LMFDB - Embedded newform 207.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2009704.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 27x^{2} - 6x + 112 $ x^4 - 27x^2 - 6x + 112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.45983$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.45983 q^{2} +3.97043 q^{4} -7.89053 q^{5} +4.57766 q^{7} +13.9416 q^{8} +27.2999 q^{10} +7.16805 q^{11} +9.66367 q^{13} -15.8379 q^{14} -79.9991 q^{16} +69.1871 q^{17} -21.2382 q^{19} -31.3288 q^{20} -24.8002 q^{22} -23.0000 q^{23} -62.7395 q^{25} -33.4347 q^{26} +18.1753 q^{28} -39.5338 q^{29} +28.5625 q^{31} +165.251 q^{32} -239.376 q^{34} -36.1202 q^{35} -170.923 q^{37} +73.4807 q^{38} -110.007 q^{40} -395.474 q^{41} -214.499 q^{43} +28.4603 q^{44} +79.5761 q^{46} -387.800 q^{47} -322.045 q^{49} +217.068 q^{50} +38.3690 q^{52} -268.151 q^{53} -56.5597 q^{55} +63.8200 q^{56} +136.780 q^{58} -552.468 q^{59} +354.991 q^{61} -98.8214 q^{62} +68.2540 q^{64} -76.2515 q^{65} +293.680 q^{67} +274.703 q^{68} +124.970 q^{70} -505.691 q^{71} +1048.26 q^{73} +591.365 q^{74} -84.3250 q^{76} +32.8129 q^{77} -38.4080 q^{79} +631.236 q^{80} +1368.27 q^{82} +111.501 q^{83} -545.923 q^{85} +742.130 q^{86} +99.9342 q^{88} +1479.10 q^{89} +44.2370 q^{91} -91.3200 q^{92} +1341.72 q^{94} +167.581 q^{95} -1274.61 q^{97} +1114.22 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 4 q^{5} - 14 q^{7} - 84 q^{8} - 100 q^{10} - 70 q^{11} - 12 q^{13} + 18 q^{14} + 130 q^{16} - 178 q^{17} + 96 q^{19} + 296 q^{20} - 326 q^{22} - 92 q^{23} - 80 q^{25} + 464 q^{26}+ \cdots + 2784 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 4 * q^5 - 14 * q^7 - 84 * q^8 - 100 * q^10 - 70 * q^11 - 12 * q^13 + 18 * q^14 + 130 * q^16 - 178 * q^17 + 96 * q^19 + 296 * q^20 - 326 * q^22 - 92 * q^23 - 80 * q^25 + 464 * q^26 - 774 * q^28 + 24 * q^29 - 400 * q^31 - 636 * q^32 - 18 * q^34 - 224 * q^35 - 358 * q^37 + 136 * q^38 - 1268 * q^40 - 152 * q^41 - 196 * q^43 - 70 * q^44 + 92 * q^46 - 260 * q^47 + 476 * q^49 - 876 * q^50 - 876 * q^52 - 968 * q^53 + 684 * q^55 + 1234 * q^56 + 904 * q^58 - 1236 * q^59 - 482 * q^61 + 1304 * q^62 + 1658 * q^64 - 1424 * q^65 + 620 * q^67 - 838 * q^68 + 2636 * q^70 + 88 * q^71 - 948 * q^73 - 710 * q^74 + 3344 * q^76 + 892 * q^77 + 254 * q^79 + 5144 * q^80 + 1568 * q^82 + 550 * q^83 - 1208 * q^85 + 1236 * q^86 + 202 * q^88 - 202 * q^89 - 356 * q^91 - 598 * q^92 - 1328 * q^94 + 460 * q^95 - 1800 * q^97 + 2784 * q^98

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 $

$2$	−3.45983	−1.22324	−0.611618	−	0.791154i	$-0.709481\pi$
$2$	−3.45983	−1.22324	−0.611618	+	0.791154i	$0.709481\pi$
$3$	0	0
$3$	0	0
$4$	3.97043	0.496304
$5$	−7.89053	−0.705751	−0.352875	−	0.935670i	$-0.614796\pi$
$5$	−7.89053	−0.705751	−0.352875	+	0.935670i	$0.614796\pi$
$6$	0	0
$7$	4.57766	0.247170	0.123585	−	0.992334i	$-0.460561\pi$
$7$	4.57766	0.247170	0.123585	+	0.992334i	$0.460561\pi$
$8$	13.9416	0.616138
$9$	0	0
$10$	27.2999	0.863299
$11$	7.16805	0.196477	0.0982386	−	0.995163i	$-0.468679\pi$
$11$	7.16805	0.196477	0.0982386	+	0.995163i	$0.468679\pi$
$12$	0	0
$13$	9.66367	0.206171	0.103085	−	0.994673i	$-0.467129\pi$
$13$	9.66367	0.206171	0.103085	+	0.994673i	$0.467129\pi$
$14$	−15.8379	−0.302348
$15$	0	0
$16$	−79.9991	−1.24999
$17$	69.1871	0.987078	0.493539	−	0.869724i	$-0.335703\pi$
$17$	69.1871	0.987078	0.493539	+	0.869724i	$0.335703\pi$
$18$	0	0
$19$	−21.2382	−0.256441	−0.128221	−	0.991746i	$-0.540927\pi$
$19$	−21.2382	−0.256441	−0.128221	+	0.991746i	$0.540927\pi$
$20$	−31.3288	−0.350267
$21$	0	0
$22$	−24.8002	−0.240338
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−62.7395	−0.501916
$26$	−33.4347	−0.252195
$27$	0	0
$28$	18.1753	0.122672
$29$	−39.5338	−0.253146	−0.126573	−	0.991957i	$-0.540398\pi$
$29$	−39.5338	−0.253146	−0.126573	+	0.991957i	$0.540398\pi$
$30$	0	0
$31$	28.5625	0.165483	0.0827415	−	0.996571i	$-0.473632\pi$
$31$	28.5625	0.165483	0.0827415	+	0.996571i	$0.473632\pi$
$32$	165.251	0.912889
$33$	0	0
$34$	−239.376	−1.20743
$35$	−36.1202	−0.174441
$36$	0	0
$37$	−170.923	−0.759448	−0.379724	−	0.925100i	$-0.623981\pi$
$37$	−170.923	−0.759448	−0.379724	+	0.925100i	$0.623981\pi$
$38$	73.4807	0.313688
$39$	0	0
$40$	−110.007	−0.434840
$41$	−395.474	−1.50641	−0.753203	−	0.657788i	$-0.771492\pi$
$41$	−395.474	−1.50641	−0.753203	+	0.657788i	$0.771492\pi$
$42$	0	0
$43$	−214.499	−0.760716	−0.380358	−	0.924839i	$-0.624199\pi$
$43$	−214.499	−0.760716	−0.380358	+	0.924839i	$0.624199\pi$
$44$	28.4603	0.0975124
$45$	0	0
$46$	79.5761	0.255062
$47$	−387.800	−1.20354	−0.601771	−	0.798669i	$-0.705538\pi$
$47$	−387.800	−1.20354	−0.601771	+	0.798669i	$0.705538\pi$
$48$	0	0
$49$	−322.045	−0.938907
$50$	217.068	0.613961
$51$	0	0
$52$	38.3690	0.102323
$53$	−268.151	−0.694969	−0.347484	−	0.937686i	$-0.612964\pi$
$53$	−268.151	−0.694969	−0.347484	+	0.937686i	$0.612964\pi$
$54$	0	0
$55$	−56.5597	−0.138664
$56$	63.8200	0.152291
$57$	0	0
$58$	136.780	0.309658
$59$	−552.468	−1.21907	−0.609535	−	0.792759i	$-0.708644\pi$
$59$	−552.468	−1.21907	−0.609535	+	0.792759i	$0.708644\pi$
$60$	0	0
