LMFDB - Embedded newform 4005.2.a.u.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.24327$ 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)

$f(q)$	$=$	$q-2.24327 q^{2} +3.03225 q^{4} +1.00000 q^{5} +1.09855 q^{7} -2.31562 q^{8} +O(q^{10})$	\(q-2.24327 q^{2} +3.03225 q^{4} +1.00000 q^{5} +1.09855 q^{7} -2.31562 q^{8} -2.24327 q^{10} +3.76806 q^{11} -6.02124 q^{13} -2.46435 q^{14} -0.869946 q^{16} -3.14554 q^{17} +0.176908 q^{19} +3.03225 q^{20} -8.45276 q^{22} +0.808666 q^{23} +1.00000 q^{25} +13.5072 q^{26} +3.33109 q^{28} +5.42525 q^{29} +3.31845 q^{31} +6.58277 q^{32} +7.05629 q^{34} +1.09855 q^{35} -11.5119 q^{37} -0.396852 q^{38} -2.31562 q^{40} +1.30517 q^{41} -0.0238550 q^{43} +11.4257 q^{44} -1.81405 q^{46} +2.23397 q^{47} -5.79318 q^{49} -2.24327 q^{50} -18.2579 q^{52} -7.11120 q^{53} +3.76806 q^{55} -2.54384 q^{56} -12.1703 q^{58} -9.73330 q^{59} -14.5050 q^{61} -7.44417 q^{62} -13.0270 q^{64} -6.02124 q^{65} -3.60188 q^{67} -9.53807 q^{68} -2.46435 q^{70} +3.36845 q^{71} +6.82666 q^{73} +25.8242 q^{74} +0.536430 q^{76} +4.13942 q^{77} +6.52986 q^{79} -0.869946 q^{80} -2.92785 q^{82} -3.43016 q^{83} -3.14554 q^{85} +0.0535131 q^{86} -8.72540 q^{88} +1.00000 q^{89} -6.61465 q^{91} +2.45208 q^{92} -5.01140 q^{94} +0.176908 q^{95} -12.9743 q^{97} +12.9957 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * 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 $

$2$	−2.24327	−1.58623	−0.793115	−	0.609072i	$-0.791542\pi$
$2$	−2.24327	−1.58623	−0.793115	+	0.609072i	$0.791542\pi$
$3$	0	0
$3$	0	0
$4$	3.03225	1.51613
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.09855	0.415214	0.207607	−	0.978212i	$-0.433432\pi$
$7$	1.09855	0.415214	0.207607	+	0.978212i	$0.433432\pi$
$8$	−2.31562	−0.818696
$9$	0	0
$10$	−2.24327	−0.709384
$11$	3.76806	1.13611	0.568056	−	0.822990i	$-0.307696\pi$
$11$	3.76806	1.13611	0.568056	+	0.822990i	$0.307696\pi$
$12$	0	0
$13$	−6.02124	−1.66999	−0.834995	−	0.550257i	$-0.814530\pi$
$13$	−6.02124	−1.66999	−0.834995	+	0.550257i	$0.814530\pi$
$14$	−2.46435	−0.658626
$15$	0	0
$16$	−0.869946	−0.217487
$17$	−3.14554	−0.762905	−0.381453	−	0.924388i	$-0.624576\pi$
$17$	−3.14554	−0.762905	−0.381453	+	0.924388i	$0.624576\pi$
$18$	0	0
$19$	0.176908	0.0405855	0.0202927	−	0.999794i	$-0.493540\pi$
$19$	0.176908	0.0405855	0.0202927	+	0.999794i	$0.493540\pi$
$20$	3.03225	0.678032
$21$	0	0
$22$	−8.45276	−1.80214
$23$	0.808666	0.168618	0.0843092	−	0.996440i	$-0.473132\pi$
$23$	0.808666	0.168618	0.0843092	+	0.996440i	$0.473132\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	13.5072	2.64899
$27$	0	0
$28$	3.33109	0.629518
$29$	5.42525	1.00744	0.503721	−	0.863866i	$-0.331964\pi$
$29$	5.42525	1.00744	0.503721	+	0.863866i	$0.331964\pi$
$30$	0	0
$31$	3.31845	0.596011	0.298006	−	0.954564i	$-0.403679\pi$
$31$	3.31845	0.596011	0.298006	+	0.954564i	$0.403679\pi$
$32$	6.58277	1.16368
$33$	0	0
$34$	7.05629	1.21014
$35$	1.09855	0.185690
$36$	0	0
$37$	−11.5119	−1.89254	−0.946270	−	0.323379i	$-0.895181\pi$
$37$	−11.5119	−1.89254	−0.946270	+	0.323379i	$0.895181\pi$
$38$	−0.396852	−0.0643779
$39$	0	0
$40$	−2.31562	−0.366132
$41$	1.30517	0.203834	0.101917	−	0.994793i	$-0.467502\pi$
$41$	1.30517	0.203834	0.101917	+	0.994793i	$0.467502\pi$
$42$	0	0
$43$	−0.0238550	−0.00363785	−0.00181892	−	0.999998i	$-0.500579\pi$
$43$	−0.0238550	−0.00363785	−0.00181892	+	0.999998i	$0.500579\pi$
$44$	11.4257	1.72249
$45$	0	0
$46$	−1.81405	−0.267468
$47$	2.23397	0.325858	0.162929	−	0.986638i	$-0.447906\pi$
$47$	2.23397	0.325858	0.162929	+	0.986638i	$0.447906\pi$
$48$	0	0
$49$	−5.79318	−0.827597
$50$	−2.24327	−0.317246
$51$	0	0
$52$	−18.2579	−2.53192
$53$	−7.11120	−0.976799	−0.488399	−	0.872620i	$-0.662419\pi$
$53$	−7.11120	−0.976799	−0.488399	+	0.872620i	$0.662419\pi$
$54$	0	0
$55$	3.76806	0.508085
$56$	−2.54384	−0.339934
$57$	0	0
$58$	−12.1703	−1.59804
$59$	−9.73330	−1.26717	−0.633584	−	0.773674i	$-0.718417\pi$
$59$	−9.73330	−1.26717	−0.633584	+	0.773674i	$0.718417\pi$
$60$	0	0
$61$	−14.5050	−1.85718	−0.928589	−	0.371111i	$-0.878977\pi$
$61$	−14.5050	−1.85718	−0.928589	+	0.371111i	$0.878977\pi$
$62$	−7.44417	−0.945411
$63$	0	0
$64$	−13.0270	−1.62838
$65$	−6.02124	−0.746842
$66$	0	0
$67$	−3.60188	−0.440040	−0.220020	−	0.975495i	$-0.570612\pi$
$67$	−3.60188	−0.440040	−0.220020	+	0.975495i	$0.570612\pi$
$68$	−9.53807	−1.15666
$69$	0	0
$70$	−2.46435	−0.294546
$71$	3.36845	0.399762	0.199881	−	0.979820i	$-0.435944\pi$
$71$	3.36845	0.399762	0.199881	+	0.979820i	$0.435944\pi$
$72$	0	0
$73$	6.82666	0.799000	0.399500	−	0.916733i	$-0.369184\pi$
$73$	6.82666	0.799000	0.399500	+	0.916733i	$0.369184\pi$
$74$	25.8242	3.00200
$75$	0	0
$76$	0.536430	0.0615327
$77$	4.13942	0.471730
$78$	0	0
$79$	6.52986	0.734667	0.367333	−	0.930089i	$-0.380271\pi$
$79$	6.52986	0.734667	0.367333	+	0.930089i	$0.380271\pi$
