LMFDB - Embedded newform 4005.2.a.j.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q+0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -0.198062 q^{7} -2.04892 q^{8} +O(q^{10})$	\(q+0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -0.198062 q^{7} -2.04892 q^{8} -0.554958 q^{10} +3.60388 q^{11} -3.24698 q^{13} -0.109916 q^{14} +2.24698 q^{16} -1.15883 q^{17} -1.50604 q^{19} +1.69202 q^{20} +2.00000 q^{22} +4.49396 q^{23} +1.00000 q^{25} -1.80194 q^{26} +0.335126 q^{28} +5.33513 q^{29} +1.38404 q^{31} +5.34481 q^{32} -0.643104 q^{34} +0.198062 q^{35} -1.15883 q^{37} -0.835790 q^{38} +2.04892 q^{40} +0.814019 q^{41} -3.55496 q^{43} -6.09783 q^{44} +2.49396 q^{46} +1.18598 q^{47} -6.96077 q^{49} +0.554958 q^{50} +5.49396 q^{52} +1.52781 q^{53} -3.60388 q^{55} +0.405813 q^{56} +2.96077 q^{58} -7.78448 q^{59} -2.39612 q^{61} +0.768086 q^{62} -1.52781 q^{64} +3.24698 q^{65} +4.71379 q^{67} +1.96077 q^{68} +0.109916 q^{70} -11.9215 q^{71} -14.5918 q^{73} -0.643104 q^{74} +2.54825 q^{76} -0.713792 q^{77} -16.6679 q^{79} -2.24698 q^{80} +0.451747 q^{82} +13.4034 q^{83} +1.15883 q^{85} -1.97285 q^{86} -7.38404 q^{88} +1.00000 q^{89} +0.643104 q^{91} -7.60388 q^{92} +0.658170 q^{94} +1.50604 q^{95} -10.4155 q^{97} -3.86294 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8} - 2 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} + 2 q^{16} + 5 q^{17} - 14 q^{19} + 6 q^{22} + 4 q^{23} + 3 q^{25} - q^{26} + 15 q^{29} - 6 q^{31} - 7 q^{32} - 6 q^{34} + 5 q^{35} + 5 q^{37} - 14 q^{38} - 3 q^{40} + 17 q^{41} - 11 q^{43} - 2 q^{46} - 11 q^{47} - 8 q^{49} + 2 q^{50} + 7 q^{52} + 11 q^{53} - 2 q^{55} - 12 q^{56} - 4 q^{58} - 3 q^{59} - 16 q^{61} - 18 q^{62} - 11 q^{64} + 5 q^{65} + 6 q^{67} - 7 q^{68} + q^{70} - 10 q^{71} - 16 q^{73} - 6 q^{74} - 14 q^{76} + 6 q^{77} - 7 q^{79} - 2 q^{80} + 23 q^{82} - 14 q^{83} - 5 q^{85} - 12 q^{86} - 12 q^{88} + 3 q^{89} + 6 q^{91} - 14 q^{92} - 19 q^{94} + 14 q^{95} + 4 q^{97} - 17 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8 - 2 * q^10 + 2 * q^11 - 5 * q^13 - q^14 + 2 * q^16 + 5 * q^17 - 14 * q^19 + 6 * q^22 + 4 * q^23 + 3 * q^25 - q^26 + 15 * q^29 - 6 * q^31 - 7 * q^32 - 6 * q^34 + 5 * q^35 + 5 * q^37 - 14 * q^38 - 3 * q^40 + 17 * q^41 - 11 * q^43 - 2 * q^46 - 11 * q^47 - 8 * q^49 + 2 * q^50 + 7 * q^52 + 11 * q^53 - 2 * q^55 - 12 * q^56 - 4 * q^58 - 3 * q^59 - 16 * q^61 - 18 * q^62 - 11 * q^64 + 5 * q^65 + 6 * q^67 - 7 * q^68 + q^70 - 10 * q^71 - 16 * q^73 - 6 * q^74 - 14 * q^76 + 6 * q^77 - 7 * q^79 - 2 * q^80 + 23 * q^82 - 14 * q^83 - 5 * q^85 - 12 * q^86 - 12 * q^88 + 3 * q^89 + 6 * q^91 - 14 * q^92 - 19 * q^94 + 14 * q^95 + 4 * q^97 - 17 * 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 $

$2$	0.554958	0.392415	0.196207	−	0.980562i	$-0.437137\pi$
$2$	0.554958	0.392415	0.196207	+	0.980562i	$0.437137\pi$
$3$	0	0
$3$	0	0
$4$	−1.69202	−0.846011
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.198062	−0.0748605	−0.0374302	−	0.999299i	$-0.511917\pi$
$7$	−0.198062	−0.0748605	−0.0374302	+	0.999299i	$0.511917\pi$
$8$	−2.04892	−0.724402
$9$	0	0
$10$	−0.554958	−0.175493
$11$	3.60388	1.08661	0.543305	−	0.839536i	$-0.317173\pi$
$11$	3.60388	1.08661	0.543305	+	0.839536i	$0.317173\pi$
$12$	0	0
$13$	−3.24698	−0.900550	−0.450275	−	0.892890i	$-0.648674\pi$
$13$	−3.24698	−0.900550	−0.450275	+	0.892890i	$0.648674\pi$
$14$	−0.109916	−0.0293764
$15$	0	0
$16$	2.24698	0.561745
$17$	−1.15883	−0.281058	−0.140529	−	0.990077i	$-0.544880\pi$
$17$	−1.15883	−0.281058	−0.140529	+	0.990077i	$0.544880\pi$
$18$	0	0
$19$	−1.50604	−0.345509	−0.172755	−	0.984965i	$-0.555267\pi$
$19$	−1.50604	−0.345509	−0.172755	+	0.984965i	$0.555267\pi$
$20$	1.69202	0.378348
$21$	0	0
$22$	2.00000	0.426401
$23$	4.49396	0.937055	0.468528	−	0.883449i	$-0.344785\pi$
$23$	4.49396	0.937055	0.468528	+	0.883449i	$0.344785\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.80194	−0.353389
$27$	0	0
$28$	0.335126	0.0633328
$29$	5.33513	0.990708	0.495354	−	0.868691i	$-0.335038\pi$
$29$	5.33513	0.990708	0.495354	+	0.868691i	$0.335038\pi$
$30$	0	0
$31$	1.38404	0.248581	0.124291	−	0.992246i	$-0.460334\pi$
$31$	1.38404	0.248581	0.124291	+	0.992246i	$0.460334\pi$
$32$	5.34481	0.944839
$33$	0	0
$34$	−0.643104	−0.110291
$35$	0.198062	0.0334786
$36$	0	0
$37$	−1.15883	−0.190511	−0.0952555	−	0.995453i	$-0.530367\pi$
$37$	−1.15883	−0.190511	−0.0952555	+	0.995453i	$0.530367\pi$
$38$	−0.835790	−0.135583
$39$	0	0
$40$	2.04892	0.323962
$41$	0.814019	0.127128	0.0635642	−	0.997978i	$-0.479753\pi$
$41$	0.814019	0.127128	0.0635642	+	0.997978i	$0.479753\pi$
$42$	0	0
$43$	−3.55496	−0.542126	−0.271063	−	0.962562i	$-0.587375\pi$
$43$	−3.55496	−0.542126	−0.271063	+	0.962562i	$0.587375\pi$
$44$	−6.09783	−0.919283
$45$	0	0
$46$	2.49396	0.367714
$47$	1.18598	0.172993	0.0864965	−	0.996252i	$-0.472433\pi$
$47$	1.18598	0.172993	0.0864965	+	0.996252i	$0.472433\pi$
$48$	0	0
$49$	−6.96077	−0.994396
$50$	0.554958	0.0784829
$51$	0	0
$52$	5.49396	0.761875
$53$	1.52781	0.209861	0.104930	−	0.994480i	$-0.466538\pi$
$53$	1.52781	0.209861	0.104930	+	0.994480i	$0.466538\pi$
$54$	0	0
$55$	−3.60388	−0.485946
$56$	0.405813	0.0542291
$57$	0	0
