LMFDB - Embedded newform 2205.4.a.be.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.236068 q^{2} -7.94427 q^{4} -5.00000 q^{5} +3.76393 q^{8} +O(q^{10})$	$q-0.236068 q^{2} -7.94427 q^{4} -5.00000 q^{5} +3.76393 q^{8} +1.18034 q^{10} +50.4721 q^{11} +80.9706 q^{13} +62.6656 q^{16} +76.3870 q^{17} -4.13777 q^{19} +39.7214 q^{20} -11.9149 q^{22} +204.721 q^{23} +25.0000 q^{25} -19.1146 q^{26} +91.1672 q^{29} -198.079 q^{31} -44.9048 q^{32} -18.0325 q^{34} +155.666 q^{37} +0.976794 q^{38} -18.8197 q^{40} -156.885 q^{41} +354.217 q^{43} -400.964 q^{44} -48.3282 q^{46} -175.659 q^{47} -5.90170 q^{50} -643.252 q^{52} -200.302 q^{53} -252.361 q^{55} -21.5217 q^{58} -312.498 q^{59} +154.170 q^{61} +46.7601 q^{62} -490.724 q^{64} -404.853 q^{65} +734.715 q^{67} -606.839 q^{68} +678.577 q^{71} +60.8003 q^{73} -36.7477 q^{74} +32.8715 q^{76} -1286.26 q^{79} -313.328 q^{80} +37.0356 q^{82} +116.170 q^{83} -381.935 q^{85} -83.6192 q^{86} +189.974 q^{88} -916.440 q^{89} -1626.36 q^{92} +41.4676 q^{94} +20.6888 q^{95} +1416.30 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{4} - 10 q^{5} + 12 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 2 * q^4 - 10 * q^5 + 12 * q^8	$ 2 q + 4 q^{2} + 2 q^{4} - 10 q^{5} + 12 q^{8} - 20 q^{10} + 92 q^{11} - 8 q^{13} + 18 q^{16} - 44 q^{17} + 108 q^{19} - 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} - 396 q^{26} + 236 q^{29} + 60 q^{31} - 300 q^{32} - 528 q^{34} + 204 q^{37} + 476 q^{38} - 60 q^{40} + 44 q^{41} + 136 q^{43} + 12 q^{44} + 440 q^{46} + 400 q^{47} + 100 q^{50} - 1528 q^{52} - 16 q^{53} - 460 q^{55} + 592 q^{58} - 464 q^{59} + 684 q^{61} + 1140 q^{62} - 1214 q^{64} + 40 q^{65} + 736 q^{67} - 1804 q^{68} + 740 q^{71} - 424 q^{73} + 168 q^{74} + 1148 q^{76} - 408 q^{79} - 90 q^{80} + 888 q^{82} + 608 q^{83} + 220 q^{85} - 1008 q^{86} + 532 q^{88} - 1332 q^{89} - 480 q^{92} + 2480 q^{94} - 540 q^{95} + 2448 q^{97}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^4 - 10 * q^5 + 12 * q^8 - 20 * q^10 + 92 * q^11 - 8 * q^13 + 18 * q^16 - 44 * q^17 + 108 * q^19 - 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 - 396 * q^26 + 236 * q^29 + 60 * q^31 - 300 * q^32 - 528 * q^34 + 204 * q^37 + 476 * q^38 - 60 * q^40 + 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 + 400 * q^47 + 100 * q^50 - 1528 * q^52 - 16 * q^53 - 460 * q^55 + 592 * q^58 - 464 * q^59 + 684 * q^61 + 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 - 1804 * q^68 + 740 * q^71 - 424 * q^73 + 168 * q^74 + 1148 * q^76 - 408 * q^79 - 90 * q^80 + 888 * q^82 + 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 - 1332 * q^89 - 480 * q^92 + 2480 * q^94 - 540 * q^95 + 2448 * 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 $

$2$	−0.236068	−0.0834626	−0.0417313	−	0.999129i	$-0.513287\pi$
$2$	−0.236068	−0.0834626	−0.0417313	+	0.999129i	$0.513287\pi$
$3$	0	0
$3$	0	0
$4$	−7.94427	−0.993034
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.76393	0.166344
$9$	0	0
$10$	1.18034	0.0373256
$11$	50.4721	1.38345	0.691724	−	0.722162i	$-0.256852\pi$
$11$	50.4721	1.38345	0.691724	+	0.722162i	$0.256852\pi$
$12$	0	0
$13$	80.9706	1.72748	0.863738	−	0.503940i	$-0.168117\pi$
$13$	80.9706	1.72748	0.863738	+	0.503940i	$0.168117\pi$
$14$	0	0
$15$	0	0
$16$	62.6656	0.979150
$17$	76.3870	1.08980	0.544899	−	0.838502i	$-0.316568\pi$
$17$	76.3870	1.08980	0.544899	+	0.838502i	$0.316568\pi$
$18$	0	0
$19$	−4.13777	−0.0499615	−0.0249808	−	0.999688i	$-0.507952\pi$
$19$	−4.13777	−0.0499615	−0.0249808	+	0.999688i	$0.507952\pi$
$20$	39.7214	0.444098
$21$	0	0
$22$	−11.9149	−0.115466
$23$	204.721	1.85597	0.927986	−	0.372615i	$-0.121539\pi$
$23$	204.721	1.85597	0.927986	+	0.372615i	$0.121539\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−19.1146	−0.144180
$27$	0	0
$28$	0	0
$29$	91.1672	0.583770	0.291885	−	0.956453i	$-0.405718\pi$
$29$	91.1672	0.583770	0.291885	+	0.956453i	$0.405718\pi$
$30$	0	0
$31$	−198.079	−1.14761	−0.573807	−	0.818991i	$-0.694534\pi$
$31$	−198.079	−1.14761	−0.573807	+	0.818991i	$0.694534\pi$
$32$	−44.9048	−0.248066
$33$	0	0
$34$	−18.0325	−0.0909574
$35$	0	0
$36$	0	0
$37$	155.666	0.691656	0.345828	−	0.938298i	$-0.387598\pi$
$37$	155.666	0.691656	0.345828	+	0.938298i	$0.387598\pi$
$38$	0.976794	0.00416992
$39$	0	0
$40$	−18.8197	−0.0743912
$41$	−156.885	−0.597595	−0.298797	−	0.954317i	$-0.596585\pi$
$41$	−156.885	−0.597595	−0.298797	+	0.954317i	$0.596585\pi$
$42$	0	0
$43$	354.217	1.25622	0.628111	−	0.778124i	$-0.283828\pi$
$43$	354.217	1.25622	0.628111	+	0.778124i	$0.283828\pi$
$44$	−400.964	−1.37381
$45$	0	0
$46$	−48.3282	−0.154904
$47$	−175.659	−0.545161	−0.272580	−	0.962133i	$-0.587877\pi$
$47$	−175.659	−0.545161	−0.272580	+	0.962133i	$0.587877\pi$
$48$	0	0
$49$	0	0
$50$	−5.90170	−0.0166925
$51$	0	0
$52$	−643.252	−1.71544
$53$	−200.302	−0.519124	−0.259562	−	0.965726i	$-0.583578\pi$
$53$	−200.302	−0.519124	−0.259562	+	0.965726i	$0.583578\pi$
$54$	0	0
$55$	−252.361	−0.618696
$56$	0	0
$57$	0	0
$58$	−21.5217	−0.0487230
$59$	−312.498	−0.689556	−0.344778	−	0.938684i	$-0.612046\pi$
