LMFDB - Embedded newform 224.2.b.b.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(113,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("224.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x + 4 $ x^4 - x^3 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$-0.780776 - 1.17915i$ of defining polynomial
Character	$\chi$	$=$	224.113
Dual form			224.2.b.b.113.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02045i q^{3} -1.69614i q^{5} +1.00000 q^{7} -6.12311 q^{9} +O(q^{10})$	$q-3.02045i q^{3} -1.69614i q^{5} +1.00000 q^{7} -6.12311 q^{9} +1.32431i q^{11} +1.69614i q^{13} -5.12311 q^{15} +2.00000 q^{17} -3.02045i q^{19} -3.02045i q^{21} -5.12311 q^{23} +2.12311 q^{25} +9.43318i q^{27} -6.04090i q^{29} +10.2462 q^{31} +4.00000 q^{33} -1.69614i q^{35} +6.04090i q^{37} +5.12311 q^{39} +4.24621 q^{41} +1.32431i q^{43} +10.3857i q^{45} +1.00000 q^{49} -6.04090i q^{51} -2.64861i q^{53} +2.24621 q^{55} -9.12311 q^{57} +0.371834i q^{59} +1.69614i q^{61} -6.12311 q^{63} +2.87689 q^{65} +11.5012i q^{67} +15.4741i q^{69} +8.00000 q^{71} -6.00000 q^{73} -6.41273i q^{75} +1.32431i q^{77} +10.1231 q^{81} +5.66906i q^{83} -3.39228i q^{85} -18.2462 q^{87} -16.2462 q^{89} +1.69614i q^{91} -30.9481i q^{93} -5.12311 q^{95} +12.2462 q^{97} -8.10887i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 - 8 * q^9	$ 4 q + 4 q^{7} - 8 q^{9} - 4 q^{15} + 8 q^{17} - 4 q^{23} - 8 q^{25} + 8 q^{31} + 16 q^{33} + 4 q^{39} - 16 q^{41} + 4 q^{49} - 24 q^{55} - 20 q^{57} - 8 q^{63} + 28 q^{65} + 32 q^{71} - 24 q^{73} + 24 q^{81} - 40 q^{87} - 32 q^{89} - 4 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 8 * q^9 - 4 * q^15 + 8 * q^17 - 4 * q^23 - 8 * q^25 + 8 * q^31 + 16 * q^33 + 4 * q^39 - 16 * q^41 + 4 * q^49 - 24 * q^55 - 20 * q^57 - 8 * q^63 + 28 * q^65 + 32 * q^71 - 24 * q^73 + 24 * q^81 - 40 * q^87 - 32 * q^89 - 4 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$1$	$-1$

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	0
$2$	0	0
$3$	− 3.02045i	− 1.74386i	−0.489634	−	0.871928i	$-0.662870\pi$
$3$	− 3.02045i	− 1.74386i	0.489634	−	0.871928i	$-0.337130\pi$
$4$	0	0
$5$	− 1.69614i	− 0.758537i	−0.925287	−	0.379269i	$-0.876176\pi$
$5$	− 1.69614i	− 0.758537i	0.925287	−	0.379269i	$-0.123824\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−6.12311	−2.04104
$10$	0	0
$11$	1.32431i	0.399294i	0.979868	+	0.199647i	$0.0639795\pi$
$11$	1.32431i	0.399294i	−0.979868	+	0.199647i	$0.936021\pi$
$12$	0	0
$13$	1.69614i	0.470425i	0.971944	+	0.235212i	$0.0755786\pi$
$13$	1.69614i	0.470425i	−0.971944	+	0.235212i	$0.924421\pi$
$14$	0	0
$15$	−5.12311	−1.32278
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 3.02045i	− 0.692938i	−0.938061	−	0.346469i	$-0.887381\pi$
$19$	− 3.02045i	− 0.692938i	0.938061	−	0.346469i	$-0.112619\pi$
$20$	0	0
$21$	− 3.02045i	− 0.659116i
$22$	0	0
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	0	0
$25$	2.12311	0.424621
$26$	0	0
$27$	9.43318i	1.81542i
$28$	0	0
$29$	− 6.04090i	− 1.12177i	−0.827895	−	0.560883i	$-0.810462\pi$
$29$	− 6.04090i	− 1.12177i	0.827895	−	0.560883i	$-0.189538\pi$
$30$	0	0
$31$	10.2462	1.84027	0.920137	−	0.391597i	$-0.128077\pi$
$31$	10.2462	1.84027	0.920137	+	0.391597i	$0.128077\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	− 1.69614i	− 0.286700i
$36$	0	0
$37$	6.04090i	0.993117i	0.868003	+	0.496559i	$0.165403\pi$
$37$	6.04090i	0.993117i	−0.868003	+	0.496559i	$0.834597\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	4.24621	0.663147	0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621	0.663147	0.331573	+	0.943429i	$0.392421\pi$
$42$	0	0
$43$	1.32431i	0.201955i	0.994889	+	0.100977i	$0.0321970\pi$
$43$	1.32431i	0.201955i	−0.994889	+	0.100977i	$0.967803\pi$
$44$	0	0
$45$	10.3857i	1.54820i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 6.04090i	− 0.845895i
$52$	0	0
$53$	− 2.64861i	− 0.363815i	−0.983316	−	0.181908i	$-0.941773\pi$
$53$	− 2.64861i	− 0.363815i	0.983316	−	0.181908i	$-0.0582272\pi$
$54$	0	0
$55$	2.24621	0.302879
$56$	0	0
$57$	−9.12311	−1.20838
$58$	0	0
$59$	0.371834i	0.0484087i	0.999707	+	0.0242043i	$0.00770523\pi$
$59$	0.371834i	0.0484087i	−0.999707	+	0.0242043i	$0.992295\pi$
$60$	0	0
$61$	1.69614i	0.217169i	0.994087	+	0.108584i	$0.0346317\pi$
$61$	1.69614i	0.217169i	−0.994087	+	0.108584i	$0.965368\pi$
$62$	0	0
$63$	−6.12311	−0.771439
$64$	0	0
$65$	2.87689	0.356835
$66$	0	0
$67$	11.5012i	1.40509i	0.711640	+	0.702545i	$0.247953\pi$
$67$	11.5012i	1.40509i	−0.711640	+	0.702545i	$0.752047\pi$
$68$	0	0
$69$	15.4741i	1.86286i
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	− 6.41273i	− 0.740478i
$76$	0	0
$77$	1.32431i	0.150919i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	10.1231	1.12479
$82$	0	0
$83$	5.66906i	0.622260i	0.950367	+	0.311130i	$0.100708\pi$
