LMFDB - Embedded newform 2205.4.a.cd.1.3

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:	$1$
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			1.3
Root			$2.78988$ 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)

$f(q)$	$=$	$q-2.78988 q^{2} -0.216543 q^{4} -5.00000 q^{5} +22.9232 q^{8} +O(q^{10})$	$q-2.78988 q^{2} -0.216543 q^{4} -5.00000 q^{5} +22.9232 q^{8} +13.9494 q^{10} -0.731118 q^{11} +63.9983 q^{13} -62.2208 q^{16} +29.6307 q^{17} +43.7843 q^{19} +1.08271 q^{20} +2.03973 q^{22} -157.920 q^{23} +25.0000 q^{25} -178.548 q^{26} -192.402 q^{29} -24.2100 q^{31} -9.79689 q^{32} -82.6662 q^{34} -62.1980 q^{37} -122.153 q^{38} -114.616 q^{40} +65.8313 q^{41} +97.8275 q^{43} +0.158318 q^{44} +440.577 q^{46} -432.364 q^{47} -69.7471 q^{50} -13.8584 q^{52} -301.048 q^{53} +3.65559 q^{55} +536.780 q^{58} +350.512 q^{59} +405.997 q^{61} +67.5432 q^{62} +525.098 q^{64} -319.992 q^{65} +853.677 q^{67} -6.41631 q^{68} +214.879 q^{71} -1132.86 q^{73} +173.525 q^{74} -9.48117 q^{76} +51.4032 q^{79} +311.104 q^{80} -183.662 q^{82} +1411.04 q^{83} -148.153 q^{85} -272.928 q^{86} -16.7596 q^{88} +266.685 q^{89} +34.1963 q^{92} +1206.24 q^{94} -218.921 q^{95} -546.792 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.78988	−0.986373	−0.493187	−	0.869924i	$-0.664168\pi$
$2$	−2.78988	−0.986373	−0.493187	+	0.869924i	$0.664168\pi$
$3$	0	0
$3$	0	0
$4$	−0.216543	−0.0270678
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.9232	1.01307
$9$	0	0
$10$	13.9494	0.441120
$11$	−0.731118	−0.0200400	−0.0100200	−	0.999950i	$-0.503190\pi$
$11$	−0.731118	−0.0200400	−0.0100200	+	0.999950i	$0.503190\pi$
$12$	0	0
$13$	63.9983	1.36538	0.682690	−	0.730708i	$-0.260810\pi$
$13$	63.9983	1.36538	0.682690	+	0.730708i	$0.260810\pi$
$14$	0	0
$15$	0	0
$16$	−62.2208	−0.972199
$17$	29.6307	0.422735	0.211368	−	0.977407i	$-0.432208\pi$
$17$	29.6307	0.422735	0.211368	+	0.977407i	$0.432208\pi$
$18$	0	0
$19$	43.7843	0.528674	0.264337	−	0.964430i	$-0.414847\pi$
$19$	43.7843	0.528674	0.264337	+	0.964430i	$0.414847\pi$
$20$	1.08271	0.0121051
$21$	0	0
$22$	2.03973	0.0197669
$23$	−157.920	−1.43167	−0.715837	−	0.698268i	$-0.753955\pi$
$23$	−157.920	−1.43167	−0.715837	+	0.698268i	$0.753955\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−178.548	−1.34677
$27$	0	0
$28$	0	0
$29$	−192.402	−1.23201	−0.616003	−	0.787744i	$-0.711249\pi$
$29$	−192.402	−1.23201	−0.616003	+	0.787744i	$0.711249\pi$
$30$	0	0
$31$	−24.2100	−0.140266	−0.0701330	−	0.997538i	$-0.522342\pi$
$31$	−24.2100	−0.140266	−0.0701330	+	0.997538i	$0.522342\pi$
$32$	−9.79689	−0.0541207
$33$	0	0
$34$	−82.6662	−0.416975
$35$	0	0
$36$	0	0
$37$	−62.1980	−0.276359	−0.138180	−	0.990407i	$-0.544125\pi$
$37$	−62.1980	−0.276359	−0.138180	+	0.990407i	$0.544125\pi$
$38$	−122.153	−0.521470
$39$	0	0
$40$	−114.616	−0.453060
$41$	65.8313	0.250759	0.125380	−	0.992109i	$-0.459985\pi$
$41$	65.8313	0.250759	0.125380	+	0.992109i	$0.459985\pi$
$42$	0	0
$43$	97.8275	0.346943	0.173472	−	0.984839i	$-0.444502\pi$
$43$	97.8275	0.346943	0.173472	+	0.984839i	$0.444502\pi$
$44$	0.158318	0.000542440 0
$45$	0	0
$46$	440.577	1.41216
$47$	−432.364	−1.34184	−0.670922	−	0.741528i	$-0.734102\pi$
$47$	−432.364	−1.34184	−0.670922	+	0.741528i	$0.734102\pi$
$48$	0	0
$49$	0	0
$50$	−69.7471	−0.197275
$51$	0	0
$52$	−13.8584	−0.0369579
$53$	−301.048	−0.780228	−0.390114	−	0.920767i	$-0.627564\pi$
$53$	−301.048	−0.780228	−0.390114	+	0.920767i	$0.627564\pi$
$54$	0	0
$55$	3.65559	0.00896217
$56$	0	0
$57$	0	0
$58$	536.780	1.21522
$59$	350.512	0.773438	0.386719	−	0.922198i	$-0.373608\pi$
$59$	350.512	0.773438	0.386719	+	0.922198i	$0.373608\pi$
$60$	0	0
$61$	405.997	0.852173	0.426087	−	0.904682i	$-0.359892\pi$
$61$	405.997	0.852173	0.426087	+	0.904682i	$0.359892\pi$
$62$	67.5432	0.138355
$63$	0	0
$64$	525.098	1.02558
$65$	−319.992	−0.610616
$66$	0	0
$67$	853.677	1.55662	0.778308	−	0.627883i	$-0.216078\pi$
$67$	853.677	1.55662	0.778308	+	0.627883i	$0.216078\pi$
$68$	−6.41631	−0.0114425
$69$	0	0
$70$	0	0
$71$	214.879	0.359175	0.179588	−	0.983742i	$-0.442524\pi$
$71$	214.879	0.359175	0.179588	+	0.983742i	$0.442524\pi$
$72$	0	0
$73$	−1132.86	−1.81632	−0.908162	−	0.418618i	$-0.862515\pi$
$73$	−1132.86	−1.81632	−0.908162	+	0.418618i	$0.862515\pi$
$74$	173.525	0.272593
$75$	0	0
$76$	−9.48117	−0.0143101
$77$	0	0
$78$	0	0
$79$	51.4032	0.0732064	0.0366032	−	0.999330i	$-0.488346\pi$
$79$	51.4032	0.0732064	0.0366032	+	0.999330i	$0.488346\pi$
$80$	311.104	0.434781
$81$	0	0
$82$	−183.662	−0.247342
$83$	1411.04	1.86604	0.933022	−	0.359820i	$-0.117162\pi$
$83$	1411.04	1.86604	0.933022	+	0.359820i	$0.117162\pi$
$84$	0	0