$58$	2.96077	0.388768
$59$	−7.78448	−1.01345	−0.506726	−	0.862107i	$-0.669145\pi$
$59$	−7.78448	−1.01345	−0.506726	+	0.862107i	$0.669145\pi$
$60$	0	0
$61$	−2.39612	−0.306792	−0.153396	−	0.988165i	$-0.549021\pi$
$61$	−2.39612	−0.306792	−0.153396	+	0.988165i	$0.549021\pi$
$62$	0.768086	0.0975470
$63$	0	0
$64$	−1.52781	−0.190976
$65$	3.24698	0.402738
$66$	0	0
$67$	4.71379	0.575881	0.287941	−	0.957648i	$-0.407029\pi$
$67$	4.71379	0.575881	0.287941	+	0.957648i	$0.407029\pi$
$68$	1.96077	0.237778
$69$	0	0
$70$	0.109916	0.0131375
$71$	−11.9215	−1.41483	−0.707413	−	0.706800i	$-0.750138\pi$
$71$	−11.9215	−1.41483	−0.707413	+	0.706800i	$0.750138\pi$
$72$	0	0
$73$	−14.5918	−1.70784	−0.853920	−	0.520404i	$-0.825781\pi$
$73$	−14.5918	−1.70784	−0.853920	+	0.520404i	$0.825781\pi$
$74$	−0.643104	−0.0747593
$75$	0	0
$76$	2.54825	0.292305
$77$	−0.713792	−0.0813441
$78$	0	0
$79$	−16.6679	−1.87528	−0.937640	−	0.347607i	$-0.886994\pi$
$79$	−16.6679	−1.87528	−0.937640	+	0.347607i	$0.886994\pi$
$80$	−2.24698	−0.251220
$81$	0	0
$82$	0.451747	0.0498871
$83$	13.4034	1.47122	0.735608	−	0.677407i	$-0.236896\pi$
$83$	13.4034	1.47122	0.735608	+	0.677407i	$0.236896\pi$
$84$	0	0
$85$	1.15883	0.125693
$86$	−1.97285	−0.212738
$87$	0	0
$88$	−7.38404	−0.787142
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	0.643104	0.0674156
$92$	−7.60388	−0.792759
$93$	0	0
$94$	0.658170	0.0678850
$95$	1.50604	0.154517
$96$	0	0
$97$	−10.4155	−1.05753	−0.528767	−	0.848767i	$-0.677345\pi$
$97$	−10.4155	−1.05753	−0.528767	+	0.848767i	$0.677345\pi$
$98$	−3.86294	−0.390216
$99$	0	0
$100$	−1.69202	−0.169202
$101$	−0.807315	−0.0803308	−0.0401654	−	0.999193i	$-0.512788\pi$
$101$	−0.807315	−0.0803308	−0.0401654	+	0.999193i	$0.512788\pi$
$102$	0	0
$103$	8.90515	0.877450	0.438725	−	0.898621i	$-0.355430\pi$
$103$	8.90515	0.877450	0.438725	+	0.898621i	$0.355430\pi$
$104$	6.65279	0.652360
$105$	0	0
$106$	0.847871	0.0823525
$107$	4.54288	0.439176	0.219588	−	0.975593i	$-0.429529\pi$
$107$	4.54288	0.439176	0.219588	+	0.975593i	$0.429529\pi$
$108$	0	0
$109$	−15.6625	−1.50019	−0.750097	−	0.661328i	$-0.769993\pi$
$109$	−15.6625	−1.50019	−0.750097	+	0.661328i	$0.769993\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	−0.445042	−0.0420525
$113$	−7.48188	−0.703836	−0.351918	−	0.936031i	$-0.614470\pi$
$113$	−7.48188	−0.703836	−0.351918	+	0.936031i	$0.614470\pi$
$114$	0	0
$115$	−4.49396	−0.419064
$116$	−9.02715	−0.838150
$117$	0	0
$118$	−4.32006	−0.397694
$119$	0.229521	0.0210402
$120$	0	0
$121$	1.98792	0.180720
$122$	−1.32975	−0.120390
$123$	0	0
$124$	−2.34183	−0.210303
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.6286	1.20935	0.604673	−	0.796474i	$-0.293304\pi$
$127$	13.6286	1.20935	0.604673	+	0.796474i	$0.293304\pi$
$128$	−11.5375	−1.01978
$129$	0	0
$130$	1.80194	0.158040
$131$	0.537500	0.0469616	0.0234808	−	0.999724i	$-0.492525\pi$
$131$	0.537500	0.0469616	0.0234808	+	0.999724i	$0.492525\pi$
$132$	0	0
$133$	0.298290	0.0258650
$134$	2.61596	0.225984
$135$	0	0
$136$	2.37435	0.203599
$137$	15.0422	1.28514	0.642571	−	0.766226i	$-0.277868\pi$
$137$	15.0422	1.28514	0.642571	+	0.766226i	$0.277868\pi$
$138$	0	0
$139$	−12.4112	−1.05270	−0.526352	−	0.850267i	$-0.676440\pi$
$139$	−12.4112	−1.05270	−0.526352	+	0.850267i	$0.676440\pi$
$140$	−0.335126	−0.0283233
$141$	0	0
$142$	−6.61596	−0.555199
$143$	−11.7017	−0.978546
$144$	0	0
$145$	−5.33513	−0.443058
$146$	−8.09783	−0.670182
$147$	0	0
$148$	1.96077	0.161174
$149$	6.43967	0.527558	0.263779	−	0.964583i	$-0.415031\pi$
$149$	6.43967	0.527558	0.263779	+	0.964583i	$0.415031\pi$
$150$	0	0
$151$	2.31767	0.188609	0.0943045	−	0.995543i	$-0.469937\pi$
$151$	2.31767	0.188609	0.0943045	+	0.995543i	$0.469937\pi$
$152$	3.08575	0.250288
$153$	0	0
$154$	−0.396125	−0.0319206
$155$	−1.38404	−0.111169
$156$	0	0
$157$	5.95646	0.475377	0.237689	−	0.971341i	$-0.423610\pi$
$157$	5.95646	0.475377	0.237689	+	0.971341i	$0.423610\pi$
$158$	−9.24996	−0.735888
$159$	0	0
$160$	−5.34481	−0.422545
$161$	−0.890084	−0.0701484
$162$	0	0
$163$	7.64310	0.598654	0.299327	−	0.954151i	$-0.403238\pi$
$163$	7.64310	0.598654	0.299327	+	0.954151i	$0.403238\pi$
$164$	−1.37734	−0.107552
$165$	0	0
$166$	7.43834	0.577327
$167$	−7.12498	−0.551348	−0.275674	−	0.961251i	$-0.588901\pi$
$167$	−7.12498	−0.551348	−0.275674	+	0.961251i	$0.588901\pi$
$168$	0	0
$169$	−2.45712	−0.189009
$170$	0.643104	0.0493238
$171$	0	0
$172$	6.01507	0.458644
$173$	−18.7289	−1.42393	−0.711964	−	0.702216i	$-0.752194\pi$
$173$	−18.7289	−1.42393	−0.711964	+	0.702216i	$0.752194\pi$
$174$	0	0
$175$	−0.198062	−0.0149721
$176$	8.09783	0.610397
$177$	0	0
$178$	0.554958	0.0415959
$179$	−22.7875	−1.70321	−0.851607	−	0.524180i	$-0.824372\pi$
$179$	−22.7875	−1.70321	−0.851607	+	0.524180i	$0.824372\pi$
$180$	0	0
$181$	4.01938	0.298758	0.149379	−	0.988780i	$-0.452273\pi$
$181$	4.01938	0.298758	0.149379	+	0.988780i	$0.452273\pi$
$182$	0.356896	0.0264549
$183$	0	0
$184$	−9.20775	−0.678804
$185$	1.15883	0.0851991
$186$	0	0
$187$	−4.17629	−0.305401
$188$	−2.00670	−0.146354
$189$	0	0
$190$	0.835790	0.0606345
$191$	−2.43535	−0.176216	−0.0881080	−	0.996111i	$-0.528082\pi$
$191$	−2.43535	−0.176216	−0.0881080	+	0.996111i	$0.528082\pi$
