LMFDB - Embedded newform 4005.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8069.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 5x + 1 $ x^4 - x^3 - 5x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.14281$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14281 q^{2} +2.59164 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.26778 q^{8} +O(q^{10})$	\(q-2.14281 q^{2} +2.59164 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.26778 q^{8} +2.14281 q^{10} +3.41059 q^{11} -3.71661 q^{13} -6.42843 q^{14} -2.46668 q^{16} +1.81895 q^{17} +1.34171 q^{19} -2.59164 q^{20} -7.30825 q^{22} -1.28339 q^{23} +1.00000 q^{25} +7.96399 q^{26} +7.77493 q^{28} -7.51293 q^{29} -0.379953 q^{31} +7.82118 q^{32} -3.89766 q^{34} -3.00000 q^{35} +8.06055 q^{37} -2.87503 q^{38} +1.26778 q^{40} -8.02486 q^{41} -7.75230 q^{43} +8.83902 q^{44} +2.75007 q^{46} -0.948969 q^{47} +2.00000 q^{49} -2.14281 q^{50} -9.63211 q^{52} -7.14982 q^{53} -3.41059 q^{55} -3.80333 q^{56} +16.0988 q^{58} -7.00446 q^{59} -14.1473 q^{61} +0.814168 q^{62} -11.8260 q^{64} +3.71661 q^{65} -2.90567 q^{67} +4.71406 q^{68} +6.42843 q^{70} +7.24993 q^{71} -9.61172 q^{73} -17.2722 q^{74} +3.47724 q^{76} +10.2318 q^{77} +7.86165 q^{79} +2.46668 q^{80} +17.1958 q^{82} -8.71183 q^{83} -1.81895 q^{85} +16.6117 q^{86} -4.32386 q^{88} +1.00000 q^{89} -11.1498 q^{91} -3.32610 q^{92} +2.03346 q^{94} -1.34171 q^{95} -8.77493 q^{97} -4.28562 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} - 7 q^{13} - 3 q^{14} - 3 q^{16} - q^{17} - q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 4 q^{25} - 7 q^{26} + 9 q^{28} - 14 q^{29} - 11 q^{31} - 16 q^{34} - 12 q^{35} - 5 q^{37} - 12 q^{38} - 3 q^{40} - 9 q^{41} - 9 q^{43} - 3 q^{44} + 12 q^{46} - 6 q^{47} + 8 q^{49} - q^{50} - 24 q^{52} - 5 q^{53} + 2 q^{55} + 9 q^{56} + 43 q^{58} + 18 q^{59} - 3 q^{61} - 12 q^{62} - 23 q^{64} + 7 q^{65} + 13 q^{67} - 19 q^{68} + 3 q^{70} + 28 q^{71} - q^{73} - 15 q^{74} + 12 q^{76} - 6 q^{77} - 7 q^{79} + 3 q^{80} - 17 q^{82} - 20 q^{83} + q^{85} + 29 q^{86} - 18 q^{88} + 4 q^{89} - 21 q^{91} + 9 q^{92} + 17 q^{94} + q^{95} - 13 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8 + q^10 - 2 * q^11 - 7 * q^13 - 3 * q^14 - 3 * q^16 - q^17 - q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 4 * q^25 - 7 * q^26 + 9 * q^28 - 14 * q^29 - 11 * q^31 - 16 * q^34 - 12 * q^35 - 5 * q^37 - 12 * q^38 - 3 * q^40 - 9 * q^41 - 9 * q^43 - 3 * q^44 + 12 * q^46 - 6 * q^47 + 8 * q^49 - q^50 - 24 * q^52 - 5 * q^53 + 2 * q^55 + 9 * q^56 + 43 * q^58 + 18 * q^59 - 3 * q^61 - 12 * q^62 - 23 * q^64 + 7 * q^65 + 13 * q^67 - 19 * q^68 + 3 * q^70 + 28 * q^71 - q^73 - 15 * q^74 + 12 * q^76 - 6 * q^77 - 7 * q^79 + 3 * q^80 - 17 * q^82 - 20 * q^83 + q^85 + 29 * q^86 - 18 * q^88 + 4 * q^89 - 21 * q^91 + 9 * q^92 + 17 * q^94 + q^95 - 13 * q^97 - 2 * q^98

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 $

$2$	−2.14281	−1.51520	−0.757598	−	0.652721i	$-0.773627\pi$
$2$	−2.14281	−1.51520	−0.757598	+	0.652721i	$0.773627\pi$
$3$	0	0
$3$	0	0
$4$	2.59164	1.29582
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.26778	−0.448227
$9$	0	0
$10$	2.14281	0.677617
$11$	3.41059	1.02833	0.514166	−	0.857691i	$-0.328102\pi$
$11$	3.41059	1.02833	0.514166	+	0.857691i	$0.328102\pi$
$12$	0	0
$13$	−3.71661	−1.03080	−0.515401	−	0.856949i	$-0.672357\pi$
$13$	−3.71661	−1.03080	−0.515401	+	0.856949i	$0.672357\pi$
$14$	−6.42843	−1.71807
$15$	0	0
$16$	−2.46668	−0.616669
$17$	1.81895	0.441159	0.220580	−	0.975369i	$-0.429205\pi$
$17$	1.81895	0.441159	0.220580	+	0.975369i	$0.429205\pi$
$18$	0	0
$19$	1.34171	0.307810	0.153905	−	0.988086i	$-0.450815\pi$
$19$	1.34171	0.307810	0.153905	+	0.988086i	$0.450815\pi$
$20$	−2.59164	−0.579509
$21$	0	0
$22$	−7.30825	−1.55812
$23$	−1.28339	−0.267606	−0.133803	−	0.991008i	$-0.542719\pi$
$23$	−1.28339	−0.267606	−0.133803	+	0.991008i	$0.542719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	7.96399	1.56187
$27$	0	0
$28$	7.77493	1.46932
$29$	−7.51293	−1.39512	−0.697558	−	0.716528i	$-0.745730\pi$
$29$	−7.51293	−1.39512	−0.697558	+	0.716528i	$0.745730\pi$
$30$	0	0
$31$	−0.379953	−0.0682416	−0.0341208	−	0.999418i	$-0.510863\pi$
$31$	−0.379953	−0.0682416	−0.0341208	+	0.999418i	$0.510863\pi$
$32$	7.82118	1.38260
$33$	0	0
$34$	−3.89766	−0.668443
$35$	−3.00000	−0.507093
$36$	0	0
$37$	8.06055	1.32515	0.662573	−	0.748997i	$-0.269464\pi$
$37$	8.06055	1.32515	0.662573	+	0.748997i	$0.269464\pi$
$38$	−2.87503	−0.466392
$39$	0	0
$40$	1.26778	0.200453
$41$	−8.02486	−1.25327	−0.626636	−	0.779312i	$-0.715569\pi$
$41$	−8.02486	−1.25327	−0.626636	+	0.779312i	$0.715569\pi$
$42$	0	0
$43$	−7.75230	−1.18221	−0.591107	−	0.806593i	$-0.701309\pi$
$43$	−7.75230	−1.18221	−0.591107	+	0.806593i	$0.701309\pi$
$44$	8.83902	1.33253
$45$	0	0
$46$	2.75007	0.405476
$47$	−0.948969	−0.138421	−0.0692107	−	0.997602i	$-0.522048\pi$
$47$	−0.948969	−0.138421	−0.0692107	+	0.997602i	$0.522048\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−2.14281	−0.303039
$51$	0	0
$52$	−9.63211	−1.33573
$53$	−7.14982	−0.982103	−0.491052	−	0.871130i	$-0.663387\pi$
$53$	−7.14982	−0.982103	−0.491052	+	0.871130i	$0.663387\pi$
