LMFDB - Embedded newform 9027.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9027,2,Mod(1,9027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9027, 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("9027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9027 = 3^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9027.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:	$72.0809579046$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1003)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +O(q^{10})$

$q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} -4.00000 q^{10} +3.00000 q^{11} +4.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} +4.00000 q^{20} -6.00000 q^{22} -1.00000 q^{23} -1.00000 q^{25} -8.00000 q^{26} -4.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} +8.00000 q^{32} +2.00000 q^{34} -4.00000 q^{35} -2.00000 q^{37} +2.00000 q^{38} +4.00000 q^{43} +6.00000 q^{44} +2.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} +2.00000 q^{50} +8.00000 q^{52} +3.00000 q^{53} +6.00000 q^{55} -12.0000 q^{58} -1.00000 q^{59} +5.00000 q^{61} +16.0000 q^{62} -8.00000 q^{64} +8.00000 q^{65} +10.0000 q^{67} -2.00000 q^{68} +8.00000 q^{70} +4.00000 q^{71} +15.0000 q^{73} +4.00000 q^{74} -2.00000 q^{76} -6.00000 q^{77} +6.00000 q^{79} -8.00000 q^{80} -2.00000 q^{83} -2.00000 q^{85} -8.00000 q^{86} +14.0000 q^{89} -8.00000 q^{91} -2.00000 q^{92} +24.0000 q^{94} -2.00000 q^{95} -19.0000 q^{97} +6.00000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	−4.00000	−1.26491
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−8.00000	−1.56893
$27$	0	0
$28$	−4.00000	−0.755929
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	2.00000	0.294884
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	2.00000	0.282843
$51$	0	0
$52$	8.00000	1.10940
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	−12.0000	−1.57568
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	16.0000	2.03200
$63$	0	0
$64$	−8.00000	−1.00000
$65$	8.00000	0.992278
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	8.00000	0.956183
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−2.00000	−0.229416
$77$	−6.00000	−0.683763
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−8.00000	−0.894427
$81$	0	0
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	−2.00000	−0.208514
$93$	0	0
$94$	24.0000	2.47541
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−19.0000	−1.92916	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−19.0000	−1.92916	−0.964579	+	0.263795i	$0.915026\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	−2.00000	−0.200000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	−12.0000	−1.14416
$111$	0	0
$112$	8.00000	0.755929
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	12.0000	1.11417
$117$	0	0
$118$	2.00000	0.184115
$119$	2.00000	0.183340
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−16.0000	−1.43684
$125$	−12.0000	−1.07331
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	0	0
$130$	−16.0000	−1.40329
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−20.0000	−1.72774
$135$	0	0
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−8.00000	−0.676123
$141$	0	0
$142$	−8.00000	−0.671345
$143$	12.0000	1.00349
$144$	0	0
$145$	12.0000	0.996546
$146$	−30.0000	−2.48282
$147$	0	0
$148$	−4.00000	−0.328798
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	12.0000	0.966988
$155$	−16.0000	−1.28515
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	16.0000	1.26491
$161$	2.00000	0.157622
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	0	0
$166$	4.00000	0.310460
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	4.00000	0.306786
$171$	0	0
$172$	8.00000	0.609994
$173$	5.00000	0.380143	0.190071	−	0.981770i	$-0.439128\pi$
$173$	5.00000	0.380143	0.190071	+	0.981770i	$0.439128\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−12.0000	−0.904534
$177$	0	0
$178$	−28.0000	−2.09869
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	16.0000	1.18600
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−3.00000	−0.219382
