LMFDB - Embedded newform 2205.4.a.ce.1.8

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			1.8
Root			$-4.86877$ 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)

$f(q)$	$=$	$q+4.86877 q^{2} +15.7049 q^{4} +5.00000 q^{5} +37.5134 q^{8} +24.3438 q^{10} +8.38312 q^{11} +92.3062 q^{13} +57.0048 q^{16} -48.7212 q^{17} -50.3375 q^{19} +78.5245 q^{20} +40.8155 q^{22} +55.5570 q^{23} +25.0000 q^{25} +449.417 q^{26} -159.685 q^{29} +232.222 q^{31} -22.5640 q^{32} -237.212 q^{34} +297.537 q^{37} -245.082 q^{38} +187.567 q^{40} +433.727 q^{41} +59.2917 q^{43} +131.656 q^{44} +270.494 q^{46} +349.673 q^{47} +121.719 q^{50} +1449.66 q^{52} -267.144 q^{53} +41.9156 q^{55} -777.468 q^{58} +515.530 q^{59} -664.827 q^{61} +1130.63 q^{62} -565.897 q^{64} +461.531 q^{65} -556.295 q^{67} -765.162 q^{68} +226.340 q^{71} +337.515 q^{73} +1448.64 q^{74} -790.546 q^{76} -305.677 q^{79} +285.024 q^{80} +2111.71 q^{82} +323.816 q^{83} -243.606 q^{85} +288.678 q^{86} +314.479 q^{88} -542.923 q^{89} +872.517 q^{92} +1702.48 q^{94} -251.688 q^{95} +1429.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	4.86877	1.72137	0.860685	−	0.509138i	$-0.170036\pi$
$2$	4.86877	1.72137	0.860685	+	0.509138i	$0.170036\pi$
$3$	0	0
$3$	0	0
$4$	15.7049	1.96311
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	37.5134	1.65787
$9$	0	0
$10$	24.3438	0.769820
$11$	8.38312	0.229782	0.114891	−	0.993378i	$-0.463348\pi$
$11$	8.38312	0.229782	0.114891	+	0.993378i	$0.463348\pi$
$12$	0	0
$13$	92.3062	1.96932	0.984659	−	0.174490i	$-0.0558277\pi$
$13$	92.3062	1.96932	0.984659	+	0.174490i	$0.0558277\pi$
$14$	0	0
$15$	0	0
$16$	57.0048	0.890700
$17$	−48.7212	−0.695096	−0.347548	−	0.937662i	$-0.612986\pi$
$17$	−48.7212	−0.695096	−0.347548	+	0.937662i	$0.612986\pi$
$18$	0	0
$19$	−50.3375	−0.607801	−0.303900	−	0.952704i	$-0.598289\pi$
$19$	−50.3375	−0.607801	−0.303900	+	0.952704i	$0.598289\pi$
$20$	78.5245	0.877931
$21$	0	0
$22$	40.8155	0.395540
$23$	55.5570	0.503671	0.251835	−	0.967770i	$-0.418966\pi$
$23$	55.5570	0.503671	0.251835	+	0.967770i	$0.418966\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	449.417	3.38992
$27$	0	0
$28$	0	0
$29$	−159.685	−1.02251	−0.511253	−	0.859430i	$-0.670819\pi$
$29$	−159.685	−1.02251	−0.511253	+	0.859430i	$0.670819\pi$
$30$	0	0
$31$	232.222	1.34543	0.672714	−	0.739903i	$-0.265128\pi$
$31$	232.222	1.34543	0.672714	+	0.739903i	$0.265128\pi$
$32$	−22.5640	−0.124650
$33$	0	0
$34$	−237.212	−1.19652
$35$	0	0
$36$	0	0
$37$	297.537	1.32202	0.661010	−	0.750377i	$-0.270128\pi$
$37$	297.537	1.32202	0.661010	+	0.750377i	$0.270128\pi$
$38$	−245.082	−1.04625
$39$	0	0
$40$	187.567	0.741424
$41$	433.727	1.65212	0.826058	−	0.563586i	$-0.190579\pi$
$41$	433.727	1.65212	0.826058	+	0.563586i	$0.190579\pi$
$42$	0	0
$43$	59.2917	0.210277	0.105138	−	0.994458i	$-0.466471\pi$
$43$	59.2917	0.210277	0.105138	+	0.994458i	$0.466471\pi$
$44$	131.656	0.451089
$45$	0	0
$46$	270.494	0.867004
$47$	349.673	1.08521	0.542606	−	0.839987i	$-0.317437\pi$
$47$	349.673	1.08521	0.542606	+	0.839987i	$0.317437\pi$
$48$	0	0
$49$	0	0
$50$	121.719	0.344274
$51$	0	0
$52$	1449.66	3.86599
$53$	−267.144	−0.692359	−0.346179	−	0.938168i	$-0.612521\pi$
$53$	−267.144	−0.692359	−0.346179	+	0.938168i	$0.612521\pi$
$54$	0	0
$55$	41.9156	0.102762
$56$	0	0
$57$	0	0
$58$	−777.468	−1.76011
$59$	515.530	1.13756	0.568782	−	0.822488i	$-0.307415\pi$
$59$	515.530	1.13756	0.568782	+	0.822488i	$0.307415\pi$
$60$	0	0
$61$	−664.827	−1.39545	−0.697724	−	0.716366i	$-0.745804\pi$
$61$	−664.827	−1.39545	−0.697724	+	0.716366i	$0.745804\pi$
$62$	1130.63	2.31598
$63$	0	0
$64$	−565.897	−1.10527
$65$	461.531	0.880706
$66$	0	0
$67$	−556.295	−1.01436	−0.507181	−	0.861840i	$-0.669312\pi$
$67$	−556.295	−1.01436	−0.507181	+	0.861840i	$0.669312\pi$
$68$	−765.162	−1.36455
$69$	0	0
$70$	0	0
$71$	226.340	0.378332	0.189166	−	0.981945i	$-0.439422\pi$
$71$	226.340	0.378332	0.189166	+	0.981945i	$0.439422\pi$
$72$	0	0
$73$	337.515	0.541139	0.270570	−	0.962700i	$-0.412788\pi$
$73$	337.515	0.541139	0.270570	+	0.962700i	$0.412788\pi$
$74$	1448.64	2.27569
$75$	0	0
$76$	−790.546	−1.19318
$77$	0	0
$78$	0	0
$79$	−305.677	−0.435333	−0.217667	−	0.976023i	$-0.569845\pi$
$79$	−305.677	−0.435333	−0.217667	+	0.976023i	$0.569845\pi$
$80$	285.024	0.398333
$81$	0	0
$82$	2111.71	2.84390
$83$	323.816	0.428233	0.214117	−	0.976808i	$-0.431313\pi$
$83$	323.816	0.428233	0.214117	+	0.976808i	$0.431313\pi$
$84$	0	0
$85$	−243.606	−0.310856
$86$	288.678	0.361964
$87$	0	0
$88$	314.479	0.380950
$89$	−542.923	−0.646626	−0.323313	−	0.946292i	$-0.604797\pi$
