LMFDB - Embedded newform 4005.2.a.s.1.4

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.4
Root			$0.970031$ 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)

$f(q)$	$=$	$q-0.970031 q^{2} -1.05904 q^{4} -1.00000 q^{5} -4.84165 q^{7} +2.96736 q^{8} +O(q^{10})$	\(q-0.970031 q^{2} -1.05904 q^{4} -1.00000 q^{5} -4.84165 q^{7} +2.96736 q^{8} +0.970031 q^{10} -1.26097 q^{11} -0.957447 q^{13} +4.69655 q^{14} -0.760356 q^{16} -2.50226 q^{17} +7.26642 q^{19} +1.05904 q^{20} +1.22318 q^{22} +2.94416 q^{23} +1.00000 q^{25} +0.928753 q^{26} +5.12750 q^{28} -2.51044 q^{29} -1.30054 q^{31} -5.19716 q^{32} +2.42727 q^{34} +4.84165 q^{35} +10.6521 q^{37} -7.04865 q^{38} -2.96736 q^{40} -10.7945 q^{41} +1.68646 q^{43} +1.33541 q^{44} -2.85593 q^{46} +7.95583 q^{47} +16.4416 q^{49} -0.970031 q^{50} +1.01397 q^{52} +10.9614 q^{53} +1.26097 q^{55} -14.3669 q^{56} +2.43520 q^{58} -5.13128 q^{59} -0.773449 q^{61} +1.26157 q^{62} +6.56212 q^{64} +0.957447 q^{65} +8.58486 q^{67} +2.64999 q^{68} -4.69655 q^{70} -3.16973 q^{71} -0.179310 q^{73} -10.3329 q^{74} -7.69543 q^{76} +6.10515 q^{77} -11.8490 q^{79} +0.760356 q^{80} +10.4710 q^{82} +0.924640 q^{83} +2.50226 q^{85} -1.63592 q^{86} -3.74174 q^{88} +1.00000 q^{89} +4.63562 q^{91} -3.11798 q^{92} -7.71741 q^{94} -7.26642 q^{95} +11.3807 q^{97} -15.9488 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * 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 $

$2$	−0.970031	−0.685916	−0.342958	−	0.939351i	$-0.611429\pi$
$2$	−0.970031	−0.685916	−0.342958	+	0.939351i	$0.611429\pi$
$3$	0	0
$3$	0	0
$4$	−1.05904	−0.529520
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.84165	−1.82997	−0.914985	−	0.403487i	$-0.867798\pi$
$7$	−4.84165	−1.82997	−0.914985	+	0.403487i	$0.867798\pi$
$8$	2.96736	1.04912
$9$	0	0
$10$	0.970031	0.306751
$11$	−1.26097	−0.380195	−0.190098	−	0.981765i	$-0.560880\pi$
$11$	−1.26097	−0.380195	−0.190098	+	0.981765i	$0.560880\pi$
$12$	0	0
$13$	−0.957447	−0.265548	−0.132774	−	0.991146i	$-0.542388\pi$
$13$	−0.957447	−0.265548	−0.132774	+	0.991146i	$0.542388\pi$
$14$	4.69655	1.25521
$15$	0	0
$16$	−0.760356	−0.190089
$17$	−2.50226	−0.606887	−0.303444	−	0.952849i	$-0.598136\pi$
$17$	−2.50226	−0.606887	−0.303444	+	0.952849i	$0.598136\pi$
$18$	0	0
$19$	7.26642	1.66703	0.833516	−	0.552496i	$-0.186325\pi$
$19$	7.26642	1.66703	0.833516	+	0.552496i	$0.186325\pi$
$20$	1.05904	0.236808
$21$	0	0
$22$	1.22318	0.260782
$23$	2.94416	0.613900	0.306950	−	0.951726i	$-0.400692\pi$
$23$	2.94416	0.613900	0.306950	+	0.951726i	$0.400692\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.928753	0.182144
$27$	0	0
$28$	5.12750	0.969006
$29$	−2.51044	−0.466176	−0.233088	−	0.972456i	$-0.574883\pi$
$29$	−2.51044	−0.466176	−0.233088	+	0.972456i	$0.574883\pi$
$30$	0	0
$31$	−1.30054	−0.233584	−0.116792	−	0.993156i	$-0.537261\pi$
$31$	−1.30054	−0.233584	−0.116792	+	0.993156i	$0.537261\pi$
$32$	−5.19716	−0.918736
$33$	0	0
$34$	2.42727	0.416274
$35$	4.84165	0.818388
$36$	0	0
$37$	10.6521	1.75120	0.875598	−	0.483040i	$-0.160468\pi$
$37$	10.6521	1.75120	0.875598	+	0.483040i	$0.160468\pi$
$38$	−7.04865	−1.14344
$39$	0	0
$40$	−2.96736	−0.469181
$41$	−10.7945	−1.68582	−0.842909	−	0.538057i	$-0.819159\pi$
$41$	−10.7945	−1.68582	−0.842909	+	0.538057i	$0.819159\pi$
$42$	0	0
$43$	1.68646	0.257183	0.128591	−	0.991698i	$-0.458954\pi$
$43$	1.68646	0.257183	0.128591	+	0.991698i	$0.458954\pi$
$44$	1.33541	0.201321
$45$	0	0
$46$	−2.85593	−0.421084
$47$	7.95583	1.16048	0.580239	−	0.814446i	$-0.302959\pi$
$47$	7.95583	1.16048	0.580239	+	0.814446i	$0.302959\pi$
$48$	0	0
$49$	16.4416	2.34879
$50$	−0.970031	−0.137183
$51$	0	0
$52$	1.01397	0.140613
$53$	10.9614	1.50566	0.752830	−	0.658214i	$-0.228688\pi$
$53$	10.9614	1.50566	0.752830	+	0.658214i	$0.228688\pi$
$54$	0	0
$55$	1.26097	0.170029
$56$	−14.3669	−1.91986
$57$	0	0
$58$	2.43520	0.319758
$59$	−5.13128	−0.668035	−0.334018	−	0.942567i	$-0.608404\pi$
$59$	−5.13128	−0.668035	−0.334018	+	0.942567i	$0.608404\pi$
$60$	0	0
$61$	−0.773449	−0.0990300	−0.0495150	−	0.998773i	$-0.515768\pi$
$61$	−0.773449	−0.0990300	−0.0495150	+	0.998773i	$0.515768\pi$
$62$	1.26157	0.160219
$63$	0	0
$64$	6.56212	0.820265
$65$	0.957447	0.118757
$66$	0	0
$67$	8.58486	1.04881	0.524404	−	0.851470i	$-0.324288\pi$
$67$	8.58486	1.04881	0.524404	+	0.851470i	$0.324288\pi$
$68$	2.64999	0.321359
$69$	0	0
$70$	−4.69655	−0.561345
$71$	−3.16973	−0.376178	−0.188089	−	0.982152i	$-0.560229\pi$
$71$	−3.16973	−0.376178	−0.188089	+	0.982152i	$0.560229\pi$
$72$	0	0
$73$	−0.179310	−0.0209867	−0.0104933	−	0.999945i	$-0.503340\pi$
$73$	−0.179310	−0.0209867	−0.0104933	+	0.999945i	$0.503340\pi$
$74$	−10.3329	−1.20117
$75$	0	0
$76$	−7.69543	−0.882726
$77$	6.10515	0.695746
$78$	0	0
$79$	−11.8490	−1.33311	−0.666557	−	0.745454i	$-0.732233\pi$
$79$	−11.8490	−1.33311	−0.666557	+	0.745454i	$0.732233\pi$
