LMFDB - Embedded newform 4005.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
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			1.1
Root			$-1.50848$ 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)

$f(q)$	$=$	$q-1.78400 q^{2} +1.18264 q^{4} +1.00000 q^{5} -3.50848 q^{7} +1.45816 q^{8} +O(q^{10})$	\(q-1.78400 q^{2} +1.18264 q^{4} +1.00000 q^{5} -3.50848 q^{7} +1.45816 q^{8} -1.78400 q^{10} +3.56799 q^{11} -2.64080 q^{13} +6.25912 q^{14} -4.96664 q^{16} +2.54184 q^{17} +1.01696 q^{19} +1.18264 q^{20} -6.36529 q^{22} -6.70809 q^{23} +1.00000 q^{25} +4.71119 q^{26} -4.14929 q^{28} +6.16625 q^{29} -2.65167 q^{31} +5.94415 q^{32} -4.53463 q^{34} -3.50848 q^{35} +5.49208 q^{37} -1.81426 q^{38} +1.45816 q^{40} +0.182644 q^{41} -10.2894 q^{43} +4.21967 q^{44} +11.9672 q^{46} -1.69422 q^{47} +5.30944 q^{49} -1.78400 q^{50} -3.12313 q^{52} -1.58738 q^{53} +3.56799 q^{55} -5.11593 q^{56} -11.0006 q^{58} -7.89992 q^{59} +6.66386 q^{61} +4.73058 q^{62} -0.671065 q^{64} -2.64080 q^{65} +0.893830 q^{67} +3.00609 q^{68} +6.25912 q^{70} +2.91632 q^{71} -2.85991 q^{73} -9.79786 q^{74} +1.20270 q^{76} -12.5182 q^{77} +0.666957 q^{79} -4.96664 q^{80} -0.325837 q^{82} +15.2930 q^{83} +2.54184 q^{85} +18.3562 q^{86} +5.20270 q^{88} -1.00000 q^{89} +9.26521 q^{91} -7.93328 q^{92} +3.02249 q^{94} +1.01696 q^{95} -7.42648 q^{97} -9.47202 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8	$ 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{13} + q^{14} - 10 q^{16} + 13 q^{17} - 10 q^{19} + 2 q^{20} - 20 q^{22} - 3 q^{23} + 4 q^{25} + 3 q^{26} - 4 q^{28} - 2 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} - 7 q^{35} - 9 q^{37} - 2 q^{38} + 3 q^{40} - 2 q^{41} - 19 q^{43} - 6 q^{44} - 5 q^{47} - 7 q^{49} - 17 q^{52} + 3 q^{53} + 2 q^{56} - 6 q^{58} - 2 q^{59} - 4 q^{61} + 8 q^{62} + q^{64} - 5 q^{65} - 15 q^{67} + q^{68} + q^{70} + 6 q^{71} - 21 q^{73} - 10 q^{74} - 4 q^{76} - 2 q^{77} - 20 q^{79} - 10 q^{80} + 3 q^{82} + 9 q^{83} + 13 q^{85} - 16 q^{86} + 12 q^{88} - 4 q^{89} + 2 q^{91} - 12 q^{92} + 25 q^{94} - 10 q^{95} - 17 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8 - 5 * q^13 + q^14 - 10 * q^16 + 13 * q^17 - 10 * q^19 + 2 * q^20 - 20 * q^22 - 3 * q^23 + 4 * q^25 + 3 * q^26 - 4 * q^28 - 2 * q^29 - 2 * q^31 + q^32 + 6 * q^34 - 7 * q^35 - 9 * q^37 - 2 * q^38 + 3 * q^40 - 2 * q^41 - 19 * q^43 - 6 * q^44 - 5 * q^47 - 7 * q^49 - 17 * q^52 + 3 * q^53 + 2 * q^56 - 6 * q^58 - 2 * q^59 - 4 * q^61 + 8 * q^62 + q^64 - 5 * q^65 - 15 * q^67 + q^68 + q^70 + 6 * q^71 - 21 * q^73 - 10 * q^74 - 4 * q^76 - 2 * q^77 - 20 * q^79 - 10 * q^80 + 3 * q^82 + 9 * q^83 + 13 * q^85 - 16 * q^86 + 12 * q^88 - 4 * q^89 + 2 * q^91 - 12 * q^92 + 25 * q^94 - 10 * q^95 - 17 * q^97 - 13 * 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 $

$2$	−1.78400	−1.26148	−0.630738	−	0.775996i	$-0.717248\pi$
$2$	−1.78400	−1.26148	−0.630738	+	0.775996i	$0.717248\pi$
$3$	0	0
$3$	0	0
$4$	1.18264	0.591322
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.50848	−1.32608	−0.663041	−	0.748583i	$-0.730734\pi$
$7$	−3.50848	−1.32608	−0.663041	+	0.748583i	$0.730734\pi$
$8$	1.45816	0.515537
$9$	0	0
$10$	−1.78400	−0.564149
$11$	3.56799	1.07579	0.537895	−	0.843012i	$-0.319220\pi$
$11$	3.56799	1.07579	0.537895	+	0.843012i	$0.319220\pi$
$12$	0	0
$13$	−2.64080	−0.732427	−0.366214	−	0.930531i	$-0.619346\pi$
$13$	−2.64080	−0.732427	−0.366214	+	0.930531i	$0.619346\pi$
$14$	6.25912	1.67282
$15$	0	0
$16$	−4.96664	−1.24166
$17$	2.54184	0.616487	0.308243	−	0.951308i	$-0.400259\pi$
$17$	2.54184	0.616487	0.308243	+	0.951308i	$0.400259\pi$
$18$	0	0
$19$	1.01696	0.233307	0.116654	−	0.993173i	$-0.462783\pi$
$19$	1.01696	0.233307	0.116654	+	0.993173i	$0.462783\pi$
$20$	1.18264	0.264447
$21$	0	0
$22$	−6.36529	−1.35708
$23$	−6.70809	−1.39873	−0.699367	−	0.714763i	$-0.746534\pi$
$23$	−6.70809	−1.39873	−0.699367	+	0.714763i	$0.746534\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.71119	0.923940
$27$	0	0
$28$	−4.14929	−0.784141
$29$	6.16625	1.14504	0.572522	−	0.819890i	$-0.305965\pi$
$29$	6.16625	1.14504	0.572522	+	0.819890i	$0.305965\pi$
$30$	0	0
$31$	−2.65167	−0.476255	−0.238127	−	0.971234i	$-0.576534\pi$
$31$	−2.65167	−0.476255	−0.238127	+	0.971234i	$0.576534\pi$
$32$	5.94415	1.05079
$33$	0	0
$34$	−4.53463	−0.777683
$35$	−3.50848	−0.593042
$36$	0	0
$37$	5.49208	0.902893	0.451447	−	0.892298i	$-0.350908\pi$
$37$	5.49208	0.902893	0.451447	+	0.892298i	$0.350908\pi$
$38$	−1.81426	−0.294311
$39$	0	0
$40$	1.45816	0.230555
$41$	0.182644	0.0285243	0.0142621	−	0.999898i	$-0.495460\pi$
$41$	0.182644	0.0285243	0.0142621	+	0.999898i	$0.495460\pi$
$42$	0	0
$43$	−10.2894	−1.56912	−0.784558	−	0.620056i	$-0.787110\pi$
$43$	−10.2894	−1.56912	−0.784558	+	0.620056i	$0.787110\pi$
$44$	4.21967	0.636139
$45$	0	0
$46$	11.9672	1.76447
$47$	−1.69422	−0.247128	−0.123564	−	0.992337i	$-0.539432\pi$
$47$	−1.69422	−0.247128	−0.123564	+	0.992337i	$0.539432\pi$
$48$	0	0
$49$	5.30944	0.758491
$50$	−1.78400	−0.252295
$51$	0	0
$52$	−3.12313	−0.433100
$53$	−1.58738	−0.218044	−0.109022	−	0.994039i	$-0.534772\pi$
$53$	−1.58738	−0.218044	−0.109022	+	0.994039i	$0.534772\pi$