$89$	−542.923	−0.646626	−0.323313	+	0.946292i	$0.604797\pi$
$90$	0	0
$91$	0	0
$92$	872.517	0.988763
$93$	0	0
$94$	1702.48	1.86805
$95$	−251.688	−0.271817
$96$	0	0
$97$	1429.82	1.49667	0.748333	−	0.663324i	$-0.230855\pi$
$97$	1429.82	1.49667	0.748333	+	0.663324i	$0.230855\pi$
$98$	0	0
$99$	0	0
$100$	392.623	0.392623
$101$	219.517	0.216265	0.108132	−	0.994137i	$-0.465513\pi$
$101$	219.517	0.216265	0.108132	+	0.994137i	$0.465513\pi$
$102$	0	0
$103$	−564.707	−0.540216	−0.270108	−	0.962830i	$-0.587059\pi$
$103$	−564.707	−0.540216	−0.270108	+	0.962830i	$0.587059\pi$
$104$	3462.72	3.26488
$105$	0	0
$106$	−1300.66	−1.19181
$107$	−2101.90	−1.89905	−0.949526	−	0.313688i	$-0.898435\pi$
$107$	−2101.90	−1.89905	−0.949526	+	0.313688i	$0.898435\pi$
$108$	0	0
$109$	1777.22	1.56171	0.780857	−	0.624710i	$-0.214783\pi$
$109$	1777.22	1.56171	0.780857	+	0.624710i	$0.214783\pi$
$110$	204.077	0.176891
$111$	0	0
$112$	0	0
$113$	−947.884	−0.789110	−0.394555	−	0.918872i	$-0.629101\pi$
$113$	−947.884	−0.789110	−0.394555	+	0.918872i	$0.629101\pi$
$114$	0	0
$115$	277.785	0.225248
$116$	−2507.83	−2.00730
$117$	0	0
$118$	2510.00	1.95817
$119$	0	0
$120$	0	0
$121$	−1260.72	−0.947200
$122$	−3236.89	−2.40208
$123$	0	0
$124$	3647.02	2.64123
$125$	125.000	0.0894427
$126$	0	0
$127$	34.5433	0.0241356	0.0120678	−	0.999927i	$-0.496159\pi$
$127$	34.5433	0.0241356	0.0120678	+	0.999927i	$0.496159\pi$
$128$	−2574.71	−1.77793
$129$	0	0
$130$	2247.09	1.51602
$131$	2878.64	1.91991	0.959953	−	0.280162i	$-0.0903883\pi$
$131$	2878.64	1.91991	0.959953	+	0.280162i	$0.0903883\pi$
$132$	0	0
$133$	0	0
$134$	−2708.47	−1.74609
$135$	0	0
$136$	−1827.70	−1.15238
$137$	1583.15	0.987282	0.493641	−	0.869666i	$-0.335666\pi$
$137$	1583.15	0.987282	0.493641	+	0.869666i	$0.335666\pi$
$138$	0	0
$139$	−1889.32	−1.15288	−0.576438	−	0.817141i	$-0.695558\pi$
$139$	−1889.32	−1.15288	−0.576438	+	0.817141i	$0.695558\pi$
$140$	0	0
$141$	0	0
$142$	1102.00	0.651249
$143$	773.814	0.452514
$144$	0	0
$145$	−798.423	−0.457279
$146$	1643.28	0.931501
$147$	0	0
$148$	4672.79	2.59528
$149$	−2890.67	−1.58935	−0.794673	−	0.607037i	$-0.792358\pi$
$149$	−2890.67	−1.58935	−0.794673	+	0.607037i	$0.792358\pi$
$150$	0	0
$151$	2298.15	1.23855	0.619274	−	0.785175i	$-0.287427\pi$
$151$	2298.15	1.23855	0.619274	+	0.785175i	$0.287427\pi$
$152$	−1888.33	−1.00766
$153$	0	0
$154$	0	0
$155$	1161.11	0.601694
$156$	0	0
$157$	981.415	0.498888	0.249444	−	0.968389i	$-0.419752\pi$
$157$	981.415	0.498888	0.249444	+	0.968389i	$0.419752\pi$
$158$	−1488.27	−0.749369
$159$	0	0
$160$	−112.820	−0.0557450
$161$	0	0
$162$	0	0
$163$	3546.80	1.70434	0.852168	−	0.523268i	$-0.175287\pi$
$163$	3546.80	1.70434	0.852168	+	0.523268i	$0.175287\pi$
$164$	6811.64	3.24329
$165$	0	0
$166$	1576.58	0.737148
$167$	−3812.18	−1.76644	−0.883219	−	0.468962i	$-0.844628\pi$
$167$	−3812.18	−1.76644	−0.883219	+	0.468962i	$0.844628\pi$
$168$	0	0
$169$	6323.43	2.87821
$170$	−1186.06	−0.535099
$171$	0	0
$172$	931.171	0.412797
$173$	−837.375	−0.368002	−0.184001	−	0.982926i	$-0.558905\pi$
$173$	−837.375	−0.368002	−0.184001	+	0.982926i	$0.558905\pi$
$174$	0	0
$175$	0	0
$176$	477.878	0.204667
$177$	0	0
$178$	−2643.37	−1.11308
$179$	2249.67	0.939376	0.469688	−	0.882833i	$-0.344366\pi$
$179$	2249.67	0.939376	0.469688	+	0.882833i	$0.344366\pi$
$180$	0	0
$181$	3639.55	1.49462	0.747308	−	0.664478i	$-0.231346\pi$
$181$	3639.55	1.49462	0.747308	+	0.664478i	$0.231346\pi$
$182$	0	0
$183$	0	0
$184$	2084.13	0.835023
$185$	1487.68	0.591226
$186$	0	0
$187$	−408.436	−0.159721
$188$	5491.58	2.13040
$189$	0	0
$190$	−1225.41	−0.467897
$191$	−3001.20	−1.13696	−0.568481	−	0.822697i	$-0.692469\pi$
$191$	−3001.20	−1.13696	−0.568481	+	0.822697i	$0.692469\pi$
$192$	0	0
$193$	−4206.48	−1.56886	−0.784428	−	0.620220i	$-0.787043\pi$
$193$	−4206.48	−1.56886	−0.784428	+	0.620220i	$0.787043\pi$
$194$	6961.48	2.57631
$195$	0	0
$196$	0	0
$197$	1399.81	0.506254	0.253127	−	0.967433i	$-0.418541\pi$
$197$	1399.81	0.506254	0.253127	+	0.967433i	$0.418541\pi$
$198$	0	0
$199$	−595.416	−0.212100	−0.106050	−	0.994361i	$-0.533820\pi$
$199$	−595.416	−0.212100	−0.106050	+	0.994361i	$0.533820\pi$
$200$	937.835	0.331575
$201$	0	0
$202$	1068.78	0.372271
$203$	0	0
$204$	0	0
$205$	2168.63	0.738848
$206$	−2749.43	−0.929911
$207$	0	0
$208$	5261.90	1.75407
$209$	−421.985	−0.139662
$210$	0	0
$211$	4808.62	1.56890	0.784452	−	0.620189i	$-0.212944\pi$
$211$	4808.62	1.56890	0.784452	+	0.620189i	$0.212944\pi$
$212$	−4195.47	−1.35918
$213$	0	0
$214$	−10233.7	−3.26897
$215$	296.458	0.0940386
$216$	0	0
$217$	0	0
$218$	8652.87	2.68829
$219$	0	0
$220$	658.280	0.201733
