LMFDB - Embedded newform 2205.4.a.ce.1.7

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			1.7
Root			$-4.22998$ 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)

$f(q)$	$=$	$q+4.22998 q^{2} +9.89271 q^{4} +5.00000 q^{5} +8.00613 q^{8} +O(q^{10})$	$q+4.22998 q^{2} +9.89271 q^{4} +5.00000 q^{5} +8.00613 q^{8} +21.1499 q^{10} +65.5972 q^{11} -7.45468 q^{13} -45.2759 q^{16} +73.6871 q^{17} +140.023 q^{19} +49.4636 q^{20} +277.475 q^{22} -58.4492 q^{23} +25.0000 q^{25} -31.5331 q^{26} +22.2547 q^{29} -283.524 q^{31} -255.565 q^{32} +311.695 q^{34} -264.051 q^{37} +592.293 q^{38} +40.0306 q^{40} +392.992 q^{41} +231.853 q^{43} +648.934 q^{44} -247.239 q^{46} +90.0518 q^{47} +105.749 q^{50} -73.7470 q^{52} +199.330 q^{53} +327.986 q^{55} +94.1369 q^{58} +425.005 q^{59} -423.758 q^{61} -1199.30 q^{62} -718.828 q^{64} -37.2734 q^{65} -57.8548 q^{67} +728.965 q^{68} -201.982 q^{71} +380.180 q^{73} -1116.93 q^{74} +1385.20 q^{76} -927.621 q^{79} -226.380 q^{80} +1662.35 q^{82} +152.787 q^{83} +368.435 q^{85} +980.732 q^{86} +525.179 q^{88} +552.480 q^{89} -578.221 q^{92} +380.917 q^{94} +700.114 q^{95} +254.755 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.22998	1.49552	0.747761	−	0.663967i	$-0.231129\pi$
$2$	4.22998	1.49552	0.747761	+	0.663967i	$0.231129\pi$
$3$	0	0
$3$	0	0
$4$	9.89271	1.23659
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00613	0.353824
$9$	0	0
$10$	21.1499	0.668818
$11$	65.5972	1.79803	0.899013	−	0.437922i	$-0.144285\pi$
$11$	65.5972	1.79803	0.899013	+	0.437922i	$0.144285\pi$
$12$	0	0
$13$	−7.45468	−0.159043	−0.0795214	−	0.996833i	$-0.525339\pi$
$13$	−7.45468	−0.159043	−0.0795214	+	0.996833i	$0.525339\pi$
$14$	0	0
$15$	0	0
$16$	−45.2759	−0.707437
$17$	73.6871	1.05128	0.525639	−	0.850708i	$-0.323826\pi$
$17$	73.6871	1.05128	0.525639	+	0.850708i	$0.323826\pi$
$18$	0	0
$19$	140.023	1.69071	0.845353	−	0.534208i	$-0.179390\pi$
$19$	140.023	1.69071	0.845353	+	0.534208i	$0.179390\pi$
$20$	49.4636	0.553019
$21$	0	0
$22$	277.475	2.68899
$23$	−58.4492	−0.529891	−0.264946	−	0.964263i	$-0.585354\pi$
$23$	−58.4492	−0.529891	−0.264946	+	0.964263i	$0.585354\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−31.5331	−0.237852
$27$	0	0
$28$	0	0
$29$	22.2547	0.142503	0.0712516	−	0.997458i	$-0.477301\pi$
$29$	22.2547	0.142503	0.0712516	+	0.997458i	$0.477301\pi$
$30$	0	0
$31$	−283.524	−1.64266	−0.821329	−	0.570455i	$-0.806767\pi$
$31$	−283.524	−1.64266	−0.821329	+	0.570455i	$0.806767\pi$
$32$	−255.565	−1.41181
$33$	0	0
$34$	311.695	1.57221
$35$	0	0
$36$	0	0
$37$	−264.051	−1.17324	−0.586618	−	0.809864i	$-0.699541\pi$
$37$	−264.051	−1.17324	−0.586618	+	0.809864i	$0.699541\pi$
$38$	592.293	2.52849
$39$	0	0
$40$	40.0306	0.158235
$41$	392.992	1.49695	0.748476	−	0.663161i	$-0.230786\pi$
$41$	392.992	1.49695	0.748476	+	0.663161i	$0.230786\pi$
$42$	0	0
$43$	231.853	0.822261	0.411130	−	0.911577i	$-0.365134\pi$
$43$	231.853	0.822261	0.411130	+	0.911577i	$0.365134\pi$
$44$	648.934	2.22342
$45$	0	0
$46$	−247.239	−0.792464
$47$	90.0518	0.279477	0.139738	−	0.990188i	$-0.455374\pi$
$47$	90.0518	0.279477	0.139738	+	0.990188i	$0.455374\pi$
$48$	0	0
$49$	0	0
$50$	105.749	0.299105
$51$	0	0
$52$	−73.7470	−0.196671
$53$	199.330	0.516606	0.258303	−	0.966064i	$-0.416837\pi$
$53$	199.330	0.516606	0.258303	+	0.966064i	$0.416837\pi$
$54$	0	0
$55$	327.986	0.804102
$56$	0	0
$57$	0	0
$58$	94.1369	0.213117
$59$	425.005	0.937813	0.468907	−	0.883248i	$-0.344648\pi$
$59$	425.005	0.937813	0.468907	+	0.883248i	$0.344648\pi$
$60$	0	0
$61$	−423.758	−0.889454	−0.444727	−	0.895666i	$-0.646699\pi$
$61$	−423.758	−0.889454	−0.444727	+	0.895666i	$0.646699\pi$
$62$	−1199.30	−2.45663
$63$	0	0
$64$	−718.828	−1.40396
$65$	−37.2734	−0.0711261
$66$	0	0
$67$	−57.8548	−0.105494	−0.0527470	−	0.998608i	$-0.516798\pi$
$67$	−57.8548	−0.105494	−0.0527470	+	0.998608i	$0.516798\pi$
$68$	728.965	1.30000
$69$	0	0
$70$	0	0
$71$	−201.982	−0.337618	−0.168809	−	0.985649i	$-0.553992\pi$
$71$	−201.982	−0.337618	−0.168809	+	0.985649i	$0.553992\pi$
$72$	0	0
$73$	380.180	0.609544	0.304772	−	0.952425i	$-0.401420\pi$
$73$	380.180	0.609544	0.304772	+	0.952425i	$0.401420\pi$
$74$	−1116.93	−1.75460
$75$	0	0
$76$	1385.20	2.09071
$77$	0	0
$78$	0	0
$79$	−927.621	−1.32108	−0.660541	−	0.750790i	$-0.729673\pi$
$79$	−927.621	−1.32108	−0.660541	+	0.750790i	$0.729673\pi$
$80$	−226.380	−0.316375
$81$	0	0
$82$	1662.35	2.23873
$83$	152.787	0.202054	0.101027	−	0.994884i	$-0.467787\pi$
$83$	152.787	0.202054	0.101027	+	0.994884i	$0.467787\pi$
$84$	0	0
$85$	368.435	0.470146
