LMFDB - Embedded newform 4600.2.e.u.4049.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 $ x^10 + 28x^8 + 260x^6 + 897x^4 + 1056x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.5
Root			$-0.568386i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.u.4049.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.568386i q^{3} -4.73770i q^{7} +2.67694 q^{9} +O(q^{10})$	$q-0.568386i q^{3} -4.73770i q^{7} +2.67694 q^{9} -0.360532 q^{11} +5.26123i q^{13} -0.370852i q^{17} +4.60586 q^{19} -2.69284 q^{21} -1.00000i q^{23} -3.22669i q^{27} -0.939238 q^{29} +9.66662 q^{31} +0.204921i q^{33} -3.26862i q^{37} +2.99041 q^{39} +5.29977 q^{41} +1.25491i q^{47} -15.4458 q^{49} -0.210787 q^{51} +10.9278i q^{53} -2.61790i q^{57} +9.66955 q^{59} +9.71441 q^{61} -12.6825i q^{63} +7.07001i q^{67} -0.568386 q^{69} +11.3747 q^{71} +0.745086i q^{73} +1.70809i q^{77} -0.415709 q^{79} +6.19680 q^{81} -9.26862i q^{83} +0.533850i q^{87} -12.6122 q^{89} +24.9261 q^{91} -5.49437i q^{93} -14.0404i q^{97} -0.965121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 26 q^{9}+O(q^{10}) $ 10 * q - 26 * q^9	$ 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) $ 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 0.568386i	− 0.328158i	−0.986447	−	0.164079i	$-0.947535\pi$
$3$	− 0.568386i	− 0.328158i	0.986447	−	0.164079i	$-0.0524652\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.73770i	− 1.79068i	−0.445381	−	0.895341i	$-0.646932\pi$
$7$	− 4.73770i	− 1.79068i	0.445381	−	0.895341i	$-0.353068\pi$
$8$	0	0
$9$	2.67694	0.892312
$10$	0	0
$11$	−0.360532	−0.108704	−0.0543522	−	0.998522i	$-0.517309\pi$
$11$	−0.360532	−0.108704	−0.0543522	+	0.998522i	$0.517309\pi$
$12$	0	0
$13$	5.26123i	1.45920i	0.683873	+	0.729601i	$0.260294\pi$
$13$	5.26123i	1.45920i	−0.683873	+	0.729601i	$0.739706\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.370852i	− 0.0899449i	−0.998988	−	0.0449724i	$-0.985680\pi$
$17$	− 0.370852i	− 0.0899449i	0.998988	−	0.0449724i	$-0.0143200\pi$
$18$	0	0
$19$	4.60586	1.05666	0.528328	−	0.849040i	$-0.322819\pi$
$19$	4.60586	1.05666	0.528328	+	0.849040i	$0.322819\pi$
$20$	0	0
$21$	−2.69284	−0.587626
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.22669i	− 0.620977i
$28$	0	0
$29$	−0.939238	−0.174412	−0.0872061	−	0.996190i	$-0.527794\pi$
$29$	−0.939238	−0.174412	−0.0872061	+	0.996190i	$0.527794\pi$
$30$	0	0
$31$	9.66662	1.73618	0.868088	−	0.496411i	$-0.165349\pi$
$31$	9.66662	1.73618	0.868088	+	0.496411i	$0.165349\pi$
$32$	0	0
$33$	0.204921i	0.0356722i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.26862i	− 0.537357i	−0.963230	−	0.268679i	$-0.913413\pi$
$37$	− 3.26862i	− 0.537357i	0.963230	−	0.268679i	$-0.0865869\pi$
$38$	0	0
$39$	2.99041	0.478849
$40$	0	0
$41$	5.29977	0.827685	0.413843	−	0.910348i	$-0.364186\pi$
$41$	5.29977	0.827685	0.413843	+	0.910348i	$0.364186\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.25491i	0.183048i	0.995803	+	0.0915240i	$0.0291738\pi$
$47$	1.25491i	0.183048i	−0.995803	+	0.0915240i	$0.970826\pi$
$48$	0	0
$49$	−15.4458	−2.20654
$50$	0	0
$51$	−0.210787	−0.0295161
$52$	0	0
$53$	10.9278i	1.50106i	0.660839	+	0.750528i	$0.270201\pi$
$53$	10.9278i	1.50106i	−0.660839	+	0.750528i	$0.729799\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.61790i	− 0.346750i
$58$	0	0
$59$	9.66955	1.25887	0.629434	−	0.777054i	$-0.283287\pi$
$59$	9.66955	1.25887	0.629434	+	0.777054i	$0.283287\pi$
$60$	0	0
$61$	9.71441	1.24380	0.621901	−	0.783096i	$-0.286361\pi$
$61$	9.71441	1.24380	0.621901	+	0.783096i	$0.286361\pi$
$62$	0	0
$63$	− 12.6825i	− 1.59785i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.07001i	0.863739i	0.901936	+	0.431870i	$0.142146\pi$
$67$	7.07001i	0.863739i	−0.901936	+	0.431870i	$0.857854\pi$
$68$	0	0
$69$	−0.568386	−0.0684257
$70$	0	0
$71$	11.3747	1.34993	0.674965	−	0.737850i	$-0.264159\pi$
$71$	11.3747	1.34993	0.674965	+	0.737850i	$0.264159\pi$
$72$	0	0
$73$	0.745086i	0.0872057i	0.999049	+	0.0436029i	$0.0138836\pi$
$73$	0.745086i	0.0872057i	−0.999049	+	0.0436029i	$0.986116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.70809i	0.194655i
$78$	0	0
$79$	−0.415709	−0.0467709	−0.0233854	−	0.999727i	$-0.507444\pi$
$79$	−0.415709	−0.0467709	−0.0233854	+	0.999727i	$0.507444\pi$
$80$	0	0
$81$	6.19680	0.688534
$82$	0	0
$83$	− 9.26862i	− 1.01736i	−0.860955	−	0.508681i	$-0.830133\pi$
$83$	− 9.26862i	− 1.01736i	0.860955	−	0.508681i	$-0.169867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.533850i	0.0572347i
$88$	0	0
$89$	−12.6122	−1.33689	−0.668444	−	0.743763i	$-0.733039\pi$
