LMFDB - Embedded newform 2151.4.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.41344 q^{2} +11.4784 q^{4} -16.6195 q^{5} -34.6158 q^{7} -15.3518 q^{8} +O(q^{10})$	\(q-4.41344 q^{2} +11.4784 q^{4} -16.6195 q^{5} -34.6158 q^{7} -15.3518 q^{8} +73.3492 q^{10} +53.4703 q^{11} +52.0679 q^{13} +152.775 q^{14} -24.0733 q^{16} +22.2133 q^{17} +89.6753 q^{19} -190.766 q^{20} -235.988 q^{22} +123.698 q^{23} +151.208 q^{25} -229.798 q^{26} -397.335 q^{28} +160.769 q^{29} +11.9376 q^{31} +229.060 q^{32} -98.0368 q^{34} +575.298 q^{35} +320.535 q^{37} -395.776 q^{38} +255.139 q^{40} +263.340 q^{41} +275.780 q^{43} +613.755 q^{44} -545.934 q^{46} -468.006 q^{47} +855.253 q^{49} -667.348 q^{50} +597.657 q^{52} -594.250 q^{53} -888.651 q^{55} +531.414 q^{56} -709.544 q^{58} +538.732 q^{59} -335.134 q^{61} -52.6859 q^{62} -818.356 q^{64} -865.343 q^{65} -408.017 q^{67} +254.973 q^{68} -2539.04 q^{70} +353.397 q^{71} +663.280 q^{73} -1414.66 q^{74} +1029.33 q^{76} -1850.92 q^{77} +320.302 q^{79} +400.086 q^{80} -1162.23 q^{82} -569.206 q^{83} -369.174 q^{85} -1217.14 q^{86} -820.865 q^{88} -366.934 q^{89} -1802.37 q^{91} +1419.86 q^{92} +2065.51 q^{94} -1490.36 q^{95} +1063.27 q^{97} -3774.61 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

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 $

$2$	−4.41344	−1.56039	−0.780193	−	0.625539i	$-0.784879\pi$
$2$	−4.41344	−1.56039	−0.780193	+	0.625539i	$0.784879\pi$
$3$	0	0
$3$	0	0
$4$	11.4784	1.43480
$5$	−16.6195	−1.48649	−0.743247	−	0.669017i	$-0.766715\pi$
$5$	−16.6195	−1.48649	−0.743247	+	0.669017i	$0.766715\pi$
$6$	0	0
$7$	−34.6158	−1.86908	−0.934539	−	0.355862i	$-0.884187\pi$
$7$	−34.6158	−1.86908	−0.934539	+	0.355862i	$0.884187\pi$
$8$	−15.3518	−0.678459
$9$	0	0
$10$	73.3492	2.31950
$11$	53.4703	1.46563	0.732814	−	0.680429i	$-0.238206\pi$
$11$	53.4703	1.46563	0.732814	+	0.680429i	$0.238206\pi$
$12$	0	0
$13$	52.0679	1.11085	0.555424	−	0.831567i	$-0.312556\pi$
$13$	52.0679	1.11085	0.555424	+	0.831567i	$0.312556\pi$
$14$	152.775	2.91648
$15$	0	0
$16$	−24.0733	−0.376145
$17$	22.2133	0.316912	0.158456	−	0.987366i	$-0.449348\pi$
$17$	22.2133	0.316912	0.158456	+	0.987366i	$0.449348\pi$
$18$	0	0
$19$	89.6753	1.08279	0.541393	−	0.840770i	$-0.317897\pi$
$19$	89.6753	1.08279	0.541393	+	0.840770i	$0.317897\pi$
$20$	−190.766	−2.13283
$21$	0	0
$22$	−235.988	−2.28694
$23$	123.698	1.12143	0.560714	−	0.828009i	$-0.310527\pi$
$23$	123.698	1.12143	0.560714	+	0.828009i	$0.310527\pi$
$24$	0	0
$25$	151.208	1.20967
$26$	−229.798	−1.73335
$27$	0	0
$28$	−397.335	−2.68176
$29$	160.769	1.02945	0.514725	−	0.857355i	$-0.327894\pi$
$29$	160.769	1.02945	0.514725	+	0.857355i	$0.327894\pi$
$30$	0	0
$31$	11.9376	0.0691632	0.0345816	−	0.999402i	$-0.488990\pi$
$31$	11.9376	0.0691632	0.0345816	+	0.999402i	$0.488990\pi$
$32$	229.060	1.26539
$33$	0	0
$34$	−98.0368	−0.494505
$35$	575.298	2.77837
$36$	0	0
$37$	320.535	1.42421	0.712104	−	0.702074i	$-0.247742\pi$
$37$	320.535	1.42421	0.712104	+	0.702074i	$0.247742\pi$
$38$	−395.776	−1.68956
$39$	0	0
$40$	255.139	1.00853
$41$	263.340	1.00309	0.501546	−	0.865131i	$-0.332765\pi$
$41$	263.340	1.00309	0.501546	+	0.865131i	$0.332765\pi$
$42$	0	0
$43$	275.780	0.978047	0.489024	−	0.872271i	$-0.337353\pi$
$43$	275.780	0.978047	0.489024	+	0.872271i	$0.337353\pi$
$44$	613.755	2.10289
$45$	0	0
$46$	−545.934	−1.74986
$47$	−468.006	−1.45246	−0.726231	−	0.687451i	$-0.758729\pi$
$47$	−468.006	−1.45246	−0.726231	+	0.687451i	$0.758729\pi$
$48$	0	0
$49$	855.253	2.49345
$50$	−667.348	−1.88755
$51$	0	0
$52$	597.657	1.59385
$53$	−594.250	−1.54012	−0.770061	−	0.637970i	$-0.779774\pi$
$53$	−594.250	−1.54012	−0.770061	+	0.637970i	$0.779774\pi$
$54$	0	0
$55$	−888.651	−2.17865
$56$	531.414	1.26809
$57$	0	0
$58$	−709.544	−1.60634
$59$	538.732	1.18876	0.594381	−	0.804184i	$-0.297397\pi$
$59$	538.732	1.18876	0.594381	+	0.804184i	$0.297397\pi$
$60$	0	0
$61$	−335.134	−0.703434	−0.351717	−	0.936106i	$-0.614402\pi$
$61$	−335.134	−0.703434	−0.351717	+	0.936106i	$0.614402\pi$
$62$	−52.6859	−0.107921
$63$	0	0
$64$	−818.356	−1.59835
$65$	−865.343	−1.65127
$66$	0	0
$67$	−408.017	−0.743989	−0.371994	−	0.928235i	$-0.621326\pi$
$67$	−408.017	−0.743989	−0.371994	+	0.928235i	$0.621326\pi$
$68$	254.973	0.454706
$69$	0	0
$70$	−2539.04	−4.33533
$71$	353.397	0.590711	0.295356	−	0.955387i	$-0.404562\pi$
$71$	353.397	0.590711	0.295356	+	0.955387i	$0.404562\pi$
$72$	0	0
$73$	663.280	1.06344	0.531720	−	0.846920i	$-0.321546\pi$
$73$	663.280	1.06344	0.531720	+	0.846920i	$0.321546\pi$
$74$	−1414.66	−2.22231
$75$	0	0
$76$	1029.33	1.55358
$77$	−1850.92	−2.73937
$78$	0	0
$79$	320.302	0.456162	0.228081	−	0.973642i	$-0.426755\pi$
$79$	320.302	0.456162	0.228081	+	0.973642i	$0.426755\pi$
$80$	400.086	0.559137
$81$	0	0