$83$	5.66906i	0.622260i	−0.950367	+	0.311130i	$0.899292\pi$
$84$	0	0
$85$	− 3.39228i	− 0.367945i
$86$	0	0
$87$	−18.2462	−1.95620
$88$	0	0
$89$	−16.2462	−1.72209	−0.861047	−	0.508525i	$-0.830191\pi$
$89$	−16.2462	−1.72209	−0.861047	+	0.508525i	$0.830191\pi$
$90$	0	0
$91$	1.69614i	0.177804i
$92$	0	0
$93$	− 30.9481i	− 3.20917i
$94$	0	0
$95$	−5.12311	−0.525620
$96$	0	0
$97$	12.2462	1.24341	0.621707	−	0.783250i	$-0.286439\pi$
$97$	12.2462	1.24341	0.621707	+	0.783250i	$0.286439\pi$
$98$	0	0
$99$	− 8.10887i	− 0.814972i
$100$	0	0
$101$	− 10.3857i	− 1.03341i	−0.856163	−	0.516705i	$-0.827158\pi$
$101$	− 10.3857i	− 1.03341i	0.856163	−	0.516705i	$-0.172842\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	−5.12311	−0.499964
$106$	0	0
$107$	− 14.1498i	− 1.36791i	−0.729524	−	0.683955i	$-0.760259\pi$
$107$	− 14.1498i	− 1.36791i	0.729524	−	0.683955i	$-0.239741\pi$
$108$	0	0
$109$	18.1227i	1.73584i	0.496705	+	0.867919i	$0.334543\pi$
$109$	18.1227i	1.73584i	−0.496705	+	0.867919i	$0.665457\pi$
$110$	0	0
$111$	18.2462	1.73185
$112$	0	0
$113$	4.87689	0.458780	0.229390	−	0.973335i	$-0.426327\pi$
$113$	4.87689	0.458780	0.229390	+	0.973335i	$0.426327\pi$
$114$	0	0
$115$	8.68951i	0.810301i
$116$	0	0
$117$	− 10.3857i	− 0.960154i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	9.24621	0.840565
$122$	0	0
$123$	− 12.8255i	− 1.15643i
$124$	0	0
$125$	− 12.0818i	− 1.08063i
$126$	0	0
$127$	−13.1231	−1.16449	−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231	−1.16449	−0.582244	+	0.813014i	$0.697825\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	12.4536i	1.08808i	0.839060	+	0.544039i	$0.183106\pi$
$131$	12.4536i	1.08808i	−0.839060	+	0.544039i	$0.816894\pi$
$132$	0	0
$133$	− 3.02045i	− 0.261906i
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	−16.2462	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$137$	−16.2462	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$138$	0	0
$139$	− 8.31768i	− 0.705496i	−0.935718	−	0.352748i	$-0.885247\pi$
$139$	− 8.31768i	− 0.705496i	0.935718	−	0.352748i	$-0.114753\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.24621	−0.187838
$144$	0	0
$145$	−10.2462	−0.850902
$146$	0	0
$147$	− 3.02045i	− 0.249122i
$148$	0	0
$149$	14.7304i	1.20676i	0.797453	+	0.603381i	$0.206180\pi$
$149$	14.7304i	1.20676i	−0.797453	+	0.603381i	$0.793820\pi$
$150$	0	0
$151$	−10.8769	−0.885149	−0.442575	−	0.896732i	$-0.645935\pi$
$151$	−10.8769	−0.885149	−0.442575	+	0.896732i	$0.645935\pi$
$152$	0	0
$153$	−12.2462	−0.990048
$154$	0	0
$155$	− 17.3790i	− 1.39592i
$156$	0	0
$157$	8.48071i	0.676834i	0.940996	+	0.338417i	$0.109891\pi$
$157$	8.48071i	0.676834i	−0.940996	+	0.338417i	$0.890109\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−5.12311	−0.403757
$162$	0	0
$163$	13.4061i	1.05005i	0.851088	+	0.525023i	$0.175943\pi$
$163$	13.4061i	1.05005i	−0.851088	+	0.525023i	$0.824057\pi$
$164$	0	0
$165$	− 6.78456i	− 0.528178i
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	10.1231	0.778700
$170$	0	0
$171$	18.4945i	1.41431i
$172$	0	0
$173$	19.0752i	1.45026i	0.688613	+	0.725129i	$0.258220\pi$
$173$	19.0752i	1.45026i	−0.688613	+	0.725129i	$0.741780\pi$
$174$	0	0
$175$	2.12311	0.160492
$176$	0	0
$177$	1.12311	0.0844178
$178$	0	0
$179$	4.71659i	0.352534i	0.984342	+	0.176267i	$0.0564023\pi$
$179$	4.71659i	0.352534i	−0.984342	+	0.176267i	$0.943598\pi$
$180$	0	0
$181$	6.99337i	0.519813i	0.965634	+	0.259906i	$0.0836917\pi$
$181$	6.99337i	0.519813i	−0.965634	+	0.259906i	$0.916308\pi$
$182$	0	0
$183$	5.12311	0.378711
$184$	0	0
$185$	10.2462	0.753316
$186$	0	0
$187$	2.64861i	0.193686i
$188$	0	0
$189$	9.43318i	0.686163i
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−11.1231	−0.800659	−0.400329	−	0.916371i	$-0.631104\pi$
$193$	−11.1231	−0.800659	−0.400329	+	0.916371i	$0.631104\pi$
$194$	0	0
$195$	− 8.68951i	− 0.622269i
$196$	0	0
$197$	21.5150i	1.53288i	0.642317	+	0.766439i	$0.277973\pi$
$197$	21.5150i	1.53288i	−0.642317	+	0.766439i	$0.722027\pi$
$198$	0	0
$199$	18.2462	1.29344	0.646720	−	0.762728i	$-0.276140\pi$
$199$	18.2462	1.29344	0.646720	+	0.762728i	$0.276140\pi$
$200$	0	0
$201$	34.7386	2.45027
$202$	0	0
$203$	− 6.04090i	− 0.423988i
$204$	0	0
$205$	− 7.20217i	− 0.503022i
$206$	0	0
$207$	31.3693	2.18032
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	4.71659i	0.324703i	0.986733	+	0.162352i	$0.0519079\pi$
$211$	4.71659i	0.324703i	−0.986733	+	0.162352i	$0.948092\pi$
$212$	0	0
$213$	− 24.1636i	− 1.65566i
$214$	0	0
$215$	2.24621	0.153190
$216$	0	0
$217$	10.2462	0.695558
$218$	0	0
$219$	18.1227i	1.22462i
$220$	0	0
$221$	3.39228i	0.228190i
$222$	0	0
$223$	5.75379	0.385302	0.192651	−	0.981267i	$-0.438291\pi$
$223$	5.75379	0.385302	0.192651	+	0.981267i	$0.438291\pi$
$224$	0	0
$225$	−13.0000	−0.866667
$226$	0	0