$89$	−12.6122	−1.33689	−0.668444	+	0.743763i	$0.733039\pi$
$90$	0	0
$91$	24.9261	2.61297
$92$	0	0
$93$	− 5.49437i	− 0.569740i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.0404i	− 1.42559i	−0.701374	−	0.712793i	$-0.747430\pi$
$97$	− 14.0404i	− 1.42559i	0.701374	−	0.712793i	$-0.252570\pi$
$98$	0	0
$99$	−0.965121	−0.0969983
$100$	0	0
$101$	−12.7440	−1.26808	−0.634038	−	0.773302i	$-0.718604\pi$
$101$	−12.7440	−1.26808	−0.634038	+	0.773302i	$0.718604\pi$
$102$	0	0
$103$	− 3.31347i	− 0.326486i	−0.986586	−	0.163243i	$-0.947805\pi$
$103$	− 3.31347i	− 0.326486i	0.986586	−	0.163243i	$-0.0521955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.13184i	0.786135i	0.919510	+	0.393067i	$0.128586\pi$
$107$	8.13184i	0.786135i	−0.919510	+	0.393067i	$0.871414\pi$
$108$	0	0
$109$	5.79508	0.555068	0.277534	−	0.960716i	$-0.410483\pi$
$109$	5.79508	0.555068	0.277534	+	0.960716i	$0.410483\pi$
$110$	0	0
$111$	−1.85784	−0.176338
$112$	0	0
$113$	− 7.60724i	− 0.715629i	−0.933793	−	0.357815i	$-0.883522\pi$
$113$	− 7.60724i	− 0.715629i	0.933793	−	0.357815i	$-0.116478\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.0840i	1.30206i
$118$	0	0
$119$	−1.75699	−0.161063
$120$	0	0
$121$	−10.8700	−0.988183
$122$	0	0
$123$	− 3.01232i	− 0.271611i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.13077i	0.366547i	0.983062	+	0.183273i	$0.0586693\pi$
$127$	4.13077i	0.366547i	−0.983062	+	0.183273i	$0.941331\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.7582	−1.55154	−0.775770	−	0.631016i	$-0.782638\pi$
$131$	−17.7582	−1.55154	−0.775770	+	0.631016i	$0.782638\pi$
$132$	0	0
$133$	− 21.8212i	− 1.89213i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.86954i	− 0.586905i	−0.955974	−	0.293452i	$-0.905196\pi$
$137$	− 6.86954i	− 0.586905i	0.955974	−	0.293452i	$-0.0948042\pi$
$138$	0	0
$139$	−20.6635	−1.75266	−0.876330	−	0.481712i	$-0.840015\pi$
$139$	−20.6635	−1.75266	−0.876330	+	0.481712i	$0.840015\pi$
$140$	0	0
$141$	0.713276	0.0600686
$142$	0	0
$143$	− 1.89684i	− 0.158622i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.77917i	0.724094i
$148$	0	0
$149$	16.4763	1.34979	0.674896	−	0.737912i	$-0.264188\pi$
$149$	16.4763	1.34979	0.674896	+	0.737912i	$0.264188\pi$
$150$	0	0
$151$	−8.89224	−0.723640	−0.361820	−	0.932248i	$-0.617844\pi$
$151$	−8.89224	−0.723640	−0.361820	+	0.932248i	$0.617844\pi$
$152$	0	0
$153$	− 0.992748i	− 0.0802589i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.62249i	0.528532i	0.964450	+	0.264266i	$0.0851297\pi$
$157$	6.62249i	0.528532i	−0.964450	+	0.264266i	$0.914870\pi$
$158$	0	0
$159$	6.21124	0.492583
$160$	0	0
$161$	−4.73770	−0.373383
$162$	0	0
$163$	− 16.0779i	− 1.25932i	−0.776872	−	0.629658i	$-0.783195\pi$
$163$	− 16.0779i	− 1.25932i	0.776872	−	0.629658i	$-0.216805\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.8763i	− 1.07378i	−0.843651	−	0.536891i	$-0.819598\pi$
$167$	− 13.8763i	− 1.07378i	0.843651	−	0.536891i	$-0.180402\pi$
$168$	0	0
$169$	−14.6805	−1.12927
$170$	0	0
$171$	12.3296	0.942867
$172$	0	0
$173$	− 2.05518i	− 0.156252i	−0.996943	−	0.0781261i	$-0.975106\pi$
$173$	− 2.05518i	− 0.156252i	0.996943	−	0.0781261i	$-0.0248937\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5.49604i	− 0.413108i
$178$	0	0
$179$	−17.6478	−1.31906	−0.659531	−	0.751678i	$-0.729245\pi$
$179$	−17.6478	−1.31906	−0.659531	+	0.751678i	$0.729245\pi$
$180$	0	0
$181$	2.85477	0.212193	0.106097	−	0.994356i	$-0.466165\pi$
$181$	2.85477	0.212193	0.106097	+	0.994356i	$0.466165\pi$
$182$	0	0
$183$	− 5.52153i	− 0.408164i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.133704i	0.00977741i
$188$	0	0
$189$	−15.2871	−1.11197
$190$	0	0
$191$	1.13677	0.0822540	0.0411270	−	0.999154i	$-0.486905\pi$
$191$	1.13677	0.0822540	0.0411270	+	0.999154i	$0.486905\pi$
$192$	0	0
$193$	− 26.4694i	− 1.90531i	−0.304055	−	0.952654i	$-0.598341\pi$
$193$	− 26.4694i	− 1.90531i	0.304055	−	0.952654i	$-0.401659\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.44424i	− 0.387886i	−0.981013	−	0.193943i	$-0.937872\pi$
$197$	− 5.44424i	− 0.387886i	0.981013	−	0.193943i	$-0.0621277\pi$
$198$	0	0
$199$	18.6122	1.31938	0.659691	−	0.751537i	$-0.270687\pi$
$199$	18.6122	1.31938	0.659691	+	0.751537i	$0.270687\pi$
$200$	0	0
$201$	4.01850	0.283443
$202$	0	0
$203$	4.44983i	0.312317i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.67694i	− 0.186060i
$208$	0	0
$209$	−1.66056	−0.114863
$210$	0	0
$211$	6.10003	0.419944	0.209972	−	0.977707i	$-0.432663\pi$
$211$	6.10003	0.419944	0.209972	+	0.977707i	$0.432663\pi$
$212$	0	0