$54$	0	0
$55$	3.56799	0.481108
$56$	−5.11593	−0.683644
$57$	0	0
$58$	−11.0006	−1.44445
$59$	−7.89992	−1.02848	−0.514241	−	0.857646i	$-0.671926\pi$
$59$	−7.89992	−1.02848	−0.514241	+	0.857646i	$0.671926\pi$
$60$	0	0
$61$	6.66386	0.853220	0.426610	−	0.904436i	$-0.359708\pi$
$61$	6.66386	0.853220	0.426610	+	0.904436i	$0.359708\pi$
$62$	4.73058	0.600784
$63$	0	0
$64$	−0.671065	−0.0838831
$65$	−2.64080	−0.327551
$66$	0	0
$67$	0.893830	0.109199	0.0545994	−	0.998508i	$-0.482612\pi$
$67$	0.893830	0.109199	0.0545994	+	0.998508i	$0.482612\pi$
$68$	3.00609	0.364542
$69$	0	0
$70$	6.25912	0.748108
$71$	2.91632	0.346103	0.173052	−	0.984913i	$-0.444637\pi$
$71$	2.91632	0.346103	0.173052	+	0.984913i	$0.444637\pi$
$72$	0	0
$73$	−2.85991	−0.334727	−0.167363	−	0.985895i	$-0.553525\pi$
$73$	−2.85991	−0.334727	−0.167363	+	0.985895i	$0.553525\pi$
$74$	−9.79786	−1.13898
$75$	0	0
$76$	1.20270	0.137960
$77$	−12.5182	−1.42659
$78$	0	0
$79$	0.666957	0.0750386	0.0375193	−	0.999296i	$-0.488054\pi$
$79$	0.666957	0.0750386	0.0375193	+	0.999296i	$0.488054\pi$
$80$	−4.96664	−0.555287
$81$	0	0
$82$	−0.325837	−0.0359827
$83$	15.2930	1.67863	0.839315	−	0.543646i	$-0.182956\pi$
$83$	15.2930	1.67863	0.839315	+	0.543646i	$0.182956\pi$
$84$	0	0
$85$	2.54184	0.275701
$86$	18.3562	1.97940
$87$	0	0
$88$	5.20270	0.554610
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	9.26521	0.971258
$92$	−7.93328	−0.827102
$93$	0	0
$94$	3.02249	0.311746
$95$	1.01696	0.104338
$96$	0	0
$97$	−7.42648	−0.754045	−0.377022	−	0.926204i	$-0.623052\pi$
$97$	−7.42648	−0.754045	−0.377022	+	0.926204i	$0.623052\pi$
$98$	−9.47202	−0.956819
$99$	0	0
$100$	1.18264	0.118264
$101$	10.5074	1.04552	0.522761	−	0.852479i	$-0.324902\pi$
$101$	10.5074	1.04552	0.522761	+	0.852479i	$0.324902\pi$
$102$	0	0
$103$	1.10674	0.109050	0.0545249	−	0.998512i	$-0.482636\pi$
$103$	1.10674	0.109050	0.0545249	+	0.998512i	$0.482636\pi$
$104$	−3.85071	−0.377594
$105$	0	0
$106$	2.83189	0.275057
$107$	−13.3465	−1.29025	−0.645126	−	0.764076i	$-0.723195\pi$
$107$	−13.3465	−1.29025	−0.645126	+	0.764076i	$0.723195\pi$
$108$	0	0
$109$	−4.79786	−0.459552	−0.229776	−	0.973244i	$-0.573799\pi$
$109$	−4.79786	−0.459552	−0.229776	+	0.973244i	$0.573799\pi$
$110$	−6.36529	−0.606906
$111$	0	0
$112$	17.4254	1.64654
$113$	15.6244	1.46982	0.734910	−	0.678164i	$-0.237224\pi$
$113$	15.6244	1.46982	0.734910	+	0.678164i	$0.237224\pi$
$114$	0	0
$115$	−6.70809	−0.625532
$116$	7.29248	0.677090
$117$	0	0
$118$	14.0934	1.29741
$119$	−8.91800	−0.817512
$120$	0	0
$121$	1.73058	0.157325
$122$	−11.8883	−1.07632
$123$	0	0
$124$	−3.13599	−0.281620
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.3233	−1.09352	−0.546758	−	0.837291i	$-0.684138\pi$
$127$	−12.3233	−1.09352	−0.546758	+	0.837291i	$0.684138\pi$
$128$	−10.6911	−0.944971
$129$	0	0
$130$	4.71119	0.413198
$131$	−15.8216	−1.38234	−0.691169	−	0.722693i	$-0.742904\pi$
$131$	−15.8216	−1.38234	−0.691169	+	0.722693i	$0.742904\pi$
$132$	0	0
$133$	−3.56799	−0.309384
$134$	−1.59459	−0.137752
$135$	0	0
$136$	3.70641	0.317822
$137$	−5.78766	−0.494473	−0.247237	−	0.968955i	$-0.579522\pi$
$137$	−5.78766	−0.494473	−0.247237	+	0.968955i	$0.579522\pi$
$138$	0	0
$139$	−11.8148	−1.00212	−0.501060	−	0.865412i	$-0.667056\pi$
$139$	−11.8148	−1.00212	−0.501060	+	0.865412i	$0.667056\pi$
$140$	−4.14929	−0.350679
$141$	0	0
$142$	−5.20270	−0.436601
$143$	−9.42237	−0.787938
$144$	0	0
$145$	6.16625	0.512079
$146$	5.10206	0.422250
$147$	0	0
$148$	6.49518	0.533901
$149$	−12.1087	−0.991985	−0.495993	−	0.868327i	$-0.665196\pi$
$149$	−12.1087	−0.991985	−0.495993	+	0.868327i	$0.665196\pi$
$150$	0	0
$151$	16.1869	1.31727	0.658635	−	0.752463i	$-0.271134\pi$
$151$	16.1869	1.31727	0.658635	+	0.752463i	$0.271134\pi$
$152$	1.48289	0.120279
$153$	0	0
$154$	22.3325	1.79960
$155$	−2.65167	−0.212988
$156$	0	0
$157$	−6.12313	−0.488679	−0.244340	−	0.969690i	$-0.578571\pi$
$157$	−6.12313	−0.488679	−0.244340	+	0.969690i	$0.578571\pi$
$158$	−1.18985	−0.0946594
$159$	0	0
$160$	5.94415	0.469926
$161$	23.5352	1.85483
$162$	0	0
$163$	−7.02783	−0.550462	−0.275231	−	0.961378i	$-0.588754\pi$
$163$	−7.02783	−0.550462	−0.275231	+	0.961378i	$0.588754\pi$
$164$	0.216003	0.0168670
$165$	0	0
$166$	−27.2827	−2.11755
$167$	−6.56190	−0.507775	−0.253888	−	0.967234i	$-0.581709\pi$
$167$	−6.56190	−0.507775	−0.253888	+	0.967234i	$0.581709\pi$
$168$	0	0
$169$	−6.02615	−0.463550
$170$	−4.53463	−0.347791
$171$	0	0
$172$	−12.1687	−0.927853
$173$	19.1746	1.45782	0.728908	−	0.684611i	$-0.240028\pi$
$173$	19.1746	1.45782	0.728908	+	0.684611i	$0.240028\pi$
$174$	0	0
$175$	−3.50848	−0.265216
$176$	−17.7209	−1.33577
$177$	0	0
$178$	1.78400	0.133716
$179$	−2.06672	−0.154474	−0.0772369	−	0.997013i	$-0.524610\pi$
$179$	−2.06672	−0.154474	−0.0772369	+	0.997013i	$0.524610\pi$
$180$	0	0
$181$	10.7184	0.796692	0.398346	−	0.917235i	$-0.369584\pi$
$181$	10.7184	0.796692	0.398346	+	0.917235i	$0.369584\pi$
$182$	−16.5291	−1.22522
$183$	0	0
$184$	−9.78146	−0.721099
$185$	5.49208	0.403786
$186$	0	0
$187$	9.06927	0.663211
$188$	−2.00366	−0.146132
$189$	0	0
$190$	−1.81426	−0.131620
$191$	22.9909	1.66357	0.831783	−	0.555101i	$-0.187321\pi$
