LMFDB - Embedded newform 2205.4.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 55x^{6} + 80x^{5} + 969x^{4} - 866x^{3} - 5783x^{2} + 2328x + 9992 $ x^8 - 2x^7 - 55x^6 + 80x^5 + 969x^4 - 866x^3 - 5783x^2 + 2328*x + 9992
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.95150$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.95150 q^{2} +16.5174 q^{4} +5.00000 q^{5} -42.1739 q^{8} +O(q^{10})$	$q-4.95150 q^{2} +16.5174 q^{4} +5.00000 q^{5} -42.1739 q^{8} -24.7575 q^{10} -25.0967 q^{11} +5.55771 q^{13} +76.6851 q^{16} -70.4574 q^{17} -57.3875 q^{19} +82.5870 q^{20} +124.266 q^{22} -154.529 q^{23} +25.0000 q^{25} -27.5190 q^{26} -245.033 q^{29} -55.1885 q^{31} -42.3154 q^{32} +348.870 q^{34} +218.748 q^{37} +284.155 q^{38} -210.869 q^{40} -415.577 q^{41} -501.230 q^{43} -414.531 q^{44} +765.150 q^{46} -531.063 q^{47} -123.788 q^{50} +91.7989 q^{52} +233.234 q^{53} -125.483 q^{55} +1213.28 q^{58} +253.572 q^{59} +260.398 q^{61} +273.266 q^{62} -403.956 q^{64} +27.7886 q^{65} +198.991 q^{67} -1163.77 q^{68} -150.449 q^{71} +145.556 q^{73} -1083.13 q^{74} -947.892 q^{76} +802.321 q^{79} +383.425 q^{80} +2057.73 q^{82} +766.712 q^{83} -352.287 q^{85} +2481.84 q^{86} +1058.42 q^{88} -1259.25 q^{89} -2552.41 q^{92} +2629.56 q^{94} -286.938 q^{95} -49.7105 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 50 q^{4} + 40 q^{5} - 66 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 50 * q^4 + 40 * q^5 - 66 * q^8	$ 8 q - 2 q^{2} + 50 q^{4} + 40 q^{5} - 66 q^{8} - 10 q^{10} - 64 q^{11} + 206 q^{16} + 48 q^{17} - 80 q^{19} + 250 q^{20} + 452 q^{22} - 120 q^{23} + 200 q^{25} + 272 q^{26} - 76 q^{29} - 20 q^{31} - 770 q^{32} - 320 q^{34} + 348 q^{37} - 236 q^{38} - 330 q^{40} + 944 q^{41} + 1116 q^{43} - 172 q^{44} + 496 q^{46} - 208 q^{47} - 50 q^{50} + 2272 q^{52} - 1144 q^{53} - 320 q^{55} + 560 q^{58} + 596 q^{59} - 740 q^{61} - 1184 q^{62} + 1298 q^{64} + 1964 q^{67} + 96 q^{68} + 4 q^{71} + 1500 q^{73} - 3368 q^{74} + 2912 q^{76} - 460 q^{79} + 1030 q^{80} + 5644 q^{82} + 700 q^{83} + 240 q^{85} + 1396 q^{86} + 6892 q^{88} - 644 q^{92} + 6692 q^{94} - 400 q^{95} + 2052 q^{97}+O(q^{100}) $ 8 * q - 2 * q^2 + 50 * q^4 + 40 * q^5 - 66 * q^8 - 10 * q^10 - 64 * q^11 + 206 * q^16 + 48 * q^17 - 80 * q^19 + 250 * q^20 + 452 * q^22 - 120 * q^23 + 200 * q^25 + 272 * q^26 - 76 * q^29 - 20 * q^31 - 770 * q^32 - 320 * q^34 + 348 * q^37 - 236 * q^38 - 330 * q^40 + 944 * q^41 + 1116 * q^43 - 172 * q^44 + 496 * q^46 - 208 * q^47 - 50 * q^50 + 2272 * q^52 - 1144 * q^53 - 320 * q^55 + 560 * q^58 + 596 * q^59 - 740 * q^61 - 1184 * q^62 + 1298 * q^64 + 1964 * q^67 + 96 * q^68 + 4 * q^71 + 1500 * q^73 - 3368 * q^74 + 2912 * q^76 - 460 * q^79 + 1030 * q^80 + 5644 * q^82 + 700 * q^83 + 240 * q^85 + 1396 * q^86 + 6892 * q^88 - 644 * q^92 + 6692 * q^94 - 400 * q^95 + 2052 * q^97

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 $

$2$	−4.95150	−1.75062	−0.875311	−	0.483561i	$-0.839343\pi$
$2$	−4.95150	−1.75062	−0.875311	+	0.483561i	$0.839343\pi$
$3$	0	0
$3$	0	0
$4$	16.5174	2.06467
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−42.1739	−1.86384
$9$	0	0
$10$	−24.7575	−0.782902
$11$	−25.0967	−0.687902	−0.343951	−	0.938988i	$-0.611765\pi$
$11$	−25.0967	−0.687902	−0.343951	+	0.938988i	$0.611765\pi$
$12$	0	0
$13$	5.55771	0.118572	0.0592858	−	0.998241i	$-0.481118\pi$
$13$	5.55771	0.118572	0.0592858	+	0.998241i	$0.481118\pi$
$14$	0	0
$15$	0	0
$16$	76.6851	1.19820
$17$	−70.4574	−1.00520	−0.502601	−	0.864518i	$-0.667624\pi$
$17$	−70.4574	−1.00520	−0.502601	+	0.864518i	$0.667624\pi$
$18$	0	0
$19$	−57.3875	−0.692926	−0.346463	−	0.938064i	$-0.612617\pi$
$19$	−57.3875	−0.692926	−0.346463	+	0.938064i	$0.612617\pi$
$20$	82.5870	0.923350
$21$	0	0
$22$	124.266	1.20426
$23$	−154.529	−1.40093	−0.700467	−	0.713685i	$-0.747025\pi$
$23$	−154.529	−1.40093	−0.700467	+	0.713685i	$0.747025\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−27.5190	−0.207574
$27$	0	0
$28$	0	0
$29$	−245.033	−1.56902	−0.784508	−	0.620118i	$-0.787085\pi$
$29$	−245.033	−1.56902	−0.784508	+	0.620118i	$0.787085\pi$
$30$	0	0
$31$	−55.1885	−0.319747	−0.159873	−	0.987138i	$-0.551109\pi$
$31$	−55.1885	−0.319747	−0.159873	+	0.987138i	$0.551109\pi$
$32$	−42.3154	−0.233762
$33$	0	0
$34$	348.870	1.75973
$35$	0	0
$36$	0	0
$37$	218.748	0.971945	0.485973	−	0.873974i	$-0.338465\pi$
$37$	218.748	0.971945	0.485973	+	0.873974i	$0.338465\pi$
$38$	284.155	1.21305
$39$	0	0
$40$	−210.869	−0.833535
$41$	−415.577	−1.58298	−0.791490	−	0.611182i	$-0.790694\pi$
$41$	−415.577	−1.58298	−0.791490	+	0.611182i	$0.790694\pi$
$42$	0	0
$43$	−501.230	−1.77760	−0.888801	−	0.458293i	$-0.848461\pi$
$43$	−501.230	−1.77760	−0.888801	+	0.458293i	$0.848461\pi$
$44$	−414.531	−1.42029
$45$	0	0
$46$	765.150	2.45250
$47$	−531.063	−1.64816	−0.824080	−	0.566473i	$-0.808307\pi$
$47$	−531.063	−1.64816	−0.824080	+	0.566473i	$0.808307\pi$
$48$	0	0
$49$	0	0
$50$	−123.788	−0.350124
$51$	0	0
$52$	91.7989	0.244812