$80$	−0.869946	−0.0972629
$81$	0	0
$82$	−2.92785	−0.323327
$83$	−3.43016	−0.376509	−0.188254	−	0.982120i	$-0.560283\pi$
$83$	−3.43016	−0.376509	−0.188254	+	0.982120i	$0.560283\pi$
$84$	0	0
$85$	−3.14554	−0.341182
$86$	0.0535131	0.00577046
$87$	0	0
$88$	−8.72540	−0.930130
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−6.61465	−0.693404
$92$	2.45208	0.255647
$93$	0	0
$94$	−5.01140	−0.516886
$95$	0.176908	0.0181504
$96$	0	0
$97$	−12.9743	−1.31734	−0.658671	−	0.752431i	$-0.728881\pi$
$97$	−12.9743	−1.31734	−0.658671	+	0.752431i	$0.728881\pi$
$98$	12.9957	1.31276
$99$	0	0
$100$	3.03225	0.303225
$101$	−13.2861	−1.32201	−0.661006	−	0.750381i	$-0.729870\pi$
$101$	−13.2861	−1.32201	−0.661006	+	0.750381i	$0.729870\pi$
$102$	0	0
$103$	4.95552	0.488282	0.244141	−	0.969740i	$-0.421494\pi$
$103$	4.95552	0.488282	0.244141	+	0.969740i	$0.421494\pi$
$104$	13.9429	1.36721
$105$	0	0
$106$	15.9523	1.54943
$107$	4.45267	0.430455	0.215228	−	0.976564i	$-0.430951\pi$
$107$	4.45267	0.430455	0.215228	+	0.976564i	$0.430951\pi$
$108$	0	0
$109$	−5.67293	−0.543368	−0.271684	−	0.962386i	$-0.587581\pi$
$109$	−5.67293	−0.543368	−0.271684	+	0.962386i	$0.587581\pi$
$110$	−8.45276	−0.805940
$111$	0	0
$112$	−0.955683	−0.0903035
$113$	5.31742	0.500221	0.250110	−	0.968217i	$-0.419533\pi$
$113$	5.31742	0.500221	0.250110	+	0.968217i	$0.419533\pi$
$114$	0	0
$115$	0.808666	0.0754085
$116$	16.4507	1.52741
$117$	0	0
$118$	21.8344	2.01002
$119$	−3.45555	−0.316769
$120$	0	0
$121$	3.19826	0.290751
$122$	32.5387	2.94591
$123$	0	0
$124$	10.0624	0.903629
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.0446	1.33499	0.667495	−	0.744615i	$-0.267367\pi$
$127$	15.0446	1.33499	0.667495	+	0.744615i	$0.267367\pi$
$128$	16.0576	1.41930
$129$	0	0
$130$	13.5072	1.18466
$131$	6.38769	0.558095	0.279047	−	0.960277i	$-0.409981\pi$
$131$	6.38769	0.558095	0.279047	+	0.960277i	$0.409981\pi$
$132$	0	0
$133$	0.194343	0.0168517
$134$	8.07999	0.698005
$135$	0	0
$136$	7.28388	0.624588
$137$	−1.98104	−0.169252	−0.0846259	−	0.996413i	$-0.526970\pi$
$137$	−1.98104	−0.169252	−0.0846259	+	0.996413i	$0.526970\pi$
$138$	0	0
$139$	−18.4186	−1.56225	−0.781124	−	0.624377i	$-0.785353\pi$
$139$	−18.4186	−1.56225	−0.781124	+	0.624377i	$0.785353\pi$
$140$	3.33109	0.281529
$141$	0	0
$142$	−7.55634	−0.634114
$143$	−22.6884	−1.89730
$144$	0	0
$145$	5.42525	0.450542
$146$	−15.3140	−1.26740
$147$	0	0
$148$	−34.9069	−2.86933
$149$	−1.29782	−0.106322	−0.0531610	−	0.998586i	$-0.516930\pi$
$149$	−1.29782	−0.106322	−0.0531610	+	0.998586i	$0.516930\pi$
$150$	0	0
$151$	17.3206	1.40953	0.704764	−	0.709442i	$-0.251053\pi$
$151$	17.3206	1.40953	0.704764	+	0.709442i	$0.251053\pi$
$152$	−0.409652	−0.0332272
$153$	0	0
$154$	−9.28582	−0.748273
$155$	3.31845	0.266544
$156$	0	0
$157$	−15.0846	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$157$	−15.0846	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$158$	−14.6482	−1.16535
$159$	0	0
$160$	6.58277	0.520413
$161$	0.888363	0.0700128
$162$	0	0
$163$	−20.9193	−1.63853	−0.819264	−	0.573416i	$-0.805618\pi$
$163$	−20.9193	−1.63853	−0.819264	+	0.573416i	$0.805618\pi$
$164$	3.95762	0.309038
$165$	0	0
$166$	7.69476	0.597230
$167$	5.09773	0.394474	0.197237	−	0.980356i	$-0.436803\pi$
$167$	5.09773	0.394474	0.197237	+	0.980356i	$0.436803\pi$
$168$	0	0
$169$	23.2553	1.78887
$170$	7.05629	0.541193
$171$	0	0
$172$	−0.0723343	−0.00551544
$173$	−13.8676	−1.05433	−0.527165	−	0.849763i	$-0.676745\pi$
$173$	−13.8676	−1.05433	−0.527165	+	0.849763i	$0.676745\pi$
$174$	0	0
$175$	1.09855	0.0830429
$176$	−3.27801	−0.247089
$177$	0	0
$178$	−2.24327	−0.168140
$179$	0.358924	0.0268272	0.0134136	−	0.999910i	$-0.495730\pi$
$179$	0.358924	0.0268272	0.0134136	+	0.999910i	$0.495730\pi$
$180$	0	0
$181$	−6.46572	−0.480593	−0.240297	−	0.970700i	$-0.577245\pi$
$181$	−6.46572	−0.480593	−0.240297	+	0.970700i	$0.577245\pi$
$182$	14.8384	1.09990
$183$	0	0
$184$	−1.87256	−0.138047
$185$	−11.5119	−0.846369
$186$	0	0
$187$	−11.8526	−0.866746
$188$	6.77397	0.494043
$189$	0	0
$190$	−0.396852	−0.0287907
$191$	19.3854	1.40268	0.701338	−	0.712829i	$-0.252586\pi$
$191$	19.3854	1.40268	0.701338	+	0.712829i	$0.252586\pi$
$192$	0	0
$193$	12.1449	0.874213	0.437106	−	0.899410i	$-0.356003\pi$
$193$	12.1449	0.874213	0.437106	+	0.899410i	$0.356003\pi$
$194$	29.1049	2.08961
$195$	0	0
$196$	−17.5664	−1.25474
$197$	−11.1021	−0.790990	−0.395495	−	0.918468i	$-0.629427\pi$
$197$	−11.1021	−0.790990	−0.395495	+	0.918468i	$0.629427\pi$
$198$	0	0
$199$	11.7641	0.833937	0.416969	−	0.908921i	$-0.363092\pi$
$199$	11.7641	0.833937	0.416969	+	0.908921i	$0.363092\pi$
$200$	−2.31562	−0.163739
$201$	0	0
$202$	29.8042	2.09702
$203$	5.95993	0.418305
$204$	0	0
$205$	1.30517	0.0911573
$206$	−11.1166	−0.774528
$207$	0	0
$208$	5.23815	0.363200