$191$	22.9909	1.66357	0.831783	+	0.555101i	$0.187321\pi$
$192$	0	0
$193$	−24.7790	−1.78363	−0.891817	−	0.452396i	$-0.850569\pi$
$193$	−24.7790	−1.78363	−0.891817	+	0.452396i	$0.850569\pi$
$194$	13.2488	0.951210
$195$	0	0
$196$	6.27918	0.448513
$197$	3.99380	0.284547	0.142273	−	0.989827i	$-0.454559\pi$
$197$	3.99380	0.284547	0.142273	+	0.989827i	$0.454559\pi$
$198$	0	0
$199$	−5.57308	−0.395065	−0.197532	−	0.980296i	$-0.563293\pi$
$199$	−5.57308	−0.395065	−0.197532	+	0.980296i	$0.563293\pi$
$200$	1.45816	0.103107
$201$	0	0
$202$	−18.7451	−1.31890
$203$	−21.6342	−1.51842
$204$	0	0
$205$	0.182644	0.0127564
$206$	−1.97441	−0.137564
$207$	0	0
$208$	13.1159	0.909426
$209$	3.62852	0.250990
$210$	0	0
$211$	6.98417	0.480810	0.240405	−	0.970673i	$-0.422720\pi$
$211$	6.98417	0.480810	0.240405	+	0.970673i	$0.422720\pi$
$212$	−1.87731	−0.128934
$213$	0	0
$214$	23.8100	1.62762
$215$	−10.2894	−0.701730
$216$	0	0
$217$	9.30335	0.631552
$218$	8.55937	0.579714
$219$	0	0
$220$	4.21967	0.284490
$221$	−6.71250	−0.451532
$222$	0	0
$223$	5.94472	0.398088	0.199044	−	0.979991i	$-0.436216\pi$
$223$	5.94472	0.398088	0.199044	+	0.979991i	$0.436216\pi$
$224$	−20.8549	−1.39343
$225$	0	0
$226$	−27.8739	−1.85414
$227$	−14.7016	−0.975782	−0.487891	−	0.872905i	$-0.662234\pi$
$227$	−14.7016	−0.975782	−0.487891	+	0.872905i	$0.662234\pi$
$228$	0	0
$229$	8.75232	0.578369	0.289185	−	0.957273i	$-0.406616\pi$
$229$	8.75232	0.578369	0.289185	+	0.957273i	$0.406616\pi$
$230$	11.9672	0.789094
$231$	0	0
$232$	8.99137	0.590313
$233$	24.0536	1.57581	0.787903	−	0.615800i	$-0.211167\pi$
$233$	24.0536	1.57581	0.787903	+	0.615800i	$0.211167\pi$
$234$	0	0
$235$	−1.69422	−0.110519
$236$	−9.34280	−0.608164
$237$	0	0
$238$	15.9097	1.03127
$239$	−6.32341	−0.409027	−0.204514	−	0.978864i	$-0.565561\pi$
$239$	−6.32341	−0.409027	−0.204514	+	0.978864i	$0.565561\pi$
$240$	0	0
$241$	−11.9672	−0.770876	−0.385438	−	0.922734i	$-0.625950\pi$
$241$	−11.9672	−0.770876	−0.385438	+	0.922734i	$0.625950\pi$
$242$	−3.08734	−0.198462
$243$	0	0
$244$	7.88098	0.504528
$245$	5.30944	0.339208
$246$	0	0
$247$	−2.68560	−0.170881
$248$	−3.86656	−0.245527
$249$	0	0
$250$	−1.78400	−0.112830
$251$	21.3011	1.34451	0.672257	−	0.740317i	$-0.265325\pi$
$251$	21.3011	1.34451	0.672257	+	0.740317i	$0.265325\pi$
$252$	0	0
$253$	−23.9344	−1.50474
$254$	21.9847	1.37944
$255$	0	0
$256$	20.4151	1.27594
$257$	−28.9164	−1.80376	−0.901879	−	0.431989i	$-0.857812\pi$
$257$	−28.9164	−1.80376	−0.901879	+	0.431989i	$0.857812\pi$
$258$	0	0
$259$	−19.2689	−1.19731
$260$	−3.12313	−0.193688
$261$	0	0
$262$	28.2257	1.74379
$263$	−9.50427	−0.586058	−0.293029	−	0.956103i	$-0.594663\pi$
$263$	−9.50427	−0.586058	−0.293029	+	0.956103i	$0.594663\pi$
$264$	0	0
$265$	−1.58738	−0.0975123
$266$	6.36529	0.390281
$267$	0	0
$268$	1.05708	0.0645716
$269$	−27.9794	−1.70593	−0.852967	−	0.521965i	$-0.825199\pi$
$269$	−27.9794	−1.70593	−0.852967	+	0.521965i	$0.825199\pi$
$270$	0	0
$271$	−20.0728	−1.21934	−0.609669	−	0.792656i	$-0.708698\pi$
$271$	−20.0728	−1.21934	−0.609669	+	0.792656i	$0.708698\pi$
$272$	−12.6244	−0.765467
$273$	0	0
$274$	10.3252	0.623766
$275$	3.56799	0.215158
$276$	0	0
$277$	7.18620	0.431777	0.215889	−	0.976418i	$-0.430735\pi$
$277$	7.18620	0.431777	0.215889	+	0.976418i	$0.430735\pi$
$278$	21.0776	1.26415
$279$	0	0
$280$	−5.11593	−0.305735
$281$	0.894943	0.0533878	0.0266939	−	0.999644i	$-0.491502\pi$
$281$	0.894943	0.0533878	0.0266939	+	0.999644i	$0.491502\pi$
$282$	0	0
$283$	−14.8157	−0.880701	−0.440350	−	0.897826i	$-0.645146\pi$
$283$	−14.8157	−0.880701	−0.440350	+	0.897826i	$0.645146\pi$
$284$	3.44897	0.204659
$285$	0	0
$286$	16.8095	0.993965
$287$	−0.640804	−0.0378255
$288$	0	0
$289$	−10.5390	−0.619944
$290$	−11.0006	−0.645975
$291$	0	0
$292$	−3.38225	−0.197931
$293$	12.6354	0.738167	0.369083	−	0.929396i	$-0.379672\pi$
$293$	12.6354	0.738167	0.369083	+	0.929396i	$0.379672\pi$
$294$	0	0
$295$	−7.89992	−0.459951
$296$	8.00834	0.465475
$297$	0	0
$298$	21.6019	1.25137
$299$	17.7147	1.02447
$300$	0	0
$301$	36.1001	2.08077
$302$	−28.8773	−1.66170
$303$	0	0
$304$	−5.05089	−0.289688
$305$	6.66386	0.381571
$306$	0	0
$307$	27.3731	1.56226	0.781132	−	0.624366i	$-0.214643\pi$
$307$	27.3731	1.56226	0.781132	+	0.624366i	$0.214643\pi$
$308$	−14.8046	−0.843572
$309$	0	0
$310$	4.73058	0.268679
$311$	−8.20748	−0.465404	−0.232702	−	0.972548i	$-0.574757\pi$
$311$	−8.20748	−0.465404	−0.232702	+	0.972548i	$0.574757\pi$
$312$	0	0
$313$	−9.75673	−0.551483	−0.275742	−	0.961232i	$-0.588923\pi$
$313$	−9.75673	−0.551483	−0.275742	+	0.961232i	$0.588923\pi$
$314$	10.9236	0.616457
$315$	0	0
$316$	0.788773	0.0443720
$317$	−16.4689	−0.924987	−0.462494	−	0.886623i	$-0.653045\pi$
$317$	−16.4689	−0.924987	−0.462494	+	0.886623i	$0.653045\pi$
$318$	0	0
$319$	22.0011	1.23183
$320$	−0.671065	−0.0375137
$321$	0	0
$322$	−41.9867	−2.33983
$323$	2.58496	0.143831
$324$	0	0
$325$	−2.64080	−0.146485
$326$	12.5376	0.694395
$327$	0	0
$328$	0.266325	0.0147053
$329$	5.94415	0.327712
$330$	0	0
$331$	1.92241	0.105665	0.0528327	−	0.998603i	$-0.483175\pi$
$331$	1.92241	0.105665	0.0528327	+	0.998603i	$0.483175\pi$
$332$	18.0862	0.992611
$333$	0	0