$61$	354.991	0.745113	0.372557	−	0.928009i	$-0.378481\pi$
$61$	354.991	0.745113	0.372557	+	0.928009i	$0.378481\pi$
$62$	−98.8214	−0.202425
$63$	0	0
$64$	68.2540	0.133309
$65$	−76.2515	−0.145505
$66$	0	0
$67$	293.680	0.535504	0.267752	−	0.963488i	$-0.413719\pi$
$67$	293.680	0.535504	0.267752	+	0.963488i	$0.413719\pi$
$68$	274.703	0.489891
$69$	0	0
$70$	124.970	0.213382
$71$	−505.691	−0.845274	−0.422637	−	0.906299i	$-0.638895\pi$
$71$	−505.691	−0.845274	−0.422637	+	0.906299i	$0.638895\pi$
$72$	0	0
$73$	1048.26	1.68069	0.840343	−	0.542055i	$-0.182354\pi$
$73$	1048.26	1.68069	0.840343	+	0.542055i	$0.182354\pi$
$74$	591.365	0.928984
$75$	0	0
$76$	−84.3250	−0.127273
$77$	32.8129	0.0485633
$78$	0	0
$79$	−38.4080	−0.0546992	−0.0273496	−	0.999626i	$-0.508707\pi$
$79$	−38.4080	−0.0546992	−0.0273496	+	0.999626i	$0.508707\pi$
$80$	631.236	0.882179
$81$	0	0
$82$	1368.27	1.84269
$83$	111.501	0.147455	0.0737276	−	0.997278i	$-0.476510\pi$
$83$	111.501	0.147455	0.0737276	+	0.997278i	$0.476510\pi$
$84$	0	0
$85$	−545.923	−0.696631
$86$	742.130	0.930534
$87$	0	0
$88$	99.9342	0.121057
$89$	1479.10	1.76163	0.880813	−	0.473464i	$-0.156996\pi$
$89$	1479.10	1.76163	0.880813	+	0.473464i	$0.156996\pi$
$90$	0	0
$91$	44.2370	0.0509593
$92$	−91.3200	−0.103487
$93$	0	0
$94$	1341.72	1.47222
$95$	167.581	0.180984
$96$	0	0
$97$	−1274.61	−1.33420	−0.667099	−	0.744969i	$-0.732464\pi$
$97$	−1274.61	−1.33420	−0.667099	+	0.744969i	$0.732464\pi$
$98$	1114.22	1.14850
$99$	0	0
$100$	−249.103	−0.249103
$101$	1030.08	1.01482	0.507412	−	0.861704i	$-0.330602\pi$
$101$	1030.08	1.01482	0.507412	+	0.861704i	$0.330602\pi$
$102$	0	0
$103$	−1261.15	−1.20646	−0.603229	−	0.797568i	$-0.706119\pi$
$103$	−1261.15	−1.20646	−0.603229	+	0.797568i	$0.706119\pi$
$104$	134.727	0.127030
$105$	0	0
$106$	927.757	0.850110
$107$	−314.435	−0.284089	−0.142045	−	0.989860i	$-0.545368\pi$
$107$	−314.435	−0.284089	−0.142045	+	0.989860i	$0.545368\pi$
$108$	0	0
$109$	−594.442	−0.522360	−0.261180	−	0.965290i	$-0.584112\pi$
$109$	−594.442	−0.522360	−0.261180	+	0.965290i	$0.584112\pi$
$110$	195.687	0.169618
$111$	0	0
$112$	−366.209	−0.308960
$113$	−705.117	−0.587008	−0.293504	−	0.955958i	$-0.594821\pi$
$113$	−705.117	−0.587008	−0.293504	+	0.955958i	$0.594821\pi$
$114$	0	0
$115$	181.482	0.147159
$116$	−156.966	−0.125638
$117$	0	0
$118$	1911.45	1.49121
$119$	316.715	0.243977
$120$	0	0
$121$	−1279.62	−0.961397
$122$	−1228.21	−0.911449
$123$	0	0
$124$	113.405	0.0821300
$125$	1481.36	1.05998
$126$	0	0
$127$	365.035	0.255052	0.127526	−	0.991835i	$-0.459296\pi$
$127$	365.035	0.255052	0.127526	+	0.991835i	$0.459296\pi$
$128$	−1558.15	−1.07596
$129$	0	0
$130$	263.817	0.177987
$131$	−841.502	−0.561239	−0.280620	−	0.959819i	$-0.590540\pi$
$131$	−841.502	−0.561239	−0.280620	+	0.959819i	$0.590540\pi$
$132$	0	0
$133$	−97.2214	−0.0633847
$134$	−1016.08	−0.655047
$135$	0	0
$136$	964.580	0.608177
$137$	−2767.12	−1.72563	−0.862814	−	0.505521i	$-0.831300\pi$
$137$	−2767.12	−1.72563	−0.862814	+	0.505521i	$0.831300\pi$
$138$	0	0
$139$	−1201.78	−0.733336	−0.366668	−	0.930352i	$-0.619502\pi$
$139$	−1201.78	−0.733336	−0.366668	+	0.930352i	$0.619502\pi$
$140$	−143.413	−0.0865757
$141$	0	0
$142$	1749.61	1.03397
$143$	69.2696	0.0405078
$144$	0	0
$145$	311.943	0.178658
$146$	−3626.82	−2.05587
$147$	0	0
$148$	−678.639	−0.376917
$149$	2411.55	1.32592	0.662961	−	0.748654i	$-0.269300\pi$
$149$	2411.55	1.32592	0.662961	+	0.748654i	$0.269300\pi$
$150$	0	0
$151$	2352.41	1.26779	0.633897	−	0.773418i	$-0.281454\pi$
$151$	2352.41	1.26779	0.633897	+	0.773418i	$0.281454\pi$
$152$	−296.095	−0.158003
$153$	0	0
$154$	−113.527	−0.0594044
$155$	−225.373	−0.116790
$156$	0	0
$157$	−2720.30	−1.38283	−0.691414	−	0.722459i	$-0.743012\pi$
$157$	−2720.30	−1.38283	−0.691414	+	0.722459i	$0.743012\pi$
$158$	132.885	0.0669099
$159$	0	0
$160$	−1303.91	−0.644272
$161$	−105.286	−0.0515386
$162$	0	0
$163$	910.107	0.437332	0.218666	−	0.975800i	$-0.429830\pi$
$163$	910.107	0.437332	0.218666	+	0.975800i	$0.429830\pi$
$164$	−1570.20	−0.747636
$165$	0	0
$166$	−385.773	−0.180372
$167$	−370.185	−0.171532	−0.0857658	−	0.996315i	$-0.527334\pi$
$167$	−370.185	−0.171532	−0.0857658	+	0.996315i	$0.527334\pi$
$168$	0	0
$169$	−2103.61	−0.957494
$170$	1888.80	0.852144
$171$	0	0
$172$	−851.654	−0.377546
$173$	1674.46	0.735879	0.367940	−	0.929850i	$-0.380063\pi$
$173$	1674.46	0.735879	0.367940	+	0.929850i	$0.380063\pi$
$174$	0	0
$175$	−287.200	−0.124059
$176$	−573.438	−0.245594
$177$	0	0
$178$	−5117.45	−2.15488
$179$	1890.08	0.789225	0.394613	−	0.918848i	$-0.370879\pi$
$179$	1890.08	0.789225	0.394613	+	0.918848i	$0.370879\pi$
$180$	0	0
$181$	−1249.68	−0.513191	−0.256596	−	0.966519i	$-0.582601\pi$
$181$	−1249.68	−0.513191	−0.256596	+	0.966519i	$0.582601\pi$
$182$	−153.053	−0.0623352
$183$	0	0
$184$	−320.657	−0.128474
$185$	1348.67	0.535981
$186$	0	0
$187$	495.936	0.193938
$188$	−1539.74	−0.597323
$189$	0	0
$190$	−579.802	−0.221386
$191$	−4304.02	−1.63051	−0.815257	−	0.579100i	$-0.803404\pi$
$191$	−4304.02	−1.63051	−0.815257	+	0.579100i	$0.803404\pi$
$192$	0	0