$53$	233.234	0.604473	0.302237	−	0.953233i	$-0.402267\pi$
$53$	233.234	0.604473	0.302237	+	0.953233i	$0.402267\pi$
$54$	0	0
$55$	−125.483	−0.307639
$56$	0	0
$57$	0	0
$58$	1213.28	2.74675
$59$	253.572	0.559530	0.279765	−	0.960069i	$-0.409743\pi$
$59$	253.572	0.559530	0.279765	+	0.960069i	$0.409743\pi$
$60$	0	0
$61$	260.398	0.546567	0.273283	−	0.961934i	$-0.411890\pi$
$61$	260.398	0.546567	0.273283	+	0.961934i	$0.411890\pi$
$62$	273.266	0.559755
$63$	0	0
$64$	−403.956	−0.788977
$65$	27.7886	0.0530269
$66$	0	0
$67$	198.991	0.362844	0.181422	−	0.983405i	$-0.441930\pi$
$67$	198.991	0.362844	0.181422	+	0.983405i	$0.441930\pi$
$68$	−1163.77	−2.07541
$69$	0	0
$70$	0	0
$71$	−150.449	−0.251479	−0.125739	−	0.992063i	$-0.540130\pi$
$71$	−150.449	−0.251479	−0.125739	+	0.992063i	$0.540130\pi$
$72$	0	0
$73$	145.556	0.233370	0.116685	−	0.993169i	$-0.462773\pi$
$73$	145.556	0.233370	0.116685	+	0.993169i	$0.462773\pi$
$74$	−1083.13	−1.70151
$75$	0	0
$76$	−947.892	−1.43067
$77$	0	0
$78$	0	0
$79$	802.321	1.14263	0.571317	−	0.820729i	$-0.306433\pi$
$79$	802.321	1.14263	0.571317	+	0.820729i	$0.306433\pi$
$80$	383.425	0.535853
$81$	0	0
$82$	2057.73	2.77120
$83$	766.712	1.01395	0.506973	−	0.861962i	$-0.330764\pi$
$83$	766.712	1.01395	0.506973	+	0.861962i	$0.330764\pi$
$84$	0	0
$85$	−352.287	−0.449540
$86$	2481.84	3.11191
$87$	0	0
$88$	1058.42	1.28214
$89$	−1259.25	−1.49978	−0.749890	−	0.661562i	$-0.769894\pi$
$89$	−1259.25	−1.49978	−0.749890	+	0.661562i	$0.769894\pi$
$90$	0	0
$91$	0	0
$92$	−2552.41	−2.89247
$93$	0	0
$94$	2629.56	2.88530
$95$	−286.938	−0.309886
$96$	0	0
$97$	−49.7105	−0.0520344	−0.0260172	−	0.999661i	$-0.508282\pi$
$97$	−49.7105	−0.0520344	−0.0260172	+	0.999661i	$0.508282\pi$
$98$	0	0
$99$	0	0
$100$	412.935	0.412935
$101$	1472.94	1.45112	0.725560	−	0.688159i	$-0.241581\pi$
$101$	1472.94	1.45112	0.725560	+	0.688159i	$0.241581\pi$
$102$	0	0
$103$	1301.04	1.24461	0.622307	−	0.782773i	$-0.286196\pi$
$103$	1301.04	1.24461	0.622307	+	0.782773i	$0.286196\pi$
$104$	−234.390	−0.220999
$105$	0	0
$106$	−1154.86	−1.05820
$107$	1696.03	1.53235	0.766175	−	0.642632i	$-0.222157\pi$
$107$	1696.03	1.53235	0.766175	+	0.642632i	$0.222157\pi$
$108$	0	0
$109$	−1034.79	−0.909314	−0.454657	−	0.890667i	$-0.650238\pi$
$109$	−1034.79	−0.909314	−0.454657	+	0.890667i	$0.650238\pi$
$110$	621.331	0.538560
$111$	0	0
$112$	0	0
$113$	854.674	0.711513	0.355756	−	0.934579i	$-0.384223\pi$
$113$	854.674	0.711513	0.355756	+	0.934579i	$0.384223\pi$
$114$	0	0
$115$	−772.644	−0.626517
$116$	−4047.31	−3.23951
$117$	0	0
$118$	−1255.56	−0.979525
$119$	0	0
$120$	0	0
$121$	−701.158	−0.526790
$122$	−1289.36	−0.956831
$123$	0	0
$124$	−911.570	−0.660172
$125$	125.000	0.0894427
$126$	0	0
$127$	863.384	0.603252	0.301626	−	0.953426i	$-0.402471\pi$
$127$	863.384	0.603252	0.301626	+	0.953426i	$0.402471\pi$
$128$	2338.71	1.61496
$129$	0	0
$130$	−137.595	−0.0928300
$131$	2738.48	1.82643	0.913213	−	0.407482i	$-0.133593\pi$
$131$	2738.48	1.82643	0.913213	+	0.407482i	$0.133593\pi$
$132$	0	0
$133$	0	0
$134$	−985.303	−0.635203
$135$	0	0
$136$	2971.46	1.87354
$137$	2854.86	1.78034	0.890171	−	0.455626i	$-0.150584\pi$
$137$	2854.86	1.78034	0.890171	+	0.455626i	$0.150584\pi$
$138$	0	0
$139$	1406.45	0.858227	0.429114	−	0.903251i	$-0.358826\pi$
$139$	1406.45	0.858227	0.429114	+	0.903251i	$0.358826\pi$
$140$	0	0
$141$	0	0
$142$	744.948	0.440244
$143$	−139.480	−0.0815658
$144$	0	0
$145$	−1225.16	−0.701686
$146$	−720.721	−0.408543
$147$	0	0
$148$	3613.15	2.00675
$149$	643.308	0.353704	0.176852	−	0.984237i	$-0.443409\pi$
$149$	643.308	0.353704	0.176852	+	0.984237i	$0.443409\pi$
$150$	0	0
$151$	940.387	0.506805	0.253403	−	0.967361i	$-0.418450\pi$
$151$	940.387	0.506805	0.253403	+	0.967361i	$0.418450\pi$
$152$	2420.26	1.29150
$153$	0	0
$154$	0	0
$155$	−275.942	−0.142995
$156$	0	0
$157$	16.1762	0.00822294	0.00411147	−	0.999992i	$-0.498691\pi$
$157$	16.1762	0.00822294	0.00411147	+	0.999992i	$0.498691\pi$
$158$	−3972.69	−2.00032
$159$	0	0
$160$	−211.577	−0.104541
$161$	0	0
$162$	0	0
$163$	329.147	0.158164	0.0790822	−	0.996868i	$-0.474801\pi$
$163$	329.147	0.158164	0.0790822	+	0.996868i	$0.474801\pi$
$164$	−6864.25	−3.26834
$165$	0	0
$166$	−3796.38	−1.77504
$167$	246.996	0.114450	0.0572250	−	0.998361i	$-0.481775\pi$
$167$	246.996	0.114450	0.0572250	+	0.998361i	$0.481775\pi$
$168$	0	0
$169$	−2166.11	−0.985941
$170$	1744.35	0.786974
$171$	0	0
$172$	−8279.02	−3.67017
$173$	−3994.14	−1.75531	−0.877655	−	0.479293i	$-0.840893\pi$
$173$	−3994.14	−1.75531	−0.877655	+	0.479293i	$0.840893\pi$
$174$	0	0
$175$	0	0
$176$	−1924.54	−0.824248
$177$	0	0
$178$	6235.19	2.62555
$179$	−2224.16	−0.928722	−0.464361	−	0.885646i	$-0.653716\pi$
$179$	−2224.16	−0.928722	−0.464361	+	0.885646i	$0.653716\pi$
$180$	0	0
$181$	−4727.90	−1.94156	−0.970779	−	0.239975i	$-0.922861\pi$
$181$	−4727.90	−1.94156	−0.970779	+	0.239975i	$0.922861\pi$
$182$	0	0
$183$	0	0
$184$	6517.08	2.61112
$185$	1093.74	0.434667
$186$	0	0
$187$	1768.25	0.691481
$188$	−8771.78	−3.40291