$85$	−148.153	−0.189053
$86$	−272.928	−0.342216
$87$	0	0
$88$	−16.7596	−0.0203020
$89$	266.685	0.317624	0.158812	−	0.987309i	$-0.449234\pi$
$89$	266.685	0.317624	0.158812	+	0.987309i	$0.449234\pi$
$90$	0	0
$91$	0	0
$92$	34.1963	0.0387523
$93$	0	0
$94$	1206.24	1.32356
$95$	−218.921	−0.236430
$96$	0	0
$97$	−546.792	−0.572354	−0.286177	−	0.958177i	$-0.592385\pi$
$97$	−546.792	−0.572354	−0.286177	+	0.958177i	$0.592385\pi$
$98$	0	0
$99$	0	0
$100$	−5.41357	−0.00541357
$101$	−968.034	−0.953693	−0.476847	−	0.878987i	$-0.658220\pi$
$101$	−968.034	−0.953693	−0.476847	+	0.878987i	$0.658220\pi$
$102$	0	0
$103$	585.116	0.559740	0.279870	−	0.960038i	$-0.409709\pi$
$103$	585.116	0.559740	0.279870	+	0.960038i	$0.409709\pi$
$104$	1467.05	1.38323
$105$	0	0
$106$	839.889	0.769596
$107$	−1673.12	−1.51165	−0.755825	−	0.654774i	$-0.772764\pi$
$107$	−1673.12	−1.51165	−0.755825	+	0.654774i	$0.772764\pi$
$108$	0	0
$109$	1742.46	1.53117	0.765585	−	0.643334i	$-0.222450\pi$
$109$	1742.46	1.53117	0.765585	+	0.643334i	$0.222450\pi$
$110$	−10.1987	−0.00884005
$111$	0	0
$112$	0	0
$113$	1436.28	1.19570	0.597850	−	0.801608i	$-0.296022\pi$
$113$	1436.28	1.19570	0.597850	+	0.801608i	$0.296022\pi$
$114$	0	0
$115$	789.598	0.640264
$116$	41.6633	0.0333478
$117$	0	0
$118$	−977.889	−0.762898
$119$	0	0
$120$	0	0
$121$	−1330.47	−0.999598
$122$	−1132.68	−0.840561
$123$	0	0
$124$	5.24250	0.00379670
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−733.011	−0.512159	−0.256079	−	0.966656i	$-0.582431\pi$
$127$	−733.011	−0.512159	−0.256079	+	0.966656i	$0.582431\pi$
$128$	−1386.59	−0.957487
$129$	0	0
$130$	892.740	0.602296
$131$	660.369	0.440433	0.220217	−	0.975451i	$-0.429324\pi$
$131$	660.369	0.440433	0.220217	+	0.975451i	$0.429324\pi$
$132$	0	0
$133$	0	0
$134$	−2381.66	−1.53540
$135$	0	0
$136$	679.230	0.428261
$137$	−1102.19	−0.687344	−0.343672	−	0.939090i	$-0.611671\pi$
$137$	−1102.19	−0.687344	−0.343672	+	0.939090i	$0.611671\pi$
$138$	0	0
$139$	−959.854	−0.585711	−0.292855	−	0.956157i	$-0.594605\pi$
$139$	−959.854	−0.585711	−0.292855	+	0.956157i	$0.594605\pi$
$140$	0	0
$141$	0	0
$142$	−599.487	−0.354281
$143$	−46.7903	−0.0273622
$144$	0	0
$145$	962.011	0.550970
$146$	3160.56	1.79157
$147$	0	0
$148$	13.4685	0.00748045
$149$	1408.20	0.774257	0.387129	−	0.922026i	$-0.373467\pi$
$149$	1408.20	0.774257	0.387129	+	0.922026i	$0.373467\pi$
$150$	0	0
$151$	−926.239	−0.499180	−0.249590	−	0.968352i	$-0.580296\pi$
$151$	−926.239	−0.499180	−0.249590	+	0.968352i	$0.580296\pi$
$152$	1003.68	0.535585
$153$	0	0
$154$	0	0
$155$	121.050	0.0627289
$156$	0	0
$157$	2579.44	1.31122	0.655610	−	0.755099i	$-0.272411\pi$
$157$	2579.44	1.31122	0.655610	+	0.755099i	$0.272411\pi$
$158$	−143.409	−0.0722088
$159$	0	0
$160$	48.9844	0.0242035
$161$	0	0
$162$	0	0
$163$	−2004.72	−0.963325	−0.481662	−	0.876357i	$-0.659967\pi$
$163$	−2004.72	−0.963325	−0.481662	+	0.876357i	$0.659967\pi$
$164$	−14.2553	−0.00678751
$165$	0	0
$166$	−3936.64	−1.84062
$167$	895.797	0.415083	0.207541	−	0.978226i	$-0.433454\pi$
$167$	895.797	0.415083	0.207541	+	0.978226i	$0.433454\pi$
$168$	0	0
$169$	1898.78	0.864262
$170$	413.331	0.186477
$171$	0	0
$172$	−21.1838	−0.00939101
$173$	−1252.14	−0.550279	−0.275140	−	0.961404i	$-0.588724\pi$
$173$	−1252.14	−0.550279	−0.275140	+	0.961404i	$0.588724\pi$
$174$	0	0
$175$	0	0
$176$	45.4907	0.0194829
$177$	0	0
$178$	−744.020	−0.313296
$179$	4175.60	1.74357	0.871785	−	0.489889i	$-0.162963\pi$
$179$	4175.60	1.74357	0.871785	+	0.489889i	$0.162963\pi$
$180$	0	0
$181$	−1773.00	−0.728098	−0.364049	−	0.931380i	$-0.618606\pi$
$181$	−1773.00	−0.728098	−0.364049	+	0.931380i	$0.618606\pi$
$182$	0	0
$183$	0	0
$184$	−3620.02	−1.45039
$185$	310.990	0.123592
$186$	0	0
$187$	−21.6635	−0.00847162
$188$	93.6252	0.0363208
$189$	0	0
$190$	610.766	0.233208
$191$	−2864.67	−1.08524	−0.542619	−	0.839979i	$-0.682567\pi$
$191$	−2864.67	−1.08524	−0.542619	+	0.839979i	$0.682567\pi$
$192$	0	0
$193$	1638.56	0.611119	0.305559	−	0.952173i	$-0.401157\pi$
$193$	1638.56	0.611119	0.305559	+	0.952173i	$0.401157\pi$
$194$	1525.49	0.564555
$195$	0	0
$196$	0	0
$197$	4927.04	1.78192	0.890958	−	0.454086i	$-0.150034\pi$
$197$	4927.04	1.78192	0.890958	+	0.454086i	$0.150034\pi$
$198$	0	0
$199$	−4727.29	−1.68396	−0.841982	−	0.539506i	$-0.818611\pi$
$199$	−4727.29	−1.68396	−0.841982	+	0.539506i	$0.818611\pi$
$200$	573.080	0.202614
$201$	0	0
$202$	2700.70	0.940697
$203$	0	0
$204$	0	0
$205$	−329.157	−0.112143
$206$	−1632.41	−0.552113
$207$	0	0
$208$	−3982.02	−1.32742
$209$	−32.0115	−0.0105946
$210$	0	0
$211$	−4450.18	−1.45196	−0.725978	−	0.687718i	$-0.758613\pi$
$211$	−4450.18	−1.45196	−0.725978	+	0.687718i	$0.758613\pi$