$192$	0	0
$193$	−6.47219	−0.465878	−0.232939	−	0.972491i	$-0.574834\pi$
$193$	−6.47219	−0.465878	−0.232939	+	0.972491i	$0.574834\pi$
$194$	−5.78017	−0.414992
$195$	0	0
$196$	11.7778	0.841270
$197$	−22.1172	−1.57579	−0.787893	−	0.615812i	$-0.788828\pi$
$197$	−22.1172	−1.57579	−0.787893	+	0.615812i	$0.788828\pi$
$198$	0	0
$199$	−8.80625	−0.624258	−0.312129	−	0.950040i	$-0.601042\pi$
$199$	−8.80625	−0.624258	−0.312129	+	0.950040i	$0.601042\pi$
$200$	−2.04892	−0.144880
$201$	0	0
$202$	−0.448026	−0.0315230
$203$	−1.05669	−0.0741649
$204$	0	0
$205$	−0.814019	−0.0568536
$206$	4.94198	0.344324
$207$	0	0
$208$	−7.29590	−0.505879
$209$	−5.42758	−0.375434
$210$	0	0
$211$	−1.82371	−0.125549	−0.0627746	−	0.998028i	$-0.519995\pi$
$211$	−1.82371	−0.125549	−0.0627746	+	0.998028i	$0.519995\pi$
$212$	−2.58509	−0.177545
$213$	0	0
$214$	2.52111	0.172339
$215$	3.55496	0.242446
$216$	0	0
$217$	−0.274127	−0.0186089
$218$	−8.69202	−0.588698
$219$	0	0
$220$	6.09783	0.411116
$221$	3.76271	0.253107
$222$	0	0
$223$	−7.26205	−0.486303	−0.243151	−	0.969988i	$-0.578181\pi$
$223$	−7.26205	−0.486303	−0.243151	+	0.969988i	$0.578181\pi$
$224$	−1.05861	−0.0707311
$225$	0	0
$226$	−4.15213	−0.276196
$227$	−17.1860	−1.14067	−0.570337	−	0.821411i	$-0.693187\pi$
$227$	−17.1860	−1.14067	−0.570337	+	0.821411i	$0.693187\pi$
$228$	0	0
$229$	−18.6461	−1.23217	−0.616084	−	0.787680i	$-0.711282\pi$
$229$	−18.6461	−1.23217	−0.616084	+	0.787680i	$0.711282\pi$
$230$	−2.49396	−0.164447
$231$	0	0
$232$	−10.9312	−0.717670
$233$	−10.0151	−0.656109	−0.328054	−	0.944659i	$-0.606393\pi$
$233$	−10.0151	−0.656109	−0.328054	+	0.944659i	$0.606393\pi$
$234$	0	0
$235$	−1.18598	−0.0773648
$236$	13.1715	0.857392
$237$	0	0
$238$	0.127375	0.00825647
$239$	12.7114	0.822232	0.411116	−	0.911583i	$-0.365139\pi$
$239$	12.7114	0.822232	0.411116	+	0.911583i	$0.365139\pi$
$240$	0	0
$241$	−7.82371	−0.503969	−0.251985	−	0.967731i	$-0.581083\pi$
$241$	−7.82371	−0.503969	−0.251985	+	0.967731i	$0.581083\pi$
$242$	1.10321	0.0709171
$243$	0	0
$244$	4.05429	0.259550
$245$	6.96077	0.444707
$246$	0	0
$247$	4.89008	0.311149
$248$	−2.83579	−0.180073
$249$	0	0
$250$	−0.554958	−0.0350986
$251$	2.76809	0.174720	0.0873600	−	0.996177i	$-0.472157\pi$
$251$	2.76809	0.174720	0.0873600	+	0.996177i	$0.472157\pi$
$252$	0	0
$253$	16.1957	1.01821
$254$	7.56332	0.474565
$255$	0	0
$256$	−3.34721	−0.209200
$257$	11.5036	0.717578	0.358789	−	0.933419i	$-0.383190\pi$
$257$	11.5036	0.717578	0.358789	+	0.933419i	$0.383190\pi$
$258$	0	0
$259$	0.229521	0.0142618
$260$	−5.49396	−0.340721
$261$	0	0
$262$	0.298290	0.0184284
$263$	7.64204	0.471228	0.235614	−	0.971847i	$-0.424290\pi$
$263$	7.64204	0.471228	0.235614	+	0.971847i	$0.424290\pi$
$264$	0	0
$265$	−1.52781	−0.0938527
$266$	0.165538	0.0101498
$267$	0	0
$268$	−7.97584	−0.487202
$269$	−15.9651	−0.973408	−0.486704	−	0.873567i	$-0.661801\pi$
$269$	−15.9651	−0.973408	−0.486704	+	0.873567i	$0.661801\pi$
$270$	0	0
$271$	−10.4480	−0.634672	−0.317336	−	0.948313i	$-0.602788\pi$
$271$	−10.4480	−0.634672	−0.317336	+	0.948313i	$0.602788\pi$
$272$	−2.60388	−0.157883
$273$	0	0
$274$	8.34780	0.504309
$275$	3.60388	0.217322
$276$	0	0
$277$	10.1414	0.609336	0.304668	−	0.952459i	$-0.401454\pi$
$277$	10.1414	0.609336	0.304668	+	0.952459i	$0.401454\pi$
$278$	−6.88769	−0.413096
$279$	0	0
$280$	−0.405813	−0.0242520
$281$	−12.2989	−0.733690	−0.366845	−	0.930282i	$-0.619562\pi$
$281$	−12.2989	−0.733690	−0.366845	+	0.930282i	$0.619562\pi$
$282$	0	0
$283$	22.7875	1.35457	0.677287	−	0.735719i	$-0.263156\pi$
$283$	22.7875	1.35457	0.677287	+	0.735719i	$0.263156\pi$
$284$	20.1715	1.19696
$285$	0	0
$286$	−6.49396	−0.383996
$287$	−0.161227	−0.00951690
$288$	0	0
$289$	−15.6571	−0.921006
$290$	−2.96077	−0.173862
$291$	0	0
$292$	24.6896	1.44485
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	7.78448	0.453230
$296$	2.37435	0.138007
$297$	0	0
$298$	3.57374	0.207021
$299$	−14.5918	−0.843865
$300$	0	0
$301$	0.704103	0.0405838
$302$	1.28621	0.0740130
$303$	0	0
$304$	−3.38404	−0.194088
$305$	2.39612	0.137202
$306$	0	0
$307$	24.7332	1.41160	0.705798	−	0.708413i	$-0.250588\pi$
$307$	24.7332	1.41160	0.705798	+	0.708413i	$0.250588\pi$
$308$	1.20775	0.0688180
$309$	0	0
$310$	−0.768086	−0.0436243
$311$	−21.8780	−1.24059	−0.620294	−	0.784370i	$-0.712987\pi$
$311$	−21.8780	−1.24059	−0.620294	+	0.784370i	$0.712987\pi$
$312$	0	0
$313$	−19.6829	−1.11254	−0.556272	−	0.831000i	$-0.687769\pi$
$313$	−19.6829	−1.11254	−0.556272	+	0.831000i	$0.687769\pi$
$314$	3.30559	0.186545
$315$	0	0
$316$	28.2024	1.58651
$317$	−25.9898	−1.45973	−0.729867	−	0.683589i	$-0.760418\pi$
$317$	−25.9898	−1.45973	−0.729867	+	0.683589i	$0.760418\pi$
$318$	0	0
$319$	19.2271	1.07651
$320$	1.52781	0.0854072
$321$	0	0
$322$	−0.493959	−0.0275273
$323$	1.74525	0.0971083
$324$	0	0
$325$	−3.24698	−0.180110
$326$	4.24160	0.234921
$327$	0	0
$328$	−1.66786	−0.0920921
$329$	−0.234898	−0.0129503
$330$	0	0
$331$	14.9922	0.824048	0.412024	−	0.911173i	$-0.364822\pi$
$331$	14.9922	0.824048	0.412024	+	0.911173i	$0.364822\pi$
$332$	−22.6789	−1.24466
$333$	0	0
$334$	−3.95407	−0.216357
$335$	−4.71379	−0.257542
$336$	0	0
