822
$866$	0	0
$867$	9500.17	0.372137
$868$	0	0
$869$	−10187.6	−0.397687
$870$	0	0
$871$	−8746.51	−0.340257
$872$	0	0
$873$	10421.5	0.404027
$874$	0	0
$875$	−11613.5	−0.448696
$876$	0	0
$877$	−856.756	−0.0329881	−0.0164941	−	0.999864i	$-0.505250\pi$
$877$	−856.756	−0.0329881	−0.0164941	+	0.999864i	$0.505250\pi$
$878$	0	0
$879$	5032.06	0.193091
$880$	0	0
$881$	33638.6	1.28640	0.643198	−	0.765700i	$-0.277607\pi$
$881$	33638.6	1.28640	0.643198	+	0.765700i	$0.277607\pi$
$882$	0	0
$883$	31109.1	1.18562	0.592811	−	0.805342i	$-0.298018\pi$
$883$	31109.1	1.18562	0.592811	+	0.805342i	$0.298018\pi$
$884$	0	0
$885$	−39715.1	−1.50848
$886$	0	0
$887$	−26080.3	−0.987248	−0.493624	−	0.869675i	$-0.664328\pi$
$887$	−26080.3	−0.987248	−0.493624	+	0.869675i	$0.664328\pi$
$888$	0	0
$889$	−5466.35	−0.206227
$890$	0	0
$891$	4946.89	0.186001
$892$	0	0
$893$	−34818.8	−1.30478
$894$	0	0
$895$	52042.8	1.94369
$896$	0	0
$897$	1112.04	0.0413934
$898$	0	0
$899$	16459.3	0.610621
$900$	0	0
$901$	1049.09	0.0387905
$902$	0	0
$903$	1032.57	0.0380529
$904$	0	0
$905$	−84536.9	−3.10508
$906$	0	0
$907$	−20169.0	−0.738369	−0.369184	−	0.929356i	$-0.620363\pi$
$907$	−20169.0	−0.738369	−0.369184	+	0.929356i	$0.620363\pi$
$908$	0	0
$909$	15324.7	0.559174
$910$	0	0
$911$	−19982.2	−0.726716	−0.363358	−	0.931650i	$-0.618370\pi$
$911$	−19982.2	−0.726716	−0.363358	+	0.931650i	$0.618370\pi$
$912$	0	0
$913$	12281.3	0.445184
$914$	0	0
$915$	−17877.4	−0.645910
$916$	0	0
$917$	−4128.52	−0.148676
$918$	0	0
$919$	−38513.1	−1.38241	−0.691203	−	0.722661i	$-0.742919\pi$
$919$	−38513.1	−1.38241	−0.691203	+	0.722661i	$0.742919\pi$
$920$	0	0
$921$	21623.1	0.773621
$922$	0	0
$923$	4248.88	0.151521
$924$	0	0
$925$	104103.	3.70043
$926$	0	0
$927$	13096.3	0.464010
$928$	0	0
$929$	23218.9	0.820009	0.410005	−	0.912083i	$-0.365527\pi$
$929$	23218.9	0.820009	0.410005	+	0.912083i	$0.365527\pi$
$930$	0	0
$931$	−34372.3	−1.21000
$932$	0	0
$933$	1238.89	0.0434721
$934$	0	0
$935$	−49358.3	−1.72640
$936$	0	0
$937$	−11112.9	−0.387452	−0.193726	−	0.981056i	$-0.562057\pi$
$937$	−11112.9	−0.387452	−0.193726	+	0.981056i	$0.562057\pi$
$938$	0	0
$939$	−8809.17	−0.306152
$940$	0	0
$941$	45570.4	1.57869	0.789347	−	0.613947i	$-0.210419\pi$
$941$	45570.4	1.57869	0.789347	+	0.613947i	$0.210419\pi$
$942$	0	0
$943$	4068.16	0.140485
$944$	0	0
$945$	2528.06	0.0870240
$946$	0	0
$947$	−34903.9	−1.19770	−0.598852	−	0.800860i	$-0.704376\pi$
$947$	−34903.9	−1.19770	−0.598852	+	0.800860i	$0.704376\pi$
$948$	0	0
$949$	315.975	0.0108082
$950$	0	0
$951$	−1133.87	−0.0386626
$952$	0	0
$953$	−2886.52	−0.0981151	−0.0490575	−	0.998796i	$-0.515622\pi$
$953$	−2886.52	−0.0981151	−0.0490575	+	0.998796i	$0.515622\pi$
$954$	0	0
$955$	−27239.8	−0.922994
$956$	0	0
$957$	16469.2	0.556296
$958$	0	0
$959$	2366.96	0.0797008
$960$	0	0
$961$	3737.42	0.125455
$962$	0	0
$963$	7404.85	0.247786
$964$	0	0
$965$	−79906.8	−2.66559
$966$	0	0
$967$	−11593.8	−0.385556	−0.192778	−	0.981242i	$-0.561750\pi$
$967$	−11593.8	−0.385556	−0.192778	+	0.981242i	$0.561750\pi$
$968$	0	0
$969$	13484.4	0.447041
$970$	0	0
$971$	4952.12	0.163667	0.0818337	−	0.996646i	$-0.473922\pi$
$971$	4952.12	0.163667	0.0818337	+	0.996646i	$0.473922\pi$
$972$	0	0
$973$	1971.03	0.0649418
$974$	0	0
$975$	−9712.33	−0.319019
$976$	0	0
$977$	−19650.1	−0.643462	−0.321731	−	0.946831i	$-0.604265\pi$
$977$	−19650.1	−0.643462	−0.321731	+	0.946831i	$0.604265\pi$
$978$	0	0
$979$	6646.77	0.216988
$980$	0	0
$981$	4115.38	0.133939
$982$	0	0
$983$	−56818.4	−1.84357	−0.921783	−	0.387707i	$-0.873267\pi$
$983$	−56818.4	−1.84357	−0.921783	+	0.387707i	$0.873267\pi$