$334$	11.7064	0.640546
$335$	0.893830	0.0488352
$336$	0	0
$337$	−23.8472	−1.29904	−0.649519	−	0.760345i	$-0.725030\pi$
$337$	−23.8472	−1.29904	−0.649519	+	0.760345i	$0.725030\pi$
$338$	10.7506	0.584758
$339$	0	0
$340$	3.00609	0.163028
$341$	−9.46115	−0.512350
$342$	0	0
$343$	5.93130	0.320260
$344$	−15.0036	−0.808938
$345$	0	0
$346$	−34.2074	−1.83900
$347$	2.14675	0.115244	0.0576219	−	0.998338i	$-0.481648\pi$
$347$	2.14675	0.115244	0.0576219	+	0.998338i	$0.481648\pi$
$348$	0	0
$349$	−1.32173	−0.0707505	−0.0353753	−	0.999374i	$-0.511263\pi$
$349$	−1.32173	−0.0707505	−0.0353753	+	0.999374i	$0.511263\pi$
$350$	6.25912	0.334564
$351$	0	0
$352$	21.2087	1.13043
$353$	−25.7929	−1.37282	−0.686409	−	0.727216i	$-0.740814\pi$
$353$	−25.7929	−1.37282	−0.686409	+	0.727216i	$0.740814\pi$
$354$	0	0
$355$	2.91632	0.154782
$356$	−1.18264	−0.0626800
$357$	0	0
$358$	3.68702	0.194865
$359$	−22.8845	−1.20780	−0.603900	−	0.797060i	$-0.706387\pi$
$359$	−22.8845	−1.20780	−0.603900	+	0.797060i	$0.706387\pi$
$360$	0	0
$361$	−17.9658	−0.945568
$362$	−19.1216	−1.00501
$363$	0	0
$364$	10.9575	0.574326
$365$	−2.85991	−0.149694
$366$	0	0
$367$	−27.1216	−1.41573	−0.707867	−	0.706345i	$-0.750343\pi$
$367$	−27.1216	−1.41573	−0.707867	+	0.706345i	$0.750343\pi$
$368$	33.3167	1.73675
$369$	0	0
$370$	−9.79786	−0.509367
$371$	5.56931	0.289144
$372$	0	0
$373$	−31.9897	−1.65636	−0.828182	−	0.560459i	$-0.810625\pi$
$373$	−31.9897	−1.65636	−0.828182	+	0.560459i	$0.810625\pi$
$374$	−16.1795	−0.836624
$375$	0	0
$376$	−2.47045	−0.127404
$377$	−16.2839	−0.838661
$378$	0	0
$379$	−4.43314	−0.227715	−0.113858	−	0.993497i	$-0.536321\pi$
$379$	−4.43314	−0.227715	−0.113858	+	0.993497i	$0.536321\pi$
$380$	1.20270	0.0616975
$381$	0	0
$382$	−41.0157	−2.09855
$383$	0.637262	0.0325626	0.0162813	−	0.999867i	$-0.494817\pi$
$383$	0.637262	0.0325626	0.0162813	+	0.999867i	$0.494817\pi$
$384$	0	0
$385$	−12.5182	−0.637988
$386$	44.2057	2.25001
$387$	0	0
$388$	−8.78288	−0.445883
$389$	3.53508	0.179236	0.0896178	−	0.995976i	$-0.471435\pi$
$389$	3.53508	0.179236	0.0896178	+	0.995976i	$0.471435\pi$
$390$	0	0
$391$	−17.0509	−0.862300
$392$	7.74201	0.391031
$393$	0	0
$394$	−7.12493	−0.358949
$395$	0.666957	0.0335583
$396$	0	0
$397$	9.48674	0.476126	0.238063	−	0.971250i	$-0.423488\pi$
$397$	9.48674	0.476126	0.238063	+	0.971250i	$0.423488\pi$
$398$	9.94235	0.498365
$399$	0	0
$400$	−4.96664	−0.248332
$401$	−24.9997	−1.24843	−0.624213	−	0.781254i	$-0.714580\pi$
$401$	−24.9997	−1.24843	−0.624213	+	0.781254i	$0.714580\pi$
$402$	0	0
$403$	7.00255	0.348822
$404$	12.4265	0.618240
$405$	0	0
$406$	38.5953	1.91545
$407$	19.5957	0.971324
$408$	0	0
$409$	−21.9231	−1.08403	−0.542014	−	0.840370i	$-0.682338\pi$
$409$	−21.9231	−1.08403	−0.542014	+	0.840370i	$0.682338\pi$
$410$	−0.325837	−0.0160919
$411$	0	0
$412$	1.30887	0.0644836
$413$	27.7167	1.36385
$414$	0	0
$415$	15.2930	0.750706
$416$	−15.6973	−0.769625
$417$	0	0
$418$	−6.47326	−0.316617
$419$	−11.2016	−0.547234	−0.273617	−	0.961839i	$-0.588220\pi$
$419$	−11.2016	−0.547234	−0.273617	+	0.961839i	$0.588220\pi$
$420$	0	0
$421$	26.9358	1.31277	0.656386	−	0.754425i	$-0.272084\pi$
$421$	26.9358	1.31277	0.656386	+	0.754425i	$0.272084\pi$
$422$	−12.4597	−0.606530
$423$	0	0
$424$	−2.31466	−0.112410
$425$	2.54184	0.123297
$426$	0	0
$427$	−23.3800	−1.13144
$428$	−15.7841	−0.762954
$429$	0	0
$430$	18.3562	0.885216
$431$	−31.2580	−1.50565	−0.752823	−	0.658223i	$-0.771308\pi$
$431$	−31.2580	−1.50565	−0.752823	+	0.658223i	$0.771308\pi$
$432$	0	0
$433$	−26.5209	−1.27451	−0.637256	−	0.770652i	$-0.719931\pi$
$433$	−26.5209	−1.27451	−0.637256	+	0.770652i	$0.719931\pi$
$434$	−16.5971	−0.796688
$435$	0	0
$436$	−5.67416	−0.271743
$437$	−6.82187	−0.326334
$438$	0	0
$439$	−0.264646	−0.0126309	−0.00631543	−	0.999980i	$-0.502010\pi$
$439$	−0.264646	−0.0126309	−0.00631543	+	0.999980i	$0.502010\pi$
$440$	5.20270	0.248029
$441$	0	0
$442$	11.9751	0.569597
$443$	−27.8679	−1.32404	−0.662021	−	0.749485i	$-0.730301\pi$
$443$	−27.8679	−1.32404	−0.662021	+	0.749485i	$0.730301\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−10.6054	−0.502178
$447$	0	0
$448$	2.35442	0.111236
$449$	−23.8433	−1.12524	−0.562618	−	0.826717i	$-0.690206\pi$
$449$	−23.8433	−1.12524	−0.562618	+	0.826717i	$0.690206\pi$
$450$	0	0
$451$	0.651674	0.0306861
$452$	18.4781	0.869138
$453$	0	0
$454$	26.2277	1.23093
$455$	9.26521	0.434360
$456$	0	0
$457$	−31.8923	−1.49186	−0.745929	−	0.666026i	$-0.767994\pi$
$457$	−31.8923	−1.49186	−0.745929	+	0.666026i	$0.767994\pi$
$458$	−15.6141	−0.729599
$459$	0	0
$460$	−7.93328	−0.369891
$461$	−18.3531	−0.854789	−0.427395	−	0.904065i	$-0.640568\pi$
$461$	−18.3531	−0.854789	−0.427395	+	0.904065i	$0.640568\pi$
$462$	0	0
$463$	−20.0718	−0.932817	−0.466408	−	0.884570i	$-0.654452\pi$
$463$	−20.0718	−0.932817	−0.466408	+	0.884570i	$0.654452\pi$
$464$	−30.6255	−1.42175
$465$	0	0
$466$	−42.9116	−1.98784
$467$	30.8011	1.42530	0.712652	−	0.701518i	$-0.247494\pi$
$467$	30.8011	1.42530	0.712652	+	0.701518i	$0.247494\pi$
$468$	0	0
$469$	−3.13599	−0.144806
$470$	3.02249	0.139417
$471$	0	0
$472$	−11.5194	−0.530221
$473$	−36.7124	−1.68804
$474$	0	0
$475$	1.01696	0.0466614