$193$	−2955.56	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$193$	−2955.56	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$194$	4409.94	1.63204
$195$	0	0
$196$	−1278.66	−0.465983
$197$	−2937.68	−1.06244	−0.531222	−	0.847233i	$-0.678267\pi$
$197$	−2937.68	−1.06244	−0.531222	+	0.847233i	$0.678267\pi$
$198$	0	0
$199$	4241.65	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$199$	4241.65	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$200$	−874.690	−0.309250
$201$	0	0
$202$	−3563.92	−1.24137
$203$	−180.972	−0.0625703
$204$	0	0
$205$	3120.50	1.06315
$206$	4363.38	1.47578
$207$	0	0
$208$	−773.085	−0.257711
$209$	−152.237	−0.0503848
$210$	0	0
$211$	5021.48	1.63835	0.819177	−	0.573541i	$-0.194431\pi$
$211$	5021.48	1.63835	0.819177	+	0.573541i	$0.194431\pi$
$212$	−1064.68	−0.344916
$213$	0	0
$214$	1087.89	0.347508
$215$	1692.51	0.536875
$216$	0	0
$217$	130.749	0.0409025
$218$	2056.67	0.638969
$219$	0	0
$220$	−224.567	−0.0688195
$221$	668.601	0.203507
$222$	0	0
$223$	−1211.64	−0.363843	−0.181922	−	0.983313i	$-0.558232\pi$
$223$	−1211.64	−0.363843	−0.181922	+	0.983313i	$0.558232\pi$
$224$	756.461	0.225639
$225$	0	0
$226$	2439.59	0.718048
$227$	−4524.63	−1.32295	−0.661476	−	0.749966i	$-0.730070\pi$
$227$	−4524.63	−1.32295	−0.661476	+	0.749966i	$0.730070\pi$
$228$	0	0
$229$	−6614.12	−1.90862	−0.954309	−	0.298821i	$-0.903407\pi$
$229$	−6614.12	−1.90862	−0.954309	+	0.298821i	$0.903407\pi$
$230$	−627.898	−0.180010
$231$	0	0
$232$	−551.166	−0.155973
$233$	2017.22	0.567179	0.283589	−	0.958946i	$-0.408475\pi$
$233$	2017.22	0.567179	0.283589	+	0.958946i	$0.408475\pi$
$234$	0	0
$235$	3059.95	0.849401
$236$	−2193.54	−0.605030
$237$	0	0
$238$	−1095.78	−0.298441
$239$	2191.56	0.593140	0.296570	−	0.955011i	$-0.404157\pi$
$239$	2191.56	0.593140	0.296570	+	0.955011i	$0.404157\pi$
$240$	0	0
$241$	4000.34	1.06923	0.534615	−	0.845096i	$-0.320457\pi$
$241$	4000.34	1.06923	0.534615	+	0.845096i	$0.320457\pi$
$242$	4427.27	1.17601
$243$	0	0
$244$	1409.47	0.369803
$245$	2541.11	0.662634
$246$	0	0
$247$	−205.239	−0.0528707
$248$	398.207	0.101960
$249$	0	0
$250$	−5125.27	−1.29660
$251$	2219.23	0.558075	0.279037	−	0.960280i	$-0.409985\pi$
$251$	2219.23	0.558075	0.279037	+	0.960280i	$0.409985\pi$
$252$	0	0
$253$	−164.865	−0.0409683
$254$	−1262.96	−0.311989
$255$	0	0
$256$	4844.91	1.18284
$257$	548.420	0.133111	0.0665554	−	0.997783i	$-0.478799\pi$
$257$	548.420	0.133111	0.0665554	+	0.997783i	$0.478799\pi$
$258$	0	0
$259$	−782.428	−0.187713
$260$	−302.751	−0.0722148
$261$	0	0
$262$	2911.46	0.686528
$263$	4180.82	0.980230	0.490115	−	0.871658i	$-0.336955\pi$
$263$	4180.82	0.980230	0.490115	+	0.871658i	$0.336955\pi$
$264$	0	0
$265$	2115.85	0.490475
$266$	336.370	0.0775344
$267$	0	0
$268$	1166.04	0.265773
$269$	−4339.45	−0.983573	−0.491787	−	0.870716i	$-0.663656\pi$
$269$	−4339.45	−0.983573	−0.491787	+	0.870716i	$0.663656\pi$
$270$	0	0
$271$	−1512.65	−0.339066	−0.169533	−	0.985524i	$-0.554226\pi$
$271$	−1512.65	−0.339066	−0.169533	+	0.985524i	$0.554226\pi$
$272$	−5534.91	−1.23383
$273$	0	0
$274$	9573.77	2.11085
$275$	−449.720	−0.0986150
$276$	0	0
$277$	537.755	0.116645	0.0583223	−	0.998298i	$-0.481425\pi$
$277$	537.755	0.116645	0.0583223	+	0.998298i	$0.481425\pi$
$278$	4157.96	0.897042
$279$	0	0
$280$	−503.574	−0.107480
$281$	−8146.34	−1.72943	−0.864715	−	0.502262i	$-0.832501\pi$
$281$	−8146.34	−1.72943	−0.864715	+	0.502262i	$0.832501\pi$
$282$	0	0
$283$	5231.91	1.09896	0.549479	−	0.835508i	$-0.314826\pi$
$283$	5231.91	1.09896	0.549479	+	0.835508i	$0.314826\pi$
$284$	−2007.81	−0.419513
$285$	0	0
$286$	−239.661	−0.0495506
$287$	−1810.35	−0.372339
$288$	0	0
$289$	−126.148	−0.0256764
$290$	−1079.27	−0.218541
$291$	0	0
$292$	4162.06	0.834131
$293$	1540.05	0.307067	0.153534	−	0.988143i	$-0.450935\pi$
$293$	1540.05	0.307067	0.153534	+	0.988143i	$0.450935\pi$
$294$	0	0
$295$	4359.26	0.860360
$296$	−2382.95	−0.467925
$297$	0	0
$298$	−8343.57	−1.62191
$299$	−222.264	−0.0429896
$300$	0	0
$301$	−981.903	−0.188026
$302$	−8138.96	−1.55081
$303$	0	0
$304$	1699.04	0.320548
$305$	−2801.07	−0.525864
$306$	0	0
$307$	−8549.38	−1.58938	−0.794689	−	0.607017i	$-0.792366\pi$
$307$	−8549.38	−1.58938	−0.794689	+	0.607017i	$0.792366\pi$
$308$	130.281	0.0241022
$309$	0	0
$310$	779.753	0.142861
$311$	3678.75	0.670749	0.335374	−	0.942085i	$-0.391137\pi$
$311$	3678.75	0.670749	0.335374	+	0.942085i	$0.391137\pi$
$312$	0	0
$313$	−4382.07	−0.791339	−0.395669	−	0.918393i	$-0.629487\pi$
$313$	−4382.07	−0.791339	−0.395669	+	0.918393i	$0.629487\pi$
$314$	9411.80	1.69152
$315$	0	0
$316$	−152.496	−0.0271474
$317$	3015.16	0.534221	0.267111	−	0.963666i	$-0.413931\pi$
$317$	3015.16	0.534221	0.267111	+	0.963666i	$0.413931\pi$
$318$	0	0
$319$	−283.380	−0.0497375
$320$	−538.560	−0.0940826
$321$	0	0
$322$	364.273	0.0630438
$323$	−1469.41	−0.253128
$324$	0	0
$325$	−606.294	−0.103480
$326$	−3148.82	−0.534959
$327$	0	0
$328$	−5513.55	−0.928155
$329$	−1775.22	−0.297480
$330$	0	0
$331$	8098.73	1.34485	0.672426	−	0.740164i	$-0.265252\pi$
$331$	8098.73	1.34485	0.672426	+	0.740164i	$0.265252\pi$
$332$	442.706	0.0731827
$333$	0	0
$334$	1280.78	0.209824
$335$	−2317.29	−0.377932