$188$	−24.0000	−1.75038
$189$	0	0
$190$	4.00000	0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	38.0000	2.72824
$195$	0	0
$196$	−6.00000	−0.428571
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	−8.00000	−0.562878
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	20.0000	1.39347
$207$	0	0
$208$	−16.0000	−1.10940
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−20.0000	−1.36717
$215$	8.00000	0.545595
$216$	0	0
$217$	16.0000	1.08615
$218$	−6.00000	−0.406371
$219$	0	0
$220$	12.0000	0.809040
$221$	−4.00000	−0.269069
$222$	0	0
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	−4.00000	−0.266076
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	−2.00000	−0.130189
$237$	0	0
$238$	−4.00000	−0.259281
$239$	7.00000	0.452792	0.226396	−	0.974035i	$-0.427306\pi$
$239$	7.00000	0.452792	0.226396	+	0.974035i	$0.427306\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	4.00000	0.257130
$243$	0	0
$244$	10.0000	0.640184
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	24.0000	1.51789
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	−40.0000	−2.50982
$255$	0	0
$256$	16.0000	1.00000
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	16.0000	0.992278
$261$	0	0
$262$	−40.0000	−2.47121
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−4.00000	−0.245256
$267$	0	0
$268$	20.0000	1.22169
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	18.0000	1.08742
$275$	−3.00000	−0.180907
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	−24.0000	−1.40933
$291$	0	0
$292$	30.0000	1.75562
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	0	0
$297$	0	0
$298$	−44.0000	−2.54885
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−8.00000	−0.461112
$302$	32.0000	1.84139
$303$	0	0
$304$	4.00000	0.229416
$305$	10.0000	0.572598
$306$	0	0
$307$	−15.0000	−0.856095	−0.428048	−	0.903756i	$-0.640798\pi$
$307$	−15.0000	−0.856095	−0.428048	+	0.903756i	$0.640798\pi$
$308$	−12.0000	−0.683763
$309$	0	0
$310$	32.0000	1.81748
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−20.0000	−1.12867
$315$	0	0
$316$	12.0000	0.675053
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	−16.0000	−0.894427
$321$	0	0
$322$	−4.00000	−0.222911
$323$	1.00000	0.0556415
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−24.0000	−1.32924
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	−44.0000	−2.40757
$335$	20.0000	1.09272
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−6.00000	−0.326357
$339$	0	0
$340$	−4.00000	−0.216930
$341$	−24.0000	−1.29967
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−5.00000	−0.268414	−0.134207	−	0.990953i	$-0.542849\pi$
$347$	−5.00000	−0.268414	−0.134207	+	0.990953i	$0.542849\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	24.0000	1.27920
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	28.0000	1.48400
$357$	0	0
$358$	24.0000	1.26844
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	16.0000	0.840941
$363$	0	0
$364$	−16.0000	−0.838628
$365$	30.0000	1.57027
$366$	0	0
$367$	−5.00000	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$367$	−5.00000	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	8.00000	0.415900
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−3.00000	−0.155334	−0.0776671	−	0.996979i	$-0.524747\pi$
$373$	−3.00000	−0.155334	−0.0776671	+	0.996979i	$0.524747\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	16.0000	0.818631
$383$	3.00000	0.153293	0.0766464	−	0.997058i	$-0.475579\pi$
$383$	3.00000	0.153293	0.0766464	+	0.997058i	$0.475579\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−38.0000	−1.92916
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	0	0
$393$	0	0
$394$	−24.0000	−1.20910
$395$	12.0000	0.603786
$396$	0	0
$397$	−9.00000	−0.451697	−0.225849	−	0.974162i	$-0.572515\pi$
$397$	−9.00000	−0.451697	−0.225849	+	0.974162i	$0.572515\pi$
$398$	40.0000	2.00502
$399$	0	0
$400$	4.00000	0.200000
$401$	−7.00000	−0.349563	−0.174782	−	0.984607i	$-0.555922\pi$
$401$	−7.00000	−0.349563	−0.174782	+	0.984607i	$0.555922\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	8.00000	0.398015