$80$	0.760356	0.0850104
$81$	0	0
$82$	10.4710	1.15633
$83$	0.924640	0.101492	0.0507462	−	0.998712i	$-0.483840\pi$
$83$	0.924640	0.101492	0.0507462	+	0.998712i	$0.483840\pi$
$84$	0	0
$85$	2.50226	0.271408
$86$	−1.63592	−0.176406
$87$	0	0
$88$	−3.74174	−0.398871
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	4.63562	0.485945
$92$	−3.11798	−0.325072
$93$	0	0
$94$	−7.71741	−0.795990
$95$	−7.26642	−0.745519
$96$	0	0
$97$	11.3807	1.15553	0.577767	−	0.816201i	$-0.303924\pi$
$97$	11.3807	1.15553	0.577767	+	0.816201i	$0.303924\pi$
$98$	−15.9488	−1.61107
$99$	0	0
$100$	−1.05904	−0.105904
$101$	0.540046	0.0537366	0.0268683	−	0.999639i	$-0.491447\pi$
$101$	0.540046	0.0537366	0.0268683	+	0.999639i	$0.491447\pi$
$102$	0	0
$103$	−5.16142	−0.508570	−0.254285	−	0.967129i	$-0.581840\pi$
$103$	−5.16142	−0.508570	−0.254285	+	0.967129i	$0.581840\pi$
$104$	−2.84109	−0.278592
$105$	0	0
$106$	−10.6329	−1.03276
$107$	−9.22424	−0.891741	−0.445871	−	0.895097i	$-0.647106\pi$
$107$	−9.22424	−0.891741	−0.445871	+	0.895097i	$0.647106\pi$
$108$	0	0
$109$	0.907932	0.0869641	0.0434821	−	0.999054i	$-0.486155\pi$
$109$	0.907932	0.0869641	0.0434821	+	0.999054i	$0.486155\pi$
$110$	−1.22318	−0.116625
$111$	0	0
$112$	3.68138	0.347858
$113$	−6.56595	−0.617673	−0.308836	−	0.951115i	$-0.599940\pi$
$113$	−6.56595	−0.617673	−0.308836	+	0.951115i	$0.599940\pi$
$114$	0	0
$115$	−2.94416	−0.274544
$116$	2.65865	0.246850
$117$	0	0
$118$	4.97750	0.458216
$119$	12.1151	1.11059
$120$	0	0
$121$	−9.40997	−0.855452
$122$	0.750270	0.0679262
$123$	0	0
$124$	1.37733	0.123688
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.25118	−0.465966	−0.232983	−	0.972481i	$-0.574849\pi$
$127$	−5.25118	−0.465966	−0.232983	+	0.972481i	$0.574849\pi$
$128$	4.02886	0.356104
$129$	0	0
$130$	−0.928753	−0.0814571
$131$	−10.8386	−0.946972	−0.473486	−	0.880801i	$-0.657004\pi$
$131$	−10.8386	−0.946972	−0.473486	+	0.880801i	$0.657004\pi$
$132$	0	0
$133$	−35.1814	−3.05062
$134$	−8.32758	−0.719394
$135$	0	0
$136$	−7.42512	−0.636699
$137$	−3.29433	−0.281454	−0.140727	−	0.990048i	$-0.544944\pi$
$137$	−3.29433	−0.281454	−0.140727	+	0.990048i	$0.544944\pi$
$138$	0	0
$139$	8.14346	0.690719	0.345360	−	0.938470i	$-0.387757\pi$
$139$	8.14346	0.690719	0.345360	+	0.938470i	$0.387757\pi$
$140$	−5.12750	−0.433352
$141$	0	0
$142$	3.07474	0.258026
$143$	1.20731	0.100960
$144$	0	0
$145$	2.51044	0.208480
$146$	0.173936	0.0143951
$147$	0	0
$148$	−11.2810	−0.927293
$149$	−0.0801450	−0.00656574	−0.00328287	−	0.999995i	$-0.501045\pi$
$149$	−0.0801450	−0.00656574	−0.00328287	+	0.999995i	$0.501045\pi$
$150$	0	0
$151$	8.88114	0.722737	0.361369	−	0.932423i	$-0.382310\pi$
$151$	8.88114	0.722737	0.361369	+	0.932423i	$0.382310\pi$
$152$	21.5621	1.74892
$153$	0	0
$154$	−5.92219	−0.477223
$155$	1.30054	0.104462
$156$	0	0
$157$	17.5406	1.39989	0.699946	−	0.714196i	$-0.253207\pi$
$157$	17.5406	1.39989	0.699946	+	0.714196i	$0.253207\pi$
$158$	11.4939	0.914404
$159$	0	0
$160$	5.19716	0.410871
$161$	−14.2546	−1.12342
$162$	0	0
$163$	12.9358	1.01321	0.506605	−	0.862179i	$-0.330900\pi$
$163$	12.9358	1.01321	0.506605	+	0.862179i	$0.330900\pi$
$164$	11.4318	0.892673
$165$	0	0
$166$	−0.896930	−0.0696153
$167$	−10.4471	−0.808423	−0.404211	−	0.914666i	$-0.632454\pi$
$167$	−10.4471	−0.808423	−0.404211	+	0.914666i	$0.632454\pi$
$168$	0	0
$169$	−12.0833	−0.929484
$170$	−2.42727	−0.186163
$171$	0	0
$172$	−1.78603	−0.136183
$173$	−16.5164	−1.25572	−0.627859	−	0.778327i	$-0.716069\pi$
$173$	−16.5164	−1.25572	−0.627859	+	0.778327i	$0.716069\pi$
$174$	0	0
$175$	−4.84165	−0.365994
$176$	0.958783	0.0722710
$177$	0	0
$178$	−0.970031	−0.0727069
$179$	−0.591766	−0.0442307	−0.0221154	−	0.999755i	$-0.507040\pi$
$179$	−0.591766	−0.0442307	−0.0221154	+	0.999755i	$0.507040\pi$
$180$	0	0
$181$	6.57081	0.488404	0.244202	−	0.969724i	$-0.421474\pi$
$181$	6.57081	0.488404	0.244202	+	0.969724i	$0.421474\pi$
$182$	−4.49670	−0.333317
$183$	0	0
$184$	8.73640	0.644056
$185$	−10.6521	−0.783159
$186$	0	0
$187$	3.15526	0.230736
$188$	−8.42554	−0.614496
$189$	0	0
$190$	7.04865	0.511363
$191$	17.5759	1.27175	0.635876	−	0.771792i	$-0.280639\pi$
$191$	17.5759	1.27175	0.635876	+	0.771792i	$0.280639\pi$
$192$	0	0
$193$	−22.1071	−1.59131	−0.795653	−	0.605753i	$-0.792872\pi$
$193$	−22.1071	−1.59131	−0.795653	+	0.605753i	$0.792872\pi$
$194$	−11.0396	−0.792600
$195$	0	0
$196$	−17.4123	−1.24373
$197$	−23.5198	−1.67572	−0.837860	−	0.545886i	$-0.816193\pi$
$197$	−23.5198	−1.67572	−0.837860	+	0.545886i	$0.816193\pi$
$198$	0	0
$199$	0.406042	0.0287835	0.0143918	−	0.999896i	$-0.495419\pi$
$199$	0.406042	0.0287835	0.0143918	+	0.999896i	$0.495419\pi$
$200$	2.96736	0.209824
$201$	0	0
$202$	−0.523861	−0.0368588
$203$	12.1546	0.853089
$204$	0	0
$205$	10.7945	0.753920
$206$	5.00674	0.348836
$207$	0	0