$336$	0	0
$337$	−1279.84	−0.206876	−0.103438	−	0.994636i	$-0.532984\pi$
$337$	−1279.84	−0.206876	−0.103438	+	0.994636i	$0.532984\pi$
$338$	7278.15	1.17124
$339$	0	0
$340$	−2167.55	−0.345741
$341$	204.737	0.0325136
$342$	0	0
$343$	−3044.35	−0.479240
$344$	−2990.46	−0.468706
$345$	0	0
$346$	−5793.36	−0.900153
$347$	6965.46	1.07760	0.538798	−	0.842435i	$-0.318879\pi$
$347$	6965.46	1.07760	0.538798	+	0.842435i	$0.318879\pi$
$348$	0	0
$349$	11580.9	1.77625	0.888123	−	0.459606i	$-0.152009\pi$
$349$	11580.9	1.77625	0.888123	+	0.459606i	$0.152009\pi$
$350$	993.664	0.151753
$351$	0	0
$352$	1184.52	0.179362
$353$	1359.33	0.204958	0.102479	−	0.994735i	$-0.467323\pi$
$353$	1359.33	0.204958	0.102479	+	0.994735i	$0.467323\pi$
$354$	0	0
$355$	3990.17	0.596553
$356$	5872.69	0.874303
$357$	0	0
$358$	−6539.36	−0.965408
$359$	11315.1	1.66348	0.831738	−	0.555168i	$-0.187346\pi$
$359$	11315.1	1.66348	0.831738	+	0.555168i	$0.187346\pi$
$360$	0	0
$361$	−6407.94	−0.934238
$362$	4323.67	0.627754
$363$	0	0
$364$	175.640	0.0252913
$365$	−8271.36	−1.18614
$366$	0	0
$367$	−9260.84	−1.31720	−0.658600	−	0.752494i	$-0.728851\pi$
$367$	−9260.84	−1.31720	−0.658600	+	0.752494i	$0.728851\pi$
$368$	1839.98	0.260640
$369$	0	0
$370$	−4666.19	−0.655631
$371$	−1227.50	−0.171776
$372$	0	0
$373$	7448.35	1.03394	0.516972	−	0.856002i	$-0.327059\pi$
$373$	7448.35	1.03394	0.516972	+	0.856002i	$0.327059\pi$
$374$	−1715.86	−0.237232
$375$	0	0
$376$	−5406.56	−0.741549
$377$	−382.042	−0.0521914
$378$	0	0
$379$	13428.8	1.82003	0.910017	−	0.414570i	$-0.136068\pi$
$379$	13428.8	1.82003	0.910017	+	0.414570i	$0.136068\pi$
$380$	665.369	0.0898229
$381$	0	0
$382$	14891.2	1.99450
$383$	12530.8	1.67178	0.835891	−	0.548896i	$-0.184952\pi$
$383$	12530.8	1.67178	0.835891	+	0.548896i	$0.184952\pi$
$384$	0	0
$385$	−258.911	−0.0342736
$386$	10225.8	1.34839
$387$	0	0
$388$	−5060.76	−0.662168
$389$	2037.90	0.265619	0.132809	−	0.991142i	$-0.457600\pi$
$389$	2037.90	0.265619	0.132809	+	0.991142i	$0.457600\pi$
$390$	0	0
$391$	−1591.30	−0.205820
$392$	−4489.83	−0.578496
$393$	0	0
$394$	10163.9	1.29962
$395$	303.059	0.0386040
$396$	0	0
$397$	−12445.6	−1.57337	−0.786683	−	0.617357i	$-0.788203\pi$
$397$	−12445.6	−1.57337	−0.786683	+	0.617357i	$0.788203\pi$
$398$	−14675.4	−1.84827
$399$	0	0
$400$	5019.11	0.627388
$401$	−7515.90	−0.935976	−0.467988	−	0.883735i	$-0.655021\pi$
$401$	−7515.90	−0.935976	−0.467988	+	0.883735i	$0.655021\pi$
$402$	0	0
$403$	276.018	0.0341178
$404$	4089.88	0.503661
$405$	0	0
$406$	626.134	0.0765382
$407$	−1225.19	−0.149214
$408$	0	0
$409$	4458.77	0.539051	0.269526	−	0.962993i	$-0.413133\pi$
$409$	4458.77	0.539051	0.269526	+	0.962993i	$0.413133\pi$
$410$	−10796.4	−1.30048
$411$	0	0
$412$	−5007.33	−0.598770
$413$	−2529.01	−0.301318
$414$	0	0
$415$	−879.799	−0.104067
$416$	1596.93	0.188211
$417$	0	0
$418$	526.713	0.0616325
$419$	−4506.59	−0.525445	−0.262722	−	0.964871i	$-0.584620\pi$
$419$	−4506.59	−0.525445	−0.262722	+	0.964871i	$0.584620\pi$
$420$	0	0
$421$	4391.61	0.508395	0.254197	−	0.967152i	$-0.418189\pi$
$421$	4391.61	0.508395	0.254197	+	0.967152i	$0.418189\pi$
$422$	−17373.5	−2.00409
$423$	0	0
$424$	−3738.46	−0.428197
$425$	−4340.76	−0.495431
$426$	0	0
$427$	1625.03	0.184170
$428$	−1248.44	−0.140995
$429$	0	0
$430$	−5855.80	−0.656725
$431$	4034.85	0.450932	0.225466	−	0.974251i	$-0.427610\pi$
$431$	4034.85	0.450932	0.225466	+	0.974251i	$0.427610\pi$
$432$	0	0
$433$	−2378.67	−0.263999	−0.131999	−	0.991250i	$-0.542140\pi$
$433$	−2378.67	−0.263999	−0.131999	+	0.991250i	$0.542140\pi$
$434$	−452.371	−0.0500334
$435$	0	0
$436$	−2360.19	−0.259249
$437$	488.479	0.0534717
$438$	0	0
$439$	6767.22	0.735722	0.367861	−	0.929881i	$-0.380090\pi$
$439$	6767.22	0.735722	0.367861	+	0.929881i	$0.380090\pi$
$440$	−788.534	−0.0854361
$441$	0	0
$442$	−2313.25	−0.248936
$443$	−2767.13	−0.296773	−0.148386	−	0.988929i	$-0.547408\pi$
$443$	−2767.13	−0.296773	−0.148386	+	0.988929i	$0.547408\pi$
$444$	0	0
$445$	−11670.9	−1.24327
$446$	4192.05	0.445066
$447$	0	0
$448$	312.443	0.0329499
$449$	−3616.87	−0.380158	−0.190079	−	0.981769i	$-0.560874\pi$
$449$	−3616.87	−0.380158	−0.190079	+	0.981769i	$0.560874\pi$
$450$	0	0
$451$	−2834.78	−0.295974
$452$	−2799.62	−0.291334
$453$	0	0
$454$	15654.5	1.61828
$455$	−349.053	−0.0359646
$456$	0	0
$457$	−5912.55	−0.605202	−0.302601	−	0.953117i	$-0.597855\pi$
$457$	−5912.55	−0.605202	−0.302601	+	0.953117i	$0.597855\pi$
$458$	22883.8	2.33469
$459$	0	0
$460$	720.563	0.0730357
$461$	−3865.90	−0.390570	−0.195285	−	0.980747i	$-0.562563\pi$
$461$	−3865.90	−0.390570	−0.195285	+	0.980747i	$0.562563\pi$
$462$	0	0
$463$	−11789.4	−1.18337	−0.591687	−	0.806168i	$-0.701538\pi$
$463$	−11789.4	−1.18337	−0.591687	+	0.806168i	$0.701538\pi$
$464$	3162.67	0.316430
$465$	0	0
$466$	−6979.25	−0.693793
$467$	−3867.34	−0.383210	−0.191605	−	0.981472i	$-0.561369\pi$
$467$	−3867.34	−0.383210	−0.191605	+	0.981472i	$0.561369\pi$
$468$	0	0
$469$	1344.37	0.132361
$470$	−10586.9	−1.03902
$471$	0	0
$472$	−7702.30	−0.751116
$473$	−1537.54	−0.149463
$474$	0	0
$475$	1332.48	0.128712
$476$	1257.50	0.121087
$477$	0	0