$86$	980.732	1.22971
$87$	0	0
$88$	525.179	0.636185
$89$	552.480	0.658008	0.329004	−	0.944328i	$-0.393287\pi$
$89$	552.480	0.658008	0.329004	+	0.944328i	$0.393287\pi$
$90$	0	0
$91$	0	0
$92$	−578.221	−0.655257
$93$	0	0
$94$	380.917	0.417964
$95$	700.114	0.756107
$96$	0	0
$97$	254.755	0.266664	0.133332	−	0.991071i	$-0.457432\pi$
$97$	254.755	0.266664	0.133332	+	0.991071i	$0.457432\pi$
$98$	0	0
$99$	0	0
$100$	247.318	0.247318
$101$	1602.19	1.57845	0.789226	−	0.614103i	$-0.210482\pi$
$101$	1602.19	1.57845	0.789226	+	0.614103i	$0.210482\pi$
$102$	0	0
$103$	1158.38	1.10814	0.554072	−	0.832469i	$-0.313073\pi$
$103$	1158.38	1.10814	0.554072	+	0.832469i	$0.313073\pi$
$104$	−59.6831	−0.0562732
$105$	0	0
$106$	843.163	0.772596
$107$	1497.05	1.35257	0.676286	−	0.736639i	$-0.263589\pi$
$107$	1497.05	1.35257	0.676286	+	0.736639i	$0.263589\pi$
$108$	0	0
$109$	1177.24	1.03449	0.517243	−	0.855839i	$-0.326958\pi$
$109$	1177.24	1.03449	0.517243	+	0.855839i	$0.326958\pi$
$110$	1387.37	1.20255
$111$	0	0
$112$	0	0
$113$	−1074.13	−0.894207	−0.447103	−	0.894482i	$-0.647544\pi$
$113$	−1074.13	−0.894207	−0.447103	+	0.894482i	$0.647544\pi$
$114$	0	0
$115$	−292.246	−0.236974
$116$	220.159	0.176218
$117$	0	0
$118$	1797.76	1.40252
$119$	0	0
$120$	0	0
$121$	2971.99	2.23290
$122$	−1792.49	−1.33020
$123$	0	0
$124$	−2804.82	−2.03129
$125$	125.000	0.0894427
$126$	0	0
$127$	2219.33	1.55066	0.775330	−	0.631556i	$-0.217584\pi$
$127$	2219.33	1.55066	0.775330	+	0.631556i	$0.217584\pi$
$128$	−996.104	−0.687843
$129$	0	0
$130$	−157.666	−0.106371
$131$	220.241	0.146889	0.0734447	−	0.997299i	$-0.476601\pi$
$131$	220.241	0.146889	0.0734447	+	0.997299i	$0.476601\pi$
$132$	0	0
$133$	0	0
$134$	−244.725	−0.157769
$135$	0	0
$136$	589.948	0.371968
$137$	−2812.97	−1.75422	−0.877109	−	0.480291i	$-0.840531\pi$
$137$	−2812.97	−1.75422	−0.877109	+	0.480291i	$0.840531\pi$
$138$	0	0
$139$	1199.68	0.732055	0.366027	−	0.930604i	$-0.380718\pi$
$139$	1199.68	0.732055	0.366027	+	0.930604i	$0.380718\pi$
$140$	0	0
$141$	0	0
$142$	−854.381	−0.504916
$143$	−489.006	−0.285963
$144$	0	0
$145$	111.274	0.0637294
$146$	1608.15	0.911588
$147$	0	0
$148$	−2612.18	−1.45081
$149$	−2096.19	−1.15253	−0.576263	−	0.817265i	$-0.695489\pi$
$149$	−2096.19	−1.15253	−0.576263	+	0.817265i	$0.695489\pi$
$150$	0	0
$151$	−645.643	−0.347958	−0.173979	−	0.984749i	$-0.555662\pi$
$151$	−645.643	−0.347958	−0.173979	+	0.984749i	$0.555662\pi$
$152$	1121.04	0.598213
$153$	0	0
$154$	0	0
$155$	−1417.62	−0.734619
$156$	0	0
$157$	2220.44	1.12873	0.564366	−	0.825525i	$-0.309121\pi$
$157$	2220.44	1.12873	0.564366	+	0.825525i	$0.309121\pi$
$158$	−3923.81	−1.97571
$159$	0	0
$160$	−1277.83	−0.631382
$161$	0	0
$162$	0	0
$163$	3116.88	1.49775	0.748875	−	0.662712i	$-0.230595\pi$
$163$	3116.88	1.49775	0.748875	+	0.662712i	$0.230595\pi$
$164$	3887.76	1.85112
$165$	0	0
$166$	646.284	0.302177
$167$	3586.15	1.66170	0.830851	−	0.556495i	$-0.187854\pi$
$167$	3586.15	1.66170	0.830851	+	0.556495i	$0.187854\pi$
$168$	0	0
$169$	−2141.43	−0.974705
$170$	1558.47	0.703114
$171$	0	0
$172$	2293.65	1.01680
$173$	2076.12	0.912395	0.456197	−	0.889879i	$-0.349211\pi$
$173$	2076.12	0.912395	0.456197	+	0.889879i	$0.349211\pi$
$174$	0	0
$175$	0	0
$176$	−2969.97	−1.27199
$177$	0	0
$178$	2336.98	0.984067
$179$	−1343.20	−0.560869	−0.280434	−	0.959873i	$-0.590478\pi$
$179$	−1343.20	−0.560869	−0.280434	+	0.959873i	$0.590478\pi$
$180$	0	0
$181$	−372.433	−0.152943	−0.0764717	−	0.997072i	$-0.524365\pi$
$181$	−372.433	−0.152943	−0.0764717	+	0.997072i	$0.524365\pi$
$182$	0	0
$183$	0	0
$184$	−467.952	−0.187488
$185$	−1320.25	−0.524687
$186$	0	0
$187$	4833.66	1.89023
$188$	890.857	0.345598
$189$	0	0
$190$	2961.47	1.13078
$191$	−2597.37	−0.983973	−0.491987	−	0.870603i	$-0.663729\pi$
$191$	−2597.37	−0.983973	−0.491987	+	0.870603i	$0.663729\pi$
$192$	0	0
$193$	4416.53	1.64719	0.823597	−	0.567175i	$-0.191964\pi$
$193$	4416.53	1.64719	0.823597	+	0.567175i	$0.191964\pi$
$194$	1077.61	0.398803
$195$	0	0
$196$	0	0
$197$	−4859.63	−1.75753	−0.878767	−	0.477252i	$-0.841633\pi$
$197$	−4859.63	−1.75753	−0.878767	+	0.477252i	$0.841633\pi$
$198$	0	0
$199$	−2851.29	−1.01569	−0.507845	−	0.861449i	$-0.669558\pi$
$199$	−2851.29	−1.01569	−0.507845	+	0.861449i	$0.669558\pi$
$200$	200.153	0.0707649
$201$	0	0
$202$	6777.22	2.36061
$203$	0	0
$204$	0	0
$205$	1964.96	0.669458
$206$	4899.93	1.65725
$207$	0	0
$208$	337.518	0.112513
$209$	9185.09	3.03993
$210$	0	0
$211$	−5859.67	−1.91183	−0.955915	−	0.293643i	$-0.905132\pi$
$211$	−5859.67	−1.91183	−0.955915	+	0.293643i	$0.905132\pi$