$208$	0.728001	0.0504778
$209$	−9.16270	−0.633797
$210$	0	0
$211$	−8.95696	−0.616623	−0.308311	−	0.951285i	$-0.599764\pi$
$211$	−8.95696	−0.616623	−0.308311	+	0.951285i	$0.599764\pi$
$212$	−11.6085	−0.797277
$213$	0	0
$214$	8.94781	0.611659
$215$	−1.68646	−0.115016
$216$	0	0
$217$	6.29677	0.427453
$218$	−0.880722	−0.0596500
$219$	0	0
$220$	−1.33541	−0.0900335
$221$	2.39578	0.161158
$222$	0	0
$223$	−13.0602	−0.874578	−0.437289	−	0.899321i	$-0.644061\pi$
$223$	−13.0602	−0.874578	−0.437289	+	0.899321i	$0.644061\pi$
$224$	25.1628	1.68126
$225$	0	0
$226$	6.36918	0.423672
$227$	2.54072	0.168634	0.0843169	−	0.996439i	$-0.473129\pi$
$227$	2.54072	0.168634	0.0843169	+	0.996439i	$0.473129\pi$
$228$	0	0
$229$	3.14557	0.207865	0.103932	−	0.994584i	$-0.466857\pi$
$229$	3.14557	0.207865	0.103932	+	0.994584i	$0.466857\pi$
$230$	2.85593	0.188314
$231$	0	0
$232$	−7.44938	−0.489076
$233$	19.5193	1.27875	0.639377	−	0.768893i	$-0.279192\pi$
$233$	19.5193	1.27875	0.639377	+	0.768893i	$0.279192\pi$
$234$	0	0
$235$	−7.95583	−0.518981
$236$	5.43422	0.353738
$237$	0	0
$238$	−11.7520	−0.761768
$239$	−14.4309	−0.933459	−0.466730	−	0.884400i	$-0.654568\pi$
$239$	−14.4309	−0.933459	−0.466730	+	0.884400i	$0.654568\pi$
$240$	0	0
$241$	−27.1333	−1.74781	−0.873905	−	0.486097i	$-0.838420\pi$
$241$	−27.1333	−1.74781	−0.873905	+	0.486097i	$0.838420\pi$
$242$	9.12796	0.586768
$243$	0	0
$244$	0.819113	0.0524383
$245$	−16.4416	−1.05041
$246$	0	0
$247$	−6.95721	−0.442677
$248$	−3.85918	−0.245058
$249$	0	0
$250$	0.970031	0.0613502
$251$	−17.1350	−1.08155	−0.540777	−	0.841166i	$-0.681870\pi$
$251$	−17.1350	−1.08155	−0.540777	+	0.841166i	$0.681870\pi$
$252$	0	0
$253$	−3.71248	−0.233402
$254$	5.09380	0.319614
$255$	0	0
$256$	−17.0324	−1.06452
$257$	−21.1235	−1.31765	−0.658824	−	0.752297i	$-0.728946\pi$
$257$	−21.1235	−1.31765	−0.658824	+	0.752297i	$0.728946\pi$
$258$	0	0
$259$	−51.5738	−3.20464
$260$	−1.01397	−0.0628840
$261$	0	0
$262$	10.5138	0.649543
$263$	20.0742	1.23783	0.618913	−	0.785459i	$-0.287573\pi$
$263$	20.0742	1.23783	0.618913	+	0.785459i	$0.287573\pi$
$264$	0	0
$265$	−10.9614	−0.673352
$266$	34.1271	2.09247
$267$	0	0
$268$	−9.09171	−0.555364
$269$	10.9077	0.665052	0.332526	−	0.943094i	$-0.392099\pi$
$269$	10.9077	0.665052	0.332526	+	0.943094i	$0.392099\pi$
$270$	0	0
$271$	−20.4886	−1.24459	−0.622297	−	0.782782i	$-0.713800\pi$
$271$	−20.4886	−1.24459	−0.622297	+	0.782782i	$0.713800\pi$
$272$	1.90261	0.115363
$273$	0	0
$274$	3.19560	0.193053
$275$	−1.26097	−0.0760391
$276$	0	0
$277$	27.0530	1.62546	0.812728	−	0.582643i	$-0.197982\pi$
$277$	27.0530	1.62546	0.812728	+	0.582643i	$0.197982\pi$
$278$	−7.89941	−0.473775
$279$	0	0
$280$	14.3669	0.858588
$281$	−16.3438	−0.974990	−0.487495	−	0.873126i	$-0.662089\pi$
$281$	−16.3438	−0.974990	−0.487495	+	0.873126i	$0.662089\pi$
$282$	0	0
$283$	−15.5600	−0.924946	−0.462473	−	0.886633i	$-0.653038\pi$
$283$	−15.5600	−0.924946	−0.462473	+	0.886633i	$0.653038\pi$
$284$	3.35687	0.199194
$285$	0	0
$286$	−1.17113	−0.0692501
$287$	52.2631	3.08500
$288$	0	0
$289$	−10.7387	−0.631688
$290$	−2.43520	−0.143000
$291$	0	0
$292$	0.189896	0.0111128
$293$	17.7412	1.03645	0.518226	−	0.855244i	$-0.326593\pi$
$293$	17.7412	1.03645	0.518226	+	0.855244i	$0.326593\pi$
$294$	0	0
$295$	5.13128	0.298754
$296$	31.6087	1.83722
$297$	0	0
$298$	0.0777432	0.00450354
$299$	−2.81888	−0.163020
$300$	0	0
$301$	−8.16525	−0.470637
$302$	−8.61499	−0.495737
$303$	0	0
$304$	−5.52507	−0.316884
$305$	0.773449	0.0442876
$306$	0	0
$307$	−14.5519	−0.830522	−0.415261	−	0.909702i	$-0.636310\pi$
$307$	−14.5519	−0.830522	−0.415261	+	0.909702i	$0.636310\pi$
$308$	−6.46559	−0.368411
$309$	0	0
$310$	−1.26157	−0.0716522
$311$	11.2035	0.635292	0.317646	−	0.948209i	$-0.397108\pi$
$311$	11.2035	0.635292	0.317646	+	0.948209i	$0.397108\pi$
$312$	0	0
$313$	21.4370	1.21169	0.605846	−	0.795582i	$-0.292835\pi$
$313$	21.4370	1.21169	0.605846	+	0.795582i	$0.292835\pi$
$314$	−17.0149	−0.960208
$315$	0	0
$316$	12.5485	0.705910
$317$	21.0534	1.18248	0.591239	−	0.806496i	$-0.298639\pi$
$317$	21.0534	1.18248	0.591239	+	0.806496i	$0.298639\pi$
$318$	0	0
$319$	3.16557	0.177238
$320$	−6.56212	−0.366834
$321$	0	0
$322$	13.8274	0.770571
$323$	−18.1825	−1.01170
$324$	0	0
$325$	−0.957447	−0.0531096
$326$	−12.5481	−0.694976
$327$	0	0
$328$	−32.0312	−1.76863
$329$	−38.5193	−2.12364
$330$	0	0
$331$	29.5188	1.62250	0.811249	−	0.584700i	$-0.198788\pi$
$331$	29.5188	1.62250	0.811249	+	0.584700i	$0.198788\pi$
$332$	−0.979231	−0.0537423
$333$	0	0
$334$	10.1340	0.554510
$335$	−8.58486	−0.469041
$336$	0	0
$337$	29.4025	1.60165	0.800827	−	0.598895i	$-0.204394\pi$
$337$	29.4025	1.60165	0.800827	+	0.598895i	$0.204394\pi$
$338$	11.7212	0.637548
$339$	0	0
$340$	−2.64999	−0.143716
$341$	1.63994	0.0888077
$342$	0	0
$343$	−45.7127	−2.46825
$344$	5.00434	0.269816
$345$	0	0