$478$	−7582.44	−0.725549
$479$	17656.8	1.68425	0.842127	−	0.539279i	$-0.181303\pi$
$479$	17656.8	1.68425	0.842127	+	0.539279i	$0.181303\pi$
$480$	0	0
$481$	−1651.74	−0.156576
$482$	−13840.5	−1.30792
$483$	0	0
$484$	−5080.64	−0.477145
$485$	10057.4	0.941611
$486$	0	0
$487$	−5750.71	−0.535092	−0.267546	−	0.963545i	$-0.586213\pi$
$487$	−5750.71	−0.535092	−0.267546	+	0.963545i	$0.586213\pi$
$488$	4949.14	0.459093
$489$	0	0
$490$	−8791.80	−0.810557
$491$	−12090.0	−1.11123	−0.555616	−	0.831439i	$-0.687517\pi$
$491$	−12090.0	−1.11123	−0.555616	+	0.831439i	$0.687517\pi$
$492$	0	0
$493$	−2735.23	−0.249875
$494$	710.093	0.0646733
$495$	0	0
$496$	−2284.97	−0.206852
$497$	−2314.88	−0.208927
$498$	0	0
$499$	14759.4	1.32410	0.662048	−	0.749462i	$-0.269688\pi$
$499$	14759.4	1.32410	0.662048	+	0.749462i	$0.269688\pi$
$500$	5881.66	0.526072
$501$	0	0
$502$	−7678.17	−0.682657
$503$	−15790.7	−1.39975	−0.699875	−	0.714265i	$-0.746761\pi$
$503$	−15790.7	−1.39975	−0.699875	+	0.714265i	$0.746761\pi$
$504$	0	0
$505$	−8127.91	−0.716212
$506$	570.406	0.0501139
$507$	0	0
$508$	1449.35	0.126584
$509$	4800.46	0.418029	0.209014	−	0.977913i	$-0.432974\pi$
$509$	4800.46	0.418029	0.209014	+	0.977913i	$0.432974\pi$
$510$	0	0
$511$	4798.60	0.415416
$512$	−4297.36	−0.370934
$513$	0	0
$514$	−1897.44	−0.162826
$515$	9951.17	0.851458
$516$	0	0
$517$	−2779.77	−0.236469
$518$	2707.07	0.229617
$519$	0	0
$520$	−1063.07	−0.0896513
$521$	5068.16	0.426180	0.213090	−	0.977033i	$-0.431647\pi$
$521$	5068.16	0.426180	0.213090	+	0.977033i	$0.431647\pi$
$522$	0	0
$523$	−4316.59	−0.360901	−0.180451	−	0.983584i	$-0.557756\pi$
$523$	−4316.59	−0.360901	−0.180451	+	0.983584i	$0.557756\pi$
$524$	−3341.13	−0.278546
$525$	0	0
$526$	−14464.9	−1.19905
$527$	1976.16	0.163345
$528$	0	0
$529$	529.000	0.0434783
$530$	−7320.49	−0.599966
$531$	0	0
$532$	−386.011	−0.0314581
$533$	−3821.73	−0.310577
$534$	0	0
$535$	2481.06	0.200496
$536$	4094.38	0.329944
$537$	0	0
$538$	15013.8	1.20314
$539$	−2308.43	−0.184474
$540$	0	0
$541$	19439.9	1.54490	0.772448	−	0.635079i	$-0.219032\pi$
$541$	19439.9	1.54490	0.772448	+	0.635079i	$0.219032\pi$
$542$	5233.51	0.414758
$543$	0	0
$544$	11433.2	0.901093
$545$	4690.46	0.368656
$546$	0	0
$547$	17162.2	1.34150	0.670752	−	0.741682i	$-0.265972\pi$
$547$	17162.2	1.34150	0.670752	+	0.741682i	$0.265972\pi$
$548$	−10986.7	−0.856437
$549$	0	0
$550$	1555.96	0.120629
$551$	839.629	0.0649172
$552$	0	0
$553$	−175.819	−0.0135200
$554$	−1860.54	−0.142684
$555$	0	0
$556$	−4771.59	−0.363958
$557$	−9533.83	−0.725245	−0.362622	−	0.931936i	$-0.618119\pi$
$557$	−9533.83	−0.725245	−0.362622	+	0.931936i	$0.618119\pi$
$558$	0	0
$559$	−2072.85	−0.156837
$560$	2889.58	0.218048
$561$	0	0
$562$	28185.0	2.11550
$563$	14499.5	1.08540	0.542702	−	0.839925i	$-0.317401\pi$
$563$	14499.5	1.08540	0.542702	+	0.839925i	$0.317401\pi$
$564$	0	0
$565$	5563.75	0.414281
$566$	−18101.5	−1.34428
$567$	0	0
$568$	−7050.15	−0.520806
$569$	3689.09	0.271801	0.135900	−	0.990723i	$-0.456607\pi$
$569$	3689.09	0.271801	0.135900	+	0.990723i	$0.456607\pi$
$570$	0	0
$571$	−18230.6	−1.33612	−0.668061	−	0.744106i	$-0.732876\pi$
$571$	−18230.6	−1.33612	−0.668061	+	0.744106i	$0.732876\pi$
$572$	275.031	0.0201042
$573$	0	0
$574$	6263.49	0.455458
$575$	1443.01	0.104657
$576$	0	0
$577$	20164.4	1.45486	0.727432	−	0.686180i	$-0.240714\pi$
$577$	20164.4	1.45486	0.727432	+	0.686180i	$0.240714\pi$
$578$	436.451	0.0314083
$579$	0	0
$580$	1238.55	0.0886689
$581$	510.412	0.0364466
$582$	0	0
$583$	−1922.12	−0.136545
$584$	14614.5	1.03553
$585$	0	0
$586$	−5328.32	−0.375616
$587$	−3604.51	−0.253448	−0.126724	−	0.991938i	$-0.540446\pi$
$587$	−3604.51	−0.253448	−0.126724	+	0.991938i	$0.540446\pi$
$588$	0	0
$589$	−606.617	−0.0424367
$590$	−15082.3	−1.05242
$591$	0	0
$592$	13673.7	0.949300
$593$	−21451.6	−1.48551	−0.742757	−	0.669561i	$-0.766482\pi$
$593$	−21451.6	−1.48551	−0.742757	+	0.669561i	$0.766482\pi$
$594$	0	0
$595$	−2499.05	−0.172187
$596$	9574.92	0.658060
$597$	0	0
$598$	768.997	0.0525863
$599$	14824.4	1.01120	0.505599	−	0.862769i	$-0.331272\pi$
$599$	14824.4	1.01120	0.505599	+	0.862769i	$0.331272\pi$
$600$	0	0
$601$	−18786.5	−1.27507	−0.637535	−	0.770422i	$-0.720046\pi$
$601$	−18786.5	−1.27507	−0.637535	+	0.770422i	$0.720046\pi$
$602$	3397.22	0.230001
$603$	0	0
$604$	9340.11	0.629211
$605$	10096.9	0.678506
$606$	0	0
$607$	−2160.73	−0.144484	−0.0722418	−	0.997387i	$-0.523015\pi$
$607$	−2160.73	−0.144484	−0.0722418	+	0.997387i	$0.523015\pi$
$608$	−3509.63	−0.234102
$609$	0	0
$610$	9691.21	0.643255
$611$	−3747.57	−0.248135
$612$	0	0
$613$	20705.6	1.36426	0.682131	−	0.731230i	$-0.261053\pi$
$613$	20705.6	1.36426	0.682131	+	0.731230i	$0.261053\pi$
$614$	29579.4	1.94418
$615$	0	0
$616$	457.465	0.0299217
$617$	19079.1	1.24488	0.622442	−	0.782666i	$-0.286140\pi$
$617$	19079.1	1.24488	0.622442	+	0.782666i	$0.286140\pi$
$618$	0	0
$619$	18244.3	1.18465	0.592325	−	0.805699i	$-0.298210\pi$
$619$	18244.3	1.18465	0.592325	+	0.805699i	$0.298210\pi$
$620$	−894.830	−0.0579633
$621$	0	0
$622$	−12727.9	−0.820484
$623$	6770.84	0.435422
$624$	0	0
$625$	−3846.31	−0.246164