$212$	1971.92	0.638829
$213$	0	0
$214$	6332.48	2.02280
$215$	1159.26	0.367726
$216$	0	0
$217$	0	0
$218$	4979.69	1.54710
$219$	0	0
$220$	3244.67	0.994343
$221$	−549.314	−0.167198
$222$	0	0
$223$	−4695.81	−1.41011	−0.705055	−	0.709153i	$-0.749078\pi$
$223$	−4695.81	−1.41011	−0.705055	+	0.709153i	$0.749078\pi$
$224$	0	0
$225$	0	0
$226$	−4543.53	−1.33731
$227$	−5144.16	−1.50410	−0.752049	−	0.659107i	$-0.770934\pi$
$227$	−5144.16	−1.50410	−0.752049	+	0.659107i	$0.770934\pi$
$228$	0	0
$229$	1322.64	0.381671	0.190836	−	0.981622i	$-0.438880\pi$
$229$	1322.64	0.381671	0.190836	+	0.981622i	$0.438880\pi$
$230$	−1236.19	−0.354401
$231$	0	0
$232$	178.174	0.0504211
$233$	2910.11	0.818230	0.409115	−	0.912483i	$-0.365837\pi$
$233$	2910.11	0.818230	0.409115	+	0.912483i	$0.365837\pi$
$234$	0	0
$235$	450.259	0.124986
$236$	4204.46	1.15969
$237$	0	0
$238$	0	0
$239$	−1589.34	−0.430151	−0.215076	−	0.976597i	$-0.569000\pi$
$239$	−1589.34	−0.430151	−0.215076	+	0.976597i	$0.569000\pi$
$240$	0	0
$241$	2419.85	0.646790	0.323395	−	0.946264i	$-0.395176\pi$
$241$	2419.85	0.646790	0.323395	+	0.946264i	$0.395176\pi$
$242$	12571.4	3.33935
$243$	0	0
$244$	−4192.12	−1.09989
$245$	0	0
$246$	0	0
$247$	−1043.83	−0.268895
$248$	−2269.93	−0.581212
$249$	0	0
$250$	528.747	0.133764
$251$	−5504.73	−1.38428	−0.692142	−	0.721761i	$-0.743333\pi$
$251$	−5504.73	−1.38428	−0.692142	+	0.721761i	$0.743333\pi$
$252$	0	0
$253$	−3834.10	−0.952758
$254$	9387.72	2.31905
$255$	0	0
$256$	1537.13	0.375275
$257$	−3103.29	−0.753221	−0.376610	−	0.926372i	$-0.622910\pi$
$257$	−3103.29	−0.753221	−0.376610	+	0.926372i	$0.622910\pi$
$258$	0	0
$259$	0	0
$260$	−368.735	−0.0879538
$261$	0	0
$262$	931.613	0.219677
$263$	5358.16	1.25627	0.628134	−	0.778106i	$-0.283819\pi$
$263$	5358.16	1.25627	0.628134	+	0.778106i	$0.283819\pi$
$264$	0	0
$265$	996.651	0.231033
$266$	0	0
$267$	0	0
$268$	−572.341	−0.130453
$269$	−3005.31	−0.681179	−0.340590	−	0.940212i	$-0.610627\pi$
$269$	−3005.31	−0.681179	−0.340590	+	0.940212i	$0.610627\pi$
$270$	0	0
$271$	−383.108	−0.0858751	−0.0429375	−	0.999078i	$-0.513672\pi$
$271$	−383.108	−0.0858751	−0.0429375	+	0.999078i	$0.513672\pi$
$272$	−3336.25	−0.743713
$273$	0	0
$274$	−11898.8	−2.62347
$275$	1639.93	0.359605
$276$	0	0
$277$	−4135.68	−0.897072	−0.448536	−	0.893765i	$-0.648054\pi$
$277$	−4135.68	−0.897072	−0.448536	+	0.893765i	$0.648054\pi$
$278$	5074.62	1.09480
$279$	0	0
$280$	0	0
$281$	−8131.23	−1.72622	−0.863111	−	0.505015i	$-0.831487\pi$
$281$	−8131.23	−1.72622	−0.863111	+	0.505015i	$0.831487\pi$
$282$	0	0
$283$	−6230.29	−1.30867	−0.654333	−	0.756206i	$-0.727051\pi$
$283$	−6230.29	−1.30867	−0.654333	+	0.756206i	$0.727051\pi$
$284$	−1998.15	−0.417495
$285$	0	0
$286$	−2068.48	−0.427664
$287$	0	0
$288$	0	0
$289$	516.783	0.105187
$290$	470.685	0.0953088
$291$	0	0
$292$	3761.01	0.753756
$293$	−3709.25	−0.739578	−0.369789	−	0.929116i	$-0.620570\pi$
$293$	−3709.25	−0.739578	−0.369789	+	0.929116i	$0.620570\pi$
$294$	0	0
$295$	2125.03	0.419403
$296$	−2114.03	−0.415119
$297$	0	0
$298$	−8866.82	−1.72363
$299$	435.720	0.0842754
$300$	0	0
$301$	0	0
$302$	−2731.05	−0.520379
$303$	0	0
$304$	−6339.66	−1.19607
$305$	−2118.79	−0.397776
$306$	0	0
$307$	−211.395	−0.0392995	−0.0196498	−	0.999807i	$-0.506255\pi$
$307$	−211.395	−0.0392995	−0.0196498	+	0.999807i	$0.506255\pi$
$308$	0	0
$309$	0	0
$310$	−5996.50	−1.09864
$311$	2412.08	0.439797	0.219898	−	0.975523i	$-0.429427\pi$
$311$	2412.08	0.439797	0.219898	+	0.975523i	$0.429427\pi$
$312$	0	0
$313$	2732.09	0.493376	0.246688	−	0.969095i	$-0.420658\pi$
$313$	2732.09	0.493376	0.246688	+	0.969095i	$0.420658\pi$
$314$	9392.43	1.68804
$315$	0	0
$316$	−9176.68	−1.63364
$317$	4310.17	0.763669	0.381834	−	0.924231i	$-0.375292\pi$
$317$	4310.17	0.763669	0.381834	+	0.924231i	$0.375292\pi$
$318$	0	0
$319$	1459.85	0.256225
$320$	−3594.14	−0.627870
$321$	0	0
$322$	0	0
$323$	10317.9	1.77740
$324$	0	0
$325$	−186.367	−0.0318086
$326$	13184.3	2.23992
$327$	0	0
$328$	3146.35	0.529658
$329$	0	0
$330$	0	0
$331$	706.087	0.117251	0.0586254	−	0.998280i	$-0.481328\pi$
$331$	706.087	0.117251	0.0586254	+	0.998280i	$0.481328\pi$
$332$	1511.47	0.249858
$333$	0	0
$334$	15169.3	2.48511
$335$	−289.274	−0.0471783
$336$	0	0
$337$	1249.46	0.201965	0.100983	−	0.994888i	$-0.467801\pi$
$337$	1249.46	0.201965	0.100983	+	0.994888i	$0.467801\pi$
$338$	−9058.19	−1.45769
$339$	0	0
$340$	3644.82	0.581377
$341$	−18598.4	−2.95354
$342$	0	0
$343$	0	0
$344$	1856.24	0.290936
$345$	0	0
$346$	8781.93	1.36451
$347$	9952.93	1.53977	0.769886	−	0.638181i	$-0.220313\pi$