$346$	16.0214	0.861317
$347$	16.5560	0.888774	0.444387	−	0.895835i	$-0.353421\pi$
$347$	16.5560	0.888774	0.444387	+	0.895835i	$0.353421\pi$
$348$	0	0
$349$	15.2845	0.818160	0.409080	−	0.912499i	$-0.365850\pi$
$349$	15.2845	0.818160	0.409080	+	0.912499i	$0.365850\pi$
$350$	4.69655	0.251041
$351$	0	0
$352$	6.55344	0.349299
$353$	−27.6140	−1.46975	−0.734873	−	0.678204i	$-0.762758\pi$
$353$	−27.6140	−1.46975	−0.734873	+	0.678204i	$0.762758\pi$
$354$	0	0
$355$	3.16973	0.168232
$356$	−1.05904	−0.0561290
$357$	0	0
$358$	0.574032	0.0303385
$359$	−23.9146	−1.26216	−0.631082	−	0.775716i	$-0.717389\pi$
$359$	−23.9146	−1.26216	−0.631082	+	0.775716i	$0.717389\pi$
$360$	0	0
$361$	33.8009	1.77899
$362$	−6.37389	−0.335004
$363$	0	0
$364$	−4.90930	−0.257317
$365$	0.179310	0.00938552
$366$	0	0
$367$	−23.5377	−1.22866	−0.614329	−	0.789050i	$-0.710573\pi$
$367$	−23.5377	−1.22866	−0.614329	+	0.789050i	$0.710573\pi$
$368$	−2.23861	−0.116696
$369$	0	0
$370$	10.3329	0.537181
$371$	−53.0711	−2.75532
$372$	0	0
$373$	5.05202	0.261583	0.130792	−	0.991410i	$-0.458248\pi$
$373$	5.05202	0.261583	0.130792	+	0.991410i	$0.458248\pi$
$374$	−3.06070	−0.158265
$375$	0	0
$376$	23.6078	1.21748
$377$	2.40361	0.123792
$378$	0	0
$379$	17.2544	0.886297	0.443149	−	0.896448i	$-0.353861\pi$
$379$	17.2544	0.886297	0.443149	+	0.896448i	$0.353861\pi$
$380$	7.69543	0.394767
$381$	0	0
$382$	−17.0492	−0.872314
$383$	−27.2695	−1.39340	−0.696702	−	0.717361i	$-0.745350\pi$
$383$	−27.2695	−1.39340	−0.696702	+	0.717361i	$0.745350\pi$
$384$	0	0
$385$	−6.10515	−0.311147
$386$	21.4446	1.09150
$387$	0	0
$388$	−12.0526	−0.611879
$389$	−24.7299	−1.25386	−0.626928	−	0.779077i	$-0.715688\pi$
$389$	−24.7299	−1.25386	−0.626928	+	0.779077i	$0.715688\pi$
$390$	0	0
$391$	−7.36706	−0.372568
$392$	48.7881	2.46417
$393$	0	0
$394$	22.8150	1.14940
$395$	11.8490	0.596187
$396$	0	0
$397$	−2.48853	−0.124896	−0.0624480	−	0.998048i	$-0.519891\pi$
$397$	−2.48853	−0.124896	−0.0624480	+	0.998048i	$0.519891\pi$
$398$	−0.393873	−0.0197431
$399$	0	0
$400$	−0.760356	−0.0380178
$401$	12.0280	0.600648	0.300324	−	0.953837i	$-0.402905\pi$
$401$	12.0280	0.600648	0.300324	+	0.953837i	$0.402905\pi$
$402$	0	0
$403$	1.24520	0.0620278
$404$	−0.571930	−0.0284546
$405$	0	0
$406$	−11.7904	−0.585147
$407$	−13.4319	−0.665797
$408$	0	0
$409$	−3.76565	−0.186199	−0.0930997	−	0.995657i	$-0.529678\pi$
$409$	−3.76565	−0.186199	−0.0930997	+	0.995657i	$0.529678\pi$
$410$	−10.4710	−0.517126
$411$	0	0
$412$	5.46615	0.269298
$413$	24.8438	1.22248
$414$	0	0
$415$	−0.924640	−0.0453888
$416$	4.97600	0.243969
$417$	0	0
$418$	8.88811	0.434732
$419$	7.01225	0.342571	0.171285	−	0.985221i	$-0.445208\pi$
$419$	7.01225	0.342571	0.171285	+	0.985221i	$0.445208\pi$
$420$	0	0
$421$	14.2678	0.695371	0.347686	−	0.937611i	$-0.386968\pi$
$421$	14.2678	0.695371	0.347686	+	0.937611i	$0.386968\pi$
$422$	8.68854	0.422951
$423$	0	0
$424$	32.5264	1.57962
$425$	−2.50226	−0.121377
$426$	0	0
$427$	3.74477	0.181222
$428$	9.76884	0.472195
$429$	0	0
$430$	1.63592	0.0788910
$431$	−18.8215	−0.906600	−0.453300	−	0.891358i	$-0.649753\pi$
$431$	−18.8215	−0.906600	−0.453300	+	0.891358i	$0.649753\pi$
$432$	0	0
$433$	24.4512	1.17505	0.587524	−	0.809207i	$-0.300103\pi$
$433$	24.4512	1.17505	0.587524	+	0.809207i	$0.300103\pi$
$434$	−6.10806	−0.293196
$435$	0	0
$436$	−0.961536	−0.0460492
$437$	21.3935	1.02339
$438$	0	0
$439$	−38.8547	−1.85443	−0.927217	−	0.374524i	$-0.877806\pi$
$439$	−38.8547	−1.85443	−0.927217	+	0.374524i	$0.877806\pi$
$440$	3.74174	0.178381
$441$	0	0
$442$	−2.32398	−0.110541
$443$	−31.5057	−1.49688	−0.748440	−	0.663202i	$-0.769197\pi$
$443$	−31.5057	−1.49688	−0.748440	+	0.663202i	$0.769197\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	12.6688	0.599887
$447$	0	0
$448$	−31.7715	−1.50106
$449$	−0.0428453	−0.00202199	−0.00101100	−	0.999999i	$-0.500322\pi$
$449$	−0.0428453	−0.00202199	−0.00101100	+	0.999999i	$0.500322\pi$
$450$	0	0
$451$	13.6115	0.640940
$452$	6.95360	0.327070
$453$	0	0
$454$	−2.46458	−0.115669
$455$	−4.63562	−0.217321
$456$	0	0
$457$	−6.01306	−0.281279	−0.140640	−	0.990061i	$-0.544916\pi$
$457$	−6.01306	−0.281279	−0.140640	+	0.990061i	$0.544916\pi$
$458$	−3.05130	−0.142578
$459$	0	0
$460$	3.11798	0.145377
$461$	−14.1798	−0.660421	−0.330211	−	0.943907i	$-0.607120\pi$
$461$	−14.1798	−0.660421	−0.330211	+	0.943907i	$0.607120\pi$
$462$	0	0
$463$	17.0659	0.793120	0.396560	−	0.918009i	$-0.370204\pi$
$463$	17.0659	0.793120	0.396560	+	0.918009i	$0.370204\pi$
$464$	1.90883	0.0886150
$465$	0	0
$466$	−18.9344	−0.877118
$467$	16.7557	0.775362	0.387681	−	0.921794i	$-0.373276\pi$
$467$	16.7557	0.775362	0.387681	+	0.921794i	$0.373276\pi$
$468$	0	0
$469$	−41.5649	−1.91929
$470$	7.71741	0.355977
$471$	0	0
$472$	−15.2264	−0.700850
$473$	−2.12657	−0.0977797
$474$	0	0
$475$	7.26642	0.333406