$626$	15161.2	0.967993
$627$	0	0
$628$	−10800.8	−0.686303
$629$	−11825.7	−0.749635
$630$	0	0
$631$	4623.60	0.291700	0.145850	−	0.989307i	$-0.453408\pi$
$631$	4623.60	0.291700	0.145850	+	0.989307i	$0.453408\pi$
$632$	−535.469	−0.0337022
$633$	0	0
$634$	−10431.9	−0.653478
$635$	−2880.32	−0.180003
$636$	0	0
$637$	−3112.14	−0.193575
$638$	980.449	0.0608406
$639$	0	0
$640$	12294.6	0.759357
$641$	−21059.1	−1.29763	−0.648816	−	0.760945i	$-0.724736\pi$
$641$	−21059.1	−1.29763	−0.648816	+	0.760945i	$0.724736\pi$
$642$	0	0
$643$	18828.5	1.15478	0.577390	−	0.816468i	$-0.304071\pi$
$643$	18828.5	1.15478	0.577390	+	0.816468i	$0.304071\pi$
$644$	−418.032	−0.0255788
$645$	0	0
$646$	5083.92	0.309635
$647$	18686.1	1.13543	0.567716	−	0.823224i	$-0.307827\pi$
$647$	18686.1	1.13543	0.567716	+	0.823224i	$0.307827\pi$
$648$	0	0
$649$	−3960.12	−0.239520
$650$	2097.67	0.126581
$651$	0	0
$652$	3613.52	0.217050
$653$	13267.2	0.795078	0.397539	−	0.917585i	$-0.369864\pi$
$653$	13267.2	0.795078	0.397539	+	0.917585i	$0.369864\pi$
$654$	0	0
$655$	6639.90	0.396095
$656$	31637.6	1.88299
$657$	0	0
$658$	6141.96	0.363888
$659$	−9430.56	−0.557455	−0.278727	−	0.960370i	$-0.589913\pi$
$659$	−9430.56	−0.557455	−0.278727	+	0.960370i	$0.589913\pi$
$660$	0	0
$661$	−8433.69	−0.496267	−0.248134	−	0.968726i	$-0.579817\pi$
$661$	−8433.69	−0.496267	−0.248134	+	0.968726i	$0.579817\pi$
$662$	−28020.2	−1.64507
$663$	0	0
$664$	1554.50	0.0908528
$665$	767.129	0.0447338
$666$	0	0
$667$	909.278	0.0527847
$668$	−1469.80	−0.0851319
$669$	0	0
$670$	8017.44	0.462300
$671$	2544.59	0.146398
$672$	0	0
$673$	1486.31	0.0851310	0.0425655	−	0.999094i	$-0.486447\pi$
$673$	1486.31	0.0851310	0.0425655	+	0.999094i	$0.486447\pi$
$674$	4428.03	0.253058
$675$	0	0
$676$	−8352.26	−0.475208
$677$	−3996.72	−0.226893	−0.113446	−	0.993544i	$-0.536189\pi$
$677$	−3996.72	−0.226893	−0.113446	+	0.993544i	$0.536189\pi$
$678$	0	0
$679$	−5834.74	−0.329774
$680$	−7611.05	−0.429221
$681$	0	0
$682$	−708.357	−0.0397718
$683$	4484.96	0.251262	0.125631	−	0.992077i	$-0.459904\pi$
$683$	4484.96	0.251262	0.125631	+	0.992077i	$0.459904\pi$
$684$	0	0
$685$	21834.0	1.21786
$686$	10532.9	0.586224
$687$	0	0
$688$	17159.7	0.950884
$689$	−2591.32	−0.143282
$690$	0	0
$691$	8534.58	0.469856	0.234928	−	0.972013i	$-0.424515\pi$
$691$	8534.58	0.469856	0.234928	+	0.972013i	$0.424515\pi$
$692$	6648.35	0.365220
$693$	0	0
$694$	−24099.3	−1.31815
$695$	9482.69	0.517552
$696$	0	0
$697$	−27361.7	−1.48694
$698$	−40067.9	−2.17277
$699$	0	0
$700$	−1140.31	−0.0615709
$701$	−5562.07	−0.299681	−0.149841	−	0.988710i	$-0.547876\pi$
$701$	−5562.07	−0.299681	−0.149841	+	0.988710i	$0.547876\pi$
$702$	0	0
$703$	3630.11	0.194754
$704$	489.248	0.0261921
$705$	0	0
$706$	−4703.07	−0.250711
$707$	4715.38	0.250834
$708$	0	0
$709$	8050.95	0.426460	0.213230	−	0.977002i	$-0.431602\pi$
$709$	8050.95	0.426460	0.213230	+	0.977002i	$0.431602\pi$
$710$	−13805.3	−0.729724
$711$	0	0
$712$	20621.1	1.08541
$713$	−656.937	−0.0345056
$714$	0	0
$715$	−546.574	−0.0285884
$716$	7504.45	0.391696
$717$	0	0
$718$	−39148.3	−2.03482
$719$	−5103.08	−0.264691	−0.132345	−	0.991204i	$-0.542251\pi$
$719$	−5103.08	−0.264691	−0.132345	+	0.991204i	$0.542251\pi$
$720$	0	0
$721$	−5773.13	−0.298201
$722$	22170.4	1.14279
$723$	0	0
$724$	−4961.75	−0.254699
$725$	2480.33	0.127058
$726$	0	0
$727$	−19611.5	−1.00048	−0.500240	−	0.865887i	$-0.666755\pi$
$727$	−19611.5	−1.00048	−0.500240	+	0.865887i	$0.666755\pi$
$728$	616.735	0.0313980
$729$	0	0
$730$	28617.5	1.45093
$731$	−14840.5	−0.750886
$732$	0	0
$733$	−13236.7	−0.666998	−0.333499	−	0.942750i	$-0.608229\pi$
$733$	−13236.7	−0.666998	−0.333499	+	0.942750i	$0.608229\pi$
$734$	32041.0	1.61124
$735$	0	0
$736$	−3800.76	−0.190350
$737$	2105.11	0.105214
$738$	0	0
$739$	−25308.9	−1.25981	−0.629906	−	0.776671i	$-0.716907\pi$
$739$	−25308.9	−1.25981	−0.629906	+	0.776671i	$0.716907\pi$
$740$	5354.82	0.266010
$741$	0	0
$742$	4246.95	0.210122
$743$	−15214.4	−0.751230	−0.375615	−	0.926776i	$-0.622568\pi$
$743$	−15214.4	−0.751230	−0.375615	+	0.926776i	$0.622568\pi$
$744$	0	0
$745$	−19028.4	−0.935770
$746$	−25770.0	−1.26476
$747$	0	0
$748$	1969.08	0.0962524
$749$	−1439.38	−0.0702185
$750$	0	0
$751$	−2600.26	−0.126345	−0.0631724	−	0.998003i	$-0.520122\pi$
$751$	−2600.26	−0.126345	−0.0631724	+	0.998003i	$0.520122\pi$
$752$	31023.7	1.50441
$753$	0	0
$754$	1321.80	0.0638423
$755$	−18561.8	−0.894746
$756$	0	0
$757$	−24500.2	−1.17632	−0.588160	−	0.808745i	$-0.700147\pi$
$757$	−24500.2	−1.17632	−0.588160	+	0.808745i	$0.700147\pi$
$758$	−46461.5	−2.22633
$759$	0	0
$760$	2336.35	0.111511
$761$	−7901.08	−0.376365	−0.188183	−	0.982134i	$-0.560260\pi$
$761$	−7901.08	−0.376365	−0.188183	+	0.982134i	$0.560260\pi$
$762$	0	0
$763$	−2721.15	−0.129112
$764$	−17088.8	−0.809231
$765$	0	0
$766$	−43354.3	−2.04498
$767$	−5338.86	−0.251337
$768$	0	0
$769$	−33754.8	−1.58287	−0.791436	−	0.611252i	$-0.790666\pi$
$769$	−33754.8	−1.58287	−0.791436	+	0.611252i	$0.790666\pi$
$770$	895.789	0.0419247
$771$	0	0
$772$	−11734.9	−0.547082
$773$	−9500.18	−0.442041	−0.221021	−	0.975269i	$-0.570939\pi$