$347$	9952.93	1.53977	0.769886	+	0.638181i	$0.220313\pi$
$348$	0	0
$349$	−7558.20	−1.15926	−0.579629	−	0.814881i	$-0.696802\pi$
$349$	−7558.20	−1.15926	−0.579629	+	0.814881i	$0.696802\pi$
$350$	0	0
$351$	0	0
$352$	−16764.4	−2.53848
$353$	−2187.44	−0.329818	−0.164909	−	0.986309i	$-0.552733\pi$
$353$	−2187.44	−0.329818	−0.164909	+	0.986309i	$0.552733\pi$
$354$	0	0
$355$	−1009.91	−0.150988
$356$	5465.52	0.813686
$357$	0	0
$358$	−5681.71	−0.838792
$359$	−12418.7	−1.82571	−0.912857	−	0.408279i	$-0.866129\pi$
$359$	−12418.7	−1.82571	−0.912857	+	0.408279i	$0.866129\pi$
$360$	0	0
$361$	12747.4	1.85849
$362$	−1575.38	−0.228730
$363$	0	0
$364$	0	0
$365$	1900.90	0.272597
$366$	0	0
$367$	2915.98	0.414750	0.207375	−	0.978262i	$-0.433508\pi$
$367$	2915.98	0.414750	0.207375	+	0.978262i	$0.433508\pi$
$368$	2646.34	0.374864
$369$	0	0
$370$	−5584.65	−0.784681
$371$	0	0
$372$	0	0
$373$	11263.9	1.56360	0.781800	−	0.623529i	$-0.214302\pi$
$373$	11263.9	1.56360	0.781800	+	0.623529i	$0.214302\pi$
$374$	20446.3	2.82688
$375$	0	0
$376$	720.966	0.0988856
$377$	−165.902	−0.0226641
$378$	0	0
$379$	10281.4	1.39346	0.696728	−	0.717335i	$-0.254638\pi$
$379$	10281.4	1.39346	0.696728	+	0.717335i	$0.254638\pi$
$380$	6926.02	0.934993
$381$	0	0
$382$	−10986.8	−1.47155
$383$	5716.34	0.762640	0.381320	−	0.924443i	$-0.375470\pi$
$383$	5716.34	0.762640	0.381320	+	0.924443i	$0.375470\pi$
$384$	0	0
$385$	0	0
$386$	18681.8	2.46342
$387$	0	0
$388$	2520.22	0.329754
$389$	−4808.09	−0.626683	−0.313342	−	0.949640i	$-0.601449\pi$
$389$	−4808.09	−0.626683	−0.313342	+	0.949640i	$0.601449\pi$
$390$	0	0
$391$	−4306.95	−0.557063
$392$	0	0
$393$	0	0
$394$	−20556.1	−2.62843
$395$	−4638.10	−0.590806
$396$	0	0
$397$	−1636.89	−0.206935	−0.103468	−	0.994633i	$-0.532994\pi$
$397$	−1636.89	−0.206935	−0.103468	+	0.994633i	$0.532994\pi$
$398$	−12060.9	−1.51899
$399$	0	0
$400$	−1131.90	−0.141487
$401$	3219.90	0.400982	0.200491	−	0.979696i	$-0.435746\pi$
$401$	3219.90	0.400982	0.200491	+	0.979696i	$0.435746\pi$
$402$	0	0
$403$	2113.58	0.261253
$404$	15850.0	1.95190
$405$	0	0
$406$	0	0
$407$	−17321.0	−2.10951
$408$	0	0
$409$	−2143.81	−0.259180	−0.129590	−	0.991568i	$-0.541366\pi$
$409$	−2143.81	−0.259180	−0.129590	+	0.991568i	$0.541366\pi$
$410$	8311.74	1.00119
$411$	0	0
$412$	11459.5	1.37032
$413$	0	0
$414$	0	0
$415$	763.933	0.0903614
$416$	1905.16	0.224539
$417$	0	0
$418$	38852.7	4.54629
$419$	−9576.92	−1.11662	−0.558309	−	0.829633i	$-0.688550\pi$
$419$	−9576.92	−1.11662	−0.558309	+	0.829633i	$0.688550\pi$
$420$	0	0
$421$	2893.25	0.334936	0.167468	−	0.985877i	$-0.446441\pi$
$421$	2893.25	0.334936	0.167468	+	0.985877i	$0.446441\pi$
$422$	−24786.3	−2.85919
$423$	0	0
$424$	1595.86	0.182788
$425$	1842.18	0.210256
$426$	0	0
$427$	0	0
$428$	14809.9	1.67257
$429$	0	0
$430$	4903.66	0.549943
$431$	−8923.24	−0.997256	−0.498628	−	0.866816i	$-0.666163\pi$
$431$	−8923.24	−0.997256	−0.498628	+	0.866816i	$0.666163\pi$
$432$	0	0
$433$	7697.81	0.854349	0.427174	−	0.904169i	$-0.359509\pi$
$433$	7697.81	0.854349	0.427174	+	0.904169i	$0.359509\pi$
$434$	0	0
$435$	0	0
$436$	11646.1	1.27923
$437$	−8184.21	−0.895890
$438$	0	0
$439$	−13101.9	−1.42442	−0.712208	−	0.701968i	$-0.752305\pi$
$439$	−13101.9	−1.42442	−0.712208	+	0.701968i	$0.752305\pi$
$440$	2625.90	0.284511
$441$	0	0
$442$	−2323.58	−0.250049
$443$	3088.33	0.331221	0.165610	−	0.986191i	$-0.447041\pi$
$443$	3088.33	0.331221	0.165610	+	0.986191i	$0.447041\pi$
$444$	0	0
$445$	2762.40	0.294270
$446$	−19863.2	−2.10885
$447$	0	0
$448$	0	0
$449$	5502.74	0.578375	0.289187	−	0.957272i	$-0.406615\pi$
$449$	5502.74	0.578375	0.289187	+	0.957272i	$0.406615\pi$
$450$	0	0
$451$	25779.2	2.69156
$452$	−10626.0	−1.10577
$453$	0	0
$454$	−21759.7	−2.24941
$455$	0	0
$456$	0	0
$457$	1830.34	0.187352	0.0936759	−	0.995603i	$-0.470138\pi$
$457$	1830.34	0.187352	0.0936759	+	0.995603i	$0.470138\pi$
$458$	5594.75	0.570798
$459$	0	0
$460$	−2891.10	−0.293040
$461$	9525.08	0.962315	0.481157	−	0.876634i	$-0.340217\pi$
$461$	9525.08	0.962315	0.481157	+	0.876634i	$0.340217\pi$
$462$	0	0
$463$	9268.79	0.930361	0.465181	−	0.885216i	$-0.345989\pi$
$463$	9268.79	0.930361	0.465181	+	0.885216i	$0.345989\pi$
$464$	−1007.60	−0.100812
$465$	0	0
$466$	12309.7	1.22368
$467$	−1724.09	−0.170838	−0.0854188	−	0.996345i	$-0.527223\pi$
$467$	−1724.09	−0.170838	−0.0854188	+	0.996345i	$0.527223\pi$
$468$	0	0
$469$	0	0
$470$	1904.59	0.186919
$471$	0	0
$472$	3402.65	0.331821
$473$	15208.9	1.47845
$474$	0	0
$475$	3500.57	0.338141
$476$	0	0
$477$	0	0
$478$	−6722.89	−0.643301
