LMFDB - Embedded newform 4284.2.d.e.3025.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4284,2,Mod(3025,4284)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4284, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4284.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4284.d (of order $2$, degree $1$, 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:	$34.2079122259$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.980441344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 $ x^8 + 8x^6 + 18x^4 + 9*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 476)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.5
Root			$2.06150i$ of defining polynomial
Character	$\chi$	$=$	4284.3025
Dual form			4284.2.d.e.3025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15845i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+1.15845i q^{5} -1.00000i q^{7} +2.12300i q^{11} -1.15283 q^{13} +(4.12300 - 0.0298321i) q^{17} +0.653275 q^{19} +7.65238i q^{23} +3.65800 q^{25} +2.49956i q^{29} -0.841553i q^{31} +1.15845 q^{35} -2.97017i q^{37} -9.81083i q^{41} -7.03456 q^{43} -6.55922 q^{47} -1.00000 q^{49} -6.90489 q^{53} -2.45938 q^{55} +0.929992 q^{59} +5.49394i q^{61} -1.33549i q^{65} +11.5871 q^{67} +5.72328i q^{71} +6.99805i q^{73} +2.12300 q^{77} +12.4883i q^{79} +11.3988 q^{83} +(0.0345589 + 4.77627i) q^{85} -6.75678 q^{89} +1.15283i q^{91} +0.756784i q^{95} +13.7810i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 20 q^{13} + 12 q^{19} - 12 q^{25} + 4 q^{35} - 12 q^{43} - 24 q^{47} - 8 q^{49} - 20 q^{53} - 4 q^{55} + 24 q^{59} - 4 q^{67} - 16 q^{77} - 4 q^{83} - 44 q^{85}+O(q^{100}) $ 8 * q + 20 * q^13 + 12 * q^19 - 12 * q^25 + 4 * q^35 - 12 * q^43 - 24 * q^47 - 8 * q^49 - 20 * q^53 - 4 * q^55 + 24 * q^59 - 4 * q^67 - 16 * q^77 - 4 * q^83 - 44 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4284\mathbb{Z}\right)^\times$.

$n$	$1261$	$1837$	$2143$	$3809$
$\chi(n)$	$-1$	$1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$			1.15845i			0.518073i	0.965868	+	0.259037i	$0.0834050\pi$
$5$			1.15845i			0.518073i	−0.965868	+	0.259037i	$0.916595\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.12300i			0.640108i	0.947399	+	0.320054i	$0.103701\pi$
$11$			2.12300i			0.640108i	−0.947399	+	0.320054i	$0.896299\pi$
$12$		0			0
$13$	−1.15283			−0.319737			−0.159869	−	0.987138i	$-0.551107\pi$
$13$	−1.15283			−0.319737			−0.159869	+	0.987138i	$0.551107\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.12300	−	0.0298321i	0.999974	−	0.00723535i
$17$	4.12300	−	0.0298321i	0.999974	−	0.00723535i
$18$		0			0
$19$	0.653275			0.149872			0.0749358	−	0.997188i	$-0.476125\pi$
$19$	0.653275			0.149872			0.0749358	+	0.997188i	$0.476125\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.65238i			1.59563i	0.602901	+	0.797816i	$0.294012\pi$
$23$			7.65238i			1.59563i	−0.602901	+	0.797816i	$0.705988\pi$
$24$		0			0
$25$	3.65800			0.731600
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.49956i			0.464156i	0.972697	+	0.232078i	$0.0745524\pi$
$29$			2.49956i			0.464156i	−0.972697	+	0.232078i	$0.925448\pi$
$30$		0			0
$31$		−	0.841553i		−	0.151147i	−0.997140	−	0.0755737i	$-0.975921\pi$
$31$		−	0.841553i		−	0.151147i	0.997140	−	0.0755737i	$-0.0240788\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.15845			0.195813
$36$		0			0
$37$		−	2.97017i		−	0.488293i	−0.969738	−	0.244146i	$-0.921492\pi$