$59$	−312.498	−0.689556	−0.344778	+	0.938684i	$0.612046\pi$
$60$	0	0
$61$	154.170	0.323598	0.161799	−	0.986824i	$-0.448270\pi$
$61$	154.170	0.323598	0.161799	+	0.986824i	$0.448270\pi$
$62$	46.7601	0.0957829
$63$	0	0
$64$	−490.724	−0.958446
$65$	−404.853	−0.772551
$66$	0	0
$67$	734.715	1.33970	0.669849	−	0.742498i	$-0.266359\pi$
$67$	734.715	1.33970	0.669849	+	0.742498i	$0.266359\pi$
$68$	−606.839	−1.08221
$69$	0	0
$70$	0	0
$71$	678.577	1.13426	0.567129	−	0.823629i	$-0.308054\pi$
$71$	678.577	1.13426	0.567129	+	0.823629i	$0.308054\pi$
$72$	0	0
$73$	60.8003	0.0974813	0.0487407	−	0.998811i	$-0.484479\pi$
$73$	60.8003	0.0974813	0.0487407	+	0.998811i	$0.484479\pi$
$74$	−36.7477	−0.0577274
$75$	0	0
$76$	32.8715	0.0496135
$77$	0	0
$78$	0	0
$79$	−1286.26	−1.83184	−0.915919	−	0.401363i	$-0.868537\pi$
$79$	−1286.26	−1.83184	−0.915919	+	0.401363i	$0.868537\pi$
$80$	−313.328	−0.437889
$81$	0	0
$82$	37.0356	0.0498768
$83$	116.170	0.153631	0.0768153	−	0.997045i	$-0.475525\pi$
$83$	116.170	0.153631	0.0768153	+	0.997045i	$0.475525\pi$
$84$	0	0
$85$	−381.935	−0.487372
$86$	−83.6192	−0.104848
$87$	0	0
$88$	189.974	0.230128
$89$	−916.440	−1.09149	−0.545744	−	0.837952i	$-0.683753\pi$
$89$	−916.440	−1.09149	−0.545744	+	0.837952i	$0.683753\pi$
$90$	0	0
$91$	0	0
$92$	−1626.36	−1.84304
$93$	0	0
$94$	41.4676	0.0455006
$95$	20.6888	0.0223435
$96$	0	0
$97$	1416.30	1.48251	0.741256	−	0.671222i	$-0.234230\pi$
$97$	1416.30	1.48251	0.741256	+	0.671222i	$0.234230\pi$
$98$	0	0
$99$	0	0
$100$	−198.607	−0.198607
$101$	−1379.19	−1.35876	−0.679381	−	0.733785i	$-0.737752\pi$
$101$	−1379.19	−1.35876	−0.679381	+	0.733785i	$0.737752\pi$
$102$	0	0
$103$	1308.43	1.25169	0.625844	−	0.779949i	$-0.284755\pi$
$103$	1308.43	1.25169	0.625844	+	0.779949i	$0.284755\pi$
$104$	304.768	0.287355
$105$	0	0
$106$	47.2849	0.0433275
$107$	−1265.65	−1.14351	−0.571754	−	0.820425i	$-0.693737\pi$
$107$	−1265.65	−1.14351	−0.571754	+	0.820425i	$0.693737\pi$
$108$	0	0
$109$	2069.32	1.81839	0.909196	−	0.416368i	$-0.136697\pi$
$109$	2069.32	1.81839	0.909196	+	0.416368i	$0.136697\pi$
$110$	59.5743	0.0516380
$111$	0	0
$112$	0	0
$113$	1953.89	1.62661	0.813303	−	0.581840i	$-0.197667\pi$
$113$	1953.89	1.62661	0.813303	+	0.581840i	$0.197667\pi$
$114$	0	0
$115$	−1023.61	−0.830016
$116$	−724.257	−0.579703
$117$	0	0
$118$	73.7709	0.0575522
$119$	0	0
$120$	0	0
$121$	1216.44	0.913927
$122$	−36.3947	−0.0270083
$123$	0	0
$124$	1573.59	1.13962
$125$	−125.000	−0.0894427
$126$	0	0
$127$	224.251	0.156685	0.0783426	−	0.996926i	$-0.475037\pi$
$127$	224.251	0.156685	0.0783426	+	0.996926i	$0.475037\pi$
$128$	475.083	0.328061
$129$	0	0
$130$	95.5728	0.0644792
$131$	−490.898	−0.327404	−0.163702	−	0.986510i	$-0.552344\pi$
$131$	−490.898	−0.327404	−0.163702	+	0.986510i	$0.552344\pi$
$132$	0	0
$133$	0	0
$134$	−173.443	−0.111815
$135$	0	0
$136$	287.515	0.181281
$137$	−1831.12	−1.14192	−0.570961	−	0.820977i	$-0.693429\pi$
$137$	−1831.12	−1.14192	−0.570961	+	0.820977i	$0.693429\pi$
$138$	0	0
$139$	3050.84	1.86165	0.930823	−	0.365469i	$-0.119091\pi$
$139$	3050.84	1.86165	0.930823	+	0.365469i	$0.119091\pi$
$140$	0	0
$141$	0	0
$142$	−160.190	−0.0946682
$143$	4086.76	2.38987
$144$	0	0
$145$	−455.836	−0.261070
$146$	−14.3530	−0.00813605
$147$	0	0
$148$	−1236.65	−0.686838
$149$	−2246.55	−1.23520	−0.617599	−	0.786493i	$-0.711895\pi$
$149$	−2246.55	−1.23520	−0.617599	+	0.786493i	$0.711895\pi$
$150$	0	0
$151$	−1311.53	−0.706826	−0.353413	−	0.935467i	$-0.614979\pi$
$151$	−1311.53	−0.706826	−0.353413	+	0.935467i	$0.614979\pi$
$152$	−15.5743	−0.00831079
$153$	0	0
$154$	0	0
$155$	990.395	0.513228
$156$	0	0
$157$	−1790.94	−0.910398	−0.455199	−	0.890390i	$-0.650432\pi$
$157$	−1790.94	−0.910398	−0.455199	+	0.890390i	$0.650432\pi$
$158$	303.644	0.152890
$159$	0	0
$160$	224.524	0.110939
$161$	0	0
$162$	0	0
$163$	−491.108	−0.235991	−0.117996	−	0.993014i	$-0.537647\pi$
$163$	−491.108	−0.235991	−0.117996	+	0.993014i	$0.537647\pi$
$164$	1246.34	0.593432
$165$	0	0
$166$	−27.4241	−0.0128224
$167$	−826.059	−0.382769	−0.191384	−	0.981515i	$-0.561298\pi$
$167$	−826.059	−0.382769	−0.191384	+	0.981515i	$0.561298\pi$
$168$	0	0
$169$	4359.24	1.98418
$170$	90.1626	0.0406774
$171$	0	0
$172$	−2813.99	−1.24747
$173$	2918.00	1.28238	0.641190	−	0.767382i	$-0.278441\pi$
$173$	2918.00	1.28238	0.641190	+	0.767382i	$0.278441\pi$
$174$	0	0
$175$	0	0
$176$	3162.87	1.35460
$177$	0	0
$178$	216.342	0.0910984
$179$	−955.745	−0.399082	−0.199541	−	0.979889i	$-0.563945\pi$
$179$	−955.745	−0.399082	−0.199541	+	0.979889i	$0.563945\pi$
$180$	0	0
$181$	−206.080	−0.0846289	−0.0423145	−	0.999104i	$-0.513473\pi$
$181$	−206.080	−0.0846289	−0.0423145	+	0.999104i	$0.513473\pi$
$182$	0	0
$183$	0	0
$184$	770.557	0.308730
$185$	−778.328	−0.309318
$186$	0	0
$187$	3855.41	1.50768
$188$	1395.49	0.541363
$189$	0	0
$190$	−4.88397	−0.00186485
$191$	2018.63	0.764727	0.382364	−	0.924012i	$-0.375110\pi$
$191$	2018.63	0.764727	0.382364	+	0.924012i	$0.375110\pi$
$192$	0	0
$193$	1031.69	0.384781	0.192390	−	0.981318i	$-0.438376\pi$