$405$	0	0
$406$	24.0000	1.19110
$407$	−6.00000	−0.297409
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	0	0
$412$	−20.0000	−0.985329
$413$	2.00000	0.0984136
$414$	0	0
$415$	−4.00000	−0.196352
$416$	32.0000	1.56893
$417$	0	0
$418$	6.00000	0.293470
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	2.00000	0.0973585
$423$	0	0
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	−10.0000	−0.483934
$428$	20.0000	0.966736
$429$	0	0
$430$	−16.0000	−0.771589
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	6.00000	0.287348
$437$	1.00000	0.0478365
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	0	0
$442$	8.00000	0.380521
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	28.0000	1.32733
$446$	−42.0000	−1.98876
$447$	0	0
$448$	16.0000	0.755929
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	0	0
$452$	4.00000	0.188144
$453$	0	0
$454$	24.0000	1.12638
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	48.0000	2.21407
$471$	0	0
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	1.00000	0.0458831
$476$	4.00000	0.183340
$477$	0	0
$478$	−14.0000	−0.640345
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	−24.0000	−1.09317
$483$	0	0
$484$	−4.00000	−0.181818
$485$	−38.0000	−1.72549
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	0	0
$490$	12.0000	0.542105
$491$	−17.0000	−0.767199	−0.383600	−	0.923499i	$-0.625316\pi$
$491$	−17.0000	−0.767199	−0.383600	+	0.923499i	$0.625316\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	8.00000	0.359937
$495$	0	0
$496$	32.0000	1.43684
$497$	−8.00000	−0.358849
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	−42.0000	−1.87455
$503$	−27.0000	−1.20387	−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000	−1.20387	−0.601935	+	0.798545i	$0.705603\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	6.00000	0.266733
$507$	0	0
$508$	40.0000	1.77471
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−30.0000	−1.32712
$512$	−32.0000	−1.41421
$513$	0	0
$514$	−30.0000	−1.32324
$515$	−20.0000	−0.881305
$516$	0	0
$517$	−36.0000	−1.58328
$518$	−8.00000	−0.351500
$519$	0	0
$520$	0	0
$521$	−32.0000	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$521$	−32.0000	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$522$	0	0
$523$	9.00000	0.393543	0.196771	−	0.980449i	$-0.436954\pi$
$523$	9.00000	0.393543	0.196771	+	0.980449i	$0.436954\pi$
$524$	40.0000	1.74741
$525$	0	0
$526$	46.0000	2.00570
$527$	8.00000	0.348485
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−12.0000	−0.521247
$531$	0	0
$532$	4.00000	0.173422
$533$	0	0
$534$	0	0
$535$	20.0000	0.864675
$536$	0	0
$537$	0	0
$538$	−34.0000	−1.46584
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	50.0000	2.14768
$543$	0	0
$544$	−8.00000	−0.342997
$545$	6.00000	0.257012
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	6.00000	0.255841
$551$	−6.00000	−0.255609
$552$	0	0
$553$	−12.0000	−0.510292
$554$	−16.0000	−0.679775
$555$	0	0
$556$	−16.0000	−0.678551
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	16.0000	0.676123
$561$	0	0
$562$	−54.0000	−2.27785
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	24.0000	1.00349
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	−2.00000	−0.0831890
$579$	0	0
$580$	24.0000	0.996546
$581$	4.00000	0.165948
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	0	0
$586$	18.0000	0.743573
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	4.00000	0.164677
$591$	0	0
$592$	8.00000	0.328798
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	44.0000	1.80231
$597$	0	0
$598$	8.00000	0.327144
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	−32.0000	−1.30206
$605$	−4.00000	−0.162623
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	−20.0000	−0.809776
$611$	−48.0000	−1.94187
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	30.0000	1.21070
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	−32.0000	−1.28515
$621$	0	0
$622$	20.0000	0.801927
$623$	−28.0000	−1.12180
$624$	0	0
$625$	−19.0000	−0.760000
$626$	20.0000	0.799361