$221$	−4497.27	−1.36886
$222$	0	0
$223$	−245.603	−0.0737525	−0.0368763	−	0.999320i	$-0.511741\pi$
$223$	−245.603	−0.0737525	−0.0368763	+	0.999320i	$0.511741\pi$
$224$	0	0
$225$	0	0
$226$	−4615.03	−1.35835
$227$	−3058.49	−0.894270	−0.447135	−	0.894466i	$-0.647556\pi$
$227$	−3058.49	−0.894270	−0.447135	+	0.894466i	$0.647556\pi$
$228$	0	0
$229$	1630.15	0.470408	0.235204	−	0.971946i	$-0.424424\pi$
$229$	1630.15	0.470408	0.235204	+	0.971946i	$0.424424\pi$
$230$	1352.47	0.387736
$231$	0	0
$232$	−5990.31	−1.69519
$233$	1562.51	0.439329	0.219665	−	0.975575i	$-0.429504\pi$
$233$	1562.51	0.439329	0.219665	+	0.975575i	$0.429504\pi$
$234$	0	0
$235$	1748.36	0.485322
$236$	8096.35	2.23317
$237$	0	0
$238$	0	0
$239$	927.209	0.250946	0.125473	−	0.992097i	$-0.459955\pi$
$239$	927.209	0.250946	0.125473	+	0.992097i	$0.459955\pi$
$240$	0	0
$241$	−4418.97	−1.18112	−0.590562	−	0.806992i	$-0.701094\pi$
$241$	−4418.97	−1.18112	−0.590562	+	0.806992i	$0.701094\pi$
$242$	−6138.17	−1.63048
$243$	0	0
$244$	−10441.0	−2.73942
$245$	0	0
$246$	0	0
$247$	−4646.46	−1.19695
$248$	8711.43	2.23055
$249$	0	0
$250$	608.596	0.153964
$251$	−1537.14	−0.386547	−0.193274	−	0.981145i	$-0.561911\pi$
$251$	−1537.14	−0.386547	−0.193274	+	0.981145i	$0.561911\pi$
$252$	0	0
$253$	465.741	0.115735
$254$	168.184	0.0415463
$255$	0	0
$256$	−8008.49	−1.95520
$257$	2256.13	0.547600	0.273800	−	0.961787i	$-0.411719\pi$
$257$	2256.13	0.547600	0.273800	+	0.961787i	$0.411719\pi$
$258$	0	0
$259$	0	0
$260$	7248.30	1.72892
$261$	0	0
$262$	14015.4	3.30487
$263$	−1044.02	−0.244779	−0.122389	−	0.992482i	$-0.539056\pi$
$263$	−1044.02	−0.244779	−0.122389	+	0.992482i	$0.539056\pi$
$264$	0	0
$265$	−1335.72	−0.309632
$266$	0	0
$267$	0	0
$268$	−8736.56	−1.99131
$269$	−1736.18	−0.393519	−0.196760	−	0.980452i	$-0.563042\pi$
$269$	−1736.18	−0.393519	−0.196760	+	0.980452i	$0.563042\pi$
$270$	0	0
$271$	8200.59	1.83819	0.919096	−	0.394033i	$-0.128920\pi$
$271$	8200.59	1.83819	0.919096	+	0.394033i	$0.128920\pi$
$272$	−2777.34	−0.619122
$273$	0	0
$274$	7707.99	1.69948
$275$	209.578	0.0459565
$276$	0	0
$277$	−1837.78	−0.398633	−0.199317	−	0.979935i	$-0.563872\pi$
$277$	−1837.78	−0.398633	−0.199317	+	0.979935i	$0.563872\pi$
$278$	−9198.65	−1.98453
$279$	0	0
$280$	0	0
$281$	−5595.24	−1.18784	−0.593922	−	0.804523i	$-0.702421\pi$
$281$	−5595.24	−1.18784	−0.593922	+	0.804523i	$0.702421\pi$
$282$	0	0
$283$	−7362.80	−1.54655	−0.773274	−	0.634072i	$-0.781382\pi$
$283$	−7362.80	−1.54655	−0.773274	+	0.634072i	$0.781382\pi$
$284$	3554.64	0.742708
$285$	0	0
$286$	3767.52	0.778945
$287$	0	0
$288$	0	0
$289$	−2539.24	−0.516842
$290$	−3887.34	−0.787146
$291$	0	0
$292$	5300.64	1.06232
$293$	3553.92	0.708609	0.354304	−	0.935130i	$-0.384718\pi$
$293$	3553.92	0.708609	0.354304	+	0.935130i	$0.384718\pi$
$294$	0	0
$295$	2577.65	0.508734
$296$	11161.6	2.19174
$297$	0	0
$298$	−14074.0	−2.73585
$299$	5128.25	0.991888
$300$	0	0
$301$	0	0
$302$	11189.2	2.13200
$303$	0	0
$304$	−2869.48	−0.541368
$305$	−3324.13	−0.624064
$306$	0	0
$307$	2458.13	0.456980	0.228490	−	0.973546i	$-0.426621\pi$
$307$	2458.13	0.456980	0.228490	+	0.973546i	$0.426621\pi$
$308$	0	0
$309$	0	0
$310$	5653.17	1.03574
$311$	−7814.26	−1.42478	−0.712389	−	0.701785i	$-0.752387\pi$
$311$	−7814.26	−1.42478	−0.712389	+	0.701785i	$0.752387\pi$
$312$	0	0
$313$	−9637.00	−1.74030	−0.870152	−	0.492783i	$-0.835980\pi$
$313$	−9637.00	−1.74030	−0.870152	+	0.492783i	$0.835980\pi$
$314$	4778.28	0.858771
$315$	0	0
$316$	−4800.62	−0.854608
$317$	−8845.80	−1.56729	−0.783643	−	0.621212i	$-0.786641\pi$
$317$	−8845.80	−1.56729	−0.783643	+	0.621212i	$0.786641\pi$
$318$	0	0
$319$	−1338.66	−0.234954
$320$	−2829.49	−0.494291
$321$	0	0
$322$	0	0
$323$	2452.50	0.422480
$324$	0	0
$325$	2307.65	0.393864
$326$	17268.5	2.93379
$327$	0	0
$328$	16270.6	2.73900
$329$	0	0
$330$	0	0
$331$	3521.33	0.584743	0.292372	−	0.956305i	$-0.405556\pi$
$331$	3521.33	0.584743	0.292372	+	0.956305i	$0.405556\pi$
$332$	5085.49	0.840671
$333$	0	0
$334$	−18560.6	−3.04069
$335$	−2781.47	−0.453636
$336$	0	0
$337$	−2391.23	−0.386523	−0.193262	−	0.981147i	$-0.561907\pi$
$337$	−2391.23	−0.386523	−0.193262	+	0.981147i	$0.561907\pi$
$338$	30787.3	4.95447
$339$	0	0
$340$	−3825.81	−0.610246
$341$	1946.74	0.309156
$342$	0	0
$343$	0	0
$344$	2224.23	0.348612
$345$	0	0
$346$	−4076.98	−0.633468
$347$	−7505.62	−1.16116	−0.580580	−	0.814203i	$-0.697174\pi$
$347$	−7505.62	−1.16116	−0.580580	+	0.814203i	$0.697174\pi$
$348$	0	0
$349$	−862.288	−0.132256	−0.0661278	−	0.997811i	$-0.521065\pi$
$349$	−862.288	−0.132256	−0.0661278	+	0.997811i	$0.521065\pi$
$350$	0	0
$351$	0	0