$193$	1031.69	0.384781	0.192390	+	0.981318i	$0.438376\pi$
$194$	−334.344	−0.123734
$195$	0	0
$196$	0	0
$197$	205.955	0.0744857	0.0372429	−	0.999306i	$-0.488142\pi$
$197$	205.955	0.0744857	0.0372429	+	0.999306i	$0.488142\pi$
$198$	0	0
$199$	1831.38	0.652376	0.326188	−	0.945305i	$-0.394236\pi$
$199$	1831.38	0.652376	0.326188	+	0.945305i	$0.394236\pi$
$200$	94.0983	0.0332688
$201$	0	0
$202$	325.584	0.113406
$203$	0	0
$204$	0	0
$205$	784.427	0.267253
$206$	−308.879	−0.104469
$207$	0	0
$208$	5074.07	1.69146
$209$	−208.842	−0.0691191
$210$	0	0
$211$	1030.19	0.336119	0.168060	−	0.985777i	$-0.446250\pi$
$211$	1030.19	0.336119	0.168060	+	0.985777i	$0.446250\pi$
$212$	1591.25	0.515508
$213$	0	0
$214$	298.780	0.0954402
$215$	−1771.08	−0.561800
$216$	0	0
$217$	0	0
$218$	−488.500	−0.151768
$219$	0	0
$220$	2004.82	0.614387
$221$	6185.10	1.88260
$222$	0	0
$223$	−5368.67	−1.61217	−0.806083	−	0.591803i	$-0.798416\pi$
$223$	−5368.67	−1.61217	−0.806083	+	0.591803i	$0.798416\pi$
$224$	0	0
$225$	0	0
$226$	−461.251	−0.135761
$227$	−932.121	−0.272542	−0.136271	−	0.990672i	$-0.543512\pi$
$227$	−932.121	−0.272542	−0.136271	+	0.990672i	$0.543512\pi$
$228$	0	0
$229$	−3163.05	−0.912752	−0.456376	−	0.889787i	$-0.650853\pi$
$229$	−3163.05	−0.912752	−0.456376	+	0.889787i	$0.650853\pi$
$230$	241.641	0.0692753
$231$	0	0
$232$	343.147	0.0971065
$233$	−436.562	−0.122747	−0.0613737	−	0.998115i	$-0.519548\pi$
$233$	−436.562	−0.122747	−0.0613737	+	0.998115i	$0.519548\pi$
$234$	0	0
$235$	878.297	0.243803
$236$	2482.57	0.684753
$237$	0	0
$238$	0	0
$239$	1980.82	0.536103	0.268051	−	0.963405i	$-0.413620\pi$
$239$	1980.82	0.536103	0.268051	+	0.963405i	$0.413620\pi$
$240$	0	0
$241$	−5303.55	−1.41756	−0.708780	−	0.705430i	$-0.750754\pi$
$241$	−5303.55	−1.41756	−0.708780	+	0.705430i	$0.750754\pi$
$242$	−287.162	−0.0762787
$243$	0	0
$244$	−1224.77	−0.321344
$245$	0	0
$246$	0	0
$247$	−335.037	−0.0863074
$248$	−745.556	−0.190899
$249$	0	0
$250$	29.5085	0.00746512
$251$	−2996.04	−0.753420	−0.376710	−	0.926331i	$-0.622945\pi$
$251$	−2996.04	−0.753420	−0.376710	+	0.926331i	$0.622945\pi$
$252$	0	0
$253$	10332.7	2.56764
$254$	−52.9384	−0.0130774
$255$	0	0
$256$	3813.64	0.931065
$257$	968.861	0.235159	0.117580	−	0.993063i	$-0.462487\pi$
$257$	968.861	0.235159	0.117580	+	0.993063i	$0.462487\pi$
$258$	0	0
$259$	0	0
$260$	3216.26	0.767170
$261$	0	0
$262$	115.885	0.0273260
$263$	4830.18	1.13248	0.566239	−	0.824241i	$-0.308398\pi$
$263$	4830.18	1.13248	0.566239	+	0.824241i	$0.308398\pi$
$264$	0	0
$265$	1001.51	0.232159
$266$	0	0
$267$	0	0
$268$	−5836.78	−1.33037
$269$	−4774.97	−1.08229	−0.541143	−	0.840930i	$-0.682008\pi$
$269$	−4774.97	−1.08229	−0.541143	+	0.840930i	$0.682008\pi$
$270$	0	0
$271$	−141.909	−0.0318094	−0.0159047	−	0.999874i	$-0.505063\pi$
$271$	−141.909	−0.0318094	−0.0159047	+	0.999874i	$0.505063\pi$
$272$	4786.84	1.06708
$273$	0	0
$274$	432.269	0.0953078
$275$	1261.80	0.276689
$276$	0	0
$277$	−3621.13	−0.785460	−0.392730	−	0.919654i	$-0.628469\pi$
$277$	−3621.13	−0.785460	−0.392730	+	0.919654i	$0.628469\pi$
$278$	−720.206	−0.155378
$279$	0	0
$280$	0	0
$281$	−5790.87	−1.22937	−0.614687	−	0.788771i	$-0.710718\pi$
$281$	−5790.87	−1.22937	−0.614687	+	0.788771i	$0.710718\pi$
$282$	0	0
$283$	−526.043	−0.110495	−0.0552474	−	0.998473i	$-0.517595\pi$
$283$	−526.043	−0.110495	−0.0552474	+	0.998473i	$0.517595\pi$
$284$	−5390.80	−1.12636
$285$	0	0
$286$	−964.753	−0.199465
$287$	0	0
$288$	0	0
$289$	921.972	0.187660
$290$	107.608	0.0217896
$291$	0	0
$292$	−483.014	−0.0968023
$293$	1914.88	0.381803	0.190901	−	0.981609i	$-0.438859\pi$
$293$	1914.88	0.381803	0.190901	+	0.981609i	$0.438859\pi$
$294$	0	0
$295$	1562.49	0.308379
$296$	585.915	0.115053
$297$	0	0
$298$	530.339	0.103093
$299$	16576.4	3.20615
$300$	0	0
$301$	0	0
$302$	309.610	0.0589936
$303$	0	0
$304$	−259.296	−0.0489199
$305$	−770.851	−0.144717
$306$	0	0
$307$	6244.17	1.16083	0.580413	−	0.814322i	$-0.302891\pi$
$307$	6244.17	1.16083	0.580413	+	0.814322i	$0.302891\pi$
$308$	0	0
$309$	0	0
$310$	−233.800	−0.0428354
$311$	−9658.78	−1.76109	−0.880545	−	0.473962i	$-0.842823\pi$
$311$	−9658.78	−1.76109	−0.880545	+	0.473962i	$0.842823\pi$
$312$	0	0
$313$	−2198.34	−0.396988	−0.198494	−	0.980102i	$-0.563605\pi$
$313$	−2198.34	−0.396988	−0.198494	+	0.980102i	$0.563605\pi$
$314$	422.783	0.0759842
$315$	0	0
$316$	10218.4	1.81908
$317$	−3030.78	−0.536990	−0.268495	−	0.963281i	$-0.586526\pi$
$317$	−3030.78	−0.536990	−0.268495	+	0.963281i	$0.586526\pi$
$318$	0	0
$319$	4601.40	0.807615
$320$	2453.62	0.428630
$321$	0	0
$322$	0	0
$323$	−316.072	−0.0544480
$324$	0	0
$325$	2024.26	0.345495
$326$	115.935	0.0196965
$327$	0	0
$328$	−590.506	−0.0994062
$329$	0	0
$330$	0	0
$331$	4753.74	0.789393	0.394696	−	0.918812i	$-0.370850\pi$
$331$	4753.74	0.789393	0.394696	+	0.918812i	$0.370850\pi$
$332$	−922.888	−0.152560
$333$	0	0
$334$	195.006	0.0319469
$335$	−3673.58	−0.599131
$336$	0	0
$337$	8824.40	1.42640	0.713199	−	0.700962i	$-0.247246\pi$
$337$	8824.40	1.42640	0.713199	+	0.700962i	$0.247246\pi$
$338$	−1029.08	−0.165605
$339$	0	0