$82$	−1162.23	−1.56521
$83$	−569.206	−0.752752	−0.376376	−	0.926467i	$-0.622830\pi$
$83$	−569.206	−0.752752	−0.376376	+	0.926467i	$0.622830\pi$
$84$	0	0
$85$	−369.174	−0.471088
$86$	−1217.14	−1.52613
$87$	0	0
$88$	−820.865	−0.994369
$89$	−366.934	−0.437022	−0.218511	−	0.975834i	$-0.570120\pi$
$89$	−366.934	−0.437022	−0.218511	+	0.975834i	$0.570120\pi$
$90$	0	0
$91$	−1802.37	−2.07626
$92$	1419.86	1.60903
$93$	0	0
$94$	2065.51	2.26640
$95$	−1490.36	−1.60955
$96$	0	0
$97$	1063.27	1.11298	0.556488	−	0.830856i	$-0.312148\pi$
$97$	1063.27	1.11298	0.556488	+	0.830856i	$0.312148\pi$
$98$	−3774.61	−3.89074
$99$	0	0
$100$	1735.63	1.73563
$101$	993.554	0.978835	0.489418	−	0.872050i	$-0.337209\pi$
$101$	993.554	0.978835	0.489418	+	0.872050i	$0.337209\pi$
$102$	0	0
$103$	−1255.90	−1.20144	−0.600718	−	0.799461i	$-0.705119\pi$
$103$	−1255.90	−1.20144	−0.600718	+	0.799461i	$0.705119\pi$
$104$	−799.335	−0.753666
$105$	0	0
$106$	2622.68	2.40318
$107$	59.0070	0.0533124	0.0266562	−	0.999645i	$-0.491514\pi$
$107$	59.0070	0.0533124	0.0266562	+	0.999645i	$0.491514\pi$
$108$	0	0
$109$	195.064	0.171410	0.0857052	−	0.996321i	$-0.472686\pi$
$109$	195.064	0.171410	0.0857052	+	0.996321i	$0.472686\pi$
$110$	3922.00	3.39953
$111$	0	0
$112$	833.315	0.703044
$113$	154.846	0.128909	0.0644543	−	0.997921i	$-0.479469\pi$
$113$	154.846	0.128909	0.0644543	+	0.997921i	$0.479469\pi$
$114$	0	0
$115$	−2055.80	−1.66700
$116$	1845.37	1.47706
$117$	0	0
$118$	−2377.66	−1.85493
$119$	−768.929	−0.592333
$120$	0	0
$121$	1528.08	1.14807
$122$	1479.09	1.09763
$123$	0	0
$124$	137.025	0.0992355
$125$	−435.569	−0.311668
$126$	0	0
$127$	1696.53	1.18537	0.592687	−	0.805433i	$-0.298067\pi$
$127$	1696.53	1.18537	0.592687	+	0.805433i	$0.298067\pi$
$128$	1779.28	1.22865
$129$	0	0
$130$	3819.14	2.57662
$131$	2145.02	1.43062	0.715311	−	0.698806i	$-0.246285\pi$
$131$	2145.02	1.43062	0.715311	+	0.698806i	$0.246285\pi$
$132$	0	0
$133$	−3104.18	−2.02381
$134$	1800.76	1.16091
$135$	0	0
$136$	−341.013	−0.215012
$137$	−1540.85	−0.960905	−0.480453	−	0.877021i	$-0.659528\pi$
$137$	−1540.85	−0.960905	−0.480453	+	0.877021i	$0.659528\pi$
$138$	0	0
$139$	1069.51	0.652622	0.326311	−	0.945262i	$-0.394194\pi$
$139$	1069.51	0.652622	0.326311	+	0.945262i	$0.394194\pi$
$140$	6603.51	3.98642
$141$	0	0
$142$	−1559.69	−0.921737
$143$	2784.09	1.62809
$144$	0	0
$145$	−2671.90	−1.53027
$146$	−2927.34	−1.65937
$147$	0	0
$148$	3679.24	2.04346
$149$	1622.58	0.892128	0.446064	−	0.895001i	$-0.352825\pi$
$149$	1622.58	0.892128	0.446064	+	0.895001i	$0.352825\pi$
$150$	0	0
$151$	−2297.63	−1.23827	−0.619133	−	0.785286i	$-0.712516\pi$
$151$	−2297.63	−1.23827	−0.619133	+	0.785286i	$0.712516\pi$
$152$	−1376.67	−0.734625
$153$	0	0
$154$	8168.91	4.27448
$155$	−198.397	−0.102811
$156$	0	0
$157$	309.454	0.157307	0.0786534	−	0.996902i	$-0.474938\pi$
$157$	309.454	0.157307	0.0786534	+	0.996902i	$0.474938\pi$
$158$	−1413.63	−0.711789
$159$	0	0
$160$	−3806.87	−1.88100
$161$	−4281.91	−2.09604
$162$	0	0
$163$	−504.118	−0.242243	−0.121121	−	0.992638i	$-0.538649\pi$
$163$	−504.118	−0.242243	−0.121121	+	0.992638i	$0.538649\pi$
$164$	3022.72	1.43924
$165$	0	0
$166$	2512.15	1.17458
$167$	1608.48	0.745317	0.372658	−	0.927969i	$-0.378446\pi$
$167$	1608.48	0.745317	0.372658	+	0.927969i	$0.378446\pi$
$168$	0	0
$169$	514.066	0.233985
$170$	1629.32	0.735079
$171$	0	0
$172$	3165.52	1.40330
$173$	1936.50	0.851038	0.425519	−	0.904949i	$-0.360092\pi$
$173$	1936.50	0.851038	0.425519	+	0.904949i	$0.360092\pi$
$174$	0	0
$175$	−5234.20	−2.26096
$176$	−1287.21	−0.551288
$177$	0	0
$178$	1619.44	0.681923
$179$	−3368.29	−1.40647	−0.703234	−	0.710958i	$-0.748261\pi$
$179$	−3368.29	−1.40647	−0.703234	+	0.710958i	$0.748261\pi$
$180$	0	0
$181$	−179.757	−0.0738192	−0.0369096	−	0.999319i	$-0.511751\pi$
$181$	−179.757	−0.0738192	−0.0369096	+	0.999319i	$0.511751\pi$
$182$	7954.65	3.23977
$183$	0	0
$184$	−1898.99	−0.760843
$185$	−5327.14	−2.11708
$186$	0	0
$187$	1187.75	0.464475
$188$	−5371.97	−2.08399
$189$	0	0
$190$	6577.61	2.51153
$191$	−709.591	−0.268818	−0.134409	−	0.990926i	$-0.542914\pi$
$191$	−709.591	−0.268818	−0.134409	+	0.990926i	$0.542914\pi$
$192$	0	0
$193$	−766.487	−0.285870	−0.142935	−	0.989732i	$-0.545654\pi$
$193$	−766.487	−0.285870	−0.142935	+	0.989732i	$0.545654\pi$
$194$	−4692.67	−1.73667
$195$	0	0
$196$	9816.95	3.57761
$197$	584.702	0.211463	0.105732	−	0.994395i	$-0.466282\pi$
$197$	584.702	0.211463	0.105732	+	0.994395i	$0.466282\pi$
$198$	0	0
$199$	1683.75	0.599790	0.299895	−	0.953972i	$-0.403048\pi$
$199$	1683.75	0.599790	0.299895	+	0.953972i	$0.403048\pi$
$200$	−2321.32	−0.820709
$201$	0	0
$202$	−4384.99	−1.52736
$203$	−5565.15	−1.92412
$204$	0	0
$205$	−4376.58	−1.49109
$206$	5542.85	1.87470
$207$	0	0
$208$	−1253.44	−0.417840