$189$	0	0
$190$	1420.77	0.542493
$191$	−110.697	−0.0419358	−0.0209679	−	0.999780i	$-0.506675\pi$
$191$	−110.697	−0.0419358	−0.0209679	+	0.999780i	$0.506675\pi$
$192$	0	0
$193$	3202.70	1.19448	0.597241	−	0.802062i	$-0.296263\pi$
$193$	3202.70	1.19448	0.597241	+	0.802062i	$0.296263\pi$
$194$	246.142	0.0910925
$195$	0	0
$196$	0	0
$197$	−4737.52	−1.71337	−0.856685	−	0.515839i	$-0.827480\pi$
$197$	−4737.52	−1.71337	−0.856685	+	0.515839i	$0.827480\pi$
$198$	0	0
$199$	−3037.53	−1.08203	−0.541017	−	0.841012i	$-0.681960\pi$
$199$	−3037.53	−1.08203	−0.541017	+	0.841012i	$0.681960\pi$
$200$	−1054.35	−0.372768
$201$	0	0
$202$	−7293.27	−2.54036
$203$	0	0
$204$	0	0
$205$	−2077.88	−0.707930
$206$	−6442.10	−2.17885
$207$	0	0
$208$	426.194	0.142073
$209$	1440.24	0.476666
$210$	0	0
$211$	−4126.01	−1.34619	−0.673096	−	0.739555i	$-0.735036\pi$
$211$	−4126.01	−1.34619	−0.673096	+	0.739555i	$0.735036\pi$
$212$	3852.41	1.24804
$213$	0	0
$214$	−8397.90	−2.68256
$215$	−2506.15	−0.794968
$216$	0	0
$217$	0	0
$218$	5123.78	1.59186
$219$	0	0
$220$	−2072.66	−0.635175
$221$	−391.582	−0.119189
$222$	0	0
$223$	152.782	0.0458790	0.0229395	−	0.999737i	$-0.492697\pi$
$223$	152.782	0.0458790	0.0229395	+	0.999737i	$0.492697\pi$
$224$	0	0
$225$	0	0
$226$	−4231.92	−1.24559
$227$	2776.18	0.811726	0.405863	−	0.913934i	$-0.366971\pi$
$227$	2776.18	0.811726	0.405863	+	0.913934i	$0.366971\pi$
$228$	0	0
$229$	2346.97	0.677260	0.338630	−	0.940920i	$-0.390037\pi$
$229$	2346.97	0.677260	0.338630	+	0.940920i	$0.390037\pi$
$230$	3825.75	1.09679
$231$	0	0
$232$	10334.0	2.92440
$233$	5752.57	1.61744	0.808720	−	0.588194i	$-0.200161\pi$
$233$	5752.57	1.61744	0.808720	+	0.588194i	$0.200161\pi$
$234$	0	0
$235$	−2655.32	−0.737080
$236$	4188.35	1.15525
$237$	0	0
$238$	0	0
$239$	5895.40	1.59557	0.797786	−	0.602940i	$-0.206004\pi$
$239$	5895.40	1.59557	0.797786	+	0.602940i	$0.206004\pi$
$240$	0	0
$241$	−3826.81	−1.02285	−0.511424	−	0.859328i	$-0.670882\pi$
$241$	−3826.81	−1.02285	−0.511424	+	0.859328i	$0.670882\pi$
$242$	3471.79	0.922210
$243$	0	0
$244$	4301.10	1.12848
$245$	0	0
$246$	0	0
$247$	−318.943	−0.0821615
$248$	2327.51	0.595957
$249$	0	0
$250$	−618.938	−0.156580
$251$	−4675.70	−1.17581	−0.587903	−	0.808931i	$-0.700046\pi$
$251$	−4675.70	−1.17581	−0.587903	+	0.808931i	$0.700046\pi$
$252$	0	0
$253$	3878.16	0.963706
$254$	−4275.05	−1.05606
$255$	0	0
$256$	−8348.50	−2.03821
$257$	−2274.11	−0.551964	−0.275982	−	0.961163i	$-0.589003\pi$
$257$	−2274.11	−0.551964	−0.275982	+	0.961163i	$0.589003\pi$
$258$	0	0
$259$	0	0
$260$	458.995	0.109483
$261$	0	0
$262$	−13559.6	−3.19738
$263$	3807.02	0.892590	0.446295	−	0.894886i	$-0.352743\pi$
$263$	3807.02	0.892590	0.446295	+	0.894886i	$0.352743\pi$
$264$	0	0
$265$	1166.17	0.270329
$266$	0	0
$267$	0	0
$268$	3286.81	0.749155
$269$	665.674	0.150881	0.0754403	−	0.997150i	$-0.475964\pi$
$269$	665.674	0.150881	0.0754403	+	0.997150i	$0.475964\pi$
$270$	0	0
$271$	5710.59	1.28005	0.640025	−	0.768354i	$-0.278924\pi$
$271$	5710.59	1.28005	0.640025	+	0.768354i	$0.278924\pi$
$272$	−5403.03	−1.20444
$273$	0	0
$274$	−14135.8	−3.11671
$275$	−627.416	−0.137580
$276$	0	0
$277$	1780.03	0.386107	0.193053	−	0.981188i	$-0.438161\pi$
$277$	1780.03	0.386107	0.193053	+	0.981188i	$0.438161\pi$
$278$	−6964.04	−1.50243
$279$	0	0
$280$	0	0
$281$	−666.478	−0.141490	−0.0707450	−	0.997494i	$-0.522538\pi$
$281$	−666.478	−0.141490	−0.0707450	+	0.997494i	$0.522538\pi$
$282$	0	0
$283$	−607.927	−0.127694	−0.0638472	−	0.997960i	$-0.520337\pi$
$283$	−607.927	−0.127694	−0.0638472	+	0.997960i	$0.520337\pi$
$284$	−2485.02	−0.519221
$285$	0	0
$286$	690.636	0.142791
$287$	0	0
$288$	0	0
$289$	51.2488	0.0104313
$290$	6066.41	1.22839
$291$	0	0
$292$	2404.20	0.481834
$293$	1329.54	0.265094	0.132547	−	0.991177i	$-0.457684\pi$
$293$	1329.54	0.265094	0.132547	+	0.991177i	$0.457684\pi$
$294$	0	0
$295$	1267.86	0.250229
$296$	−9225.46	−1.81155
$297$	0	0
$298$	−3185.34	−0.619201
$299$	−858.826	−0.166111
$300$	0	0
$301$	0	0
$302$	−4656.33	−0.887224
$303$	0	0
$304$	−4400.77	−0.830268
$305$	1301.99	0.244432
$306$	0	0
$307$	−5121.84	−0.952179	−0.476090	−	0.879397i	$-0.657946\pi$
$307$	−5121.84	−0.952179	−0.476090	+	0.879397i	$0.657946\pi$
$308$	0	0
$309$	0	0
$310$	1366.33	0.250330
$311$	4492.03	0.819034	0.409517	−	0.912303i	$-0.365697\pi$
$311$	4492.03	0.819034	0.409517	+	0.912303i	$0.365697\pi$
$312$	0	0
$313$	3035.96	0.548251	0.274125	−	0.961694i	$-0.411612\pi$
$313$	3035.96	0.548251	0.274125	+	0.961694i	$0.411612\pi$
$314$	−80.0966	−0.0143953
$315$	0	0
$316$	13252.2	2.35917
$317$	−4233.27	−0.750045	−0.375023	−	0.927016i	$-0.622365\pi$
$317$	−4233.27	−0.750045	−0.375023	+	0.927016i	$0.622365\pi$
$318$	0	0
$319$	6149.51	1.07933
$320$	−2019.78	−0.352841
$321$	0	0
$322$	0	0
$323$	4043.38	0.696531
$324$	0	0
$325$	138.943	0.0237143
$326$	−1629.77	−0.276886
$327$	0	0
$328$	17526.5	2.95042
$329$	0	0
$330$	0	0
$331$	−3196.70	−0.530836	−0.265418	−	0.964133i	$-0.585510\pi$