$352$	−189.157	−0.0286423
$353$	1594.04	0.240347	0.120173	−	0.992753i	$-0.461655\pi$
$353$	1594.04	0.240347	0.120173	+	0.992753i	$0.461655\pi$
$354$	0	0
$355$	1131.70	0.169195
$356$	−8526.55	−1.26940
$357$	0	0
$358$	10953.1	1.61701
$359$	−191.248	−0.0281161	−0.0140581	−	0.999901i	$-0.504475\pi$
$359$	−191.248	−0.0281161	−0.0140581	+	0.999901i	$0.504475\pi$
$360$	0	0
$361$	−4325.14	−0.630578
$362$	17720.1	2.57279
$363$	0	0
$364$	0	0
$365$	1687.58	0.242005
$366$	0	0
$367$	2390.78	0.340048	0.170024	−	0.985440i	$-0.445615\pi$
$367$	2390.78	0.340048	0.170024	+	0.985440i	$0.445615\pi$
$368$	3167.01	0.448620
$369$	0	0
$370$	7243.19	1.01772
$371$	0	0
$372$	0	0
$373$	−1.45578	−0.000202085 0	−0.000101042	−	1.00000i	$-0.500032\pi$
$373$	−1.45578	−0.000202085 0	−0.000101042		1.00000i	$0.500032\pi$
$374$	−1988.58	−0.274938
$375$	0	0
$376$	13117.4	1.79915
$377$	−14739.9	−2.01364
$378$	0	0
$379$	−6257.71	−0.848119	−0.424060	−	0.905634i	$-0.639395\pi$
$379$	−6257.71	−0.848119	−0.424060	+	0.905634i	$0.639395\pi$
$380$	−3952.73	−0.533607
$381$	0	0
$382$	−14612.2	−1.95713
$383$	−8645.03	−1.15337	−0.576684	−	0.816967i	$-0.695654\pi$
$383$	−8645.03	−1.15337	−0.576684	+	0.816967i	$0.695654\pi$
$384$	0	0
$385$	0	0
$386$	−20480.4	−2.70058
$387$	0	0
$388$	22455.2	2.93812
$389$	−44.1024	−0.00574827	−0.00287414	−	0.999996i	$-0.500915\pi$
$389$	−44.1024	−0.00574827	−0.00287414	+	0.999996i	$0.500915\pi$
$390$	0	0
$391$	−2706.80	−0.350100
$392$	0	0
$393$	0	0
$394$	6815.33	0.871451
$395$	−1528.38	−0.194687
$396$	0	0
$397$	−4665.73	−0.589839	−0.294920	−	0.955522i	$-0.595293\pi$
$397$	−4665.73	−0.589839	−0.294920	+	0.955522i	$0.595293\pi$
$398$	−2898.94	−0.365103
$399$	0	0
$400$	1425.12	0.178140
$401$	−11797.4	−1.46916	−0.734582	−	0.678520i	$-0.762622\pi$
$401$	−11797.4	−1.46916	−0.734582	+	0.678520i	$0.762622\pi$
$402$	0	0
$403$	21435.5	2.64958
$404$	3447.49	0.424552
$405$	0	0
$406$	0	0
$407$	2494.29	0.303777
$408$	0	0
$409$	−13031.0	−1.57540	−0.787702	−	0.616056i	$-0.788729\pi$
$409$	−13031.0	−1.57540	−0.787702	+	0.616056i	$0.788729\pi$
$410$	10558.6	1.27183
$411$	0	0
$412$	−8868.67	−1.06051
$413$	0	0
$414$	0	0
$415$	1619.08	0.191512
$416$	−2082.80	−0.245475
$417$	0	0
$418$	−2054.55	−0.240410
$419$	−6990.53	−0.815059	−0.407529	−	0.913192i	$-0.633610\pi$
$419$	−6990.53	−0.815059	−0.407529	+	0.913192i	$0.633610\pi$
$420$	0	0
$421$	3044.58	0.352455	0.176228	−	0.984349i	$-0.443611\pi$
$421$	3044.58	0.352455	0.176228	+	0.984349i	$0.443611\pi$
$422$	23412.0	2.70066
$423$	0	0
$424$	−10021.5	−1.14784
$425$	−1218.03	−0.139019
$426$	0	0
$427$	0	0
$428$	−33010.2	−3.72805
$429$	0	0
$430$	1443.39	0.161875
$431$	2719.16	0.303891	0.151946	−	0.988389i	$-0.451446\pi$
$431$	2719.16	0.303891	0.151946	+	0.988389i	$0.451446\pi$
$432$	0	0
$433$	7382.83	0.819391	0.409695	−	0.912222i	$-0.365635\pi$
$433$	7382.83	0.819391	0.409695	+	0.912222i	$0.365635\pi$
$434$	0	0
$435$	0	0
$436$	27911.1	3.06582
$437$	−2796.60	−0.306132
$438$	0	0
$439$	−563.983	−0.0613154	−0.0306577	−	0.999530i	$-0.509760\pi$
$439$	−563.983	−0.0613154	−0.0306577	+	0.999530i	$0.509760\pi$
$440$	1572.40	0.170366
$441$	0	0
$442$	−21896.2	−2.35632
$443$	6884.63	0.738371	0.369186	−	0.929356i	$-0.379637\pi$
$443$	6884.63	0.738371	0.369186	+	0.929356i	$0.379637\pi$
$444$	0	0
$445$	−2714.61	−0.289180
$446$	−1195.79	−0.126955
$447$	0	0
$448$	0	0
$449$	1334.99	0.140316	0.0701580	−	0.997536i	$-0.477650\pi$
$449$	1334.99	0.140316	0.0701580	+	0.997536i	$0.477650\pi$
$450$	0	0
$451$	3635.98	0.379627
$452$	−14886.4	−1.54911
$453$	0	0
$454$	−14891.1	−1.53937
$455$	0	0
$456$	0	0
$457$	5142.27	0.526357	0.263178	−	0.964747i	$-0.415229\pi$
$457$	5142.27	0.526357	0.263178	+	0.964747i	$0.415229\pi$
$458$	7936.84	0.809747
$459$	0	0
$460$	4362.59	0.442188
$461$	1855.57	0.187467	0.0937337	−	0.995597i	$-0.470120\pi$
$461$	1855.57	0.187467	0.0937337	+	0.995597i	$0.470120\pi$
$462$	0	0
$463$	1548.49	0.155431	0.0777155	−	0.996976i	$-0.475237\pi$
$463$	1548.49	0.155431	0.0777155	+	0.996976i	$0.475237\pi$
$464$	−9102.79	−0.910747
$465$	0	0
$466$	7607.52	0.756248
$467$	−12575.3	−1.24607	−0.623036	−	0.782193i	$-0.714101\pi$
$467$	−12575.3	−1.24607	−0.623036	+	0.782193i	$0.714101\pi$
$468$	0	0
$469$	0	0
$470$	8512.38	0.835418
$471$	0	0
$472$	19339.3	1.88594
$473$	497.049	0.0483179
$474$	0	0
$475$	−1258.44	−0.121560
$476$	0	0
$477$	0	0
$478$	4514.37	0.431971
$479$	9898.57	0.944211	0.472106	−	0.881542i	$-0.343494\pi$
$479$	9898.57	0.944211	0.472106	+	0.881542i	$0.343494\pi$
$480$	0	0
$481$	27464.5	2.60348
$482$	−21514.9	−2.03315
$483$	0	0
$484$	−19799.5	−1.85946
$485$	7149.11	0.669329