$773$	−9500.18	−0.442041	−0.221021	+	0.975269i	$0.570939\pi$
$774$	0	0
$775$	−1792.00	−0.0830586
$776$	−17770.2	−0.822051
$777$	0	0
$778$	−7050.79	−0.324914
$779$	8399.17	0.386305
$780$	0	0
$781$	−3624.82	−0.166077
$782$	5505.64	0.251766
$783$	0	0
$784$	25763.3	1.17362
$785$	21464.6	0.975931
$786$	0	0
$787$	−37524.0	−1.69960	−0.849800	−	0.527105i	$-0.823277\pi$
$787$	−37524.0	−1.69960	−0.849800	+	0.527105i	$0.823277\pi$
$788$	−11663.9	−0.527295
$789$	0	0
$790$	−1048.53	−0.0472217
$791$	−3227.79	−0.145091
$792$	0	0
$793$	3430.51	0.153620
$794$	43059.6	1.92460
$795$	0	0
$796$	16841.2	0.749899
$797$	−33330.8	−1.48135	−0.740675	−	0.671864i	$-0.765494\pi$
$797$	−33330.8	−1.48135	−0.740675	+	0.671864i	$0.765494\pi$
$798$	0	0
$799$	−26830.8	−1.18799
$800$	−10367.7	−0.458194
$801$	0	0
$802$	26003.8	1.14492
$803$	7514.01	0.330216
$804$	0	0
$805$	830.764	0.0363734
$806$	−954.977	−0.0417340
$807$	0	0
$808$	14361.0	0.625272
$809$	−16696.0	−0.725586	−0.362793	−	0.931870i	$-0.618177\pi$
$809$	−16696.0	−0.725586	−0.362793	+	0.931870i	$0.618177\pi$
$810$	0	0
$811$	27187.8	1.17718	0.588589	−	0.808432i	$-0.299684\pi$
$811$	27187.8	1.17718	0.588589	+	0.808432i	$0.299684\pi$
$812$	−718.539	−0.0310539
$813$	0	0
$814$	4238.94	0.182524
$815$	−7181.22	−0.308647
$816$	0	0
$817$	4555.58	0.195079
$818$	−15426.6	−0.659387
$819$	0	0
$820$	12389.7	0.527645
$821$	14087.5	0.598852	0.299426	−	0.954119i	$-0.403205\pi$
$821$	14087.5	0.598852	0.299426	+	0.954119i	$0.403205\pi$
$822$	0	0
$823$	10841.3	0.459178	0.229589	−	0.973288i	$-0.426262\pi$
$823$	10841.3	0.459178	0.229589	+	0.973288i	$0.426262\pi$
$824$	−17582.5	−0.743345
$825$	0	0
$826$	8749.95	0.368583
$827$	−11420.3	−0.480195	−0.240097	−	0.970749i	$-0.577179\pi$
$827$	−11420.3	−0.480195	−0.240097	+	0.970749i	$0.577179\pi$
$828$	0	0
$829$	20555.8	0.861196	0.430598	−	0.902544i	$-0.358303\pi$
$829$	20555.8	0.861196	0.430598	+	0.902544i	$0.358303\pi$
$830$	3043.96	0.127298
$831$	0	0
$832$	659.583	0.0274843
$833$	−22281.4	−0.926775
$834$	0	0
$835$	2920.96	0.121059
$836$	−604.446	−0.0250062
$837$	0	0
$838$	15592.0	0.642742
$839$	12.5459	0.000516247 0	0.000258123	−	1.00000i	$-0.499918\pi$
$839$	12.5459	0.000516247 0	0.000258123		1.00000i	$0.499918\pi$
$840$	0	0
$841$	−22826.1	−0.935917
$842$	−15194.2	−0.621886
$843$	0	0
$844$	19937.4	0.813122
$845$	16598.6	0.675752
$846$	0	0
$847$	−5857.66	−0.237629
$848$	21451.8	0.868701
$849$	0	0
$850$	15018.3	0.606028
$851$	3931.23	0.158356
$852$	0	0
$853$	−19060.7	−0.765096	−0.382548	−	0.923936i	$-0.624953\pi$
$853$	−19060.7	−0.765096	−0.382548	+	0.923936i	$0.624953\pi$
$854$	−5622.32	−0.225283
$855$	0	0
$856$	−4383.73	−0.175038
$857$	15239.1	0.607418	0.303709	−	0.952765i	$-0.401775\pi$
$857$	15239.1	0.607418	0.303709	+	0.952765i	$0.401775\pi$
$858$	0	0
$859$	9970.27	0.396020	0.198010	−	0.980200i	$-0.436552\pi$
$859$	9970.27	0.396020	0.198010	+	0.980200i	$0.436552\pi$
$860$	6720.00	0.266454
$861$	0	0
$862$	−13959.9	−0.551596
$863$	−35257.3	−1.39070	−0.695350	−	0.718671i	$-0.744751\pi$
$863$	−35257.3	−1.39070	−0.695350	+	0.718671i	$0.744751\pi$
$864$	0	0
$865$	−13212.4	−0.519347
$866$	8229.79	0.322932
$867$	0	0
$868$	519.132	0.0203001
$869$	−275.310	−0.0107471
$870$	0	0
$871$	2838.03	0.110405
$872$	−8287.48	−0.321846
$873$	0	0
$874$	−1690.06	−0.0654085
$875$	6781.18	0.261995
$876$	0	0
$877$	−19721.8	−0.759358	−0.379679	−	0.925118i	$-0.623966\pi$
$877$	−19721.8	−0.759358	−0.379679	+	0.925118i	$0.623966\pi$
$878$	−23413.4	−0.899960
$879$	0	0
$880$	4524.73	0.173328
$881$	−27621.7	−1.05630	−0.528150	−	0.849151i	$-0.677114\pi$
$881$	−27621.7	−1.05630	−0.528150	+	0.849151i	$0.677114\pi$
$882$	0	0
$883$	35195.3	1.34135	0.670677	−	0.741750i	$-0.266004\pi$
$883$	35195.3	1.34135	0.670677	+	0.741750i	$0.266004\pi$
$884$	2654.64	0.101001
$885$	0	0
$886$	9573.80	0.363023
$887$	37874.1	1.43369	0.716847	−	0.697231i	$-0.245585\pi$
$887$	37874.1	1.43369	0.716847	+	0.697231i	$0.245585\pi$
$888$	0	0
$889$	1671.01	0.0630414
$890$	40379.4	1.52081
$891$	0	0
$892$	−4810.72	−0.180577
$893$	8236.19	0.308638
$894$	0	0
$895$	−14913.7	−0.556996
$896$	−7132.69	−0.265945
$897$	0	0
$898$	12513.8	0.465022
$899$	−1129.18	−0.0418915
$900$	0	0
$901$	−18552.6	−0.685989
$902$	9807.85	0.362046
$903$	0	0
$904$	−9830.48	−0.361678
$905$	9860.60	0.362185
$906$	0	0
$907$	−23648.3	−0.865744	−0.432872	−	0.901455i	$-0.642500\pi$
$907$	−23648.3	−0.865744	−0.432872	+	0.901455i	$0.642500\pi$
$908$	−17964.7	−0.656587
$909$	0	0
$910$	1207.67	0.0439931
$911$	24936.9	0.906911	0.453456	−	0.891279i	$-0.350191\pi$
$911$	24936.9	0.906911	0.453456	+	0.891279i	$0.350191\pi$
$912$	0	0
$913$	799.242	0.0289716
$914$	20456.4	0.740304
$915$	0	0
$916$	−26260.9	−0.947255
$917$	−3852.11	−0.138722
$918$	0	0
$919$	38348.8	1.37651	0.688253	−	0.725470i	$-0.258378\pi$
$919$	38348.8	1.37651	0.688253	+	0.725470i	$0.258378\pi$
$920$	2530.16	0.0906704
$921$	0	0
$922$	13375.4	0.477759
$923$	−4886.83	−0.174271
$924$	0	0
$925$	10723.6	0.381179
$926$	40789.5	1.44754
$927$	0	0
$928$	−6532.99	−0.231095
$929$	28584.5	1.00950	0.504751	−	0.863265i	$-0.331584\pi$
$929$	28584.5	1.00950	0.504751	+	0.863265i	$0.331584\pi$