$476$	−10.5468	−0.483413
$477$	0	0
$478$	11.2809	0.515978
$479$	3.16991	0.144837	0.0724185	−	0.997374i	$-0.476928\pi$
$479$	3.16991	0.144837	0.0724185	+	0.997374i	$0.476928\pi$
$480$	0	0
$481$	−14.5035	−0.661304
$482$	21.3495	0.972441
$483$	0	0
$484$	2.04666	0.0930299
$485$	−7.42648	−0.337219
$486$	0	0
$487$	−10.8972	−0.493799	−0.246899	−	0.969041i	$-0.579412\pi$
$487$	−10.8972	−0.493799	−0.246899	+	0.969041i	$0.579412\pi$
$488$	9.71697	0.439867
$489$	0	0
$490$	−9.47202	−0.427902
$491$	8.15086	0.367843	0.183922	−	0.982941i	$-0.441121\pi$
$491$	8.15086	0.367843	0.183922	+	0.982941i	$0.441121\pi$
$492$	0	0
$493$	15.6736	0.705904
$494$	4.79110	0.215562
$495$	0	0
$496$	13.1699	0.591346
$497$	−10.2319	−0.458961
$498$	0	0
$499$	10.3387	0.462823	0.231412	−	0.972856i	$-0.425666\pi$
$499$	10.3387	0.462823	0.231412	+	0.972856i	$0.425666\pi$
$500$	1.18264	0.0528895
$501$	0	0
$502$	−38.0011	−1.69607
$503$	−9.52168	−0.424551	−0.212275	−	0.977210i	$-0.568087\pi$
$503$	−9.52168	−0.424551	−0.212275	+	0.977210i	$0.568087\pi$
$504$	0	0
$505$	10.5074	0.467572
$506$	42.6989	1.89820
$507$	0	0
$508$	−14.5741	−0.646621
$509$	−18.3693	−0.814203	−0.407102	−	0.913383i	$-0.633460\pi$
$509$	−18.3693	−0.814203	−0.407102	+	0.913383i	$0.633460\pi$
$510$	0	0
$511$	10.0339	0.443875
$512$	−15.0382	−0.664599
$513$	0	0
$514$	51.5868	2.27540
$515$	1.10674	0.0487686
$516$	0	0
$517$	−6.04498	−0.265858
$518$	34.3756	1.51038
$519$	0	0
$520$	−3.85071	−0.168865
$521$	20.3459	0.891370	0.445685	−	0.895190i	$-0.352960\pi$
$521$	20.3459	0.891370	0.445685	+	0.895190i	$0.352960\pi$
$522$	0	0
$523$	−14.7040	−0.642960	−0.321480	−	0.946916i	$-0.604180\pi$
$523$	−14.7040	−0.642960	−0.321480	+	0.946916i	$0.604180\pi$
$524$	−18.7113	−0.817407
$525$	0	0
$526$	16.9556	0.739299
$527$	−6.74013	−0.293605
$528$	0	0
$529$	21.9984	0.956454
$530$	2.83189	0.123009
$531$	0	0
$532$	−4.21967	−0.182946
$533$	−0.482328	−0.0208919
$534$	0	0
$535$	−13.3465	−0.577018
$536$	1.30335	0.0562960
$537$	0	0
$538$	49.9151	2.15199
$539$	18.9440	0.815978
$540$	0	0
$541$	−7.27825	−0.312916	−0.156458	−	0.987685i	$-0.550008\pi$
$541$	−7.27825	−0.312916	−0.156458	+	0.987685i	$0.550008\pi$
$542$	35.8099	1.53817
$543$	0	0
$544$	15.1091	0.647797
$545$	−4.79786	−0.205518
$546$	0	0
$547$	−24.4564	−1.04568	−0.522840	−	0.852431i	$-0.675128\pi$
$547$	−24.4564	−1.04568	−0.522840	+	0.852431i	$0.675128\pi$
$548$	−6.84474	−0.292393
$549$	0	0
$550$	−6.36529	−0.271417
$551$	6.27084	0.267147
$552$	0	0
$553$	−2.34001	−0.0995073
$554$	−12.8202	−0.544677
$555$	0	0
$556$	−13.9727	−0.592576
$557$	28.6999	1.21605	0.608026	−	0.793917i	$-0.291962\pi$
$557$	28.6999	1.21605	0.608026	+	0.793917i	$0.291962\pi$
$558$	0	0
$559$	27.1722	1.14926
$560$	17.4254	0.736356
$561$	0	0
$562$	−1.59658	−0.0673475
$563$	−15.7168	−0.662386	−0.331193	−	0.943563i	$-0.607451\pi$
$563$	−15.7168	−0.662386	−0.331193	+	0.943563i	$0.607451\pi$
$564$	0	0
$565$	15.6244	0.657324
$566$	26.4311	1.11098
$567$	0	0
$568$	4.25246	0.178429
$569$	28.9958	1.21557	0.607783	−	0.794103i	$-0.292059\pi$
$569$	28.9958	1.21557	0.607783	+	0.794103i	$0.292059\pi$
$570$	0	0
$571$	24.5623	1.02790	0.513951	−	0.857820i	$-0.328181\pi$
$571$	24.5623	1.02790	0.513951	+	0.857820i	$0.328181\pi$
$572$	−11.1433	−0.465925
$573$	0	0
$574$	1.14319	0.0477159
$575$	−6.70809	−0.279747
$576$	0	0
$577$	16.7341	0.696648	0.348324	−	0.937374i	$-0.386751\pi$
$577$	16.7341	0.696648	0.348324	+	0.937374i	$0.386751\pi$
$578$	18.8016	0.782045
$579$	0	0
$580$	7.29248	0.302804
$581$	−53.6554	−2.22600
$582$	0	0
$583$	−5.66378	−0.234570
$584$	−4.17020	−0.172564
$585$	0	0
$586$	−22.5415	−0.931180
$587$	−7.14420	−0.294873	−0.147436	−	0.989072i	$-0.547102\pi$
$587$	−7.14420	−0.294873	−0.147436	+	0.989072i	$0.547102\pi$
$588$	0	0
$589$	−2.69665	−0.111114
$590$	14.0934	0.580218
$591$	0	0
$592$	−27.2772	−1.12109
$593$	−25.8829	−1.06288	−0.531441	−	0.847095i	$-0.678349\pi$
$593$	−25.8829	−1.06288	−0.531441	+	0.847095i	$0.678349\pi$
$594$	0	0
$595$	−8.91800	−0.365602
$596$	−14.3203	−0.586583
$597$	0	0
$598$	−31.6030	−1.29234
$599$	−14.0185	−0.572781	−0.286390	−	0.958113i	$-0.592455\pi$
$599$	−14.0185	−0.572781	−0.286390	+	0.958113i	$0.592455\pi$
$600$	0	0
$601$	32.7184	1.33461	0.667306	−	0.744784i	$-0.267447\pi$
$601$	32.7184	1.33461	0.667306	+	0.744784i	$0.267447\pi$
$602$	−64.4025	−2.62485
$603$	0	0
$604$	19.1433	0.778930
$605$	1.73058	0.0703580
$606$	0	0
$607$	30.9473	1.25611	0.628055	−	0.778169i	$-0.283851\pi$
$607$	30.9473	1.25611	0.628055	+	0.778169i	$0.283851\pi$
$608$	6.04498	0.245156
$609$	0	0
$610$	−11.8883	−0.481343
$611$	4.47411	0.181003
$612$	0	0
$613$	−31.0005	−1.25210	−0.626048	−	0.779784i	$-0.715329\pi$
$613$	−31.0005	−1.25210	−0.626048	+	0.779784i	$0.715329\pi$
$614$	−48.8335	−1.97076
$615$	0	0
$616$	−18.2536	−0.735458
$617$	25.3937	1.02231	0.511156	−	0.859488i	$-0.329218\pi$
$617$	25.3937	1.02231	0.511156	+	0.859488i	$0.329218\pi$
$618$	0	0
$619$	−41.7651	−1.67868	−0.839340	−	0.543606i	$-0.817058\pi$
$619$	−41.7651	−1.67868	−0.839340	+	0.543606i	$0.817058\pi$
$620$	−3.13599	−0.125944
$621$	0	0
$622$	14.6421	0.587096
$623$	3.50848	0.140564
$624$	0	0
$625$	1.00000	0.0400000