$486$	0	0
$487$	15374.1	1.43053	0.715266	−	0.698853i	$-0.246306\pi$
$487$	15374.1	1.43053	0.715266	+	0.698853i	$0.246306\pi$
$488$	−24939.9	−2.31348
$489$	0	0
$490$	0	0
$491$	7569.63	0.695749	0.347874	−	0.937541i	$-0.386904\pi$
$491$	7569.63	0.695749	0.347874	+	0.937541i	$0.386904\pi$
$492$	0	0
$493$	7780.03	0.710740
$494$	−22622.6	−2.06040
$495$	0	0
$496$	13237.8	1.19837
$497$	0	0
$498$	0	0
$499$	−1979.62	−0.177595	−0.0887974	−	0.996050i	$-0.528302\pi$
$499$	−1979.62	−0.177595	−0.0887974	+	0.996050i	$0.528302\pi$
$500$	1963.11	0.175586
$501$	0	0
$502$	−7483.97	−0.665391
$503$	−10518.7	−0.932419	−0.466209	−	0.884674i	$-0.654381\pi$
$503$	−10518.7	−0.932419	−0.466209	+	0.884674i	$0.654381\pi$
$504$	0	0
$505$	1097.58	0.0967164
$506$	2267.58	0.199222
$507$	0	0
$508$	542.500	0.0473810
$509$	−14783.6	−1.28737	−0.643685	−	0.765291i	$-0.722595\pi$
$509$	−14783.6	−1.28737	−0.643685	+	0.765291i	$0.722595\pi$
$510$	0	0
$511$	0	0
$512$	−18393.8	−1.58769
$513$	0	0
$514$	10984.6	0.942622
$515$	−2823.54	−0.241592
$516$	0	0
$517$	2931.35	0.249363
$518$	0	0
$519$	0	0
$520$	17313.6	1.46010
$521$	2115.59	0.177900	0.0889499	−	0.996036i	$-0.471649\pi$
$521$	2115.59	0.177900	0.0889499	+	0.996036i	$0.471649\pi$
$522$	0	0
$523$	1268.62	0.106066	0.0530332	−	0.998593i	$-0.483111\pi$
$523$	1268.62	0.106066	0.0530332	+	0.998593i	$0.483111\pi$
$524$	45208.7	3.76899
$525$	0	0
$526$	−5083.07	−0.421355
$527$	−11314.1	−0.935201
$528$	0	0
$529$	−9080.42	−0.746316
$530$	−6503.31	−0.532992
$531$	0	0
$532$	0	0
$533$	40035.7	3.25354
$534$	0	0
$535$	−10509.5	−0.849282
$536$	−20868.5	−1.68168
$537$	0	0
$538$	−8453.06	−0.677392
$539$	0	0
$540$	0	0
$541$	17392.6	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$541$	17392.6	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$542$	39926.8	3.16421
$543$	0	0
$544$	1099.35	0.0866435
$545$	8886.10	0.698419
$546$	0	0
$547$	−7651.05	−0.598053	−0.299027	−	0.954245i	$-0.596662\pi$
$547$	−7651.05	−0.598053	−0.299027	+	0.954245i	$0.596662\pi$
$548$	24863.2	1.93815
$549$	0	0
$550$	1020.39	0.0791081
$551$	8038.13	0.621480
$552$	0	0
$553$	0	0
$554$	−8947.72	−0.686195
$555$	0	0
$556$	−29671.6	−2.26323
$557$	−259.664	−0.0197528	−0.00987641	−	0.999951i	$-0.503144\pi$
$557$	−259.664	−0.0197528	−0.00987641	+	0.999951i	$0.503144\pi$
$558$	0	0
$559$	5472.99	0.414102
$560$	0	0
$561$	0	0
$562$	−27241.9	−2.04472
$563$	24994.6	1.87104	0.935520	−	0.353274i	$-0.114932\pi$
$563$	24994.6	1.87104	0.935520	+	0.353274i	$0.114932\pi$
$564$	0	0
$565$	−4739.42	−0.352901
$566$	−35847.8	−2.66218
$567$	0	0
$568$	8490.77	0.627227
$569$	−15283.2	−1.12602	−0.563008	−	0.826451i	$-0.690356\pi$
$569$	−15283.2	−1.12602	−0.563008	+	0.826451i	$0.690356\pi$
$570$	0	0
$571$	−14452.4	−1.05922	−0.529610	−	0.848241i	$-0.677662\pi$
$571$	−14452.4	−1.05922	−0.529610	+	0.848241i	$0.677662\pi$
$572$	12152.7	0.888337
$573$	0	0
$574$	0	0
$575$	1388.92	0.100734
$576$	0	0
$577$	−9021.22	−0.650881	−0.325441	−	0.945563i	$-0.605513\pi$
$577$	−9021.22	−0.650881	−0.325441	+	0.945563i	$0.605513\pi$
$578$	−12363.0	−0.889676
$579$	0	0
$580$	−12539.2	−0.897690
$581$	0	0
$582$	0	0
$583$	−2239.50	−0.159092
$584$	12661.3	0.897140
$585$	0	0
$586$	17303.2	1.21978
$587$	11540.7	0.811472	0.405736	−	0.913990i	$-0.367015\pi$
$587$	11540.7	0.811472	0.405736	+	0.913990i	$0.367015\pi$
$588$	0	0
$589$	−11689.5	−0.817752
$590$	12550.0	0.875720
$591$	0	0
$592$	16961.0	1.17752
$593$	−2670.07	−0.184901	−0.0924507	−	0.995717i	$-0.529470\pi$
$593$	−2670.07	−0.184901	−0.0924507	+	0.995717i	$0.529470\pi$
$594$	0	0
$595$	0	0
$596$	−45397.6	−3.12007
$597$	0	0
$598$	24968.3	1.70741
$599$	−20092.2	−1.37053	−0.685264	−	0.728295i	$-0.740313\pi$
$599$	−20092.2	−1.37053	−0.685264	+	0.728295i	$0.740313\pi$
$600$	0	0
$601$	4796.65	0.325556	0.162778	−	0.986663i	$-0.447955\pi$
$601$	4796.65	0.325556	0.162778	+	0.986663i	$0.447955\pi$
$602$	0	0
$603$	0	0
$604$	36092.2	2.43141
$605$	−6303.62	−0.423601
$606$	0	0
$607$	8710.29	0.582438	0.291219	−	0.956656i	$-0.405939\pi$
$607$	8710.29	0.582438	0.291219	+	0.956656i	$0.405939\pi$
$608$	1135.82	0.0757622
$609$	0	0
$610$	−16184.4	−1.07424
$611$	32277.0	2.13713
$612$	0	0
$613$	4910.86	0.323569	0.161785	−	0.986826i	$-0.448275\pi$
$613$	4910.86	0.323569	0.161785	+	0.986826i	$0.448275\pi$
$614$	11968.1	0.786631
$615$	0	0
$616$	0	0
$617$	−21878.0	−1.42751	−0.713756	−	0.700394i	$-0.753008\pi$
$617$	−21878.0	−1.42751	−0.713756	+	0.700394i	$0.753008\pi$
$618$	0	0
$619$	−2275.14	−0.147731	−0.0738657	−	0.997268i	$-0.523534\pi$
$619$	−2275.14	−0.147731	−0.0738657	+	0.997268i	$0.523534\pi$