$227$	− 9.80501i	− 0.650782i	−0.945580	−	0.325391i	$-0.894504\pi$
$227$	− 9.80501i	− 0.650782i	0.945580	−	0.325391i	$-0.105496\pi$
$228$	0	0
$229$	− 25.8597i	− 1.70886i	−0.519568	−	0.854429i	$-0.673907\pi$
$229$	− 25.8597i	− 1.70886i	0.519568	−	0.854429i	$-0.326093\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−16.2462	−1.06432	−0.532162	−	0.846642i	$-0.678620\pi$
$233$	−16.2462	−1.06432	−0.532162	+	0.846642i	$0.678620\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.6155	1.13945	0.569727	−	0.821834i	$-0.307049\pi$
$239$	17.6155	1.13945	0.569727	+	0.821834i	$0.307049\pi$
$240$	0	0
$241$	−3.75379	−0.241803	−0.120901	−	0.992665i	$-0.538578\pi$
$241$	−3.75379	−0.241803	−0.120901	+	0.992665i	$0.538578\pi$
$242$	0	0
$243$	− 2.27678i	− 0.146055i
$244$	0	0
$245$	− 1.69614i	− 0.108362i
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	17.1231	1.08513
$250$	0	0
$251$	10.9663i	0.692186i	0.938200	+	0.346093i	$0.112492\pi$
$251$	10.9663i	0.692186i	−0.938200	+	0.346093i	$0.887508\pi$
$252$	0	0
$253$	− 6.78456i	− 0.426542i
$254$	0	0
$255$	−10.2462	−0.641643
$256$	0	0
$257$	22.4924	1.40304	0.701519	−	0.712650i	$-0.252505\pi$
$257$	22.4924	1.40304	0.701519	+	0.712650i	$0.252505\pi$
$258$	0	0
$259$	6.04090i	0.375363i
$260$	0	0
$261$	36.9890i	2.28956i
$262$	0	0
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	0	0
$265$	−4.49242	−0.275967
$266$	0	0
$267$	49.0708i	3.00309i
$268$	0	0
$269$	− 11.8730i	− 0.723909i	−0.932196	−	0.361954i	$-0.882110\pi$
$269$	− 11.8730i	− 0.723909i	0.932196	−	0.361954i	$-0.117890\pi$
$270$	0	0
$271$	−10.2462	−0.622413	−0.311207	−	0.950342i	$-0.600733\pi$
$271$	−10.2462	−0.622413	−0.311207	+	0.950342i	$0.600733\pi$
$272$	0	0
$273$	5.12311	0.310064
$274$	0	0
$275$	2.81164i	0.169548i
$276$	0	0
$277$	− 2.64861i	− 0.159140i	−0.996829	−	0.0795699i	$-0.974645\pi$
$277$	− 2.64861i	− 0.159140i	0.996829	−	0.0795699i	$-0.0253547\pi$
$278$	0	0
$279$	−62.7386	−3.75606
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 21.8868i	− 1.30104i	−0.759491	−	0.650518i	$-0.774552\pi$
$283$	− 21.8868i	− 1.30104i	0.759491	−	0.650518i	$-0.225448\pi$
$284$	0	0
$285$	15.4741i	0.916605i
$286$	0	0
$287$	4.24621	0.250646
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 36.9890i	− 2.16834i
$292$	0	0
$293$	− 10.3857i	− 0.606736i	−0.952873	−	0.303368i	$-0.901889\pi$
$293$	− 10.3857i	− 0.606736i	0.952873	−	0.303368i	$-0.0981112\pi$
$294$	0	0
$295$	0.630683	0.0367198
$296$	0	0
$297$	−12.4924	−0.724884
$298$	0	0
$299$	− 8.68951i	− 0.502527i
$300$	0	0
$301$	1.32431i	0.0763318i
$302$	0	0
$303$	−31.3693	−1.80212
$304$	0	0
$305$	2.87689	0.164730
$306$	0	0
$307$	− 25.2791i	− 1.44275i	−0.692543	−	0.721377i	$-0.743510\pi$
$307$	− 25.2791i	− 1.44275i	0.692543	−	0.721377i	$-0.256490\pi$
$308$	0	0
$309$	6.78456i	0.385960i
$310$	0	0
$311$	12.4924	0.708380	0.354190	−	0.935173i	$-0.384757\pi$
$311$	12.4924	0.708380	0.354190	+	0.935173i	$0.384757\pi$
$312$	0	0
$313$	−20.7386	−1.17222	−0.586108	−	0.810233i	$-0.699341\pi$
$313$	−20.7386	−1.17222	−0.586108	+	0.810233i	$0.699341\pi$
$314$	0	0
$315$	10.3857i	0.585165i
$316$	0	0
$317$	− 4.13595i	− 0.232298i	−0.993232	−	0.116149i	$-0.962945\pi$
$317$	− 4.13595i	− 0.232298i	0.993232	−	0.116149i	$-0.0370550\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−42.7386	−2.38544
$322$	0	0
$323$	− 6.04090i	− 0.336124i
$324$	0	0
$325$	3.60109i	0.199752i
$326$	0	0
$327$	54.7386	3.02705
$328$	0	0
$329$	0	0
$330$	0	0
$331$	14.8934i	0.818617i	0.912396	+	0.409309i	$0.134230\pi$
$331$	14.8934i	0.818617i	−0.912396	+	0.409309i	$0.865770\pi$
$332$	0	0
$333$	− 36.9890i	− 2.02699i
$334$	0	0
$335$	19.5076	1.06581
$336$	0	0
$337$	−0.876894	−0.0477675	−0.0238837	−	0.999715i	$-0.507603\pi$
$337$	−0.876894	−0.0477675	−0.0238837	+	0.999715i	$0.507603\pi$
$338$	0	0
$339$	− 14.7304i	− 0.800046i
$340$	0	0
$341$	13.5691i	0.734809i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	26.2462	1.41305
$346$	0	0
$347$	8.10887i	0.435307i	0.976026	+	0.217654i	$0.0698403\pi$
$347$	8.10887i	0.435307i	−0.976026	+	0.217654i	$0.930160\pi$
$348$	0	0
$349$	− 27.3471i	− 1.46385i	−0.681383	−	0.731927i	$-0.738621\pi$
$349$	− 27.3471i	− 1.46385i	0.681383	−	0.731927i	$-0.261379\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	7.75379	0.412693	0.206346	−	0.978479i	$-0.433843\pi$
$353$	7.75379	0.412693	0.206346	+	0.978479i	$0.433843\pi$
$354$	0	0
$355$	− 13.5691i	− 0.720175i
$356$	0	0
$357$	− 6.04090i	− 0.319718i
$358$	0	0
$359$	31.3693	1.65561	0.827805	−	0.561017i	$-0.189590\pi$
$359$	31.3693	1.65561	0.827805	+	0.561017i	$0.189590\pi$
$360$	0	0
$361$	9.87689	0.519837
$362$	0	0
$363$	− 27.9277i	− 1.46582i
$364$	0	0
$365$	10.1768i	0.532680i
$366$	0	0
$367$	−10.2462	−0.534848	−0.267424	−	0.963579i	$-0.586172\pi$