$209$	4794.97	1.58696
$210$	0	0
$211$	2821.83	0.920677	0.460339	−	0.887743i	$-0.347728\pi$
$211$	2821.83	0.920677	0.460339	+	0.887743i	$0.347728\pi$
$212$	−6821.05	−2.20977
$213$	0	0
$214$	−260.424	−0.0831878
$215$	−4583.33	−1.45386
$216$	0	0
$217$	−413.230	−0.129271
$218$	−860.902	−0.267466
$219$	0	0
$220$	−10200.3	−3.12593
$221$	1156.60	0.352041
$222$	0	0
$223$	716.246	0.215083	0.107541	−	0.994201i	$-0.465702\pi$
$223$	716.246	0.215083	0.107541	+	0.994201i	$0.465702\pi$
$224$	−7929.09	−2.36511
$225$	0	0
$226$	−683.402	−0.201147
$227$	4521.35	1.32199	0.660997	−	0.750389i	$-0.270134\pi$
$227$	4521.35	1.32199	0.660997	+	0.750389i	$0.270134\pi$
$228$	0	0
$229$	−12.5347	−0.00361711	−0.00180856	−	0.999998i	$-0.500576\pi$
$229$	−12.5347	−0.00361711	−0.00180856	+	0.999998i	$0.500576\pi$
$230$	9073.16	2.60116
$231$	0	0
$232$	−2468.09	−0.698440
$233$	−2716.88	−0.763901	−0.381951	−	0.924183i	$-0.624748\pi$
$233$	−2716.88	−0.763901	−0.381951	+	0.924183i	$0.624748\pi$
$234$	0	0
$235$	7778.03	2.15908
$236$	6183.79	1.70564
$237$	0	0
$238$	3393.62	0.924268
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	2853.95	0.762817	0.381409	−	0.924407i	$-0.375439\pi$
$241$	2853.95	0.762817	0.381409	+	0.924407i	$0.375439\pi$
$242$	−6744.07	−1.79143
$243$	0	0
$244$	−3846.80	−1.00929
$245$	−14213.9	−3.70650
$246$	0	0
$247$	4669.20	1.20281
$248$	−183.264	−0.0469244
$249$	0	0
$250$	1922.36	0.486322
$251$	−3758.27	−0.945100	−0.472550	−	0.881304i	$-0.656666\pi$
$251$	−3758.27	−0.945100	−0.472550	+	0.881304i	$0.656666\pi$
$252$	0	0
$253$	6614.18	1.64360
$254$	−7487.51	−1.84964
$255$	0	0
$256$	−1305.89	−0.318822
$257$	1290.73	0.313281	0.156641	−	0.987656i	$-0.449934\pi$
$257$	1290.73	0.313281	0.156641	+	0.987656i	$0.449934\pi$
$258$	0	0
$259$	−11095.6	−2.66195
$260$	−9932.77	−2.36925
$261$	0	0
$262$	−9466.92	−2.23232
$263$	−1507.26	−0.353391	−0.176695	−	0.984266i	$-0.556541\pi$
$263$	−1507.26	−0.353391	−0.176695	+	0.984266i	$0.556541\pi$
$264$	0	0
$265$	9876.15	2.28938
$266$	13700.1	3.15792
$267$	0	0
$268$	−4683.39	−1.06748
$269$	−4336.11	−0.982814	−0.491407	−	0.870930i	$-0.663517\pi$
$269$	−4336.11	−0.982814	−0.491407	+	0.870930i	$0.663517\pi$
$270$	0	0
$271$	−3629.17	−0.813492	−0.406746	−	0.913541i	$-0.633337\pi$
$271$	−3629.17	−0.813492	−0.406746	+	0.913541i	$0.633337\pi$
$272$	−534.746	−0.119205
$273$	0	0
$274$	6800.46	1.49938
$275$	8085.16	1.77292
$276$	0	0
$277$	2675.85	0.580419	0.290210	−	0.956963i	$-0.406275\pi$
$277$	2675.85	0.580419	0.290210	+	0.956963i	$0.406275\pi$
$278$	−4720.20	−1.01834
$279$	0	0
$280$	−8831.84	−1.88501
$281$	4759.08	1.01033	0.505165	−	0.863022i	$-0.331431\pi$
$281$	4759.08	1.01033	0.505165	+	0.863022i	$0.331431\pi$
$282$	0	0
$283$	−4306.73	−0.904624	−0.452312	−	0.891860i	$-0.649401\pi$
$283$	−4306.73	−0.904624	−0.452312	+	0.891860i	$0.649401\pi$
$284$	4056.44	0.847554
$285$	0	0
$286$	−12287.4	−2.54045
$287$	−9115.71	−1.87486
$288$	0	0
$289$	−4419.57	−0.899567
$290$	11792.3	2.38782
$291$	0	0
$292$	7613.41	1.52582
$293$	−7826.05	−1.56042	−0.780210	−	0.625518i	$-0.784888\pi$
$293$	−7826.05	−1.56042	−0.780210	+	0.625518i	$0.784888\pi$
$294$	0	0
$295$	−8953.46	−1.76709
$296$	−4920.79	−0.966267
$297$	0	0
$298$	−7161.16	−1.39206
$299$	6440.70	1.24574
$300$	0	0
$301$	−9546.34	−1.82805
$302$	10140.4	1.93217
$303$	0	0
$304$	−2158.78	−0.407284
$305$	5569.76	1.04565
$306$	0	0
$307$	3609.63	0.671050	0.335525	−	0.942031i	$-0.391086\pi$
$307$	3609.63	0.671050	0.335525	+	0.942031i	$0.391086\pi$
$308$	−21245.6	−3.93046
$309$	0	0
$310$	875.614	0.160424
$311$	−3030.05	−0.552470	−0.276235	−	0.961090i	$-0.589087\pi$
$311$	−3030.05	−0.552470	−0.276235	+	0.961090i	$0.589087\pi$
$312$	0	0
$313$	892.924	0.161249	0.0806246	−	0.996745i	$-0.474308\pi$
$313$	892.924	0.161249	0.0806246	+	0.996745i	$0.474308\pi$
$314$	−1365.76	−0.245459
$315$	0	0
$316$	3676.56	0.654503
$317$	7327.80	1.29833	0.649164	−	0.760648i	$-0.275119\pi$
$317$	7327.80	1.29833	0.649164	+	0.760648i	$0.275119\pi$
$318$	0	0
$319$	8596.38	1.50879
$320$	13600.7	2.37594
$321$	0	0
$322$	18897.9	3.27062
$323$	1991.98	0.343148
$324$	0	0
$325$	7873.10	1.34376
$326$	2224.89	0.377992
$327$	0	0
$328$	−4042.73	−0.680556
$329$	16200.4	2.71476
$330$	0	0
$331$	7093.31	1.17790	0.588948	−	0.808171i	$-0.299542\pi$
$331$	7093.31	1.17790	0.588948	+	0.808171i	$0.299542\pi$
$332$	−6533.58	−1.08005
$333$	0	0
$334$	−7098.92	−1.16298
$335$	6781.05	1.10594
$336$	0	0
$337$	−597.503	−0.0965818	−0.0482909	−	0.998833i	$-0.515377\pi$
$337$	−597.503	−0.0965818	−0.0482909	+	0.998833i	$0.515377\pi$
$338$	−2268.80	−0.365107
$339$	0	0
$340$	−4237.53	−0.675918
$341$	638.308	0.101368
$342$	0	0
$343$	−17732.1	−2.79137
$344$	−4233.71	−0.663565
$345$	0	0
$346$	−8546.63	−1.32795
$347$	4676.58	0.723493	0.361747	−	0.932276i	$-0.382181\pi$