$331$	−3196.70	−0.530836	−0.265418	+	0.964133i	$0.585510\pi$
$332$	12664.1	2.09347
$333$	0	0
$334$	−1223.00	−0.200359
$335$	994.953	0.162269
$336$	0	0
$337$	464.150	0.0750263	0.0375132	−	0.999296i	$-0.488056\pi$
$337$	464.150	0.0750263	0.0375132	+	0.999296i	$0.488056\pi$
$338$	10725.5	1.72601
$339$	0	0
$340$	−5818.86	−0.928154
$341$	1385.05	0.219954
$342$	0	0
$343$	0	0
$344$	21138.8	3.31317
$345$	0	0
$346$	19777.0	3.07288
$347$	−9544.77	−1.47663	−0.738315	−	0.674456i	$-0.764378\pi$
$347$	−9544.77	−1.47663	−0.738315	+	0.674456i	$0.764378\pi$
$348$	0	0
$349$	4387.55	0.672952	0.336476	−	0.941692i	$-0.390765\pi$
$349$	4387.55	0.672952	0.336476	+	0.941692i	$0.390765\pi$
$350$	0	0
$351$	0	0
$352$	1061.97	0.160805
$353$	8014.76	1.20845	0.604224	−	0.796814i	$-0.293483\pi$
$353$	8014.76	1.20845	0.604224	+	0.796814i	$0.293483\pi$
$354$	0	0
$355$	−752.244	−0.112465
$356$	−20799.6	−3.09656
$357$	0	0
$358$	11012.9	1.62584
$359$	−819.972	−0.120547	−0.0602736	−	0.998182i	$-0.519197\pi$
$359$	−819.972	−0.120547	−0.0602736	+	0.998182i	$0.519197\pi$
$360$	0	0
$361$	−3565.67	−0.519853
$362$	23410.2	3.39893
$363$	0	0
$364$	0	0
$365$	727.780	0.104366
$366$	0	0
$367$	−4563.91	−0.649139	−0.324570	−	0.945862i	$-0.605219\pi$
$367$	−4563.91	−0.649139	−0.324570	+	0.945862i	$0.605219\pi$
$368$	−11850.1	−1.67861
$369$	0	0
$370$	−5415.66	−0.760937
$371$	0	0
$372$	0	0
$373$	14065.4	1.95248	0.976242	−	0.216681i	$-0.0695232\pi$
$373$	14065.4	1.95248	0.976242	+	0.216681i	$0.0695232\pi$
$374$	−8755.48	−1.21052
$375$	0	0
$376$	22397.0	3.07191
$377$	−1361.82	−0.186041
$378$	0	0
$379$	1418.52	0.192254	0.0961271	−	0.995369i	$-0.469354\pi$
$379$	1418.52	0.192254	0.0961271	+	0.995369i	$0.469354\pi$
$380$	−4739.46	−0.639814
$381$	0	0
$382$	548.115	0.0734137
$383$	−2025.77	−0.270266	−0.135133	−	0.990827i	$-0.543146\pi$
$383$	−2025.77	−0.270266	−0.135133	+	0.990827i	$0.543146\pi$
$384$	0	0
$385$	0	0
$386$	−15858.2	−2.09109
$387$	0	0
$388$	−821.088	−0.107434
$389$	−5111.36	−0.666212	−0.333106	−	0.942889i	$-0.608097\pi$
$389$	−5111.36	−0.666212	−0.333106	+	0.942889i	$0.608097\pi$
$390$	0	0
$391$	10887.7	1.40822
$392$	0	0
$393$	0	0
$394$	23457.8	2.99946
$395$	4011.60	0.511002
$396$	0	0
$397$	−3065.20	−0.387502	−0.193751	−	0.981051i	$-0.562065\pi$
$397$	−3065.20	−0.387502	−0.193751	+	0.981051i	$0.562065\pi$
$398$	15040.3	1.89423
$399$	0	0
$400$	1917.13	0.239641
$401$	−5344.81	−0.665604	−0.332802	−	0.942997i	$-0.607994\pi$
$401$	−5344.81	−0.665604	−0.332802	+	0.942997i	$0.607994\pi$
$402$	0	0
$403$	−306.722	−0.0379129
$404$	24329.1	2.99609
$405$	0	0
$406$	0	0
$407$	−5489.85	−0.668603
$408$	0	0
$409$	4389.81	0.530714	0.265357	−	0.964150i	$-0.414510\pi$
$409$	4389.81	0.530714	0.265357	+	0.964150i	$0.414510\pi$
$410$	10288.7	1.23932
$411$	0	0
$412$	21489.8	2.56972
$413$	0	0
$414$	0	0
$415$	3833.56	0.453451
$416$	−235.177	−0.0277175
$417$	0	0
$418$	−7131.33	−0.834461
$419$	4975.01	0.580060	0.290030	−	0.957018i	$-0.406335\pi$
$419$	4975.01	0.580060	0.290030	+	0.957018i	$0.406335\pi$
$420$	0	0
$421$	−8507.30	−0.984847	−0.492424	−	0.870356i	$-0.663889\pi$
$421$	−8507.30	−0.984847	−0.492424	+	0.870356i	$0.663889\pi$
$422$	20430.0	2.35667
$423$	0	0
$424$	−9836.37	−1.12664
$425$	−1761.44	−0.201040
$426$	0	0
$427$	0	0
$428$	28014.0	3.16380
$429$	0	0
$430$	12409.2	1.39169
$431$	7532.06	0.841779	0.420889	−	0.907112i	$-0.361718\pi$
$431$	7532.06	0.841779	0.420889	+	0.907112i	$0.361718\pi$
$432$	0	0
$433$	12960.1	1.43839	0.719196	−	0.694807i	$-0.244510\pi$
$433$	12960.1	1.43839	0.719196	+	0.694807i	$0.244510\pi$
$434$	0	0
$435$	0	0
$436$	−17092.1	−1.87744
$437$	8868.02	0.970744
$438$	0	0
$439$	4918.84	0.534769	0.267385	−	0.963590i	$-0.413841\pi$
$439$	4918.84	0.534769	0.267385	+	0.963590i	$0.413841\pi$
$440$	5292.12	0.573391
$441$	0	0
$442$	1938.92	0.208654
$443$	−10632.9	−1.14038	−0.570188	−	0.821514i	$-0.693130\pi$
$443$	−10632.9	−1.14038	−0.570188	+	0.821514i	$0.693130\pi$
$444$	0	0
$445$	−6296.26	−0.670722
$446$	−756.499	−0.0803167
$447$	0	0
$448$	0	0
$449$	−11645.2	−1.22399	−0.611996	−	0.790861i	$-0.709633\pi$
$449$	−11645.2	−1.22399	−0.611996	+	0.790861i	$0.709633\pi$
$450$	0	0
$451$	10429.6	1.08894
$452$	14117.0	1.46904
$453$	0	0
$454$	−13746.3	−1.42102
$455$	0	0
$456$	0	0
$457$	1323.40	0.135461	0.0677307	−	0.997704i	$-0.478424\pi$
$457$	1323.40	0.135461	0.0677307	+	0.997704i	$0.478424\pi$
$458$	−11621.1	−1.18562
$459$	0	0
$460$	−12762.1	−1.29355
$461$	7449.27	0.752597	0.376298	−	0.926499i	$-0.377197\pi$
$461$	7449.27	0.752597	0.376298	+	0.926499i	$0.377197\pi$
$462$	0	0
$463$	−15123.6	−1.51805	−0.759023	−	0.651064i	$-0.774323\pi$
$463$	−15123.6	−1.51805	−0.759023	+	0.651064i	$0.774323\pi$
$464$	−18790.4	−1.88000
$465$	0	0
$466$	−28483.9	−2.83152
$467$	14066.9	1.39387	0.696935	−	0.717135i	$-0.254547\pi$
$467$	14066.9	1.39387	0.696935	+	0.717135i	$0.254547\pi$
$468$	0	0
$469$	0	0
$470$	13147.8	1.29035
$471$	0	0
$472$	−10694.1	−1.04287
$473$	12579.2	1.22282
$474$	0	0
$475$	−1434.69	−0.138585
$476$	0	0