$367$	−10.2462	−0.534848	−0.267424	+	0.963579i	$0.586172\pi$
$368$	0	0
$369$	−26.0000	−1.35351
$370$	0	0
$371$	− 2.64861i	− 0.137509i
$372$	0	0
$373$	14.7304i	0.762711i	0.924428	+	0.381356i	$0.124543\pi$
$373$	14.7304i	0.762711i	−0.924428	+	0.381356i	$0.875457\pi$
$374$	0	0
$375$	−36.4924	−1.88446
$376$	0	0
$377$	10.2462	0.527707
$378$	0	0
$379$	− 36.4084i	− 1.87017i	−0.354418	−	0.935087i	$-0.615321\pi$
$379$	− 36.4084i	− 1.87017i	0.354418	−	0.935087i	$-0.384679\pi$
$380$	0	0
$381$	39.6377i	2.03070i
$382$	0	0
$383$	4.49242	0.229552	0.114776	−	0.993391i	$-0.463385\pi$
$383$	4.49242	0.229552	0.114776	+	0.993391i	$0.463385\pi$
$384$	0	0
$385$	2.24621	0.114478
$386$	0	0
$387$	− 8.10887i	− 0.412197i
$388$	0	0
$389$	− 24.9073i	− 1.26285i	−0.775438	−	0.631424i	$-0.782471\pi$
$389$	− 24.9073i	− 1.26285i	0.775438	−	0.631424i	$-0.217529\pi$
$390$	0	0
$391$	−10.2462	−0.518173
$392$	0	0
$393$	37.6155	1.89745
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.5625i	− 1.03200i	−0.856588	−	0.516001i	$-0.827420\pi$
$397$	− 20.5625i	− 1.03200i	0.856588	−	0.516001i	$-0.172580\pi$
$398$	0	0
$399$	−9.12311	−0.456727
$400$	0	0
$401$	−0.876894	−0.0437900	−0.0218950	−	0.999760i	$-0.506970\pi$
$401$	−0.876894	−0.0437900	−0.0218950	+	0.999760i	$0.506970\pi$
$402$	0	0
$403$	17.3790i	0.865711i
$404$	0	0
$405$	− 17.1702i	− 0.853195i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	49.0708i	2.42049i
$412$	0	0
$413$	0.371834i	0.0182968i
$414$	0	0
$415$	9.61553	0.472008
$416$	0	0
$417$	−25.1231	−1.23028
$418$	0	0
$419$	27.9277i	1.36436i	0.731185	+	0.682179i	$0.238967\pi$
$419$	27.9277i	1.36436i	−0.731185	+	0.682179i	$0.761033\pi$
$420$	0	0
$421$	− 26.8122i	− 1.30675i	−0.757036	−	0.653373i	$-0.773353\pi$
$421$	− 26.8122i	− 1.30675i	0.757036	−	0.653373i	$-0.226647\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.24621	0.205971
$426$	0	0
$427$	1.69614i	0.0820820i
$428$	0	0
$429$	6.78456i	0.327562i
$430$	0	0
$431$	−17.6155	−0.848510	−0.424255	−	0.905543i	$-0.639464\pi$
$431$	−17.6155	−0.848510	−0.424255	+	0.905543i	$0.639464\pi$
$432$	0	0
$433$	−18.4924	−0.888689	−0.444345	−	0.895856i	$-0.646563\pi$
$433$	−18.4924	−0.888689	−0.444345	+	0.895856i	$0.646563\pi$
$434$	0	0
$435$	30.9481i	1.48385i
$436$	0	0
$437$	15.4741i	0.740225i
$438$	0	0
$439$	−22.7386	−1.08526	−0.542628	−	0.839973i	$-0.682571\pi$
$439$	−22.7386	−1.08526	−0.542628	+	0.839973i	$0.682571\pi$
$440$	0	0
$441$	−6.12311	−0.291576
$442$	0	0
$443$	− 7.36520i	− 0.349931i	−0.984575	−	0.174966i	$-0.944019\pi$
$443$	− 7.36520i	− 0.349931i	0.984575	−	0.174966i	$-0.0559814\pi$
$444$	0	0
$445$	27.5559i	1.30627i
$446$	0	0
$447$	44.4924	2.10442
$448$	0	0
$449$	16.7386	0.789945	0.394972	−	0.918693i	$-0.370754\pi$
$449$	16.7386	0.789945	0.394972	+	0.918693i	$0.370754\pi$
$450$	0	0
$451$	5.62329i	0.264790i
$452$	0	0
$453$	32.8531i	1.54357i
$454$	0	0
$455$	2.87689	0.134871
$456$	0	0
$457$	17.3693	0.812502	0.406251	−	0.913761i	$-0.366836\pi$
$457$	17.3693	0.812502	0.406251	+	0.913761i	$0.366836\pi$
$458$	0	0
$459$	18.8664i	0.880606i
$460$	0	0
$461$	− 24.3724i	− 1.13514i	−0.823327	−	0.567568i	$-0.807885\pi$
$461$	− 24.3724i	− 1.13514i	0.823327	−	0.567568i	$-0.192115\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−52.4924	−2.43428
$466$	0	0
$467$	− 3.02045i	− 0.139770i	−0.997555	−	0.0698848i	$-0.977737\pi$
$467$	− 3.02045i	− 0.139770i	0.997555	−	0.0698848i	$-0.0222632\pi$
$468$	0	0
$469$	11.5012i	0.531074i
$470$	0	0
$471$	25.6155	1.18030
$472$	0	0
$473$	−1.75379	−0.0806393
$474$	0	0
$475$	− 6.41273i	− 0.294236i
$476$	0	0
$477$	16.2177i	0.742559i
$478$	0	0
$479$	−10.2462	−0.468161	−0.234081	−	0.972217i	$-0.575208\pi$
$479$	−10.2462	−0.468161	−0.234081	+	0.972217i	$0.575208\pi$
$480$	0	0
$481$	−10.2462	−0.467187
$482$	0	0
$483$	15.4741i	0.704095i
$484$	0	0
$485$	− 20.7713i	− 0.943176i
$486$	0	0
$487$	−0.630683	−0.0285790	−0.0142895	−	0.999898i	$-0.504549\pi$
$487$	−0.630683	−0.0285790	−0.0142895	+	0.999898i	$0.504549\pi$
$488$	0	0
$489$	40.4924	1.83113
$490$	0	0
$491$	34.1774i	1.54240i	0.636590	+	0.771202i	$0.280344\pi$
$491$	34.1774i	1.54240i	−0.636590	+	0.771202i	$0.719656\pi$
$492$	0	0
$493$	− 12.0818i	− 0.544137i
$494$	0	0
$495$	−13.7538	−0.618187
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	− 8.85254i	− 0.396294i	−0.980172	−	0.198147i	$-0.936508\pi$
$499$	− 8.85254i	− 0.396294i	0.980172	−	0.198147i	$-0.0634924\pi$
$500$	0	0
$501$	24.1636i	1.07955i
$502$	0	0
$503$	−13.7538	−0.613251	−0.306626	−	0.951830i	$-0.599200\pi$
$503$	−13.7538	−0.613251	−0.306626	+	0.951830i	$0.599200\pi$
$504$	0	0
$505$	−17.6155	−0.783881
$506$	0	0
$507$	− 30.5763i	− 1.35794i
$508$	0	0
$509$	30.7393i	1.36250i	0.732052	+	0.681249i	$0.238563\pi$
$509$	30.7393i	1.36250i	−0.732052	+	0.681249i	$0.761437\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	28.4924	1.25797