$347$	4676.58	0.723493	0.361747	+	0.932276i	$0.382181\pi$
$348$	0	0
$349$	−1622.04	−0.248785	−0.124392	−	0.992233i	$-0.539698\pi$
$349$	−1622.04	−0.248785	−0.124392	+	0.992233i	$0.539698\pi$
$350$	23100.8	3.52797
$351$	0	0
$352$	12247.9	1.85459
$353$	−10824.9	−1.63216	−0.816080	−	0.577939i	$-0.803857\pi$
$353$	−10824.9	−1.63216	−0.816080	+	0.577939i	$0.803857\pi$
$354$	0	0
$355$	−5873.28	−0.878089
$356$	−4211.82	−0.627040
$357$	0	0
$358$	14865.7	2.19463
$359$	5633.38	0.828185	0.414093	−	0.910235i	$-0.364099\pi$
$359$	5633.38	0.828185	0.414093	+	0.910235i	$0.364099\pi$
$360$	0	0
$361$	1182.66	0.172424
$362$	793.348	0.115186
$363$	0	0
$364$	−20688.4	−2.97903
$365$	−11023.4	−1.58080
$366$	0	0
$367$	−7985.81	−1.13585	−0.567924	−	0.823081i	$-0.692253\pi$
$367$	−7985.81	−1.13585	−0.567924	+	0.823081i	$0.692253\pi$
$368$	−2977.82	−0.421819
$369$	0	0
$370$	23511.0	3.30346
$371$	20570.4	2.87861
$372$	0	0
$373$	3415.67	0.474146	0.237073	−	0.971492i	$-0.423812\pi$
$373$	3415.67	0.474146	0.237073	+	0.971492i	$0.423812\pi$
$374$	−5242.06	−0.724760
$375$	0	0
$376$	7184.72	0.985435
$377$	8370.91	1.14356
$378$	0	0
$379$	6966.89	0.944236	0.472118	−	0.881535i	$-0.343490\pi$
$379$	6966.89	0.944236	0.472118	+	0.881535i	$0.343490\pi$
$380$	−17107.0	−2.30939
$381$	0	0
$382$	3131.73	0.419459
$383$	9821.61	1.31034	0.655171	−	0.755481i	$-0.272597\pi$
$383$	9821.61	1.31034	0.655171	+	0.755481i	$0.272597\pi$
$384$	0	0
$385$	30761.4	4.07206
$386$	3382.84	0.446068
$387$	0	0
$388$	12204.7	1.59690
$389$	−961.787	−0.125359	−0.0626794	−	0.998034i	$-0.519965\pi$
$389$	−961.787	−0.125359	−0.0626794	+	0.998034i	$0.519965\pi$
$390$	0	0
$391$	2747.74	0.355394
$392$	−13129.7	−1.69170
$393$	0	0
$394$	−2580.55	−0.329965
$395$	−5323.27	−0.678083
$396$	0	0
$397$	−6253.63	−0.790581	−0.395291	−	0.918556i	$-0.629356\pi$
$397$	−6253.63	−0.790581	−0.395291	+	0.918556i	$0.629356\pi$
$398$	−7431.14	−0.935903
$399$	0	0
$400$	−3640.08	−0.455010
$401$	−4974.26	−0.619458	−0.309729	−	0.950825i	$-0.600238\pi$
$401$	−4974.26	−0.619458	−0.309729	+	0.950825i	$0.600238\pi$
$402$	0	0
$403$	621.567	0.0768299
$404$	11404.4	1.40443
$405$	0	0
$406$	24561.4	3.00237
$407$	17139.1	2.08736
$408$	0	0
$409$	−12286.8	−1.48544	−0.742719	−	0.669603i	$-0.766464\pi$
$409$	−12286.8	−1.48544	−0.742719	+	0.669603i	$0.766464\pi$
$410$	19315.7	2.32667
$411$	0	0
$412$	−14415.8	−1.72382
$413$	−18648.6	−2.22189
$414$	0	0
$415$	9459.92	1.11896
$416$	11926.7	1.40566
$417$	0	0
$418$	−21162.3	−2.47627
$419$	15073.4	1.75748	0.878742	−	0.477297i	$-0.158383\pi$
$419$	15073.4	1.75748	0.878742	+	0.477297i	$0.158383\pi$
$420$	0	0
$421$	4464.36	0.516816	0.258408	−	0.966036i	$-0.416802\pi$
$421$	4464.36	0.516816	0.258408	+	0.966036i	$0.416802\pi$
$422$	−12454.0	−1.43661
$423$	0	0
$424$	9122.79	1.04491
$425$	3358.83	0.383358
$426$	0	0
$427$	11600.9	1.31477
$428$	677.307	0.0764927
$429$	0	0
$430$	20228.2	2.26858
$431$	−16865.6	−1.88489	−0.942445	−	0.334360i	$-0.891480\pi$
$431$	−16865.6	−1.88489	−0.942445	+	0.334360i	$0.891480\pi$
$432$	0	0
$433$	12886.4	1.43021	0.715107	−	0.699015i	$-0.246378\pi$
$433$	12886.4	1.43021	0.715107	+	0.699015i	$0.246378\pi$
$434$	1823.76	0.201713
$435$	0	0
$436$	2239.03	0.245940
$437$	11092.7	1.21427
$438$	0	0
$439$	−7592.13	−0.825405	−0.412702	−	0.910866i	$-0.635415\pi$
$439$	−7592.13	−0.825405	−0.412702	+	0.910866i	$0.635415\pi$
$440$	13642.4	1.47812
$441$	0	0
$442$	−5104.57	−0.549320
$443$	−16133.5	−1.73031	−0.865155	−	0.501504i	$-0.832780\pi$
$443$	−16133.5	−1.73031	−0.865155	+	0.501504i	$0.832780\pi$
$444$	0	0
$445$	6098.27	0.649631
$446$	−3161.11	−0.335612
$447$	0	0
$448$	28328.0	2.98744
$449$	15279.0	1.60593	0.802964	−	0.596028i	$-0.203255\pi$
$449$	15279.0	1.60593	0.802964	+	0.596028i	$0.203255\pi$
$450$	0	0
$451$	14080.9	1.47016
$452$	1777.38	0.184958
$453$	0	0
$454$	−19954.7	−2.06282
$455$	29954.5	3.08635
$456$	0	0
$457$	−4143.25	−0.424098	−0.212049	−	0.977259i	$-0.568014\pi$
$457$	−4143.25	−0.424098	−0.212049	+	0.977259i	$0.568014\pi$
$458$	55.3213	0.00564409
$459$	0	0
$460$	−23597.4	−2.39181
$461$	13032.3	1.31665	0.658324	−	0.752734i	$-0.271265\pi$
$461$	13032.3	1.31665	0.658324	+	0.752734i	$0.271265\pi$
$462$	0	0
$463$	−9076.50	−0.911060	−0.455530	−	0.890220i	$-0.650550\pi$
$463$	−9076.50	−0.911060	−0.455530	+	0.890220i	$0.650550\pi$
$464$	−3870.24	−0.387222
$465$	0	0
$466$	11990.8	1.19198
$467$	−5910.85	−0.585699	−0.292850	−	0.956159i	$-0.594603\pi$
$467$	−5910.85	−0.585699	−0.292850	+	0.956159i	$0.594603\pi$
$468$	0	0
$469$	14123.8	1.39057
$470$	−34327.8	−3.36899
$471$	0	0
$472$	−8270.49	−0.806526
$473$	14746.0	1.43345
$474$	0	0
$475$	13559.6	1.30981
$476$	−8826.09	−0.849881
$477$	0	0
$478$	−1054.81	−0.100933
$479$	−2208.82	−0.210696	−0.105348	−	0.994435i	$-0.533596\pi$