$340$	3034.20	0.483977
$341$	−9997.47	−1.58766
$342$	0	0
$343$	0	0
$344$	1333.25	0.208965
$345$	0	0
$346$	−688.847	−0.107031
$347$	3413.97	0.528160	0.264080	−	0.964501i	$-0.414932\pi$
$347$	3413.97	0.528160	0.264080	+	0.964501i	$0.414932\pi$
$348$	0	0
$349$	5676.32	0.870621	0.435310	−	0.900280i	$-0.356639\pi$
$349$	5676.32	0.870621	0.435310	+	0.900280i	$0.356639\pi$
$350$	0	0
$351$	0	0
$352$	−2266.44	−0.343187
$353$	6225.80	0.938713	0.469357	−	0.883009i	$-0.344486\pi$
$353$	6225.80	0.938713	0.469357	+	0.883009i	$0.344486\pi$
$354$	0	0
$355$	−3392.89	−0.507256
$356$	7280.45	1.08388
$357$	0	0
$358$	225.621	0.0333084
$359$	4907.73	0.721505	0.360752	−	0.932662i	$-0.382520\pi$
$359$	4907.73	0.721505	0.360752	+	0.932662i	$0.382520\pi$
$360$	0	0
$361$	−6841.88	−0.997504
$362$	48.6490	0.00706335
$363$	0	0
$364$	0	0
$365$	−304.001	−0.0435950
$366$	0	0
$367$	3906.48	0.555631	0.277816	−	0.960634i	$-0.410390\pi$
$367$	3906.48	0.555631	0.277816	+	0.960634i	$0.410390\pi$
$368$	12829.0	1.81728
$369$	0	0
$370$	183.738	0.0258165
$371$	0	0
$372$	0	0
$373$	−2102.52	−0.291862	−0.145931	−	0.989295i	$-0.546618\pi$
$373$	−2102.52	−0.291862	−0.145931	+	0.989295i	$0.546618\pi$
$374$	−910.140	−0.125835
$375$	0	0
$376$	−661.170	−0.0906842
$377$	7381.86	1.00845
$378$	0	0
$379$	−6612.76	−0.896239	−0.448120	−	0.893974i	$-0.647906\pi$
$379$	−6612.76	−0.896239	−0.448120	+	0.893974i	$0.647906\pi$
$380$	−164.358	−0.0221878
$381$	0	0
$382$	−476.534	−0.0638262
$383$	2.37457	0.000316801 0	0.000158401	−	1.00000i	$-0.499950\pi$
$383$	2.37457	0.000316801 0	0.000158401		1.00000i	$0.499950\pi$
$384$	0	0
$385$	0	0
$386$	−243.549	−0.0321148
$387$	0	0
$388$	−11251.5	−1.47218
$389$	−7716.98	−1.00583	−0.502913	−	0.864337i	$-0.667739\pi$
$389$	−7716.98	−1.00583	−0.502913	+	0.864337i	$0.667739\pi$
$390$	0	0
$391$	15638.0	2.02263
$392$	0	0
$393$	0	0
$394$	−48.6194	−0.00621678
$395$	6431.28	0.819223
$396$	0	0
$397$	−6403.95	−0.809584	−0.404792	−	0.914409i	$-0.632656\pi$
$397$	−6403.95	−0.809584	−0.404792	+	0.914409i	$0.632656\pi$
$398$	−432.329	−0.0544490
$399$	0	0
$400$	1566.64	0.195830
$401$	10969.9	1.36611	0.683054	−	0.730368i	$-0.260651\pi$
$401$	10969.9	1.36611	0.683054	+	0.730368i	$0.260651\pi$
$402$	0	0
$403$	−16038.6	−1.98248
$404$	10956.7	1.34930
$405$	0	0
$406$	0	0
$407$	7856.78	0.956870
$408$	0	0
$409$	400.353	0.0484015	0.0242007	−	0.999707i	$-0.492296\pi$
$409$	400.353	0.0484015	0.0242007	+	0.999707i	$0.492296\pi$
$410$	−185.178	−0.0223056
$411$	0	0
$412$	−10394.6	−1.24297
$413$	0	0
$414$	0	0
$415$	−580.851	−0.0687057
$416$	−3635.97	−0.428529
$417$	0	0
$418$	49.3009	0.00576887
$419$	−15815.4	−1.84399	−0.921995	−	0.387201i	$-0.873442\pi$
$419$	−15815.4	−1.84399	−0.921995	+	0.387201i	$0.873442\pi$
$420$	0	0
$421$	1936.53	0.224182	0.112091	−	0.993698i	$-0.464245\pi$
$421$	1936.53	0.224182	0.112091	+	0.993698i	$0.464245\pi$
$422$	−243.195	−0.0280534
$423$	0	0
$424$	−753.923	−0.0863531
$425$	1909.67	0.217960
$426$	0	0
$427$	0	0
$428$	10054.7	1.13554
$429$	0	0
$430$	418.096	0.0468893
$431$	−2030.91	−0.226973	−0.113487	−	0.993540i	$-0.536202\pi$
$431$	−2030.91	−0.226973	−0.113487	+	0.993540i	$0.536202\pi$
$432$	0	0
$433$	10784.1	1.19689	0.598443	−	0.801165i	$-0.295786\pi$
$433$	10784.1	1.19689	0.598443	+	0.801165i	$0.295786\pi$
$434$	0	0
$435$	0	0
$436$	−16439.2	−1.80573
$437$	−847.089	−0.0927272
$438$	0	0
$439$	6304.19	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$439$	6304.19	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$440$	−949.868	−0.102916
$441$	0	0
$442$	−1460.10	−0.157127
$443$	−15494.8	−1.66181	−0.830905	−	0.556414i	$-0.812177\pi$
$443$	−15494.8	−1.66181	−0.830905	+	0.556414i	$0.812177\pi$
$444$	0	0
$445$	4582.20	0.488128
$446$	1267.37	0.134556
$447$	0	0
$448$	0	0
$449$	242.018	0.0254377	0.0127189	−	0.999919i	$-0.495951\pi$
$449$	242.018	0.0254377	0.0127189	+	0.999919i	$0.495951\pi$
$450$	0	0
$451$	−7918.34	−0.826741
$452$	−15522.2	−1.61528
$453$	0	0
$454$	220.044	0.0227471
$455$	0	0
$456$	0	0
$457$	−11670.4	−1.19457	−0.597283	−	0.802030i	$-0.703753\pi$
$457$	−11670.4	−1.19457	−0.597283	+	0.802030i	$0.703753\pi$
$458$	746.695	0.0761807
$459$	0	0
$460$	8131.81	0.824234
$461$	12128.1	1.22530	0.612649	−	0.790355i	$-0.290104\pi$
$461$	12128.1	1.22530	0.612649	+	0.790355i	$0.290104\pi$
$462$	0	0
$463$	5161.64	0.518103	0.259051	−	0.965864i	$-0.416590\pi$
$463$	5161.64	0.518103	0.259051	+	0.965864i	$0.416590\pi$
$464$	5713.05	0.571598
$465$	0	0
$466$	103.058	0.0102448
$467$	11680.2	1.15738	0.578691	−	0.815547i	$-0.303564\pi$
$467$	11680.2	1.15738	0.578691	+	0.815547i	$0.303564\pi$
$468$	0	0
$469$	0	0
$470$	−207.338	−0.0203485
$471$	0	0
$472$	−1176.22	−0.114703
$473$	17878.1	1.73792
$474$	0	0
$475$	−103.444	−0.00999230
$476$	0	0
$477$	0	0
$478$	−467.608	−0.0447445
$479$	−18458.7	−1.76075	−0.880373	−	0.474282i	$-0.842708\pi$
$479$	−18458.7	−1.76075	−0.880373	+	0.474282i	$0.842708\pi$
$480$	0	0
$481$	12604.3	1.19482
$482$	1252.00	0.118313
$483$	0	0
$484$	−9663.70	−0.907560
$485$	−7081.51	−0.663000
$486$	0	0