$337$	10.6243	0.578743	0.289372	−	0.957217i	$-0.406554\pi$
$337$	10.6243	0.578743	0.289372	+	0.957217i	$0.406554\pi$
$338$	−1.36360	−0.0741701
$339$	0	0
$340$	−1.96077	−0.106338
$341$	4.98792	0.270111
$342$	0	0
$343$	2.76510	0.149301
$344$	7.28382	0.392717
$345$	0	0
$346$	−10.3937	−0.558770
$347$	−11.4034	−0.612168	−0.306084	−	0.952005i	$-0.599019\pi$
$347$	−11.4034	−0.612168	−0.306084	+	0.952005i	$0.599019\pi$
$348$	0	0
$349$	13.4034	0.717469	0.358734	−	0.933440i	$-0.383208\pi$
$349$	13.4034	0.717469	0.358734	+	0.933440i	$0.383208\pi$
$350$	−0.109916	−0.00587527
$351$	0	0
$352$	19.2620	1.02667
$353$	−25.6775	−1.36668	−0.683339	−	0.730101i	$-0.739473\pi$
$353$	−25.6775	−1.36668	−0.683339	+	0.730101i	$0.739473\pi$
$354$	0	0
$355$	11.9215	0.632730
$356$	−1.69202	−0.0896770
$357$	0	0
$358$	−12.6461	−0.668367
$359$	−3.52781	−0.186191	−0.0930954	−	0.995657i	$-0.529676\pi$
$359$	−3.52781	−0.186191	−0.0930954	+	0.995657i	$0.529676\pi$
$360$	0	0
$361$	−16.7318	−0.880623
$362$	2.23059	0.117237
$363$	0	0
$364$	−1.08815	−0.0570343
$365$	14.5918	0.763769
$366$	0	0
$367$	2.24400	0.117136	0.0585678	−	0.998283i	$-0.481347\pi$
$367$	2.24400	0.117136	0.0585678	+	0.998283i	$0.481347\pi$
$368$	10.0978	0.526386
$369$	0	0
$370$	0.643104	0.0334334
$371$	−0.302602	−0.0157103
$372$	0	0
$373$	17.4711	0.904621	0.452310	−	0.891861i	$-0.350600\pi$
$373$	17.4711	0.904621	0.452310	+	0.891861i	$0.350600\pi$
$374$	−2.31767	−0.119844
$375$	0	0
$376$	−2.42998	−0.125316
$377$	−17.3230	−0.892182
$378$	0	0
$379$	33.3793	1.71458	0.857289	−	0.514836i	$-0.172147\pi$
$379$	33.3793	1.71458	0.857289	+	0.514836i	$0.172147\pi$
$380$	−2.54825	−0.130723
$381$	0	0
$382$	−1.35152	−0.0691497
$383$	10.3961	0.531217	0.265609	−	0.964081i	$-0.414427\pi$
$383$	10.3961	0.531217	0.265609	+	0.964081i	$0.414427\pi$
$384$	0	0
$385$	0.713792	0.0363782
$386$	−3.59179	−0.182817
$387$	0	0
$388$	17.6233	0.894685
$389$	31.9734	1.62112	0.810559	−	0.585657i	$-0.199163\pi$
$389$	31.9734	1.62112	0.810559	+	0.585657i	$0.199163\pi$
$390$	0	0
$391$	−5.20775	−0.263367
$392$	14.2620	0.720342
$393$	0	0
$394$	−12.2741	−0.618362
$395$	16.6679	0.838651
$396$	0	0
$397$	−19.9433	−1.00093	−0.500463	−	0.865758i	$-0.666837\pi$
$397$	−19.9433	−1.00093	−0.500463	+	0.865758i	$0.666837\pi$
$398$	−4.88710	−0.244968
$399$	0	0
$400$	2.24698	0.112349
$401$	29.2379	1.46007	0.730035	−	0.683410i	$-0.239504\pi$
$401$	29.2379	1.46007	0.730035	+	0.683410i	$0.239504\pi$
$402$	0	0
$403$	−4.49396	−0.223860
$404$	1.36599	0.0679607
$405$	0	0
$406$	−0.586417	−0.0291034
$407$	−4.17629	−0.207011
$408$	0	0
$409$	7.22952	0.357477	0.178738	−	0.983897i	$-0.442798\pi$
$409$	7.22952	0.357477	0.178738	+	0.983897i	$0.442798\pi$
$410$	−0.451747	−0.0223102
$411$	0	0
$412$	−15.0677	−0.742332
$413$	1.54181	0.0758676
$414$	0	0
$415$	−13.4034	−0.657948
$416$	−17.3545	−0.850875
$417$	0	0
$418$	−3.01208	−0.147326
$419$	−10.4015	−0.508147	−0.254073	−	0.967185i	$-0.581770\pi$
$419$	−10.4015	−0.508147	−0.254073	+	0.967185i	$0.581770\pi$
$420$	0	0
$421$	8.95300	0.436343	0.218171	−	0.975910i	$-0.429991\pi$
$421$	8.95300	0.436343	0.218171	+	0.975910i	$0.429991\pi$
$422$	−1.01208	−0.0492674
$423$	0	0
$424$	−3.13036	−0.152024
$425$	−1.15883	−0.0562117
$426$	0	0
$427$	0.474582	0.0229666
$428$	−7.68664	−0.371548
$429$	0	0
$430$	1.97285	0.0951394
$431$	17.5007	0.842977	0.421489	−	0.906834i	$-0.361508\pi$
$431$	17.5007	0.842977	0.421489	+	0.906834i	$0.361508\pi$
$432$	0	0
$433$	21.8582	1.05044	0.525218	−	0.850968i	$-0.323984\pi$
$433$	21.8582	1.05044	0.525218	+	0.850968i	$0.323984\pi$
$434$	−0.152129	−0.00730242
$435$	0	0
$436$	26.5013	1.26918
$437$	−6.76809	−0.323761
$438$	0	0
$439$	−21.9517	−1.04770	−0.523848	−	0.851812i	$-0.675504\pi$
$439$	−21.9517	−1.04770	−0.523848	+	0.851812i	$0.675504\pi$
$440$	7.38404	0.352020
$441$	0	0
$442$	2.08815	0.0993230
$443$	5.69740	0.270692	0.135346	−	0.990798i	$-0.456785\pi$
$443$	5.69740	0.270692	0.135346	+	0.990798i	$0.456785\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−4.03013	−0.190832
$447$	0	0
$448$	0.302602	0.0142966
$449$	8.73317	0.412144	0.206072	−	0.978537i	$-0.433932\pi$
$449$	8.73317	0.412144	0.206072	+	0.978537i	$0.433932\pi$
$450$	0	0
$451$	2.93362	0.138139
$452$	12.6595	0.595453
$453$	0	0
$454$	−9.53750	−0.447617
$455$	−0.643104	−0.0301492
$456$	0	0
$457$	−1.91292	−0.0894826	−0.0447413	−	0.998999i	$-0.514246\pi$
$457$	−1.91292	−0.0894826	−0.0447413	+	0.998999i	$0.514246\pi$
$458$	−10.3478	−0.483521
$459$	0	0
$460$	7.60388	0.354533
$461$	−2.56166	−0.119309	−0.0596543	−	0.998219i	$-0.519000\pi$
$461$	−2.56166	−0.119309	−0.0596543	+	0.998219i	$0.519000\pi$
$462$	0	0
$463$	−6.06292	−0.281768	−0.140884	−	0.990026i	$-0.544994\pi$
$463$	−6.06292	−0.281768	−0.140884	+	0.990026i	$0.544994\pi$
$464$	11.9879	0.556525
$465$	0	0
$466$	−5.55794	−0.257467
$467$	−1.14914	−0.0531761	−0.0265880	−	0.999646i	$-0.508464\pi$
$467$	−1.14914	−0.0531761	−0.0265880	+	0.999646i	$0.508464\pi$
$468$	0	0
$469$	−0.933624	−0.0431108
$470$	−0.658170	−0.0303591
$471$	0	0
$472$	15.9498	0.734147
$473$	−12.8116	−0.589079
$474$	0	0
$475$	−1.50604	−0.0691019
$476$	−0.388355	−0.0178002
$477$	0	0
$478$	7.05429	0.322656