$479$	−2208.82	−0.210696	−0.105348	+	0.994435i	$0.533596\pi$
$480$	0	0
$481$	16689.6	1.58208
$482$	−12595.7	−1.19029
$483$	0	0
$484$	17539.9	1.64725
$485$	−17671.0	−1.65443
$486$	0	0
$487$	−10364.3	−0.964373	−0.482186	−	0.876069i	$-0.660157\pi$
$487$	−10364.3	−0.964373	−0.482186	+	0.876069i	$0.660157\pi$
$488$	5144.90	0.477251
$489$	0	0
$490$	62732.1	5.78357
$491$	9858.61	0.906136	0.453068	−	0.891476i	$-0.350329\pi$
$491$	9858.61	0.906136	0.453068	+	0.891476i	$0.350329\pi$
$492$	0	0
$493$	3571.21	0.326245
$494$	−20607.2	−1.87685
$495$	0	0
$496$	−287.377	−0.0260154
$497$	−12233.1	−1.10408
$498$	0	0
$499$	21169.1	1.89911	0.949557	−	0.313595i	$-0.101533\pi$
$499$	21169.1	1.89911	0.949557	+	0.313595i	$0.101533\pi$
$500$	−4999.64	−0.447182
$501$	0	0
$502$	16586.9	1.47472
$503$	3584.90	0.317779	0.158889	−	0.987296i	$-0.449209\pi$
$503$	3584.90	0.317779	0.158889	+	0.987296i	$0.449209\pi$
$504$	0	0
$505$	−16512.4	−1.45503
$506$	−29191.3	−2.56464
$507$	0	0
$508$	19473.4	1.70078
$509$	−6337.55	−0.551880	−0.275940	−	0.961175i	$-0.588989\pi$
$509$	−6337.55	−0.551880	−0.275940	+	0.961175i	$0.588989\pi$
$510$	0	0
$511$	−22960.0	−1.98765
$512$	−8470.76	−0.731169
$513$	0	0
$514$	−5696.54	−0.488839
$515$	20872.5	1.78593
$516$	0	0
$517$	−25024.4	−2.12877
$518$	48969.7	4.15368
$519$	0	0
$520$	13284.6	1.12032
$521$	4244.23	0.356897	0.178448	−	0.983949i	$-0.442892\pi$
$521$	4244.23	0.356897	0.178448	+	0.983949i	$0.442892\pi$
$522$	0	0
$523$	11727.2	0.980489	0.490245	−	0.871585i	$-0.336907\pi$
$523$	11727.2	0.980489	0.490245	+	0.871585i	$0.336907\pi$
$524$	24621.5	2.05266
$525$	0	0
$526$	6652.20	0.551426
$527$	265.173	0.0219187
$528$	0	0
$529$	3134.24	0.257601
$530$	−43587.7	−3.57232
$531$	0	0
$532$	−35631.1	−2.90377
$533$	13711.5	1.11428
$534$	0	0
$535$	−980.668	−0.0792485
$536$	6263.79	0.504766
$537$	0	0
$538$	19137.1	1.53357
$539$	45730.7	3.65447
$540$	0	0
$541$	−150.338	−0.0119474	−0.00597369	−	0.999982i	$-0.501901\pi$
$541$	−150.338	−0.0119474	−0.00597369	+	0.999982i	$0.501901\pi$
$542$	16017.1	1.26936
$543$	0	0
$544$	5088.17	0.401017
$545$	−3241.87	−0.254801
$546$	0	0
$547$	−8075.69	−0.631246	−0.315623	−	0.948885i	$-0.602214\pi$
$547$	−8075.69	−0.631246	−0.315623	+	0.948885i	$0.602214\pi$
$548$	−17686.6	−1.37871
$549$	0	0
$550$	−35683.3	−2.76644
$551$	14417.0	1.11467
$552$	0	0
$553$	−11087.5	−0.852603
$554$	−11809.7	−0.905678
$555$	0	0
$556$	12276.3	0.936384
$557$	−9258.02	−0.704264	−0.352132	−	0.935950i	$-0.614543\pi$
$557$	−9258.02	−0.704264	−0.352132	+	0.935950i	$0.614543\pi$
$558$	0	0
$559$	14359.3	1.08646
$560$	−13849.3	−1.04507
$561$	0	0
$562$	−21003.9	−1.57651
$563$	−155.617	−0.0116492	−0.00582458	−	0.999983i	$-0.501854\pi$
$563$	−155.617	−0.0116492	−0.00582458	+	0.999983i	$0.501854\pi$
$564$	0	0
$565$	−2573.46	−0.191622
$566$	19007.5	1.41156
$567$	0	0
$568$	−5425.27	−0.400773
$569$	−4094.08	−0.301639	−0.150820	−	0.988561i	$-0.548191\pi$
$569$	−4094.08	−0.301639	−0.150820	+	0.988561i	$0.548191\pi$
$570$	0	0
$571$	18574.3	1.36132	0.680658	−	0.732601i	$-0.261694\pi$
$571$	18574.3	1.36132	0.680658	+	0.732601i	$0.261694\pi$
$572$	31956.9	2.33599
$573$	0	0
$574$	40231.6	2.92550
$575$	18704.2	1.35655
$576$	0	0
$577$	−2687.01	−0.193868	−0.0969340	−	0.995291i	$-0.530904\pi$
$577$	−2687.01	−0.193868	−0.0969340	+	0.995291i	$0.530904\pi$
$578$	19505.5	1.40367
$579$	0	0
$580$	−30669.2	−2.19564
$581$	19703.5	1.40695
$582$	0	0
$583$	−31774.7	−2.25725
$584$	−10182.5	−0.721500
$585$	0	0
$586$	34539.8	2.43486
$587$	−7901.78	−0.555607	−0.277804	−	0.960638i	$-0.589606\pi$
$587$	−7901.78	−0.555607	−0.277804	+	0.960638i	$0.589606\pi$
$588$	0	0
$589$	1070.51	0.0748889
$590$	39515.5	2.75734
$591$	0	0
$592$	−7716.33	−0.535708
$593$	6736.96	0.466532	0.233266	−	0.972413i	$-0.425059\pi$
$593$	6736.96	0.466532	0.233266	+	0.972413i	$0.425059\pi$
$594$	0	0
$595$	12779.2	0.880500
$596$	18624.7	1.28003
$597$	0	0
$598$	−28425.6	−1.94383
$599$	−21879.7	−1.49246	−0.746228	−	0.665691i	$-0.768137\pi$
$599$	−21879.7	−1.49246	−0.746228	+	0.665691i	$0.768137\pi$
$600$	0	0
$601$	8678.15	0.589000	0.294500	−	0.955651i	$-0.404847\pi$
$601$	8678.15	0.589000	0.294500	+	0.955651i	$0.404847\pi$
$602$	42132.2	2.85246
$603$	0	0
$604$	−26373.1	−1.77667
$605$	−25395.9	−1.70659
$606$	0	0
$607$	−26370.8	−1.76336	−0.881679	−	0.471849i	$-0.843587\pi$
$607$	−26370.8	−1.76336	−0.881679	+	0.471849i	$0.843587\pi$
$608$	20541.0	1.37015
$609$	0	0
$610$	−24581.8	−1.63162
$611$	−24368.1	−1.61346
$612$	0	0
$613$	−1507.25	−0.0993106	−0.0496553	−	0.998766i	$-0.515812\pi$
$613$	−1507.25	−0.0993106	−0.0496553	+	0.998766i	$0.515812\pi$
$614$	−15930.9	−1.04710
$615$	0	0
$616$	28414.9	1.85855
$617$	10403.9	0.678839	0.339419	−	0.940635i	$-0.389769\pi$