$930$	0	0
$931$	6839.67	0.240774
$932$	8009.25	0.281493
$933$	0	0
$934$	13380.3	0.468756
$935$	−3913.20	−0.136872
$936$	0	0
$937$	−6982.17	−0.243434	−0.121717	−	0.992565i	$-0.538840\pi$
$937$	−6982.17	−0.243434	−0.121717	+	0.992565i	$0.538840\pi$
$938$	−4651.29	−0.161908
$939$	0	0
$940$	12149.3	0.421561
$941$	34182.7	1.18419	0.592095	−	0.805868i	$-0.298301\pi$
$941$	34182.7	1.18419	0.592095	+	0.805868i	$0.298301\pi$
$942$	0	0
$943$	9095.90	0.314108
$944$	44196.9	1.52382
$945$	0	0
$946$	5319.62	0.182829
$947$	44105.7	1.51346	0.756729	−	0.653729i	$-0.226796\pi$
$947$	44105.7	1.51346	0.756729	+	0.653729i	$0.226796\pi$
$948$	0	0
$949$	10130.1	0.346508
$950$	−4610.14	−0.157445
$951$	0	0
$952$	4415.52	0.150323
$953$	−22993.3	−0.781560	−0.390780	−	0.920484i	$-0.627795\pi$
$953$	−22993.3	−0.781560	−0.390780	+	0.920484i	$0.627795\pi$
$954$	0	0
$955$	33961.0	1.15074
$956$	8701.45	0.294378
$957$	0	0
$958$	−61089.4	−2.06024
$959$	−12666.9	−0.426524
$960$	0	0
$961$	−28975.2	−0.972615
$962$	5714.76	0.191529
$963$	0	0
$964$	15883.1	0.530664
$965$	23321.0	0.777957
$966$	0	0
$967$	30170.2	1.00332	0.501658	−	0.865066i	$-0.332724\pi$
$967$	30170.2	1.00332	0.501658	+	0.865066i	$0.332724\pi$
$968$	−17840.0	−0.592353
$969$	0	0
$970$	−34796.8	−1.15181
$971$	−59927.1	−1.98059	−0.990295	−	0.138984i	$-0.955616\pi$
$971$	−59927.1	−1.98059	−0.990295	+	0.138984i	$0.955616\pi$
$972$	0	0
$973$	−5501.34	−0.181259
$974$	19896.5	0.654543
$975$	0	0
$976$	−28398.9	−0.931381
$977$	22546.4	0.738304	0.369152	−	0.929369i	$-0.379648\pi$
$977$	22546.4	0.738304	0.369152	+	0.929369i	$0.379648\pi$
$978$	0	0
$979$	10602.3	0.346119
$980$	10089.3	0.328868
$981$	0	0
$982$	41829.4	1.35930
$983$	−1932.66	−0.0627084	−0.0313542	−	0.999508i	$-0.509982\pi$
$983$	−1932.66	−0.0627084	−0.0313542	+	0.999508i	$0.509982\pi$
$984$	0	0
$985$	23179.9	0.749820
$986$	9463.44	0.305656
$987$	0	0
$988$	−814.889	−0.0262399
$989$	4933.47	0.158620
$990$	0	0
$991$	−47353.0	−1.51788	−0.758939	−	0.651161i	$-0.774282\pi$
$991$	−47353.0	−1.51788	−0.758939	+	0.651161i	$0.774282\pi$
$992$	4719.97	0.151068
$993$	0	0
$994$	8009.10	0.255567
$995$	−33468.8	−1.06637
$996$	0	0
$997$	20613.3	0.654795	0.327397	−	0.944887i	$-0.393828\pi$
$997$	20613.3	0.654795	0.327397	+	0.944887i	$0.393828\pi$
$998$	−51065.2	−1.61968
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.d.1.2		4
3.2	odd	2		69.4.a.d.1.3	✓	4
12.11	even	2		1104.4.a.t.1.3		4
15.14	odd	2		1725.4.a.p.1.2		4
69.68	even	2		1587.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.d.1.3	✓	4	3.2	odd	2
207.4.a.d.1.2		4	1.1	even	1	trivial
1104.4.a.t.1.3		4	12.11	even	2
1587.4.a.g.1.3		4	69.68	even	2
1725.4.a.p.1.2		4	15.14	odd	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-3.45983 q^{2} +3.97043 q^{4} -7.89053 q^{5} +4.57766 q^{7} +13.9416 q^{8} +27.2999 q^{10} +7.16805 q^{11} +9.66367 q^{13} -15.8379 q^{14} -79.9991 q^{16} +69.1871 q^{17} -21.2382 q^{19} -31.3288 q^{20} -24.8002 q^{22} -23.0000 q^{23} -62.7395 q^{25} -33.4347 q^{26} +18.1753 q^{28} -39.5338 q^{29} +28.5625 q^{31} +165.251 q^{32} -239.376 q^{34} -36.1202 q^{35} -170.923 q^{37} +73.4807 q^{38} -110.007 q^{40} -395.474 q^{41} -214.499 q^{43} +28.4603 q^{44} +79.5761 q^{46} -387.800 q^{47} -322.045 q^{49} +217.068 q^{50} +38.3690 q^{52} -268.151 q^{53} -56.5597 q^{55} +63.8200 q^{56} +136.780 q^{58} -552.468 q^{59} +354.991 q^{61} -98.8214 q^{62} +68.2540 q^{64} -76.2515 q^{65} +293.680 q^{67} +274.703 q^{68} +124.970 q^{70} -505.691 q^{71} +1048.26 q^{73} +591.365 q^{74} -84.3250 q^{76} +32.8129 q^{77} -38.4080 q^{79} +631.236 q^{80} +1368.27 q^{82} +111.501 q^{83} -545.923 q^{85} +742.130 q^{86} +99.9342 q^{88} +1479.10 q^{89} +44.2370 q^{91} -91.3200 q^{92} +1341.72 q^{94} +167.581 q^{95} -1274.61 q^{97} +1114.22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 4 q^{5} - 14 q^{7} - 84 q^{8} - 100 q^{10} - 70 q^{11} - 12 q^{13} + 18 q^{14} + 130 q^{16} - 178 q^{17} + 96 q^{19} + 296 q^{20} - 326 q^{22} - 92 q^{23} - 80 q^{25} + 464 q^{26}+ \cdots + 2784 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 4 * q^5 - 14 * q^7 - 84 * q^8 - 100 * q^10 - 70 * q^11 - 12 * q^13 + 18 * q^14 + 130 * q^16 - 178 * q^17 + 96 * q^19 + 296 * q^20 - 326 * q^22 - 92 * q^23 - 80 * q^25 + 464 * q^26 - 774 * q^28 + 24 * q^29 - 400 * q^31 - 636 * q^32 - 18 * q^34 - 224 * q^35 - 358 * q^37 + 136 * q^38 - 1268 * q^40 - 152 * q^41 - 196 * q^43 - 70 * q^44 + 92 * q^46 - 260 * q^47 + 476 * q^49 - 876 * q^50 - 876 * q^52 - 968 * q^53 + 684 * q^55 + 1234 * q^56 + 904 * q^58 - 1236 * q^59 - 482 * q^61 + 1304 * q^62 + 1658 * q^64 - 1424 * q^65 + 620 * q^67 - 838 * q^68 + 2636 * q^70 + 88 * q^71 - 948 * q^73 - 710 * q^74 + 3344 * q^76 + 892 * q^77 + 254 * q^79 + 5144 * q^80 + 1568 * q^82 + 550 * q^83 - 1208 * q^85 + 1236 * q^86 + 202 * q^88 - 202 * q^89 - 356 * q^91 - 598 * q^92 - 1328 * q^94 + 460 * q^95 - 1800 * q^97 + 2784 * q^98

Introduction

L-functions

Modular forms

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Representations

Groups

Database

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