$487$	8630.11	0.803014	0.401507	−	0.915856i	$-0.368487\pi$
$487$	8630.11	0.803014	0.401507	+	0.915856i	$0.368487\pi$
$488$	580.286	0.0538286
$489$	0	0
$490$	0	0
$491$	−17801.6	−1.63621	−0.818103	−	0.575072i	$-0.804974\pi$
$491$	−17801.6	−1.63621	−0.818103	+	0.575072i	$0.804974\pi$
$492$	0	0
$493$	6963.99	0.636191
$494$	79.0916	0.00720344
$495$	0	0
$496$	−12412.7	−1.12369
$497$	0	0
$498$	0	0
$499$	200.167	0.0179574	0.00897868	−	0.999960i	$-0.497142\pi$
$499$	200.167	0.0179574	0.00897868	+	0.999960i	$0.497142\pi$
$500$	993.034	0.0888197
$501$	0	0
$502$	707.269	0.0628824
$503$	16400.4	1.45379	0.726896	−	0.686747i	$-0.240962\pi$
$503$	16400.4	1.45379	0.726896	+	0.686747i	$0.240962\pi$
$504$	0	0
$505$	6895.97	0.607657
$506$	−2439.23	−0.214302
$507$	0	0
$508$	−1781.51	−0.155594
$509$	−15006.6	−1.30679	−0.653394	−	0.757018i	$-0.726656\pi$
$509$	−15006.6	−1.30679	−0.653394	+	0.757018i	$0.726656\pi$
$510$	0	0
$511$	0	0
$512$	−4700.94	−0.405770
$513$	0	0
$514$	−228.717	−0.0196270
$515$	−6542.17	−0.559772
$516$	0	0
$517$	−8865.91	−0.754201
$518$	0	0
$519$	0	0
$520$	−1523.84	−0.128509
$521$	7113.18	0.598146	0.299073	−	0.954230i	$-0.403323\pi$
$521$	7113.18	0.598146	0.299073	+	0.954230i	$0.403323\pi$
$522$	0	0
$523$	8888.46	0.743146	0.371573	−	0.928404i	$-0.378819\pi$
$523$	8888.46	0.743146	0.371573	+	0.928404i	$0.378819\pi$
$524$	3899.83	0.325123
$525$	0	0
$526$	−1140.25	−0.0945196
$527$	−15130.7	−1.25067
$528$	0	0
$529$	29743.8	2.44463
$530$	−236.424	−0.0193766
$531$	0	0
$532$	0	0
$533$	−12703.1	−1.03233
$534$	0	0
$535$	6328.27	0.511392
$536$	2765.42	0.222850
$537$	0	0
$538$	1127.22	0.0903305
$539$	0	0
$540$	0	0
$541$	−653.827	−0.0519597	−0.0259799	−	0.999662i	$-0.508271\pi$
$541$	−653.827	−0.0519597	−0.0259799	+	0.999662i	$0.508271\pi$
$542$	33.5001	0.00265489
$543$	0	0
$544$	−3430.14	−0.270342
$545$	−10346.6	−0.813210
$546$	0	0
$547$	1138.52	0.0889940	0.0444970	−	0.999010i	$-0.485831\pi$
$547$	1138.52	0.0889940	0.0444970	+	0.999010i	$0.485831\pi$
$548$	14546.9	1.13397
$549$	0	0
$550$	−297.871	−0.0230932
$551$	−377.229	−0.0291660
$552$	0	0
$553$	0	0
$554$	854.832	0.0655566
$555$	0	0
$556$	−24236.7	−1.84868
$557$	19804.8	1.50657	0.753283	−	0.657696i	$-0.228469\pi$
$557$	19804.8	1.50657	0.753283	+	0.657696i	$0.228469\pi$
$558$	0	0
$559$	28681.1	2.17009
$560$	0	0
$561$	0	0
$562$	1367.04	0.102607
$563$	10276.3	0.769265	0.384632	−	0.923070i	$-0.374328\pi$
$563$	10276.3	0.769265	0.384632	+	0.923070i	$0.374328\pi$
$564$	0	0
$565$	−9769.45	−0.727440
$566$	124.182	0.00922219
$567$	0	0
$568$	2554.12	0.188677
$569$	−4139.03	−0.304951	−0.152475	−	0.988307i	$-0.548724\pi$
$569$	−4139.03	−0.304951	−0.152475	+	0.988307i	$0.548724\pi$
$570$	0	0
$571$	−4486.81	−0.328839	−0.164420	−	0.986390i	$-0.552575\pi$
$571$	−4486.81	−0.328839	−0.164420	+	0.986390i	$0.552575\pi$
$572$	−32466.3	−2.37323
$573$	0	0
$574$	0	0
$575$	5118.03	0.371194
$576$	0	0
$577$	1104.77	0.0797093	0.0398547	−	0.999205i	$-0.487311\pi$
$577$	1104.77	0.0797093	0.0398547	+	0.999205i	$0.487311\pi$
$578$	−217.648	−0.0156626
$579$	0	0
$580$	3621.28	0.259251
$581$	0	0
$582$	0	0
$583$	−10109.7	−0.718181
$584$	228.848	0.0162154
$585$	0	0
$586$	−452.041	−0.0318663
$587$	−10413.2	−0.732199	−0.366100	−	0.930576i	$-0.619307\pi$
$587$	−10413.2	−0.732199	−0.366100	+	0.930576i	$0.619307\pi$
$588$	0	0
$589$	819.605	0.0573365
$590$	−368.854	−0.0257381
$591$	0	0
$592$	9754.89	0.677235
$593$	−3235.16	−0.224034	−0.112017	−	0.993706i	$-0.535731\pi$
$593$	−3235.16	−0.224034	−0.112017	+	0.993706i	$0.535731\pi$
$594$	0	0
$595$	0	0
$596$	17847.2	1.22659
$597$	0	0
$598$	−3913.16	−0.267594
$599$	−569.048	−0.0388158	−0.0194079	−	0.999812i	$-0.506178\pi$
$599$	−569.048	−0.0388158	−0.0194079	+	0.999812i	$0.506178\pi$
$600$	0	0
$601$	−3760.89	−0.255258	−0.127629	−	0.991822i	$-0.540737\pi$
$601$	−3760.89	−0.255258	−0.127629	+	0.991822i	$0.540737\pi$
$602$	0	0
$603$	0	0
$604$	10419.1	0.701902
$605$	−6082.18	−0.408720
$606$	0	0
$607$	2224.05	0.148717	0.0743585	−	0.997232i	$-0.476309\pi$
$607$	2224.05	0.148717	0.0743585	+	0.997232i	$0.476309\pi$
$608$	185.806	0.0123938
$609$	0	0
$610$	181.973	0.0120785
$611$	−14223.2	−0.941753
$612$	0	0
$613$	5914.50	0.389697	0.194849	−	0.980833i	$-0.437578\pi$
$613$	5914.50	0.389697	0.194849	+	0.980833i	$0.437578\pi$
$614$	−1474.05	−0.0968856
$615$	0	0
$616$	0	0
$617$	18591.2	1.21306	0.606528	−	0.795062i	$-0.292562\pi$
$617$	18591.2	1.21306	0.606528	+	0.795062i	$0.292562\pi$
$618$	0	0
$619$	−5125.97	−0.332844	−0.166422	−	0.986055i	$-0.553221\pi$
$619$	−5125.97	−0.332844	−0.166422	+	0.986055i	$0.553221\pi$
$620$	−7867.96	−0.509653
$621$	0	0
$622$	2280.13	0.146985
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	518.957	0.0331337
$627$	0	0
$628$	14227.7	0.904056
$629$	11890.8	0.753765
$630$	0	0
$631$	−10649.0	−0.671839	−0.335919	−	0.941891i	$-0.609047\pi$
$631$	−10649.0	−0.671839	−0.335919	+	0.941891i	$0.609047\pi$
$632$	−4841.38	−0.304715
$633$	0	0
$634$	715.471	0.0448186
$635$	−1121.25	−0.0700718
$636$	0	0
$637$	0	0
$638$	−1086.24	−0.0674056
$639$	0	0