$620$	18235.1	1.18119
$621$	0	0
$622$	−38045.8	−2.45257
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−46920.3	−2.99571
$627$	0	0
$628$	15413.0	0.979374
$629$	−14496.4	−0.918931
$630$	0	0
$631$	21451.6	1.35337	0.676684	−	0.736274i	$-0.263417\pi$
$631$	21451.6	1.35337	0.676684	+	0.736274i	$0.263417\pi$
$632$	−11467.0	−0.721727
$633$	0	0
$634$	−43068.1	−2.69788
$635$	172.717	0.0107938
$636$	0	0
$637$	0	0
$638$	−6517.60	−0.404443
$639$	0	0
$640$	−12873.6	−0.795112
$641$	−14268.0	−0.879174	−0.439587	−	0.898200i	$-0.644875\pi$
$641$	−14268.0	−0.879174	−0.439587	+	0.898200i	$0.644875\pi$
$642$	0	0
$643$	−14862.8	−0.911555	−0.455778	−	0.890094i	$-0.650639\pi$
$643$	−14862.8	−0.911555	−0.455778	+	0.890094i	$0.650639\pi$
$644$	0	0
$645$	0	0
$646$	11940.7	0.727244
$647$	−20479.1	−1.24439	−0.622193	−	0.782864i	$-0.713758\pi$
$647$	−20479.1	−1.24439	−0.622193	+	0.782864i	$0.713758\pi$
$648$	0	0
$649$	4321.75	0.261392
$650$	11235.4	0.677985
$651$	0	0
$652$	55702.2	3.34580
$653$	−5425.53	−0.325141	−0.162571	−	0.986697i	$-0.551979\pi$
$653$	−5425.53	−0.325141	−0.162571	+	0.986697i	$0.551979\pi$
$654$	0	0
$655$	14393.2	0.858608
$656$	24724.5	1.47154
$657$	0	0
$658$	0	0
$659$	22882.3	1.35261	0.676304	−	0.736623i	$-0.263581\pi$
$659$	22882.3	1.35261	0.676304	+	0.736623i	$0.263581\pi$
$660$	0	0
$661$	−15720.0	−0.925018	−0.462509	−	0.886615i	$-0.653051\pi$
$661$	−15720.0	−0.925018	−0.462509	+	0.886615i	$0.653051\pi$
$662$	17144.6	1.00656
$663$	0	0
$664$	12147.4	0.709957
$665$	0	0
$666$	0	0
$667$	−8871.60	−0.515007
$668$	−59869.9	−3.46772
$669$	0	0
$670$	−13542.4	−0.780876
$671$	−5573.32	−0.320649
$672$	0	0
$673$	24945.2	1.42877	0.714387	−	0.699751i	$-0.246706\pi$
$673$	24945.2	1.42877	0.714387	+	0.699751i	$0.246706\pi$
$674$	−11642.3	−0.665350
$675$	0	0
$676$	99308.9	5.65026
$677$	−15431.8	−0.876059	−0.438030	−	0.898961i	$-0.644323\pi$
$677$	−15431.8	−0.876059	−0.438030	+	0.898961i	$0.644323\pi$
$678$	0	0
$679$	0	0
$680$	−9138.49	−0.515360
$681$	0	0
$682$	9478.24	0.532171
$683$	−28177.2	−1.57858	−0.789289	−	0.614021i	$-0.789551\pi$
$683$	−28177.2	−1.57858	−0.789289	+	0.614021i	$0.789551\pi$
$684$	0	0
$685$	7915.75	0.441526
$686$	0	0
$687$	0	0
$688$	3379.91	0.187293
$689$	−24659.0	−1.36347
$690$	0	0
$691$	−5614.31	−0.309086	−0.154543	−	0.987986i	$-0.549391\pi$
$691$	−5614.31	−0.309086	−0.154543	+	0.987986i	$0.549391\pi$
$692$	−13150.9	−0.722430
$693$	0	0
$694$	−36543.1	−1.99879
$695$	−9446.59	−0.515582
$696$	0	0
$697$	−21131.7	−1.14838
$698$	−4198.28	−0.227661
$699$	0	0
$700$	0	0
$701$	−5331.75	−0.287272	−0.143636	−	0.989631i	$-0.545879\pi$
$701$	−5331.75	−0.287272	−0.143636	+	0.989631i	$0.545879\pi$
$702$	0	0
$703$	−14977.3	−0.803525
$704$	−4743.98	−0.253971
$705$	0	0
$706$	7761.03	0.413725
$707$	0	0
$708$	0	0
$709$	−31881.9	−1.68879	−0.844393	−	0.535725i	$-0.820039\pi$
$709$	−31881.9	−1.68879	−0.844393	+	0.535725i	$0.820039\pi$
$710$	5509.98	0.291247
$711$	0	0
$712$	−20366.9	−1.07202
$713$	12901.5	0.677653
$714$	0	0
$715$	3869.07	0.202371
$716$	35330.9	1.84410
$717$	0	0
$718$	−931.143	−0.0483983
$719$	−21491.4	−1.11473	−0.557367	−	0.830266i	$-0.688188\pi$
$719$	−21491.4	−1.11473	−0.557367	+	0.830266i	$0.688188\pi$
$720$	0	0
$721$	0	0
$722$	−21058.1	−1.08546
$723$	0	0
$724$	57158.8	2.93410
$725$	−3992.12	−0.204501
$726$	0	0
$727$	−15984.6	−0.815454	−0.407727	−	0.913104i	$-0.633679\pi$
$727$	−15984.6	−0.815454	−0.407727	+	0.913104i	$0.633679\pi$
$728$	0	0
$729$	0	0
$730$	8216.42	0.416580
$731$	−2888.76	−0.146162
$732$	0	0
$733$	−9051.99	−0.456129	−0.228065	−	0.973646i	$-0.573240\pi$
$733$	−9051.99	−0.456129	−0.228065	+	0.973646i	$0.573240\pi$
$734$	11640.1	0.585348
$735$	0	0
$736$	−1253.59	−0.0627824
$737$	−4663.48	−0.233082
$738$	0	0
$739$	−24946.7	−1.24179	−0.620893	−	0.783895i	$-0.713230\pi$
$739$	−24946.7	−1.24179	−0.620893	+	0.783895i	$0.713230\pi$
$740$	23363.9	1.16064
$741$	0	0
$742$	0	0
$743$	32100.6	1.58500	0.792500	−	0.609872i	$-0.208779\pi$
$743$	32100.6	1.58500	0.792500	+	0.609872i	$0.208779\pi$
$744$	0	0
$745$	−14453.3	−0.710777
$746$	−7.08787	−0.000347863 0
$747$	0	0
$748$	−6414.44	−0.313550
$749$	0	0
$750$	0	0
$751$	2604.17	0.126535	0.0632673	−	0.997997i	$-0.479848\pi$
$751$	2604.17	0.126535	0.0632673	+	0.997997i	$0.479848\pi$
$752$	19933.0	0.966599
$753$	0	0
$754$	−71765.1	−3.46622
$755$	11490.7	0.553896
$756$	0	0
$757$	31576.5	1.51607	0.758037	−	0.652212i	$-0.226159\pi$
$757$	31576.5	1.51607	0.758037	+	0.652212i	$0.226159\pi$
$758$	−30467.4	−1.45993
$759$	0	0
$760$	−9441.65	−0.450638
$761$	−22563.4	−1.07480	−0.537400	−	0.843328i	$-0.680593\pi$