$476$	−12.8303	−0.588077
$477$	0	0
$478$	13.9984	0.640274
$479$	−17.5127	−0.800176	−0.400088	−	0.916477i	$-0.631020\pi$
$479$	−17.5127	−0.800176	−0.400088	+	0.916477i	$0.631020\pi$
$480$	0	0
$481$	−10.1988	−0.465027
$482$	26.3202	1.19885
$483$	0	0
$484$	9.96553	0.452978
$485$	−11.3807	−0.516771
$486$	0	0
$487$	−40.4312	−1.83211	−0.916056	−	0.401050i	$-0.868645\pi$
$487$	−40.4312	−1.83211	−0.916056	+	0.401050i	$0.868645\pi$
$488$	−2.29511	−0.103895
$489$	0	0
$490$	15.9488	0.720494
$491$	−42.9414	−1.93792	−0.968959	−	0.247220i	$-0.920483\pi$
$491$	−42.9414	−1.93792	−0.968959	+	0.247220i	$0.920483\pi$
$492$	0	0
$493$	6.28177	0.282917
$494$	6.74871	0.303639
$495$	0	0
$496$	0.988876	0.0444018
$497$	15.3467	0.688394
$498$	0	0
$499$	35.4613	1.58747	0.793733	−	0.608266i	$-0.208135\pi$
$499$	35.4613	1.58747	0.793733	+	0.608266i	$0.208135\pi$
$500$	1.05904	0.0473617
$501$	0	0
$502$	16.6215	0.741855
$503$	−10.6586	−0.475241	−0.237621	−	0.971358i	$-0.576368\pi$
$503$	−10.6586	−0.475241	−0.237621	+	0.971358i	$0.576368\pi$
$504$	0	0
$505$	−0.540046	−0.0240317
$506$	3.60123	0.160094
$507$	0	0
$508$	5.56120	0.246738
$509$	15.3253	0.679282	0.339641	−	0.940555i	$-0.389694\pi$
$509$	15.3253	0.679282	0.339641	+	0.940555i	$0.389694\pi$
$510$	0	0
$511$	0.868156	0.0384050
$512$	8.46420	0.374068
$513$	0	0
$514$	20.4905	0.903795
$515$	5.16142	0.227440
$516$	0	0
$517$	−10.0320	−0.441208
$518$	50.0282	2.19811
$519$	0	0
$520$	2.84109	0.124590
$521$	1.42539	0.0624475	0.0312237	−	0.999512i	$-0.490060\pi$
$521$	1.42539	0.0624475	0.0312237	+	0.999512i	$0.490060\pi$
$522$	0	0
$523$	−22.8141	−0.997590	−0.498795	−	0.866720i	$-0.666224\pi$
$523$	−22.8141	−0.997590	−0.498795	+	0.866720i	$0.666224\pi$
$524$	11.4785	0.501440
$525$	0	0
$526$	−19.4726	−0.849044
$527$	3.25430	0.141759
$528$	0	0
$529$	−14.3319	−0.623127
$530$	10.6329	0.461863
$531$	0	0
$532$	37.2585	1.61536
$533$	10.3352	0.447665
$534$	0	0
$535$	9.22424	0.398799
$536$	25.4744	1.10033
$537$	0	0
$538$	−10.5808	−0.456170
$539$	−20.7322	−0.893000
$540$	0	0
$541$	−13.4937	−0.580141	−0.290070	−	0.957005i	$-0.593679\pi$
$541$	−13.4937	−0.580141	−0.290070	+	0.957005i	$0.593679\pi$
$542$	19.8746	0.853686
$543$	0	0
$544$	13.0046	0.557570
$545$	−0.907932	−0.0388915
$546$	0	0
$547$	−28.2383	−1.20738	−0.603691	−	0.797219i	$-0.706304\pi$
$547$	−28.2383	−1.20738	−0.603691	+	0.797219i	$0.706304\pi$
$548$	3.48883	0.149035
$549$	0	0
$550$	1.22318	0.0521564
$551$	−18.2419	−0.777130
$552$	0	0
$553$	57.3686	2.43956
$554$	−26.2422	−1.11493
$555$	0	0
$556$	−8.62425	−0.365749
$557$	−33.2231	−1.40771	−0.703855	−	0.710344i	$-0.748539\pi$
$557$	−33.2231	−1.40771	−0.703855	+	0.710344i	$0.748539\pi$
$558$	0	0
$559$	−1.61470	−0.0682944
$560$	−3.68138	−0.155567
$561$	0	0
$562$	15.8540	0.668761
$563$	28.9628	1.22064	0.610319	−	0.792156i	$-0.291041\pi$
$563$	28.9628	1.22064	0.610319	+	0.792156i	$0.291041\pi$
$564$	0	0
$565$	6.56595	0.276232
$566$	15.0937	0.634435
$567$	0	0
$568$	−9.40574	−0.394656
$569$	12.0048	0.503268	0.251634	−	0.967822i	$-0.419032\pi$
$569$	12.0048	0.503268	0.251634	+	0.967822i	$0.419032\pi$
$570$	0	0
$571$	−11.7190	−0.490426	−0.245213	−	0.969469i	$-0.578858\pi$
$571$	−11.7190	−0.490426	−0.245213	+	0.969469i	$0.578858\pi$
$572$	−1.27859	−0.0534604
$573$	0	0
$574$	−50.6969	−2.11605
$575$	2.94416	0.122780
$576$	0	0
$577$	−28.8315	−1.20027	−0.600135	−	0.799899i	$-0.704886\pi$
$577$	−28.8315	−1.20027	−0.600135	+	0.799899i	$0.704886\pi$
$578$	10.4169	0.433285
$579$	0	0
$580$	−2.65865	−0.110394
$581$	−4.47678	−0.185728
$582$	0	0
$583$	−13.8219	−0.572445
$584$	−0.532078	−0.0220176
$585$	0	0
$586$	−17.2095	−0.710918
$587$	−47.1515	−1.94615	−0.973075	−	0.230488i	$-0.925968\pi$
$587$	−47.1515	−1.94615	−0.973075	+	0.230488i	$0.925968\pi$
$588$	0	0
$589$	−9.45029	−0.389392
$590$	−4.97750	−0.204920
$591$	0	0
$592$	−8.09940	−0.332883
$593$	−12.5235	−0.514280	−0.257140	−	0.966374i	$-0.582780\pi$
$593$	−12.5235	−0.514280	−0.257140	+	0.966374i	$0.582780\pi$
$594$	0	0
$595$	−12.1151	−0.496669
$596$	0.0848768	0.00347669
$597$	0	0
$598$	2.73440	0.111818
$599$	−42.1633	−1.72275	−0.861374	−	0.507972i	$-0.830395\pi$
$599$	−42.1633	−1.72275	−0.861374	+	0.507972i	$0.830395\pi$
$600$	0	0
$601$	−19.9645	−0.814369	−0.407185	−	0.913346i	$-0.633489\pi$
$601$	−19.9645	−0.814369	−0.407185	+	0.913346i	$0.633489\pi$
$602$	7.92055	0.322817
$603$	0	0
$604$	−9.40548	−0.382704
$605$	9.40997	0.382570
$606$	0	0
$607$	26.8828	1.09114	0.545570	−	0.838065i	$-0.316313\pi$
$607$	26.8828	1.09114	0.545570	+	0.838065i	$0.316313\pi$
$608$	−37.7647	−1.53156
$609$	0	0
$610$	−0.750270	−0.0303775
$611$	−7.61729	−0.308162
$612$	0	0
$613$	28.8740	1.16621	0.583105	−	0.812397i	$-0.301838\pi$
$613$	28.8740	1.16621	0.583105	+	0.812397i	$0.301838\pi$
$614$	14.1158	0.569668
$615$	0	0
$616$	18.1162	0.729922