$640$	−2375.41	−0.146713
$641$	−24025.7	−1.48044	−0.740218	−	0.672367i	$-0.765278\pi$
$641$	−24025.7	−1.48044	−0.740218	+	0.672367i	$0.765278\pi$
$642$	0	0
$643$	−14929.3	−0.915638	−0.457819	−	0.889045i	$-0.651369\pi$
$643$	−14929.3	−0.915638	−0.457819	+	0.889045i	$0.651369\pi$
$644$	0	0
$645$	0	0
$646$	74.6144	0.00454437
$647$	14479.1	0.879801	0.439901	−	0.898046i	$-0.355014\pi$
$647$	14479.1	0.879801	0.439901	+	0.898046i	$0.355014\pi$
$648$	0	0
$649$	−15772.5	−0.953965
$650$	−477.864	−0.0288360
$651$	0	0
$652$	3901.50	0.234347
$653$	−898.168	−0.0538255	−0.0269127	−	0.999638i	$-0.508568\pi$
$653$	−898.168	−0.0538255	−0.0269127	+	0.999638i	$0.508568\pi$
$654$	0	0
$655$	2454.49	0.146420
$656$	−9831.33	−0.585135
$657$	0	0
$658$	0	0
$659$	30198.0	1.78505	0.892526	−	0.450997i	$-0.148931\pi$
$659$	30198.0	1.78505	0.892526	+	0.450997i	$0.148931\pi$
$660$	0	0
$661$	19337.8	1.13790	0.568952	−	0.822371i	$-0.307349\pi$
$661$	19337.8	1.13790	0.568952	+	0.822371i	$0.307349\pi$
$662$	−1122.20	−0.0658848
$663$	0	0
$664$	437.257	0.0255555
$665$	0	0
$666$	0	0
$667$	18663.9	1.08346
$668$	6562.44	0.380102
$669$	0	0
$670$	867.214	0.0500051
$671$	7781.30	0.447681
$672$	0	0
$673$	10132.2	0.580336	0.290168	−	0.956976i	$-0.406289\pi$
$673$	10132.2	0.580336	0.290168	+	0.956976i	$0.406289\pi$
$674$	−2083.16	−0.119051
$675$	0	0
$676$	−34631.0	−1.97035
$677$	−33177.3	−1.88347	−0.941733	−	0.336361i	$-0.890804\pi$
$677$	−33177.3	−1.88347	−0.941733	+	0.336361i	$0.890804\pi$
$678$	0	0
$679$	0	0
$680$	−1437.58	−0.0810714
$681$	0	0
$682$	2360.08	0.132511
$683$	11423.6	0.639987	0.319994	−	0.947420i	$-0.396319\pi$
$683$	11423.6	0.639987	0.319994	+	0.947420i	$0.396319\pi$
$684$	0	0
$685$	9155.61	0.510683
$686$	0	0
$687$	0	0
$688$	22197.2	1.23003
$689$	−16218.6	−0.896775
$690$	0	0
$691$	19737.4	1.08661	0.543304	−	0.839536i	$-0.317173\pi$
$691$	19737.4	1.08661	0.543304	+	0.839536i	$0.317173\pi$
$692$	−23181.4	−1.27345
$693$	0	0
$694$	−805.929	−0.0440816
$695$	−15254.2	−0.832554
$696$	0	0
$697$	−11984.0	−0.651258
$698$	−1340.00	−0.0726643
$699$	0	0
$700$	0	0
$701$	−11307.8	−0.609258	−0.304629	−	0.952471i	$-0.598532\pi$
$701$	−11307.8	−0.609258	−0.304629	+	0.952471i	$0.598532\pi$
$702$	0	0
$703$	−644.108	−0.0345562
$704$	−24767.9	−1.32596
$705$	0	0
$706$	−1469.71	−0.0783475
$707$	0	0
$708$	0	0
$709$	30859.2	1.63461	0.817307	−	0.576202i	$-0.195466\pi$
$709$	30859.2	1.63461	0.817307	+	0.576202i	$0.195466\pi$
$710$	800.952	0.0423369
$711$	0	0
$712$	−3449.42	−0.181562
$713$	−40551.0	−2.12994
$714$	0	0
$715$	−20433.8	−1.06878
$716$	7592.69	0.396302
$717$	0	0
$718$	−1158.56	−0.0602187
$719$	33152.4	1.71958	0.859789	−	0.510650i	$-0.170595\pi$
$719$	33152.4	1.71958	0.859789	+	0.510650i	$0.170595\pi$
$720$	0	0
$721$	0	0
$722$	1615.15	0.0832543
$723$	0	0
$724$	1637.16	0.0840394
$725$	2279.18	0.116754
$726$	0	0
$727$	16743.0	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$727$	16743.0	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$728$	0	0
$729$	0	0
$730$	71.7650	0.00363855
$731$	27057.5	1.36903
$732$	0	0
$733$	8827.55	0.444820	0.222410	−	0.974953i	$-0.428608\pi$
$733$	8827.55	0.444820	0.222410	+	0.974953i	$0.428608\pi$
$734$	−922.195	−0.0463744
$735$	0	0
$736$	−9192.97	−0.460404
$737$	37082.6	1.85340
$738$	0	0
$739$	36154.0	1.79966	0.899829	−	0.436243i	$-0.143691\pi$
$739$	36154.0	1.79966	0.899829	+	0.436243i	$0.143691\pi$
$740$	6183.25	0.307163
$741$	0	0
$742$	0	0
$743$	1820.69	0.0898987	0.0449494	−	0.998989i	$-0.485687\pi$
$743$	1820.69	0.0898987	0.0449494	+	0.998989i	$0.485687\pi$
$744$	0	0
$745$	11232.8	0.552398
$746$	496.338	0.0243596
$747$	0	0
$748$	−30628.5	−1.49718
$749$	0	0
$750$	0	0
$751$	27764.4	1.34905	0.674526	−	0.738251i	$-0.264348\pi$
$751$	27764.4	1.34905	0.674526	+	0.738251i	$0.264348\pi$
$752$	−11007.8	−0.533795
$753$	0	0
$754$	−1742.62	−0.0841678
$755$	6557.65	0.316102
$756$	0	0
$757$	−13518.3	−0.649050	−0.324525	−	0.945877i	$-0.605205\pi$
$757$	−13518.3	−0.649050	−0.324525	+	0.945877i	$0.605205\pi$
$758$	1561.06	0.0748025
$759$	0	0
$760$	77.8714	0.00371670
$761$	30695.2	1.46216	0.731078	−	0.682294i	$-0.239017\pi$
$761$	30695.2	1.46216	0.731078	+	0.682294i	$0.239017\pi$
$762$	0	0
$763$	0	0
$764$	−16036.5	−0.759400
$765$	0	0
$766$	−0.560560	−2.64410e−5 0
$767$	−25303.2	−1.19119
$768$	0	0
$769$	−4536.39	−0.212726	−0.106363	−	0.994327i	$-0.533921\pi$
$769$	−4536.39	−0.212726	−0.106363	+	0.994327i	$0.533921\pi$
$770$	0	0
$771$	0	0
$772$	−8196.03	−0.382100
$773$	31238.9	1.45354	0.726768	−	0.686883i	$-0.241022\pi$
$773$	31238.9	1.45354	0.726768	+	0.686883i	$0.241022\pi$
$774$	0	0
$775$	−4951.97	−0.229523
$776$	5330.86	0.246607
$777$	0	0
$778$	1821.73	0.0839490
$779$	649.155	0.0298567
$780$	0	0
$781$	34249.2	1.56919
$782$	−3691.64	−0.168814
$783$	0	0
$784$	0	0
$785$	8954.70	0.407143
$786$	0	0
$787$	−39597.2	−1.79350	−0.896752	−	0.442534i	$-0.854079\pi$
$787$	−39597.2	−1.79350	−0.896752	+	0.442534i	$0.854079\pi$
$788$	−1636.16	−0.0739669
$789$	0	0
$790$	−1518.22	−0.0683745
$791$	0	0
$792$	0	0
$793$	12483.3	0.559008