$626$	17.4060	0.695683
$627$	0	0
$628$	−7.24149	−0.288967
$629$	13.9600	0.556622
$630$	0	0
$631$	7.25445	0.288795	0.144397	−	0.989520i	$-0.453876\pi$
$631$	7.25445	0.288795	0.144397	+	0.989520i	$0.453876\pi$
$632$	0.972531	0.0386852
$633$	0	0
$634$	29.3805	1.16685
$635$	−12.3233	−0.489035
$636$	0	0
$637$	−14.0212	−0.555540
$638$	−39.2499	−1.55392
$639$	0	0
$640$	−10.6911	−0.422604
$641$	35.0026	1.38252	0.691259	−	0.722607i	$-0.257056\pi$
$641$	35.0026	1.38252	0.691259	+	0.722607i	$0.257056\pi$
$642$	0	0
$643$	−29.0773	−1.14670	−0.573349	−	0.819311i	$-0.694356\pi$
$643$	−29.0773	−1.14670	−0.573349	+	0.819311i	$0.694356\pi$
$644$	27.8338	1.09680
$645$	0	0
$646$	−4.61155	−0.181439
$647$	33.1392	1.30284	0.651418	−	0.758719i	$-0.274174\pi$
$647$	33.1392	1.30284	0.651418	+	0.758719i	$0.274174\pi$
$648$	0	0
$649$	−28.1869	−1.10643
$650$	4.71119	0.184788
$651$	0	0
$652$	−8.31143	−0.325501
$653$	41.8113	1.63620	0.818101	−	0.575075i	$-0.195027\pi$
$653$	41.8113	1.63620	0.818101	+	0.575075i	$0.195027\pi$
$654$	0	0
$655$	−15.8216	−0.618200
$656$	−0.907129	−0.0354174
$657$	0	0
$658$	−10.6043	−0.413400
$659$	−33.7077	−1.31306	−0.656532	−	0.754298i	$-0.727977\pi$
$659$	−33.7077	−1.31306	−0.656532	+	0.754298i	$0.727977\pi$
$660$	0	0
$661$	25.9638	1.00988	0.504938	−	0.863155i	$-0.331515\pi$
$661$	25.9638	1.00988	0.504938	+	0.863155i	$0.331515\pi$
$662$	−3.42958	−0.133294
$663$	0	0
$664$	22.2997	0.865396
$665$	−3.56799	−0.138361
$666$	0	0
$667$	−41.3637	−1.60161
$668$	−7.76039	−0.300259
$669$	0	0
$670$	−1.59459	−0.0616044
$671$	23.7766	0.917886
$672$	0	0
$673$	−13.5599	−0.522696	−0.261348	−	0.965245i	$-0.584167\pi$
$673$	−13.5599	−0.522696	−0.261348	+	0.965245i	$0.584167\pi$
$674$	42.5433	1.63871
$675$	0	0
$676$	−7.12680	−0.274108
$677$	−46.0778	−1.77091	−0.885457	−	0.464721i	$-0.846155\pi$
$677$	−46.0778	−1.77091	−0.885457	+	0.464721i	$0.846155\pi$
$678$	0	0
$679$	26.0557	0.999925
$680$	3.70641	0.142134
$681$	0	0
$682$	16.8787	0.646318
$683$	6.15220	0.235407	0.117704	−	0.993049i	$-0.462447\pi$
$683$	6.15220	0.235407	0.117704	+	0.993049i	$0.462447\pi$
$684$	0	0
$685$	−5.78766	−0.221135
$686$	−10.5814	−0.404000
$687$	0	0
$688$	51.1037	1.94831
$689$	4.19197	0.159701
$690$	0	0
$691$	−29.0053	−1.10342	−0.551708	−	0.834038i	$-0.686024\pi$
$691$	−29.0053	−1.10342	−0.551708	+	0.834038i	$0.686024\pi$
$692$	22.6767	0.862039
$693$	0	0
$694$	−3.82980	−0.145377
$695$	−11.8148	−0.448162
$696$	0	0
$697$	0.464253	0.0175848
$698$	2.35796	0.0892501
$699$	0	0
$700$	−4.14929	−0.156828
$701$	46.3859	1.75197	0.875986	−	0.482336i	$-0.160212\pi$
$701$	46.3859	1.75197	0.875986	+	0.482336i	$0.160212\pi$
$702$	0	0
$703$	5.58524	0.210651
$704$	−2.39436	−0.0902407
$705$	0	0
$706$	46.0144	1.73178
$707$	−36.8649	−1.38645
$708$	0	0
$709$	43.6893	1.64079	0.820393	−	0.571801i	$-0.193755\pi$
$709$	43.6893	1.64079	0.820393	+	0.571801i	$0.193755\pi$
$710$	−5.20270	−0.195254
$711$	0	0
$712$	−1.45816	−0.0546469
$713$	17.7877	0.666153
$714$	0	0
$715$	−9.42237	−0.352377
$716$	−2.44419	−0.0913438
$717$	0	0
$718$	40.8259	1.52361
$719$	−36.3623	−1.35609	−0.678043	−	0.735022i	$-0.737172\pi$
$719$	−36.3623	−1.35609	−0.678043	+	0.735022i	$0.737172\pi$
$720$	0	0
$721$	−3.88296	−0.144609
$722$	32.0509	1.19281
$723$	0	0
$724$	12.6760	0.471102
$725$	6.16625	0.229009
$726$	0	0
$727$	7.34646	0.272465	0.136233	−	0.990677i	$-0.456501\pi$
$727$	7.34646	0.272465	0.136233	+	0.990677i	$0.456501\pi$
$728$	13.5102	0.500720
$729$	0	0
$730$	5.10206	0.188836
$731$	−26.1540	−0.967339
$732$	0	0
$733$	−12.3394	−0.455768	−0.227884	−	0.973688i	$-0.573181\pi$
$733$	−12.3394	−0.455768	−0.227884	+	0.973688i	$0.573181\pi$
$734$	48.3848	1.78592
$735$	0	0
$736$	−39.8739	−1.46977
$737$	3.18918	0.117475
$738$	0	0
$739$	−2.22586	−0.0818797	−0.0409399	−	0.999162i	$-0.513035\pi$
$739$	−2.22586	−0.0818797	−0.0409399	+	0.999162i	$0.513035\pi$
$740$	6.49518	0.238768
$741$	0	0
$742$	−9.93563	−0.364748
$743$	−40.7739	−1.49585	−0.747925	−	0.663783i	$-0.768950\pi$
$743$	−40.7739	−1.49585	−0.747925	+	0.663783i	$0.768950\pi$
$744$	0	0
$745$	−12.1087	−0.443629
$746$	57.0695	2.08946
$747$	0	0
$748$	10.7257	0.392171
$749$	46.8258	1.71098
$750$	0	0
$751$	−19.3369	−0.705614	−0.352807	−	0.935696i	$-0.614773\pi$
$751$	−19.3369	−0.705614	−0.352807	+	0.935696i	$0.614773\pi$
$752$	8.41460	0.306849
$753$	0	0
$754$	29.0503	1.05795
$755$	16.1869	0.589101
$756$	0	0
$757$	−7.29304	−0.265070	−0.132535	−	0.991178i	$-0.542312\pi$
$757$	−7.29304	−0.265070	−0.132535	+	0.991178i	$0.542312\pi$
$758$	7.90870	0.287257
$759$	0	0
$760$	1.48289	0.0537902
$761$	41.8043	1.51541	0.757703	−	0.652600i	$-0.226322\pi$
$761$	41.8043	1.51541	0.757703	+	0.652600i	$0.226322\pi$
$762$	0	0
$763$	16.8332	0.609403
$764$	27.1901	0.983703
$765$	0	0
$766$	−1.13687	−0.0410769
$767$	20.8622	0.753289
$768$	0	0
$769$	0.274587	0.00990186	0.00495093	−	0.999988i	$-0.498424\pi$
$769$	0.274587	0.00990186	0.00495093	+	0.999988i	$0.498424\pi$
$770$	22.3325	0.804807
$771$	0	0
$772$	−29.3048	−1.05470
$773$	47.7833	1.71865	0.859324	−	0.511432i	$-0.170885\pi$
$773$	47.7833	1.71865	0.859324	+	0.511432i	$0.170885\pi$
$774$	0	0
$775$	−2.65167	−0.0952509