$209$	0.666599	0.0461096
$210$	0	0
$211$	26.1785	1.80220	0.901101	−	0.433610i	$-0.142761\pi$
$211$	26.1785	1.80220	0.901101	+	0.433610i	$0.142761\pi$
$212$	−21.5630	−1.48095
$213$	0	0
$214$	−9.98853	−0.682802
$215$	−0.0238550	−0.00162690
$216$	0	0
$217$	3.64550	0.247472
$218$	12.7259	0.861907
$219$	0	0
$220$	11.4257	0.770321
$221$	18.9400	1.27404
$222$	0	0
$223$	−9.50225	−0.636318	−0.318159	−	0.948037i	$-0.603065\pi$
$223$	−9.50225	−0.636318	−0.318159	+	0.948037i	$0.603065\pi$
$224$	7.23153	0.483177
$225$	0	0
$226$	−11.9284	−0.793466
$227$	−24.0391	−1.59553	−0.797764	−	0.602970i	$-0.793984\pi$
$227$	−24.0391	−1.59553	−0.797764	+	0.602970i	$0.793984\pi$
$228$	0	0
$229$	14.2933	0.944530	0.472265	−	0.881457i	$-0.343437\pi$
$229$	14.2933	0.944530	0.472265	+	0.881457i	$0.343437\pi$
$230$	−1.81405	−0.119615
$231$	0	0
$232$	−12.5628	−0.824789
$233$	−18.0516	−1.18260	−0.591300	−	0.806452i	$-0.701385\pi$
$233$	−18.0516	−1.18260	−0.591300	+	0.806452i	$0.701385\pi$
$234$	0	0
$235$	2.23397	0.145728
$236$	−29.5138	−1.92119
$237$	0	0
$238$	7.75172	0.502469
$239$	−14.2590	−0.922341	−0.461171	−	0.887311i	$-0.652570\pi$
$239$	−14.2590	−0.922341	−0.461171	+	0.887311i	$0.652570\pi$
$240$	0	0
$241$	−3.81330	−0.245637	−0.122818	−	0.992429i	$-0.539193\pi$
$241$	−3.81330	−0.245637	−0.122818	+	0.992429i	$0.539193\pi$
$242$	−7.17455	−0.461198
$243$	0	0
$244$	−43.9829	−2.81572
$245$	−5.79318	−0.370113
$246$	0	0
$247$	−1.06520	−0.0677773
$248$	−7.68427	−0.487952
$249$	0	0
$250$	−2.24327	−0.141877
$251$	21.2668	1.34235	0.671176	−	0.741298i	$-0.265790\pi$
$251$	21.2668	1.34235	0.671176	+	0.741298i	$0.265790\pi$
$252$	0	0
$253$	3.04710	0.191569
$254$	−33.7490	−2.11760
$255$	0	0
$256$	−9.96740	−0.622963
$257$	15.2059	0.948519	0.474260	−	0.880385i	$-0.342716\pi$
$257$	15.2059	0.948519	0.474260	+	0.880385i	$0.342716\pi$
$258$	0	0
$259$	−12.6464	−0.785810
$260$	−18.2579	−1.13231
$261$	0	0
$262$	−14.3293	−0.885267
$263$	−19.2341	−1.18602	−0.593012	−	0.805194i	$-0.702061\pi$
$263$	−19.2341	−1.18602	−0.593012	+	0.805194i	$0.702061\pi$
$264$	0	0
$265$	−7.11120	−0.436838
$266$	−0.435963	−0.0267306
$267$	0	0
$268$	−10.9218	−0.667157
$269$	−22.3617	−1.36342	−0.681709	−	0.731623i	$-0.738763\pi$
$269$	−22.3617	−1.36342	−0.681709	+	0.731623i	$0.738763\pi$
$270$	0	0
$271$	6.56421	0.398747	0.199374	−	0.979924i	$-0.436109\pi$
$271$	6.56421	0.398747	0.199374	+	0.979924i	$0.436109\pi$
$272$	2.73645	0.165922
$273$	0	0
$274$	4.44401	0.268472
$275$	3.76806	0.227222
$276$	0	0
$277$	11.5967	0.696776	0.348388	−	0.937350i	$-0.386729\pi$
$277$	11.5967	0.696776	0.348388	+	0.937350i	$0.386729\pi$
$278$	41.3179	2.47808
$279$	0	0
$280$	−2.54384	−0.152023
$281$	4.02983	0.240399	0.120200	−	0.992750i	$-0.461646\pi$
$281$	4.02983	0.240399	0.120200	+	0.992750i	$0.461646\pi$
$282$	0	0
$283$	1.36836	0.0813407	0.0406703	−	0.999173i	$-0.487051\pi$
$283$	1.36836	0.0813407	0.0406703	+	0.999173i	$0.487051\pi$
$284$	10.2140	0.606089
$285$	0	0
$286$	50.8961	3.00955
$287$	1.43380	0.0846348
$288$	0	0
$289$	−7.10558	−0.417975
$290$	−12.1703	−0.714664
$291$	0	0
$292$	20.7002	1.21139
$293$	−2.01728	−0.117851	−0.0589255	−	0.998262i	$-0.518767\pi$
$293$	−2.01728	−0.117851	−0.0589255	+	0.998262i	$0.518767\pi$
$294$	0	0
$295$	−9.73330	−0.566695
$296$	26.6571	1.54941
$297$	0	0
$298$	2.91137	0.168651
$299$	−4.86917	−0.281591
$300$	0	0
$301$	−0.0262060	−0.00151049
$302$	−38.8547	−2.23584
$303$	0	0
$304$	−0.153900	−0.00882679
$305$	−14.5050	−0.830555
$306$	0	0
$307$	−16.7819	−0.957796	−0.478898	−	0.877871i	$-0.658964\pi$
$307$	−16.7819	−0.957796	−0.478898	+	0.877871i	$0.658964\pi$
$308$	12.5518	0.715203
$309$	0	0
$310$	−7.44417	−0.422801
$311$	14.0795	0.798376	0.399188	−	0.916869i	$-0.369292\pi$
$311$	14.0795	0.798376	0.399188	+	0.916869i	$0.369292\pi$
$312$	0	0
$313$	−17.5261	−0.990636	−0.495318	−	0.868712i	$-0.664949\pi$
$313$	−17.5261	−0.990636	−0.495318	+	0.868712i	$0.664949\pi$
$314$	33.8389	1.90964
$315$	0	0
$316$	19.8002	1.11385
$317$	24.7281	1.38887	0.694435	−	0.719555i	$-0.255654\pi$
$317$	24.7281	1.38887	0.694435	+	0.719555i	$0.255654\pi$
$318$	0	0
$319$	20.4426	1.14457
$320$	−13.0270	−0.728232
$321$	0	0
$322$	−1.99284	−0.111056
$323$	−0.556471	−0.0309629
$324$	0	0
$325$	−6.02124	−0.333998
$326$	46.9277	2.59908
$327$	0	0
$328$	−3.02229	−0.166878
$329$	2.45414	0.135301
$330$	0	0
$331$	−10.3046	−0.566390	−0.283195	−	0.959062i	$-0.591394\pi$
$331$	−10.3046	−0.566390	−0.283195	+	0.959062i	$0.591394\pi$
$332$	−10.4011	−0.570835
$333$	0	0
$334$	−11.4356	−0.625727
$335$	−3.60188	−0.196792
$336$	0	0
$337$	−18.5574	−1.01089	−0.505443	−	0.862860i	$-0.668671\pi$
$337$	−18.5574	−1.01089	−0.505443	+	0.862860i	$0.668671\pi$
$338$	−52.1678	−2.83756
$339$	0	0
$340$	−9.53807	−0.517275
$341$	12.5041	0.677136
$342$	0	0
$343$	−14.0540	−0.758845
$344$	0.0552391	0.00297829