$761$	−22563.4	−1.07480	−0.537400	+	0.843328i	$0.680593\pi$
$762$	0	0
$763$	0	0
$764$	−47133.6	−2.23198
$765$	0	0
$766$	−42090.6	−1.98537
$767$	47586.6	2.24023
$768$	0	0
$769$	8008.29	0.375535	0.187767	−	0.982214i	$-0.439875\pi$
$769$	8008.29	0.375535	0.187767	+	0.982214i	$0.439875\pi$
$770$	0	0
$771$	0	0
$772$	−66062.4	−3.07984
$773$	−31038.9	−1.44423	−0.722117	−	0.691771i	$-0.756831\pi$
$773$	−31038.9	−1.44423	−0.722117	+	0.691771i	$0.756831\pi$
$774$	0	0
$775$	5805.54	0.269086
$776$	53637.5	2.48128
$777$	0	0
$778$	−214.724	−0.00989491
$779$	−21832.7	−1.00416
$780$	0	0
$781$	1897.43	0.0869340
$782$	−13178.8	−0.602651
$783$	0	0
$784$	0	0
$785$	4907.07	0.223110
$786$	0	0
$787$	31693.7	1.43553	0.717764	−	0.696287i	$-0.245166\pi$
$787$	31693.7	1.43553	0.717764	+	0.696287i	$0.245166\pi$
$788$	21983.8	0.993834
$789$	0	0
$790$	−7441.35	−0.335128
$791$	0	0
$792$	0	0
$793$	−61367.6	−2.74808
$794$	−22716.3	−1.01533
$795$	0	0
$796$	−9350.96	−0.416377
$797$	−24049.7	−1.06886	−0.534432	−	0.845211i	$-0.679475\pi$
$797$	−24049.7	−1.06886	−0.534432	+	0.845211i	$0.679475\pi$
$798$	0	0
$799$	−17036.5	−0.754327
$800$	−564.100	−0.0249299
$801$	0	0
$802$	−57438.9	−2.52897
$803$	2829.43	0.124344
$804$	0	0
$805$	0	0
$806$	104365.	4.56090
$807$	0	0
$808$	8234.81	0.358539
$809$	11709.6	0.508884	0.254442	−	0.967088i	$-0.418108\pi$
$809$	11709.6	0.508884	0.254442	+	0.967088i	$0.418108\pi$
$810$	0	0
$811$	−18961.4	−0.820991	−0.410495	−	0.911863i	$-0.634644\pi$
$811$	−18961.4	−0.820991	−0.410495	+	0.911863i	$0.634644\pi$
$812$	0	0
$813$	0	0
$814$	12144.1	0.522913
$815$	17734.0	0.762202
$816$	0	0
$817$	−2984.60	−0.127806
$818$	−63444.8	−2.71185
$819$	0	0
$820$	34058.2	1.45044
$821$	−36561.4	−1.55420	−0.777102	−	0.629375i	$-0.783311\pi$
$821$	−36561.4	−1.55420	−0.777102	+	0.629375i	$0.783311\pi$
$822$	0	0
$823$	11532.0	0.488433	0.244217	−	0.969721i	$-0.421469\pi$
$823$	11532.0	0.488433	0.244217	+	0.969721i	$0.421469\pi$
$824$	−21184.1	−0.895610
$825$	0	0
$826$	0	0
$827$	20058.3	0.843403	0.421702	−	0.906735i	$-0.361433\pi$
$827$	20058.3	0.843403	0.421702	+	0.906735i	$0.361433\pi$
$828$	0	0
$829$	18879.7	0.790975	0.395487	−	0.918471i	$-0.370576\pi$
$829$	18879.7	0.790975	0.395487	+	0.918471i	$0.370576\pi$
$830$	7882.91	0.329663
$831$	0	0
$832$	−52235.8	−2.17662
$833$	0	0
$834$	0	0
$835$	−19060.9	−0.789975
$836$	−6627.24	−0.274172
$837$	0	0
$838$	−34035.3	−1.40302
$839$	37711.0	1.55176	0.775880	−	0.630880i	$-0.217306\pi$
$839$	37711.0	1.55176	0.775880	+	0.630880i	$0.217306\pi$
$840$	0	0
$841$	1110.19	0.0455200
$842$	14823.3	0.606705
$843$	0	0
$844$	75518.9	3.07994
$845$	31617.2	1.28718
$846$	0	0
$847$	0	0
$848$	−15228.5	−0.616684
$849$	0	0
$850$	−5930.31	−0.239303
$851$	16530.3	0.665864
$852$	0	0
$853$	10230.5	0.410651	0.205326	−	0.978694i	$-0.434175\pi$
$853$	10230.5	0.410651	0.205326	+	0.978694i	$0.434175\pi$
$854$	0	0
$855$	0	0
$856$	−78849.5	−3.14839
$857$	10261.1	0.409000	0.204500	−	0.978867i	$-0.434443\pi$
$857$	10261.1	0.409000	0.204500	+	0.978867i	$0.434443\pi$
$858$	0	0
$859$	42672.7	1.69496	0.847481	−	0.530826i	$-0.178118\pi$
$859$	42672.7	1.69496	0.847481	+	0.530826i	$0.178118\pi$
$860$	4655.85	0.184608
$861$	0	0
$862$	13238.9	0.523109
$863$	13931.6	0.549521	0.274760	−	0.961513i	$-0.411401\pi$
$863$	13931.6	0.549521	0.274760	+	0.961513i	$0.411401\pi$
$864$	0	0
$865$	−4186.87	−0.164576
$866$	35945.3	1.41047
$867$	0	0
$868$	0	0
$869$	−2562.52	−0.100032
$870$	0	0
$871$	−51349.4	−1.99760
$872$	66669.5	2.58912
$873$	0	0
$874$	−13616.0	−0.526966
$875$	0	0
$876$	0	0
$877$	−14810.0	−0.570238	−0.285119	−	0.958492i	$-0.592033\pi$
$877$	−14810.0	−0.570238	−0.285119	+	0.958492i	$0.592033\pi$
$878$	−2745.90	−0.105546
$879$	0	0
$880$	2389.39	0.0915299
$881$	12814.7	0.490054	0.245027	−	0.969516i	$-0.421203\pi$
$881$	12814.7	0.490054	0.245027	+	0.969516i	$0.421203\pi$
$882$	0	0
$883$	−25182.6	−0.959753	−0.479877	−	0.877336i	$-0.659319\pi$
$883$	−25182.6	−0.959753	−0.479877	+	0.877336i	$0.659319\pi$
$884$	−70629.2	−2.68724
$885$	0	0
$886$	33519.7	1.27101
$887$	−3364.76	−0.127370	−0.0636851	−	0.997970i	$-0.520285\pi$
$887$	−3364.76	−0.127370	−0.0636851	+	0.997970i	$0.520285\pi$
$888$	0	0
$889$	0	0
$890$	−13216.8	−0.497785
$891$	0	0
$892$	−3857.18	−0.144785
$893$	−17601.7	−0.659593
$894$	0	0
$895$	11248.3	0.420102
$896$	0	0
$897$	0	0
$898$	6499.73	0.241536
$899$	−37082.3	−1.37571
$900$	0	0
$901$	13015.6	0.481256
$902$	17702.8	0.653478
$903$	0	0
$904$	−35558.3	−1.30824
$905$	18197.7	0.668413
$906$	0	0
$907$	−48494.8	−1.77535	−0.887676	−	0.460469i	$-0.847681\pi$
$907$	−48494.8	−1.77535	−0.887676	+	0.460469i	$0.847681\pi$