$627$	0	0
$628$	20.0000	0.798087
$629$	2.00000	0.0797452
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	0	0
$634$	−56.0000	−2.22404
$635$	40.0000	1.58735
$636$	0	0
$637$	−12.0000	−0.475457
$638$	−36.0000	−1.42525
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	43.0000	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$647$	43.0000	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	8.00000	0.313786
$651$	0	0
$652$	24.0000	0.939913
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	40.0000	1.56293
$656$	0	0
$657$	0	0
$658$	−48.0000	−1.87123
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	16.0000	0.621858
$663$	0	0
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	−6.00000	−0.232321
$668$	44.0000	1.70241
$669$	0	0
$670$	−40.0000	−1.54533
$671$	15.0000	0.579069
$672$	0	0
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	6.00000	0.230769
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	38.0000	1.45831
$680$	0	0
$681$	0	0
$682$	48.0000	1.83801
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	−40.0000	−1.52721
$687$	0	0
$688$	−16.0000	−0.609994
$689$	12.0000	0.457164
$690$	0	0
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	10.0000	0.379595
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	−12.0000	−0.454207
$699$	0	0
$700$	4.00000	0.151186
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	−24.0000	−0.904534
$705$	0	0
$706$	−12.0000	−0.451626
$707$	−8.00000	−0.300871
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	−16.0000	−0.600469
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	24.0000	0.897549
$716$	−24.0000	−0.896922
$717$	0	0
$718$	30.0000	1.11959
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	36.0000	1.33978
$723$	0	0
$724$	−16.0000	−0.594635
$725$	−6.00000	−0.222834
$726$	0	0
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	0	0
$729$	0	0
$730$	−60.0000	−2.22070
$731$	−4.00000	−0.147945
$732$	0	0
$733$	37.0000	1.36663	0.683313	−	0.730125i	$-0.260538\pi$
$733$	37.0000	1.36663	0.683313	+	0.730125i	$0.260538\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−8.00000	−0.294884
$737$	30.0000	1.10506
$738$	0	0
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	12.0000	0.440534
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	44.0000	1.61204
$746$	6.00000	0.219676
$747$	0	0
$748$	−6.00000	−0.219382
$749$	−20.0000	−0.730784
$750$	0	0
$751$	53.0000	1.93400	0.966999	−	0.254781i	$-0.0820034\pi$
$751$	53.0000	1.93400	0.966999	+	0.254781i	$0.0820034\pi$
$752$	48.0000	1.75038
$753$	0	0
$754$	−48.0000	−1.74806
$755$	−32.0000	−1.16460
$756$	0	0
$757$	1.00000	0.0363456	0.0181728	−	0.999835i	$-0.494215\pi$
$757$	1.00000	0.0363456	0.0181728	+	0.999835i	$0.494215\pi$
$758$	−56.0000	−2.03401
$759$	0	0
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−6.00000	−0.216789
$767$	−4.00000	−0.144432
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	24.0000	0.864900
$771$	0	0
$772$	4.00000	0.143963
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	12.0000	0.428571
$785$	20.0000	0.713831
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	−24.0000	−0.853882
$791$	−4.00000	−0.142224
$792$	0	0
$793$	20.0000	0.710221
$794$	18.0000	0.638796
$795$	0	0
$796$	−40.0000	−1.41776
$797$	44.0000	1.55856	0.779280	−	0.626676i	$-0.215585\pi$
$797$	44.0000	1.55856	0.779280	+	0.626676i	$0.215585\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	−8.00000	−0.282843
$801$	0	0
$802$	14.0000	0.494357
$803$	45.0000	1.58802
$804$	0	0
$805$	4.00000	0.140981
$806$	64.0000	2.25430
$807$	0	0
$808$	0	0
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	13.0000	0.456492	0.228246	−	0.973604i	$-0.426701\pi$
$811$	13.0000	0.456492	0.228246	+	0.973604i	$0.426701\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	12.0000	0.420600
$815$	24.0000	0.840683
$816$	0	0
$817$	−4.00000	−0.139942
$818$	−40.0000	−1.39857
$819$	0	0
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	0	0
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	0	0
$825$	0	0
$826$	−4.00000	−0.139178