$345$	0	0
$346$	31.1086	1.67241
$347$	−3.26527	−0.175289	−0.0876444	−	0.996152i	$-0.527934\pi$
$347$	−3.26527	−0.175289	−0.0876444	+	0.996152i	$0.527934\pi$
$348$	0	0
$349$	−19.9316	−1.06692	−0.533458	−	0.845827i	$-0.679108\pi$
$349$	−19.9316	−1.06692	−0.533458	+	0.845827i	$0.679108\pi$
$350$	−2.46435	−0.131725
$351$	0	0
$352$	24.8042	1.32207
$353$	11.1973	0.595970	0.297985	−	0.954571i	$-0.403685\pi$
$353$	11.1973	0.595970	0.297985	+	0.954571i	$0.403685\pi$
$354$	0	0
$355$	3.36845	0.178779
$356$	3.03225	0.160709
$357$	0	0
$358$	−0.805163	−0.0425542
$359$	−0.488764	−0.0257960	−0.0128980	−	0.999917i	$-0.504106\pi$
$359$	−0.488764	−0.0257960	−0.0128980	+	0.999917i	$0.504106\pi$
$360$	0	0
$361$	−18.9687	−0.998353
$362$	14.5043	0.762331
$363$	0	0
$364$	−20.0573	−1.05129
$365$	6.82666	0.357324
$366$	0	0
$367$	−22.2854	−1.16329	−0.581643	−	0.813444i	$-0.697590\pi$
$367$	−22.2854	−1.16329	−0.581643	+	0.813444i	$0.697590\pi$
$368$	−0.703495	−0.0366722
$369$	0	0
$370$	25.8242	1.34254
$371$	−7.81204	−0.405581
$372$	0	0
$373$	16.6035	0.859699	0.429849	−	0.902901i	$-0.358567\pi$
$373$	16.6035	0.859699	0.429849	+	0.902901i	$0.358567\pi$
$374$	26.5885	1.37486
$375$	0	0
$376$	−5.17303	−0.266779
$377$	−32.6667	−1.68242
$378$	0	0
$379$	−3.53568	−0.181616	−0.0908079	−	0.995868i	$-0.528945\pi$
$379$	−3.53568	−0.181616	−0.0908079	+	0.995868i	$0.528945\pi$
$380$	0.536430	0.0275183
$381$	0	0
$382$	−43.4866	−2.22497
$383$	23.6074	1.20628	0.603141	−	0.797635i	$-0.293916\pi$
$383$	23.6074	1.20628	0.603141	+	0.797635i	$0.293916\pi$
$384$	0	0
$385$	4.13942	0.210964
$386$	−27.2444	−1.38670
$387$	0	0
$388$	−39.3414	−1.99726
$389$	−31.5276	−1.59851	−0.799257	−	0.600989i	$-0.794773\pi$
$389$	−31.5276	−1.59851	−0.799257	+	0.600989i	$0.794773\pi$
$390$	0	0
$391$	−2.54369	−0.128640
$392$	13.4148	0.677550
$393$	0	0
$394$	24.9049	1.25469
$395$	6.52986	0.328553
$396$	0	0
$397$	−7.59785	−0.381325	−0.190663	−	0.981656i	$-0.561064\pi$
$397$	−7.59785	−0.381325	−0.190663	+	0.981656i	$0.561064\pi$
$398$	−26.3901	−1.32282
$399$	0	0
$400$	−0.869946	−0.0434973
$401$	25.3882	1.26783	0.633914	−	0.773404i	$-0.281447\pi$
$401$	25.3882	1.26783	0.633914	+	0.773404i	$0.281447\pi$
$402$	0	0
$403$	−19.9812	−0.995333
$404$	−40.2867	−2.00434
$405$	0	0
$406$	−13.3697	−0.663528
$407$	−43.3774	−2.15014
$408$	0	0
$409$	−35.0702	−1.73411	−0.867055	−	0.498212i	$-0.833990\pi$
$409$	−35.0702	−1.73411	−0.867055	+	0.498212i	$0.833990\pi$
$410$	−2.92785	−0.144596
$411$	0	0
$412$	15.0264	0.740297
$413$	−10.6926	−0.526146
$414$	0	0
$415$	−3.43016	−0.168380
$416$	−39.6364	−1.94333
$417$	0	0
$418$	−1.49536	−0.0731405
$419$	36.5977	1.78792	0.893958	−	0.448151i	$-0.147917\pi$
$419$	36.5977	1.78792	0.893958	+	0.448151i	$0.147917\pi$
$420$	0	0
$421$	28.2058	1.37467	0.687334	−	0.726342i	$-0.258781\pi$
$421$	28.2058	1.37467	0.687334	+	0.726342i	$0.258781\pi$
$422$	−58.7254	−2.85871
$423$	0	0
$424$	16.4669	0.799701
$425$	−3.14554	−0.152581
$426$	0	0
$427$	−15.9345	−0.771127
$428$	13.5016	0.652625
$429$	0	0
$430$	0.0535131	0.00258063
$431$	−33.1065	−1.59468	−0.797341	−	0.603529i	$-0.793761\pi$
$431$	−33.1065	−1.59468	−0.797341	+	0.603529i	$0.793761\pi$
$432$	0	0
$433$	16.1496	0.776100	0.388050	−	0.921638i	$-0.373149\pi$
$433$	16.1496	0.776100	0.388050	+	0.921638i	$0.373149\pi$
$434$	−8.17783	−0.392548
$435$	0	0
$436$	−17.2018	−0.823815
$437$	0.143059	0.00684346
$438$	0	0
$439$	−37.6294	−1.79595	−0.897977	−	0.440043i	$-0.854963\pi$
$439$	−37.6294	−1.79595	−0.897977	+	0.440043i	$0.854963\pi$
$440$	−8.72540	−0.415967
$441$	0	0
$442$	−42.4876	−2.02093
$443$	5.81502	0.276280	0.138140	−	0.990413i	$-0.455888\pi$
$443$	5.81502	0.276280	0.138140	+	0.990413i	$0.455888\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	21.3161	1.00935
$447$	0	0
$448$	−14.3109	−0.676126
$449$	−34.4209	−1.62442	−0.812212	−	0.583363i	$-0.801737\pi$
$449$	−34.4209	−1.62442	−0.812212	+	0.583363i	$0.801737\pi$
$450$	0	0
$451$	4.91797	0.231578
$452$	16.1238	0.758398
$453$	0	0
$454$	53.9261	2.53088
$455$	−6.61465	−0.310100
$456$	0	0
$457$	−27.7343	−1.29736	−0.648679	−	0.761062i	$-0.724678\pi$
$457$	−27.7343	−1.29736	−0.648679	+	0.761062i	$0.724678\pi$
$458$	−32.0638	−1.49824
$459$	0	0
$460$	2.45208	0.114329
$461$	−13.2563	−0.617409	−0.308705	−	0.951158i	$-0.599895\pi$
$461$	−13.2563	−0.617409	−0.308705	+	0.951158i	$0.599895\pi$
$462$	0	0
$463$	20.6779	0.960983	0.480491	−	0.876999i	$-0.340458\pi$
$463$	20.6779	0.960983	0.480491	+	0.876999i	$0.340458\pi$
$464$	−4.71967	−0.219105
$465$	0	0
$466$	40.4946	1.87587
$467$	−8.67046	−0.401221	−0.200611	−	0.979671i	$-0.564293\pi$
$467$	−8.67046	−0.401221	−0.200611	+	0.979671i	$0.564293\pi$
$468$	0	0
$469$	−3.95687	−0.182711
$470$	−5.01140	−0.231159
$471$	0	0
$472$	22.5386	1.03743
$473$	−0.0898869	−0.00413300
$474$	0	0