$54$	0	0
$55$	−3.41059	−0.459884
$56$	−3.80333	−0.508241
$57$	0	0
$58$	16.0988	2.11387
$59$	−7.00446	−0.911903	−0.455952	−	0.890005i	$-0.650701\pi$
$59$	−7.00446	−0.911903	−0.455952	+	0.890005i	$0.650701\pi$
$60$	0	0
$61$	−14.1473	−1.81137	−0.905686	−	0.423949i	$-0.860644\pi$
$61$	−14.1473	−1.81137	−0.905686	+	0.423949i	$0.860644\pi$
$62$	0.814168	0.103399
$63$	0	0
$64$	−11.8260	−1.47824
$65$	3.71661	0.460988
$66$	0	0
$67$	−2.90567	−0.354984	−0.177492	−	0.984122i	$-0.556798\pi$
$67$	−2.90567	−0.354984	−0.177492	+	0.984122i	$0.556798\pi$
$68$	4.71406	0.571664
$69$	0	0
$70$	6.42843	0.768345
$71$	7.24993	0.860408	0.430204	−	0.902732i	$-0.358442\pi$
$71$	7.24993	0.860408	0.430204	+	0.902732i	$0.358442\pi$
$72$	0	0
$73$	−9.61172	−1.12497	−0.562483	−	0.826809i	$-0.690154\pi$
$73$	−9.61172	−1.12497	−0.562483	+	0.826809i	$0.690154\pi$
$74$	−17.2722	−2.00786
$75$	0	0
$76$	3.47724	0.398866
$77$	10.2318	1.16602
$78$	0	0
$79$	7.86165	0.884505	0.442252	−	0.896891i	$-0.354180\pi$
$79$	7.86165	0.884505	0.442252	+	0.896891i	$0.354180\pi$
$80$	2.46668	0.275783
$81$	0	0
$82$	17.1958	1.89895
$83$	−8.71183	−0.956247	−0.478124	−	0.878293i	$-0.658683\pi$
$83$	−8.71183	−0.956247	−0.478124	+	0.878293i	$0.658683\pi$
$84$	0	0
$85$	−1.81895	−0.197292
$86$	16.6117	1.79129
$87$	0	0
$88$	−4.32386	−0.460926
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−11.1498	−1.16882
$92$	−3.32610	−0.346769
$93$	0	0
$94$	2.03346	0.209736
$95$	−1.34171	−0.137657
$96$	0	0
$97$	−8.77493	−0.890959	−0.445479	−	0.895292i	$-0.646967\pi$
$97$	−8.77493	−0.890959	−0.445479	+	0.895292i	$0.646967\pi$
$98$	−4.28562	−0.432913
$99$	0	0
$100$	2.59164	0.259164
$101$	−13.4128	−1.33463	−0.667313	−	0.744778i	$-0.732556\pi$
$101$	−13.4128	−1.33463	−0.667313	+	0.744778i	$0.732556\pi$
$102$	0	0
$103$	13.6879	1.34871	0.674354	−	0.738409i	$-0.264422\pi$
$103$	13.6879	1.34871	0.674354	+	0.738409i	$0.264422\pi$
$104$	4.71183	0.462033
$105$	0	0
$106$	15.3207	1.48808
$107$	2.65219	0.256397	0.128198	−	0.991749i	$-0.459081\pi$
$107$	2.65219	0.256397	0.128198	+	0.991749i	$0.459081\pi$
$108$	0	0
$109$	4.79500	0.459278	0.229639	−	0.973276i	$-0.426245\pi$
$109$	4.79500	0.459278	0.229639	+	0.973276i	$0.426245\pi$
$110$	7.30825	0.696814
$111$	0	0
$112$	−7.40003	−0.699237
$113$	16.1677	1.52093	0.760463	−	0.649381i	$-0.224972\pi$
$113$	16.1677	1.52093	0.760463	+	0.649381i	$0.224972\pi$
$114$	0	0
$115$	1.28339	0.119677
$116$	−19.4708	−1.80782
$117$	0	0
$118$	15.0092	1.38171
$119$	5.45684	0.500228
$120$	0	0
$121$	0.632114	0.0574649
$122$	30.3149	2.74459
$123$	0	0
$124$	−0.984702	−0.0884289
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.4583	1.10550	0.552750	−	0.833347i	$-0.313579\pi$
$127$	12.4583	1.10550	0.552750	+	0.833347i	$0.313579\pi$
$128$	9.69844	0.857229
$129$	0	0
$130$	−7.96399	−0.698488
$131$	4.14727	0.362349	0.181174	−	0.983451i	$-0.442010\pi$
$131$	4.14727	0.362349	0.181174	+	0.983451i	$0.442010\pi$
$132$	0	0
$133$	4.02513	0.349023
$134$	6.22630	0.537871
$135$	0	0
$136$	−2.30602	−0.197739
$137$	4.62155	0.394846	0.197423	−	0.980318i	$-0.436743\pi$
$137$	4.62155	0.394846	0.197423	+	0.980318i	$0.436743\pi$
$138$	0	0
$139$	5.22954	0.443563	0.221782	−	0.975096i	$-0.428813\pi$
$139$	5.22954	0.443563	0.221782	+	0.975096i	$0.428813\pi$
$140$	−7.77493	−0.657101
$141$	0	0
$142$	−15.5352	−1.30369
$143$	−12.6758	−1.06001
$144$	0	0
$145$	7.51293	0.623915
$146$	20.5961	1.70455
$147$	0	0
$148$	20.8901	1.71715
$149$	−18.1983	−1.49086	−0.745432	−	0.666582i	$-0.767757\pi$
$149$	−18.1983	−1.49086	−0.745432	+	0.666582i	$0.767757\pi$
$150$	0	0
$151$	−16.2722	−1.32422	−0.662108	−	0.749408i	$-0.730338\pi$
$151$	−16.2722	−1.32422	−0.662108	+	0.749408i	$0.730338\pi$
$152$	−1.70099	−0.137969
$153$	0	0
$154$	−21.9247	−1.76675
$155$	0.379953	0.0305186
$156$	0	0
$157$	5.17627	0.413112	0.206556	−	0.978435i	$-0.433774\pi$
$157$	5.17627	0.413112	0.206556	+	0.978435i	$0.433774\pi$
$158$	−16.8460	−1.34020
$159$	0	0
$160$	−7.82118	−0.618318
$161$	−3.85018	−0.303437
$162$	0	0
$163$	22.6694	1.77561	0.887804	−	0.460222i	$-0.152230\pi$
$163$	22.6694	1.77561	0.887804	+	0.460222i	$0.152230\pi$
$164$	−20.7976	−1.62402
$165$	0	0
$166$	18.6678	1.44890
$167$	−11.1138	−0.860012	−0.430006	−	0.902826i	$-0.641489\pi$
$167$	−11.1138	−0.860012	−0.430006	+	0.902826i	$0.641489\pi$
$168$	0	0
$169$	0.813167	0.0625513
$170$	3.89766	0.298937
$171$	0	0
$172$	−20.0912	−1.53194
$173$	3.19607	0.242993	0.121496	−	0.992592i	$-0.461231\pi$
$173$	3.19607	0.242993	0.121496	+	0.992592i	$0.461231\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	−8.41282	−0.634140
$177$	0	0
$178$	−2.14281	−0.160611
$179$	12.7545	0.953318	0.476659	−	0.879088i	$-0.341848\pi$
$179$	12.7545	0.953318	0.476659	+	0.879088i	$0.341848\pi$
$180$	0	0
$181$	−23.4351	−1.74192	−0.870960	−	0.491355i	$-0.836502\pi$
$181$	−23.4351	−1.74192	−0.870960	+	0.491355i	$0.836502\pi$
$182$	23.8920	1.77099
$183$	0	0
$184$	1.62706	0.119948
$185$	−8.06055	−0.592623
$186$	0	0
$187$	6.20368	0.453658
$188$	−2.45939	−0.179369
$189$	0	0
$190$	2.87503	0.208577
$191$	11.0653	0.800659	0.400329	−	0.916371i	$-0.368896\pi$
$191$	11.0653	0.800659	0.400329	+	0.916371i	$0.368896\pi$