$514$	0	0
$515$	3.80989i	0.167884i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	57.6155	2.52904
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	− 41.1708i	− 1.80027i	−0.435609	−	0.900136i	$-0.643467\pi$
$523$	− 41.1708i	− 1.80027i	0.435609	−	0.900136i	$-0.356533\pi$
$524$	0	0
$525$	− 6.41273i	− 0.279874i
$526$	0	0
$527$	20.4924	0.892664
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	− 2.27678i	− 0.0988038i
$532$	0	0
$533$	7.20217i	0.311961i
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	14.2462	0.614769
$538$	0	0
$539$	1.32431i	0.0570419i
$540$	0	0
$541$	13.2431i	0.569364i	0.958622	+	0.284682i	$0.0918880\pi$
$541$	13.2431i	0.569364i	−0.958622	+	0.284682i	$0.908112\pi$
$542$	0	0
$543$	21.1231	0.906479
$544$	0	0
$545$	30.7386	1.31670
$546$	0	0
$547$	9.59621i	0.410304i	0.978730	+	0.205152i	$0.0657689\pi$
$547$	9.59621i	0.410304i	−0.978730	+	0.205152i	$0.934231\pi$
$548$	0	0
$549$	− 10.3857i	− 0.443249i
$550$	0	0
$551$	−18.2462	−0.777315
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 30.9481i	− 1.31368i
$556$	0	0
$557$	2.64861i	0.112225i	0.998424	+	0.0561127i	$0.0178706\pi$
$557$	2.64861i	0.112225i	−0.998424	+	0.0561127i	$0.982129\pi$
$558$	0	0
$559$	−2.24621	−0.0950046
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	29.8326i	1.25730i	0.777690	+	0.628648i	$0.216391\pi$
$563$	29.8326i	1.25730i	−0.777690	+	0.628648i	$0.783609\pi$
$564$	0	0
$565$	− 8.27190i	− 0.348001i
$566$	0	0
$567$	10.1231	0.425130
$568$	0	0
$569$	−13.3693	−0.560471	−0.280235	−	0.959931i	$-0.590413\pi$
$569$	−13.3693	−0.560471	−0.280235	+	0.959931i	$0.590413\pi$
$570$	0	0
$571$	− 9.27015i	− 0.387944i	−0.981007	−	0.193972i	$-0.937863\pi$
$571$	− 9.27015i	− 0.387944i	0.981007	−	0.193972i	$-0.0621370\pi$
$572$	0	0
$573$	48.3272i	2.01890i
$574$	0	0
$575$	−10.8769	−0.453598
$576$	0	0
$577$	7.75379	0.322794	0.161397	−	0.986890i	$-0.448400\pi$
$577$	7.75379	0.322794	0.161397	+	0.986890i	$0.448400\pi$
$578$	0	0
$579$	33.5968i	1.39623i
$580$	0	0
$581$	5.66906i	0.235192i
$582$	0	0
$583$	3.50758	0.145269
$584$	0	0
$585$	−17.6155	−0.728312
$586$	0	0
$587$	− 21.8868i	− 0.903365i	−0.892179	−	0.451683i	$-0.850824\pi$
$587$	− 21.8868i	− 0.903365i	0.892179	−	0.451683i	$-0.149176\pi$
$588$	0	0
$589$	− 30.9481i	− 1.27520i
$590$	0	0
$591$	64.9848	2.67312
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	− 3.39228i	− 0.139070i
$596$	0	0
$597$	− 55.1117i	− 2.25557i
$598$	0	0
$599$	12.4924	0.510427	0.255213	−	0.966885i	$-0.417854\pi$
$599$	12.4924	0.510427	0.255213	+	0.966885i	$0.417854\pi$
$600$	0	0
$601$	−16.2462	−0.662697	−0.331348	−	0.943508i	$-0.607504\pi$
$601$	−16.2462	−0.662697	−0.331348	+	0.943508i	$0.607504\pi$
$602$	0	0
$603$	− 70.4228i	− 2.86784i
$604$	0	0
$605$	− 15.6829i	− 0.637600i
$606$	0	0
$607$	−40.9848	−1.66352	−0.831762	−	0.555133i	$-0.812667\pi$
$607$	−40.9848	−1.66352	−0.831762	+	0.555133i	$0.812667\pi$
$608$	0	0
$609$	−18.2462	−0.739374
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.23100i	0.0901094i	0.998985	+	0.0450547i	$0.0143462\pi$
$613$	2.23100i	0.0901094i	−0.998985	+	0.0450547i	$0.985654\pi$
$614$	0	0
$615$	−21.7538	−0.877197
$616$	0	0
$617$	−29.3693	−1.18236	−0.591182	−	0.806538i	$-0.701339\pi$
$617$	−29.3693	−1.18236	−0.591182	+	0.806538i	$0.701339\pi$
$618$	0	0
$619$	19.6558i	0.790033i	0.918674	+	0.395017i	$0.129261\pi$
$619$	19.6558i	0.790033i	−0.918674	+	0.395017i	$0.870739\pi$
$620$	0	0
$621$	− 48.3272i	− 1.93930i
$622$	0	0
$623$	−16.2462	−0.650891
$624$	0	0
$625$	−9.87689	−0.395076
$626$	0	0
$627$	− 12.0818i	− 0.482500i
$628$	0	0
$629$	12.0818i	0.481733i
$630$	0	0
$631$	3.50758	0.139634	0.0698172	−	0.997560i	$-0.477758\pi$
$631$	3.50758	0.139634	0.0698172	+	0.997560i	$0.477758\pi$
$632$	0	0
$633$	14.2462	0.566236
$634$	0	0
$635$	22.2586i	0.883307i
$636$	0	0
$637$	1.69614i	0.0672036i
$638$	0	0
$639$	−48.9848	−1.93781
$640$	0	0
$641$	−5.36932	−0.212075	−0.106038	−	0.994362i	$-0.533816\pi$
$641$	−5.36932	−0.212075	−0.106038	+	0.994362i	$0.533816\pi$
$642$	0	0
$643$	5.66906i	0.223566i	0.993733	+	0.111783i	$0.0356562\pi$
$643$	5.66906i	0.223566i	−0.993733	+	0.111783i	$0.964344\pi$
$644$	0	0
$645$	− 6.78456i	− 0.267142i
$646$	0	0
$647$	−43.2311	−1.69959	−0.849794	−	0.527115i	$-0.823274\pi$
$647$	−43.2311	−1.69959	−0.849794	+	0.527115i	$0.823274\pi$
$648$	0	0
$649$	−0.492423	−0.0193293
$650$	0	0
$651$	− 30.9481i	− 1.21295i
$652$	0	0
$653$	31.6918i	1.24020i	0.784524	+	0.620098i	$0.212907\pi$
$653$	31.6918i	1.24020i	−0.784524	+	0.620098i	$0.787093\pi$
$654$	0	0
$655$	21.1231	0.825348
$656$	0	0
$657$	36.7386	1.43331
$658$	0	0
$659$	6.62153i	0.257938i	0.991649	+	0.128969i	$0.0411668\pi$
$659$	6.62153i	0.257938i	−0.991649	+	0.128969i	$0.958833\pi$
$660$	0	0