$475$	0.176908	0.00811709
$476$	−10.4781	−0.480263
$477$	0	0
$478$	31.9869	1.46305
$479$	−18.5707	−0.848518	−0.424259	−	0.905541i	$-0.639465\pi$
$479$	−18.5707	−0.848518	−0.424259	+	0.905541i	$0.639465\pi$
$480$	0	0
$481$	69.3157	3.16052
$482$	8.55427	0.389636
$483$	0	0
$484$	9.69793	0.440815
$485$	−12.9743	−0.589134
$486$	0	0
$487$	19.9652	0.904710	0.452355	−	0.891838i	$-0.350584\pi$
$487$	19.9652	0.904710	0.452355	+	0.891838i	$0.350584\pi$
$488$	33.5881	1.52046
$489$	0	0
$490$	12.9957	0.587084
$491$	3.79090	0.171081	0.0855405	−	0.996335i	$-0.472738\pi$
$491$	3.79090	0.171081	0.0855405	+	0.996335i	$0.472738\pi$
$492$	0	0
$493$	−17.0653	−0.768584
$494$	2.38954	0.107510
$495$	0	0
$496$	−2.88687	−0.129624
$497$	3.70043	0.165987
$498$	0	0
$499$	−20.6849	−0.925985	−0.462993	−	0.886362i	$-0.653224\pi$
$499$	−20.6849	−0.925985	−0.462993	+	0.886362i	$0.653224\pi$
$500$	3.03225	0.135606
$501$	0	0
$502$	−47.7072	−2.12928
$503$	−15.5947	−0.695335	−0.347667	−	0.937618i	$-0.613026\pi$
$503$	−15.5947	−0.695335	−0.347667	+	0.937618i	$0.613026\pi$
$504$	0	0
$505$	−13.2861	−0.591222
$506$	−6.83546	−0.303873
$507$	0	0
$508$	45.6189	2.02401
$509$	7.17747	0.318136	0.159068	−	0.987268i	$-0.449151\pi$
$509$	7.17747	0.318136	0.159068	+	0.987268i	$0.449151\pi$
$510$	0	0
$511$	7.49945	0.331756
$512$	−9.75558	−0.431140
$513$	0	0
$514$	−34.1110	−1.50457
$515$	4.95552	0.218366
$516$	0	0
$517$	8.41774	0.370212
$518$	28.3693	1.24648
$519$	0	0
$520$	13.9429	0.611437
$521$	−11.7594	−0.515190	−0.257595	−	0.966253i	$-0.582930\pi$
$521$	−11.7594	−0.515190	−0.257595	+	0.966253i	$0.582930\pi$
$522$	0	0
$523$	3.26789	0.142895	0.0714474	−	0.997444i	$-0.477238\pi$
$523$	3.26789	0.142895	0.0714474	+	0.997444i	$0.477238\pi$
$524$	19.3691	0.846143
$525$	0	0
$526$	43.1472	1.88131
$527$	−10.4383	−0.454700
$528$	0	0
$529$	−22.3461	−0.971568
$530$	15.9523	0.692925
$531$	0	0
$532$	0.589297	0.0255493
$533$	−7.85876	−0.340400
$534$	0	0
$535$	4.45267	0.192506
$536$	8.34060	0.360259
$537$	0	0
$538$	50.1634	2.16270
$539$	−21.8290	−0.940243
$540$	0	0
$541$	−21.7791	−0.936355	−0.468178	−	0.883634i	$-0.655089\pi$
$541$	−21.7791	−0.936355	−0.468178	+	0.883634i	$0.655089\pi$
$542$	−14.7253	−0.632505
$543$	0	0
$544$	−20.7064	−0.887778
$545$	−5.67293	−0.243002
$546$	0	0
$547$	−4.65276	−0.198938	−0.0994688	−	0.995041i	$-0.531714\pi$
$547$	−4.65276	−0.198938	−0.0994688	+	0.995041i	$0.531714\pi$
$548$	−6.00702	−0.256607
$549$	0	0
$550$	−8.45276	−0.360427
$551$	0.959769	0.0408875
$552$	0	0
$553$	7.17341	0.305044
$554$	−26.0144	−1.10525
$555$	0	0
$556$	−55.8499	−2.36856
$557$	−34.1381	−1.44648	−0.723239	−	0.690598i	$-0.757348\pi$
$557$	−34.1381	−1.44648	−0.723239	+	0.690598i	$0.757348\pi$
$558$	0	0
$559$	0.143636	0.00607517
$560$	−0.955683	−0.0403850
$561$	0	0
$562$	−9.03999	−0.381329
$563$	−13.5274	−0.570113	−0.285056	−	0.958511i	$-0.592012\pi$
$563$	−13.5274	−0.570113	−0.285056	+	0.958511i	$0.592012\pi$
$564$	0	0
$565$	5.31742	0.223706
$566$	−3.06960	−0.129025
$567$	0	0
$568$	−7.80006	−0.327283
$569$	−6.61413	−0.277279	−0.138639	−	0.990343i	$-0.544273\pi$
$569$	−6.61413	−0.277279	−0.138639	+	0.990343i	$0.544273\pi$
$570$	0	0
$571$	6.84541	0.286471	0.143236	−	0.989689i	$-0.454249\pi$
$571$	6.84541	0.286471	0.143236	+	0.989689i	$0.454249\pi$
$572$	−68.7969	−2.87654
$573$	0	0
$574$	−3.21641	−0.134250
$575$	0.808666	0.0337237
$576$	0	0
$577$	22.8018	0.949252	0.474626	−	0.880188i	$-0.342583\pi$
$577$	22.8018	0.949252	0.474626	+	0.880188i	$0.342583\pi$
$578$	15.9397	0.663005
$579$	0	0
$580$	16.4507	0.683079
$581$	−3.76821	−0.156332
$582$	0	0
$583$	−26.7954	−1.10975
$584$	−15.8080	−0.654138
$585$	0	0
$586$	4.52531	0.186939
$587$	16.4123	0.677409	0.338704	−	0.940893i	$-0.390011\pi$
$587$	16.4123	0.677409	0.338704	+	0.940893i	$0.390011\pi$
$588$	0	0
$589$	0.587060	0.0241894
$590$	21.8344	0.898908
$591$	0	0
$592$	10.0147	0.411602
$593$	−9.53659	−0.391621	−0.195810	−	0.980642i	$-0.562734\pi$
$593$	−9.53659	−0.391621	−0.195810	+	0.980642i	$0.562734\pi$
$594$	0	0
$595$	−3.45555	−0.141664
$596$	−3.93533	−0.161198
$597$	0	0
$598$	10.9228	0.446668
$599$	36.4279	1.48840	0.744201	−	0.667955i	$-0.232830\pi$
$599$	36.4279	1.48840	0.744201	+	0.667955i	$0.232830\pi$
$600$	0	0
$601$	9.62716	0.392700	0.196350	−	0.980534i	$-0.437091\pi$
$601$	9.62716	0.392700	0.196350	+	0.980534i	$0.437091\pi$
$602$	0.0587870	0.00239598
$603$	0	0
$604$	52.5204	2.13702
$605$	3.19826	0.130028
$606$	0	0
$607$	14.4400	0.586104	0.293052	−	0.956097i	$-0.405329\pi$
$607$	14.4400	0.586104	0.293052	+	0.956097i	$0.405329\pi$
$608$	1.16454	0.0472285
$609$	0	0
$610$	32.5387	1.31745
$611$	−13.4513	−0.544180
$612$	0	0
$613$	32.6904	1.32035	0.660176	−	0.751111i	$-0.270482\pi$
$613$	32.6904	1.32035	0.660176	+	0.751111i	$0.270482\pi$
$614$	37.6464	1.51929
$615$	0	0
$616$	−9.58532	−0.386204