$776$	−10.8290	−0.388738
$777$	0	0
$778$	−6.30657	−0.226101
$779$	0.185742	0.00665491
$780$	0	0
$781$	10.4054	0.372335
$782$	30.4187	1.08777
$783$	0	0
$784$	−26.3701	−0.941789
$785$	−6.12313	−0.218544
$786$	0	0
$787$	46.1319	1.64442	0.822211	−	0.569182i	$-0.192740\pi$
$787$	46.1319	1.64442	0.822211	+	0.569182i	$0.192740\pi$
$788$	4.72325	0.168259
$789$	0	0
$790$	−1.18985	−0.0423330
$791$	−54.8179	−1.94910
$792$	0	0
$793$	−17.5979	−0.624921
$794$	−16.9243	−0.600622
$795$	0	0
$796$	−6.59097	−0.233611
$797$	39.7050	1.40642	0.703212	−	0.710980i	$-0.251749\pi$
$797$	39.7050	1.40642	0.703212	+	0.710980i	$0.251749\pi$
$798$	0	0
$799$	−4.30645	−0.152351
$800$	5.94415	0.210157
$801$	0	0
$802$	44.5994	1.57486
$803$	−10.2041	−0.360096
$804$	0	0
$805$	23.5352	0.829507
$806$	−12.4925	−0.440031
$807$	0	0
$808$	15.3214	0.539006
$809$	16.5306	0.581186	0.290593	−	0.956847i	$-0.406147\pi$
$809$	16.5306	0.581186	0.290593	+	0.956847i	$0.406147\pi$
$810$	0	0
$811$	1.70651	0.0599238	0.0299619	−	0.999551i	$-0.490461\pi$
$811$	1.70651	0.0599238	0.0299619	+	0.999551i	$0.490461\pi$
$812$	−25.5855	−0.897876
$813$	0	0
$814$	−34.9587	−1.22530
$815$	−7.02783	−0.246174
$816$	0	0
$817$	−10.4639	−0.366086
$818$	39.1107	1.36747
$819$	0	0
$820$	0.216003	0.00754316
$821$	43.2867	1.51072	0.755358	−	0.655312i	$-0.227463\pi$
$821$	43.2867	1.51072	0.755358	+	0.655312i	$0.227463\pi$
$822$	0	0
$823$	27.9775	0.975235	0.487617	−	0.873057i	$-0.337866\pi$
$823$	27.9775	0.975235	0.487617	+	0.873057i	$0.337866\pi$
$824$	1.61380	0.0562193
$825$	0	0
$826$	−49.4466	−1.72047
$827$	17.0361	0.592403	0.296202	−	0.955125i	$-0.404280\pi$
$827$	17.0361	0.592403	0.296202	+	0.955125i	$0.404280\pi$
$828$	0	0
$829$	−2.10663	−0.0731664	−0.0365832	−	0.999331i	$-0.511647\pi$
$829$	−2.10663	−0.0731664	−0.0365832	+	0.999331i	$0.511647\pi$
$830$	−27.2827	−0.946998
$831$	0	0
$832$	1.77215	0.0614383
$833$	13.4957	0.467600
$834$	0	0
$835$	−6.56190	−0.227084
$836$	4.29124	0.148416
$837$	0	0
$838$	19.9836	0.690322
$839$	−27.6740	−0.955412	−0.477706	−	0.878520i	$-0.658532\pi$
$839$	−27.6740	−0.955412	−0.477706	+	0.878520i	$0.658532\pi$
$840$	0	0
$841$	9.02261	0.311125
$842$	−48.0534	−1.65603
$843$	0	0
$844$	8.25979	0.284314
$845$	−6.02615	−0.207306
$846$	0	0
$847$	−6.07170	−0.208626
$848$	7.88397	0.270737
$849$	0	0
$850$	−4.53463	−0.155537
$851$	−36.8414	−1.26291
$852$	0	0
$853$	7.52751	0.257737	0.128869	−	0.991662i	$-0.458865\pi$
$853$	7.52751	0.257737	0.128869	+	0.991662i	$0.458865\pi$
$854$	41.7099	1.42728
$855$	0	0
$856$	−19.4613	−0.665173
$857$	−19.7447	−0.674466	−0.337233	−	0.941421i	$-0.609491\pi$
$857$	−19.7447	−0.674466	−0.337233	+	0.941421i	$0.609491\pi$
$858$	0	0
$859$	20.2075	0.689470	0.344735	−	0.938700i	$-0.387969\pi$
$859$	20.2075	0.689470	0.344735	+	0.938700i	$0.387969\pi$
$860$	−12.1687	−0.414948
$861$	0	0
$862$	55.7642	1.89934
$863$	−14.5448	−0.495112	−0.247556	−	0.968874i	$-0.579627\pi$
$863$	−14.5448	−0.495112	−0.247556	+	0.968874i	$0.579627\pi$
$864$	0	0
$865$	19.1746	0.651955
$866$	47.3132	1.60777
$867$	0	0
$868$	11.0026	0.373451
$869$	2.37970	0.0807258
$870$	0	0
$871$	−2.36043	−0.0799801
$872$	−6.99605	−0.236916
$873$	0	0
$874$	12.1702	0.411663
$875$	−3.50848	−0.118608
$876$	0	0
$877$	13.4029	0.452582	0.226291	−	0.974060i	$-0.427340\pi$
$877$	13.4029	0.452582	0.226291	+	0.974060i	$0.427340\pi$
$878$	0.472128	0.0159335
$879$	0	0
$880$	−17.7209	−0.597373
$881$	−22.6494	−0.763079	−0.381540	−	0.924352i	$-0.624606\pi$
$881$	−22.6494	−0.763079	−0.381540	+	0.924352i	$0.624606\pi$
$882$	0	0
$883$	34.3103	1.15463	0.577316	−	0.816521i	$-0.304100\pi$
$883$	34.3103	1.15463	0.577316	+	0.816521i	$0.304100\pi$
$884$	−7.93850	−0.267001
$885$	0	0
$886$	49.7162	1.67025
$887$	36.1037	1.21224	0.606121	−	0.795372i	$-0.292725\pi$
$887$	36.1037	1.21224	0.606121	+	0.795372i	$0.292725\pi$
$888$	0	0
$889$	43.2361	1.45009
$890$	1.78400	0.0597997
$891$	0	0
$892$	7.03049	0.235398
$893$	−1.72296	−0.0576567
$894$	0	0
$895$	−2.06672	−0.0690828
$896$	37.5096	1.25311
$897$	0	0
$898$	42.5364	1.41946
$899$	−16.3509	−0.545332
$900$	0	0
$901$	−4.03488	−0.134421
$902$	−1.16258	−0.0387098
$903$	0	0
$904$	22.7829	0.757747
$905$	10.7184	0.356291
$906$	0	0
$907$	−30.5673	−1.01497	−0.507486	−	0.861660i	$-0.669425\pi$
$907$	−30.5673	−1.01497	−0.507486	+	0.861660i	$0.669425\pi$
$908$	−17.3868	−0.577001
$909$	0	0
$910$	−16.5291	−0.547935
$911$	−6.28974	−0.208388	−0.104194	−	0.994557i	$-0.533226\pi$
$911$	−6.28974	−0.208388	−0.104194	+	0.994557i	$0.533226\pi$
$912$	0	0
$913$	54.5655	1.80585
$914$	56.8957	1.88194
$915$	0	0
$916$	10.3509	0.342003
$917$	55.5097	1.83309
$918$	0	0
$919$	−39.8315	−1.31392	−0.656961	−	0.753925i	$-0.728158\pi$
$919$	−39.8315	−1.31392	−0.656961	+	0.753925i	$0.728158\pi$
$920$	−9.78146	−0.322485
$921$	0	0
$922$	32.7419	1.07830
$923$	−7.70143	−0.253496
$924$	0	0
$925$	5.49208	0.180579
$926$	35.8081	1.17673
$927$	0	0
$928$	36.6531	1.20320
$929$	−15.5075	−0.508784	−0.254392	−	0.967101i	$-0.581875\pi$
$929$	−15.5075	−0.508784	−0.254392	+	0.967101i	$0.581875\pi$
$930$	0	0
$931$	5.39950	0.176962
$932$	28.4469	0.931809
$933$	0	0
$934$	−54.9490	−1.79799
$935$	9.06927	0.296597