$661$	44.7261i	1.73964i	0.493367	+	0.869821i	$0.335766\pi$
$661$	44.7261i	1.73964i	−0.493367	+	0.869821i	$0.664234\pi$
$662$	0	0
$663$	10.2462	0.397930
$664$	0	0
$665$	−5.12311	−0.198666
$666$	0	0
$667$	30.9481i	1.19832i
$668$	0	0
$669$	− 17.3790i	− 0.671912i
$670$	0	0
$671$	−2.24621	−0.0867140
$672$	0	0
$673$	−38.9848	−1.50276	−0.751378	−	0.659872i	$-0.770610\pi$
$673$	−38.9848	−1.50276	−0.751378	+	0.659872i	$0.770610\pi$
$674$	0	0
$675$	20.0276i	0.770864i
$676$	0	0
$677$	− 17.1702i	− 0.659905i	−0.943998	−	0.329952i	$-0.892967\pi$
$677$	− 17.1702i	− 0.659905i	0.943998	−	0.329952i	$-0.107033\pi$
$678$	0	0
$679$	12.2462	0.469966
$680$	0	0
$681$	−29.6155	−1.13487
$682$	0	0
$683$	− 0.580639i	− 0.0222175i	−0.999938	−	0.0111088i	$-0.996464\pi$
$683$	− 0.580639i	− 0.0222175i	0.999938	−	0.0111088i	$-0.00353610\pi$
$684$	0	0
$685$	27.5559i	1.05286i
$686$	0	0
$687$	−78.1080	−2.98000
$688$	0	0
$689$	4.49242	0.171148
$690$	0	0
$691$	− 38.8482i	− 1.47786i	−0.673785	−	0.738928i	$-0.735332\pi$
$691$	− 38.8482i	− 1.47786i	0.673785	−	0.738928i	$-0.264668\pi$
$692$	0	0
$693$	− 8.10887i	− 0.308031i
$694$	0	0
$695$	−14.1080	−0.535145
$696$	0	0
$697$	8.49242	0.321673
$698$	0	0
$699$	49.0708i	1.85603i
$700$	0	0
$701$	− 2.23100i	− 0.0842639i	−0.999112	−	0.0421319i	$-0.986585\pi$
$701$	− 2.23100i	− 0.0842639i	0.999112	−	0.0421319i	$-0.0134150\pi$
$702$	0	0
$703$	18.2462	0.688169
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.3857i	− 0.390593i
$708$	0	0
$709$	− 28.7171i	− 1.07849i	−0.842147	−	0.539247i	$-0.818709\pi$
$709$	− 28.7171i	− 1.07849i	0.842147	−	0.539247i	$-0.181291\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−52.4924	−1.96586
$714$	0	0
$715$	3.80989i	0.142482i
$716$	0	0
$717$	− 53.2068i	− 1.98704i
$718$	0	0
$719$	52.4924	1.95764	0.978819	−	0.204730i	$-0.0656316\pi$
$719$	52.4924	1.95764	0.978819	+	0.204730i	$0.0656316\pi$
$720$	0	0
$721$	−2.24621	−0.0836533
$722$	0	0
$723$	11.3381i	0.421669i
$724$	0	0
$725$	− 12.8255i	− 0.476326i
$726$	0	0
$727$	16.9848	0.629933	0.314967	−	0.949103i	$-0.398007\pi$
$727$	16.9848	0.629933	0.314967	+	0.949103i	$0.398007\pi$
$728$	0	0
$729$	23.4924	0.870090
$730$	0	0
$731$	2.64861i	0.0979625i
$732$	0	0
$733$	− 29.2520i	− 1.08045i	−0.841521	−	0.540224i	$-0.818340\pi$
$733$	− 29.2520i	− 1.08045i	0.841521	−	0.540224i	$-0.181660\pi$
$734$	0	0
$735$	−5.12311	−0.188969
$736$	0	0
$737$	−15.2311	−0.561043
$738$	0	0
$739$	− 21.3519i	− 0.785444i	−0.919657	−	0.392722i	$-0.871533\pi$
$739$	− 21.3519i	− 0.785444i	0.919657	−	0.392722i	$-0.128467\pi$
$740$	0	0
$741$	− 15.4741i	− 0.568454i
$742$	0	0
$743$	−0.630683	−0.0231375	−0.0115688	−	0.999933i	$-0.503683\pi$
$743$	−0.630683	−0.0231375	−0.0115688	+	0.999933i	$0.503683\pi$
$744$	0	0
$745$	24.9848	0.915374
$746$	0	0
$747$	− 34.7123i	− 1.27006i
$748$	0	0
$749$	− 14.1498i	− 0.517021i
$750$	0	0
$751$	−8.63068	−0.314938	−0.157469	−	0.987524i	$-0.550333\pi$
$751$	−8.63068	−0.314938	−0.157469	+	0.987524i	$0.550333\pi$
$752$	0	0
$753$	33.1231	1.20707
$754$	0	0
$755$	18.4487i	0.671419i
$756$	0	0
$757$	26.3946i	0.959328i	0.877452	+	0.479664i	$0.159241\pi$
$757$	26.3946i	0.959328i	−0.877452	+	0.479664i	$0.840759\pi$
$758$	0	0
$759$	−20.4924	−0.743828
$760$	0	0
$761$	8.73863	0.316775	0.158388	−	0.987377i	$-0.449370\pi$
$761$	8.73863	0.316775	0.158388	+	0.987377i	$0.449370\pi$
$762$	0	0
$763$	18.1227i	0.656085i
$764$	0	0
$765$	20.7713i	0.750988i
$766$	0	0
$767$	−0.630683	−0.0227726
$768$	0	0
$769$	−40.2462	−1.45132	−0.725658	−	0.688056i	$-0.758464\pi$
$769$	−40.2462	−1.45132	−0.725658	+	0.688056i	$0.758464\pi$
$770$	0	0
$771$	− 67.9372i	− 2.44670i
$772$	0	0
$773$	− 1.69614i	− 0.0610060i	−0.999535	−	0.0305030i	$-0.990289\pi$
$773$	− 1.69614i	− 0.0610060i	0.999535	−	0.0305030i	$-0.00971091\pi$
$774$	0	0
$775$	21.7538	0.781419
$776$	0	0
$777$	18.2462	0.654579
$778$	0	0
$779$	− 12.8255i	− 0.459520i
$780$	0	0
$781$	10.5945i	0.379099i
$782$	0	0
$783$	56.9848	2.03647
$784$	0	0
$785$	14.3845	0.513404
$786$	0	0
$787$	10.5487i	0.376020i	0.982167	+	0.188010i	$0.0602037\pi$
$787$	10.5487i	0.376020i	−0.982167	+	0.188010i	$0.939796\pi$
$788$	0	0
$789$	37.7327i	1.34332i
$790$	0	0
$791$	4.87689	0.173402
$792$	0	0
$793$	−2.87689	−0.102162
$794$	0	0
$795$	13.5691i	0.481247i
$796$	0	0
$797$	25.8597i	0.915998i	0.888953	+	0.457999i	$0.151434\pi$
$797$	25.8597i	0.915998i	−0.888953	+	0.457999i	$0.848566\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	99.4773	3.51486
$802$	0	0
$803$	− 7.94584i	− 0.280403i
$804$	0	0
$805$	8.68951i	0.306265i
$806$	0	0
$807$	−35.8617	−1.26239
$808$	0	0
$809$	37.8617	1.33115	0.665574	−	0.746332i	$-0.268187\pi$
$809$	37.8617	1.33115	0.665574	+	0.746332i	$0.268187\pi$
$810$	0	0
$811$	15.8459i	0.556425i	0.960520	+	0.278213i	$0.0897420\pi$
$811$	15.8459i	0.556425i	−0.960520	+	0.278213i	$0.910258\pi$