$479$	26.6025	1.21550	0.607751	−	0.794128i	$-0.292072\pi$
$479$	26.6025	1.21550	0.607751	+	0.794128i	$0.292072\pi$
$480$	0	0
$481$	3.76271	0.171565
$482$	−4.34183	−0.197765
$483$	0	0
$484$	−3.36360	−0.152891
$485$	10.4155	0.472944
$486$	0	0
$487$	15.4082	0.698212	0.349106	−	0.937083i	$-0.386485\pi$
$487$	15.4082	0.698212	0.349106	+	0.937083i	$0.386485\pi$
$488$	4.90946	0.222241
$489$	0	0
$490$	3.86294	0.174510
$491$	4.35258	0.196429	0.0982147	−	0.995165i	$-0.468687\pi$
$491$	4.35258	0.196429	0.0982147	+	0.995165i	$0.468687\pi$
$492$	0	0
$493$	−6.18252	−0.278447
$494$	2.71379	0.122099
$495$	0	0
$496$	3.10992	0.139639
$497$	2.36121	0.105915
$498$	0	0
$499$	12.1715	0.544871	0.272436	−	0.962174i	$-0.412171\pi$
$499$	12.1715	0.544871	0.272436	+	0.962174i	$0.412171\pi$
$500$	1.69202	0.0756695
$501$	0	0
$502$	1.53617	0.0685627
$503$	1.46250	0.0652097	0.0326048	−	0.999468i	$-0.489620\pi$
$503$	1.46250	0.0652097	0.0326048	+	0.999468i	$0.489620\pi$
$504$	0	0
$505$	0.807315	0.0359250
$506$	8.98792	0.399562
$507$	0	0
$508$	−23.0599	−1.02312
$509$	−10.2547	−0.454534	−0.227267	−	0.973833i	$-0.572979\pi$
$509$	−10.2547	−0.454534	−0.227267	+	0.973833i	$0.572979\pi$
$510$	0	0
$511$	2.89008	0.127850
$512$	21.2174	0.937687
$513$	0	0
$514$	6.38404	0.281588
$515$	−8.90515	−0.392408
$516$	0	0
$517$	4.27413	0.187976
$518$	0.127375	0.00559652
$519$	0	0
$520$	−6.65279	−0.291744
$521$	35.0726	1.53656	0.768279	−	0.640115i	$-0.221113\pi$
$521$	35.0726	1.53656	0.768279	+	0.640115i	$0.221113\pi$
$522$	0	0
$523$	−24.9879	−1.09265	−0.546323	−	0.837575i	$-0.683973\pi$
$523$	−24.9879	−1.09265	−0.546323	+	0.837575i	$0.683973\pi$
$524$	−0.909461	−0.0397300
$525$	0	0
$526$	4.24101	0.184917
$527$	−1.60388	−0.0698659
$528$	0	0
$529$	−2.80433	−0.121927
$530$	−0.847871	−0.0368292
$531$	0	0
$532$	−0.504713	−0.0218821
$533$	−2.64310	−0.114486
$534$	0	0
$535$	−4.54288	−0.196406
$536$	−9.65817	−0.417169
$537$	0	0
$538$	−8.85995	−0.381980
$539$	−25.0858	−1.08052
$540$	0	0
$541$	−11.4034	−0.490271	−0.245136	−	0.969489i	$-0.578833\pi$
$541$	−11.4034	−0.490271	−0.245136	+	0.969489i	$0.578833\pi$
$542$	−5.79822	−0.249055
$543$	0	0
$544$	−6.19375	−0.265555
$545$	15.6625	0.670907
$546$	0	0
$547$	−23.2121	−0.992476	−0.496238	−	0.868186i	$-0.665286\pi$
$547$	−23.2121	−0.992476	−0.496238	+	0.868186i	$0.665286\pi$
$548$	−25.4517	−1.08724
$549$	0	0
$550$	2.00000	0.0852803
$551$	−8.03492	−0.342299
$552$	0	0
$553$	3.30127	0.140384
$554$	5.62804	0.239112
$555$	0	0
$556$	21.0000	0.890598
$557$	4.49396	0.190415	0.0952076	−	0.995457i	$-0.469649\pi$
$557$	4.49396	0.190415	0.0952076	+	0.995457i	$0.469649\pi$
$558$	0	0
$559$	11.5429	0.488212
$560$	0.445042	0.0188065
$561$	0	0
$562$	−6.82536	−0.287911
$563$	9.78017	0.412185	0.206092	−	0.978533i	$-0.433925\pi$
$563$	9.78017	0.412185	0.206092	+	0.978533i	$0.433925\pi$
$564$	0	0
$565$	7.48188	0.314765
$566$	12.6461	0.531555
$567$	0	0
$568$	24.4263	1.02490
$569$	19.4099	0.813704	0.406852	−	0.913494i	$-0.366627\pi$
$569$	19.4099	0.813704	0.406852	+	0.913494i	$0.366627\pi$
$570$	0	0
$571$	−32.2935	−1.35144	−0.675721	−	0.737158i	$-0.736167\pi$
$571$	−32.2935	−1.35144	−0.675721	+	0.737158i	$0.736167\pi$
$572$	19.7995	0.827861
$573$	0	0
$574$	−0.0894740	−0.00373457
$575$	4.49396	0.187411
$576$	0	0
$577$	36.0954	1.50267	0.751336	−	0.659919i	$-0.229410\pi$
$577$	36.0954	1.50267	0.751336	+	0.659919i	$0.229410\pi$
$578$	−8.68904	−0.361416
$579$	0	0
$580$	9.02715	0.374832
$581$	−2.65471	−0.110136
$582$	0	0
$583$	5.50604	0.228037
$584$	29.8974	1.23716
$585$	0	0
$586$	−7.76941	−0.320952
$587$	12.4397	0.513440	0.256720	−	0.966486i	$-0.417358\pi$
$587$	12.4397	0.513440	0.256720	+	0.966486i	$0.417358\pi$
$588$	0	0
$589$	−2.08443	−0.0858872
$590$	4.32006	0.177854
$591$	0	0
$592$	−2.60388	−0.107019
$593$	−1.37329	−0.0563942	−0.0281971	−	0.999602i	$-0.508977\pi$
$593$	−1.37329	−0.0563942	−0.0281971	+	0.999602i	$0.508977\pi$
$594$	0	0
$595$	−0.229521	−0.00940945
$596$	−10.8961	−0.446320
$597$	0	0
$598$	−8.09783	−0.331145
$599$	−20.4722	−0.836471	−0.418235	−	0.908339i	$-0.637351\pi$
$599$	−20.4722	−0.836471	−0.418235	+	0.908339i	$0.637351\pi$
$600$	0	0
$601$	−5.13946	−0.209643	−0.104821	−	0.994491i	$-0.533427\pi$
$601$	−5.13946	−0.209643	−0.104821	+	0.994491i	$0.533427\pi$
$602$	0.390748	0.0159257
$603$	0	0
$604$	−3.92154	−0.159565
$605$	−1.98792	−0.0808204
$606$	0	0
$607$	−3.36467	−0.136568	−0.0682838	−	0.997666i	$-0.521752\pi$
$607$	−3.36467	−0.136568	−0.0682838	+	0.997666i	$0.521752\pi$
$608$	−8.04951	−0.326451
$609$	0	0
$610$	1.32975	0.0538400
$611$	−3.85086	−0.155789
$612$	0	0
$613$	−4.76809	−0.192581	−0.0962906	−	0.995353i	$-0.530698\pi$
$613$	−4.76809	−0.192581	−0.0962906	+	0.995353i	$0.530698\pi$
$614$	13.7259	0.553931
$615$	0	0
$616$	1.46250	0.0589258
$617$	36.4698	1.46822	0.734109	−	0.679031i	$-0.237600\pi$
$617$	36.4698	1.46822	0.734109	+	0.679031i	$0.237600\pi$
$618$	0	0
$619$	−45.1075	−1.81302	−0.906512	−	0.422180i	$-0.861265\pi$
$619$	−45.1075	−1.81302	−0.906512	+	0.422180i	$0.861265\pi$
$620$	2.34183	0.0940502
$621$	0	0
$622$	−12.1414	−0.486825
$623$	−0.198062	−0.00793520
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.9232	−0.436579
$627$	0	0