$936$	0	0
$937$	−51.0805	−1.66873	−0.834364	−	0.551215i	$-0.814165\pi$
$937$	−51.0805	−1.66873	−0.834364	+	0.551215i	$0.814165\pi$
$938$	5.59459	0.182670
$939$	0	0
$940$	−2.00366	−0.0653523
$941$	−29.3251	−0.955971	−0.477986	−	0.878368i	$-0.658633\pi$
$941$	−29.3251	−0.955971	−0.477986	+	0.878368i	$0.658633\pi$
$942$	0	0
$943$	−1.22519	−0.0398978
$944$	39.2361	1.27703
$945$	0	0
$946$	65.4949	2.12942
$947$	17.8336	0.579514	0.289757	−	0.957100i	$-0.406426\pi$
$947$	17.8336	0.579514	0.289757	+	0.957100i	$0.406426\pi$
$948$	0	0
$949$	7.55245	0.245163
$950$	−1.81426	−0.0588623
$951$	0	0
$952$	−13.0039	−0.421458
$953$	36.2229	1.17337	0.586687	−	0.809814i	$-0.300432\pi$
$953$	36.2229	1.17337	0.586687	+	0.809814i	$0.300432\pi$
$954$	0	0
$955$	22.9909	0.743969
$956$	−7.47834	−0.241867
$957$	0	0
$958$	−5.65511	−0.182708
$959$	20.3059	0.655712
$960$	0	0
$961$	−23.9686	−0.773181
$962$	25.8742	0.834219
$963$	0	0
$964$	−14.1529	−0.455836
$965$	−24.7790	−0.797665
$966$	0	0
$967$	20.0275	0.644042	0.322021	−	0.946732i	$-0.395638\pi$
$967$	20.0275	0.644042	0.322021	+	0.946732i	$0.395638\pi$
$968$	2.52346	0.0811070
$969$	0	0
$970$	13.2488	0.425394
$971$	−55.2949	−1.77450	−0.887249	−	0.461290i	$-0.847387\pi$
$971$	−55.2949	−1.77450	−0.887249	+	0.461290i	$0.847387\pi$
$972$	0	0
$973$	41.4521	1.32889
$974$	19.4405	0.622915
$975$	0	0
$976$	−33.0970	−1.05941
$977$	9.66965	0.309359	0.154680	−	0.987965i	$-0.450565\pi$
$977$	9.66965	0.309359	0.154680	+	0.987965i	$0.450565\pi$
$978$	0	0
$979$	−3.56799	−0.114034
$980$	6.27918	0.200581
$981$	0	0
$982$	−14.5411	−0.464025
$983$	26.4073	0.842262	0.421131	−	0.907000i	$-0.361633\pi$
$983$	26.4073	0.842262	0.421131	+	0.907000i	$0.361633\pi$
$984$	0	0
$985$	3.99380	0.127253
$986$	−27.9617	−0.890481
$987$	0	0
$988$	−3.17611	−0.101045
$989$	69.0221	2.19477
$990$	0	0
$991$	−11.9805	−0.380574	−0.190287	−	0.981729i	$-0.560942\pi$
$991$	−11.9805	−0.380574	−0.190287	+	0.981729i	$0.560942\pi$
$992$	−15.7620	−0.500442
$993$	0	0
$994$	18.2536	0.578969
$995$	−5.57308	−0.176678
$996$	0	0
$997$	−18.8211	−0.596071	−0.298035	−	0.954555i	$-0.596331\pi$
$997$	−18.8211	−0.596071	−0.298035	+	0.954555i	$0.596331\pi$
$998$	−18.4442	−0.583841
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.m.1.1		4
3.2	odd	2		1335.2.a.f.1.4	✓	4
15.14	odd	2		6675.2.a.r.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.f.1.4	✓	4	3.2	odd	2
4005.2.a.m.1.1		4	1.1	even	1	trivial
6675.2.a.r.1.1		4	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)

\(f(q)\)	\(=\)	\(q-1.78400 q^{2} +1.18264 q^{4} +1.00000 q^{5} -3.50848 q^{7} +1.45816 q^{8} +O(q^{10})\)	\(q-1.78400 q^{2} +1.18264 q^{4} +1.00000 q^{5} -3.50848 q^{7} +1.45816 q^{8} -1.78400 q^{10} +3.56799 q^{11} -2.64080 q^{13} +6.25912 q^{14} -4.96664 q^{16} +2.54184 q^{17} +1.01696 q^{19} +1.18264 q^{20} -6.36529 q^{22} -6.70809 q^{23} +1.00000 q^{25} +4.71119 q^{26} -4.14929 q^{28} +6.16625 q^{29} -2.65167 q^{31} +5.94415 q^{32} -4.53463 q^{34} -3.50848 q^{35} +5.49208 q^{37} -1.81426 q^{38} +1.45816 q^{40} +0.182644 q^{41} -10.2894 q^{43} +4.21967 q^{44} +11.9672 q^{46} -1.69422 q^{47} +5.30944 q^{49} -1.78400 q^{50} -3.12313 q^{52} -1.58738 q^{53} +3.56799 q^{55} -5.11593 q^{56} -11.0006 q^{58} -7.89992 q^{59} +6.66386 q^{61} +4.73058 q^{62} -0.671065 q^{64} -2.64080 q^{65} +0.893830 q^{67} +3.00609 q^{68} +6.25912 q^{70} +2.91632 q^{71} -2.85991 q^{73} -9.79786 q^{74} +1.20270 q^{76} -12.5182 q^{77} +0.666957 q^{79} -4.96664 q^{80} -0.325837 q^{82} +15.2930 q^{83} +2.54184 q^{85} +18.3562 q^{86} +5.20270 q^{88} -1.00000 q^{89} +9.26521 q^{91} -7.93328 q^{92} +3.02249 q^{94} +1.01696 q^{95} -7.42648 q^{97} -9.47202 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8	\( 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{13} + q^{14} - 10 q^{16} + 13 q^{17} - 10 q^{19} + 2 q^{20} - 20 q^{22} - 3 q^{23} + 4 q^{25} + 3 q^{26} - 4 q^{28} - 2 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} - 7 q^{35} - 9 q^{37} - 2 q^{38} + 3 q^{40} - 2 q^{41} - 19 q^{43} - 6 q^{44} - 5 q^{47} - 7 q^{49} - 17 q^{52} + 3 q^{53} + 2 q^{56} - 6 q^{58} - 2 q^{59} - 4 q^{61} + 8 q^{62} + q^{64} - 5 q^{65} - 15 q^{67} + q^{68} + q^{70} + 6 q^{71} - 21 q^{73} - 10 q^{74} - 4 q^{76} - 2 q^{77} - 20 q^{79} - 10 q^{80} + 3 q^{82} + 9 q^{83} + 13 q^{85} - 16 q^{86} + 12 q^{88} - 4 q^{89} + 2 q^{91} - 12 q^{92} + 25 q^{94} - 10 q^{95} - 17 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8 - 5 * q^13 + q^14 - 10 * q^16 + 13 * q^17 - 10 * q^19 + 2 * q^20 - 20 * q^22 - 3 * q^23 + 4 * q^25 + 3 * q^26 - 4 * q^28 - 2 * q^29 - 2 * q^31 + q^32 + 6 * q^34 - 7 * q^35 - 9 * q^37 - 2 * q^38 + 3 * q^40 - 2 * q^41 - 19 * q^43 - 6 * q^44 - 5 * q^47 - 7 * q^49 - 17 * q^52 + 3 * q^53 + 2 * q^56 - 6 * q^58 - 2 * q^59 - 4 * q^61 + 8 * q^62 + q^64 - 5 * q^65 - 15 * q^67 + q^68 + q^70 + 6 * q^71 - 21 * q^73 - 10 * q^74 - 4 * q^76 - 2 * q^77 - 20 * q^79 - 10 * q^80 + 3 * q^82 + 9 * q^83 + 13 * q^85 - 16 * q^86 + 12 * q^88 - 4 * q^89 + 2 * q^91 - 12 * q^92 + 25 * q^94 - 10 * q^95 - 17 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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