$812$	0	0
$813$	30.9481i	1.08540i
$814$	0	0
$815$	22.7386	0.796500
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	− 10.3857i	− 0.362904i
$820$	0	0
$821$	− 33.5968i	− 1.17254i	−0.810118	−	0.586268i	$-0.800597\pi$
$821$	− 33.5968i	− 1.17254i	0.810118	−	0.586268i	$-0.199403\pi$
$822$	0	0
$823$	32.9848	1.14978	0.574890	−	0.818231i	$-0.305045\pi$
$823$	32.9848	1.14978	0.574890	+	0.818231i	$0.305045\pi$
$824$	0	0
$825$	8.49242	0.295668
$826$	0	0
$827$	6.20393i	0.215732i	0.994165	+	0.107866i	$0.0344017\pi$
$827$	6.20393i	0.215732i	−0.994165	+	0.107866i	$0.965598\pi$
$828$	0	0
$829$	− 46.6310i	− 1.61956i	−0.586732	−	0.809781i	$-0.699586\pi$
$829$	− 46.6310i	− 1.61956i	0.586732	−	0.809781i	$-0.300414\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	13.5691i	0.469579i
$836$	0	0
$837$	96.6543i	3.34086i
$838$	0	0
$839$	6.73863	0.232643	0.116322	−	0.993212i	$-0.462890\pi$
$839$	6.73863	0.232643	0.116322	+	0.993212i	$0.462890\pi$
$840$	0	0
$841$	−7.49242	−0.258359
$842$	0	0
$843$	18.1227i	0.624179i
$844$	0	0
$845$	− 17.1702i	− 0.590673i
$846$	0	0
$847$	9.24621	0.317704
$848$	0	0
$849$	−66.1080	−2.26882
$850$	0	0
$851$	− 30.9481i	− 1.06089i
$852$	0	0
$853$	25.4421i	0.871121i	0.900159	+	0.435561i	$0.143450\pi$
$853$	25.4421i	0.871121i	−0.900159	+	0.435561i	$0.856550\pi$
$854$	0	0
$855$	31.3693	1.07281
$856$	0	0
$857$	−46.9848	−1.60497	−0.802486	−	0.596671i	$-0.796490\pi$
$857$	−46.9848	−1.60497	−0.802486	+	0.596671i	$0.796490\pi$
$858$	0	0
$859$	28.3453i	0.967129i	0.875309	+	0.483565i	$0.160658\pi$
$859$	28.3453i	0.967129i	−0.875309	+	0.483565i	$0.839342\pi$
$860$	0	0
$861$	− 12.8255i	− 0.437091i
$862$	0	0
$863$	−36.4924	−1.24222	−0.621108	−	0.783725i	$-0.713317\pi$
$863$	−36.4924	−1.24222	−0.621108	+	0.783725i	$0.713317\pi$
$864$	0	0
$865$	32.3542	1.10007
$866$	0	0
$867$	39.2658i	1.33354i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−19.5076	−0.660989
$872$	0	0
$873$	−74.9848	−2.53785
$874$	0	0
$875$	− 12.0818i	− 0.408439i
$876$	0	0
$877$	35.5017i	1.19881i	0.800447	+	0.599404i	$0.204596\pi$
$877$	35.5017i	1.19881i	−0.800447	+	0.599404i	$0.795404\pi$
$878$	0	0
$879$	−31.3693	−1.05806
$880$	0	0
$881$	44.2462	1.49069	0.745346	−	0.666677i	$-0.232284\pi$
$881$	44.2462	1.49069	0.745346	+	0.666677i	$0.232284\pi$
$882$	0	0
$883$	4.71659i	0.158726i	0.996846	+	0.0793629i	$0.0252886\pi$
$883$	4.71659i	0.158726i	−0.996846	+	0.0793629i	$0.974711\pi$
$884$	0	0
$885$	− 1.90495i	− 0.0640340i
$886$	0	0
$887$	28.4924	0.956682	0.478341	−	0.878174i	$-0.341238\pi$
$887$	28.4924	0.956682	0.478341	+	0.878174i	$0.341238\pi$
$888$	0	0
$889$	−13.1231	−0.440135
$890$	0	0
$891$	13.4061i	0.449121i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	−26.2462	−0.876335
$898$	0	0
$899$	− 61.8963i	− 2.06436i
$900$	0	0
$901$	− 5.29723i	− 0.176476i
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	11.8617	0.394298
$906$	0	0
$907$	51.5564i	1.71190i	0.517056	+	0.855951i	$0.327028\pi$
$907$	51.5564i	1.71190i	−0.517056	+	0.855951i	$0.672972\pi$
$908$	0	0
$909$	63.5924i	2.10923i
$910$	0	0
$911$	−11.8617	−0.392997	−0.196498	−	0.980504i	$-0.562957\pi$
$911$	−11.8617	−0.392997	−0.196498	+	0.980504i	$0.562957\pi$
$912$	0	0
$913$	−7.50758	−0.248465
$914$	0	0
$915$	− 8.68951i	− 0.287266i
$916$	0	0
$917$	12.4536i	0.411255i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−76.3542	−2.51596
$922$	0	0
$923$	13.5691i	0.446633i
$924$	0	0
$925$	12.8255i	0.421699i
$926$	0	0
$927$	13.7538	0.451734
$928$	0	0
$929$	−2.49242	−0.0817737	−0.0408869	−	0.999164i	$-0.513018\pi$
$929$	−2.49242	−0.0817737	−0.0408869	+	0.999164i	$0.513018\pi$
$930$	0	0
$931$	− 3.02045i	− 0.0989912i
$932$	0	0
$933$	− 37.7327i	− 1.23531i
$934$	0	0
$935$	4.49242	0.146918
$936$	0	0
$937$	34.9848	1.14291	0.571453	−	0.820635i	$-0.306380\pi$
$937$	34.9848	1.14291	0.571453	+	0.820635i	$0.306380\pi$
$938$	0	0
$939$	62.6400i	2.04418i
$940$	0	0
$941$	30.7393i	1.00207i	0.865426	+	0.501037i	$0.167048\pi$
$941$	30.7393i	1.00207i	−0.865426	+	0.501037i	$0.832952\pi$
$942$	0	0
$943$	−21.7538	−0.708401
$944$	0	0
$945$	16.0000	0.520480
$946$	0	0
$947$	− 41.7056i	− 1.35525i	−0.735407	−	0.677625i	$-0.763009\pi$
$947$	− 41.7056i	− 1.35525i	0.735407	−	0.677625i	$-0.236991\pi$
$948$	0	0
$949$	− 10.1768i	− 0.330354i
$950$	0	0
$951$	−12.4924	−0.405095
$952$	0	0
$953$	−17.5076	−0.567126	−0.283563	−	0.958954i	$-0.591517\pi$
$953$	−17.5076	−0.567126	−0.283563	+	0.958954i	$0.591517\pi$
$954$	0	0
$955$	27.1383i	0.878173i
$956$	0	0
$957$	− 24.1636i	− 0.781098i
$958$	0	0
$959$	−16.2462	−0.524618
$960$	0	0
$961$	73.9848	2.38661
$962$	0	0
$963$	86.6405i	2.79195i
$964$	0	0
$965$	18.8664i	0.607329i
$966$	0	0
$967$	10.8769	0.349777	0.174889	−	0.984588i	$-0.444043\pi$
$967$	10.8769	0.349777	0.174889	+	0.984588i	$0.444043\pi$