$617$	10403.9	0.678839	0.339419	+	0.940635i	$0.389769\pi$
$618$	0	0
$619$	−10613.6	−0.689170	−0.344585	−	0.938755i	$-0.611980\pi$
$619$	−10613.6	−0.689170	−0.344585	+	0.938755i	$0.611980\pi$
$620$	−2277.29	−0.147513
$621$	0	0
$622$	13372.9	0.862066
$623$	12701.7	0.816828
$624$	0	0
$625$	−11662.1	−0.746374
$626$	−3940.86	−0.251611
$627$	0	0
$628$	3552.05	0.225704
$629$	7120.14	0.451349
$630$	0	0
$631$	−10520.8	−0.663752	−0.331876	−	0.943323i	$-0.607682\pi$
$631$	−10520.8	−0.663752	−0.331876	+	0.943323i	$0.607682\pi$
$632$	−4917.21	−0.309487
$633$	0	0
$634$	−32340.8	−2.02589
$635$	−28195.5	−1.76205
$636$	0	0
$637$	44531.2	2.76985
$638$	−37939.6	−2.35430
$639$	0	0
$640$	−29570.8	−1.82639
$641$	−27260.0	−1.67973	−0.839865	−	0.542795i	$-0.817366\pi$
$641$	−27260.0	−1.67973	−0.839865	+	0.542795i	$0.817366\pi$
$642$	0	0
$643$	27351.3	1.67749	0.838747	−	0.544521i	$-0.183289\pi$
$643$	27351.3	1.67749	0.838747	+	0.544521i	$0.183289\pi$
$644$	−49149.6	−3.00740
$645$	0	0
$646$	−8791.48	−0.535443
$647$	6691.05	0.406573	0.203286	−	0.979119i	$-0.434838\pi$
$647$	6691.05	0.406573	0.203286	+	0.979119i	$0.434838\pi$
$648$	0	0
$649$	28806.2	1.74228
$650$	−34747.4	−2.09678
$651$	0	0
$652$	−5786.47	−0.347570
$653$	10097.5	0.605121	0.302560	−	0.953130i	$-0.402159\pi$
$653$	10097.5	0.605121	0.302560	+	0.953130i	$0.402159\pi$
$654$	0	0
$655$	−35649.2	−2.12661
$656$	−6339.44	−0.377308
$657$	0	0
$658$	−71499.4	−4.23607
$659$	16684.4	0.986240	0.493120	−	0.869961i	$-0.335856\pi$
$659$	16684.4	0.986240	0.493120	+	0.869961i	$0.335856\pi$
$660$	0	0
$661$	29113.2	1.71312	0.856559	−	0.516049i	$-0.172598\pi$
$661$	29113.2	1.71312	0.856559	+	0.516049i	$0.172598\pi$
$662$	−31305.8	−1.83797
$663$	0	0
$664$	8738.32	0.510711
$665$	51590.0	3.00838
$666$	0	0
$667$	19886.8	1.15446
$668$	18462.8	1.06938
$669$	0	0
$670$	−29927.7	−1.72569
$671$	−17919.7	−1.03097
$672$	0	0
$673$	−31392.8	−1.79807	−0.899036	−	0.437874i	$-0.855732\pi$
$673$	−31392.8	−1.79807	−0.899036	+	0.437874i	$0.855732\pi$
$674$	2637.04	0.150705
$675$	0	0
$676$	5900.66	0.335723
$677$	27165.1	1.54216	0.771079	−	0.636740i	$-0.219717\pi$
$677$	27165.1	1.54216	0.771079	+	0.636740i	$0.219717\pi$
$678$	0	0
$679$	−36805.9	−2.08024
$680$	5667.47	0.319614
$681$	0	0
$682$	−2817.13	−0.158172
$683$	−26865.6	−1.50510	−0.752551	−	0.658534i	$-0.771177\pi$
$683$	−26865.6	−1.50510	−0.752551	+	0.658534i	$0.771177\pi$
$684$	0	0
$685$	25608.2	1.42838
$686$	78259.3	4.35562
$687$	0	0
$688$	−6638.92	−0.367887
$689$	−30941.3	−1.71084
$690$	0	0
$691$	−17236.9	−0.948946	−0.474473	−	0.880270i	$-0.657361\pi$
$691$	−17236.9	−0.948946	−0.474473	+	0.880270i	$0.657361\pi$
$692$	22228.0	1.22107
$693$	0	0
$694$	−20639.8	−1.12893
$695$	−17774.7	−0.970119
$696$	0	0
$697$	5849.63	0.317892
$698$	7158.78	0.388200
$699$	0	0
$700$	−60080.3	−3.24403
$701$	31733.1	1.70976	0.854881	−	0.518824i	$-0.173630\pi$
$701$	31733.1	1.70976	0.854881	+	0.518824i	$0.173630\pi$
$702$	0	0
$703$	28744.1	1.54211
$704$	−43757.7	−2.34259
$705$	0	0
$706$	47775.1	2.54680
$707$	−34392.7	−1.82952
$708$	0	0
$709$	−10854.0	−0.574936	−0.287468	−	0.957790i	$-0.592814\pi$
$709$	−10854.0	−0.574936	−0.287468	+	0.957790i	$0.592814\pi$
$710$	25921.4	1.37016
$711$	0	0
$712$	5633.09	0.296501
$713$	1476.66	0.0775616
$714$	0	0
$715$	−46270.2	−2.42015
$716$	−38662.6	−2.01800
$717$	0	0
$718$	−24862.6	−1.29229
$719$	13437.5	0.696985	0.348493	−	0.937312i	$-0.386694\pi$
$719$	13437.5	0.696985	0.348493	+	0.937312i	$0.386694\pi$
$720$	0	0
$721$	43474.1	2.24558
$722$	−5219.58	−0.269048
$723$	0	0
$724$	−2063.33	−0.105916
$725$	24309.6	1.24529
$726$	0	0
$727$	12621.8	0.643901	0.321951	−	0.946756i	$-0.395662\pi$
$727$	12621.8	0.643901	0.321951	+	0.946756i	$0.395662\pi$
$728$	27669.6	1.40866
$729$	0	0
$730$	48651.0	2.46665
$731$	6125.97	0.309955
$732$	0	0
$733$	5559.89	0.280163	0.140081	−	0.990140i	$-0.455264\pi$
$733$	5559.89	0.280163	0.140081	+	0.990140i	$0.455264\pi$
$734$	35244.9	1.77236
$735$	0	0
$736$	28334.3	1.41904
$737$	−21816.8	−1.09041
$738$	0	0
$739$	−14806.4	−0.737024	−0.368512	−	0.929623i	$-0.620133\pi$
$739$	−14806.4	−0.737024	−0.368512	+	0.929623i	$0.620133\pi$
$740$	−61147.2	−3.03759
$741$	0	0
$742$	−90786.3	−4.49174
$743$	31879.6	1.57409	0.787044	−	0.616896i	$-0.211610\pi$
$743$	31879.6	1.57409	0.787044	+	0.616896i	$0.211610\pi$
$744$	0	0
$745$	−26966.5	−1.32614
$746$	−15074.8	−0.739851
$747$	0	0
$748$	13633.5	0.666430
$749$	−2042.57	−0.0996449
$750$	0	0
$751$	−10267.9	−0.498907	−0.249454	−	0.968387i	$-0.580251\pi$
$751$	−10267.9	−0.498907	−0.249454	+	0.968387i	$0.580251\pi$
$752$	11266.4	0.546336
$753$	0	0
$754$	−36944.5	−1.78440
$755$	38185.4	1.84067
$756$	0	0
$757$	−1922.82	−0.0923196	−0.0461598	−	0.998934i	$-0.514698\pi$
$757$	−1922.82	−0.0923196	−0.0461598	+	0.998934i	$0.514698\pi$