$213$	− 6.46523i	− 0.442990i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 45.7975i	− 3.10894i
$218$	0	0
$219$	0.423497	0.0286173
$220$	0	0
$221$	1.95114	0.131248
$222$	0	0
$223$	− 16.4136i	− 1.09913i	−0.835450	−	0.549567i	$-0.814793\pi$
$223$	− 16.4136i	− 1.09913i	0.835450	−	0.549567i	$-0.185207\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.21171i	− 0.0804240i	−0.999191	−	0.0402120i	$-0.987197\pi$
$227$	− 1.21171i	− 0.0804240i	0.999191	−	0.0402120i	$-0.0128033\pi$
$228$	0	0
$229$	22.8293	1.50860	0.754300	−	0.656529i	$-0.227976\pi$
$229$	22.8293	1.50860	0.754300	+	0.656529i	$0.227976\pi$
$230$	0	0
$231$	0.970856	0.0638776
$232$	0	0
$233$	21.9420i	1.43747i	0.695284	+	0.718735i	$0.255278\pi$
$233$	21.9420i	1.43747i	−0.695284	+	0.718735i	$0.744722\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.236283i	0.0153482i
$238$	0	0
$239$	8.58274	0.555172	0.277586	−	0.960701i	$-0.410466\pi$
$239$	8.58274	0.555172	0.277586	+	0.960701i	$0.410466\pi$
$240$	0	0
$241$	15.3857	0.991079	0.495540	−	0.868585i	$-0.334970\pi$
$241$	15.3857	0.991079	0.495540	+	0.868585i	$0.334970\pi$
$242$	0	0
$243$	− 13.2023i	− 0.846925i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.2325i	1.54187i
$248$	0	0
$249$	−5.26815	−0.333856
$250$	0	0
$251$	−8.85477	−0.558908	−0.279454	−	0.960159i	$-0.590153\pi$
$251$	−8.85477	−0.558908	−0.279454	+	0.960159i	$0.590153\pi$
$252$	0	0
$253$	0.360532i	0.0226664i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.6008i	1.40980i	0.709308	+	0.704899i	$0.249008\pi$
$257$	22.6008i	1.40980i	−0.709308	+	0.704899i	$0.750992\pi$
$258$	0	0
$259$	−15.4857	−0.962236
$260$	0	0
$261$	−2.51428	−0.155630
$262$	0	0
$263$	− 7.73590i	− 0.477016i	−0.971141	−	0.238508i	$-0.923342\pi$
$263$	− 7.73590i	− 0.477016i	0.971141	−	0.238508i	$-0.0766583\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.16858i	0.438710i
$268$	0	0
$269$	3.87154	0.236052	0.118026	−	0.993011i	$-0.462343\pi$
$269$	3.87154	0.236052	0.118026	+	0.993011i	$0.462343\pi$
$270$	0	0
$271$	4.24705	0.257990	0.128995	−	0.991645i	$-0.458825\pi$
$271$	4.24705	0.257990	0.128995	+	0.991645i	$0.458825\pi$
$272$	0	0
$273$	− 14.1677i	− 0.857466i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.26754i	0.436664i	0.975875	+	0.218332i	$0.0700616\pi$
$277$	7.26754i	0.436664i	−0.975875	+	0.218332i	$0.929938\pi$
$278$	0	0
$279$	25.8769	1.54921
$280$	0	0
$281$	−13.0131	−0.776297	−0.388148	−	0.921597i	$-0.626885\pi$
$281$	−13.0131	−0.776297	−0.388148	+	0.921597i	$0.626885\pi$
$282$	0	0
$283$	17.8342i	1.06013i	0.847956	+	0.530067i	$0.177833\pi$
$283$	17.8342i	1.06013i	−0.847956	+	0.530067i	$0.822167\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 25.1087i	− 1.48212i
$288$	0	0
$289$	16.8625	0.991910
$290$	0	0
$291$	−7.98037	−0.467818
$292$	0	0
$293$	7.32291i	0.427809i	0.976855	+	0.213905i	$0.0686182\pi$
$293$	7.32291i	0.427809i	−0.976855	+	0.213905i	$0.931382\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.16333i	0.0675030i
$298$	0	0
$299$	5.26123	0.304265
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.24352i	0.416129i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.1127i	1.66155i	0.556608	+	0.830775i	$0.312103\pi$
$307$	29.1127i	1.66155i	−0.556608	+	0.830775i	$0.687897\pi$
$308$	0	0
$309$	−1.88333	−0.107139
$310$	0	0
$311$	0.458912	0.0260225	0.0130113	−	0.999915i	$-0.495858\pi$
$311$	0.458912	0.0260225	0.0130113	+	0.999915i	$0.495858\pi$
$312$	0	0
$313$	− 17.8279i	− 1.00769i	−0.863794	−	0.503846i	$-0.831918\pi$
$313$	− 17.8279i	− 1.00769i	0.863794	−	0.503846i	$-0.168082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.8799i	− 1.34123i	−0.741807	−	0.670614i	$-0.766031\pi$
$317$	− 23.8799i	− 1.34123i	0.741807	−	0.670614i	$-0.233969\pi$
$318$	0	0
$319$	0.338625	0.0189594
$320$	0	0
$321$	4.62203	0.257976
$322$	0	0
$323$	− 1.70809i	− 0.0950407i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.29384i	− 0.182150i
$328$	0	0
$329$	5.94540	0.327781
$330$	0	0
$331$	30.0214	1.65013	0.825063	−	0.565041i	$-0.191140\pi$
$331$	30.0214	1.65013	0.825063	+	0.565041i	$0.191140\pi$
$332$	0	0
$333$	− 8.74988i	− 0.479490i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 11.9440i	− 0.650631i	−0.945605	−	0.325316i	$-0.894529\pi$
$337$	− 11.9440i	− 0.650631i	0.945605	−	0.325316i	$-0.105471\pi$
$338$	0	0
$339$	−4.32385	−0.234839
$340$	0	0
$341$	−3.48512	−0.188730
$342$	0	0
$343$	40.0136i	2.16053i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.873132i	0.0468722i	0.999725	+	0.0234361i	$0.00746062\pi$
$347$	0.873132i	0.0468722i	−0.999725	+	0.0234361i	$0.992539\pi$
$348$	0	0