$984$	0	0
$985$	−65779.7	−2.12783
$986$	0	0
$987$	−4701.61	−0.151625
$988$	0	0
$989$	−2027.15	−0.0651764
$990$	0	0
$991$	19120.4	0.612897	0.306448	−	0.951887i	$-0.400859\pi$
$991$	19120.4	0.612897	0.306448	+	0.951887i	$0.400859\pi$
$992$	0	0
$993$	−13284.5	−0.424543
$994$	0	0
$995$	68090.9	2.16947
$996$	0	0
$997$	38887.9	1.23530	0.617650	−	0.786453i	$-0.288085\pi$
$997$	38887.9	1.23530	0.617650	+	0.786453i	$0.288085\pi$
$998$	0	0
$999$	−11286.8	−0.357455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.t.1.3		3
3.2	odd	2		1872.4.a.bk.1.1		3
4.3	odd	2		39.4.a.c.1.2	✓	3
8.3	odd	2		2496.4.a.bl.1.1		3
8.5	even	2		2496.4.a.bp.1.1		3
12.11	even	2		117.4.a.f.1.2		3
20.19	odd	2		975.4.a.l.1.2		3
28.27	even	2		1911.4.a.k.1.2		3
52.31	even	4		507.4.b.g.337.3		6
52.47	even	4		507.4.b.g.337.4		6
52.51	odd	2		507.4.a.h.1.2		3
156.155	even	2		1521.4.a.u.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.2	✓	3	4.3	odd	2
117.4.a.f.1.2		3	12.11	even	2
507.4.a.h.1.2		3	52.51	odd	2
507.4.b.g.337.3		6	52.31	even	4
507.4.b.g.337.4		6	52.47	even	4
624.4.a.t.1.3		3	1.1	even	1	trivial
975.4.a.l.1.2		3	20.19	odd	2
1521.4.a.u.1.2		3	156.155	even	2
1872.4.a.bk.1.1		3	3.2	odd	2
1911.4.a.k.1.2		3	28.27	even	2
2496.4.a.bl.1.1		3	8.3	odd	2
2496.4.a.bp.1.1		3	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +19.3400 q^{5} -4.84136 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +19.3400 q^{5} -4.84136 q^{7} +9.00000 q^{9} +61.0728 q^{11} +13.0000 q^{13} -58.0199 q^{15} -41.7885 q^{17} +107.561 q^{19} +14.5241 q^{21} -28.5138 q^{23} +249.034 q^{25} -27.0000 q^{27} -89.8886 q^{29} -183.108 q^{31} -183.218 q^{33} -93.6318 q^{35} +418.029 q^{37} -39.0000 q^{39} -142.674 q^{41} +71.0935 q^{43} +174.060 q^{45} -323.711 q^{47} -319.561 q^{49} +125.365 q^{51} -25.1047 q^{53} +1181.15 q^{55} -322.683 q^{57} +684.508 q^{59} +308.125 q^{61} -43.5723 q^{63} +251.420 q^{65} -672.808 q^{67} +85.5413 q^{69} +326.837 q^{71} +24.3058 q^{73} -747.103 q^{75} -295.675 q^{77} -166.810 q^{79} +81.0000 q^{81} +201.093 q^{83} -808.188 q^{85} +269.666 q^{87} +108.834 q^{89} -62.9377 q^{91} +549.323 q^{93} +2080.23 q^{95} +1157.95 q^{97} +549.655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 4 * q^5 - 30 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9} + 16 q^{11} + 39 q^{13} - 12 q^{15} - 146 q^{17} - 94 q^{19} + 90 q^{21} + 48 q^{23} + 145 q^{25} - 81 q^{27} - 2 q^{29} - 302 q^{31} - 48 q^{33} - 80 q^{35} + 374 q^{37} - 117 q^{39} + 480 q^{41} + 260 q^{43} + 36 q^{45} + 24 q^{47} + 447 q^{49} + 438 q^{51} - 678 q^{53} + 1552 q^{55} + 282 q^{57} + 1788 q^{59} + 230 q^{61} - 270 q^{63} + 52 q^{65} - 74 q^{67} - 144 q^{69} + 948 q^{71} - 222 q^{73} - 435 q^{75} + 112 q^{77} + 24 q^{79} + 243 q^{81} + 796 q^{83} - 248 q^{85} + 6 q^{87} + 1436 q^{89} - 390 q^{91} + 906 q^{93} + 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 4 * q^5 - 30 * q^7 + 27 * q^9 + 16 * q^11 + 39 * q^13 - 12 * q^15 - 146 * q^17 - 94 * q^19 + 90 * q^21 + 48 * q^23 + 145 * q^25 - 81 * q^27 - 2 * q^29 - 302 * q^31 - 48 * q^33 - 80 * q^35 + 374 * q^37 - 117 * q^39 + 480 * q^41 + 260 * q^43 + 36 * q^45 + 24 * q^47 + 447 * q^49 + 438 * q^51 - 678 * q^53 + 1552 * q^55 + 282 * q^57 + 1788 * q^59 + 230 * q^61 - 270 * q^63 + 52 * q^65 - 74 * q^67 - 144 * q^69 + 948 * q^71 - 222 * q^73 - 435 * q^75 + 112 * q^77 + 24 * q^79 + 243 * q^81 + 796 * q^83 - 248 * q^85 + 6 * q^87 + 1436 * q^89 - 390 * q^91 + 906 * q^93 + 4032 * q^95 + 3242 * q^97 + 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more