$477$	0	0
$478$	−29191.1	−2.79324
$479$	12137.4	1.15777	0.578884	−	0.815410i	$-0.303488\pi$
$479$	12137.4	1.15777	0.578884	+	0.815410i	$0.303488\pi$
$480$	0	0
$481$	1215.74	0.115245
$482$	18948.5	1.79062
$483$	0	0
$484$	−11581.3	−1.08765
$485$	−248.553	−0.0232705
$486$	0	0
$487$	18571.3	1.72802	0.864010	−	0.503474i	$-0.167945\pi$
$487$	18571.3	1.72802	0.864010	+	0.503474i	$0.167945\pi$
$488$	−10982.0	−1.01871
$489$	0	0
$490$	0	0
$491$	−16626.3	−1.52817	−0.764086	−	0.645114i	$-0.776810\pi$
$491$	−16626.3	−1.52817	−0.764086	+	0.645114i	$0.776810\pi$
$492$	0	0
$493$	17264.4	1.57718
$494$	1579.25	0.143834
$495$	0	0
$496$	−4232.13	−0.383122
$497$	0	0
$498$	0	0
$499$	−9538.79	−0.855741	−0.427871	−	0.903840i	$-0.640736\pi$
$499$	−9538.79	−0.855741	−0.427871	+	0.903840i	$0.640736\pi$
$500$	2064.67	0.184670
$501$	0	0
$502$	23151.7	2.05839
$503$	15237.4	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$503$	15237.4	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$504$	0	0
$505$	7364.70	0.648960
$506$	−19202.7	−1.68708
$507$	0	0
$508$	14260.9	1.24552
$509$	4015.78	0.349698	0.174849	−	0.984595i	$-0.444056\pi$
$509$	4015.78	0.349698	0.174849	+	0.984595i	$0.444056\pi$
$510$	0	0
$511$	0	0
$512$	22627.9	1.95317
$513$	0	0
$514$	11260.2	0.966280
$515$	6505.20	0.556608
$516$	0	0
$517$	13327.9	1.13377
$518$	0	0
$519$	0	0
$520$	−1171.95	−0.0988336
$521$	9090.14	0.764388	0.382194	−	0.924082i	$-0.375169\pi$
$521$	9090.14	0.764388	0.382194	+	0.924082i	$0.375169\pi$
$522$	0	0
$523$	9574.52	0.800505	0.400253	−	0.916405i	$-0.368922\pi$
$523$	9574.52	0.800505	0.400253	+	0.916405i	$0.368922\pi$
$524$	45232.5	3.77098
$525$	0	0
$526$	−18850.5	−1.56259
$527$	3888.44	0.321410
$528$	0	0
$529$	11712.1	0.962615
$530$	−5774.28	−0.473243
$531$	0	0
$532$	0	0
$533$	−2309.66	−0.187697
$534$	0	0
$535$	8480.15	0.685288
$536$	−8392.21	−0.676284
$537$	0	0
$538$	−3296.09	−0.264135
$539$	0	0
$540$	0	0
$541$	−7891.23	−0.627118	−0.313559	−	0.949569i	$-0.601521\pi$
$541$	−7891.23	−0.627118	−0.313559	+	0.949569i	$0.601521\pi$
$542$	−28276.0	−2.24088
$543$	0	0
$544$	2981.43	0.234978
$545$	−5173.97	−0.406657
$546$	0	0
$547$	−742.224	−0.0580168	−0.0290084	−	0.999579i	$-0.509235\pi$
$547$	−742.224	−0.0580168	−0.0290084	+	0.999579i	$0.509235\pi$
$548$	47154.8	3.67583
$549$	0	0
$550$	3106.65	0.240851
$551$	14061.8	1.08721
$552$	0	0
$553$	0	0
$554$	−8813.82	−0.675926
$555$	0	0
$556$	23230.9	1.77196
$557$	385.961	0.0293603	0.0146802	−	0.999892i	$-0.495327\pi$
$557$	385.961	0.0293603	0.0146802	+	0.999892i	$0.495327\pi$
$558$	0	0
$559$	−2785.69	−0.210773
$560$	0	0
$561$	0	0
$562$	3300.07	0.247696
$563$	3243.68	0.242815	0.121407	−	0.992603i	$-0.461259\pi$
$563$	3243.68	0.242815	0.121407	+	0.992603i	$0.461259\pi$
$564$	0	0
$565$	4273.37	0.318198
$566$	3010.15	0.223544
$567$	0	0
$568$	6345.01	0.468716
$569$	−23709.2	−1.74682	−0.873410	−	0.486985i	$-0.838097\pi$
$569$	−23709.2	−1.74682	−0.873410	+	0.486985i	$0.838097\pi$
$570$	0	0
$571$	16559.4	1.21364	0.606820	−	0.794839i	$-0.292445\pi$
$571$	16559.4	1.21364	0.606820	+	0.794839i	$0.292445\pi$
$572$	−2303.85	−0.168407
$573$	0	0
$574$	0	0
$575$	−3863.22	−0.280187
$576$	0	0
$577$	−431.422	−0.0311271	−0.0155636	−	0.999879i	$-0.504954\pi$
$577$	−431.422	−0.0311271	−0.0155636	+	0.999879i	$0.504954\pi$
$578$	−253.759	−0.0182612
$579$	0	0
$580$	−20236.5	−1.44875
$581$	0	0
$582$	0	0
$583$	−5853.38	−0.415819
$584$	−6138.66	−0.434965
$585$	0	0
$586$	−6583.22	−0.464079
$587$	19822.3	1.39379	0.696895	−	0.717173i	$-0.254564\pi$
$587$	19822.3	1.39379	0.696895	+	0.717173i	$0.254564\pi$
$588$	0	0
$589$	3167.13	0.221561
$590$	−6277.81	−0.438057
$591$	0	0
$592$	16774.7	1.16459
$593$	−27219.6	−1.88495	−0.942475	−	0.334276i	$-0.891508\pi$
$593$	−27219.6	−1.88495	−0.942475	+	0.334276i	$0.891508\pi$
$594$	0	0
$595$	0	0
$596$	10625.8	0.730283
$597$	0	0
$598$	4252.48	0.290798
$599$	24621.6	1.67949	0.839744	−	0.542982i	$-0.182705\pi$
$599$	24621.6	1.67949	0.839744	+	0.542982i	$0.182705\pi$
$600$	0	0
$601$	−3061.85	−0.207813	−0.103906	−	0.994587i	$-0.533134\pi$
$601$	−3061.85	−0.207813	−0.103906	+	0.994587i	$0.533134\pi$
$602$	0	0
$603$	0	0
$604$	15532.7	1.04639
$605$	−3505.79	−0.235588
$606$	0	0
$607$	802.286	0.0536471	0.0268235	−	0.999640i	$-0.491461\pi$
$607$	802.286	0.0536471	0.0268235	+	0.999640i	$0.491461\pi$
$608$	2428.37	0.161980
$609$	0	0
$610$	−6446.81	−0.427908
$611$	−2951.50	−0.195425
$612$	0	0
$613$	−20200.6	−1.33099	−0.665495	−	0.746403i	$-0.731779\pi$
$613$	−20200.6	−1.33099	−0.665495	+	0.746403i	$0.731779\pi$
$614$	25360.8	1.66691
$615$	0	0
$616$	0	0
$617$	−2395.41	−0.156297	−0.0781486	−	0.996942i	$-0.524901\pi$
$617$	−2395.41	−0.156297	−0.0781486	+	0.996942i	$0.524901\pi$
$618$	0	0
$619$	−751.635	−0.0488058	−0.0244029	−	0.999702i	$-0.507768\pi$
$619$	−751.635	−0.0488058	−0.0244029	+	0.999702i	$0.507768\pi$
$620$	−4557.85	−0.295238
$621$	0	0
$622$	−22242.3	−1.43382
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−15032.6	−0.959779
$627$	0	0