$479$	−16627.5	−1.58607	−0.793036	−	0.609175i	$-0.791501\pi$
$479$	−16627.5	−1.58607	−0.793036	+	0.609175i	$0.791501\pi$
$480$	0	0
$481$	1968.42	0.186595
$482$	10235.9	0.967289
$483$	0	0
$484$	29401.0	2.76118
$485$	1273.78	0.119256
$486$	0	0
$487$	−7805.21	−0.726258	−0.363129	−	0.931739i	$-0.618292\pi$
$487$	−7805.21	−0.726258	−0.363129	+	0.931739i	$0.618292\pi$
$488$	−3392.66	−0.314710
$489$	0	0
$490$	0	0
$491$	6793.02	0.624368	0.312184	−	0.950022i	$-0.398939\pi$
$491$	6793.02	0.624368	0.312184	+	0.950022i	$0.398939\pi$
$492$	0	0
$493$	1639.88	0.149811
$494$	−4415.36	−0.402138
$495$	0	0
$496$	12836.8	1.16208
$497$	0	0
$498$	0	0
$499$	−13772.6	−1.23556	−0.617782	−	0.786350i	$-0.711968\pi$
$499$	−13772.6	−1.23556	−0.617782	+	0.786350i	$0.711968\pi$
$500$	1236.59	0.110604
$501$	0	0
$502$	−23284.9	−2.07023
$503$	−16717.8	−1.48192	−0.740962	−	0.671547i	$-0.765630\pi$
$503$	−16717.8	−1.48192	−0.740962	+	0.671547i	$0.765630\pi$
$504$	0	0
$505$	8010.94	0.705905
$506$	−16218.2	−1.42487
$507$	0	0
$508$	21955.2	1.91753
$509$	18415.2	1.60362	0.801808	−	0.597582i	$-0.203872\pi$
$509$	18415.2	1.60362	0.801808	+	0.597582i	$0.203872\pi$
$510$	0	0
$511$	0	0
$512$	14470.8	1.24908
$513$	0	0
$514$	−13126.8	−1.12646
$515$	5791.91	0.495577
$516$	0	0
$517$	5907.14	0.502506
$518$	0	0
$519$	0	0
$520$	−298.416	−0.0251661
$521$	−6857.76	−0.576668	−0.288334	−	0.957530i	$-0.593101\pi$
$521$	−6857.76	−0.576668	−0.288334	+	0.957530i	$0.593101\pi$
$522$	0	0
$523$	−21247.1	−1.77643	−0.888213	−	0.459431i	$-0.848053\pi$
$523$	−21247.1	−1.77643	−0.888213	+	0.459431i	$0.848053\pi$
$524$	2178.78	0.181642
$525$	0	0
$526$	22664.9	1.87878
$527$	−20892.0	−1.72689
$528$	0	0
$529$	−8750.70	−0.719216
$530$	4215.81	0.345516
$531$	0	0
$532$	0	0
$533$	−2929.63	−0.238080
$534$	0	0
$535$	7485.24	0.604888
$536$	−463.193	−0.0373263
$537$	0	0
$538$	−12712.4	−1.01872
$539$	0	0
$540$	0	0
$541$	8605.77	0.683902	0.341951	−	0.939718i	$-0.388912\pi$
$541$	8605.77	0.683902	0.341951	+	0.939718i	$0.388912\pi$
$542$	−1620.54	−0.128428
$543$	0	0
$544$	−18831.9	−1.48421
$545$	5886.19	0.462636
$546$	0	0
$547$	−20369.4	−1.59220	−0.796098	−	0.605168i	$-0.793106\pi$
$547$	−20369.4	−1.59220	−0.796098	+	0.605168i	$0.793106\pi$
$548$	−27827.9	−2.16925
$549$	0	0
$550$	6936.86	0.537798
$551$	3116.16	0.240931
$552$	0	0
$553$	0	0
$554$	−17493.8	−1.34159
$555$	0	0
$556$	11868.1	0.905251
$557$	−3287.44	−0.250078	−0.125039	−	0.992152i	$-0.539906\pi$
$557$	−3287.44	−0.250078	−0.125039	+	0.992152i	$0.539906\pi$
$558$	0	0
$559$	−1728.39	−0.130775
$560$	0	0
$561$	0	0
$562$	−34394.9	−2.58160
$563$	−15715.7	−1.17645	−0.588224	−	0.808698i	$-0.700173\pi$
$563$	−15715.7	−1.17645	−0.588224	+	0.808698i	$0.700173\pi$
$564$	0	0
$565$	−5370.64	−0.399901
$566$	−26354.0	−1.95714
$567$	0	0
$568$	−1617.10	−0.119458
$569$	10232.8	0.753919	0.376960	−	0.926230i	$-0.376970\pi$
$569$	10232.8	0.753919	0.376960	+	0.926230i	$0.376970\pi$
$570$	0	0
$571$	16958.2	1.24287	0.621434	−	0.783467i	$-0.286550\pi$
$571$	16958.2	1.24287	0.621434	+	0.783467i	$0.286550\pi$
$572$	−4837.59	−0.353619
$573$	0	0
$574$	0	0
$575$	−1461.23	−0.105978
$576$	0	0
$577$	−9254.30	−0.667698	−0.333849	−	0.942627i	$-0.608348\pi$
$577$	−9254.30	−0.667698	−0.333849	+	0.942627i	$0.608348\pi$
$578$	2185.98	0.157309
$579$	0	0
$580$	1100.80	0.0788071
$581$	0	0
$582$	0	0
$583$	13075.5	0.928871
$584$	3043.77	0.215672
$585$	0	0
$586$	−15690.0	−1.10606
$587$	8986.07	0.631848	0.315924	−	0.948785i	$-0.397686\pi$
$587$	8986.07	0.631848	0.315924	+	0.948785i	$0.397686\pi$
$588$	0	0
$589$	−39699.8	−2.77725
$590$	8988.82	0.627227
$591$	0	0
$592$	11955.2	0.829990
$593$	12846.1	0.889587	0.444794	−	0.895633i	$-0.353277\pi$
$593$	12846.1	0.889587	0.444794	+	0.895633i	$0.353277\pi$
$594$	0	0
$595$	0	0
$596$	−20737.0	−1.42520
$597$	0	0
$598$	1843.09	0.126036
$599$	5479.76	0.373785	0.186892	−	0.982380i	$-0.440158\pi$
$599$	5479.76	0.373785	0.186892	+	0.982380i	$0.440158\pi$
$600$	0	0
$601$	−6440.27	−0.437112	−0.218556	−	0.975824i	$-0.570135\pi$
$601$	−6440.27	−0.437112	−0.218556	+	0.975824i	$0.570135\pi$
$602$	0	0
$603$	0	0
$604$	−6387.16	−0.430281
$605$	14859.9	0.998582
$606$	0	0
$607$	16799.3	1.12333	0.561666	−	0.827364i	$-0.310161\pi$
$607$	16799.3	1.12333	0.561666	+	0.827364i	$0.310161\pi$
$608$	−35785.0	−2.38696
$609$	0	0
$610$	−8962.44	−0.594883
$611$	−671.308	−0.0444488
$612$	0	0
$613$	−10430.0	−0.687219	−0.343609	−	0.939113i	$-0.611650\pi$
$613$	−10430.0	−0.687219	−0.343609	+	0.939113i	$0.611650\pi$
$614$	−894.196	−0.0587733
$615$	0	0
$616$	0	0