$192$	0	0
$193$	−8.63494	−0.621557	−0.310778	−	0.950482i	$-0.600590\pi$
$193$	−8.63494	−0.621557	−0.310778	+	0.950482i	$0.600590\pi$
$194$	18.8030	1.34998
$195$	0	0
$196$	5.18328	0.370235
$197$	−5.90822	−0.420943	−0.210472	−	0.977600i	$-0.567500\pi$
$197$	−5.90822	−0.420943	−0.210472	+	0.977600i	$0.567500\pi$
$198$	0	0
$199$	0.117677	0.00834193	0.00417097	−	0.999991i	$-0.498672\pi$
$199$	0.117677	0.00834193	0.00417097	+	0.999991i	$0.498672\pi$
$200$	−1.26778	−0.0896454
$201$	0	0
$202$	28.7411	2.02222
$203$	−22.5388	−1.58191
$204$	0	0
$205$	8.02486	0.560480
$206$	−29.3306	−2.04356
$207$	0	0
$208$	9.16767	0.635663
$209$	4.57603	0.316530
$210$	0	0
$211$	−5.29546	−0.364554	−0.182277	−	0.983247i	$-0.558347\pi$
$211$	−5.29546	−0.364554	−0.182277	+	0.983247i	$0.558347\pi$
$212$	−18.5298	−1.27263
$213$	0	0
$214$	−5.68314	−0.388492
$215$	7.75230	0.528702
$216$	0	0
$217$	−1.13986	−0.0773787
$218$	−10.2748	−0.695896
$219$	0	0
$220$	−8.83902	−0.595927
$221$	−6.76031	−0.454748
$222$	0	0
$223$	20.3994	1.36605	0.683024	−	0.730396i	$-0.260665\pi$
$223$	20.3994	1.36605	0.683024	+	0.730396i	$0.260665\pi$
$224$	23.4635	1.56772
$225$	0	0
$226$	−34.6443	−2.30450
$227$	−24.8566	−1.64979	−0.824895	−	0.565286i	$-0.808766\pi$
$227$	−24.8566	−1.64979	−0.824895	+	0.565286i	$0.808766\pi$
$228$	0	0
$229$	−21.6439	−1.43027	−0.715133	−	0.698988i	$-0.753634\pi$
$229$	−21.6439	−1.43027	−0.715133	+	0.698988i	$0.753634\pi$
$230$	−2.75007	−0.181334
$231$	0	0
$232$	9.52472	0.625328
$233$	21.5224	1.40998	0.704991	−	0.709216i	$-0.250951\pi$
$233$	21.5224	1.40998	0.704991	+	0.709216i	$0.250951\pi$
$234$	0	0
$235$	0.948969	0.0619039
$236$	−18.1531	−1.18166
$237$	0	0
$238$	−11.6930	−0.757943
$239$	20.2808	1.31186	0.655929	−	0.754822i	$-0.272277\pi$
$239$	20.2808	1.31186	0.655929	+	0.754822i	$0.272277\pi$
$240$	0	0
$241$	−0.610808	−0.0393456	−0.0196728	−	0.999806i	$-0.506262\pi$
$241$	−0.610808	−0.0393456	−0.0196728	+	0.999806i	$0.506262\pi$
$242$	−1.35450	−0.0870706
$243$	0	0
$244$	−36.6647	−2.34721
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−4.98661	−0.317291
$248$	0.481696	0.0305877
$249$	0	0
$250$	2.14281	0.135523
$251$	−17.6761	−1.11571	−0.557854	−	0.829939i	$-0.688375\pi$
$251$	−17.6761	−1.11571	−0.557854	+	0.829939i	$0.688375\pi$
$252$	0	0
$253$	−4.37713	−0.275188
$254$	−26.6959	−1.67505
$255$	0	0
$256$	2.86998	0.179374
$257$	−24.9171	−1.55429	−0.777144	−	0.629323i	$-0.783332\pi$
$257$	−24.9171	−1.55429	−0.777144	+	0.629323i	$0.783332\pi$
$258$	0	0
$259$	24.1816	1.50257
$260$	9.63211	0.597358
$261$	0	0
$262$	−8.88682	−0.549030
$263$	24.6800	1.52183	0.760917	−	0.648849i	$-0.224749\pi$
$263$	24.6800	1.52183	0.760917	+	0.648849i	$0.224749\pi$
$264$	0	0
$265$	7.14982	0.439210
$266$	−8.62510	−0.528839
$267$	0	0
$268$	−7.53046	−0.459996
$269$	7.08895	0.432221	0.216111	−	0.976369i	$-0.430663\pi$
$269$	7.08895	0.432221	0.216111	+	0.976369i	$0.430663\pi$
$270$	0	0
$271$	−7.39497	−0.449213	−0.224606	−	0.974450i	$-0.572110\pi$
$271$	−7.39497	−0.449213	−0.224606	+	0.974450i	$0.572110\pi$
$272$	−4.48675	−0.272049
$273$	0	0
$274$	−9.90312	−0.598269
$275$	3.41059	0.205666
$276$	0	0
$277$	7.88232	0.473603	0.236801	−	0.971558i	$-0.423901\pi$
$277$	7.88232	0.473603	0.236801	+	0.971558i	$0.423901\pi$
$278$	−11.2059	−0.672086
$279$	0	0
$280$	3.80333	0.227292
$281$	−5.05609	−0.301621	−0.150810	−	0.988563i	$-0.548188\pi$
$281$	−5.05609	−0.301621	−0.150810	+	0.988563i	$0.548188\pi$
$282$	0	0
$283$	−8.14950	−0.484438	−0.242219	−	0.970222i	$-0.577875\pi$
$283$	−8.14950	−0.484438	−0.242219	+	0.970222i	$0.577875\pi$
$284$	18.7892	1.11494
$285$	0	0
$286$	27.1619	1.60612
$287$	−24.0746	−1.42108
$288$	0	0
$289$	−13.6914	−0.805378
$290$	−16.0988	−0.945354
$291$	0	0
$292$	−24.9101	−1.45776
$293$	−5.89543	−0.344415	−0.172207	−	0.985061i	$-0.555090\pi$
$293$	−5.89543	−0.344415	−0.172207	+	0.985061i	$0.555090\pi$
$294$	0	0
$295$	7.00446	0.407815
$296$	−10.2190	−0.593966
$297$	0	0
$298$	38.9955	2.25895
$299$	4.76987	0.275849
$300$	0	0
$301$	−23.2569	−1.34051
$302$	34.8683	2.00645
$303$	0	0
$304$	−3.30957	−0.189817
$305$	14.1473	0.810070
$306$	0	0
$307$	13.0590	0.745319	0.372659	−	0.927968i	$-0.378446\pi$
$307$	13.0590	0.745319	0.372659	+	0.927968i	$0.378446\pi$
$308$	26.5171	1.51095
$309$	0	0
$310$	−0.814168	−0.0462416
$311$	−21.5476	−1.22185	−0.610926	−	0.791688i	$-0.709203\pi$
$311$	−21.5476	−1.22185	−0.610926	+	0.791688i	$0.709203\pi$
$312$	0	0
$313$	0.675134	0.0381608	0.0190804	−	0.999818i	$-0.493926\pi$
$313$	0.675134	0.0381608	0.0190804	+	0.999818i	$0.493926\pi$
$314$	−11.0918	−0.625945
$315$	0	0
$316$	20.3746	1.14616
$317$	−6.53328	−0.366946	−0.183473	−	0.983025i	$-0.558734\pi$
$317$	−6.53328	−0.366946	−0.183473	+	0.983025i	$0.558734\pi$
$318$	0	0
$319$	−25.6235	−1.43464
$320$	11.8260	0.661091
$321$	0	0
$322$	8.25021	0.459766
$323$	2.44050	0.135793
$324$	0	0
$325$	−3.71661	−0.206160
$326$	−48.5763	−2.69039
$327$	0	0
$328$	10.1737	0.561750
$329$	−2.84691	−0.156955
$330$	0	0
$331$	20.3449	1.11825	0.559127	−	0.829082i	$-0.311136\pi$
$331$	20.3449	1.11825	0.559127	+	0.829082i	$0.311136\pi$
$332$	−22.5779	−1.23913
$333$	0	0