$212$	65.1897	0.0211191
$213$	0	0
$214$	4667.81	1.49105
$215$	−489.138	−0.155158
$216$	0	0
$217$	0	0
$218$	−4861.27	−1.51031
$219$	0	0
$220$	−0.791591	−0.000242587 0
$221$	1896.31	0.577194
$222$	0	0
$223$	2400.22	0.720766	0.360383	−	0.932804i	$-0.382646\pi$
$223$	2400.22	0.720766	0.360383	+	0.932804i	$0.382646\pi$
$224$	0	0
$225$	0	0
$226$	−4007.06	−1.17941
$227$	−4793.00	−1.40142	−0.700710	−	0.713446i	$-0.747133\pi$
$227$	−4793.00	−1.40142	−0.700710	+	0.713446i	$0.747133\pi$
$228$	0	0
$229$	2051.56	0.592014	0.296007	−	0.955186i	$-0.404345\pi$
$229$	2051.56	0.592014	0.296007	+	0.955186i	$0.404345\pi$
$230$	−2202.89	−0.631539
$231$	0	0
$232$	−4410.47	−1.24811
$233$	−5459.91	−1.53515	−0.767576	−	0.640958i	$-0.778537\pi$
$233$	−5459.91	−1.53515	−0.767576	+	0.640958i	$0.778537\pi$
$234$	0	0
$235$	2161.82	0.600091
$236$	−75.9009	−0.0209353
$237$	0	0
$238$	0	0
$239$	−2381.20	−0.644465	−0.322233	−	0.946661i	$-0.604433\pi$
$239$	−2381.20	−0.644465	−0.322233	+	0.946661i	$0.604433\pi$
$240$	0	0
$241$	−1653.02	−0.441828	−0.220914	−	0.975293i	$-0.570904\pi$
$241$	−1653.02	−0.441828	−0.220914	+	0.975293i	$0.570904\pi$
$242$	3711.85	0.985977
$243$	0	0
$244$	−87.9157	−0.0230665
$245$	0	0
$246$	0	0
$247$	2802.12	0.721841
$248$	−554.971	−0.142100
$249$	0	0
$250$	348.736	0.0882239
$251$	−1231.36	−0.309652	−0.154826	−	0.987942i	$-0.549482\pi$
$251$	−1231.36	−0.309652	−0.154826	+	0.987942i	$0.549482\pi$
$252$	0	0
$253$	115.458	0.0286908
$254$	2045.02	0.505180
$255$	0	0
$256$	−332.364	−0.0811435
$257$	−671.457	−0.162974	−0.0814871	−	0.996674i	$-0.525967\pi$
$257$	−671.457	−0.162974	−0.0814871	+	0.996674i	$0.525967\pi$
$258$	0	0
$259$	0	0
$260$	69.2919	0.0165281
$261$	0	0
$262$	−1842.35	−0.434432
$263$	2703.80	0.633928	0.316964	−	0.948437i	$-0.397336\pi$
$263$	2703.80	0.633928	0.316964	+	0.948437i	$0.397336\pi$
$264$	0	0
$265$	1505.24	0.348929
$266$	0	0
$267$	0	0
$268$	−184.858	−0.0421342
$269$	3301.70	0.748358	0.374179	−	0.927357i	$-0.377925\pi$
$269$	3301.70	0.748358	0.374179	+	0.927357i	$0.377925\pi$
$270$	0	0
$271$	−2324.28	−0.520995	−0.260498	−	0.965474i	$-0.583887\pi$
$271$	−2324.28	−0.520995	−0.260498	+	0.965474i	$0.583887\pi$
$272$	−1843.64	−0.410983
$273$	0	0
$274$	3074.97	0.677977
$275$	−18.2779	−0.00400801
$276$	0	0
$277$	8571.17	1.85918	0.929588	−	0.368600i	$-0.120163\pi$
$277$	8571.17	1.85918	0.929588	+	0.368600i	$0.120163\pi$
$278$	2677.88	0.577729
$279$	0	0
$280$	0	0
$281$	1009.22	0.214252	0.107126	−	0.994245i	$-0.465835\pi$
$281$	1009.22	0.214252	0.107126	+	0.994245i	$0.465835\pi$
$282$	0	0
$283$	−9099.63	−1.91137	−0.955684	−	0.294395i	$-0.904882\pi$
$283$	−9099.63	−1.91137	−0.955684	+	0.294395i	$0.904882\pi$
$284$	−46.5305	−0.00972209
$285$	0	0
$286$	130.540	0.0269894
$287$	0	0
$288$	0	0
$289$	−4035.02	−0.821295
$290$	−2683.90	−0.543462
$291$	0	0
$292$	245.313	0.0491640
$293$	1843.00	0.367471	0.183736	−	0.982976i	$-0.441181\pi$
$293$	1843.00	0.367471	0.183736	+	0.982976i	$0.441181\pi$
$294$	0	0
$295$	−1752.56	−0.345892
$296$	−1425.78	−0.279972
$297$	0	0
$298$	−3928.72	−0.763707
$299$	−10106.6	−1.95478
$300$	0	0
$301$	0	0
$302$	2584.10	0.492378
$303$	0	0
$304$	−2724.29	−0.513977
$305$	−2029.98	−0.381103
$306$	0	0
$307$	3479.16	0.646796	0.323398	−	0.946263i	$-0.395175\pi$
$307$	3479.16	0.646796	0.323398	+	0.946263i	$0.395175\pi$
$308$	0	0
$309$	0	0
$310$	−337.716	−0.0618741
$311$	9352.18	1.70519	0.852594	−	0.522574i	$-0.175028\pi$
$311$	9352.18	1.70519	0.852594	+	0.522574i	$0.175028\pi$
$312$	0	0
$313$	−8798.47	−1.58888	−0.794439	−	0.607343i	$-0.792235\pi$
$313$	−8798.47	−1.58888	−0.794439	+	0.607343i	$0.792235\pi$
$314$	−7196.34	−1.29335
$315$	0	0
$316$	−11.1310	−0.00198154
$317$	−4292.39	−0.760519	−0.380260	−	0.924880i	$-0.624165\pi$
$317$	−4292.39	−0.760519	−0.380260	+	0.924880i	$0.624165\pi$
$318$	0	0
$319$	140.669	0.0246894
$320$	−2625.49	−0.458655
$321$	0	0
$322$	0	0
$323$	1297.36	0.223489
$324$	0	0
$325$	1599.96	0.273076
$326$	5592.94	0.950198
$327$	0	0
$328$	1509.06	0.254037
$329$	0	0
$330$	0	0
$331$	2894.70	0.480687	0.240343	−	0.970688i	$-0.422740\pi$
$331$	2894.70	0.480687	0.240343	+	0.970688i	$0.422740\pi$
$332$	−305.550	−0.0505098
$333$	0	0
$334$	−2499.17	−0.409427
$335$	−4268.38	−0.696140
$336$	0	0
$337$	−3578.67	−0.578464	−0.289232	−	0.957259i	$-0.593400\pi$
$337$	−3578.67	−0.578464	−0.289232	+	0.957259i	$0.593400\pi$
$338$	−5297.39	−0.852485
$339$	0	0
$340$	32.0816	0.00511725
$341$	17.7004	0.00281093
$342$	0	0
$343$	0	0
$344$	2242.52	0.351479
$345$	0	0
$346$	3493.32	0.542781
$347$	5633.16	0.871481	0.435741	−	0.900072i	$-0.356486\pi$
$347$	5633.16	0.871481	0.435741	+	0.900072i	$0.356486\pi$