$617$	−13.4787	−0.542631	−0.271315	−	0.962491i	$-0.587459\pi$
$617$	−13.4787	−0.542631	−0.271315	+	0.962491i	$0.587459\pi$
$618$	0	0
$619$	−23.0241	−0.925417	−0.462708	−	0.886511i	$-0.653122\pi$
$619$	−23.0241	−0.925417	−0.462708	+	0.886511i	$0.653122\pi$
$620$	10.0624	0.404115
$621$	0	0
$622$	−31.5841	−1.26641
$623$	1.09855	0.0440126
$624$	0	0
$625$	1.00000	0.0400000
$626$	39.3159	1.57138
$627$	0	0
$628$	−45.7404	−1.82524
$629$	36.2110	1.44383
$630$	0	0
$631$	19.8492	0.790183	0.395091	−	0.918642i	$-0.370713\pi$
$631$	19.8492	0.790183	0.395091	+	0.918642i	$0.370713\pi$
$632$	−15.1207	−0.601469
$633$	0	0
$634$	−55.4719	−2.20307
$635$	15.0446	0.597025
$636$	0	0
$637$	34.8821	1.38208
$638$	−45.8583	−1.81555
$639$	0	0
$640$	16.0576	0.634731
$641$	10.7996	0.426559	0.213279	−	0.976991i	$-0.431586\pi$
$641$	10.7996	0.426559	0.213279	+	0.976991i	$0.431586\pi$
$642$	0	0
$643$	−11.8100	−0.465742	−0.232871	−	0.972508i	$-0.574812\pi$
$643$	−11.8100	−0.465742	−0.232871	+	0.972508i	$0.574812\pi$
$644$	2.69374	0.106148
$645$	0	0
$646$	1.24831	0.0491143
$647$	−28.0761	−1.10379	−0.551893	−	0.833915i	$-0.686094\pi$
$647$	−28.0761	−1.10379	−0.551893	+	0.833915i	$0.686094\pi$
$648$	0	0
$649$	−36.6756	−1.43964
$650$	13.5072	0.529798
$651$	0	0
$652$	−63.4327	−2.48422
$653$	25.0500	0.980283	0.490141	−	0.871643i	$-0.336945\pi$
$653$	25.0500	0.980283	0.490141	+	0.871643i	$0.336945\pi$
$654$	0	0
$655$	6.38769	0.249588
$656$	−1.13543	−0.0443311
$657$	0	0
$658$	−5.50529	−0.214619
$659$	15.8683	0.618143	0.309071	−	0.951039i	$-0.399982\pi$
$659$	15.8683	0.618143	0.309071	+	0.951039i	$0.399982\pi$
$660$	0	0
$661$	0.800529	0.0311370	0.0155685	−	0.999879i	$-0.495044\pi$
$661$	0.800529	0.0311370	0.0155685	+	0.999879i	$0.495044\pi$
$662$	23.1159	0.898426
$663$	0	0
$664$	7.94295	0.308246
$665$	0.194343	0.00753630
$666$	0	0
$667$	4.38721	0.169873
$668$	15.4576	0.598073
$669$	0	0
$670$	8.07999	0.312157
$671$	−54.6557	−2.10996
$672$	0	0
$673$	−17.2977	−0.666776	−0.333388	−	0.942790i	$-0.608192\pi$
$673$	−17.2977	−0.666776	−0.333388	+	0.942790i	$0.608192\pi$
$674$	41.6292	1.60350
$675$	0	0
$676$	70.5159	2.71215
$677$	−16.1923	−0.622321	−0.311161	−	0.950357i	$-0.600718\pi$
$677$	−16.1923	−0.622321	−0.311161	+	0.950357i	$0.600718\pi$
$678$	0	0
$679$	−14.2530	−0.546980
$680$	7.28388	0.279324
$681$	0	0
$682$	−28.0501	−1.07409
$683$	−35.6468	−1.36399	−0.681993	−	0.731359i	$-0.738886\pi$
$683$	−35.6468	−1.36399	−0.681993	+	0.731359i	$0.738886\pi$
$684$	0	0
$685$	−1.98104	−0.0756917
$686$	31.5269	1.20370
$687$	0	0
$688$	0.0207525	0.000791183 0
$689$	42.8182	1.63124
$690$	0	0
$691$	39.8122	1.51453	0.757263	−	0.653110i	$-0.226536\pi$
$691$	39.8122	1.51453	0.757263	+	0.653110i	$0.226536\pi$
$692$	−42.0499	−1.59850
$693$	0	0
$694$	7.32487	0.278048
$695$	−18.4186	−0.698658
$696$	0	0
$697$	−4.10547	−0.155506
$698$	44.7120	1.69237
$699$	0	0
$700$	3.33109	0.125904
$701$	−5.95704	−0.224994	−0.112497	−	0.993652i	$-0.535885\pi$
$701$	−5.95704	−0.224994	−0.112497	+	0.993652i	$0.535885\pi$
$702$	0	0
$703$	−2.03654	−0.0768096
$704$	−49.0866	−1.85002
$705$	0	0
$706$	−25.1185	−0.945345
$707$	−14.5955	−0.548919
$708$	0	0
$709$	−27.4036	−1.02916	−0.514581	−	0.857442i	$-0.672053\pi$
$709$	−27.4036	−1.02916	−0.514581	+	0.857442i	$0.672053\pi$
$710$	−7.55634	−0.283584
$711$	0	0
$712$	−2.31562	−0.0867816
$713$	2.68352	0.100498
$714$	0	0
$715$	−22.6884	−0.848497
$716$	1.08835	0.0406735
$717$	0	0
$718$	1.09643	0.0409184
$719$	10.6003	0.395325	0.197662	−	0.980270i	$-0.436665\pi$
$719$	10.6003	0.395325	0.197662	+	0.980270i	$0.436665\pi$
$720$	0	0
$721$	5.44391	0.202742
$722$	42.5519	1.58362
$723$	0	0
$724$	−19.6057	−0.728640
$725$	5.42525	0.201489
$726$	0	0
$727$	48.7224	1.80701	0.903507	−	0.428573i	$-0.140983\pi$
$727$	48.7224	1.80701	0.903507	+	0.428573i	$0.140983\pi$
$728$	15.3170	0.567687
$729$	0	0
$730$	−15.3140	−0.566798
$731$	0.0750367	0.00277533
$732$	0	0
$733$	−19.0937	−0.705240	−0.352620	−	0.935767i	$-0.614709\pi$
$733$	−19.0937	−0.705240	−0.352620	+	0.935767i	$0.614709\pi$
$734$	49.9920	1.84524
$735$	0	0
$736$	5.32326	0.196218
$737$	−13.5721	−0.499935
$738$	0	0
$739$	−44.7827	−1.64736	−0.823679	−	0.567056i	$-0.808082\pi$
$739$	−44.7827	−1.64736	−0.823679	+	0.567056i	$0.808082\pi$
$740$	−34.9069	−1.28320
$741$	0	0
$742$	17.5245	0.643345
$743$	−18.4725	−0.677691	−0.338845	−	0.940842i	$-0.610036\pi$
$743$	−18.4725	−0.677691	−0.338845	+	0.940842i	$0.610036\pi$
$744$	0	0
$745$	−1.29782	−0.0475486
$746$	−37.2462	−1.36368
$747$	0	0
$748$	−35.9400	−1.31410
$749$	4.89150	0.178731
$750$	0	0
$751$	32.8072	1.19715	0.598576	−	0.801066i	$-0.295734\pi$
$751$	32.8072	1.19715	0.598576	+	0.801066i	$0.295734\pi$
$752$	−1.94344	−0.0708698
$753$	0	0
$754$	73.2801	2.66870
$755$	17.3206	0.630360
$756$	0	0
$757$	23.3854	0.849958	0.424979	−	0.905203i	$-0.360281\pi$