$37$		−	2.97017i		−	0.488293i	0.969738	−	0.244146i	$-0.0785077\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.81083i		−	1.53219i	−0.642725	−	0.766097i	$-0.722196\pi$
$41$		−	9.81083i		−	1.53219i	0.642725	−	0.766097i	$-0.277804\pi$
$42$		0			0
$43$	−7.03456			−1.07276			−0.536380	−	0.843977i	$-0.680209\pi$
$43$	−7.03456			−1.07276			−0.536380	+	0.843977i	$0.680209\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.55922			−0.956760			−0.478380	−	0.878153i	$-0.658776\pi$
$47$	−6.55922			−0.956760			−0.478380	+	0.878153i	$0.658776\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.90489			−0.948459			−0.474230	−	0.880401i	$-0.657273\pi$
$53$	−6.90489			−0.948459			−0.474230	+	0.880401i	$0.657273\pi$
$54$		0			0
$55$	−2.45938			−0.331623
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.929992			0.121075			0.0605373	−	0.998166i	$-0.480719\pi$
$59$	0.929992			0.121075			0.0605373	+	0.998166i	$0.480719\pi$
$60$		0			0
$61$			5.49394i			0.703427i	0.936108	+	0.351713i	$0.114401\pi$
$61$			5.49394i			0.703427i	−0.936108	+	0.351713i	$0.885599\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.33549i		−	0.165647i
$66$		0			0
$67$	11.5871			1.41559			0.707795	−	0.706418i	$-0.249690\pi$
$67$	11.5871			1.41559			0.707795	+	0.706418i	$0.249690\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.72328i			0.679229i	0.940565	+	0.339614i	$0.110297\pi$
$71$			5.72328i			0.679229i	−0.940565	+	0.339614i	$0.889703\pi$
$72$		0			0
$73$			6.99805i			0.819060i	0.912297	+	0.409530i	$0.134307\pi$
$73$			6.99805i			0.819060i	−0.912297	+	0.409530i	$0.865693\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.12300			0.241938
$78$		0			0
$79$			12.4883i			1.40505i	0.711661	+	0.702523i	$0.247943\pi$
$79$			12.4883i			1.40505i	−0.711661	+	0.702523i	$0.752057\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.3988			1.25118			0.625592	−	0.780151i	$-0.284858\pi$
$83$	11.3988			1.25118			0.625592	+	0.780151i	$0.284858\pi$
$84$		0			0
$85$	0.0345589	+	4.77627i	0.00374844	+	0.518059i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.75678			−0.716218			−0.358109	−	0.933680i	$-0.616578\pi$
$89$	−6.75678			−0.716218			−0.358109	+	0.933680i	$0.616578\pi$
$90$		0			0
$91$			1.15283i			0.120849i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			0.756784i			0.0776444i
$96$		0			0
$97$			13.7810i			1.39925i	0.714511	+	0.699624i	$0.246649\pi$
$97$			13.7810i			1.39925i	−0.714511	+	0.699624i	$0.753351\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.28706			0.327075			0.163537	−	0.986537i	$-0.447710\pi$
$101$	3.28706			0.327075			0.163537	+	0.986537i	$0.447710\pi$
$102$		0			0
$103$	13.2795			1.30847			0.654234	−	0.756292i	$-0.272991\pi$
$103$	13.2795			1.30847			0.654234	+	0.756292i	$0.272991\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			7.52939i			0.727893i	0.931420	+	0.363947i	$0.118571\pi$
$107$			7.52939i			0.727893i	−0.931420	+	0.363947i	$0.881429\pi$
$108$		0			0
$109$			15.9291i			1.52573i	0.646557	+	0.762866i	$0.276208\pi$
$109$			15.9291i			1.52573i	−0.646557	+	0.762866i	$0.723792\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	17.4390i		−	1.64052i	−0.571989	−	0.820262i	$-0.693828\pi$
$113$		−	17.4390i		−	1.64052i	0.571989	−	0.820262i	$-0.306172\pi$
$114$		0			0
$115$	−8.86488			−0.826654
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.0298321	−	4.12300i	−0.00273470	−	0.377955i