$348$	0	0
$349$	3643.34	0.558807	0.279404	−	0.960174i	$-0.409863\pi$
$349$	3643.34	0.558807	0.279404	+	0.960174i	$0.409863\pi$
$350$	0	0
$351$	0	0
$352$	7.16268	0.00108458
$353$	11996.2	1.80877	0.904383	−	0.426721i	$-0.140331\pi$
$353$	11996.2	1.80877	0.904383	+	0.426721i	$0.140331\pi$
$354$	0	0
$355$	−1074.39	−0.160628
$356$	−57.7486	−0.00859739
$357$	0	0
$358$	−11649.4	−1.71981
$359$	3809.64	0.560070	0.280035	−	0.959990i	$-0.409654\pi$
$359$	3809.64	0.560070	0.280035	+	0.959990i	$0.409654\pi$
$360$	0	0
$361$	−4941.94	−0.720504
$362$	4946.46	0.718177
$363$	0	0
$364$	0	0
$365$	5664.32	0.812285
$366$	0	0
$367$	−11373.0	−1.61761	−0.808807	−	0.588074i	$-0.799886\pi$
$367$	−11373.0	−1.61761	−0.808807	+	0.588074i	$0.799886\pi$
$368$	9825.87	1.39187
$369$	0	0
$370$	−867.626	−0.121907
$371$	0	0
$372$	0	0
$373$	11378.1	1.57945	0.789726	−	0.613460i	$-0.210223\pi$
$373$	11378.1	1.57945	0.789726	+	0.613460i	$0.210223\pi$
$374$	60.4387	0.00835618
$375$	0	0
$376$	−9911.16	−1.35939
$377$	−12313.4	−1.68216
$378$	0	0
$379$	6703.78	0.908575	0.454287	−	0.890855i	$-0.349894\pi$
$379$	6703.78	0.908575	0.454287	+	0.890855i	$0.349894\pi$
$380$	47.4059	0.00639966
$381$	0	0
$382$	7992.11	1.07045
$383$	−1331.05	−0.177581	−0.0887904	−	0.996050i	$-0.528300\pi$
$383$	−1331.05	−0.177581	−0.0887904	+	0.996050i	$0.528300\pi$
$384$	0	0
$385$	0	0
$386$	−4571.39	−0.602791
$387$	0	0
$388$	118.404	0.0154924
$389$	−7343.70	−0.957173	−0.478586	−	0.878040i	$-0.658851\pi$
$389$	−7343.70	−0.957173	−0.478586	+	0.878040i	$0.658851\pi$
$390$	0	0
$391$	−4679.26	−0.605219
$392$	0	0
$393$	0	0
$394$	−13745.9	−1.75763
$395$	−257.016	−0.0327389
$396$	0	0
$397$	−7022.99	−0.887843	−0.443921	−	0.896066i	$-0.646413\pi$
$397$	−7022.99	−0.887843	−0.443921	+	0.896066i	$0.646413\pi$
$398$	13188.6	1.66102
$399$	0	0
$400$	−1555.52	−0.194440
$401$	−1321.37	−0.164554	−0.0822769	−	0.996610i	$-0.526219\pi$
$401$	−1321.37	−0.164554	−0.0822769	+	0.996610i	$0.526219\pi$
$402$	0	0
$403$	−1549.40	−0.191516
$404$	209.621	0.0258144
$405$	0	0
$406$	0	0
$407$	45.4741	0.00553824
$408$	0	0
$409$	−7688.44	−0.929509	−0.464754	−	0.885440i	$-0.653857\pi$
$409$	−7688.44	−0.929509	−0.464754	+	0.885440i	$0.653857\pi$
$410$	918.309	0.110615
$411$	0	0
$412$	−126.703	−0.0151510
$413$	0	0
$414$	0	0
$415$	−7055.19	−0.834520
$416$	−626.984	−0.0738953
$417$	0	0
$418$	89.3083	0.0104503
$419$	−8557.89	−0.997805	−0.498903	−	0.866658i	$-0.666263\pi$
$419$	−8557.89	−0.997805	−0.498903	+	0.866658i	$0.666263\pi$
$420$	0	0
$421$	−5035.39	−0.582922	−0.291461	−	0.956583i	$-0.594141\pi$
$421$	−5035.39	−0.582922	−0.291461	+	0.956583i	$0.594141\pi$
$422$	12415.5	1.43217
$423$	0	0
$424$	−6900.98	−0.790428
$425$	740.767	0.0845470
$426$	0	0
$427$	0	0
$428$	362.302	0.0409171
$429$	0	0
$430$	1364.64	0.153043
$431$	−12084.9	−1.35060	−0.675300	−	0.737543i	$-0.735986\pi$
$431$	−12084.9	−1.35060	−0.675300	+	0.737543i	$0.735986\pi$
$432$	0	0
$433$	−17171.7	−1.90582	−0.952911	−	0.303251i	$-0.901928\pi$
$433$	−17171.7	−1.90582	−0.952911	+	0.303251i	$0.901928\pi$
$434$	0	0
$435$	0	0
$436$	−377.318	−0.0414455
$437$	−6914.39	−0.756889
$438$	0	0
$439$	−194.147	−0.0211074	−0.0105537	−	0.999944i	$-0.503359\pi$
$439$	−194.147	−0.0211074	−0.0105537	+	0.999944i	$0.503359\pi$
$440$	83.7978	0.00907933
$441$	0	0
$442$	−5290.50	−0.569329
$443$	9741.13	1.04473	0.522365	−	0.852722i	$-0.325050\pi$
$443$	9741.13	1.04473	0.522365	+	0.852722i	$0.325050\pi$
$444$	0	0
$445$	−1333.42	−0.142046
$446$	−6696.34	−0.710944
$447$	0	0
$448$	0	0
$449$	−11654.8	−1.22500	−0.612499	−	0.790471i	$-0.709836\pi$
$449$	−11654.8	−1.22500	−0.612499	+	0.790471i	$0.709836\pi$
$450$	0	0
$451$	−48.1304	−0.00502522
$452$	−311.017	−0.0323650
$453$	0	0
$454$	13371.9	1.38232
$455$	0	0
$456$	0	0
$457$	13348.6	1.36634	0.683172	−	0.730258i	$-0.260600\pi$
$457$	13348.6	1.36634	0.683172	+	0.730258i	$0.260600\pi$
$458$	−5723.63	−0.583947
$459$	0	0
$460$	−170.982	−0.0173306
$461$	−14731.8	−1.48835	−0.744174	−	0.667986i	$-0.767157\pi$
$461$	−14731.8	−1.48835	−0.744174	+	0.667986i	$0.767157\pi$
$462$	0	0
$463$	−6127.24	−0.615026	−0.307513	−	0.951544i	$-0.599497\pi$
$463$	−6127.24	−0.615026	−0.307513	+	0.951544i	$0.599497\pi$
$464$	11971.4	1.19776
$465$	0	0
$466$	15232.5	1.51423
$467$	−9314.99	−0.923012	−0.461506	−	0.887137i	$-0.652691\pi$
$467$	−9314.99	−0.923012	−0.461506	+	0.887137i	$0.652691\pi$
$468$	0	0
$469$	0	0
$470$	−6031.22	−0.591914
$471$	0	0
$472$	8034.87	0.783548
$473$	−71.5234	−0.00695275
$474$	0	0
$475$	1094.61	0.105735
$476$	0	0
$477$	0	0
$478$	6643.28	0.635683
$479$	−12944.7	−1.23478	−0.617388	−	0.786659i	$-0.711809\pi$