$334$	23.8148	1.30309
$335$	2.90567	0.158754
$336$	0	0
$337$	22.3335	1.21658	0.608292	−	0.793713i	$-0.291855\pi$
$337$	22.3335	1.21658	0.608292	+	0.793713i	$0.291855\pi$
$338$	−1.74246	−0.0947775
$339$	0	0
$340$	−4.71406	−0.255656
$341$	−1.29586	−0.0701749
$342$	0	0
$343$	−15.0000	−0.809924
$344$	9.82819	0.529900
$345$	0	0
$346$	−6.84858	−0.368182
$347$	11.1447	0.598280	0.299140	−	0.954209i	$-0.403300\pi$
$347$	11.1447	0.598280	0.299140	+	0.954209i	$0.403300\pi$
$348$	0	0
$349$	4.02513	0.215460	0.107730	−	0.994180i	$-0.465642\pi$
$349$	4.02513	0.215460	0.107730	+	0.994180i	$0.465642\pi$
$350$	−6.42843	−0.343614
$351$	0	0
$352$	26.6748	1.42177
$353$	−26.2085	−1.39494	−0.697470	−	0.716614i	$-0.745691\pi$
$353$	−26.2085	−1.39494	−0.697470	+	0.716614i	$0.745691\pi$
$354$	0	0
$355$	−7.24993	−0.384786
$356$	2.59164	0.137357
$357$	0	0
$358$	−27.3306	−1.44446
$359$	−31.8691	−1.68198	−0.840992	−	0.541047i	$-0.818028\pi$
$359$	−31.8691	−1.68198	−0.840992	+	0.541047i	$0.818028\pi$
$360$	0	0
$361$	−17.1998	−0.905253
$362$	50.2171	2.63935
$363$	0	0
$364$	−28.8963	−1.51458
$365$	9.61172	0.503100
$366$	0	0
$367$	12.3118	0.642671	0.321335	−	0.946965i	$-0.395868\pi$
$367$	12.3118	0.642671	0.321335	+	0.946965i	$0.395868\pi$
$368$	3.16572	0.165024
$369$	0	0
$370$	17.2722	0.897941
$371$	−21.4495	−1.11360
$372$	0	0
$373$	−33.7638	−1.74822	−0.874111	−	0.485726i	$-0.838555\pi$
$373$	−33.7638	−1.74822	−0.874111	+	0.485726i	$0.838555\pi$
$374$	−13.2933	−0.687381
$375$	0	0
$376$	1.20308	0.0620442
$377$	27.9226	1.43809
$378$	0	0
$379$	−5.64769	−0.290102	−0.145051	−	0.989424i	$-0.546335\pi$
$379$	−5.64769	−0.290102	−0.145051	+	0.989424i	$0.546335\pi$
$380$	−3.47724	−0.178378
$381$	0	0
$382$	−23.7109	−1.21316
$383$	6.20842	0.317235	0.158618	−	0.987340i	$-0.449296\pi$
$383$	6.20842	0.317235	0.158618	+	0.987340i	$0.449296\pi$
$384$	0	0
$385$	−10.2318	−0.521459
$386$	18.5031	0.941781
$387$	0	0
$388$	−22.7415	−1.15452
$389$	21.6803	1.09923	0.549617	−	0.835417i	$-0.314774\pi$
$389$	21.6803	1.09923	0.549617	+	0.835417i	$0.314774\pi$
$390$	0	0
$391$	−2.33442	−0.118057
$392$	−2.53555	−0.128065
$393$	0	0
$394$	12.6602	0.637812
$395$	−7.86165	−0.395562
$396$	0	0
$397$	−6.56073	−0.329273	−0.164637	−	0.986354i	$-0.552645\pi$
$397$	−6.56073	−0.329273	−0.164637	+	0.986354i	$0.552645\pi$
$398$	−0.252161	−0.0126397
$399$	0	0
$400$	−2.46668	−0.123334
$401$	−30.8173	−1.53894	−0.769471	−	0.638681i	$-0.779480\pi$
$401$	−30.8173	−1.53894	−0.769471	+	0.638681i	$0.779480\pi$
$402$	0	0
$403$	1.41214	0.0703435
$404$	−34.7612	−1.72944
$405$	0	0
$406$	48.2964	2.39691
$407$	27.4912	1.36269
$408$	0	0
$409$	−17.3258	−0.856706	−0.428353	−	0.903611i	$-0.640906\pi$
$409$	−17.3258	−0.856706	−0.428353	+	0.903611i	$0.640906\pi$
$410$	−17.1958	−0.849238
$411$	0	0
$412$	35.4741	1.74768
$413$	−21.0134	−1.03400
$414$	0	0
$415$	8.71183	0.427647
$416$	−29.0682	−1.42519
$417$	0	0
$418$	−9.80556	−0.479606
$419$	35.2269	1.72095	0.860473	−	0.509496i	$-0.170168\pi$
$419$	35.2269	1.72095	0.860473	+	0.509496i	$0.170168\pi$
$420$	0	0
$421$	−15.3694	−0.749058	−0.374529	−	0.927215i	$-0.622196\pi$
$421$	−15.3694	−0.749058	−0.374529	+	0.927215i	$0.622196\pi$
$422$	11.3472	0.552372
$423$	0	0
$424$	9.06438	0.440205
$425$	1.81895	0.0882319
$426$	0	0
$427$	−42.4418	−2.05390
$428$	6.87353	0.332244
$429$	0	0
$430$	−16.6117	−0.801088
$431$	40.3926	1.94564	0.972822	−	0.231555i	$-0.0743813\pi$
$431$	40.3926	1.94564	0.972822	+	0.231555i	$0.0743813\pi$
$432$	0	0
$433$	−34.0099	−1.63441	−0.817206	−	0.576345i	$-0.804478\pi$
$433$	−34.0099	−1.63441	−0.817206	+	0.576345i	$0.804478\pi$
$434$	2.44250	0.117244
$435$	0	0
$436$	12.4269	0.595142
$437$	−1.72194	−0.0823717
$438$	0	0
$439$	−3.89256	−0.185782	−0.0928909	−	0.995676i	$-0.529611\pi$
$439$	−3.89256	−0.185782	−0.0928909	+	0.995676i	$0.529611\pi$
$440$	4.32386	0.206132
$441$	0	0
$442$	14.4861	0.689032
$443$	−0.0369222	−0.00175423	−0.000877113	−	1.00000i	$-0.500279\pi$
$443$	−0.0369222	−0.00175423	−0.000877113		1.00000i	$0.500279\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−43.7121	−2.06983
$447$	0	0
$448$	−35.4779	−1.67617
$449$	32.9784	1.55635	0.778174	−	0.628049i	$-0.216146\pi$
$449$	32.9784	1.55635	0.778174	+	0.628049i	$0.216146\pi$
$450$	0	0
$451$	−27.3695	−1.28878
$452$	41.9008	1.97085
$453$	0	0
$454$	53.2630	2.49976
$455$	11.1498	0.522712
$456$	0	0
$457$	−33.5473	−1.56928	−0.784638	−	0.619954i	$-0.787151\pi$
$457$	−33.5473	−1.56928	−0.784638	+	0.619954i	$0.787151\pi$
$458$	46.3787	2.16714
$459$	0	0
$460$	3.32610	0.155080
$461$	7.04342	0.328045	0.164022	−	0.986457i	$-0.447553\pi$
$461$	7.04342	0.328045	0.164022	+	0.986457i	$0.447553\pi$
$462$	0	0
$463$	12.2901	0.571170	0.285585	−	0.958353i	$-0.407812\pi$
$463$	12.2901	0.571170	0.285585	+	0.958353i	$0.407812\pi$
$464$	18.5320	0.860325
$465$	0	0
$466$	−46.1185	−2.13640
$467$	−19.1233	−0.884922	−0.442461	−	0.896788i	$-0.645894\pi$
$467$	−19.1233	−0.884922	−0.442461	+	0.896788i	$0.645894\pi$
$468$	0	0
$469$	−8.71701	−0.402514
$470$	−2.03346	−0.0937966
$471$	0	0
$472$	8.88009	0.408739
$473$	−26.4399	−1.21571
$474$	0	0
$475$	1.34171	0.0615619
$476$	14.1422	0.648206
$477$	0	0