$628$	267.189	0.0169777
$629$	−15412.4	−0.977001
$630$	0	0
$631$	−7589.48	−0.478815	−0.239408	−	0.970919i	$-0.576953\pi$
$631$	−7589.48	−0.478815	−0.239408	+	0.970919i	$0.576953\pi$
$632$	−33837.0	−2.12969
$633$	0	0
$634$	20961.1	1.31305
$635$	4316.92	0.269782
$636$	0	0
$637$	0	0
$638$	−30449.3	−1.88950
$639$	0	0
$640$	11693.6	0.722232
$641$	−16335.5	−1.00657	−0.503285	−	0.864120i	$-0.667876\pi$
$641$	−16335.5	−1.00657	−0.503285	+	0.864120i	$0.667876\pi$
$642$	0	0
$643$	29799.4	1.82764	0.913822	−	0.406115i	$-0.133117\pi$
$643$	29799.4	1.82764	0.913822	+	0.406115i	$0.133117\pi$
$644$	0	0
$645$	0	0
$646$	−20020.8	−1.21936
$647$	−15133.3	−0.919556	−0.459778	−	0.888034i	$-0.652071\pi$
$647$	−15133.3	−0.919556	−0.459778	+	0.888034i	$0.652071\pi$
$648$	0	0
$649$	−6363.81	−0.384902
$650$	−687.976	−0.0415148
$651$	0	0
$652$	5436.66	0.326558
$653$	745.478	0.0446751	0.0223375	−	0.999750i	$-0.492889\pi$
$653$	745.478	0.0446751	0.0223375	+	0.999750i	$0.492889\pi$
$654$	0	0
$655$	13692.4	0.816803
$656$	−31868.5	−1.89673
$657$	0	0
$658$	0	0
$659$	7444.43	0.440051	0.220026	−	0.975494i	$-0.429386\pi$
$659$	7444.43	0.440051	0.220026	+	0.975494i	$0.429386\pi$
$660$	0	0
$661$	−10723.8	−0.631027	−0.315514	−	0.948921i	$-0.602177\pi$
$661$	−10723.8	−0.631027	−0.315514	+	0.948921i	$0.602177\pi$
$662$	15828.5	0.929292
$663$	0	0
$664$	−32335.2	−1.88983
$665$	0	0
$666$	0	0
$667$	37864.6	2.19809
$668$	4079.74	0.236302
$669$	0	0
$670$	−4926.51	−0.284071
$671$	−6535.12	−0.375985
$672$	0	0
$673$	31935.2	1.82914	0.914570	−	0.404428i	$-0.132529\pi$
$673$	31935.2	1.82914	0.914570	+	0.404428i	$0.132529\pi$
$674$	−2298.24	−0.131343
$675$	0	0
$676$	−35778.5	−2.03565
$677$	35170.2	1.99661	0.998303	−	0.0582347i	$-0.0185472\pi$
$677$	35170.2	1.99661	0.998303	+	0.0582347i	$0.0185472\pi$
$678$	0	0
$679$	0	0
$680$	14857.3	0.837871
$681$	0	0
$682$	−6858.06	−0.385057
$683$	−1194.88	−0.0669409	−0.0334705	−	0.999440i	$-0.510656\pi$
$683$	−1194.88	−0.0669409	−0.0334705	+	0.999440i	$0.510656\pi$
$684$	0	0
$685$	14274.3	0.796194
$686$	0	0
$687$	0	0
$688$	−38436.9	−2.12993
$689$	1296.24	0.0716734
$690$	0	0
$691$	−34727.9	−1.91189	−0.955943	−	0.293553i	$-0.905162\pi$
$691$	−34727.9	−1.91189	−0.955943	+	0.293553i	$0.905162\pi$
$692$	−65972.7	−3.62414
$693$	0	0
$694$	47261.0	2.58502
$695$	7032.25	0.383811
$696$	0	0
$697$	29280.5	1.59122
$698$	−21725.0	−1.17808
$699$	0	0
$700$	0	0
$701$	−28621.0	−1.54208	−0.771040	−	0.636786i	$-0.780263\pi$
$701$	−28621.0	−1.54208	−0.771040	+	0.636786i	$0.780263\pi$
$702$	0	0
$703$	−12553.4	−0.673487
$704$	10137.9	0.542739
$705$	0	0
$706$	−39685.1	−2.11554
$707$	0	0
$708$	0	0
$709$	25355.7	1.34310	0.671548	−	0.740961i	$-0.265630\pi$
$709$	25355.7	1.34310	0.671548	+	0.740961i	$0.265630\pi$
$710$	3724.74	0.196883
$711$	0	0
$712$	53107.6	2.79535
$713$	8528.21	0.447944
$714$	0	0
$715$	−697.400	−0.0364773
$716$	−36737.3	−1.91751
$717$	0	0
$718$	4060.09	0.211033
$719$	−7146.40	−0.370676	−0.185338	−	0.982675i	$-0.559338\pi$
$719$	−7146.40	−0.370676	−0.185338	+	0.982675i	$0.559338\pi$
$720$	0	0
$721$	0	0
$722$	17655.4	0.910065
$723$	0	0
$724$	−78092.6	−4.00868
$725$	−6125.82	−0.313803
$726$	0	0
$727$	24244.7	1.23685	0.618423	−	0.785845i	$-0.287772\pi$
$727$	24244.7	1.23685	0.618423	+	0.785845i	$0.287772\pi$
$728$	0	0
$729$	0	0
$730$	−3603.60	−0.182706
$731$	35315.4	1.78685
$732$	0	0
$733$	−11847.3	−0.596986	−0.298493	−	0.954412i	$-0.596484\pi$
$733$	−11847.3	−0.596986	−0.298493	+	0.954412i	$0.596484\pi$
$734$	22598.2	1.13640
$735$	0	0
$736$	6538.94	0.327484
$737$	−4994.00	−0.249601
$738$	0	0
$739$	9290.24	0.462445	0.231223	−	0.972901i	$-0.425727\pi$
$739$	9290.24	0.462445	0.231223	+	0.972901i	$0.425727\pi$
$740$	18065.7	0.897446
$741$	0	0
$742$	0	0
$743$	−37754.3	−1.86416	−0.932081	−	0.362250i	$-0.882009\pi$
$743$	−37754.3	−1.86416	−0.932081	+	0.362250i	$0.882009\pi$
$744$	0	0
$745$	3216.54	0.158181
$746$	−69644.7	−3.41806
$747$	0	0
$748$	29206.8	1.42768
$749$	0	0
$750$	0	0
$751$	−39407.9	−1.91480	−0.957399	−	0.288767i	$-0.906755\pi$
$751$	−39407.9	−1.91480	−0.957399	+	0.288767i	$0.906755\pi$
$752$	−40724.6	−1.97483
$753$	0	0
$754$	6743.07	0.325687
$755$	4701.93	0.226650
$756$	0	0
$757$	4083.29	0.196050	0.0980250	−	0.995184i	$-0.468747\pi$
$757$	4083.29	0.196050	0.0980250	+	0.995184i	$0.468747\pi$
$758$	−7023.79	−0.336564
$759$	0	0
$760$	12101.3	0.577578
$761$	40455.5	1.92709	0.963543	−	0.267553i	$-0.0862152\pi$
$761$	40455.5	1.92709	0.963543	+	0.267553i	$0.0862152\pi$
$762$	0	0
$763$	0	0
$764$	−1828.42	−0.0865837
$765$	0	0
$766$	10030.6	0.473133
$767$	1409.28	0.0663444
$768$	0	0
$769$	−10029.5	−0.470316	−0.235158	−	0.971957i	$-0.575561\pi$
$769$	−10029.5	−0.470316	−0.235158	+	0.971957i	$0.575561\pi$
$770$	0	0
$771$	0	0
$772$	52900.2	2.46622
$773$	−7090.10	−0.329901	−0.164950	−	0.986302i	$-0.552746\pi$
$773$	−7090.10	−0.329901	−0.164950	+	0.986302i	$0.552746\pi$
$774$	0	0
$775$	−1379.71	−0.0639493
$776$	2096.49	0.0969838
$777$	0	0
$778$	25308.9	1.16628