$479$	−12944.7	−1.23478	−0.617388	+	0.786659i	$0.711809\pi$
$480$	0	0
$481$	−3980.57	−0.377335
$482$	4611.75	0.435808
$483$	0	0
$484$	288.103	0.0270570
$485$	2733.96	0.255965
$486$	0	0
$487$	3020.84	0.281083	0.140542	−	0.990075i	$-0.455116\pi$
$487$	3020.84	0.281083	0.140542	+	0.990075i	$0.455116\pi$
$488$	9306.75	0.863313
$489$	0	0
$490$	0	0
$491$	−1787.34	−0.164280	−0.0821400	−	0.996621i	$-0.526175\pi$
$491$	−1787.34	−0.164280	−0.0821400	+	0.996621i	$0.526175\pi$
$492$	0	0
$493$	−5701.01	−0.520812
$494$	−7817.59	−0.712004
$495$	0	0
$496$	1506.37	0.136367
$497$	0	0
$498$	0	0
$499$	−4646.59	−0.416853	−0.208427	−	0.978038i	$-0.566834\pi$
$499$	−4646.59	−0.416853	−0.208427	+	0.978038i	$0.566834\pi$
$500$	27.0678	0.00242102
$501$	0	0
$502$	3435.35	0.305432
$503$	3776.03	0.334721	0.167361	−	0.985896i	$-0.446476\pi$
$503$	3776.03	0.334721	0.167361	+	0.985896i	$0.446476\pi$
$504$	0	0
$505$	4840.17	0.426505
$506$	−322.114	−0.0282998
$507$	0	0
$508$	158.728	0.0138630
$509$	−12236.2	−1.06554	−0.532772	−	0.846259i	$-0.678850\pi$
$509$	−12236.2	−1.06554	−0.532772	+	0.846259i	$0.678850\pi$
$510$	0	0
$511$	0	0
$512$	12020.0	1.03752
$513$	0	0
$514$	1873.29	0.160753
$515$	−2925.58	−0.250323
$516$	0	0
$517$	316.109	0.0268906
$518$	0	0
$519$	0	0
$520$	−7335.23	−0.618599
$521$	−6367.49	−0.535441	−0.267721	−	0.963497i	$-0.586270\pi$
$521$	−6367.49	−0.535441	−0.267721	+	0.963497i	$0.586270\pi$
$522$	0	0
$523$	−6786.42	−0.567398	−0.283699	−	0.958913i	$-0.591562\pi$
$523$	−6786.42	−0.567398	−0.283699	+	0.958913i	$0.591562\pi$
$524$	−142.998	−0.0119216
$525$	0	0
$526$	−7543.28	−0.625290
$527$	−717.359	−0.0592954
$528$	0	0
$529$	12771.6	1.04969
$530$	−4199.44	−0.344174
$531$	0	0
$532$	0	0
$533$	4213.09	0.342381
$534$	0	0
$535$	8365.59	0.676030
$536$	19569.0	1.57696
$537$	0	0
$538$	−9211.37	−0.738160
$539$	0	0
$540$	0	0
$541$	−14729.1	−1.17052	−0.585262	−	0.810844i	$-0.699008\pi$
$541$	−14729.1	−1.17052	−0.585262	+	0.810844i	$0.699008\pi$
$542$	6484.46	0.513896
$543$	0	0
$544$	−290.289	−0.0228787
$545$	−8712.31	−0.684760
$546$	0	0
$547$	12444.4	0.972734	0.486367	−	0.873755i	$-0.338322\pi$
$547$	12444.4	0.972734	0.486367	+	0.873755i	$0.338322\pi$
$548$	238.670	0.0186049
$549$	0	0
$550$	50.9934	0.00395339
$551$	−8424.19	−0.651330
$552$	0	0
$553$	0	0
$554$	−23912.6	−1.83384
$555$	0	0
$556$	207.849	0.0158539
$557$	11188.3	0.851100	0.425550	−	0.904935i	$-0.360081\pi$
$557$	11188.3	0.851100	0.425550	+	0.904935i	$0.360081\pi$
$558$	0	0
$559$	6260.80	0.473709
$560$	0	0
$561$	0	0
$562$	−2815.60	−0.211332
$563$	−22723.4	−1.70103	−0.850513	−	0.525954i	$-0.823708\pi$
$563$	−22723.4	−1.70103	−0.850513	+	0.525954i	$0.823708\pi$
$564$	0	0
$565$	−7181.41	−0.534733
$566$	25386.9	1.88532
$567$	0	0
$568$	4925.71	0.363870
$569$	16424.6	1.21011	0.605057	−	0.796182i	$-0.293150\pi$
$569$	16424.6	1.21011	0.605057	+	0.796182i	$0.293150\pi$
$570$	0	0
$571$	−24106.3	−1.76676	−0.883378	−	0.468660i	$-0.844737\pi$
$571$	−24106.3	−1.76676	−0.883378	+	0.468660i	$0.844737\pi$
$572$	10.1321	0.000740637 0
$573$	0	0
$574$	0	0
$575$	−3947.99	−0.286335
$576$	0	0
$577$	−15925.5	−1.14902	−0.574512	−	0.818496i	$-0.694808\pi$
$577$	−15925.5	−1.14902	−0.574512	+	0.818496i	$0.694808\pi$
$578$	11257.2	0.810103
$579$	0	0
$580$	−208.316	−0.0149136
$581$	0	0
$582$	0	0
$583$	220.101	0.0156358
$584$	−25968.9	−1.84007
$585$	0	0
$586$	−5141.75	−0.362464
$587$	−8033.03	−0.564836	−0.282418	−	0.959291i	$-0.591136\pi$
$587$	−8033.03	−0.564836	−0.282418	+	0.959291i	$0.591136\pi$
$588$	0	0
$589$	−1060.02	−0.0741550
$590$	4889.45	0.341179
$591$	0	0
$592$	3870.01	0.268676
$593$	5753.12	0.398402	0.199201	−	0.979959i	$-0.436165\pi$
$593$	5753.12	0.398402	0.199201	+	0.979959i	$0.436165\pi$
$594$	0	0
$595$	0	0
$596$	−304.936	−0.0209575
$597$	0	0
$598$	28196.2	1.92814
$599$	−17086.5	−1.16550	−0.582752	−	0.812650i	$-0.698024\pi$
$599$	−17086.5	−1.16550	−0.582752	+	0.812650i	$0.698024\pi$
$600$	0	0
$601$	−27943.6	−1.89658	−0.948290	−	0.317405i	$-0.897189\pi$
$601$	−27943.6	−1.89658	−0.948290	+	0.317405i	$0.897189\pi$
$602$	0	0
$603$	0	0
$604$	200.570	0.0135117
$605$	6652.33	0.447034
$606$	0	0
$607$	18259.9	1.22100	0.610500	−	0.792016i	$-0.290969\pi$
$607$	18259.9	1.22100	0.610500	+	0.792016i	$0.290969\pi$
$608$	−428.950	−0.0286122
$609$	0	0
$610$	5663.42	0.375910
$611$	−27670.5	−1.83213
$612$	0	0
$613$	17268.4	1.13779	0.568895	−	0.822410i	$-0.307371\pi$
$613$	17268.4	1.13779	0.568895	+	0.822410i	$0.307371\pi$
$614$	−9706.46	−0.637982
$615$	0	0
$616$	0	0
$617$	−27762.0	−1.81144	−0.905719	−	0.423878i	$-0.860669\pi$
$617$	−27762.0	−1.81144	−0.905719	+	0.423878i	$0.860669\pi$
$618$	0	0