$349$	18.8868	1.01099	0.505493	−	0.862831i	$-0.331311\pi$
$349$	18.8868	1.01099	0.505493	+	0.862831i	$0.331311\pi$
$350$	0	0
$351$	16.9764	0.906131
$352$	0	0
$353$	− 5.43875i	− 0.289475i	−0.989470	−	0.144738i	$-0.953766\pi$
$353$	− 5.43875i	− 0.289475i	0.989470	−	0.144738i	$-0.0462338\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.998646i	0.0528540i
$358$	0	0
$359$	23.4229	1.23622	0.618108	−	0.786093i	$-0.287899\pi$
$359$	23.4229	1.23622	0.618108	+	0.786093i	$0.287899\pi$
$360$	0	0
$361$	2.21390	0.116521
$362$	0	0
$363$	6.17837i	0.324280i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 27.2014i	− 1.41990i	−0.704252	−	0.709951i	$-0.748717\pi$
$367$	− 27.2014i	− 1.41990i	0.704252	−	0.709951i	$-0.251283\pi$
$368$	0	0
$369$	14.1872	0.738554
$370$	0	0
$371$	51.7728	2.68791
$372$	0	0
$373$	20.0431i	1.03779i	0.854837	+	0.518897i	$0.173657\pi$
$373$	20.0431i	1.03779i	−0.854837	+	0.518897i	$0.826343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.94155i	− 0.254503i
$378$	0	0
$379$	12.8364	0.659362	0.329681	−	0.944092i	$-0.393059\pi$
$379$	12.8364	0.659362	0.329681	+	0.944092i	$0.393059\pi$
$380$	0	0
$381$	2.34787	0.120285
$382$	0	0
$383$	− 30.5405i	− 1.56055i	−0.625439	−	0.780273i	$-0.715080\pi$
$383$	− 30.5405i	− 1.56055i	0.625439	−	0.780273i	$-0.284920\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.3392	−1.63966	−0.819831	−	0.572605i	$-0.805933\pi$
$389$	−32.3392	−1.63966	−0.819831	+	0.572605i	$0.805933\pi$
$390$	0	0
$391$	−0.370852	−0.0187548
$392$	0	0
$393$	10.0935i	0.509150i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 28.9147i	− 1.45119i	−0.688123	−	0.725594i	$-0.741565\pi$
$397$	− 28.9147i	− 1.45119i	0.688123	−	0.725594i	$-0.258435\pi$
$398$	0	0
$399$	−12.4028	−0.620919
$400$	0	0
$401$	24.2862	1.21279	0.606397	−	0.795162i	$-0.292614\pi$
$401$	24.2862	1.21279	0.606397	+	0.795162i	$0.292614\pi$
$402$	0	0
$403$	50.8583i	2.53343i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.17844i	0.0584131i
$408$	0	0
$409$	−31.8371	−1.57424	−0.787122	−	0.616797i	$-0.788430\pi$
$409$	−31.8371	−1.57424	−0.787122	+	0.616797i	$0.788430\pi$
$410$	0	0
$411$	−3.90455	−0.192597
$412$	0	0
$413$	− 45.8114i	− 2.25423i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.7449i	0.575149i
$418$	0	0
$419$	21.4880	1.04976	0.524879	−	0.851177i	$-0.324110\pi$
$419$	21.4880	1.04976	0.524879	+	0.851177i	$0.324110\pi$
$420$	0	0
$421$	−13.2930	−0.647859	−0.323930	−	0.946081i	$-0.605004\pi$
$421$	−13.2930	−0.647859	−0.323930	+	0.946081i	$0.605004\pi$
$422$	0	0
$423$	3.35933i	0.163336i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 46.0239i	− 2.22725i
$428$	0	0
$429$	−1.07814	−0.0520530
$430$	0	0
$431$	−0.534460	−0.0257440	−0.0128720	−	0.999917i	$-0.504097\pi$
$431$	−0.534460	−0.0257440	−0.0128720	+	0.999917i	$0.504097\pi$
$432$	0	0
$433$	16.3377i	0.785140i	0.919722	+	0.392570i	$0.128414\pi$
$433$	16.3377i	0.785140i	−0.919722	+	0.392570i	$0.871586\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.60586i	− 0.220328i
$438$	0	0
$439$	−28.7573	−1.37251	−0.686255	−	0.727361i	$-0.740746\pi$
$439$	−28.7573	−1.37251	−0.686255	+	0.727361i	$0.740746\pi$
$440$	0	0
$441$	−41.3474	−1.96892
$442$	0	0
$443$	− 15.6542i	− 0.743751i	−0.928283	−	0.371876i	$-0.878715\pi$
$443$	− 15.6542i	− 0.743751i	0.928283	−	0.371876i	$-0.121285\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 9.36491i	− 0.442945i
$448$	0	0
$449$	−38.4302	−1.81363	−0.906817	−	0.421525i	$-0.861495\pi$
$449$	−38.4302	−1.81363	−0.906817	+	0.421525i	$0.861495\pi$
$450$	0	0
$451$	−1.91074	−0.0899730
$452$	0	0
$453$	5.05422i	0.237468i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.2199i	1.74107i	0.492104	+	0.870536i	$0.336228\pi$
$457$	37.2199i	1.74107i	−0.492104	+	0.870536i	$0.663772\pi$
$458$	0	0
$459$	−1.19663	−0.0558537
$460$	0	0
$461$	−3.25491	−0.151596	−0.0757982	−	0.997123i	$-0.524150\pi$
$461$	−3.25491	−0.151596	−0.0757982	+	0.997123i	$0.524150\pi$
$462$	0	0
$463$	29.9534i	1.39205i	0.718016	+	0.696027i	$0.245050\pi$
$463$	29.9534i	1.39205i	−0.718016	+	0.696027i	$0.754950\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.3302i	1.81999i	0.414623	+	0.909993i	$0.363913\pi$
$467$	39.3302i	1.81999i	−0.414623	+	0.909993i	$0.636087\pi$
$468$	0	0
$469$	33.4956	1.54668
$470$	0	0
$471$	3.76413	0.173442
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	29.2532i	1.33941i
$478$	0	0
$479$	−8.18660	−0.374055	−0.187028	−	0.982355i	$-0.559885\pi$
$479$	−8.18660	−0.374055	−0.187028	+	0.982355i	$0.559885\pi$
$480$	0	0
$481$	17.1969	0.784113
$482$	0	0