$478$	−43.4580	−1.98772
$479$	−32.1540	−1.46915	−0.734577	−	0.678525i	$-0.762619\pi$
$479$	−32.1540	−1.46915	−0.734577	+	0.678525i	$0.762619\pi$
$480$	0	0
$481$	−29.9579	−1.36596
$482$	1.30885	0.0596163
$483$	0	0
$484$	1.63821	0.0744642
$485$	8.77493	0.398449
$486$	0	0
$487$	−8.52468	−0.386290	−0.193145	−	0.981170i	$-0.561869\pi$
$487$	−8.52468	−0.386290	−0.193145	+	0.981170i	$0.561869\pi$
$488$	17.9356	0.811906
$489$	0	0
$490$	4.28562	0.193605
$491$	−18.1712	−0.820056	−0.410028	−	0.912073i	$-0.634481\pi$
$491$	−18.1712	−0.820056	−0.410028	+	0.912073i	$0.634481\pi$
$492$	0	0
$493$	−13.6656	−0.615468
$494$	10.6854	0.480758
$495$	0	0
$496$	0.937221	0.0420825
$497$	21.7498	0.975611
$498$	0	0
$499$	−20.9408	−0.937437	−0.468719	−	0.883347i	$-0.655284\pi$
$499$	−20.9408	−0.937437	−0.468719	+	0.883347i	$0.655284\pi$
$500$	−2.59164	−0.115902
$501$	0	0
$502$	37.8766	1.69052
$503$	32.8148	1.46314	0.731569	−	0.681767i	$-0.238788\pi$
$503$	32.8148	1.46314	0.731569	+	0.681767i	$0.238788\pi$
$504$	0	0
$505$	13.4128	0.596863
$506$	9.37936	0.416963
$507$	0	0
$508$	32.2876	1.43253
$509$	0.620087	0.0274849	0.0137424	−	0.999906i	$-0.495626\pi$
$509$	0.620087	0.0274849	0.0137424	+	0.999906i	$0.495626\pi$
$510$	0	0
$511$	−28.8352	−1.27559
$512$	−25.5467	−1.12902
$513$	0	0
$514$	53.3927	2.35505
$515$	−13.6879	−0.603160
$516$	0	0
$517$	−3.23654	−0.142343
$518$	−51.8167	−2.27670
$519$	0	0
$520$	−4.71183	−0.206627
$521$	−28.5926	−1.25266	−0.626331	−	0.779557i	$-0.715444\pi$
$521$	−28.5926	−1.25266	−0.626331	+	0.779557i	$0.715444\pi$
$522$	0	0
$523$	9.78325	0.427792	0.213896	−	0.976856i	$-0.431385\pi$
$523$	9.78325	0.427792	0.213896	+	0.976856i	$0.431385\pi$
$524$	10.7482	0.469539
$525$	0	0
$526$	−52.8846	−2.30588
$527$	−0.691114	−0.0301054
$528$	0	0
$529$	−21.3529	−0.928387
$530$	−15.3207	−0.665490
$531$	0	0
$532$	10.4317	0.452272
$533$	29.8252	1.29187
$534$	0	0
$535$	−2.65219	−0.114664
$536$	3.68374	0.159113
$537$	0	0
$538$	−15.1903	−0.654900
$539$	6.82118	0.293809
$540$	0	0
$541$	−29.5732	−1.27145	−0.635725	−	0.771916i	$-0.719299\pi$
$541$	−29.5732	−1.27145	−0.635725	+	0.771916i	$0.719299\pi$
$542$	15.8460	0.680645
$543$	0	0
$544$	14.2263	0.609948
$545$	−4.79500	−0.205395
$546$	0	0
$547$	15.6127	0.667550	0.333775	−	0.942653i	$-0.391678\pi$
$547$	15.6127	0.667550	0.333775	+	0.942653i	$0.391678\pi$
$548$	11.9774	0.511650
$549$	0	0
$550$	−7.30825	−0.311625
$551$	−10.0802	−0.429430
$552$	0	0
$553$	23.5849	1.00293
$554$	−16.8903	−0.717601
$555$	0	0
$556$	13.5531	0.574779
$557$	30.6649	1.29932	0.649658	−	0.760227i	$-0.274912\pi$
$557$	30.6649	1.29932	0.649658	+	0.760227i	$0.274912\pi$
$558$	0	0
$559$	28.8123	1.21863
$560$	7.40003	0.312708
$561$	0	0
$562$	10.8342	0.457015
$563$	−30.8387	−1.29970	−0.649848	−	0.760064i	$-0.725168\pi$
$563$	−30.8387	−1.29970	−0.649848	+	0.760064i	$0.725168\pi$
$564$	0	0
$565$	−16.1677	−0.680179
$566$	17.4628	0.734018
$567$	0	0
$568$	−9.19129	−0.385658
$569$	−27.4775	−1.15192	−0.575959	−	0.817479i	$-0.695371\pi$
$569$	−27.4775	−1.15192	−0.575959	+	0.817479i	$0.695371\pi$
$570$	0	0
$571$	−27.5885	−1.15454	−0.577271	−	0.816552i	$-0.695882\pi$
$571$	−27.5885	−1.15454	−0.577271	+	0.816552i	$0.695882\pi$
$572$	−32.8512	−1.37358
$573$	0	0
$574$	51.5873	2.15321
$575$	−1.28339	−0.0535212
$576$	0	0
$577$	30.0876	1.25256	0.626282	−	0.779596i	$-0.284576\pi$
$577$	30.0876	1.25256	0.626282	+	0.779596i	$0.284576\pi$
$578$	29.3382	1.22031
$579$	0	0
$580$	19.4708	0.808482
$581$	−26.1355	−1.08428
$582$	0	0
$583$	−24.3851	−1.00993
$584$	12.1855	0.504240
$585$	0	0
$586$	12.6328	0.521856
$587$	4.94124	0.203947	0.101973	−	0.994787i	$-0.467484\pi$
$587$	4.94124	0.203947	0.101973	+	0.994787i	$0.467484\pi$
$588$	0	0
$589$	−0.509787	−0.0210054
$590$	−15.0092	−0.617921
$591$	0	0
$592$	−19.8828	−0.817177
$593$	5.33442	0.219059	0.109529	−	0.993984i	$-0.465066\pi$
$593$	5.33442	0.219059	0.109529	+	0.993984i	$0.465066\pi$
$594$	0	0
$595$	−5.45684	−0.223709
$596$	−47.1635	−1.93189
$597$	0	0
$598$	−10.2209	−0.417965
$599$	−4.64223	−0.189676	−0.0948382	−	0.995493i	$-0.530233\pi$
$599$	−4.64223	−0.189676	−0.0948382	+	0.995493i	$0.530233\pi$
$600$	0	0
$601$	19.8977	0.811642	0.405821	−	0.913952i	$-0.366986\pi$
$601$	19.8977	0.811642	0.405821	+	0.913952i	$0.366986\pi$
$602$	49.8352	2.03113
$603$	0	0
$604$	−42.1718	−1.71595
$605$	−0.632114	−0.0256991
$606$	0	0
$607$	4.41565	0.179226	0.0896128	−	0.995977i	$-0.471437\pi$
$607$	4.41565	0.179226	0.0896128	+	0.995977i	$0.471437\pi$
$608$	10.4938	0.425578
$609$	0	0
$610$	−30.3149	−1.22742
$611$	3.52695	0.142685
$612$	0	0
$613$	27.7283	1.11994	0.559968	−	0.828514i	$-0.310813\pi$
$613$	27.7283	1.11994	0.559968	+	0.828514i	$0.310813\pi$
$614$	−27.9831	−1.12930
$615$	0	0
$616$	−12.9716	−0.522640
$617$	−14.7810	−0.595060	−0.297530	−	0.954712i	$-0.596163\pi$
$617$	−14.7810	−0.595060	−0.297530	+	0.954712i	$0.596163\pi$
$618$	0	0
$619$	17.0465	0.685158	0.342579	−	0.939489i	$-0.388700\pi$
$619$	17.0465	0.685158	0.342579	+	0.939489i	$0.388700\pi$
$620$	0.984702	0.0395466
$621$	0	0
$622$	46.1724	1.85134
$623$	3.00000	0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	−1.44669	−0.0578212
$627$	0	0