$619$	−11829.9	−0.768147	−0.384074	−	0.923302i	$-0.625479\pi$
$619$	−11829.9	−0.768147	−0.384074	+	0.923302i	$0.625479\pi$
$620$	−26.2125	−0.00169794
$621$	0	0
$622$	−26091.5	−1.68195
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	24546.7	1.56723
$627$	0	0
$628$	−558.559	−0.0354919
$629$	−1842.97	−0.116827
$630$	0	0
$631$	1835.49	0.115800	0.0579000	−	0.998322i	$-0.481560\pi$
$631$	1835.49	0.115800	0.0579000	+	0.998322i	$0.481560\pi$
$632$	1178.33	0.0741634
$633$	0	0
$634$	11975.3	0.750156
$635$	3665.05	0.229044
$636$	0	0
$637$	0	0
$638$	−392.449	−0.0243530
$639$	0	0
$640$	6932.94	0.428201
$641$	−21034.5	−1.29612	−0.648060	−	0.761589i	$-0.724419\pi$
$641$	−21034.5	−1.29612	−0.648060	+	0.761589i	$0.724419\pi$
$642$	0	0
$643$	−27368.9	−1.67858	−0.839288	−	0.543687i	$-0.817028\pi$
$643$	−27368.9	−1.67858	−0.839288	+	0.543687i	$0.817028\pi$
$644$	0	0
$645$	0	0
$646$	−3619.48	−0.220444
$647$	6272.47	0.381138	0.190569	−	0.981674i	$-0.438967\pi$
$647$	6272.47	0.381138	0.190569	+	0.981674i	$0.438967\pi$
$648$	0	0
$649$	−256.266	−0.0154997
$650$	−4463.70	−0.269355
$651$	0	0
$652$	434.108	0.0260751
$653$	−14081.2	−0.843856	−0.421928	−	0.906629i	$-0.638647\pi$
$653$	−14081.2	−0.843856	−0.421928	+	0.906629i	$0.638647\pi$
$654$	0	0
$655$	−3301.85	−0.196968
$656$	−4096.07	−0.243788
$657$	0	0
$658$	0	0
$659$	−3490.76	−0.206344	−0.103172	−	0.994664i	$-0.532899\pi$
$659$	−3490.76	−0.206344	−0.103172	+	0.994664i	$0.532899\pi$
$660$	0	0
$661$	30723.5	1.80787	0.903936	−	0.427667i	$-0.140664\pi$
$661$	30723.5	1.80787	0.903936	+	0.427667i	$0.140664\pi$
$662$	−8075.89	−0.474137
$663$	0	0
$664$	32345.5	1.89044
$665$	0	0
$666$	0	0
$667$	30384.1	1.76383
$668$	−193.978	−0.0112354
$669$	0	0
$670$	11908.3	0.686653
$671$	−296.831	−0.0170776
$672$	0	0
$673$	−3435.85	−0.196794	−0.0983968	−	0.995147i	$-0.531371\pi$
$673$	−3435.85	−0.196794	−0.0983968	+	0.995147i	$0.531371\pi$
$674$	9984.07	0.570581
$675$	0	0
$676$	−411.168	−0.0233937
$677$	19600.2	1.11270	0.556350	−	0.830948i	$-0.312201\pi$
$677$	19600.2	1.11270	0.556350	+	0.830948i	$0.312201\pi$
$678$	0	0
$679$	0	0
$680$	−3396.15	−0.191524
$681$	0	0
$682$	−49.3820	−0.00277263
$683$	−28431.7	−1.59284	−0.796418	−	0.604746i	$-0.793274\pi$
$683$	−28431.7	−1.59284	−0.796418	+	0.604746i	$0.793274\pi$
$684$	0	0
$685$	5510.93	0.307389
$686$	0	0
$687$	0	0
$688$	−6086.90	−0.337298
$689$	−19266.6	−1.06531
$690$	0	0
$691$	−853.786	−0.0470037	−0.0235019	−	0.999724i	$-0.507482\pi$
$691$	−853.786	−0.0470037	−0.0235019	+	0.999724i	$0.507482\pi$
$692$	271.141	0.0148949
$693$	0	0
$694$	−15715.9	−0.859606
$695$	4799.27	0.261938
$696$	0	0
$697$	1950.63	0.106005
$698$	−10164.5	−0.551192
$699$	0	0
$700$	0	0
$701$	6952.79	0.374612	0.187306	−	0.982302i	$-0.440024\pi$
$701$	6952.79	0.374612	0.187306	+	0.982302i	$0.440024\pi$
$702$	0	0
$703$	−2723.30	−0.146104
$704$	−383.909	−0.0205527
$705$	0	0
$706$	−33468.1	−1.78412
$707$	0	0
$708$	0	0
$709$	22740.2	1.20455	0.602275	−	0.798289i	$-0.294261\pi$
$709$	22740.2	1.20455	0.602275	+	0.798289i	$0.294261\pi$
$710$	2997.44	0.158439
$711$	0	0
$712$	6113.27	0.321776
$713$	3823.23	0.200815
$714$	0	0
$715$	233.952	0.0122368
$716$	−904.196	−0.0471947
$717$	0	0
$718$	−10628.5	−0.552438
$719$	−9318.04	−0.483316	−0.241658	−	0.970361i	$-0.577691\pi$
$719$	−9318.04	−0.483316	−0.241658	+	0.970361i	$0.577691\pi$
$720$	0	0
$721$	0	0
$722$	13787.4	0.710686
$723$	0	0
$724$	383.930	0.0197080
$725$	−4810.05	−0.246401
$726$	0	0
$727$	−19866.3	−1.01348	−0.506740	−	0.862099i	$-0.669150\pi$
$727$	−19866.3	−1.01348	−0.506740	+	0.862099i	$0.669150\pi$
$728$	0	0
$729$	0	0
$730$	−15802.8	−0.801216
$731$	2898.70	0.146665
$732$	0	0
$733$	−2009.24	−0.101246	−0.0506228	−	0.998718i	$-0.516121\pi$
$733$	−2009.24	−0.101246	−0.0506228	+	0.998718i	$0.516121\pi$
$734$	31729.3	1.59557
$735$	0	0
$736$	1547.12	0.0774831
$737$	−624.138	−0.0311946
$738$	0	0
$739$	7608.18	0.378716	0.189358	−	0.981908i	$-0.439359\pi$
$739$	7608.18	0.378716	0.189358	+	0.981908i	$0.439359\pi$
$740$	−67.3426	−0.00334536
$741$	0	0
$742$	0	0
$743$	19382.7	0.957040	0.478520	−	0.878077i	$-0.341173\pi$
$743$	19382.7	0.957040	0.478520	+	0.878077i	$0.341173\pi$
$744$	0	0
$745$	−7041.01	−0.346258
$746$	−31743.6	−1.55793
$747$	0	0
$748$	4.69108	0.000229309 0
$749$	0	0
$750$	0	0
$751$	35182.8	1.70951	0.854753	−	0.519035i	$-0.173709\pi$
$751$	35182.8	1.70951	0.854753	+	0.519035i	$0.173709\pi$
$752$	26902.0	1.30454
$753$	0	0
$754$	34353.0	1.65923
$755$	4631.19	0.223240
$756$	0	0
$757$	32409.1	1.55605	0.778025	−	0.628234i	$-0.216222\pi$
$757$	32409.1	1.55605	0.778025	+	0.628234i	$0.216222\pi$
$758$	−18702.8	−0.896194
$759$	0	0