$483$	2.69284i	0.122529i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.9963i	1.44989i	0.688806	+	0.724946i	$0.258135\pi$
$487$	31.9963i	1.44989i	−0.688806	+	0.724946i	$0.741865\pi$
$488$	0	0
$489$	−9.13844	−0.413255
$490$	0	0
$491$	−7.12584	−0.321585	−0.160792	−	0.986988i	$-0.551405\pi$
$491$	−7.12584	−0.321585	−0.160792	+	0.986988i	$0.551405\pi$
$492$	0	0
$493$	0.348319i	0.0156875i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 53.8899i	− 2.41729i
$498$	0	0
$499$	−23.4219	−1.04851	−0.524254	−	0.851562i	$-0.675656\pi$
$499$	−23.4219	−1.04851	−0.524254	+	0.851562i	$0.675656\pi$
$500$	0	0
$501$	−7.88711	−0.352370
$502$	0	0
$503$	− 3.94170i	− 0.175752i	−0.996131	−	0.0878758i	$-0.971992\pi$
$503$	− 3.94170i	− 0.175752i	0.996131	−	0.0878758i	$-0.0280079\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.34421i	0.370579i
$508$	0	0
$509$	−38.5237	−1.70753	−0.853766	−	0.520656i	$-0.825687\pi$
$509$	−38.5237	−1.70753	−0.853766	+	0.520656i	$0.825687\pi$
$510$	0	0
$511$	3.52999	0.156158
$512$	0	0
$513$	− 14.8617i	− 0.656159i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.452436i	− 0.0198981i
$518$	0	0
$519$	−1.16813	−0.0512754
$520$	0	0
$521$	−39.2428	−1.71926	−0.859630	−	0.510917i	$-0.829306\pi$
$521$	−39.2428	−1.71926	−0.859630	+	0.510917i	$0.829306\pi$
$522$	0	0
$523$	− 39.0190i	− 1.70618i	−0.521763	−	0.853090i	$-0.674726\pi$
$523$	− 39.0190i	− 1.70618i	0.521763	−	0.853090i	$-0.325274\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.58489i	− 0.156160i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	25.8848	1.12330
$532$	0	0
$533$	27.8833i	1.20776i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0308i	0.432860i
$538$	0	0
$539$	5.56870	0.239861
$540$	0	0
$541$	2.54061	0.109230	0.0546148	−	0.998508i	$-0.482607\pi$
$541$	2.54061	0.109230	0.0546148	+	0.998508i	$0.482607\pi$
$542$	0	0
$543$	− 1.62261i	− 0.0696329i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6.78776i	− 0.290224i	−0.989415	−	0.145112i	$-0.953646\pi$
$547$	− 6.78776i	− 0.290224i	0.989415	−	0.145112i	$-0.0463542\pi$
$548$	0	0
$549$	26.0049	1.10986
$550$	0	0
$551$	−4.32600	−0.184294
$552$	0	0
$553$	1.96950i	0.0837517i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.7086i	1.04694i	0.852045	+	0.523468i	$0.175362\pi$
$557$	24.7086i	1.04694i	−0.852045	+	0.523468i	$0.824638\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.0759955	0.00320853
$562$	0	0
$563$	− 28.1423i	− 1.18606i	−0.805181	−	0.593029i	$-0.797932\pi$
$563$	− 28.1423i	− 1.18606i	0.805181	−	0.593029i	$-0.202068\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 29.3586i	− 1.23294i
$568$	0	0
$569$	−41.1938	−1.72694	−0.863468	−	0.504404i	$-0.831712\pi$
$569$	−41.1938	−1.72694	−0.863468	+	0.504404i	$0.831712\pi$
$570$	0	0
$571$	32.4687	1.35877	0.679387	−	0.733780i	$-0.262246\pi$
$571$	32.4687	1.35877	0.679387	+	0.733780i	$0.262246\pi$
$572$	0	0
$573$	− 0.646126i	− 0.0269923i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 6.61770i	− 0.275498i	−0.990467	−	0.137749i	$-0.956013\pi$
$577$	− 6.61770i	− 0.275498i	0.990467	−	0.137749i	$-0.0439868\pi$
$578$	0	0
$579$	−15.0448	−0.625242
$580$	0	0
$581$	−43.9119	−1.82177
$582$	0	0
$583$	− 3.93984i	− 0.163171i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.18008i	0.337628i	0.985648	+	0.168814i	$0.0539937\pi$
$587$	8.18008i	0.337628i	−0.985648	+	0.168814i	$0.946006\pi$
$588$	0	0
$589$	44.5230	1.83454
$590$	0	0
$591$	−3.09443	−0.127288
$592$	0	0
$593$	29.7489i	1.22164i	0.791768	+	0.610821i	$0.209161\pi$
$593$	29.7489i	1.22164i	−0.791768	+	0.610821i	$0.790839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.5789i	− 0.432966i
$598$	0	0
$599$	−39.4069	−1.61012	−0.805061	−	0.593192i	$-0.797868\pi$
$599$	−39.4069	−1.61012	−0.805061	+	0.593192i	$0.797868\pi$
$600$	0	0
$601$	−34.3183	−1.39987	−0.699937	−	0.714205i	$-0.746788\pi$
$601$	−34.3183	−1.39987	−0.699937	+	0.714205i	$0.746788\pi$
$602$	0	0
$603$	18.9260i	0.770725i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.7570i	0.477200i	0.971118	+	0.238600i	$0.0766885\pi$
$607$	11.7570i	0.477200i	−0.971118	+	0.238600i	$0.923312\pi$
$608$	0	0
$609$	2.52922	0.102489
$610$	0	0
$611$	−6.60239	−0.267104
$612$	0	0
$613$	5.86862i	0.237031i	0.992952	+	0.118516i	$0.0378136\pi$
$613$	5.86862i	0.237031i	−0.992952	+	0.118516i	$0.962186\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.45842i	0.340523i	0.985399	+	0.170262i	$0.0544613\pi$
$617$	8.45842i	0.340523i	−0.985399	+	0.170262i	$0.945539\pi$
$618$	0	0
$619$	−0.851047	−0.0342065	−0.0171032	−	0.999854i	$-0.505444\pi$
$619$	−0.851047	−0.0342065	−0.0171032	+	0.999854i	$0.505444\pi$