$757$	23.3854	0.849958	0.424979	+	0.905203i	$0.360281\pi$
$758$	7.93148	0.288084
$759$	0	0
$760$	−0.409652	−0.0148596
$761$	−26.6731	−0.966900	−0.483450	−	0.875372i	$-0.660616\pi$
$761$	−26.6731	−0.966900	−0.483450	+	0.875372i	$0.660616\pi$
$762$	0	0
$763$	−6.23202	−0.225614
$764$	58.7814	2.12663
$765$	0	0
$766$	−52.9577	−1.91344
$767$	58.6065	2.11616
$768$	0	0
$769$	4.94308	0.178252	0.0891261	−	0.996020i	$-0.471593\pi$
$769$	4.94308	0.178252	0.0891261	+	0.996020i	$0.471593\pi$
$770$	−9.28582	−0.334638
$771$	0	0
$772$	36.8266	1.32542
$773$	49.9641	1.79709	0.898543	−	0.438886i	$-0.144627\pi$
$773$	49.9641	1.79709	0.898543	+	0.438886i	$0.144627\pi$
$774$	0	0
$775$	3.31845	0.119202
$776$	30.0436	1.07850
$777$	0	0
$778$	70.7249	2.53561
$779$	0.230896	0.00827269
$780$	0	0
$781$	12.6925	0.454174
$782$	5.70618	0.204053
$783$	0	0
$784$	5.03975	0.179991
$785$	−15.0846	−0.538393
$786$	0	0
$787$	−31.7326	−1.13115	−0.565573	−	0.824698i	$-0.691345\pi$
$787$	−31.7326	−1.13115	−0.565573	+	0.824698i	$0.691345\pi$
$788$	−33.6643	−1.19924
$789$	0	0
$790$	−14.6482	−0.521161
$791$	5.84148	0.207699
$792$	0	0
$793$	87.3381	3.10147
$794$	17.0440	0.604870
$795$	0	0
$796$	35.6718	1.26435
$797$	52.3138	1.85305	0.926525	−	0.376233i	$-0.122781\pi$
$797$	52.3138	1.85305	0.926525	+	0.376233i	$0.122781\pi$
$798$	0	0
$799$	−7.02705	−0.248599
$800$	6.58277	0.232736
$801$	0	0
$802$	−56.9526	−2.01107
$803$	25.7232	0.907753
$804$	0	0
$805$	0.888363	0.0313107
$806$	44.8231	1.57883
$807$	0	0
$808$	30.7655	1.08233
$809$	−44.2277	−1.55496	−0.777482	−	0.628906i	$-0.783503\pi$
$809$	−44.2277	−1.55496	−0.777482	+	0.628906i	$0.783503\pi$
$810$	0	0
$811$	−37.3992	−1.31326	−0.656632	−	0.754211i	$-0.728019\pi$
$811$	−37.3992	−1.31326	−0.656632	+	0.754211i	$0.728019\pi$
$812$	18.0720	0.634203
$813$	0	0
$814$	97.3071	3.41061
$815$	−20.9193	−0.732772
$816$	0	0
$817$	−0.00422013	−0.000147644 0
$818$	78.6719	2.75070
$819$	0	0
$820$	3.95762	0.138206
$821$	28.8066	1.00536	0.502678	−	0.864474i	$-0.332348\pi$
$821$	28.8066	1.00536	0.502678	+	0.864474i	$0.332348\pi$
$822$	0	0
$823$	6.78866	0.236638	0.118319	−	0.992976i	$-0.462249\pi$
$823$	6.78866	0.236638	0.118319	+	0.992976i	$0.462249\pi$
$824$	−11.4751	−0.399754
$825$	0	0
$826$	23.9863	0.834590
$827$	−1.17992	−0.0410300	−0.0205150	−	0.999790i	$-0.506531\pi$
$827$	−1.17992	−0.0410300	−0.0205150	+	0.999790i	$0.506531\pi$
$828$	0	0
$829$	10.0987	0.350742	0.175371	−	0.984502i	$-0.443888\pi$
$829$	10.0987	0.350742	0.175371	+	0.984502i	$0.443888\pi$
$830$	7.69476	0.267089
$831$	0	0
$832$	78.4387	2.71937
$833$	18.2227	0.631378
$834$	0	0
$835$	5.09773	0.176414
$836$	2.02130	0.0699081
$837$	0	0
$838$	−82.0985	−2.83605
$839$	−27.7796	−0.959060	−0.479530	−	0.877526i	$-0.659193\pi$
$839$	−27.7796	−0.959060	−0.479530	+	0.877526i	$0.659193\pi$
$840$	0	0
$841$	0.433285	0.0149408
$842$	−63.2732	−2.18054
$843$	0	0
$844$	79.3798	2.73237
$845$	23.2553	0.800006
$846$	0	0
$847$	3.51346	0.120724
$848$	6.18636	0.212441
$849$	0	0
$850$	7.05629	0.242029
$851$	−9.30925	−0.319117
$852$	0	0
$853$	−16.6953	−0.571638	−0.285819	−	0.958284i	$-0.592266\pi$
$853$	−16.6953	−0.571638	−0.285819	+	0.958284i	$0.592266\pi$
$854$	35.7455	1.22318
$855$	0	0
$856$	−10.3107	−0.352412
$857$	11.4100	0.389757	0.194878	−	0.980827i	$-0.437569\pi$
$857$	11.4100	0.389757	0.194878	+	0.980827i	$0.437569\pi$
$858$	0	0
$859$	−5.90082	−0.201334	−0.100667	−	0.994920i	$-0.532098\pi$
$859$	−5.90082	−0.201334	−0.100667	+	0.994920i	$0.532098\pi$
$860$	−0.0723343	−0.00246658
$861$	0	0
$862$	74.2667	2.52953
$863$	−11.1157	−0.378382	−0.189191	−	0.981940i	$-0.560587\pi$
$863$	−11.1157	−0.378382	−0.189191	+	0.981940i	$0.560587\pi$
$864$	0	0
$865$	−13.8676	−0.471511
$866$	−36.2279	−1.23107
$867$	0	0
$868$	11.0541	0.375200
$869$	24.6049	0.834664
$870$	0	0
$871$	21.6878	0.734863
$872$	13.1364	0.444853
$873$	0	0
$874$	−0.320921	−0.0108553
$875$	1.09855	0.0371379
$876$	0	0
$877$	−50.9437	−1.72025	−0.860123	−	0.510086i	$-0.829614\pi$
$877$	−50.9437	−1.72025	−0.860123	+	0.510086i	$0.829614\pi$
$878$	84.4128	2.84880
$879$	0	0
$880$	−3.27801	−0.110502
$881$	2.33760	0.0787557	0.0393778	−	0.999224i	$-0.487462\pi$
$881$	2.33760	0.0787557	0.0393778	+	0.999224i	$0.487462\pi$
$882$	0	0
$883$	−9.59655	−0.322950	−0.161475	−	0.986877i	$-0.551625\pi$
$883$	−9.59655	−0.322950	−0.161475	+	0.986877i	$0.551625\pi$
$884$	57.4310	1.93161
$885$	0	0
$886$	−13.0446	−0.438244
$887$	51.9949	1.74582	0.872908	−	0.487884i	$-0.162231\pi$
$887$	51.9949	1.74582	0.872908	+	0.487884i	$0.162231\pi$
$888$	0	0
$889$	16.5273	0.554307
$890$	−2.24327	−0.0751945
$891$	0	0
$892$	−28.8132	−0.964739
$893$	0.395207	0.0132251
$894$	0	0
$895$	0.358924	0.0119975
$896$	17.6401	0.589315
$897$	0	0
$898$	77.2153	2.57671
$899$	18.0034	0.600447
$900$	0	0