$628$	−10.0785	−0.402174
$629$	1.34290	0.0535447
$630$	0	0
$631$	39.9232	1.58932	0.794659	−	0.607056i	$-0.207650\pi$
$631$	39.9232	1.58932	0.794659	+	0.607056i	$0.207650\pi$
$632$	34.1511	1.35846
$633$	0	0
$634$	−14.4233	−0.572821
$635$	−13.6286	−0.540836
$636$	0	0
$637$	22.6015	0.895503
$638$	10.6703	0.422439
$639$	0	0
$640$	11.5375	0.456060
$641$	−3.12929	−0.123600	−0.0617998	−	0.998089i	$-0.519684\pi$
$641$	−3.12929	−0.123600	−0.0617998	+	0.998089i	$0.519684\pi$
$642$	0	0
$643$	−15.2185	−0.600159	−0.300080	−	0.953914i	$-0.597013\pi$
$643$	−15.2185	−0.600159	−0.300080	+	0.953914i	$0.597013\pi$
$644$	1.50604	0.0593463
$645$	0	0
$646$	0.968541	0.0381067
$647$	7.18359	0.282416	0.141208	−	0.989980i	$-0.454901\pi$
$647$	7.18359	0.282416	0.141208	+	0.989980i	$0.454901\pi$
$648$	0	0
$649$	−28.0543	−1.10123
$650$	−1.80194	−0.0706778
$651$	0	0
$652$	−12.9323	−0.506468
$653$	0.689629	0.0269873	0.0134936	−	0.999909i	$-0.495705\pi$
$653$	0.689629	0.0269873	0.0134936	+	0.999909i	$0.495705\pi$
$654$	0	0
$655$	−0.537500	−0.0210019
$656$	1.82908	0.0714138
$657$	0	0
$658$	−0.130359	−0.00508191
$659$	26.9396	1.04942	0.524709	−	0.851282i	$-0.324174\pi$
$659$	26.9396	1.04942	0.524709	+	0.851282i	$0.324174\pi$
$660$	0	0
$661$	−39.6233	−1.54117	−0.770583	−	0.637340i	$-0.780035\pi$
$661$	−39.6233	−1.54117	−0.770583	+	0.637340i	$0.780035\pi$
$662$	8.32006	0.323368
$663$	0	0
$664$	−27.4625	−1.06575
$665$	−0.298290	−0.0115672
$666$	0	0
$667$	23.9758	0.928348
$668$	12.0556	0.466446
$669$	0	0
$670$	−2.61596	−0.101063
$671$	−8.63533	−0.333363
$672$	0	0
$673$	23.5749	0.908747	0.454373	−	0.890811i	$-0.349863\pi$
$673$	23.5749	0.908747	0.454373	+	0.890811i	$0.349863\pi$
$674$	5.89605	0.227107
$675$	0	0
$676$	4.15751	0.159904
$677$	32.8525	1.26262	0.631312	−	0.775529i	$-0.282517\pi$
$677$	32.8525	1.26262	0.631312	+	0.775529i	$0.282517\pi$
$678$	0	0
$679$	2.06292	0.0791675
$680$	−2.37435	−0.0910523
$681$	0	0
$682$	2.76809	0.105995
$683$	−18.6848	−0.714956	−0.357478	−	0.933922i	$-0.616363\pi$
$683$	−18.6848	−0.714956	−0.357478	+	0.933922i	$0.616363\pi$
$684$	0	0
$685$	−15.0422	−0.574733
$686$	1.53452	0.0585881
$687$	0	0
$688$	−7.98792	−0.304537
$689$	−4.96077	−0.188990
$690$	0	0
$691$	2.16123	0.0822169	0.0411085	−	0.999155i	$-0.486911\pi$
$691$	2.16123	0.0822169	0.0411085	+	0.999155i	$0.486911\pi$
$692$	31.6896	1.20466
$693$	0	0
$694$	−6.32842	−0.240224
$695$	12.4112	0.470783
$696$	0	0
$697$	−0.943313	−0.0357305
$698$	7.43834	0.281545
$699$	0	0
$700$	0.335126	0.0126666
$701$	−8.57242	−0.323776	−0.161888	−	0.986809i	$-0.551758\pi$
$701$	−8.57242	−0.323776	−0.161888	+	0.986809i	$0.551758\pi$
$702$	0	0
$703$	1.74525	0.0658234
$704$	−5.50604	−0.207517
$705$	0	0
$706$	−14.2500	−0.536304
$707$	0.159899	0.00601360
$708$	0	0
$709$	−25.5469	−0.959435	−0.479717	−	0.877423i	$-0.659261\pi$
$709$	−25.5469	−0.959435	−0.479717	+	0.877423i	$0.659261\pi$
$710$	6.61596	0.248292
$711$	0	0
$712$	−2.04892	−0.0767864
$713$	6.21983	0.232935
$714$	0	0
$715$	11.7017	0.437619
$716$	38.5569	1.44094
$717$	0	0
$718$	−1.95779	−0.0730640
$719$	−16.7675	−0.625322	−0.312661	−	0.949865i	$-0.601220\pi$
$719$	−16.7675	−0.625322	−0.312661	+	0.949865i	$0.601220\pi$
$720$	0	0
$721$	−1.76377	−0.0656864
$722$	−9.28547	−0.345569
$723$	0	0
$724$	−6.80087	−0.252752
$725$	5.33513	0.198142
$726$	0	0
$727$	−19.3002	−0.715805	−0.357903	−	0.933759i	$-0.616508\pi$
$727$	−19.3002	−0.715805	−0.357903	+	0.933759i	$0.616508\pi$
$728$	−1.31767	−0.0488360
$729$	0	0
$730$	8.09783	0.299714
$731$	4.11960	0.152369
$732$	0	0
$733$	18.8659	0.696829	0.348414	−	0.937341i	$-0.386720\pi$
$733$	18.8659	0.696829	0.348414	+	0.937341i	$0.386720\pi$
$734$	1.24532	0.0459657
$735$	0	0
$736$	24.0194	0.885366
$737$	16.9879	0.625758
$738$	0	0
$739$	14.1608	0.520912	0.260456	−	0.965486i	$-0.416127\pi$
$739$	14.1608	0.520912	0.260456	+	0.965486i	$0.416127\pi$
$740$	−1.96077	−0.0720794
$741$	0	0
$742$	−0.167931	−0.00616495
$743$	−3.31634	−0.121665	−0.0608323	−	0.998148i	$-0.519375\pi$
$743$	−3.31634	−0.121665	−0.0608323	+	0.998148i	$0.519375\pi$
$744$	0	0
$745$	−6.43967	−0.235931
$746$	9.69574	0.354986
$747$	0	0
$748$	7.06638	0.258372
$749$	−0.899772	−0.0328770
$750$	0	0
$751$	28.2097	1.02939	0.514693	−	0.857375i	$-0.327906\pi$
$751$	28.2097	1.02939	0.514693	+	0.857375i	$0.327906\pi$
$752$	2.66487	0.0971780
$753$	0	0
$754$	−9.61356	−0.350105
$755$	−2.31767	−0.0843485
$756$	0	0
$757$	17.2573	0.627226	0.313613	−	0.949551i	$-0.398461\pi$
$757$	17.2573	0.627226	0.313613	+	0.949551i	$0.398461\pi$
$758$	18.5241	0.672825
$759$	0	0
$760$	−3.08575	−0.111932
$761$	38.3720	1.39098	0.695491	−	0.718535i	$-0.255187\pi$
$761$	38.3720	1.39098	0.695491	+	0.718535i	$0.255187\pi$
$762$	0	0
$763$	3.10215	0.112305
$764$	4.12067	0.149081
$765$	0	0
$766$	5.76941	0.208457
$767$	25.2760	0.912665
$768$	0	0
$769$	9.68771	0.349348	0.174674	−	0.984626i	$-0.444113\pi$
$769$	9.68771	0.349348	0.174674	+	0.984626i	$0.444113\pi$
$770$	0.396125	0.0142753
$771$	0	0
$772$	10.9511	0.394138
$773$	12.7090	0.457111	0.228556	−	0.973531i	$-0.426600\pi$
$773$	12.7090	0.457111	0.228556	+	0.973531i	$0.426600\pi$
$774$	0	0
$775$	1.38404	0.0497163
$776$	21.3405	0.766079