$968$	0	0
$969$	−18.2462	−0.586153
$970$	0	0
$971$	10.9663i	0.351925i	0.984397	+	0.175962i	$0.0563037\pi$
$971$	10.9663i	0.351925i	−0.984397	+	0.175962i	$0.943696\pi$
$972$	0	0
$973$	− 8.31768i	− 0.266652i
$974$	0	0
$975$	10.8769	0.348339
$976$	0	0
$977$	23.7538	0.759951	0.379976	−	0.924997i	$-0.375932\pi$
$977$	23.7538	0.759951	0.379976	+	0.924997i	$0.375932\pi$
$978$	0	0
$979$	− 21.5150i	− 0.687621i
$980$	0	0
$981$	− 110.967i	− 3.54291i
$982$	0	0
$983$	34.2462	1.09228	0.546142	−	0.837692i	$-0.316096\pi$
$983$	34.2462	1.09228	0.546142	+	0.837692i	$0.316096\pi$
$984$	0	0
$985$	36.4924	1.16275
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 6.78456i	− 0.215737i
$990$	0	0
$991$	36.4924	1.15922	0.579610	−	0.814894i	$-0.303205\pi$
$991$	36.4924	1.15922	0.579610	+	0.814894i	$0.303205\pi$
$992$	0	0
$993$	44.9848	1.42755
$994$	0	0
$995$	− 30.9481i	− 0.981122i
$996$	0	0
$997$	33.0619i	1.04708i	0.852001	+	0.523540i	$0.175389\pi$
$997$	33.0619i	1.04708i	−0.852001	+	0.523540i	$0.824611\pi$
$998$	0	0
$999$	−56.9848	−1.80292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.b.b.113.1		4
3.2	odd	2		2016.2.c.c.1009.3		4
4.3	odd	2		56.2.b.b.29.3	✓	4
7.2	even	3		1568.2.t.d.753.4		8
7.3	odd	6		1568.2.t.e.177.4		8
7.4	even	3		1568.2.t.d.177.1		8
7.5	odd	6		1568.2.t.e.753.1		8
7.6	odd	2		1568.2.b.d.785.4		4
8.3	odd	2		56.2.b.b.29.4	yes	4
8.5	even	2	inner	224.2.b.b.113.4		4
12.11	even	2		504.2.c.d.253.2		4
16.3	odd	4		1792.2.a.x.1.1		4
16.5	even	4		1792.2.a.v.1.1		4
16.11	odd	4		1792.2.a.x.1.4		4
16.13	even	4		1792.2.a.v.1.4		4
24.5	odd	2		2016.2.c.c.1009.2		4
24.11	even	2		504.2.c.d.253.1		4
28.3	even	6		392.2.p.e.373.3		8
28.11	odd	6		392.2.p.f.373.3		8
28.19	even	6		392.2.p.e.165.1		8
28.23	odd	6		392.2.p.f.165.1		8
28.27	even	2		392.2.b.c.197.3		4
56.3	even	6		392.2.p.e.373.1		8
56.5	odd	6		1568.2.t.e.753.4		8
56.11	odd	6		392.2.p.f.373.1		8
56.13	odd	2		1568.2.b.d.785.1		4
56.19	even	6		392.2.p.e.165.3		8
56.27	even	2		392.2.b.c.197.4		4
56.37	even	6		1568.2.t.d.753.1		8
56.45	odd	6		1568.2.t.e.177.1		8
56.51	odd	6		392.2.p.f.165.3		8
56.53	even	6		1568.2.t.d.177.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.b.29.3	✓	4	4.3	odd	2
56.2.b.b.29.4	yes	4	8.3	odd	2
224.2.b.b.113.1		4	1.1	even	1	trivial
224.2.b.b.113.4		4	8.5	even	2	inner
392.2.b.c.197.3		4	28.27	even	2
392.2.b.c.197.4		4	56.27	even	2
392.2.p.e.165.1		8	28.19	even	6
392.2.p.e.165.3		8	56.19	even	6
392.2.p.e.373.1		8	56.3	even	6
392.2.p.e.373.3		8	28.3	even	6
392.2.p.f.165.1		8	28.23	odd	6
392.2.p.f.165.3		8	56.51	odd	6
392.2.p.f.373.1		8	56.11	odd	6
392.2.p.f.373.3		8	28.11	odd	6
504.2.c.d.253.1		4	24.11	even	2
504.2.c.d.253.2		4	12.11	even	2
1568.2.b.d.785.1		4	56.13	odd	2
1568.2.b.d.785.4		4	7.6	odd	2
1568.2.t.d.177.1		8	7.4	even	3
1568.2.t.d.177.4		8	56.53	even	6
1568.2.t.d.753.1		8	56.37	even	6
1568.2.t.d.753.4		8	7.2	even	3
1568.2.t.e.177.1		8	56.45	odd	6
1568.2.t.e.177.4		8	7.3	odd	6
1568.2.t.e.753.1		8	7.5	odd	6
1568.2.t.e.753.4		8	56.5	odd	6
1792.2.a.v.1.1		4	16.5	even	4
1792.2.a.v.1.4		4	16.13	even	4
1792.2.a.x.1.1		4	16.3	odd	4
1792.2.a.x.1.4		4	16.11	odd	4
2016.2.c.c.1009.2		4	24.5	odd	2
2016.2.c.c.1009.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.02045i q^{3} -1.69614i q^{5} +1.00000 q^{7} -6.12311 q^{9} +O(q^{10})\)	\(q-3.02045i q^{3} -1.69614i q^{5} +1.00000 q^{7} -6.12311 q^{9} +1.32431i q^{11} +1.69614i q^{13} -5.12311 q^{15} +2.00000 q^{17} -3.02045i q^{19} -3.02045i q^{21} -5.12311 q^{23} +2.12311 q^{25} +9.43318i q^{27} -6.04090i q^{29} +10.2462 q^{31} +4.00000 q^{33} -1.69614i q^{35} +6.04090i q^{37} +5.12311 q^{39} +4.24621 q^{41} +1.32431i q^{43} +10.3857i q^{45} +1.00000 q^{49} -6.04090i q^{51} -2.64861i q^{53} +2.24621 q^{55} -9.12311 q^{57} +0.371834i q^{59} +1.69614i q^{61} -6.12311 q^{63} +2.87689 q^{65} +11.5012i q^{67} +15.4741i q^{69} +8.00000 q^{71} -6.00000 q^{73} -6.41273i q^{75} +1.32431i q^{77} +10.1231 q^{81} +5.66906i q^{83} -3.39228i q^{85} -18.2462 q^{87} -16.2462 q^{89} +1.69614i q^{91} -30.9481i q^{93} -5.12311 q^{95} +12.2462 q^{97} -8.10887i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 - 8 * q^9	\( 4 q + 4 q^{7} - 8 q^{9} - 4 q^{15} + 8 q^{17} - 4 q^{23} - 8 q^{25} + 8 q^{31} + 16 q^{33} + 4 q^{39} - 16 q^{41} + 4 q^{49} - 24 q^{55} - 20 q^{57} - 8 q^{63} + 28 q^{65} + 32 q^{71} - 24 q^{73} + 24 q^{81} - 40 q^{87} - 32 q^{89} - 4 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 8 * q^9 - 4 * q^15 + 8 * q^17 - 4 * q^23 - 8 * q^25 + 8 * q^31 + 16 * q^33 + 4 * q^39 - 16 * q^41 + 4 * q^49 - 24 * q^55 - 20 * q^57 - 8 * q^63 + 28 * q^65 + 32 * q^71 - 24 * q^73 + 24 * q^81 - 40 * q^87 - 32 * q^89 - 4 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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