$628$	13.4150	0.535319
$629$	14.6617	0.584601
$630$	0	0
$631$	−21.6870	−0.863345	−0.431672	−	0.902030i	$-0.642076\pi$
$631$	−21.6870	−0.863345	−0.431672	+	0.902030i	$0.642076\pi$
$632$	−9.96682	−0.396459
$633$	0	0
$634$	13.9996	0.555995
$635$	−12.4583	−0.494394
$636$	0	0
$637$	−7.43321	−0.294515
$638$	54.9063	2.17376
$639$	0	0
$640$	−9.69844	−0.383365
$641$	−47.7899	−1.88759	−0.943794	−	0.330535i	$-0.892771\pi$
$641$	−47.7899	−1.88759	−0.943794	+	0.330535i	$0.892771\pi$
$642$	0	0
$643$	−32.8176	−1.29420	−0.647100	−	0.762405i	$-0.724018\pi$
$643$	−32.8176	−1.29420	−0.647100	+	0.762405i	$0.724018\pi$
$644$	−9.97829	−0.393200
$645$	0	0
$646$	−5.22954	−0.205753
$647$	16.6440	0.654341	0.327171	−	0.944965i	$-0.393905\pi$
$647$	16.6440	0.654341	0.327171	+	0.944965i	$0.393905\pi$
$648$	0	0
$649$	−23.8893	−0.937738
$650$	7.96399	0.312373
$651$	0	0
$652$	58.7511	2.30087
$653$	11.5783	0.453092	0.226546	−	0.974000i	$-0.427257\pi$
$653$	11.5783	0.453092	0.226546	+	0.974000i	$0.427257\pi$
$654$	0	0
$655$	−4.14727	−0.162047
$656$	19.7947	0.772854
$657$	0	0
$658$	6.10039	0.237818
$659$	−8.76974	−0.341621	−0.170810	−	0.985304i	$-0.554639\pi$
$659$	−8.76974	−0.341621	−0.170810	+	0.985304i	$0.554639\pi$
$660$	0	0
$661$	−11.1830	−0.434968	−0.217484	−	0.976064i	$-0.569785\pi$
$661$	−11.1830	−0.434968	−0.217484	+	0.976064i	$0.569785\pi$
$662$	−43.5952	−1.69438
$663$	0	0
$664$	11.0447	0.428616
$665$	−4.02513	−0.156088
$666$	0	0
$667$	9.64204	0.373341
$668$	−28.8030	−1.11442
$669$	0	0
$670$	−6.22630	−0.240543
$671$	−48.2505	−1.86269
$672$	0	0
$673$	−40.6477	−1.56685	−0.783427	−	0.621483i	$-0.786530\pi$
$673$	−40.6477	−1.56685	−0.783427	+	0.621483i	$0.786530\pi$
$674$	−47.8565	−1.84336
$675$	0	0
$676$	2.10744	0.0810553
$677$	−10.0075	−0.384618	−0.192309	−	0.981334i	$-0.561598\pi$
$677$	−10.0075	−0.384618	−0.192309	+	0.981334i	$0.561598\pi$
$678$	0	0
$679$	−26.3248	−1.01025
$680$	2.30602	0.0884318
$681$	0	0
$682$	2.77679	0.106329
$683$	−29.9604	−1.14640	−0.573201	−	0.819415i	$-0.694299\pi$
$683$	−29.9604	−1.14640	−0.573201	+	0.819415i	$0.694299\pi$
$684$	0	0
$685$	−4.62155	−0.176581
$686$	32.1422	1.22719
$687$	0	0
$688$	19.1224	0.729035
$689$	26.5731	1.01235
$690$	0	0
$691$	−4.32773	−0.164635	−0.0823174	−	0.996606i	$-0.526232\pi$
$691$	−4.32773	−0.164635	−0.0823174	+	0.996606i	$0.526232\pi$
$692$	8.28307	0.314875
$693$	0	0
$694$	−23.8810	−0.906512
$695$	−5.22954	−0.198368
$696$	0	0
$697$	−14.5968	−0.552893
$698$	−8.62510	−0.326465
$699$	0	0
$700$	7.77493	0.293865
$701$	−11.7133	−0.442407	−0.221203	−	0.975228i	$-0.570998\pi$
$701$	−11.7133	−0.442407	−0.221203	+	0.975228i	$0.570998\pi$
$702$	0	0
$703$	10.8149	0.407893
$704$	−40.3335	−1.52012
$705$	0	0
$706$	56.1600	2.11361
$707$	−40.2385	−1.51332
$708$	0	0
$709$	13.1744	0.494773	0.247387	−	0.968917i	$-0.420428\pi$
$709$	13.1744	0.494773	0.247387	+	0.968917i	$0.420428\pi$
$710$	15.5352	0.583027
$711$	0	0
$712$	−1.26778	−0.0475119
$713$	0.487629	0.0182619
$714$	0	0
$715$	12.6758	0.474049
$716$	33.0552	1.23533
$717$	0	0
$718$	68.2894	2.54854
$719$	1.07616	0.0401342	0.0200671	−	0.999799i	$-0.493612\pi$
$719$	1.07616	0.0401342	0.0200671	+	0.999799i	$0.493612\pi$
$720$	0	0
$721$	41.0636	1.52929
$722$	36.8560	1.37164
$723$	0	0
$724$	−60.7354	−2.25722
$725$	−7.51293	−0.279023
$726$	0	0
$727$	28.1722	1.04485	0.522425	−	0.852685i	$-0.325028\pi$
$727$	28.1722	1.04485	0.522425	+	0.852685i	$0.325028\pi$
$728$	14.1355	0.523896
$729$	0	0
$730$	−20.5961	−0.762296
$731$	−14.1010	−0.521545
$732$	0	0
$733$	37.4631	1.38373	0.691866	−	0.722026i	$-0.256789\pi$
$733$	37.4631	1.38373	0.691866	+	0.722026i	$0.256789\pi$
$734$	−26.3819	−0.973772
$735$	0	0
$736$	−10.0376	−0.369992
$737$	−9.91005	−0.365041
$738$	0	0
$739$	19.9963	0.735574	0.367787	−	0.929910i	$-0.380116\pi$
$739$	19.9963	0.735574	0.367787	+	0.929910i	$0.380116\pi$
$740$	−20.8901	−0.767934
$741$	0	0
$742$	45.9622	1.68732
$743$	−29.2429	−1.07282	−0.536409	−	0.843958i	$-0.680220\pi$
$743$	−29.2429	−1.07282	−0.536409	+	0.843958i	$0.680220\pi$
$744$	0	0
$745$	18.1983	0.666734
$746$	72.3494	2.64890
$747$	0	0
$748$	16.0777	0.587859
$749$	7.95657	0.290727
$750$	0	0
$751$	38.5253	1.40581	0.702904	−	0.711285i	$-0.251887\pi$
$751$	38.5253	1.40581	0.702904	+	0.711285i	$0.251887\pi$
$752$	2.34080	0.0853602
$753$	0	0
$754$	−59.8329	−2.17898
$755$	16.2722	0.592207
$756$	0	0
$757$	−31.2779	−1.13681	−0.568407	−	0.822747i	$-0.692440\pi$
$757$	−31.2779	−1.13681	−0.568407	+	0.822747i	$0.692440\pi$
$758$	12.1019	0.439562
$759$	0	0
$760$	1.70099	0.0617014
$761$	18.7789	0.680736	0.340368	−	0.940292i	$-0.389448\pi$
$761$	18.7789	0.680736	0.340368	+	0.940292i	$0.389448\pi$
$762$	0	0
$763$	14.3850	0.520772
$764$	28.6774	1.03751
$765$	0	0
$766$	−13.3035	−0.480674
$767$	26.0328	0.939991
$768$	0	0
$769$	−50.5140	−1.82158	−0.910790	−	0.412870i	$-0.864526\pi$
$769$	−50.5140	−1.82158	−0.910790	+	0.412870i	$0.864526\pi$
$770$	21.9247	0.790113
$771$	0	0
$772$	−22.3787	−0.805426
$773$	29.2726	1.05286	0.526432	−	0.850217i	$-0.323530\pi$
$773$	29.2726	1.05286	0.526432	+	0.850217i	$0.323530\pi$
$774$	0	0
$775$	−0.379953	−0.0136483
$776$	11.1246	0.399352
$777$	0	0