$760$	−5018.38	−0.239521
$761$	3587.94	0.170910	0.0854551	−	0.996342i	$-0.472766\pi$
$761$	3587.94	0.170910	0.0854551	+	0.996342i	$0.472766\pi$
$762$	0	0
$763$	0	0
$764$	620.324	0.0293751
$765$	0	0
$766$	3713.48	0.175161
$767$	22432.2	1.05604
$768$	0	0
$769$	36593.9	1.71601	0.858003	−	0.513644i	$-0.171705\pi$
$769$	36593.9	1.71601	0.858003	+	0.513644i	$0.171705\pi$
$770$	0	0
$771$	0	0
$772$	−354.818	−0.0165417
$773$	−3543.13	−0.164861	−0.0824306	−	0.996597i	$-0.526268\pi$
$773$	−3543.13	−0.164861	−0.0824306	+	0.996597i	$0.526268\pi$
$774$	0	0
$775$	−605.250	−0.0280532
$776$	−12534.2	−0.579836
$777$	0	0
$778$	20488.1	0.944130
$779$	2882.38	0.132570
$780$	0	0
$781$	−157.102	−0.00719788
$782$	13054.6	0.596971
$783$	0	0
$784$	0	0
$785$	−12897.2	−0.586396
$786$	0	0
$787$	−33793.1	−1.53061	−0.765307	−	0.643665i	$-0.777413\pi$
$787$	−33793.1	−1.53061	−0.765307	+	0.643665i	$0.777413\pi$
$788$	−1066.92	−0.0482326
$789$	0	0
$790$	717.044	0.0322928
$791$	0	0
$792$	0	0
$793$	25983.1	1.16354
$794$	19593.3	0.875744
$795$	0	0
$796$	1023.66	0.0455813
$797$	3931.05	0.174711	0.0873556	−	0.996177i	$-0.472158\pi$
$797$	3931.05	0.174711	0.0873556	+	0.996177i	$0.472158\pi$
$798$	0	0
$799$	−12811.2	−0.567245
$800$	−244.922	−0.0108241
$801$	0	0
$802$	3686.47	0.162311
$803$	828.257	0.0363992
$804$	0	0
$805$	0	0
$806$	4322.65	0.188907
$807$	0	0
$808$	−22190.5	−0.966160
$809$	−32120.2	−1.39590	−0.697952	−	0.716144i	$-0.745905\pi$
$809$	−32120.2	−1.39590	−0.697952	+	0.716144i	$0.745905\pi$
$810$	0	0
$811$	32260.5	1.39682	0.698410	−	0.715698i	$-0.253891\pi$
$811$	32260.5	1.39682	0.698410	+	0.715698i	$0.253891\pi$
$812$	0	0
$813$	0	0
$814$	−126.867	−0.00546278
$815$	10023.6	0.430812
$816$	0	0
$817$	4283.31	0.183420
$818$	21449.9	0.916843
$819$	0	0
$820$	71.2765	0.00303547
$821$	−36937.2	−1.57018	−0.785090	−	0.619382i	$-0.787383\pi$
$821$	−36937.2	−1.57018	−0.785090	+	0.619382i	$0.787383\pi$
$822$	0	0
$823$	−36494.5	−1.54571	−0.772854	−	0.634584i	$-0.781172\pi$
$823$	−36494.5	−1.54571	−0.772854	+	0.634584i	$0.781172\pi$
$824$	13412.7	0.567057
$825$	0	0
$826$	0	0
$827$	12606.3	0.530067	0.265034	−	0.964239i	$-0.414617\pi$
$827$	12606.3	0.530067	0.265034	+	0.964239i	$0.414617\pi$
$828$	0	0
$829$	6446.91	0.270097	0.135048	−	0.990839i	$-0.456881\pi$
$829$	6446.91	0.270097	0.135048	+	0.990839i	$0.456881\pi$
$830$	19683.2	0.823148
$831$	0	0
$832$	33605.4	1.40031
$833$	0	0
$834$	0	0
$835$	−4478.98	−0.185631
$836$	6.93185	0.000286774 0
$837$	0	0
$838$	23875.5	0.984208
$839$	17728.9	0.729523	0.364762	−	0.931101i	$-0.381150\pi$
$839$	17728.9	0.729523	0.364762	+	0.931101i	$0.381150\pi$
$840$	0	0
$841$	12629.6	0.517839
$842$	14048.2	0.574978
$843$	0	0
$844$	963.653	0.0393013
$845$	−9493.92	−0.386510
$846$	0	0
$847$	0	0
$848$	18731.4	0.758538
$849$	0	0
$850$	−2066.65	−0.0833949
$851$	9822.28	0.395656
$852$	0	0
$853$	33066.7	1.32730	0.663648	−	0.748045i	$-0.269007\pi$
$853$	33066.7	1.32730	0.663648	+	0.748045i	$0.269007\pi$
$854$	0	0
$855$	0	0
$856$	−38353.2	−1.53141
$857$	3084.25	0.122936	0.0614680	−	0.998109i	$-0.480422\pi$
$857$	3084.25	0.122936	0.0614680	+	0.998109i	$0.480422\pi$
$858$	0	0
$859$	41474.0	1.64735	0.823675	−	0.567062i	$-0.191920\pi$
$859$	41474.0	1.64735	0.823675	+	0.567062i	$0.191920\pi$
$860$	105.919	0.00419979
$861$	0	0
$862$	33715.5	1.33220
$863$	−12327.0	−0.486230	−0.243115	−	0.969998i	$-0.578169\pi$
$863$	−12327.0	−0.486230	−0.243115	+	0.969998i	$0.578169\pi$
$864$	0	0
$865$	6260.69	0.246092
$866$	47907.1	1.87985
$867$	0	0
$868$	0	0
$869$	−37.5818	−0.00146706
$870$	0	0
$871$	54633.9	2.12537
$872$	39942.8	1.55119
$873$	0	0
$874$	19290.4	0.746575
$875$	0	0
$876$	0	0
$877$	6381.94	0.245727	0.122864	−	0.992424i	$-0.460792\pi$
$877$	6381.94	0.245727	0.122864	+	0.992424i	$0.460792\pi$
$878$	541.649	0.0208198
$879$	0	0
$880$	−227.454	−0.00871302
$881$	−10276.4	−0.392985	−0.196492	−	0.980505i	$-0.562955\pi$
$881$	−10276.4	−0.392985	−0.196492	+	0.980505i	$0.562955\pi$
$882$	0	0
$883$	−38805.8	−1.47896	−0.739479	−	0.673180i	$-0.764928\pi$
$883$	−38805.8	−1.47896	−0.739479	+	0.673180i	$0.764928\pi$
$884$	−410.633	−0.0156234
$885$	0	0
$886$	−27176.6	−1.03049
$887$	35174.4	1.33150	0.665750	−	0.746175i	$-0.268112\pi$
$887$	35174.4	1.33150	0.665750	+	0.746175i	$0.268112\pi$
$888$	0	0
$889$	0	0
$890$	3720.10	0.140110
$891$	0	0
$892$	−519.751	−0.0195096
$893$	−18930.7	−0.709398
$894$	0	0
$895$	−20878.0	−0.779748
$896$	0	0
$897$	0	0
$898$	32515.6	1.20831
$899$	4658.06	0.172809
$900$	0	0
$901$	−8920.25	−0.329830
$902$	134.278	0.00495674
$903$	0	0
$904$	32924.2	1.21133
$905$	8864.98	0.325615
$906$	0	0