$620$	0	0
$621$	−3.22669	−0.129483
$622$	0	0
$623$	59.7527i	2.39394i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.943838i	0.0376933i
$628$	0	0
$629$	−1.21217	−0.0483325
$630$	0	0
$631$	19.7131	0.784768	0.392384	−	0.919802i	$-0.371650\pi$
$631$	19.7131	0.784768	0.392384	+	0.919802i	$0.371650\pi$
$632$	0	0
$633$	− 3.46717i	− 0.137808i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 81.2638i	− 3.21979i
$638$	0	0
$639$	30.4494	1.20456
$640$	0	0
$641$	−34.3593	−1.35711	−0.678555	−	0.734550i	$-0.737393\pi$
$641$	−34.3593	−1.35711	−0.678555	+	0.734550i	$0.737393\pi$
$642$	0	0
$643$	− 24.2817i	− 0.957578i	−0.877930	−	0.478789i	$-0.841076\pi$
$643$	− 24.2817i	− 0.957578i	0.877930	−	0.478789i	$-0.158924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.1465i	1.18518i	0.805504	+	0.592590i	$0.201895\pi$
$647$	30.1465i	1.18518i	−0.805504	+	0.592590i	$0.798105\pi$
$648$	0	0
$649$	−3.48618	−0.136845
$650$	0	0
$651$	−26.0307	−1.02022
$652$	0	0
$653$	− 7.17298i	− 0.280700i	−0.990102	−	0.140350i	$-0.955177\pi$
$653$	− 7.17298i	− 0.280700i	0.990102	−	0.140350i	$-0.0448228\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.99455i	0.0778148i
$658$	0	0
$659$	12.8521	0.500645	0.250323	−	0.968162i	$-0.419463\pi$
$659$	12.8521	0.500645	0.250323	+	0.968162i	$0.419463\pi$
$660$	0	0
$661$	3.72426	0.144857	0.0724285	−	0.997374i	$-0.476925\pi$
$661$	3.72426	0.144857	0.0724285	+	0.997374i	$0.476925\pi$
$662$	0	0
$663$	− 1.10900i	− 0.0430700i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.939238i	0.0363675i
$668$	0	0
$669$	−9.32924	−0.360689
$670$	0	0
$671$	−3.50235	−0.135207
$672$	0	0
$673$	13.5204i	0.521175i	0.965450	+	0.260587i	$0.0839162\pi$
$673$	13.5204i	0.521175i	−0.965450	+	0.260587i	$0.916084\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.8534i	0.686161i	0.939306	+	0.343081i	$0.111470\pi$
$677$	17.8534i	0.686161i	−0.939306	+	0.343081i	$0.888530\pi$
$678$	0	0
$679$	−66.5192	−2.55277
$680$	0	0
$681$	−0.688719	−0.0263918
$682$	0	0
$683$	15.8200i	0.605337i	0.953096	+	0.302669i	$0.0978776\pi$
$683$	15.8200i	0.605337i	−0.953096	+	0.302669i	$0.902122\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.9758i	− 0.495059i
$688$	0	0
$689$	−57.4939	−2.19034
$690$	0	0
$691$	9.45476	0.359676	0.179838	−	0.983696i	$-0.442443\pi$
$691$	9.45476	0.359676	0.179838	+	0.983696i	$0.442443\pi$
$692$	0	0
$693$	4.57245i	0.173693i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.96543i	− 0.0744460i
$698$	0	0
$699$	12.4715	0.471717
$700$	0	0
$701$	29.0096	1.09568	0.547838	−	0.836584i	$-0.315451\pi$
$701$	29.0096	1.09568	0.547838	+	0.836584i	$0.315451\pi$
$702$	0	0
$703$	− 15.0548i	− 0.567801i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	60.3773i	2.27072i
$708$	0	0
$709$	8.22423	0.308868	0.154434	−	0.988003i	$-0.450645\pi$
$709$	8.22423	0.308868	0.154434	+	0.988003i	$0.450645\pi$
$710$	0	0
$711$	−1.11283	−0.0417342
$712$	0	0
$713$	− 9.66662i	− 0.362018i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.87831i	− 0.182184i
$718$	0	0
$719$	12.5468	0.467917	0.233958	−	0.972247i	$-0.424832\pi$
$719$	12.5468	0.467917	0.233958	+	0.972247i	$0.424832\pi$
$720$	0	0
$721$	−15.6982	−0.584633
$722$	0	0
$723$	− 8.74501i	− 0.325230i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.1952i	1.08279i	0.840768	+	0.541395i	$0.182104\pi$
$727$	29.1952i	1.08279i	−0.840768	+	0.541395i	$0.817896\pi$
$728$	0	0
$729$	11.0864	0.410609
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 36.0414i	− 1.33122i	−0.746299	−	0.665611i	$-0.768171\pi$
$733$	− 36.0414i	− 1.33122i	0.746299	−	0.665611i	$-0.231829\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.54896i	− 0.0938923i
$738$	0	0
$739$	−0.728002	−0.0267800	−0.0133900	−	0.999910i	$-0.504262\pi$
$739$	−0.728002	−0.0267800	−0.0133900	+	0.999910i	$0.504262\pi$
$740$	0	0
$741$	13.7734	0.505978
$742$	0	0
$743$	− 32.8880i	− 1.20655i	−0.797535	−	0.603273i	$-0.793863\pi$
$743$	− 32.8880i	− 1.20655i	0.797535	−	0.603273i	$-0.206137\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 24.8115i	− 0.907805i
$748$	0	0
$749$	38.5262	1.40772
$750$	0	0
$751$	14.3337	0.523044	0.261522	−	0.965197i	$-0.415776\pi$
$751$	14.3337	0.523044	0.261522	+	0.965197i	$0.415776\pi$
$752$	0	0
$753$	5.03293i	0.183410i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.0587519i	0.00213537i	0.999999	+	0.00106769i	$0.000339855\pi$
$757$	0.0587519i	0.00213537i	−0.999999	+	0.00106769i	$0.999660\pi$
$758$	0	0
$759$	0.204921	0.00743817
$760$	0	0
$761$	45.7135	1.65711	0.828556	−	0.559906i	$-0.189163\pi$