$794$	1511.77	0.0675700
$795$	0	0
$796$	−14549.0	−0.647832
$797$	−16567.0	−0.736302	−0.368151	−	0.929766i	$-0.620009\pi$
$797$	−16567.0	−0.736302	−0.368151	+	0.929766i	$0.620009\pi$
$798$	0	0
$799$	−13418.1	−0.594115
$800$	−1122.62	−0.0496133
$801$	0	0
$802$	−2589.63	−0.114019
$803$	3068.72	0.134860
$804$	0	0
$805$	0	0
$806$	3786.19	0.165463
$807$	0	0
$808$	−5191.20	−0.226022
$809$	12141.6	0.527657	0.263828	−	0.964570i	$-0.415015\pi$
$809$	12141.6	0.527657	0.263828	+	0.964570i	$0.415015\pi$
$810$	0	0
$811$	−30295.6	−1.31174	−0.655870	−	0.754873i	$-0.727698\pi$
$811$	−30295.6	−1.31174	−0.655870	+	0.754873i	$0.727698\pi$
$812$	0	0
$813$	0	0
$814$	−1854.73	−0.0798629
$815$	2455.54	0.105538
$816$	0	0
$817$	−1465.67	−0.0627628
$818$	−94.5106	−0.00403971
$819$	0	0
$820$	−6231.70	−0.265391
$821$	14914.8	0.634018	0.317009	−	0.948423i	$-0.397322\pi$
$821$	14914.8	0.634018	0.317009	+	0.948423i	$0.397322\pi$
$822$	0	0
$823$	−31077.6	−1.31628	−0.658138	−	0.752897i	$-0.728656\pi$
$823$	−31077.6	−1.31628	−0.658138	+	0.752897i	$0.728656\pi$
$824$	4924.85	0.208210
$825$	0	0
$826$	0	0
$827$	−15527.8	−0.652908	−0.326454	−	0.945213i	$-0.605854\pi$
$827$	−15527.8	−0.652908	−0.326454	+	0.945213i	$0.605854\pi$
$828$	0	0
$829$	40221.5	1.68510	0.842551	−	0.538617i	$-0.181053\pi$
$829$	40221.5	1.68510	0.842551	+	0.538617i	$0.181053\pi$
$830$	137.120	0.00573436
$831$	0	0
$832$	−39734.2	−1.65569
$833$	0	0
$834$	0	0
$835$	4130.29	0.171179
$836$	1659.10	0.0686377
$837$	0	0
$838$	3733.51	0.153904
$839$	−21153.7	−0.870448	−0.435224	−	0.900322i	$-0.643331\pi$
$839$	−21153.7	−0.870448	−0.435224	+	0.900322i	$0.643331\pi$
$840$	0	0
$841$	−16077.5	−0.659213
$842$	−457.152	−0.0187108
$843$	0	0
$844$	−8184.10	−0.333778
$845$	−21796.2	−0.887351
$846$	0	0
$847$	0	0
$848$	−12552.0	−0.508301
$849$	0	0
$850$	−450.813	−0.0181915
$851$	31868.1	1.28369
$852$	0	0
$853$	−636.075	−0.0255320	−0.0127660	−	0.999919i	$-0.504064\pi$
$853$	−636.075	−0.0255320	−0.0127660	+	0.999919i	$0.504064\pi$
$854$	0	0
$855$	0	0
$856$	−4763.83	−0.190215
$857$	−3941.46	−0.157104	−0.0785518	−	0.996910i	$-0.525030\pi$
$857$	−3941.46	−0.157104	−0.0785518	+	0.996910i	$0.525030\pi$
$858$	0	0
$859$	21781.6	0.865165	0.432583	−	0.901594i	$-0.357602\pi$
$859$	21781.6	0.865165	0.432583	+	0.901594i	$0.357602\pi$
$860$	14070.0	0.557886
$861$	0	0
$862$	479.432	0.0189438
$863$	44697.9	1.76308	0.881538	−	0.472112i	$-0.156508\pi$
$863$	44697.9	1.76308	0.881538	+	0.472112i	$0.156508\pi$
$864$	0	0
$865$	−14590.0	−0.573498
$866$	−2545.79	−0.0998953
$867$	0	0
$868$	0	0
$869$	−64920.1	−2.53425
$870$	0	0
$871$	59490.3	2.31430
$872$	7788.78	0.302478
$873$	0	0
$874$	199.971	0.00773926
$875$	0	0
$876$	0	0
$877$	−20171.7	−0.776682	−0.388341	−	0.921516i	$-0.626952\pi$
$877$	−20171.7	−0.776682	−0.388341	+	0.921516i	$0.626952\pi$
$878$	−1488.22	−0.0572038
$879$	0	0
$880$	−15814.3	−0.605797
$881$	11577.6	0.442744	0.221372	−	0.975189i	$-0.428946\pi$
$881$	11577.6	0.442744	0.221372	+	0.975189i	$0.428946\pi$
$882$	0	0
$883$	−35388.3	−1.34871	−0.674355	−	0.738407i	$-0.735578\pi$
$883$	−35388.3	−1.34871	−0.674355	+	0.738407i	$0.735578\pi$
$884$	−49136.1	−1.86949
$885$	0	0
$886$	3657.83	0.138699
$887$	−41705.0	−1.57871	−0.789356	−	0.613936i	$-0.789585\pi$
$887$	−41705.0	−1.57871	−0.789356	+	0.613936i	$0.789585\pi$
$888$	0	0
$889$	0	0
$890$	−1081.71	−0.0407405
$891$	0	0
$892$	42650.2	1.60093
$893$	726.838	0.0272371
$894$	0	0
$895$	4778.72	0.178475
$896$	0	0
$897$	0	0
$898$	−57.1328	−0.00212310
$899$	−18058.3	−0.669942
$900$	0	0
$901$	−15300.5	−0.565740
$902$	1869.27	0.0690020
$903$	0	0
$904$	7354.31	0.270576
$905$	1030.40	0.0378472
$906$	0	0
$907$	−28645.1	−1.04867	−0.524336	−	0.851511i	$-0.675687\pi$
$907$	−28645.1	−1.04867	−0.524336	+	0.851511i	$0.675687\pi$
$908$	7405.02	0.270643
$909$	0	0
$910$	0	0
$911$	−13337.8	−0.485074	−0.242537	−	0.970142i	$-0.577980\pi$
$911$	−13337.8	−0.485074	−0.242537	+	0.970142i	$0.577980\pi$
$912$	0	0
$913$	5863.36	0.212540
$914$	2755.00	0.0997016
$915$	0	0
$916$	25128.1	0.906394
$917$	0	0
$918$	0	0
$919$	−28911.0	−1.03774	−0.518871	−	0.854853i	$-0.673647\pi$
$919$	−28911.0	−1.03774	−0.518871	+	0.854853i	$0.673647\pi$
$920$	−3852.79	−0.138068
$921$	0	0
$922$	−2863.06	−0.102267
$923$	54944.8	1.95940
$924$	0	0
$925$	3891.64	0.138331
$926$	−1218.50	−0.0432422
$927$	0	0
$928$	−4093.84	−0.144814
$929$	7093.88	0.250530	0.125265	−	0.992123i	$-0.460022\pi$
$929$	7093.88	0.250530	0.125265	+	0.992123i	$0.460022\pi$
$930$	0	0
$931$	0	0
$932$	3468.17	0.121892
$933$	0	0
$934$	−2757.33	−0.0965981
$935$	−19277.1	−0.674254
$936$	0	0
$937$	19271.1	0.671888	0.335944	−	0.941882i	$-0.390945\pi$
$937$	19271.1	0.671888	0.335944	+	0.941882i	$0.390945\pi$
$938$	0	0
$939$	0	0
$940$	−6977.43	−0.242105
$941$	−18115.2	−0.627563	−0.313782	−	0.949495i	$-0.601596\pi$
$941$	−18115.2	−0.627563	−0.313782	+	0.949495i	$0.601596\pi$
$942$	0	0
$943$	−32117.8	−1.10912
$944$	−19582.9	−0.675180
$945$	0	0
$946$	−4220.44	−0.145051
$947$	2475.55	0.0849468	0.0424734	−	0.999098i	$-0.486476\pi$