$907$	13194.2	0.483028	0.241514	−	0.970397i	$-0.422356\pi$
$907$	13194.2	0.483028	0.241514	+	0.970397i	$0.422356\pi$
$908$	1037.89	0.0379334
$909$	0	0
$910$	0	0
$911$	25551.3	0.929255	0.464627	−	0.885506i	$-0.346188\pi$
$911$	25551.3	0.929255	0.464627	+	0.885506i	$0.346188\pi$
$912$	0	0
$913$	−1031.64	−0.0373956
$914$	−37240.9	−1.34772
$915$	0	0
$916$	−444.251	−0.0160245
$917$	0	0
$918$	0	0
$919$	−11116.4	−0.399018	−0.199509	−	0.979896i	$-0.563935\pi$
$919$	−11116.4	−0.399018	−0.199509	+	0.979896i	$0.563935\pi$
$920$	18100.1	0.648634
$921$	0	0
$922$	41100.0	1.46807
$923$	13751.9	0.490410
$924$	0	0
$925$	−1554.95	−0.0552718
$926$	17094.3	0.606645
$927$	0	0
$928$	1884.94	0.0666770
$929$	−18019.7	−0.636390	−0.318195	−	0.948025i	$-0.603077\pi$
$929$	−18019.7	−0.636390	−0.318195	+	0.948025i	$0.603077\pi$
$930$	0	0
$931$	0	0
$932$	1182.30	0.0415533
$933$	0	0
$934$	25987.8	0.910434
$935$	108.318	0.00378862
$936$	0	0
$937$	−15659.4	−0.545968	−0.272984	−	0.962019i	$-0.588011\pi$
$937$	−15659.4	−0.545968	−0.272984	+	0.962019i	$0.588011\pi$
$938$	0	0
$939$	0	0
$940$	−468.126	−0.0162432
$941$	−632.621	−0.0219159	−0.0109579	−	0.999940i	$-0.503488\pi$
$941$	−632.621	−0.0219159	−0.0109579	+	0.999940i	$0.503488\pi$
$942$	0	0
$943$	−10396.0	−0.359005
$944$	−21809.2	−0.751936
$945$	0	0
$946$	199.542	0.00685801
$947$	47903.6	1.64378	0.821888	−	0.569649i	$-0.192921\pi$
$947$	47903.6	1.64378	0.821888	+	0.569649i	$0.192921\pi$
$948$	0	0
$949$	−72501.4	−2.47997
$950$	−3053.83	−0.104294
$951$	0	0
$952$	0	0
$953$	16683.3	0.567079	0.283539	−	0.958961i	$-0.408491\pi$
$953$	16683.3	0.567079	0.283539	+	0.958961i	$0.408491\pi$
$954$	0	0
$955$	14323.4	0.485333
$956$	515.632	0.0174443
$957$	0	0
$958$	36114.2	1.21795
$959$	0	0
$960$	0	0
$961$	−29204.9	−0.980325
$962$	11105.3	0.372193
$963$	0	0
$964$	357.950	0.0119593
$965$	−8192.79	−0.273301
$966$	0	0
$967$	46024.1	1.53054	0.765272	−	0.643707i	$-0.222604\pi$
$967$	46024.1	1.53054	0.765272	+	0.643707i	$0.222604\pi$
$968$	−30498.5	−1.01267
$969$	0	0
$970$	−7627.44	−0.252477
$971$	−44399.5	−1.46740	−0.733701	−	0.679472i	$-0.762209\pi$
$971$	−44399.5	−1.46740	−0.733701	+	0.679472i	$0.762209\pi$
$972$	0	0
$973$	0	0
$974$	−8427.80	−0.277253
$975$	0	0
$976$	−25261.4	−0.828482
$977$	42047.7	1.37689	0.688447	−	0.725287i	$-0.258293\pi$
$977$	42047.7	1.37689	0.688447	+	0.725287i	$0.258293\pi$
$978$	0	0
$979$	−194.978	−0.00636519
$980$	0	0
$981$	0	0
$982$	4986.47	0.162041
$983$	−6415.09	−0.208148	−0.104074	−	0.994570i	$-0.533188\pi$
$983$	−6415.09	−0.208148	−0.104074	+	0.994570i	$0.533188\pi$
$984$	0	0
$985$	−24635.2	−0.796897
$986$	15905.2	0.513715
$987$	0	0
$988$	−606.779	−0.0195387
$989$	−15448.9	−0.496709
$990$	0	0
$991$	−46931.6	−1.50437	−0.752186	−	0.658951i	$-0.771000\pi$
$991$	−46931.6	−1.50437	−0.752186	+	0.658951i	$0.771000\pi$
$992$	237.183	0.00759129
$993$	0	0
$994$	0	0
$995$	23636.4	0.753091
$996$	0	0
$997$	−28929.1	−0.918950	−0.459475	−	0.888191i	$-0.651962\pi$
$997$	−28929.1	−0.918950	−0.459475	+	0.888191i	$0.651962\pi$
$998$	12963.4	0.411173
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.cd.1.3		8
3.2	odd	2		735.4.a.bc.1.6	yes	8
7.6	odd	2		2205.4.a.ce.1.3		8
21.20	even	2		735.4.a.bb.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.bb.1.6	✓	8	21.20	even	2
735.4.a.bc.1.6	yes	8	3.2	odd	2
2205.4.a.cd.1.3		8	1.1	even	1	trivial
2205.4.a.ce.1.3		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.78988 q^{2} -0.216543 q^{4} -5.00000 q^{5} +22.9232 q^{8} +O(q^{10})\)	\(q-2.78988 q^{2} -0.216543 q^{4} -5.00000 q^{5} +22.9232 q^{8} +13.9494 q^{10} -0.731118 q^{11} +63.9983 q^{13} -62.2208 q^{16} +29.6307 q^{17} +43.7843 q^{19} +1.08271 q^{20} +2.03973 q^{22} -157.920 q^{23} +25.0000 q^{25} -178.548 q^{26} -192.402 q^{29} -24.2100 q^{31} -9.79689 q^{32} -82.6662 q^{34} -62.1980 q^{37} -122.153 q^{38} -114.616 q^{40} +65.8313 q^{41} +97.8275 q^{43} +0.158318 q^{44} +440.577 q^{46} -432.364 q^{47} -69.7471 q^{50} -13.8584 q^{52} -301.048 q^{53} +3.65559 q^{55} +536.780 q^{58} +350.512 q^{59} +405.997 q^{61} +67.5432 q^{62} +525.098 q^{64} -319.992 q^{65} +853.677 q^{67} -6.41631 q^{68} +214.879 q^{71} -1132.86 q^{73} +173.525 q^{74} -9.48117 q^{76} +51.4032 q^{79} +311.104 q^{80} -183.662 q^{82} +1411.04 q^{83} -148.153 q^{85} -272.928 q^{86} -16.7596 q^{88} +266.685 q^{89} +34.1963 q^{92} +1206.24 q^{94} -218.921 q^{95} -546.792 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 50 q^{4} - 40 q^{5} - 66 q^{8}+O(q^{10}) \) 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

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Database

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