$617$	27.1937	1.09478	0.547388	−	0.836879i	$-0.315622\pi$
$617$	27.1937	1.09478	0.547388	+	0.836879i	$0.315622\pi$
$618$	0	0
$619$	6.00427	0.241332	0.120666	−	0.992693i	$-0.461497\pi$
$619$	6.00427	0.241332	0.120666	+	0.992693i	$0.461497\pi$
$620$	−1.37733	−0.0553147
$621$	0	0
$622$	−10.8677	−0.435757
$623$	−4.84165	−0.193977
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.7946	−0.831119
$627$	0	0
$628$	−18.5762	−0.741271
$629$	−26.6544	−1.06278
$630$	0	0
$631$	27.2201	1.08362	0.541808	−	0.840502i	$-0.317740\pi$
$631$	27.2201	1.08362	0.541808	+	0.840502i	$0.317740\pi$
$632$	−35.1602	−1.39860
$633$	0	0
$634$	−20.4225	−0.811080
$635$	5.25118	0.208387
$636$	0	0
$637$	−15.7419	−0.623717
$638$	−3.07070	−0.121570
$639$	0	0
$640$	−4.02886	−0.159255
$641$	−3.63150	−0.143436	−0.0717179	−	0.997425i	$-0.522848\pi$
$641$	−3.63150	−0.143436	−0.0717179	+	0.997425i	$0.522848\pi$
$642$	0	0
$643$	32.5013	1.28172	0.640862	−	0.767656i	$-0.278577\pi$
$643$	32.5013	1.28172	0.640862	+	0.767656i	$0.278577\pi$
$644$	15.0962	0.594872
$645$	0	0
$646$	17.6376	0.693941
$647$	34.7958	1.36796	0.683982	−	0.729499i	$-0.260247\pi$
$647$	34.7958	1.36796	0.683982	+	0.729499i	$0.260247\pi$
$648$	0	0
$649$	6.47036	0.253984
$650$	0.928753	0.0364287
$651$	0	0
$652$	−13.6995	−0.536514
$653$	−8.75257	−0.342514	−0.171257	−	0.985226i	$-0.554783\pi$
$653$	−8.75257	−0.342514	−0.171257	+	0.985226i	$0.554783\pi$
$654$	0	0
$655$	10.8386	0.423499
$656$	8.20767	0.320455
$657$	0	0
$658$	37.3650	1.45664
$659$	33.5109	1.30540	0.652701	−	0.757616i	$-0.273636\pi$
$659$	33.5109	1.30540	0.652701	+	0.757616i	$0.273636\pi$
$660$	0	0
$661$	4.44926	0.173056	0.0865280	−	0.996249i	$-0.472423\pi$
$661$	4.44926	0.173056	0.0865280	+	0.996249i	$0.472423\pi$
$662$	−28.6341	−1.11290
$663$	0	0
$664$	2.74374	0.106478
$665$	35.1814	1.36428
$666$	0	0
$667$	−7.39113	−0.286186
$668$	11.0639	0.428076
$669$	0	0
$670$	8.32758	0.321723
$671$	0.975293	0.0376508
$672$	0	0
$673$	−14.9337	−0.575650	−0.287825	−	0.957683i	$-0.592932\pi$
$673$	−14.9337	−0.575650	−0.287825	+	0.957683i	$0.592932\pi$
$674$	−28.5213	−1.09860
$675$	0	0
$676$	12.7967	0.492180
$677$	17.3442	0.666590	0.333295	−	0.942823i	$-0.391839\pi$
$677$	17.3442	0.666590	0.333295	+	0.942823i	$0.391839\pi$
$678$	0	0
$679$	−55.1013	−2.11460
$680$	7.42512	0.284740
$681$	0	0
$682$	−1.59079	−0.0609146
$683$	36.2406	1.38671	0.693353	−	0.720598i	$-0.256133\pi$
$683$	36.2406	1.38671	0.693353	+	0.720598i	$0.256133\pi$
$684$	0	0
$685$	3.29433	0.125870
$686$	44.3427	1.69301
$687$	0	0
$688$	−1.28231	−0.0488877
$689$	−10.4949	−0.399825
$690$	0	0
$691$	−47.9692	−1.82483	−0.912417	−	0.409262i	$-0.865786\pi$
$691$	−47.9692	−1.82483	−0.912417	+	0.409262i	$0.865786\pi$
$692$	17.4915	0.664927
$693$	0	0
$694$	−16.0599	−0.609624
$695$	−8.14346	−0.308899
$696$	0	0
$697$	27.0106	1.02310
$698$	−14.8264	−0.561188
$699$	0	0
$700$	5.12750	0.193801
$701$	22.7966	0.861016	0.430508	−	0.902587i	$-0.358335\pi$
$701$	22.7966	0.861016	0.430508	+	0.902587i	$0.358335\pi$
$702$	0	0
$703$	77.4027	2.91930
$704$	−8.27460	−0.311861
$705$	0	0
$706$	26.7865	1.00812
$707$	−2.61471	−0.0983364
$708$	0	0
$709$	29.4952	1.10772	0.553858	−	0.832611i	$-0.313155\pi$
$709$	29.4952	1.10772	0.553858	+	0.832611i	$0.313155\pi$
$710$	−3.07474	−0.115393
$711$	0	0
$712$	2.96736	0.111207
$713$	−3.82901	−0.143397
$714$	0	0
$715$	−1.20731	−0.0451507
$716$	0.626704	0.0234210
$717$	0	0
$718$	23.1979	0.865738
$719$	−15.7542	−0.587534	−0.293767	−	0.955877i	$-0.594909\pi$
$719$	−15.7542	−0.587534	−0.293767	+	0.955877i	$0.594909\pi$
$720$	0	0
$721$	24.9898	0.930669
$722$	−32.7879	−1.22024
$723$	0	0
$724$	−6.95875	−0.258620
$725$	−2.51044	−0.0932353
$726$	0	0
$727$	16.2559	0.602899	0.301449	−	0.953482i	$-0.402530\pi$
$727$	16.2559	0.602899	0.301449	+	0.953482i	$0.402530\pi$
$728$	13.7556	0.509815
$729$	0	0
$730$	−0.173936	−0.00643767
$731$	−4.21996	−0.156081
$732$	0	0
$733$	14.7171	0.543590	0.271795	−	0.962355i	$-0.412383\pi$
$733$	14.7171	0.543590	0.271795	+	0.962355i	$0.412383\pi$
$734$	22.8323	0.842755
$735$	0	0
$736$	−15.3013	−0.564012
$737$	−10.8252	−0.398752
$738$	0	0
$739$	17.7624	0.653400	0.326700	−	0.945128i	$-0.394063\pi$
$739$	17.7624	0.653400	0.326700	+	0.945128i	$0.394063\pi$
$740$	11.2810	0.414698
$741$	0	0
$742$	51.4806	1.88991
$743$	5.90489	0.216629	0.108315	−	0.994117i	$-0.465455\pi$
$743$	5.90489	0.216629	0.108315	+	0.994117i	$0.465455\pi$
$744$	0	0
$745$	0.0801450	0.00293629
$746$	−4.90061	−0.179424
$747$	0	0
$748$	−3.34155	−0.122179
$749$	44.6605	1.63186
$750$	0	0
$751$	15.4581	0.564075	0.282037	−	0.959403i	$-0.408990\pi$
$751$	15.4581	0.564075	0.282037	+	0.959403i	$0.408990\pi$
$752$	−6.04927	−0.220594
$753$	0	0
$754$	−2.33158	−0.0849110
$755$	−8.88114	−0.323218
$756$	0	0
$757$	−17.8323	−0.648125	−0.324062	−	0.946036i	$-0.605049\pi$