$778$	−46.4567	−1.66556
$779$	−10.7670	−0.385769
$780$	0	0
$781$	24.7265	0.884785
$782$	5.00223	0.178879
$783$	0	0
$784$	−4.93335	−0.176191
$785$	−5.17627	−0.184749
$786$	0	0
$787$	42.8540	1.52758	0.763791	−	0.645464i	$-0.223336\pi$
$787$	42.8540	1.52758	0.763791	+	0.645464i	$0.223336\pi$
$788$	−15.3120	−0.545467
$789$	0	0
$790$	16.8460	0.599355
$791$	48.5030	1.72457
$792$	0	0
$793$	52.5798	1.86717
$794$	14.0584	0.498914
$795$	0	0
$796$	0.304978	0.0108096
$797$	1.86102	0.0659206	0.0329603	−	0.999457i	$-0.489507\pi$
$797$	1.86102	0.0659206	0.0329603	+	0.999457i	$0.489507\pi$
$798$	0	0
$799$	−1.72612	−0.0610659
$800$	7.82118	0.276520
$801$	0	0
$802$	66.0357	2.33180
$803$	−32.7816	−1.15684
$804$	0	0
$805$	3.85018	0.135701
$806$	−3.02594	−0.106584
$807$	0	0
$808$	17.0045	0.598215
$809$	17.7332	0.623467	0.311733	−	0.950170i	$-0.399090\pi$
$809$	17.7332	0.623467	0.311733	+	0.950170i	$0.399090\pi$
$810$	0	0
$811$	27.5106	0.966027	0.483013	−	0.875613i	$-0.339542\pi$
$811$	27.5106	0.966027	0.483013	+	0.875613i	$0.339542\pi$
$812$	−58.4125	−2.04988
$813$	0	0
$814$	−58.9085	−2.06474
$815$	−22.6694	−0.794076
$816$	0	0
$817$	−10.4013	−0.363897
$818$	37.1260	1.29808
$819$	0	0
$820$	20.7976	0.726282
$821$	22.5702	0.787706	0.393853	−	0.919174i	$-0.371142\pi$
$821$	22.5702	0.787706	0.393853	+	0.919174i	$0.371142\pi$
$822$	0	0
$823$	13.8597	0.483119	0.241559	−	0.970386i	$-0.422341\pi$
$823$	13.8597	0.483119	0.241559	+	0.970386i	$0.422341\pi$
$824$	−17.3532	−0.604527
$825$	0	0
$826$	45.0277	1.56671
$827$	−37.9573	−1.31990	−0.659952	−	0.751308i	$-0.729423\pi$
$827$	−37.9573	−1.31990	−0.659952	+	0.751308i	$0.729423\pi$
$828$	0	0
$829$	49.1040	1.70545	0.852726	−	0.522359i	$-0.174948\pi$
$829$	49.1040	1.70545	0.852726	+	0.522359i	$0.174948\pi$
$830$	−18.6678	−0.647969
$831$	0	0
$832$	43.9524	1.52378
$833$	3.63789	0.126046
$834$	0	0
$835$	11.1138	0.384609
$836$	11.8594	0.410167
$837$	0	0
$838$	−75.4846	−2.60757
$839$	25.9261	0.895067	0.447534	−	0.894267i	$-0.352303\pi$
$839$	25.9261	0.895067	0.447534	+	0.894267i	$0.352303\pi$
$840$	0	0
$841$	27.4441	0.946348
$842$	32.9337	1.13497
$843$	0	0
$844$	−13.7239	−0.472397
$845$	−0.813167	−0.0279738
$846$	0	0
$847$	1.89634	0.0651590
$848$	17.6363	0.605633
$849$	0	0
$850$	−3.89766	−0.133689
$851$	−10.3449	−0.354617
$852$	0	0
$853$	−36.7931	−1.25977	−0.629886	−	0.776688i	$-0.716898\pi$
$853$	−36.7931	−1.25977	−0.629886	+	0.776688i	$0.716898\pi$
$854$	90.9448	3.11207
$855$	0	0
$856$	−3.36239	−0.114924
$857$	32.2161	1.10048	0.550241	−	0.835006i	$-0.314536\pi$
$857$	32.2161	1.10048	0.550241	+	0.835006i	$0.314536\pi$
$858$	0	0
$859$	22.2008	0.757482	0.378741	−	0.925503i	$-0.376357\pi$
$859$	22.2008	0.757482	0.378741	+	0.925503i	$0.376357\pi$
$860$	20.0912	0.685104
$861$	0	0
$862$	−86.5538	−2.94803
$863$	2.64685	0.0901000	0.0450500	−	0.998985i	$-0.485655\pi$
$863$	2.64685	0.0901000	0.0450500	+	0.998985i	$0.485655\pi$
$864$	0	0
$865$	−3.19607	−0.108670
$866$	72.8769	2.47646
$867$	0	0
$868$	−2.95411	−0.100269
$869$	26.8128	0.909564
$870$	0	0
$871$	10.7992	0.365918
$872$	−6.07899	−0.205861
$873$	0	0
$874$	3.68980	0.124809
$875$	−3.00000	−0.101419
$876$	0	0
$877$	36.6449	1.23741	0.618705	−	0.785623i	$-0.287658\pi$
$877$	36.6449	1.23741	0.618705	+	0.785623i	$0.287658\pi$
$878$	8.34103	0.281496
$879$	0	0
$880$	8.41282	0.283596
$881$	22.0585	0.743170	0.371585	−	0.928399i	$-0.378814\pi$
$881$	22.0585	0.743170	0.371585	+	0.928399i	$0.378814\pi$
$882$	0	0
$883$	−53.0781	−1.78622	−0.893110	−	0.449839i	$-0.851481\pi$
$883$	−53.0781	−1.78622	−0.893110	+	0.449839i	$0.851481\pi$
$884$	−17.5203	−0.589272
$885$	0	0
$886$	0.0791173	0.00265800
$887$	1.50341	0.0504796	0.0252398	−	0.999681i	$-0.491965\pi$
$887$	1.50341	0.0504796	0.0252398	+	0.999681i	$0.491965\pi$
$888$	0	0
$889$	37.3750	1.25352
$890$	2.14281	0.0718272
$891$	0	0
$892$	52.8680	1.77015
$893$	−1.27324	−0.0426075
$894$	0	0
$895$	−12.7545	−0.426337
$896$	29.0953	0.972007
$897$	0	0
$898$	−70.6665	−2.35817
$899$	2.85456	0.0952049
$900$	0	0
$901$	−13.0051	−0.433264
$902$	58.6476	1.95275
$903$	0	0
$904$	−20.4970	−0.681720
$905$	23.4351	0.779010
$906$	0	0
$907$	36.1250	1.19951	0.599755	−	0.800184i	$-0.295265\pi$
$907$	36.1250	1.19951	0.599755	+	0.800184i	$0.295265\pi$
$908$	−64.4194	−2.13783
$909$	0	0
$910$	−23.8920	−0.792011
$911$	53.2103	1.76294	0.881468	−	0.472243i	$-0.156556\pi$
$911$	53.2103	1.76294	0.881468	+	0.472243i	$0.156556\pi$
$912$	0	0
$913$	−29.7125	−0.983339
$914$	71.8855	2.37776
$915$	0	0
$916$	−56.0931	−1.85337
$917$	12.4418	0.410865
$918$	0	0
$919$	21.9294	0.723386	0.361693	−	0.932297i	$-0.382199\pi$
$919$	21.9294	0.723386	0.361693	+	0.932297i	$0.382199\pi$
$920$	−1.62706	−0.0536424
$921$	0	0
$922$	−15.0927	−0.497053
$923$	−26.9451	−0.886910
$924$	0	0
$925$	8.06055	0.265029
$926$	−26.3354	−0.865436
$927$	0	0
$928$	−58.7599	−1.92889
$929$	27.7383	0.910065	0.455032	−	0.890475i	$-0.349628\pi$
$929$	27.7383	0.910065	0.455032	+	0.890475i	$0.349628\pi$
$930$	0	0
$931$	2.68342	0.0879456
$932$	55.7785	1.82708
$933$	0	0
$934$	40.9777	1.34083
$935$	−6.20368	−0.202882
$936$	0	0
$937$	39.5141	1.29087	0.645435	−	0.763815i	$-0.276676\pi$