$908$	−48033.4	−1.75555
$909$	0	0
$910$	0	0
$911$	24561.0	0.893240	0.446620	−	0.894724i	$-0.352628\pi$
$911$	24561.0	0.893240	0.446620	+	0.894724i	$0.352628\pi$
$912$	0	0
$913$	2714.58	0.0984005
$914$	25036.5	0.906055
$915$	0	0
$916$	25601.4	0.923465
$917$	0	0
$918$	0	0
$919$	−20884.8	−0.749647	−0.374823	−	0.927096i	$-0.622297\pi$
$919$	−20884.8	−0.749647	−0.374823	+	0.927096i	$0.622297\pi$
$920$	10420.7	0.373434
$921$	0	0
$922$	9034.34	0.322701
$923$	20892.5	0.745056
$924$	0	0
$925$	7438.42	0.264404
$926$	7539.25	0.267554
$927$	0	0
$928$	3603.13	0.127455
$929$	6728.63	0.237631	0.118816	−	0.992916i	$-0.462090\pi$
$929$	6728.63	0.237631	0.118816	+	0.992916i	$0.462090\pi$
$930$	0	0
$931$	0	0
$932$	24539.1	0.862453
$933$	0	0
$934$	−61226.3	−2.14495
$935$	−2042.18	−0.0714293
$936$	0	0
$937$	−8765.60	−0.305613	−0.152807	−	0.988256i	$-0.548831\pi$
$937$	−8765.60	−0.305613	−0.152807	+	0.988256i	$0.548831\pi$
$938$	0	0
$939$	0	0
$940$	27457.9	0.952742
$941$	1016.96	0.0352304	0.0176152	−	0.999845i	$-0.494393\pi$
$941$	1016.96	0.0352304	0.0176152	+	0.999845i	$0.494393\pi$
$942$	0	0
$943$	24096.5	0.832122
$944$	29387.7	1.01323
$945$	0	0
$946$	2420.02	0.0831729
$947$	−37392.6	−1.28310	−0.641550	−	0.767081i	$-0.721708\pi$
$947$	−37392.6	−1.28310	−0.641550	+	0.767081i	$0.721708\pi$
$948$	0	0
$949$	31154.7	1.06568
$950$	−6127.04	−0.209250
$951$	0	0
$952$	0	0
$953$	−10556.6	−0.358827	−0.179414	−	0.983774i	$-0.557420\pi$
$953$	−10556.6	−0.358827	−0.179414	+	0.983774i	$0.557420\pi$
$954$	0	0
$955$	−15006.0	−0.508464
$956$	14561.7	0.492636
$957$	0	0
$958$	48193.8	1.62534
$959$	0	0
$960$	0	0
$961$	24136.0	0.810176
$962$	133718.	4.48155
$963$	0	0
$964$	−69399.5	−2.31868
$965$	−21032.4	−0.701614
$966$	0	0
$967$	29776.6	0.990227	0.495114	−	0.868828i	$-0.335126\pi$
$967$	29776.6	0.990227	0.495114	+	0.868828i	$0.335126\pi$
$968$	−47294.0	−1.57034
$969$	0	0
$970$	34807.4	1.15216
$971$	−11807.7	−0.390245	−0.195123	−	0.980779i	$-0.562510\pi$
$971$	−11807.7	−0.390245	−0.195123	+	0.980779i	$0.562510\pi$
$972$	0	0
$973$	0	0
$974$	74853.1	2.46247
$975$	0	0
$976$	−37898.3	−1.24293
$977$	−34143.3	−1.11806	−0.559029	−	0.829148i	$-0.688826\pi$
$977$	−34143.3	−1.11806	−0.559029	+	0.829148i	$0.688826\pi$
$978$	0	0
$979$	−4551.39	−0.148583
$980$	0	0
$981$	0	0
$982$	36854.8	1.19764
$983$	12321.6	0.399794	0.199897	−	0.979817i	$-0.435939\pi$
$983$	12321.6	0.399794	0.199897	+	0.979817i	$0.435939\pi$
$984$	0	0
$985$	6999.03	0.226404
$986$	37879.2	1.22345
$987$	0	0
$988$	−72972.3	−2.34975
$989$	3294.07	0.105910
$990$	0	0
$991$	−4566.58	−0.146380	−0.0731898	−	0.997318i	$-0.523318\pi$
$991$	−4566.58	−0.146380	−0.0731898	+	0.997318i	$0.523318\pi$
$992$	−5239.85	−0.167707
$993$	0	0
$994$	0	0
$995$	−2977.08	−0.0948541
$996$	0	0
$997$	20965.2	0.665971	0.332986	−	0.942932i	$-0.391944\pi$
$997$	20965.2	0.665971	0.332986	+	0.942932i	$0.391944\pi$
$998$	−9638.29	−0.305706
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ce.1.8		8
3.2	odd	2		735.4.a.bb.1.1	✓	8
7.6	odd	2		2205.4.a.cd.1.8		8
21.20	even	2		735.4.a.bc.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.bb.1.1	✓	8	3.2	odd	2
735.4.a.bc.1.1	yes	8	21.20	even	2
2205.4.a.cd.1.8		8	7.6	odd	2
2205.4.a.ce.1.8		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)

\(f(q)\)	\(=\)	\(q+4.86877 q^{2} +15.7049 q^{4} +5.00000 q^{5} +37.5134 q^{8} +24.3438 q^{10} +8.38312 q^{11} +92.3062 q^{13} +57.0048 q^{16} -48.7212 q^{17} -50.3375 q^{19} +78.5245 q^{20} +40.8155 q^{22} +55.5570 q^{23} +25.0000 q^{25} +449.417 q^{26} -159.685 q^{29} +232.222 q^{31} -22.5640 q^{32} -237.212 q^{34} +297.537 q^{37} -245.082 q^{38} +187.567 q^{40} +433.727 q^{41} +59.2917 q^{43} +131.656 q^{44} +270.494 q^{46} +349.673 q^{47} +121.719 q^{50} +1449.66 q^{52} -267.144 q^{53} +41.9156 q^{55} -777.468 q^{58} +515.530 q^{59} -664.827 q^{61} +1130.63 q^{62} -565.897 q^{64} +461.531 q^{65} -556.295 q^{67} -765.162 q^{68} +226.340 q^{71} +337.515 q^{73} +1448.64 q^{74} -790.546 q^{76} -305.677 q^{79} +285.024 q^{80} +2111.71 q^{82} +323.816 q^{83} -243.606 q^{85} +288.678 q^{86} +314.479 q^{88} -542.923 q^{89} +872.517 q^{92} +1702.48 q^{94} -251.688 q^{95} +1429.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 50 q^{4} + 40 q^{5} - 66 q^{8} - 10 q^{10} - 64 q^{11} + 206 q^{16} + 48 q^{17} - 80 q^{19} + 250 q^{20} + 452 q^{22} - 120 q^{23} + 200 q^{25} + 272 q^{26} - 76 q^{29} - 20 q^{31} - 770 q^{32}+ \cdots + 2052 q^{97}+O(q^{100}) \) 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

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