$757$	−17.8323	−0.648125	−0.324062	+	0.946036i	$0.605049\pi$
$758$	−16.7373	−0.607925
$759$	0	0
$760$	−21.5621	−0.782140
$761$	−2.06887	−0.0749965	−0.0374983	−	0.999297i	$-0.511939\pi$
$761$	−2.06887	−0.0749965	−0.0374983	+	0.999297i	$0.511939\pi$
$762$	0	0
$763$	−4.39589	−0.159142
$764$	−18.6136	−0.673417
$765$	0	0
$766$	26.4522	0.955757
$767$	4.91292	0.177395
$768$	0	0
$769$	0.182522	0.00658191	0.00329096	−	0.999995i	$-0.498952\pi$
$769$	0.182522	0.00658191	0.00329096	+	0.999995i	$0.498952\pi$
$770$	5.92219	0.213421
$771$	0	0
$772$	23.4123	0.842628
$773$	22.5668	0.811672	0.405836	−	0.913946i	$-0.366980\pi$
$773$	22.5668	0.811672	0.405836	+	0.913946i	$0.366980\pi$
$774$	0	0
$775$	−1.30054	−0.0467169
$776$	33.7707	1.21230
$777$	0	0
$778$	23.9888	0.860040
$779$	−78.4374	−2.81031
$780$	0	0
$781$	3.99692	0.143021
$782$	7.14628	0.255550
$783$	0	0
$784$	−12.5014	−0.446480
$785$	−17.5406	−0.626051
$786$	0	0
$787$	−3.57295	−0.127362	−0.0636810	−	0.997970i	$-0.520284\pi$
$787$	−3.57295	−0.127362	−0.0636810	+	0.997970i	$0.520284\pi$
$788$	24.9084	0.887326
$789$	0	0
$790$	−11.4939	−0.408934
$791$	31.7900	1.13032
$792$	0	0
$793$	0.740536	0.0262972
$794$	2.41396	0.0856681
$795$	0	0
$796$	−0.430014	−0.0152415
$797$	38.9586	1.37998	0.689992	−	0.723817i	$-0.257614\pi$
$797$	38.9586	1.37998	0.689992	+	0.723817i	$0.257614\pi$
$798$	0	0
$799$	−19.9076	−0.704279
$800$	−5.19716	−0.183747
$801$	0	0
$802$	−11.6675	−0.411994
$803$	0.226104	0.00797903
$804$	0	0
$805$	14.2546	0.502408
$806$	−1.20788	−0.0425459
$807$	0	0
$808$	1.60251	0.0563762
$809$	−21.2303	−0.746417	−0.373208	−	0.927748i	$-0.621742\pi$
$809$	−21.2303	−0.746417	−0.373208	+	0.927748i	$0.621742\pi$
$810$	0	0
$811$	−12.4400	−0.436827	−0.218414	−	0.975856i	$-0.570088\pi$
$811$	−12.4400	−0.436827	−0.218414	+	0.975856i	$0.570088\pi$
$812$	−12.8723	−0.451727
$813$	0	0
$814$	13.0294	0.456680
$815$	−12.9358	−0.453121
$816$	0	0
$817$	12.2545	0.428732
$818$	3.65280	0.127717
$819$	0	0
$820$	−11.4318	−0.399216
$821$	7.62904	0.266255	0.133128	−	0.991099i	$-0.457498\pi$
$821$	7.62904	0.266255	0.133128	+	0.991099i	$0.457498\pi$
$822$	0	0
$823$	−50.8434	−1.77229	−0.886144	−	0.463410i	$-0.846626\pi$
$823$	−50.8434	−1.77229	−0.886144	+	0.463410i	$0.846626\pi$
$824$	−15.3158	−0.533552
$825$	0	0
$826$	−24.0993	−0.838522
$827$	−25.6950	−0.893502	−0.446751	−	0.894658i	$-0.647419\pi$
$827$	−25.6950	−0.893502	−0.446751	+	0.894658i	$0.647419\pi$
$828$	0	0
$829$	28.1275	0.976909	0.488454	−	0.872589i	$-0.337561\pi$
$829$	28.1275	0.976909	0.488454	+	0.872589i	$0.337561\pi$
$830$	0.896930	0.0311329
$831$	0	0
$832$	−6.28288	−0.217820
$833$	−41.1410	−1.42545
$834$	0	0
$835$	10.4471	0.361538
$836$	9.70367	0.335608
$837$	0	0
$838$	−6.80210	−0.234975
$839$	28.4960	0.983790	0.491895	−	0.870654i	$-0.336304\pi$
$839$	28.4960	0.983790	0.491895	+	0.870654i	$0.336304\pi$
$840$	0	0
$841$	−22.6977	−0.782680
$842$	−13.8402	−0.476966
$843$	0	0
$844$	9.48578	0.326514
$845$	12.0833	0.415678
$846$	0	0
$847$	45.5597	1.56545
$848$	−8.33455	−0.286210
$849$	0	0
$850$	2.42727	0.0832547
$851$	31.3615	1.07506
$852$	0	0
$853$	−6.37854	−0.218397	−0.109198	−	0.994020i	$-0.534828\pi$
$853$	−6.37854	−0.218397	−0.109198	+	0.994020i	$0.534828\pi$
$854$	−3.63254	−0.124303
$855$	0	0
$856$	−27.3717	−0.935545
$857$	2.79032	0.0953157	0.0476578	−	0.998864i	$-0.484824\pi$
$857$	2.79032	0.0953157	0.0476578	+	0.998864i	$0.484824\pi$
$858$	0	0
$859$	−52.1048	−1.77779	−0.888897	−	0.458108i	$-0.848527\pi$
$859$	−52.1048	−1.77779	−0.888897	+	0.458108i	$0.848527\pi$
$860$	1.78603	0.0609031
$861$	0	0
$862$	18.2574	0.621851
$863$	−11.9685	−0.407413	−0.203707	−	0.979032i	$-0.565299\pi$
$863$	−11.9685	−0.407413	−0.203707	+	0.979032i	$0.565299\pi$
$864$	0	0
$865$	16.5164	0.561574
$866$	−23.7184	−0.805984
$867$	0	0
$868$	−6.66853	−0.226345
$869$	14.9411	0.506844
$870$	0	0
$871$	−8.21955	−0.278509
$872$	2.69416	0.0912359
$873$	0	0
$874$	−20.7524	−0.701959
$875$	4.84165	0.163678
$876$	0	0
$877$	−7.55377	−0.255073	−0.127536	−	0.991834i	$-0.540707\pi$
$877$	−7.55377	−0.255073	−0.127536	+	0.991834i	$0.540707\pi$
$878$	37.6903	1.27199
$879$	0	0
$880$	−0.958783	−0.0323206
$881$	−42.2046	−1.42191	−0.710955	−	0.703238i	$-0.751737\pi$
$881$	−42.2046	−1.42191	−0.710955	+	0.703238i	$0.751737\pi$
$882$	0	0
$883$	−48.5407	−1.63352	−0.816762	−	0.576974i	$-0.804233\pi$
$883$	−48.5407	−1.63352	−0.816762	+	0.576974i	$0.804233\pi$
$884$	−2.53723	−0.0853362
$885$	0	0
$886$	30.5615	1.02673
$887$	−31.4482	−1.05593	−0.527963	−	0.849267i	$-0.677044\pi$
$887$	−31.4482	−1.05593	−0.527963	+	0.849267i	$0.677044\pi$
$888$	0	0
$889$	25.4243	0.852705
$890$	0.970031	0.0325155
$891$	0	0
$892$	13.8313	0.463106
$893$	57.8104	1.93455
$894$	0	0
$895$	0.591766	0.0197806
$896$	−19.5063	−0.651660
$897$	0	0
$898$	0.0415613	0.00138692
$899$	3.26493	0.108891
$900$	0	0