$761$	45.7135	1.65711	0.828556	+	0.559906i	$0.189163\pi$
$762$	0	0
$763$	− 27.4553i	− 0.993950i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	50.8737i	1.83694i
$768$	0	0
$769$	39.3949	1.42061	0.710307	−	0.703892i	$-0.248556\pi$
$769$	39.3949	1.42061	0.710307	+	0.703892i	$0.248556\pi$
$770$	0	0
$771$	12.8460	0.462636
$772$	0	0
$773$	21.1799i	0.761788i	0.924619	+	0.380894i	$0.124384\pi$
$773$	21.1799i	0.761788i	−0.924619	+	0.380894i	$0.875616\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.80187i	0.315765i
$778$	0	0
$779$	24.4100	0.874578
$780$	0	0
$781$	−4.10094	−0.146743
$782$	0	0
$783$	3.03063i	0.108306i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 21.5154i	− 0.766941i	−0.923553	−	0.383470i	$-0.874729\pi$
$787$	− 21.5154i	− 0.766941i	0.923553	−	0.383470i	$-0.125271\pi$
$788$	0	0
$789$	−4.39698	−0.156537
$790$	0	0
$791$	−36.0408	−1.28146
$792$	0	0
$793$	51.1097i	1.81496i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	54.7227i	1.93838i	0.246320	+	0.969189i	$0.420779\pi$
$797$	54.7227i	1.93838i	−0.246320	+	0.969189i	$0.579221\pi$
$798$	0	0
$799$	0.465388	0.0164642
$800$	0	0
$801$	−33.7620	−1.19292
$802$	0	0
$803$	− 0.268627i	− 0.00947965i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.20053i	− 0.0774623i
$808$	0	0
$809$	43.5554	1.53133	0.765663	−	0.643242i	$-0.222411\pi$
$809$	43.5554	1.53133	0.765663	+	0.643242i	$0.222411\pi$
$810$	0	0
$811$	−13.1837	−0.462944	−0.231472	−	0.972842i	$-0.574354\pi$
$811$	−13.1837	−0.462944	−0.231472	+	0.972842i	$0.574354\pi$
$812$	0	0
$813$	− 2.41397i	− 0.0846615i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	66.7256	2.33158
$820$	0	0
$821$	−3.63328	−0.126803	−0.0634013	−	0.997988i	$-0.520195\pi$
$821$	−3.63328	−0.126803	−0.0634013	+	0.997988i	$0.520195\pi$
$822$	0	0
$823$	− 39.5044i	− 1.37704i	−0.725220	−	0.688518i	$-0.758262\pi$
$823$	− 39.5044i	− 1.37704i	0.725220	−	0.688518i	$-0.241738\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 48.3819i	− 1.68240i	−0.540721	−	0.841202i	$-0.681848\pi$
$827$	− 48.3819i	− 1.68240i	0.540721	−	0.841202i	$-0.318152\pi$
$828$	0	0
$829$	44.5864	1.54855	0.774275	−	0.632850i	$-0.218115\pi$
$829$	44.5864	1.54855	0.774275	+	0.632850i	$0.218115\pi$
$830$	0	0
$831$	4.13077	0.143295
$832$	0	0
$833$	5.72810i	0.198467i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 31.1912i	− 1.07813i
$838$	0	0
$839$	29.2324	1.00921	0.504606	−	0.863350i	$-0.331638\pi$
$839$	29.2324	1.00921	0.504606	+	0.863350i	$0.331638\pi$
$840$	0	0
$841$	−28.1178	−0.969580
$842$	0	0
$843$	7.39647i	0.254748i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	51.4989i	1.76952i
$848$	0	0
$849$	10.1367	0.347891
$850$	0	0
$851$	−3.26862	−0.112047
$852$	0	0
$853$	50.5599i	1.73114i	0.500790	+	0.865569i	$0.333043\pi$
$853$	50.5599i	1.73114i	−0.500790	+	0.865569i	$0.666957\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.9568i	0.750030i	0.927019	+	0.375015i	$0.122362\pi$
$857$	21.9568i	0.750030i	−0.927019	+	0.375015i	$0.877638\pi$
$858$	0	0
$859$	−13.7671	−0.469726	−0.234863	−	0.972029i	$-0.575464\pi$
$859$	−13.7671	−0.469726	−0.234863	+	0.972029i	$0.575464\pi$
$860$	0	0
$861$	−14.2714	−0.486370
$862$	0	0
$863$	− 26.5748i	− 0.904618i	−0.891861	−	0.452309i	$-0.850600\pi$
$863$	− 26.5748i	− 0.904618i	0.891861	−	0.452309i	$-0.149400\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 9.58439i	− 0.325503i
$868$	0	0
$869$	0.149876	0.00508420
$870$	0	0
$871$	−37.1969	−1.26037
$872$	0	0
$873$	− 37.5853i	− 1.27207i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 45.9414i	− 1.55133i	−0.631145	−	0.775665i	$-0.717415\pi$
$877$	− 45.9414i	− 1.55133i	0.631145	−	0.775665i	$-0.282585\pi$
$878$	0	0
$879$	4.16224	0.140389
$880$	0	0
$881$	−23.9054	−0.805392	−0.402696	−	0.915334i	$-0.631927\pi$
$881$	−23.9054	−0.805392	−0.402696	+	0.915334i	$0.631927\pi$
$882$	0	0
$883$	52.4002i	1.76341i	0.471804	+	0.881703i	$0.343603\pi$
$883$	52.4002i	1.76341i	−0.471804	+	0.881703i	$0.656397\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.7608i	1.10000i	0.835165	+	0.550000i	$0.185372\pi$
$887$	32.7608i	1.10000i	−0.835165	+	0.550000i	$0.814628\pi$
$888$	0	0
$889$	19.5703	0.656368
$890$	0	0
$891$	−2.23415	−0.0748467
$892$	0	0
$893$	5.77995i	0.193419i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.99041i	− 0.0998469i
$898$	0	0
$899$	−9.07926	−0.302810
$900$	0	0
$901$	4.05262	0.135012
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 20.9426i	− 0.695388i	−0.937608	−	0.347694i	$-0.886965\pi$
$907$	− 20.9426i	− 0.695388i	0.937608	−	0.347694i	$-0.113035\pi$
$908$	0	0