$779$	23848.9	1.09689
$780$	0	0
$781$	3775.76	0.172993
$782$	−53910.5	−2.46526
$783$	0	0
$784$	0	0
$785$	80.8810	0.00367741
$786$	0	0
$787$	−11645.7	−0.527477	−0.263739	−	0.964594i	$-0.584956\pi$
$787$	−11645.7	−0.527477	−0.263739	+	0.964594i	$0.584956\pi$
$788$	−78251.4	−3.53755
$789$	0	0
$790$	−19863.5	−0.894570
$791$	0	0
$792$	0	0
$793$	1447.22	0.0648073
$794$	15177.4	0.678368
$795$	0	0
$796$	−50172.0	−2.23405
$797$	−25792.7	−1.14633	−0.573164	−	0.819441i	$-0.694284\pi$
$797$	−25792.7	−1.14633	−0.573164	+	0.819441i	$0.694284\pi$
$798$	0	0
$799$	37417.3	1.65673
$800$	−1057.88	−0.0467523
$801$	0	0
$802$	26464.9	1.16522
$803$	−3652.97	−0.160536
$804$	0	0
$805$	0	0
$806$	1518.73	0.0663711
$807$	0	0
$808$	−62119.6	−2.70466
$809$	1317.71	0.0572660	0.0286330	−	0.999590i	$-0.490885\pi$
$809$	1317.71	0.0572660	0.0286330	+	0.999590i	$0.490885\pi$
$810$	0	0
$811$	−16474.1	−0.713298	−0.356649	−	0.934238i	$-0.616081\pi$
$811$	−16474.1	−0.713298	−0.356649	+	0.934238i	$0.616081\pi$
$812$	0	0
$813$	0	0
$814$	27183.0	1.17047
$815$	1645.74	0.0707333
$816$	0	0
$817$	28764.4	1.23175
$818$	−21736.1	−0.929079
$819$	0	0
$820$	−34321.2	−1.46165
$821$	4052.93	0.172288	0.0861439	−	0.996283i	$-0.472546\pi$
$821$	4052.93	0.172288	0.0861439	+	0.996283i	$0.472546\pi$
$822$	0	0
$823$	−21060.3	−0.891998	−0.445999	−	0.895033i	$-0.647152\pi$
$823$	−21060.3	−0.891998	−0.445999	+	0.895033i	$0.647152\pi$
$824$	−54869.9	−2.31976
$825$	0	0
$826$	0	0
$827$	6953.04	0.292359	0.146179	−	0.989258i	$-0.453302\pi$
$827$	6953.04	0.292359	0.146179	+	0.989258i	$0.453302\pi$
$828$	0	0
$829$	−6526.12	−0.273416	−0.136708	−	0.990611i	$-0.543652\pi$
$829$	−6526.12	−0.273416	−0.136708	+	0.990611i	$0.543652\pi$
$830$	−18981.9	−0.793820
$831$	0	0
$832$	−2245.07	−0.0935503
$833$	0	0
$834$	0	0
$835$	1234.98	0.0511836
$836$	23788.9	0.984159
$837$	0	0
$838$	−24633.8	−1.01547
$839$	34156.6	1.40550	0.702751	−	0.711436i	$-0.251955\pi$
$839$	34156.6	1.40550	0.702751	+	0.711436i	$0.251955\pi$
$840$	0	0
$841$	35652.2	1.46181
$842$	42124.0	1.72409
$843$	0	0
$844$	−68151.0	−2.77945
$845$	−10830.6	−0.440926
$846$	0	0
$847$	0	0
$848$	17885.5	0.724283
$849$	0	0
$850$	8721.76	0.351946
$851$	−33802.9	−1.36163
$852$	0	0
$853$	−7853.28	−0.315230	−0.157615	−	0.987501i	$-0.550381\pi$
$853$	−7853.28	−0.315230	−0.157615	+	0.987501i	$0.550381\pi$
$854$	0	0
$855$	0	0
$856$	−71528.2	−2.85606
$857$	−21020.6	−0.837866	−0.418933	−	0.908017i	$-0.637596\pi$
$857$	−21020.6	−0.837866	−0.418933	+	0.908017i	$0.637596\pi$
$858$	0	0
$859$	15945.0	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$859$	15945.0	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$860$	−41395.1	−1.64135
$861$	0	0
$862$	−37295.0	−1.47364
$863$	−7662.76	−0.302252	−0.151126	−	0.988515i	$-0.548290\pi$
$863$	−7662.76	−0.302252	−0.151126	+	0.988515i	$0.548290\pi$
$864$	0	0
$865$	−19970.7	−0.784998
$866$	−64172.1	−2.51808
$867$	0	0
$868$	0	0
$869$	−20135.6	−0.786021
$870$	0	0
$871$	1105.93	0.0430231
$872$	43641.3	1.69482
$873$	0	0
$874$	−43910.1	−1.69940
$875$	0	0
$876$	0	0
$877$	−13622.4	−0.524512	−0.262256	−	0.964998i	$-0.584466\pi$
$877$	−13622.4	−0.524512	−0.262256	+	0.964998i	$0.584466\pi$
$878$	−24355.7	−0.936178
$879$	0	0
$880$	−9622.70	−0.368615
$881$	978.110	0.0374045	0.0187022	−	0.999825i	$-0.494047\pi$
$881$	978.110	0.0374045	0.0187022	+	0.999825i	$0.494047\pi$
$882$	0	0
$883$	−35332.8	−1.34660	−0.673298	−	0.739371i	$-0.735123\pi$
$883$	−35332.8	−1.34660	−0.673298	+	0.739371i	$0.735123\pi$
$884$	−6467.91	−0.246085
$885$	0	0
$886$	52649.1	1.99637
$887$	−26088.8	−0.987571	−0.493786	−	0.869584i	$-0.664387\pi$
$887$	−26088.8	−0.987571	−0.493786	+	0.869584i	$0.664387\pi$
$888$	0	0
$889$	0	0
$890$	31176.0	1.17418
$891$	0	0
$892$	2523.55	0.0947252
$893$	30476.4	1.14205
$894$	0	0
$895$	−11120.8	−0.415337
$896$	0	0
$897$	0	0
$898$	57661.4	2.14275
$899$	13523.0	0.501688
$900$	0	0
$901$	−16433.0	−0.607618
$902$	−51642.2	−1.90631
$903$	0	0
$904$	−36044.9	−1.32615
$905$	−23639.5	−0.868291
$906$	0	0
$907$	−34353.4	−1.25765	−0.628824	−	0.777548i	$-0.716463\pi$
$907$	−34353.4	−1.25765	−0.628824	+	0.777548i	$0.716463\pi$
$908$	45855.3	1.67595
$909$	0	0
$910$	0	0
$911$	46650.6	1.69660	0.848301	−	0.529515i	$-0.177626\pi$
$911$	46650.6	1.69660	0.848301	+	0.529515i	$0.177626\pi$
$912$	0	0
$913$	−19241.9	−0.697496
$914$	−6552.81	−0.237142
$915$	0	0
$916$	38765.9	1.39832
$917$	0	0
$918$	0	0
$919$	−699.167	−0.0250962	−0.0125481	−	0.999921i	$-0.503994\pi$
$919$	−699.167	−0.0250962	−0.0125481	+	0.999921i	$0.503994\pi$
$920$	32585.4	1.16773
$921$	0	0
$922$	−36885.1	−1.31751
$923$	−836.151	−0.0298183
$924$	0	0
$925$	5468.70	0.194389
$926$	74884.8	2.65752
$927$	0	0
$928$	10368.7	0.366776
$929$	14134.4	0.499178	0.249589	−	0.968352i	$-0.419705\pi$
$929$	14134.4	0.499178	0.249589	+	0.968352i	$0.419705\pi$
$930$	0	0
$931$	0	0
$932$	95017.5	3.33949
$933$	0	0
$934$	−69652.2	−2.44014
$935$	8841.23	0.309240
$936$	0	0
$937$	43506.0	1.51684	0.758419	−	0.651767i	$-0.225972\pi$