$901$	−27.4282	−0.913767
$902$	−13.2036	−0.439631
$903$	0	0
$904$	−19.4836	−0.648014
$905$	−6.57081	−0.218421
$906$	0	0
$907$	−29.8819	−0.992212	−0.496106	−	0.868262i	$-0.665237\pi$
$907$	−29.8819	−0.992212	−0.496106	+	0.868262i	$0.665237\pi$
$908$	−2.69073	−0.0892949
$909$	0	0
$910$	4.49670	0.149064
$911$	4.70778	0.155976	0.0779879	−	0.996954i	$-0.475150\pi$
$911$	4.70778	0.155976	0.0779879	+	0.996954i	$0.475150\pi$
$912$	0	0
$913$	−1.16594	−0.0385870
$914$	5.83286	0.192934
$915$	0	0
$916$	−3.33128	−0.110069
$917$	52.4766	1.73293
$918$	0	0
$919$	49.0844	1.61914	0.809572	−	0.587021i	$-0.199699\pi$
$919$	49.0844	1.61914	0.809572	+	0.587021i	$0.199699\pi$
$920$	−8.73640	−0.288030
$921$	0	0
$922$	13.7549	0.452993
$923$	3.03485	0.0998933
$924$	0	0
$925$	10.6521	0.350239
$926$	−16.5545	−0.544013
$927$	0	0
$928$	13.0471	0.428293
$929$	40.3235	1.32297	0.661485	−	0.749958i	$-0.269926\pi$
$929$	40.3235	1.32297	0.661485	+	0.749958i	$0.269926\pi$
$930$	0	0
$931$	119.471	3.91551
$932$	−20.6718	−0.677126
$933$	0	0
$934$	−16.2536	−0.531833
$935$	−3.15526	−0.103188
$936$	0	0
$937$	−11.9997	−0.392013	−0.196006	−	0.980603i	$-0.562797\pi$
$937$	−11.9997	−0.392013	−0.196006	+	0.980603i	$0.562797\pi$
$938$	40.3192	1.31647
$939$	0	0
$940$	8.42554	0.274811
$941$	50.4797	1.64559	0.822795	−	0.568338i	$-0.192413\pi$
$941$	50.4797	1.64559	0.822795	+	0.568338i	$0.192413\pi$
$942$	0	0
$943$	−31.7807	−1.03492
$944$	3.90160	0.126986
$945$	0	0
$946$	2.06284	0.0670686
$947$	50.8704	1.65307	0.826533	−	0.562889i	$-0.190310\pi$
$947$	50.8704	1.65307	0.826533	+	0.562889i	$0.190310\pi$
$948$	0	0
$949$	0.171680	0.00557296
$950$	−7.04865	−0.228689
$951$	0	0
$952$	35.9498	1.16514
$953$	32.0577	1.03845	0.519225	−	0.854637i	$-0.326221\pi$
$953$	32.0577	1.03845	0.519225	+	0.854637i	$0.326221\pi$
$954$	0	0
$955$	−17.5759	−0.568744
$956$	15.2829	0.494285
$957$	0	0
$958$	16.9879	0.548853
$959$	15.9500	0.515052
$960$	0	0
$961$	−29.3086	−0.945438
$962$	9.89318	0.318969
$963$	0	0
$964$	28.7352	0.925500
$965$	22.1071	0.711654
$966$	0	0
$967$	28.7578	0.924789	0.462394	−	0.886674i	$-0.346990\pi$
$967$	28.7578	0.924789	0.462394	+	0.886674i	$0.346990\pi$
$968$	−27.9228	−0.897473
$969$	0	0
$970$	11.0396	0.354461
$971$	−15.8385	−0.508280	−0.254140	−	0.967167i	$-0.581792\pi$
$971$	−15.8385	−0.508280	−0.254140	+	0.967167i	$0.581792\pi$
$972$	0	0
$973$	−39.4278	−1.26400
$974$	39.2195	1.25667
$975$	0	0
$976$	0.588097	0.0188245
$977$	4.99133	0.159687	0.0798434	−	0.996807i	$-0.474558\pi$
$977$	4.99133	0.159687	0.0798434	+	0.996807i	$0.474558\pi$
$978$	0	0
$979$	−1.26097	−0.0403006
$980$	17.4123	0.556214
$981$	0	0
$982$	41.6545	1.32925
$983$	−41.2405	−1.31537	−0.657683	−	0.753295i	$-0.728463\pi$
$983$	−41.2405	−1.31537	−0.657683	+	0.753295i	$0.728463\pi$
$984$	0	0
$985$	23.5198	0.749404
$986$	−6.09351	−0.194057
$987$	0	0
$988$	7.36796	0.234406
$989$	4.96521	0.157885
$990$	0	0
$991$	29.2806	0.930128	0.465064	−	0.885277i	$-0.346031\pi$
$991$	29.2806	0.930128	0.465064	+	0.885277i	$0.346031\pi$
$992$	6.75913	0.214602
$993$	0	0
$994$	−14.8868	−0.472181
$995$	−0.406042	−0.0128724
$996$	0	0
$997$	25.6186	0.811348	0.405674	−	0.914018i	$-0.367037\pi$
$997$	25.6186	0.811348	0.405674	+	0.914018i	$0.367037\pi$
$998$	−34.3986	−1.08887
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.4		10
3.2	odd	2		1335.2.a.j.1.7	✓	10
15.14	odd	2		6675.2.a.z.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.7	✓	10	3.2	odd	2
4005.2.a.s.1.4		10	1.1	even	1	trivial
6675.2.a.z.1.4		10	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)

\(f(q)\)	\(=\)	\(q-0.970031 q^{2} -1.05904 q^{4} -1.00000 q^{5} -4.84165 q^{7} +2.96736 q^{8} +O(q^{10})\)	\(q-0.970031 q^{2} -1.05904 q^{4} -1.00000 q^{5} -4.84165 q^{7} +2.96736 q^{8} +0.970031 q^{10} -1.26097 q^{11} -0.957447 q^{13} +4.69655 q^{14} -0.760356 q^{16} -2.50226 q^{17} +7.26642 q^{19} +1.05904 q^{20} +1.22318 q^{22} +2.94416 q^{23} +1.00000 q^{25} +0.928753 q^{26} +5.12750 q^{28} -2.51044 q^{29} -1.30054 q^{31} -5.19716 q^{32} +2.42727 q^{34} +4.84165 q^{35} +10.6521 q^{37} -7.04865 q^{38} -2.96736 q^{40} -10.7945 q^{41} +1.68646 q^{43} +1.33541 q^{44} -2.85593 q^{46} +7.95583 q^{47} +16.4416 q^{49} -0.970031 q^{50} +1.01397 q^{52} +10.9614 q^{53} +1.26097 q^{55} -14.3669 q^{56} +2.43520 q^{58} -5.13128 q^{59} -0.773449 q^{61} +1.26157 q^{62} +6.56212 q^{64} +0.957447 q^{65} +8.58486 q^{67} +2.64999 q^{68} -4.69655 q^{70} -3.16973 q^{71} -0.179310 q^{73} -10.3329 q^{74} -7.69543 q^{76} +6.10515 q^{77} -11.8490 q^{79} +0.760356 q^{80} +10.4710 q^{82} +0.924640 q^{83} +2.50226 q^{85} -1.63592 q^{86} -3.74174 q^{88} +1.00000 q^{89} +4.63562 q^{91} -3.11798 q^{92} -7.71741 q^{94} -7.26642 q^{95} +11.3807 q^{97} -15.9488 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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