$901$	22.3686	0.745205
$902$	−11.0323	−0.367336
$903$	0	0
$904$	−12.3131	−0.409529
$905$	−6.46572	−0.214928
$906$	0	0
$907$	−23.7776	−0.789521	−0.394761	−	0.918784i	$-0.629173\pi$
$907$	−23.7776	−0.789521	−0.394761	+	0.918784i	$0.629173\pi$
$908$	−72.8925	−2.41902
$909$	0	0
$910$	14.8384	0.491890
$911$	45.1846	1.49703	0.748517	−	0.663116i	$-0.230766\pi$
$911$	45.1846	1.49703	0.748517	+	0.663116i	$0.230766\pi$
$912$	0	0
$913$	−12.9250	−0.427756
$914$	62.2156	2.05791
$915$	0	0
$916$	43.3410	1.43203
$917$	7.01722	0.231729
$918$	0	0
$919$	14.9631	0.493588	0.246794	−	0.969068i	$-0.420623\pi$
$919$	14.9631	0.493588	0.246794	+	0.969068i	$0.420623\pi$
$920$	−1.87256	−0.0617366
$921$	0	0
$922$	29.7375	0.979353
$923$	−20.2822	−0.667598
$924$	0	0
$925$	−11.5119	−0.378508
$926$	−46.3861	−1.52434
$927$	0	0
$928$	35.7131	1.17234
$929$	5.40594	0.177363	0.0886815	−	0.996060i	$-0.471735\pi$
$929$	5.40594	0.177363	0.0886815	+	0.996060i	$0.471735\pi$
$930$	0	0
$931$	−1.02486	−0.0335884
$932$	−54.7370	−1.79297
$933$	0	0
$934$	19.4502	0.636429
$935$	−11.8526	−0.387621
$936$	0	0
$937$	−17.9222	−0.585494	−0.292747	−	0.956190i	$-0.594569\pi$
$937$	−17.9222	−0.585494	−0.292747	+	0.956190i	$0.594569\pi$
$938$	8.87631	0.289822
$939$	0	0
$940$	6.77397	0.220943
$941$	−38.7567	−1.26343	−0.631716	−	0.775200i	$-0.717649\pi$
$941$	−38.7567	−1.26343	−0.631716	+	0.775200i	$0.717649\pi$
$942$	0	0
$943$	1.05545	0.0343701
$944$	8.46745	0.275592
$945$	0	0
$946$	0.201640	0.00655590
$947$	−25.0905	−0.815332	−0.407666	−	0.913131i	$-0.633657\pi$
$947$	−25.0905	−0.815332	−0.407666	+	0.913131i	$0.633657\pi$
$948$	0	0
$949$	−41.1049	−1.33432
$950$	−0.396852	−0.0128756
$951$	0	0
$952$	8.00174	0.259338
$953$	34.9250	1.13133	0.565665	−	0.824635i	$-0.308619\pi$
$953$	34.9250	1.13133	0.565665	+	0.824635i	$0.308619\pi$
$954$	0	0
$955$	19.3854	0.627296
$956$	−43.2371	−1.39839
$957$	0	0
$958$	41.6591	1.34594
$959$	−2.17628	−0.0702758
$960$	0	0
$961$	−19.9879	−0.644771
$962$	−155.494	−5.01332
$963$	0	0
$964$	−11.5629	−0.372416
$965$	12.1449	0.390960
$966$	0	0
$967$	17.8447	0.573845	0.286923	−	0.957954i	$-0.407368\pi$
$967$	17.8447	0.573845	0.286923	+	0.957954i	$0.407368\pi$
$968$	−7.40596	−0.238036
$969$	0	0
$970$	29.1049	0.934502
$971$	16.2574	0.521726	0.260863	−	0.965376i	$-0.415993\pi$
$971$	16.2574	0.521726	0.260863	+	0.965376i	$0.415993\pi$
$972$	0	0
$973$	−20.2339	−0.648668
$974$	−44.7873	−1.43508
$975$	0	0
$976$	12.6186	0.403911
$977$	43.1505	1.38051	0.690254	−	0.723567i	$-0.257499\pi$
$977$	43.1505	1.38051	0.690254	+	0.723567i	$0.257499\pi$
$978$	0	0
$979$	3.76806	0.120428
$980$	−17.5664	−0.561138
$981$	0	0
$982$	−8.50401	−0.271374
$983$	−41.1549	−1.31264	−0.656318	−	0.754484i	$-0.727887\pi$
$983$	−41.1549	−1.31264	−0.656318	+	0.754484i	$0.727887\pi$
$984$	0	0
$985$	−11.1021	−0.353741
$986$	38.2821	1.21915
$987$	0	0
$988$	−3.22997	−0.102759
$989$	−0.0192907	−0.000613408 0
$990$	0	0
$991$	−14.7645	−0.469008	−0.234504	−	0.972115i	$-0.575347\pi$
$991$	−14.7645	−0.469008	−0.234504	+	0.972115i	$0.575347\pi$
$992$	21.8446	0.693566
$993$	0	0
$994$	−8.30105	−0.263293
$995$	11.7641	0.372948
$996$	0	0
$997$	−12.3613	−0.391486	−0.195743	−	0.980655i	$-0.562712\pi$
$997$	−12.3613	−0.391486	−0.195743	+	0.980655i	$0.562712\pi$
$998$	46.4019	1.46883
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.u.1.2	✓	12
3.2	odd	2		4005.2.a.v.1.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.2	✓	12	1.1	even	1	trivial
4005.2.a.v.1.11	yes	12	3.2	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)

\(f(q)\)	\(=\)	\(q-2.24327 q^{2} +3.03225 q^{4} +1.00000 q^{5} +1.09855 q^{7} -2.31562 q^{8} +O(q^{10})\)	\(q-2.24327 q^{2} +3.03225 q^{4} +1.00000 q^{5} +1.09855 q^{7} -2.31562 q^{8} -2.24327 q^{10} +3.76806 q^{11} -6.02124 q^{13} -2.46435 q^{14} -0.869946 q^{16} -3.14554 q^{17} +0.176908 q^{19} +3.03225 q^{20} -8.45276 q^{22} +0.808666 q^{23} +1.00000 q^{25} +13.5072 q^{26} +3.33109 q^{28} +5.42525 q^{29} +3.31845 q^{31} +6.58277 q^{32} +7.05629 q^{34} +1.09855 q^{35} -11.5119 q^{37} -0.396852 q^{38} -2.31562 q^{40} +1.30517 q^{41} -0.0238550 q^{43} +11.4257 q^{44} -1.81405 q^{46} +2.23397 q^{47} -5.79318 q^{49} -2.24327 q^{50} -18.2579 q^{52} -7.11120 q^{53} +3.76806 q^{55} -2.54384 q^{56} -12.1703 q^{58} -9.73330 q^{59} -14.5050 q^{61} -7.44417 q^{62} -13.0270 q^{64} -6.02124 q^{65} -3.60188 q^{67} -9.53807 q^{68} -2.46435 q^{70} +3.36845 q^{71} +6.82666 q^{73} +25.8242 q^{74} +0.536430 q^{76} +4.13942 q^{77} +6.52986 q^{79} -0.869946 q^{80} -2.92785 q^{82} -3.43016 q^{83} -3.14554 q^{85} +0.0535131 q^{86} -8.72540 q^{88} +1.00000 q^{89} -6.61465 q^{91} +2.45208 q^{92} -5.01140 q^{94} +0.176908 q^{95} -12.9743 q^{97} +12.9957 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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