$777$	0	0
$778$	17.7439	0.636150
$779$	−1.22595	−0.0439241
$780$	0	0
$781$	−42.9638	−1.53736
$782$	−2.89008	−0.103349
$783$	0	0
$784$	−15.6407	−0.558597
$785$	−5.95646	−0.212595
$786$	0	0
$787$	−17.9782	−0.640855	−0.320427	−	0.947273i	$-0.603827\pi$
$787$	−17.9782	−0.640855	−0.320427	+	0.947273i	$0.603827\pi$
$788$	37.4228	1.33313
$789$	0	0
$790$	9.24996	0.329099
$791$	1.48188	0.0526895
$792$	0	0
$793$	7.78017	0.276282
$794$	−11.0677	−0.392778
$795$	0	0
$796$	14.9004	0.528129
$797$	30.7289	1.08847	0.544236	−	0.838932i	$-0.316820\pi$
$797$	30.7289	1.08847	0.544236	+	0.838932i	$0.316820\pi$
$798$	0	0
$799$	−1.37435	−0.0486212
$800$	5.34481	0.188968
$801$	0	0
$802$	16.2258	0.572953
$803$	−52.5870	−1.85576
$804$	0	0
$805$	0.890084	0.0313713
$806$	−2.49396	−0.0878460
$807$	0	0
$808$	1.65412	0.0581918
$809$	7.25129	0.254942	0.127471	−	0.991842i	$-0.459314\pi$
$809$	7.25129	0.254942	0.127471	+	0.991842i	$0.459314\pi$
$810$	0	0
$811$	−47.0756	−1.65305	−0.826524	−	0.562902i	$-0.809685\pi$
$811$	−47.0756	−1.65305	−0.826524	+	0.562902i	$0.809685\pi$
$812$	1.78794	0.0627443
$813$	0	0
$814$	−2.31767	−0.0812342
$815$	−7.64310	−0.267726
$816$	0	0
$817$	5.35391	0.187310
$818$	4.01208	0.140279
$819$	0	0
$820$	1.37734	0.0480987
$821$	21.1970	0.739780	0.369890	−	0.929075i	$-0.379395\pi$
$821$	21.1970	0.739780	0.369890	+	0.929075i	$0.379395\pi$
$822$	0	0
$823$	2.54825	0.0888265	0.0444133	−	0.999013i	$-0.485858\pi$
$823$	2.54825	0.0888265	0.0444133	+	0.999013i	$0.485858\pi$
$824$	−18.2459	−0.635627
$825$	0	0
$826$	0.855641	0.0297716
$827$	−52.9530	−1.84136	−0.920678	−	0.390323i	$-0.872363\pi$
$827$	−52.9530	−1.84136	−0.920678	+	0.390323i	$0.872363\pi$
$828$	0	0
$829$	−11.6582	−0.404905	−0.202452	−	0.979292i	$-0.564891\pi$
$829$	−11.6582	−0.404905	−0.202452	+	0.979292i	$0.564891\pi$
$830$	−7.43834	−0.258188
$831$	0	0
$832$	4.96077	0.171984
$833$	8.06638	0.279483
$834$	0	0
$835$	7.12498	0.246570
$836$	9.18359	0.317621
$837$	0	0
$838$	−5.77240	−0.199404
$839$	−50.3782	−1.73925	−0.869624	−	0.493714i	$-0.835639\pi$
$839$	−50.3782	−1.73925	−0.869624	+	0.493714i	$0.835639\pi$
$840$	0	0
$841$	−0.536435	−0.0184978
$842$	4.96854	0.171227
$843$	0	0
$844$	3.08575	0.106216
$845$	2.45712	0.0845276
$846$	0	0
$847$	−0.393732	−0.0135288
$848$	3.43296	0.117888
$849$	0	0
$850$	−0.643104	−0.0220583
$851$	−5.20775	−0.178519
$852$	0	0
$853$	−39.7700	−1.36170	−0.680850	−	0.732423i	$-0.738389\pi$
$853$	−39.7700	−1.36170	−0.680850	+	0.732423i	$0.738389\pi$
$854$	0.263373	0.00901244
$855$	0	0
$856$	−9.30798	−0.318140
$857$	21.2707	0.726592	0.363296	−	0.931674i	$-0.381651\pi$
$857$	21.2707	0.726592	0.363296	+	0.931674i	$0.381651\pi$
$858$	0	0
$859$	−48.8504	−1.66675	−0.833377	−	0.552705i	$-0.813596\pi$
$859$	−48.8504	−1.66675	−0.833377	+	0.552705i	$0.813596\pi$
$860$	−6.01507	−0.205112
$861$	0	0
$862$	9.71214	0.330797
$863$	54.4456	1.85335	0.926675	−	0.375862i	$-0.122654\pi$
$863$	54.4456	1.85335	0.926675	+	0.375862i	$0.122654\pi$
$864$	0	0
$865$	18.7289	0.636800
$866$	12.1304	0.412206
$867$	0	0
$868$	0.463828	0.0157434
$869$	−60.0689	−2.03770
$870$	0	0
$871$	−15.3056	−0.518610
$872$	32.0911	1.08674
$873$	0	0
$874$	−3.75600	−0.127049
$875$	0.198062	0.00669573
$876$	0	0
$877$	−22.0388	−0.744196	−0.372098	−	0.928193i	$-0.621361\pi$
$877$	−22.0388	−0.744196	−0.372098	+	0.928193i	$0.621361\pi$
$878$	−12.1823	−0.411131
$879$	0	0
$880$	−8.09783	−0.272978
$881$	−43.0315	−1.44977	−0.724883	−	0.688872i	$-0.758106\pi$
$881$	−43.0315	−1.44977	−0.724883	+	0.688872i	$0.758106\pi$
$882$	0	0
$883$	−37.8213	−1.27279	−0.636394	−	0.771364i	$-0.719575\pi$
$883$	−37.8213	−1.27279	−0.636394	+	0.771364i	$0.719575\pi$
$884$	−6.36658	−0.214131
$885$	0	0
$886$	3.16182	0.106223
$887$	9.30559	0.312451	0.156225	−	0.987721i	$-0.450067\pi$
$887$	9.30559	0.312451	0.156225	+	0.987721i	$0.450067\pi$
$888$	0	0
$889$	−2.69932	−0.0905322
$890$	−0.554958	−0.0186022
$891$	0	0
$892$	12.2875	0.411417
$893$	−1.78614	−0.0597707
$894$	0	0
$895$	22.7875	0.761701
$896$	2.28514	0.0763413
$897$	0	0
$898$	4.84654	0.161731
$899$	7.38404	0.246272
$900$	0	0
$901$	−1.77048	−0.0589832
$902$	1.62804	0.0542078
$903$	0	0
$904$	15.3297	0.509860
$905$	−4.01938	−0.133609
$906$	0	0
$907$	36.7681	1.22086	0.610432	−	0.792069i	$-0.290996\pi$
$907$	36.7681	1.22086	0.610432	+	0.792069i	$0.290996\pi$
$908$	29.0790	0.965022
$909$	0	0
$910$	−0.356896	−0.0118310
$911$	−6.21983	−0.206072	−0.103036	−	0.994678i	$-0.532856\pi$
$911$	−6.21983	−0.206072	−0.103036	+	0.994678i	$0.532856\pi$
$912$	0	0
$913$	48.3043	1.59864
$914$	−1.06159	−0.0351143
$915$	0	0
$916$	31.5496	1.04243
$917$	−0.106458	−0.00351557
$918$	0	0
$919$	30.4940	1.00590	0.502951	−	0.864315i	$-0.332247\pi$
$919$	30.4940	1.00590	0.502951	+	0.864315i	$0.332247\pi$
$920$	9.20775	0.303571
$921$	0	0
$922$	−1.42162	−0.0468184
$923$	38.7090	1.27412
$924$	0	0
$925$	−1.15883	−0.0381022
$926$	−3.36467	−0.110570
$927$	0	0
$928$	28.5153	0.936059
$929$	−18.3612	−0.602412	−0.301206	−	0.953559i	$-0.597389\pi$
$929$	−18.3612	−0.602412	−0.301206	+	0.953559i	$0.597389\pi$
$930$	0	0
$931$	10.4832	0.343573
$932$	16.9457	0.555075
$933$	0	0
$934$	−0.637727	−0.0208671
$935$	4.17629	0.136579