$758$	−30747.9	−1.47337
$759$	0	0
$760$	22879.7	1.09202
$761$	17174.3	0.818090	0.409045	−	0.912514i	$-0.365862\pi$
$761$	17174.3	0.818090	0.409045	+	0.912514i	$0.365862\pi$
$762$	0	0
$763$	−6752.30	−0.320379
$764$	−8144.98	−0.385700
$765$	0	0
$766$	−43347.1	−2.04464
$767$	28050.6	1.32053
$768$	0	0
$769$	33732.2	1.58181	0.790906	−	0.611938i	$-0.209610\pi$
$769$	33732.2	1.58181	0.790906	+	0.611938i	$0.209610\pi$
$770$	−135763.	−6.35399
$771$	0	0
$772$	−8798.06	−0.410167
$773$	−31586.0	−1.46969	−0.734843	−	0.678237i	$-0.762744\pi$
$773$	−31586.0	−1.46969	−0.734843	+	0.678237i	$0.762744\pi$
$774$	0	0
$775$	1805.07	0.0836644
$776$	−16323.1	−0.755109
$777$	0	0
$778$	4244.79	0.195608
$779$	23615.1	1.08613
$780$	0	0
$781$	18896.2	0.865763
$782$	−12127.0	−0.554552
$783$	0	0
$784$	−20588.7	−0.937898
$785$	−5142.98	−0.233836
$786$	0	0
$787$	36952.8	1.67373	0.836865	−	0.547409i	$-0.184386\pi$
$787$	36952.8	1.67373	0.836865	+	0.547409i	$0.184386\pi$
$788$	6711.46	0.303408
$789$	0	0
$790$	23493.9	1.05807
$791$	−5360.11	−0.240940
$792$	0	0
$793$	−17449.7	−0.781409
$794$	27600.0	1.23361
$795$	0	0
$796$	19326.8	0.860580
$797$	20017.1	0.889638	0.444819	−	0.895620i	$-0.353268\pi$
$797$	20017.1	0.889638	0.444819	+	0.895620i	$0.353268\pi$
$798$	0	0
$799$	−10395.9	−0.460302
$800$	34635.8	1.53070
$801$	0	0
$802$	21953.6	0.966593
$803$	35465.8	1.55861
$804$	0	0
$805$	71163.3	3.11575
$806$	−2743.24	−0.119884
$807$	0	0
$808$	−15252.8	−0.664100
$809$	35285.6	1.53347	0.766735	−	0.641964i	$-0.221880\pi$
$809$	35285.6	1.53347	0.766735	+	0.641964i	$0.221880\pi$
$810$	0	0
$811$	31455.3	1.36196	0.680978	−	0.732304i	$-0.261555\pi$
$811$	31455.3	1.36196	0.680978	+	0.732304i	$0.261555\pi$
$812$	−63879.1	−2.76074
$813$	0	0
$814$	−75642.5	−3.25709
$815$	8378.19	0.360092
$816$	0	0
$817$	24730.6	1.05902
$818$	54227.1	2.31786
$819$	0	0
$820$	−50236.2	−2.13942
$821$	33919.7	1.44191	0.720953	−	0.692984i	$-0.243704\pi$
$821$	33919.7	1.44191	0.720953	+	0.692984i	$0.243704\pi$
$822$	0	0
$823$	−14426.2	−0.611016	−0.305508	−	0.952190i	$-0.598826\pi$
$823$	−14426.2	−0.611016	−0.305508	+	0.952190i	$0.598826\pi$
$824$	19280.4	0.815125
$825$	0	0
$826$	82304.6	3.46700
$827$	−19870.7	−0.835518	−0.417759	−	0.908558i	$-0.637184\pi$
$827$	−19870.7	−0.835518	−0.417759	+	0.908558i	$0.637184\pi$
$828$	0	0
$829$	879.847	0.0368617	0.0184309	−	0.999830i	$-0.494133\pi$
$829$	879.847	0.0368617	0.0184309	+	0.999830i	$0.494133\pi$
$830$	−41750.8	−1.74601
$831$	0	0
$832$	−42610.1	−1.77553
$833$	18998.0	0.790204
$834$	0	0
$835$	−26732.2	−1.10791
$836$	55038.6	2.27697
$837$	0	0
$838$	−66525.7	−2.74235
$839$	−26990.3	−1.11062	−0.555308	−	0.831645i	$-0.687400\pi$
$839$	−26990.3	−1.11062	−0.555308	+	0.831645i	$0.687400\pi$
$840$	0	0
$841$	1457.70	0.0597689
$842$	−19703.2	−0.806432
$843$	0	0
$844$	32390.2	1.32099
$845$	−8543.53	−0.347818
$846$	0	0
$847$	−52895.6	−2.14582
$848$	14305.5	0.579309
$849$	0	0
$850$	−14824.0	−0.598186
$851$	39649.6	1.59715
$852$	0	0
$853$	12572.9	0.504676	0.252338	−	0.967639i	$-0.418800\pi$
$853$	12572.9	0.504676	0.252338	+	0.967639i	$0.418800\pi$
$854$	−51199.9	−2.05155
$855$	0	0
$856$	−905.862	−0.0361702
$857$	−34377.3	−1.37025	−0.685126	−	0.728425i	$-0.740253\pi$
$857$	−34377.3	−1.37025	−0.685126	+	0.728425i	$0.740253\pi$
$858$	0	0
$859$	−4462.37	−0.177246	−0.0886229	−	0.996065i	$-0.528247\pi$
$859$	−4462.37	−0.177246	−0.0886229	+	0.996065i	$0.528247\pi$
$860$	−52609.3	−2.08600
$861$	0	0
$862$	74435.3	2.94116
$863$	25088.2	0.989585	0.494793	−	0.869011i	$-0.335244\pi$
$863$	25088.2	0.989585	0.494793	+	0.869011i	$0.335244\pi$
$864$	0	0
$865$	−32183.7	−1.26506
$866$	−56873.5	−2.23168
$867$	0	0
$868$	−4743.23	−0.185479
$869$	17126.7	0.668564
$870$	0	0
$871$	−21244.6	−0.826459
$872$	−2994.58	−0.116295
$873$	0	0
$874$	−48956.8	−1.89472
$875$	15077.6	0.582531
$876$	0	0
$877$	−50245.3	−1.93462	−0.967312	−	0.253591i	$-0.918388\pi$
$877$	−50245.3	−1.93462	−0.967312	+	0.253591i	$0.918388\pi$
$878$	33507.4	1.28795
$879$	0	0
$880$	21392.7	0.819487
$881$	4111.10	0.157215	0.0786075	−	0.996906i	$-0.474953\pi$
$881$	4111.10	0.157215	0.0786075	+	0.996906i	$0.474953\pi$
$882$	0	0
$883$	−15774.1	−0.601178	−0.300589	−	0.953754i	$-0.597183\pi$
$883$	−15774.1	−0.601178	−0.300589	+	0.953754i	$0.597183\pi$
$884$	13275.9	0.505110
$885$	0	0
$886$	71204.3	2.69995
$887$	−20248.0	−0.766472	−0.383236	−	0.923651i	$-0.625190\pi$
$887$	−20248.0	−0.766472	−0.383236	+	0.923651i	$0.625190\pi$
$888$	0	0
$889$	−58726.6	−2.21555
$890$	−26914.3	−1.01367
$891$	0	0
$892$	8221.38	0.308601
$893$	−41968.6	−1.57270
$894$	0	0
$895$	55979.4	2.09071
$896$	−61591.2	−2.29645
$897$	0	0
$898$	−67432.9	−2.50586
$899$	1919.20	0.0712001
$900$	0	0
$901$	−13200.2	−0.488084
$902$	−62145.0	−2.29401
$903$	0	0