$937$	43506.0	1.51684	0.758419	+	0.651767i	$0.225972\pi$
$938$	0	0
$939$	0	0
$940$	−43858.9	−1.52183
$941$	45033.0	1.56008	0.780039	−	0.625730i	$-0.215199\pi$
$941$	45033.0	1.56008	0.780039	+	0.625730i	$0.215199\pi$
$942$	0	0
$943$	64218.6	2.21765
$944$	19445.2	0.670431
$945$	0	0
$946$	−62286.0	−2.14069
$947$	−3911.83	−0.134232	−0.0671159	−	0.997745i	$-0.521380\pi$
$947$	−3911.83	−0.134232	−0.0671159	+	0.997745i	$0.521380\pi$
$948$	0	0
$949$	808.958	0.0276711
$950$	7103.86	0.242610
$951$	0	0
$952$	0	0
$953$	−30921.1	−1.05103	−0.525516	−	0.850784i	$-0.676128\pi$
$953$	−30921.1	−1.05103	−0.525516	+	0.850784i	$0.676128\pi$
$954$	0	0
$955$	−553.484	−0.0187543
$956$	97376.6	3.29434
$957$	0	0
$958$	−60098.3	−2.02681
$959$	0	0
$960$	0	0
$961$	−26745.2	−0.897762
$962$	−6019.74	−0.201751
$963$	0	0
$964$	−63208.9	−2.11185
$965$	16013.5	0.534189
$966$	0	0
$967$	−31264.0	−1.03969	−0.519847	−	0.854260i	$-0.674011\pi$
$967$	−31264.0	−1.03969	−0.519847	+	0.854260i	$0.674011\pi$
$968$	29570.6	0.981853
$969$	0	0
$970$	1230.71	0.0407378
$971$	−30830.5	−1.01895	−0.509474	−	0.860486i	$-0.670160\pi$
$971$	−30830.5	−1.01895	−0.509474	+	0.860486i	$0.670160\pi$
$972$	0	0
$973$	0	0
$974$	−91955.9	−3.02511
$975$	0	0
$976$	19968.7	0.654899
$977$	12890.2	0.422103	0.211052	−	0.977475i	$-0.432311\pi$
$977$	12890.2	0.422103	0.211052	+	0.977475i	$0.432311\pi$
$978$	0	0
$979$	31603.0	1.03170
$980$	0	0
$981$	0	0
$982$	82325.0	2.67525
$983$	−24611.8	−0.798571	−0.399285	−	0.916827i	$-0.630742\pi$
$983$	−24611.8	−0.798571	−0.399285	+	0.916827i	$0.630742\pi$
$984$	0	0
$985$	−23687.6	−0.766243
$986$	−85484.7	−2.76104
$987$	0	0
$988$	−5268.11	−0.169637
$989$	77454.5	2.49030
$990$	0	0
$991$	33188.8	1.06385	0.531925	−	0.846791i	$-0.321469\pi$
$991$	33188.8	1.06385	0.531925	+	0.846791i	$0.321469\pi$
$992$	2335.32	0.0747445
$993$	0	0
$994$	0	0
$995$	−15187.6	−0.483900
$996$	0	0
$997$	28219.7	0.896417	0.448208	−	0.893929i	$-0.352062\pi$
$997$	28219.7	0.896417	0.448208	+	0.893929i	$0.352062\pi$
$998$	47231.4	1.49808
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ce.1.2		8
3.2	odd	2		735.4.a.bb.1.7	✓	8
7.6	odd	2		2205.4.a.cd.1.2		8
21.20	even	2		735.4.a.bc.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.bb.1.7	✓	8	3.2	odd	2
735.4.a.bc.1.7	yes	8	21.20	even	2
2205.4.a.cd.1.2		8	7.6	odd	2
2205.4.a.ce.1.2		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-4.95150 q^{2} +16.5174 q^{4} +5.00000 q^{5} -42.1739 q^{8} +O(q^{10})\)	\(q-4.95150 q^{2} +16.5174 q^{4} +5.00000 q^{5} -42.1739 q^{8} -24.7575 q^{10} -25.0967 q^{11} +5.55771 q^{13} +76.6851 q^{16} -70.4574 q^{17} -57.3875 q^{19} +82.5870 q^{20} +124.266 q^{22} -154.529 q^{23} +25.0000 q^{25} -27.5190 q^{26} -245.033 q^{29} -55.1885 q^{31} -42.3154 q^{32} +348.870 q^{34} +218.748 q^{37} +284.155 q^{38} -210.869 q^{40} -415.577 q^{41} -501.230 q^{43} -414.531 q^{44} +765.150 q^{46} -531.063 q^{47} -123.788 q^{50} +91.7989 q^{52} +233.234 q^{53} -125.483 q^{55} +1213.28 q^{58} +253.572 q^{59} +260.398 q^{61} +273.266 q^{62} -403.956 q^{64} +27.7886 q^{65} +198.991 q^{67} -1163.77 q^{68} -150.449 q^{71} +145.556 q^{73} -1083.13 q^{74} -947.892 q^{76} +802.321 q^{79} +383.425 q^{80} +2057.73 q^{82} +766.712 q^{83} -352.287 q^{85} +2481.84 q^{86} +1058.42 q^{88} -1259.25 q^{89} -2552.41 q^{92} +2629.56 q^{94} -286.938 q^{95} -49.7105 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 50 q^{4} + 40 q^{5} - 66 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 50 * q^4 + 40 * q^5 - 66 * q^8	\( 8 q - 2 q^{2} + 50 q^{4} + 40 q^{5} - 66 q^{8} - 10 q^{10} - 64 q^{11} + 206 q^{16} + 48 q^{17} - 80 q^{19} + 250 q^{20} + 452 q^{22} - 120 q^{23} + 200 q^{25} + 272 q^{26} - 76 q^{29} - 20 q^{31} - 770 q^{32} - 320 q^{34} + 348 q^{37} - 236 q^{38} - 330 q^{40} + 944 q^{41} + 1116 q^{43} - 172 q^{44} + 496 q^{46} - 208 q^{47} - 50 q^{50} + 2272 q^{52} - 1144 q^{53} - 320 q^{55} + 560 q^{58} + 596 q^{59} - 740 q^{61} - 1184 q^{62} + 1298 q^{64} + 1964 q^{67} + 96 q^{68} + 4 q^{71} + 1500 q^{73} - 3368 q^{74} + 2912 q^{76} - 460 q^{79} + 1030 q^{80} + 5644 q^{82} + 700 q^{83} + 240 q^{85} + 1396 q^{86} + 6892 q^{88} - 644 q^{92} + 6692 q^{94} - 400 q^{95} + 2052 q^{97}+O(q^{100}) \) 8 * q - 2 * q^2 + 50 * q^4 + 40 * q^5 - 66 * q^8 - 10 * q^10 - 64 * q^11 + 206 * q^16 + 48 * q^17 - 80 * q^19 + 250 * q^20 + 452 * q^22 - 120 * q^23 + 200 * q^25 + 272 * q^26 - 76 * q^29 - 20 * q^31 - 770 * q^32 - 320 * q^34 + 348 * q^37 - 236 * q^38 - 330 * q^40 + 944 * q^41 + 1116 * q^43 - 172 * q^44 + 496 * q^46 - 208 * q^47 - 50 * q^50 + 2272 * q^52 - 1144 * q^53 - 320 * q^55 + 560 * q^58 + 596 * q^59 - 740 * q^61 - 1184 * q^62 + 1298 * q^64 + 1964 * q^67 + 96 * q^68 + 4 * q^71 + 1500 * q^73 - 3368 * q^74 + 2912 * q^76 - 460 * q^79 + 1030 * q^80 + 5644 * q^82 + 700 * q^83 + 240 * q^85 + 1396 * q^86 + 6892 * q^88 - 644 * q^92 + 6692 * q^94 - 400 * q^95 + 2052 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more