$617$	−5763.97	−0.376092	−0.188046	−	0.982160i	$-0.560215\pi$
$617$	−5763.97	−0.376092	−0.188046	+	0.982160i	$0.560215\pi$
$618$	0	0
$619$	26244.7	1.70414	0.852070	−	0.523427i	$-0.175347\pi$
$619$	26244.7	1.70414	0.852070	+	0.523427i	$0.175347\pi$
$620$	−14024.1	−0.908422
$621$	0	0
$622$	10203.1	0.657726
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	11556.7	0.737856
$627$	0	0
$628$	21966.2	1.39578
$629$	−19457.1	−1.23340
$630$	0	0
$631$	−27755.5	−1.75108	−0.875538	−	0.483150i	$-0.839493\pi$
$631$	−27755.5	−1.75108	−0.875538	+	0.483150i	$0.839493\pi$
$632$	−7426.65	−0.467431
$633$	0	0
$634$	18231.9	1.14208
$635$	11096.7	0.693476
$636$	0	0
$637$	0	0
$638$	6175.11	0.383190
$639$	0	0
$640$	−4980.52	−0.307613
$641$	−15310.5	−0.943411	−0.471706	−	0.881756i	$-0.656362\pi$
$641$	−15310.5	−0.943411	−0.471706	+	0.881756i	$0.656362\pi$
$642$	0	0
$643$	12010.3	0.736609	0.368304	−	0.929705i	$-0.379938\pi$
$643$	12010.3	0.736609	0.368304	+	0.929705i	$0.379938\pi$
$644$	0	0
$645$	0	0
$646$	43644.3	2.65815
$647$	−10605.9	−0.644452	−0.322226	−	0.946663i	$-0.604431\pi$
$647$	−10605.9	−0.644452	−0.322226	+	0.946663i	$0.604431\pi$
$648$	0	0
$649$	27879.1	1.68621
$650$	−788.328	−0.0475704
$651$	0	0
$652$	30834.4	1.85210
$653$	−13680.3	−0.819836	−0.409918	−	0.912122i	$-0.634443\pi$
$653$	−13680.3	−0.819836	−0.409918	+	0.912122i	$0.634443\pi$
$654$	0	0
$655$	1101.20	0.0656910
$656$	−17793.1	−1.05900
$657$	0	0
$658$	0	0
$659$	−17039.6	−1.00723	−0.503617	−	0.863927i	$-0.667998\pi$
$659$	−17039.6	−1.00723	−0.503617	+	0.863927i	$0.667998\pi$
$660$	0	0
$661$	8882.54	0.522679	0.261339	−	0.965247i	$-0.415836\pi$
$661$	8882.54	0.522679	0.261339	+	0.965247i	$0.415836\pi$
$662$	2986.73	0.175351
$663$	0	0
$664$	1223.23	0.0714917
$665$	0	0
$666$	0	0
$667$	−1300.77	−0.0755112
$668$	35476.7	2.05484
$669$	0	0
$670$	−1223.62	−0.0705562
$671$	−27797.3	−1.59926
$672$	0	0
$673$	26805.6	1.53533	0.767667	−	0.640849i	$-0.221418\pi$
$673$	26805.6	1.53533	0.767667	+	0.640849i	$0.221418\pi$
$674$	5285.18	0.302044
$675$	0	0
$676$	−21184.5	−1.20531
$677$	−759.188	−0.0430989	−0.0215495	−	0.999768i	$-0.506860\pi$
$677$	−759.188	−0.0430989	−0.0215495	+	0.999768i	$0.506860\pi$
$678$	0	0
$679$	0	0
$680$	2949.74	0.166349
$681$	0	0
$682$	−78670.7	−4.41709
$683$	11586.9	0.649138	0.324569	−	0.945862i	$-0.394781\pi$
$683$	11586.9	0.649138	0.324569	+	0.945862i	$0.394781\pi$
$684$	0	0
$685$	−14064.8	−0.784511
$686$	0	0
$687$	0	0
$688$	−10497.4	−0.581697
$689$	−1485.94	−0.0821625
$690$	0	0
$691$	−29026.7	−1.59802	−0.799008	−	0.601321i	$-0.794641\pi$
$691$	−29026.7	−1.59802	−0.799008	+	0.601321i	$0.794641\pi$
$692$	20538.4	1.12826
$693$	0	0
$694$	42100.7	2.30277
$695$	5998.40	0.327385
$696$	0	0
$697$	28958.4	1.57371
$698$	−31971.0	−1.73370
$699$	0	0
$700$	0	0
$701$	−30673.5	−1.65267	−0.826335	−	0.563179i	$-0.809578\pi$
$701$	−30673.5	−1.65267	−0.826335	+	0.563179i	$0.809578\pi$
$702$	0	0
$703$	−36973.1	−1.98360
$704$	−47153.1	−2.52436
$705$	0	0
$706$	−9252.82	−0.493250
$707$	0	0
$708$	0	0
$709$	−8934.80	−0.473277	−0.236639	−	0.971598i	$-0.576046\pi$
$709$	−8934.80	−0.473277	−0.236639	+	0.971598i	$0.576046\pi$
$710$	−4271.91	−0.225805
$711$	0	0
$712$	4423.22	0.232819
$713$	16571.7	0.870430
$714$	0	0
$715$	−2445.03	−0.127887
$716$	−13287.9	−0.693564
$717$	0	0
$718$	−52530.6	−2.73040
$719$	−23330.0	−1.21010	−0.605050	−	0.796188i	$-0.706847\pi$
$719$	−23330.0	−1.21010	−0.605050	+	0.796188i	$0.706847\pi$
$720$	0	0
$721$	0	0
$722$	53921.1	2.77941
$723$	0	0
$724$	−3684.38	−0.189128
$725$	556.368	0.0285007
$726$	0	0
$727$	7577.43	0.386563	0.193281	−	0.981143i	$-0.438087\pi$
$727$	7577.43	0.386563	0.193281	+	0.981143i	$0.438087\pi$
$728$	0	0
$729$	0	0
$730$	8040.77	0.407674
$731$	17084.5	0.864425
$732$	0	0
$733$	8368.88	0.421708	0.210854	−	0.977518i	$-0.432376\pi$
$733$	8368.88	0.421708	0.210854	+	0.977518i	$0.432376\pi$
$734$	12334.6	0.620268
$735$	0	0
$736$	14937.6	0.748107
$737$	−3795.11	−0.189681
$738$	0	0
$739$	−9506.35	−0.473203	−0.236601	−	0.971607i	$-0.576034\pi$
$739$	−9506.35	−0.473203	−0.236601	+	0.971607i	$0.576034\pi$
$740$	−13060.9	−0.648822
$741$	0	0
$742$	0	0
$743$	−5309.44	−0.262159	−0.131080	−	0.991372i	$-0.541844\pi$
$743$	−5309.44	−0.262159	−0.131080	+	0.991372i	$0.541844\pi$
$744$	0	0
$745$	−10480.9	−0.515425
$746$	47646.1	2.33840
$747$	0	0
$748$	47818.0	2.33743
$749$	0	0
$750$	0	0
$751$	18449.8	0.896463	0.448232	−	0.893917i	$-0.352054\pi$
$751$	18449.8	0.896463	0.448232	+	0.893917i	$0.352054\pi$
$752$	−4077.18	−0.197712
$753$	0	0
$754$	−701.761	−0.0338947