$904$	−2377.16	−0.0874592
$905$	2987.48	0.109732
$906$	0	0
$907$	19608.0	0.717830	0.358915	−	0.933370i	$-0.383147\pi$
$907$	19608.0	0.717830	0.358915	+	0.933370i	$0.383147\pi$
$908$	51897.9	1.89680
$909$	0	0
$910$	−132202.	−4.81590
$911$	−9001.04	−0.327352	−0.163676	−	0.986514i	$-0.552335\pi$
$911$	−9001.04	−0.327352	−0.163676	+	0.986514i	$0.552335\pi$
$912$	0	0
$913$	−30435.6	−1.10325
$914$	18286.0	0.661757
$915$	0	0
$916$	−143.879	−0.00518984
$917$	−74251.7	−2.67394
$918$	0	0
$919$	−40189.4	−1.44257	−0.721287	−	0.692636i	$-0.756449\pi$
$919$	−40189.4	−1.44257	−0.721287	+	0.692636i	$0.756449\pi$
$920$	31560.2	1.13099
$921$	0	0
$922$	−57517.3	−2.05448
$923$	18400.6	0.656191
$924$	0	0
$925$	48467.6	1.72282
$926$	40058.6	1.42160
$927$	0	0
$928$	36825.8	1.30266
$929$	−35284.5	−1.24612	−0.623062	−	0.782173i	$-0.714111\pi$
$929$	−35284.5	−1.24612	−0.623062	+	0.782173i	$0.714111\pi$
$930$	0	0
$931$	76695.1	2.69987
$932$	−31185.5	−1.09605
$933$	0	0
$934$	26087.2	0.913916
$935$	−19739.8	−0.690440
$936$	0	0
$937$	−20447.8	−0.712914	−0.356457	−	0.934312i	$-0.616015\pi$
$937$	−20447.8	−0.712914	−0.356457	+	0.934312i	$0.616015\pi$
$938$	−62334.7	−2.16983
$939$	0	0
$940$	89279.5	3.09785
$941$	19811.3	0.686325	0.343162	−	0.939276i	$-0.388502\pi$
$941$	19811.3	0.686325	0.343162	+	0.939276i	$0.388502\pi$
$942$	0	0
$943$	32574.6	1.12489
$944$	−12969.0	−0.447146
$945$	0	0
$946$	−65080.7	−2.23674
$947$	−24496.1	−0.840568	−0.420284	−	0.907393i	$-0.638069\pi$
$947$	−24496.1	−0.840568	−0.420284	+	0.907393i	$0.638069\pi$
$948$	0	0
$949$	34535.6	1.18132
$950$	−59844.6	−2.04381
$951$	0	0
$952$	11804.4	0.401874
$953$	−51080.3	−1.73626	−0.868129	−	0.496338i	$-0.834678\pi$
$953$	−51080.3	−1.73626	−0.868129	+	0.496338i	$0.834678\pi$
$954$	0	0
$955$	11793.1	0.399596
$956$	2743.34	0.0928096
$957$	0	0
$958$	9748.49	0.328768
$959$	53337.9	1.79601
$960$	0	0
$961$	−29648.5	−0.995216
$962$	−73658.5	−2.46865
$963$	0	0
$964$	32758.8	1.09449
$965$	12738.6	0.424945
$966$	0	0
$967$	39531.4	1.31463	0.657313	−	0.753618i	$-0.271693\pi$
$967$	39531.4	1.31463	0.657313	+	0.753618i	$0.271693\pi$
$968$	−23458.7	−0.778916
$969$	0	0
$970$	77990.0	2.58155
$971$	18854.7	0.623147	0.311573	−	0.950222i	$-0.399144\pi$
$971$	18854.7	0.623147	0.311573	+	0.950222i	$0.399144\pi$
$972$	0	0
$973$	−37021.9	−1.21980
$974$	45742.0	1.50479
$975$	0	0
$976$	8067.76	0.264593
$977$	−17078.6	−0.559256	−0.279628	−	0.960108i	$-0.590211\pi$
$977$	−17078.6	−0.559256	−0.279628	+	0.960108i	$0.590211\pi$
$978$	0	0
$979$	−19620.1	−0.640512
$980$	−163153.	−5.31809
$981$	0	0
$982$	−43510.3	−1.41392
$983$	19272.5	0.625328	0.312664	−	0.949864i	$-0.398779\pi$
$983$	19272.5	0.625328	0.312664	+	0.949864i	$0.398779\pi$
$984$	0	0
$985$	−9717.47	−0.314339
$986$	−15761.3	−0.509069
$987$	0	0
$988$	53595.1	1.72580
$989$	34113.5	1.09681
$990$	0	0
$991$	−14595.4	−0.467848	−0.233924	−	0.972255i	$-0.575157\pi$
$991$	−14595.4	−0.467848	−0.233924	+	0.972255i	$0.575157\pi$
$992$	2734.43	0.0875184
$993$	0	0
$994$	53990.1	1.72280
$995$	−27983.2	−0.891584
$996$	0	0
$997$	7579.06	0.240754	0.120377	−	0.992728i	$-0.461590\pi$
$997$	7579.06	0.240754	0.120377	+	0.992728i	$0.461590\pi$
$998$	−93428.3	−2.96335
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.7		37
3.2	odd	2		239.4.a.b.1.31	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.31	✓	37	3.2	odd	2
2151.4.a.f.1.7		37	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-4.41344 q^{2} +11.4784 q^{4} -16.6195 q^{5} -34.6158 q^{7} -15.3518 q^{8} +O(q^{10})\)	\(q-4.41344 q^{2} +11.4784 q^{4} -16.6195 q^{5} -34.6158 q^{7} -15.3518 q^{8} +73.3492 q^{10} +53.4703 q^{11} +52.0679 q^{13} +152.775 q^{14} -24.0733 q^{16} +22.2133 q^{17} +89.6753 q^{19} -190.766 q^{20} -235.988 q^{22} +123.698 q^{23} +151.208 q^{25} -229.798 q^{26} -397.335 q^{28} +160.769 q^{29} +11.9376 q^{31} +229.060 q^{32} -98.0368 q^{34} +575.298 q^{35} +320.535 q^{37} -395.776 q^{38} +255.139 q^{40} +263.340 q^{41} +275.780 q^{43} +613.755 q^{44} -545.934 q^{46} -468.006 q^{47} +855.253 q^{49} -667.348 q^{50} +597.657 q^{52} -594.250 q^{53} -888.651 q^{55} +531.414 q^{56} -709.544 q^{58} +538.732 q^{59} -335.134 q^{61} -52.6859 q^{62} -818.356 q^{64} -865.343 q^{65} -408.017 q^{67} +254.973 q^{68} -2539.04 q^{70} +353.397 q^{71} +663.280 q^{73} -1414.66 q^{74} +1029.33 q^{76} -1850.92 q^{77} +320.302 q^{79} +400.086 q^{80} -1162.23 q^{82} -569.206 q^{83} -369.174 q^{85} -1217.14 q^{86} -820.865 q^{88} -366.934 q^{89} -1802.37 q^{91} +1419.86 q^{92} +2065.51 q^{94} -1490.36 q^{95} +1063.27 q^{97} -3774.61 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

L-functions

Modular forms

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Database

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