$909$	−34.1149	−1.13152
$910$	0	0
$911$	40.6418	1.34652	0.673262	−	0.739404i	$-0.264893\pi$
$911$	40.6418	1.34652	0.673262	+	0.739404i	$0.264893\pi$
$912$	0	0
$913$	3.34163i	0.110592i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	84.1330i	2.77831i
$918$	0	0
$919$	50.3241	1.66004	0.830019	−	0.557735i	$-0.188330\pi$
$919$	50.3241	1.66004	0.830019	+	0.557735i	$0.188330\pi$
$920$	0	0
$921$	16.5473	0.545251
$922$	0	0
$923$	59.8449i	1.96982i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.86996i	− 0.291328i
$928$	0	0
$929$	−24.1954	−0.793825	−0.396912	−	0.917856i	$-0.629918\pi$
$929$	−24.1954	−0.793825	−0.396912	+	0.917856i	$0.629918\pi$
$930$	0	0
$931$	−71.1411	−2.33155
$932$	0	0
$933$	− 0.260839i	− 0.00853949i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.2988i	1.21850i	0.792978	+	0.609250i	$0.208529\pi$
$937$	37.2988i	1.21850i	−0.792978	+	0.609250i	$0.791471\pi$
$938$	0	0
$939$	−10.1331	−0.330682
$940$	0	0
$941$	16.6467	0.542667	0.271334	−	0.962485i	$-0.412535\pi$
$941$	16.6467	0.542667	0.271334	+	0.962485i	$0.412535\pi$
$942$	0	0
$943$	− 5.29977i	− 0.172584i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.1868i	0.980938i	0.871459	+	0.490469i	$0.163175\pi$
$947$	30.1868i	0.980938i	−0.871459	+	0.490469i	$0.836825\pi$
$948$	0	0
$949$	−3.92007	−0.127251
$950$	0	0
$951$	−13.5730	−0.440134
$952$	0	0
$953$	34.6992i	1.12402i	0.827132	+	0.562008i	$0.189971\pi$
$953$	34.6992i	1.12402i	−0.827132	+	0.562008i	$0.810029\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 0.192470i	− 0.00622167i
$958$	0	0
$959$	−32.5458	−1.05096
$960$	0	0
$961$	62.4435	2.01431
$962$	0	0
$963$	21.7684i	0.701478i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	11.4788i	0.369133i	0.982820	+	0.184566i	$0.0590881\pi$
$967$	11.4788i	0.369133i	−0.982820	+	0.184566i	$0.940912\pi$
$968$	0	0
$969$	−0.970856	−0.0311884
$970$	0	0
$971$	−46.8441	−1.50330	−0.751650	−	0.659563i	$-0.770742\pi$
$971$	−46.8441	−1.50330	−0.751650	+	0.659563i	$0.770742\pi$
$972$	0	0
$973$	97.8977i	3.13846i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.3737i	− 1.25968i	−0.776727	−	0.629838i	$-0.783121\pi$
$977$	− 39.3737i	− 1.25968i	0.776727	−	0.629838i	$-0.216879\pi$
$978$	0	0
$979$	4.54709	0.145326
$980$	0	0
$981$	15.5131	0.495294
$982$	0	0
$983$	42.3505i	1.35077i	0.737465	+	0.675385i	$0.236023\pi$
$983$	42.3505i	1.35077i	−0.737465	+	0.675385i	$0.763977\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.37929i	− 0.107564i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−2.00104	−0.0635652	−0.0317826	−	0.999495i	$-0.510118\pi$
$991$	−2.00104	−0.0635652	−0.0317826	+	0.999495i	$0.510118\pi$
$992$	0	0
$993$	− 17.0638i	− 0.541502i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.97936i	− 0.0626869i	−0.999509	−	0.0313435i	$-0.990021\pi$
$997$	− 1.97936i	− 0.0626869i	0.999509	−	0.0313435i	$-0.00997857\pi$
$998$	0	0
$999$	−10.5468	−0.333687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.u.4049.5		10
5.2	odd	4		920.2.a.j.1.3	✓	5
5.3	odd	4		4600.2.a.be.1.3		5
5.4	even	2	inner	4600.2.e.u.4049.6		10
15.2	even	4		8280.2.a.bs.1.5		5
20.3	even	4		9200.2.a.cu.1.3		5
20.7	even	4		1840.2.a.v.1.3		5
40.27	even	4		7360.2.a.cp.1.3		5
40.37	odd	4		7360.2.a.co.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.3	✓	5	5.2	odd	4
1840.2.a.v.1.3		5	20.7	even	4
4600.2.a.be.1.3		5	5.3	odd	4
4600.2.e.u.4049.5		10	1.1	even	1	trivial
4600.2.e.u.4049.6		10	5.4	even	2	inner
7360.2.a.co.1.3		5	40.37	odd	4
7360.2.a.cp.1.3		5	40.27	even	4
8280.2.a.bs.1.5		5	15.2	even	4
9200.2.a.cu.1.3		5	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.568386i q^{3} -4.73770i q^{7} +2.67694 q^{9} +O(q^{10})\)	\(q-0.568386i q^{3} -4.73770i q^{7} +2.67694 q^{9} -0.360532 q^{11} +5.26123i q^{13} -0.370852i q^{17} +4.60586 q^{19} -2.69284 q^{21} -1.00000i q^{23} -3.22669i q^{27} -0.939238 q^{29} +9.66662 q^{31} +0.204921i q^{33} -3.26862i q^{37} +2.99041 q^{39} +5.29977 q^{41} +1.25491i q^{47} -15.4458 q^{49} -0.210787 q^{51} +10.9278i q^{53} -2.61790i q^{57} +9.66955 q^{59} +9.71441 q^{61} -12.6825i q^{63} +7.07001i q^{67} -0.568386 q^{69} +11.3747 q^{71} +0.745086i q^{73} +1.70809i q^{77} -0.415709 q^{79} +6.19680 q^{81} -9.26862i q^{83} +0.533850i q^{87} -12.6122 q^{89} +24.9261 q^{91} -5.49437i q^{93} -14.0404i q^{97} -0.965121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 26 q^{9}+O(q^{10}) \) 10 * q - 26 * q^9	\( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) \) 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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