$755$	−3228.21	−0.155612
$756$	0	0
$757$	−30189.4	−1.44947	−0.724737	−	0.689026i	$-0.758039\pi$
$757$	−30189.4	−1.44947	−0.724737	+	0.689026i	$0.758039\pi$
$758$	43490.1	2.08395
$759$	0	0
$760$	5605.20	0.267529
$761$	15440.8	0.735519	0.367759	−	0.929921i	$-0.380125\pi$
$761$	15440.8	0.735519	0.367759	+	0.929921i	$0.380125\pi$
$762$	0	0
$763$	0	0
$764$	−25695.0	−1.21677
$765$	0	0
$766$	24180.0	1.14055
$767$	−3168.28	−0.149152
$768$	0	0
$769$	23462.7	1.10024	0.550122	−	0.835084i	$-0.314581\pi$
$769$	23462.7	1.10024	0.550122	+	0.835084i	$0.314581\pi$
$770$	0	0
$771$	0	0
$772$	43691.4	2.03690
$773$	−25409.7	−1.18231	−0.591154	−	0.806559i	$-0.701328\pi$
$773$	−25409.7	−1.18231	−0.591154	+	0.806559i	$0.701328\pi$
$774$	0	0
$775$	−7088.10	−0.328532
$776$	2039.60	0.0943524
$777$	0	0
$778$	−20338.1	−0.937219
$779$	55027.8	2.53091
$780$	0	0
$781$	−13249.5	−0.607047
$782$	−18218.3	−0.833101
$783$	0	0
$784$	0	0
$785$	11102.2	0.504784
$786$	0	0
$787$	−32730.9	−1.48250	−0.741251	−	0.671228i	$-0.765767\pi$
$787$	−32730.9	−1.48250	−0.741251	+	0.671228i	$0.765767\pi$
$788$	−48074.9	−2.17335
$789$	0	0
$790$	−19619.1	−0.883564
$791$	0	0
$792$	0	0
$793$	3158.98	0.141461
$794$	−6924.01	−0.309476
$795$	0	0
$796$	−28206.9	−1.25599
$797$	−15615.1	−0.693994	−0.346997	−	0.937866i	$-0.612799\pi$
$797$	−15615.1	−0.693994	−0.346997	+	0.937866i	$0.612799\pi$
$798$	0	0
$799$	6635.65	0.293808
$800$	−6389.13	−0.282362
$801$	0	0
$802$	13620.1	0.599678
$803$	24938.7	1.09598
$804$	0	0
$805$	0	0
$806$	8940.40	0.390710
$807$	0	0
$808$	12827.3	0.558494
$809$	−9642.75	−0.419062	−0.209531	−	0.977802i	$-0.567194\pi$
$809$	−9642.75	−0.419062	−0.209531	+	0.977802i	$0.567194\pi$
$810$	0	0
$811$	−20924.1	−0.905974	−0.452987	−	0.891517i	$-0.649642\pi$
$811$	−20924.1	−0.905974	−0.452987	+	0.891517i	$0.649642\pi$
$812$	0	0
$813$	0	0
$814$	−73267.4	−3.15482
$815$	15584.4	0.669814
$816$	0	0
$817$	32464.7	1.39020
$818$	−9068.26	−0.387609
$819$	0	0
$820$	19438.8	0.827844
$821$	34387.0	1.46177	0.730886	−	0.682499i	$-0.239107\pi$
$821$	34387.0	1.46177	0.730886	+	0.682499i	$0.239107\pi$
$822$	0	0
$823$	3181.12	0.134735	0.0673674	−	0.997728i	$-0.478540\pi$
$823$	3181.12	0.134735	0.0673674	+	0.997728i	$0.478540\pi$
$824$	9274.16	0.392088
$825$	0	0
$826$	0	0
$827$	35461.7	1.49108	0.745540	−	0.666460i	$-0.232191\pi$
$827$	35461.7	1.49108	0.745540	+	0.666460i	$0.232191\pi$
$828$	0	0
$829$	−23520.3	−0.985396	−0.492698	−	0.870200i	$-0.663989\pi$
$829$	−23520.3	−0.985396	−0.492698	+	0.870200i	$0.663989\pi$
$830$	3231.42	0.135138
$831$	0	0
$832$	5358.63	0.223290
$833$	0	0
$834$	0	0
$835$	17930.7	0.743136
$836$	90865.5	3.75915
$837$	0	0
$838$	−40510.1	−1.66993
$839$	−6496.59	−0.267327	−0.133664	−	0.991027i	$-0.542674\pi$
$839$	−6496.59	−0.267327	−0.133664	+	0.991027i	$0.542674\pi$
$840$	0	0
$841$	−23893.7	−0.979693
$842$	12238.4	0.500905
$843$	0	0
$844$	−57968.0	−2.36415
$845$	−10707.1	−0.435901
$846$	0	0
$847$	0	0
$848$	−9024.87	−0.365466
$849$	0	0
$850$	7792.37	0.314442
$851$	15433.6	0.621687
$852$	0	0
$853$	33236.8	1.33412	0.667060	−	0.745004i	$-0.267552\pi$
$853$	33236.8	1.33412	0.667060	+	0.745004i	$0.267552\pi$
$854$	0	0
$855$	0	0
$856$	11985.6	0.478573
$857$	−37755.8	−1.50492	−0.752459	−	0.658639i	$-0.771133\pi$
$857$	−37755.8	−1.50492	−0.752459	+	0.658639i	$0.771133\pi$
$858$	0	0
$859$	−29443.3	−1.16949	−0.584745	−	0.811218i	$-0.698805\pi$
$859$	−29443.3	−1.16949	−0.584745	+	0.811218i	$0.698805\pi$
$860$	11468.3	0.454726
$861$	0	0
$862$	−37745.1	−1.49142
$863$	−3594.47	−0.141781	−0.0708906	−	0.997484i	$-0.522584\pi$
$863$	−3594.47	−0.141781	−0.0708906	+	0.997484i	$0.522584\pi$
$864$	0	0
$865$	10380.6	0.408035
$866$	32561.6	1.27770
$867$	0	0
$868$	0	0
$869$	−60849.3	−2.37534
$870$	0	0
$871$	431.289	0.0167780
$872$	9425.12	0.366026
$873$	0	0
$874$	−34619.0	−1.33982
$875$	0	0
$876$	0	0
$877$	−27162.3	−1.04585	−0.522923	−	0.852380i	$-0.675158\pi$
$877$	−27162.3	−1.04585	−0.522923	+	0.852380i	$0.675158\pi$
$878$	−55420.7	−2.13025
$879$	0	0
$880$	−14849.9	−0.568851
$881$	−10081.8	−0.385544	−0.192772	−	0.981244i	$-0.561748\pi$
$881$	−10081.8	−0.385544	−0.192772	+	0.981244i	$0.561748\pi$
$882$	0	0
$883$	−46515.1	−1.77277	−0.886386	−	0.462946i	$-0.846792\pi$
$883$	−46515.1	−1.77277	−0.886386	+	0.462946i	$0.846792\pi$
$884$	−5434.20	−0.206756
$885$	0	0
$886$	13063.6	0.495348
$887$	14304.6	0.541489	0.270745	−	0.962651i	$-0.412730\pi$
$887$	14304.6	0.541489	0.270745	+	0.962651i	$0.412730\pi$
$888$	0	0
$889$	0	0
$890$	11684.9	0.440088
$891$	0	0
$892$	−46454.3	−1.74373