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	−32.0000	−1.10940
$833$	3.00000	0.103944
$834$	0	0
$835$	44.0000	1.52268
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−14.0000	−0.483622
$839$	31.0000	1.07024	0.535119	−	0.844776i	$-0.320267\pi$
$839$	31.0000	1.07024	0.535119	+	0.844776i	$0.320267\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−20.0000	−0.689246
$843$	0	0
$844$	−2.00000	−0.0688428
$845$	6.00000	0.206406
$846$	0	0
$847$	4.00000	0.137442
$848$	−12.0000	−0.412082
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	2.00000	0.0685591
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	0	0
$857$	47.0000	1.60549	0.802745	−	0.596323i	$-0.203372\pi$
$857$	47.0000	1.60549	0.802745	+	0.596323i	$0.203372\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	−46.0000	−1.56314
$867$	0	0
$868$	32.0000	1.08615
$869$	18.0000	0.610608
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	24.0000	0.811348
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−80.0000	−2.69987
$879$	0	0
$880$	−24.0000	−0.809040
$881$	19.0000	0.640126	0.320063	−	0.947396i	$-0.396296\pi$
$881$	19.0000	0.640126	0.320063	+	0.947396i	$0.396296\pi$
$882$	0	0
$883$	45.0000	1.51437	0.757185	−	0.653200i	$-0.226574\pi$
$883$	45.0000	1.51437	0.757185	+	0.653200i	$0.226574\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	32.0000	1.07506
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−40.0000	−1.34156
$890$	−56.0000	−1.87712
$891$	0	0
$892$	42.0000	1.40626
$893$	12.0000	0.401565
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	56.0000	1.86874
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.0000	−0.531858
$906$	0	0
$907$	50.0000	1.66022	0.830111	−	0.557598i	$-0.188277\pi$
$907$	50.0000	1.66022	0.830111	+	0.557598i	$0.188277\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	32.0000	1.06079
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	84.0000	2.77847
$915$	0	0
$916$	12.0000	0.396491
$917$	−40.0000	−1.32092
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	0	0
$922$	26.0000	0.856264
$923$	16.0000	0.526646
$924$	0	0
$925$	2.00000	0.0657596
$926$	−28.0000	−0.920137
$927$	0	0
$928$	48.0000	1.57568
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	6.00000	0.196537
$933$	0	0
$934$	12.0000	0.392652
$935$	−6.00000	−0.196221
$936$	0	0
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	40.0000	1.30605
$939$	0	0
$940$	−48.0000	−1.56559
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	−24.0000	−0.780307
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	−2.00000	−0.0648886
$951$	0	0
$952$	0	0
$953$	−59.0000	−1.91120	−0.955599	−	0.294671i	$-0.904790\pi$
$953$	−59.0000	−1.91120	−0.955599	+	0.294671i	$0.904790\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	14.0000	0.452792
$957$	0	0
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	33.0000	1.06452
$962$	16.0000	0.515861
$963$	0	0
$964$	24.0000	0.772988
$965$	4.00000	0.128765
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	0	0
$970$	76.0000	2.44021
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	44.0000	1.40985
$975$	0	0
$976$	−20.0000	−0.640184
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	42.0000	1.34233
$980$	−12.0000	−0.383326
$981$	0	0
$982$	34.0000	1.08498
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	12.0000	0.382158
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−4.00000	−0.127193
$990$	0	0
$991$	9.00000	0.285894	0.142947	−	0.989730i	$-0.454342\pi$
$991$	9.00000	0.285894	0.142947	+	0.989730i	$0.454342\pi$
$992$	−64.0000	−2.03200
$993$	0	0
$994$	16.0000	0.507489
$995$	−40.0000	−1.26809
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	−40.0000	−1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9027.2.a.a.1.1		1
3.2	odd	2		1003.2.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.d.1.1	✓	1	3.2	odd	2
9027.2.a.a.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9027 = 3^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9027.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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