$936$	0	0
$937$	−57.3658	−1.87406	−0.937030	−	0.349248i	$-0.886437\pi$
$937$	−57.3658	−1.87406	−0.937030	+	0.349248i	$0.886437\pi$
$938$	−0.518122	−0.0169173
$939$	0	0
$940$	2.00670	0.0654515
$941$	−14.9584	−0.487629	−0.243815	−	0.969822i	$-0.578399\pi$
$941$	−14.9584	−0.487629	−0.243815	+	0.969822i	$0.578399\pi$
$942$	0	0
$943$	3.65817	0.119126
$944$	−17.4916	−0.569302
$945$	0	0
$946$	−7.10992	−0.231163
$947$	−37.6577	−1.22371	−0.611855	−	0.790970i	$-0.709577\pi$
$947$	−37.6577	−1.22371	−0.611855	+	0.790970i	$0.709577\pi$
$948$	0	0
$949$	47.3793	1.53800
$950$	−0.835790	−0.0271166
$951$	0	0
$952$	−0.470270	−0.0152415
$953$	−14.7224	−0.476906	−0.238453	−	0.971154i	$-0.576640\pi$
$953$	−14.7224	−0.476906	−0.238453	+	0.971154i	$0.576640\pi$
$954$	0	0
$955$	2.43535	0.0788062
$956$	−21.5080	−0.695617
$957$	0	0
$958$	14.7633	0.476981
$959$	−2.97929	−0.0962064
$960$	0	0
$961$	−29.0844	−0.938207
$962$	2.08815	0.0673245
$963$	0	0
$964$	13.2379	0.426363
$965$	6.47219	0.208347
$966$	0	0
$967$	−21.6577	−0.696465	−0.348232	−	0.937408i	$-0.613218\pi$
$967$	−21.6577	−0.696465	−0.348232	+	0.937408i	$0.613218\pi$
$968$	−4.07308	−0.130914
$969$	0	0
$970$	5.78017	0.185590
$971$	11.4168	0.366384	0.183192	−	0.983077i	$-0.441357\pi$
$971$	11.4168	0.366384	0.183192	+	0.983077i	$0.441357\pi$
$972$	0	0
$973$	2.45819	0.0788059
$974$	8.55091	0.273989
$975$	0	0
$976$	−5.38404	−0.172339
$977$	−30.3368	−0.970560	−0.485280	−	0.874359i	$-0.661282\pi$
$977$	−30.3368	−0.970560	−0.485280	+	0.874359i	$0.661282\pi$
$978$	0	0
$979$	3.60388	0.115180
$980$	−11.7778	−0.376227
$981$	0	0
$982$	2.41550	0.0770818
$983$	44.8579	1.43074	0.715372	−	0.698744i	$-0.246257\pi$
$983$	44.8579	1.43074	0.715372	+	0.698744i	$0.246257\pi$
$984$	0	0
$985$	22.1172	0.704713
$986$	−3.43104	−0.109267
$987$	0	0
$988$	−8.27413	−0.263235
$989$	−15.9758	−0.508002
$990$	0	0
$991$	−25.3250	−0.804474	−0.402237	−	0.915536i	$-0.631767\pi$
$991$	−25.3250	−0.804474	−0.402237	+	0.915536i	$0.631767\pi$
$992$	7.39745	0.234869
$993$	0	0
$994$	1.31037	0.0415625
$995$	8.80625	0.279177
$996$	0	0
$997$	32.9638	1.04397	0.521986	−	0.852954i	$-0.325191\pi$
$997$	32.9638	1.04397	0.521986	+	0.852954i	$0.325191\pi$
$998$	6.75468	0.213816
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.j.1.2		3
3.2	odd	2		1335.2.a.d.1.2	✓	3
15.14	odd	2		6675.2.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.d.1.2	✓	3	3.2	odd	2
4005.2.a.j.1.2		3	1.1	even	1	trivial
6675.2.a.o.1.2		3	15.14	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -0.198062 q^{7} -2.04892 q^{8} +O(q^{10})\)	\(q+0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -0.198062 q^{7} -2.04892 q^{8} -0.554958 q^{10} +3.60388 q^{11} -3.24698 q^{13} -0.109916 q^{14} +2.24698 q^{16} -1.15883 q^{17} -1.50604 q^{19} +1.69202 q^{20} +2.00000 q^{22} +4.49396 q^{23} +1.00000 q^{25} -1.80194 q^{26} +0.335126 q^{28} +5.33513 q^{29} +1.38404 q^{31} +5.34481 q^{32} -0.643104 q^{34} +0.198062 q^{35} -1.15883 q^{37} -0.835790 q^{38} +2.04892 q^{40} +0.814019 q^{41} -3.55496 q^{43} -6.09783 q^{44} +2.49396 q^{46} +1.18598 q^{47} -6.96077 q^{49} +0.554958 q^{50} +5.49396 q^{52} +1.52781 q^{53} -3.60388 q^{55} +0.405813 q^{56} +2.96077 q^{58} -7.78448 q^{59} -2.39612 q^{61} +0.768086 q^{62} -1.52781 q^{64} +3.24698 q^{65} +4.71379 q^{67} +1.96077 q^{68} +0.109916 q^{70} -11.9215 q^{71} -14.5918 q^{73} -0.643104 q^{74} +2.54825 q^{76} -0.713792 q^{77} -16.6679 q^{79} -2.24698 q^{80} +0.451747 q^{82} +13.4034 q^{83} +1.15883 q^{85} -1.97285 q^{86} -7.38404 q^{88} +1.00000 q^{89} +0.643104 q^{91} -7.60388 q^{92} +0.658170 q^{94} +1.50604 q^{95} -10.4155 q^{97} -3.86294 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8} - 2 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} + 2 q^{16} + 5 q^{17} - 14 q^{19} + 6 q^{22} + 4 q^{23} + 3 q^{25} - q^{26} + 15 q^{29} - 6 q^{31} - 7 q^{32} - 6 q^{34} + 5 q^{35} + 5 q^{37} - 14 q^{38} - 3 q^{40} + 17 q^{41} - 11 q^{43} - 2 q^{46} - 11 q^{47} - 8 q^{49} + 2 q^{50} + 7 q^{52} + 11 q^{53} - 2 q^{55} - 12 q^{56} - 4 q^{58} - 3 q^{59} - 16 q^{61} - 18 q^{62} - 11 q^{64} + 5 q^{65} + 6 q^{67} - 7 q^{68} + q^{70} - 10 q^{71} - 16 q^{73} - 6 q^{74} - 14 q^{76} + 6 q^{77} - 7 q^{79} - 2 q^{80} + 23 q^{82} - 14 q^{83} - 5 q^{85} - 12 q^{86} - 12 q^{88} + 3 q^{89} + 6 q^{91} - 14 q^{92} - 19 q^{94} + 14 q^{95} + 4 q^{97} - 17 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8 - 2 * q^10 + 2 * q^11 - 5 * q^13 - q^14 + 2 * q^16 + 5 * q^17 - 14 * q^19 + 6 * q^22 + 4 * q^23 + 3 * q^25 - q^26 + 15 * q^29 - 6 * q^31 - 7 * q^32 - 6 * q^34 + 5 * q^35 + 5 * q^37 - 14 * q^38 - 3 * q^40 + 17 * q^41 - 11 * q^43 - 2 * q^46 - 11 * q^47 - 8 * q^49 + 2 * q^50 + 7 * q^52 + 11 * q^53 - 2 * q^55 - 12 * q^56 - 4 * q^58 - 3 * q^59 - 16 * q^61 - 18 * q^62 - 11 * q^64 + 5 * q^65 + 6 * q^67 - 7 * q^68 + q^70 - 10 * q^71 - 16 * q^73 - 6 * q^74 - 14 * q^76 + 6 * q^77 - 7 * q^79 - 2 * q^80 + 23 * q^82 - 14 * q^83 - 5 * q^85 - 12 * q^86 - 12 * q^88 + 3 * q^89 + 6 * q^91 - 14 * q^92 - 19 * q^94 + 14 * q^95 + 4 * q^97 - 17 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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