$947$	2475.55	0.0849468	0.0424734	+	0.999098i	$0.486476\pi$
$948$	0	0
$949$	4923.04	0.168397
$950$	24.4199	0.000833984 0
$951$	0	0
$952$	0	0
$953$	−12866.6	−0.437344	−0.218672	−	0.975798i	$-0.570173\pi$
$953$	−12866.6	−0.437344	−0.218672	+	0.975798i	$0.570173\pi$
$954$	0	0
$955$	−10093.2	−0.341997
$956$	−15736.2	−0.532368
$957$	0	0
$958$	4357.50	0.146957
$959$	0	0
$960$	0	0
$961$	9444.26	0.317017
$962$	−2975.48	−0.0997228
$963$	0	0
$964$	42132.9	1.40768
$965$	−5158.45	−0.172079
$966$	0	0
$967$	−2142.86	−0.0712614	−0.0356307	−	0.999365i	$-0.511344\pi$
$967$	−2142.86	−0.0712614	−0.0356307	+	0.999365i	$0.511344\pi$
$968$	4578.58	0.152026
$969$	0	0
$970$	1671.72	0.0553357
$971$	2879.06	0.0951529	0.0475765	−	0.998868i	$-0.484850\pi$
$971$	2879.06	0.0951529	0.0475765	+	0.998868i	$0.484850\pi$
$972$	0	0
$973$	0	0
$974$	−2037.29	−0.0670217
$975$	0	0
$976$	9661.18	0.316851
$977$	−48741.1	−1.59607	−0.798037	−	0.602608i	$-0.794128\pi$
$977$	−48741.1	−1.59607	−0.798037	+	0.602608i	$0.794128\pi$
$978$	0	0
$979$	−46254.7	−1.51002
$980$	0	0
$981$	0	0
$982$	4202.40	0.136562
$983$	−45756.8	−1.48466	−0.742328	−	0.670037i	$-0.766278\pi$
$983$	−45756.8	−1.48466	−0.742328	+	0.670037i	$0.766278\pi$
$984$	0	0
$985$	−1029.78	−0.0333110
$986$	−1643.97	−0.0530982
$987$	0	0
$988$	2661.63	0.0857062
$989$	72515.7	2.33151
$990$	0	0
$991$	−51552.1	−1.65248	−0.826240	−	0.563319i	$-0.809524\pi$
$991$	−51552.1	−1.65248	−0.826240	+	0.563319i	$0.809524\pi$
$992$	8894.70	0.284684
$993$	0	0
$994$	0	0
$995$	−9156.88	−0.291751
$996$	0	0
$997$	−25565.3	−0.812097	−0.406048	−	0.913852i	$-0.633094\pi$
$997$	−25565.3	−0.812097	−0.406048	+	0.913852i	$0.633094\pi$
$998$	−47.2531	−0.00149877
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.be.1.1		2
3.2	odd	2		735.4.a.m.1.2		2
7.6	odd	2		315.4.a.l.1.1		2
21.20	even	2		105.4.a.d.1.2	✓	2
35.34	odd	2		1575.4.a.n.1.2		2
84.83	odd	2		1680.4.a.bd.1.2		2
105.62	odd	4		525.4.d.k.274.3		4
105.83	odd	4		525.4.d.k.274.2		4
105.104	even	2		525.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.2	✓	2	21.20	even	2
315.4.a.l.1.1		2	7.6	odd	2
525.4.a.o.1.1		2	105.104	even	2
525.4.d.k.274.2		4	105.83	odd	4
525.4.d.k.274.3		4	105.62	odd	4
735.4.a.m.1.2		2	3.2	odd	2
1575.4.a.n.1.2		2	35.34	odd	2
1680.4.a.bd.1.2		2	84.83	odd	2
2205.4.a.be.1.1		2	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)

\(f(q)\)	\(=\)	\(q-0.236068 q^{2} -7.94427 q^{4} -5.00000 q^{5} +3.76393 q^{8} +O(q^{10})\)	\(q-0.236068 q^{2} -7.94427 q^{4} -5.00000 q^{5} +3.76393 q^{8} +1.18034 q^{10} +50.4721 q^{11} +80.9706 q^{13} +62.6656 q^{16} +76.3870 q^{17} -4.13777 q^{19} +39.7214 q^{20} -11.9149 q^{22} +204.721 q^{23} +25.0000 q^{25} -19.1146 q^{26} +91.1672 q^{29} -198.079 q^{31} -44.9048 q^{32} -18.0325 q^{34} +155.666 q^{37} +0.976794 q^{38} -18.8197 q^{40} -156.885 q^{41} +354.217 q^{43} -400.964 q^{44} -48.3282 q^{46} -175.659 q^{47} -5.90170 q^{50} -643.252 q^{52} -200.302 q^{53} -252.361 q^{55} -21.5217 q^{58} -312.498 q^{59} +154.170 q^{61} +46.7601 q^{62} -490.724 q^{64} -404.853 q^{65} +734.715 q^{67} -606.839 q^{68} +678.577 q^{71} +60.8003 q^{73} -36.7477 q^{74} +32.8715 q^{76} -1286.26 q^{79} -313.328 q^{80} +37.0356 q^{82} +116.170 q^{83} -381.935 q^{85} -83.6192 q^{86} +189.974 q^{88} -916.440 q^{89} -1626.36 q^{92} +41.4676 q^{94} +20.6888 q^{95} +1416.30 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{4} - 10 q^{5} + 12 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 2 * q^4 - 10 * q^5 + 12 * q^8	\( 2 q + 4 q^{2} + 2 q^{4} - 10 q^{5} + 12 q^{8} - 20 q^{10} + 92 q^{11} - 8 q^{13} + 18 q^{16} - 44 q^{17} + 108 q^{19} - 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} - 396 q^{26} + 236 q^{29} + 60 q^{31} - 300 q^{32} - 528 q^{34} + 204 q^{37} + 476 q^{38} - 60 q^{40} + 44 q^{41} + 136 q^{43} + 12 q^{44} + 440 q^{46} + 400 q^{47} + 100 q^{50} - 1528 q^{52} - 16 q^{53} - 460 q^{55} + 592 q^{58} - 464 q^{59} + 684 q^{61} + 1140 q^{62} - 1214 q^{64} + 40 q^{65} + 736 q^{67} - 1804 q^{68} + 740 q^{71} - 424 q^{73} + 168 q^{74} + 1148 q^{76} - 408 q^{79} - 90 q^{80} + 888 q^{82} + 608 q^{83} + 220 q^{85} - 1008 q^{86} + 532 q^{88} - 1332 q^{89} - 480 q^{92} + 2480 q^{94} - 540 q^{95} + 2448 q^{97}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^4 - 10 * q^5 + 12 * q^8 - 20 * q^10 + 92 * q^11 - 8 * q^13 + 18 * q^16 - 44 * q^17 + 108 * q^19 - 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 - 396 * q^26 + 236 * q^29 + 60 * q^31 - 300 * q^32 - 528 * q^34 + 204 * q^37 + 476 * q^38 - 60 * q^40 + 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 + 400 * q^47 + 100 * q^50 - 1528 * q^52 - 16 * q^53 - 460 * q^55 + 592 * q^58 - 464 * q^59 + 684 * q^61 + 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 - 1804 * q^68 + 740 * q^71 - 424 * q^73 + 168 * q^74 + 1148 * q^76 - 408 * q^79 - 90 * q^80 + 888 * q^82 + 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 - 1332 * q^89 - 480 * q^92 + 2480 * q^94 - 540 * q^95 + 2448 * q^97

Introduction

L-functions

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