$893$	12609.3	0.472513
$894$	0	0
$895$	−6716.00	−0.250828
$896$	0	0
$897$	0	0
$898$	23276.5	0.864973
$899$	−6309.74	−0.234084
$900$	0	0
$901$	14688.1	0.543097
$902$	109045.	4.02529
$903$	0	0
$904$	−8599.60	−0.316392
$905$	−1862.17	−0.0683984
$906$	0	0
$907$	−34092.6	−1.24810	−0.624049	−	0.781385i	$-0.714514\pi$
$907$	−34092.6	−1.24810	−0.624049	+	0.781385i	$0.714514\pi$
$908$	−50889.7	−1.85995
$909$	0	0
$910$	0	0
$911$	29631.2	1.07764	0.538818	−	0.842422i	$-0.318871\pi$
$911$	29631.2	1.07764	0.538818	+	0.842422i	$0.318871\pi$
$912$	0	0
$913$	10022.4	0.363299
$914$	7742.31	0.280189
$915$	0	0
$916$	13084.5	0.471970
$917$	0	0
$918$	0	0
$919$	4503.92	0.161666	0.0808328	−	0.996728i	$-0.474242\pi$
$919$	4503.92	0.161666	0.0808328	+	0.996728i	$0.474242\pi$
$920$	−2339.76	−0.0838473
$921$	0	0
$922$	40290.9	1.43916
$923$	1505.71	0.0536958
$924$	0	0
$925$	−6601.27	−0.234647
$926$	39206.8	1.39138
$927$	0	0
$928$	−5687.53	−0.201188
$929$	17120.0	0.604615	0.302308	−	0.953210i	$-0.402243\pi$
$929$	17120.0	0.604615	0.302308	+	0.953210i	$0.402243\pi$
$930$	0	0
$931$	0	0
$932$	28788.9	1.01181
$933$	0	0
$934$	−7292.84	−0.255492
$935$	24168.3	0.845335
$936$	0	0
$937$	−5672.48	−0.197772	−0.0988858	−	0.995099i	$-0.531528\pi$
$937$	−5672.48	−0.197772	−0.0988858	+	0.995099i	$0.531528\pi$
$938$	0	0
$939$	0	0
$940$	4454.28	0.154556
$941$	−44038.6	−1.52563	−0.762815	−	0.646616i	$-0.776183\pi$
$941$	−44038.6	−1.52563	−0.762815	+	0.646616i	$0.776183\pi$
$942$	0	0
$943$	−22970.1	−0.793222
$944$	−19242.5	−0.663444
$945$	0	0
$946$	64333.2	2.21105
$947$	31055.4	1.06564	0.532821	−	0.846228i	$-0.321132\pi$
$947$	31055.4	1.06564	0.532821	+	0.846228i	$0.321132\pi$
$948$	0	0
$949$	−2834.12	−0.0969437
$950$	14807.3	0.505698
$951$	0	0
$952$	0	0
$953$	−25702.3	−0.873639	−0.436820	−	0.899549i	$-0.643895\pi$
$953$	−25702.3	−0.873639	−0.436820	+	0.899549i	$0.643895\pi$
$954$	0	0
$955$	−12986.8	−0.440046
$956$	−15722.9	−0.531920
$957$	0	0
$958$	−70333.8	−2.37201
$959$	0	0
$960$	0	0
$961$	50594.8	1.69833
$962$	8326.35	0.279057
$963$	0	0
$964$	23938.9	0.799813
$965$	22082.6	0.736648
$966$	0	0
$967$	8898.41	0.295919	0.147959	−	0.988993i	$-0.452730\pi$
$967$	8898.41	0.295919	0.147959	+	0.988993i	$0.452730\pi$
$968$	23794.1	0.790054
$969$	0	0
$970$	5388.04	0.178350
$971$	5799.21	0.191664	0.0958319	−	0.995398i	$-0.469449\pi$
$971$	5799.21	0.191664	0.0958319	+	0.995398i	$0.469449\pi$
$972$	0	0
$973$	0	0
$974$	−33015.9	−1.08614
$975$	0	0
$976$	19186.1	0.629232
$977$	−59923.4	−1.96225	−0.981126	−	0.193369i	$-0.938058\pi$
$977$	−59923.4	−1.96225	−0.981126	+	0.193369i	$0.938058\pi$
$978$	0	0
$979$	36241.1	1.18312
$980$	0	0
$981$	0	0
$982$	28734.3	0.933757
$983$	37990.9	1.23268	0.616338	−	0.787482i	$-0.288616\pi$
$983$	37990.9	1.23268	0.616338	+	0.787482i	$0.288616\pi$
$984$	0	0
$985$	−24298.1	−0.785993
$986$	6936.67	0.224045
$987$	0	0
$988$	−10326.3	−0.332512
$989$	−13551.6	−0.435709
$990$	0	0
$991$	9402.25	0.301385	0.150692	−	0.988581i	$-0.451850\pi$
$991$	9402.25	0.301385	0.150692	+	0.988581i	$0.451850\pi$
$992$	72458.9	2.31912
$993$	0	0
$994$	0	0
$995$	−14256.4	−0.454230
$996$	0	0
$997$	−17998.5	−0.571732	−0.285866	−	0.958270i	$-0.592281\pi$
$997$	−17998.5	−0.571732	−0.285866	+	0.958270i	$0.592281\pi$
$998$	−58257.8	−1.84781
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+4.22998 q^{2} +9.89271 q^{4} +5.00000 q^{5} +8.00613 q^{8} +O(q^{10})\)	\(q+4.22998 q^{2} +9.89271 q^{4} +5.00000 q^{5} +8.00613 q^{8} +21.1499 q^{10} +65.5972 q^{11} -7.45468 q^{13} -45.2759 q^{16} +73.6871 q^{17} +140.023 q^{19} +49.4636 q^{20} +277.475 q^{22} -58.4492 q^{23} +25.0000 q^{25} -31.5331 q^{26} +22.2547 q^{29} -283.524 q^{31} -255.565 q^{32} +311.695 q^{34} -264.051 q^{37} +592.293 q^{38} +40.0306 q^{40} +392.992 q^{41} +231.853 q^{43} +648.934 q^{44} -247.239 q^{46} +90.0518 q^{47} +105.749 q^{50} -73.7470 q^{52} +199.330 q^{53} +327.986 q^{55} +94.1369 q^{58} +425.005 q^{59} -423.758 q^{61} -1199.30 q^{62} -718.828 q^{64} -37.2734 q^{65} -57.8548 q^{67} +728.965 q^{68} -201.982 q^{71} +380.180 q^{73} -1116.93 q^{74} +1385.20 q^{76} -927.621 q^{79} -226.380 q^{80} +1662.35 q^{82} +152.787 q^{83} +368.435 q^{85} +980.732 q^{86} +525.179 q^{88} +552.480 q^{89} -578.221 q^{92} +380.917 q^{94} +700.114 q^{95} +254.755 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 50 q^{4} + 40 q^{5} - 66 q^{8}+O(q^{10}) \) 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

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Database

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