$937$	39.5141	1.29087	0.645435	+	0.763815i	$0.276676\pi$
$938$	18.6789	0.609888
$939$	0	0
$940$	2.45939	0.0802164
$941$	29.5186	0.962279	0.481140	−	0.876644i	$-0.340223\pi$
$941$	29.5186	0.962279	0.481140	+	0.876644i	$0.340223\pi$
$942$	0	0
$943$	10.2990	0.335383
$944$	17.2777	0.562342
$945$	0	0
$946$	56.6557	1.84204
$947$	41.1862	1.33837	0.669186	−	0.743095i	$-0.266643\pi$
$947$	41.1862	1.33837	0.669186	+	0.743095i	$0.266643\pi$
$948$	0	0
$949$	35.7230	1.15962
$950$	−2.87503	−0.0932784
$951$	0	0
$952$	−6.91806	−0.224215
$953$	3.09612	0.100293	0.0501465	−	0.998742i	$-0.484031\pi$
$953$	3.09612	0.100293	0.0501465	+	0.998742i	$0.484031\pi$
$954$	0	0
$955$	−11.0653	−0.358066
$956$	52.5607	1.69993
$957$	0	0
$958$	68.9000	2.22606
$959$	13.8647	0.447713
$960$	0	0
$961$	−30.8556	−0.995343
$962$	64.1941	2.06970
$963$	0	0
$964$	−1.58299	−0.0509848
$965$	8.63494	0.277969
$966$	0	0
$967$	−38.9986	−1.25411	−0.627056	−	0.778974i	$-0.715740\pi$
$967$	−38.9986	−1.25411	−0.627056	+	0.778974i	$0.715740\pi$
$968$	−0.801379	−0.0257573
$969$	0	0
$970$	−18.8030	−0.603728
$971$	12.9040	0.414110	0.207055	−	0.978329i	$-0.433612\pi$
$971$	12.9040	0.414110	0.207055	+	0.978329i	$0.433612\pi$
$972$	0	0
$973$	15.6886	0.502954
$974$	18.2668	0.585305
$975$	0	0
$976$	34.8967	1.11702
$977$	−18.9374	−0.605860	−0.302930	−	0.953013i	$-0.597965\pi$
$977$	−18.9374	−0.605860	−0.302930	+	0.953013i	$0.597965\pi$
$978$	0	0
$979$	3.41059	0.109003
$980$	−5.18328	−0.165574
$981$	0	0
$982$	38.9375	1.24255
$983$	−9.16352	−0.292271	−0.146135	−	0.989265i	$-0.546684\pi$
$983$	−9.16352	−0.292271	−0.146135	+	0.989265i	$0.546684\pi$
$984$	0	0
$985$	5.90822	0.188252
$986$	29.2828	0.932556
$987$	0	0
$988$	−12.9235	−0.411152
$989$	9.94925	0.316368
$990$	0	0
$991$	5.00952	0.159133	0.0795663	−	0.996830i	$-0.474646\pi$
$991$	5.00952	0.159133	0.0795663	+	0.996830i	$0.474646\pi$
$992$	−2.97168	−0.0943509
$993$	0	0
$994$	−46.6057	−1.47824
$995$	−0.117677	−0.00373062
$996$	0	0
$997$	38.7118	1.22602	0.613008	−	0.790077i	$-0.289959\pi$
$997$	38.7118	1.22602	0.613008	+	0.790077i	$0.289959\pi$
$998$	44.8721	1.42040
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.k.1.1		4
3.2	odd	2		445.2.a.e.1.4	✓	4
12.11	even	2		7120.2.a.z.1.3		4
15.14	odd	2		2225.2.a.h.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.e.1.4	✓	4	3.2	odd	2
2225.2.a.h.1.1		4	15.14	odd	2
4005.2.a.k.1.1		4	1.1	even	1	trivial
7120.2.a.z.1.3		4	12.11	even	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14281 q^{2} +2.59164 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.26778 q^{8} +O(q^{10})\)	\(q-2.14281 q^{2} +2.59164 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.26778 q^{8} +2.14281 q^{10} +3.41059 q^{11} -3.71661 q^{13} -6.42843 q^{14} -2.46668 q^{16} +1.81895 q^{17} +1.34171 q^{19} -2.59164 q^{20} -7.30825 q^{22} -1.28339 q^{23} +1.00000 q^{25} +7.96399 q^{26} +7.77493 q^{28} -7.51293 q^{29} -0.379953 q^{31} +7.82118 q^{32} -3.89766 q^{34} -3.00000 q^{35} +8.06055 q^{37} -2.87503 q^{38} +1.26778 q^{40} -8.02486 q^{41} -7.75230 q^{43} +8.83902 q^{44} +2.75007 q^{46} -0.948969 q^{47} +2.00000 q^{49} -2.14281 q^{50} -9.63211 q^{52} -7.14982 q^{53} -3.41059 q^{55} -3.80333 q^{56} +16.0988 q^{58} -7.00446 q^{59} -14.1473 q^{61} +0.814168 q^{62} -11.8260 q^{64} +3.71661 q^{65} -2.90567 q^{67} +4.71406 q^{68} +6.42843 q^{70} +7.24993 q^{71} -9.61172 q^{73} -17.2722 q^{74} +3.47724 q^{76} +10.2318 q^{77} +7.86165 q^{79} +2.46668 q^{80} +17.1958 q^{82} -8.71183 q^{83} -1.81895 q^{85} +16.6117 q^{86} -4.32386 q^{88} +1.00000 q^{89} -11.1498 q^{91} -3.32610 q^{92} +2.03346 q^{94} -1.34171 q^{95} -8.77493 q^{97} -4.28562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} - 7 q^{13} - 3 q^{14} - 3 q^{16} - q^{17} - q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 4 q^{25} - 7 q^{26} + 9 q^{28} - 14 q^{29} - 11 q^{31} - 16 q^{34} - 12 q^{35} - 5 q^{37} - 12 q^{38} - 3 q^{40} - 9 q^{41} - 9 q^{43} - 3 q^{44} + 12 q^{46} - 6 q^{47} + 8 q^{49} - q^{50} - 24 q^{52} - 5 q^{53} + 2 q^{55} + 9 q^{56} + 43 q^{58} + 18 q^{59} - 3 q^{61} - 12 q^{62} - 23 q^{64} + 7 q^{65} + 13 q^{67} - 19 q^{68} + 3 q^{70} + 28 q^{71} - q^{73} - 15 q^{74} + 12 q^{76} - 6 q^{77} - 7 q^{79} + 3 q^{80} - 17 q^{82} - 20 q^{83} + q^{85} + 29 q^{86} - 18 q^{88} + 4 q^{89} - 21 q^{91} + 9 q^{92} + 17 q^{94} + q^{95} - 13 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8 + q^10 - 2 * q^11 - 7 * q^13 - 3 * q^14 - 3 * q^16 - q^17 - q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 4 * q^25 - 7 * q^26 + 9 * q^28 - 14 * q^29 - 11 * q^31 - 16 * q^34 - 12 * q^35 - 5 * q^37 - 12 * q^38 - 3 * q^40 - 9 * q^41 - 9 * q^43 - 3 * q^44 + 12 * q^46 - 6 * q^47 + 8 * q^49 - q^50 - 24 * q^52 - 5 * q^53 + 2 * q^55 + 9 * q^56 + 43 * q^58 + 18 * q^59 - 3 * q^61 - 12 * q^62 - 23 * q^64 + 7 * q^65 + 13 * q^67 - 19 * q^68 + 3 * q^70 + 28 * q^71 - q^73 - 15 * q^74 + 12 * q^76 - 6 * q^77 - 7 * q^79 + 3 * q^80 - 17 * q^82 - 20 * q^83 + q^85 + 29 * q^86 - 18 * q^88 + 4 * q^89 - 21 * q^91 + 9 * q^92 + 17 * q^94 + q^95 - 13 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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