$120$		0			0
$121$	6.49288			0.590262
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.0298i			0.897095i
$126$		0			0
$127$	−8.51710			−0.755770			−0.377885	−	0.925853i	$-0.623349\pi$
$127$	−8.51710			−0.755770			−0.377885	+	0.925853i	$0.623349\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.69345i		−	0.235328i	−0.993053	−	0.117664i	$-0.962459\pi$
$131$		−	2.69345i		−	0.235328i	0.993053	−	0.117664i	$-0.0375406\pi$
$132$		0			0
$133$		−	0.653275i		−	0.0566461i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.89454			−0.418169			−0.209085	−	0.977898i	$-0.567048\pi$
$137$	−4.89454			−0.418169			−0.209085	+	0.977898i	$0.567048\pi$
$138$		0			0
$139$			1.34478i			0.114063i	0.998372	+	0.0570313i	$0.0181635\pi$
$139$			1.34478i			0.114063i	−0.998372	+	0.0570313i	$0.981837\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.44746i		−	0.204666i
$144$		0			0
$145$	−2.89560			−0.240467
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.8331			−1.05133			−0.525664	−	0.850692i	$-0.676183\pi$
$149$	−12.8331			−1.05133			−0.525664	+	0.850692i	$0.676183\pi$
$150$		0			0
$151$	12.1509			0.988825			0.494412	−	0.869227i	$-0.335383\pi$
$151$	12.1509			0.988825			0.494412	+	0.869227i	$0.335383\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.974895			0.0783054
$156$		0			0
$157$	−15.3793			−1.22740			−0.613702	−	0.789537i	$-0.710321\pi$
$157$	−15.3793			−1.22740			−0.613702	+	0.789537i	$0.710321\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.65238			0.603092
$162$		0			0
$163$			7.41006i			0.580401i	0.956966	+	0.290200i	$0.0937219\pi$
$163$			7.41006i			0.580401i	−0.956966	+	0.290200i	$0.906278\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.1174i			1.16982i	0.811099	+	0.584909i	$0.198870\pi$
$167$			15.1174i			1.16982i	−0.811099	+	0.584909i	$0.801130\pi$
$168$		0			0
$169$	−11.6710			−0.897768
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.0474i			0.763888i	0.924186	+	0.381944i	$0.124745\pi$
$173$			10.0474i			0.763888i	−0.924186	+	0.381944i	$0.875255\pi$
$174$		0			0
$175$		−	3.65800i		−	0.276519i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.9533			−1.19241			−0.596203	−	0.802833i	$-0.703325\pi$
$179$	−15.9533			−1.19241			−0.596203	+	0.802833i	$0.703325\pi$
$180$		0			0
$181$		−	5.69256i		−	0.423125i	−0.977365	−	0.211562i	$-0.932145\pi$
$181$		−	5.69256i		−	0.423125i	0.977365	−	0.211562i	$-0.0678551\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.44078			0.252971
$186$		0			0
$187$	0.0633335	+	8.75311i	0.00463140	+	0.640091i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.15177			0.589842			0.294921	−	0.955522i	$-0.404707\pi$
$191$	8.15177			0.589842			0.294921	+	0.955522i	$0.404707\pi$
$192$		0			0
$193$		−	2.91807i		−	0.210047i	−0.994470	−	0.105024i	$-0.966508\pi$
$193$		−	2.91807i		−	0.210047i	0.994470	−	0.105024i	$-0.0334918\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			9.33916i			0.665388i	0.943035	+	0.332694i	$0.107958\pi$
$197$			9.33916i			0.665388i	−0.943035	+	0.332694i	$0.892042\pi$
$198$		0			0
$199$			12.9868i			0.920611i	0.887761	+	0.460306i	$0.152260\pi$
$199$			12.9868i			0.920611i	−0.887761	+	0.460306i	$0.847740\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.49956			0.175434
$204$		0			0
$205$	11.3653			0.793789
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.38690i			0.0959339i
$210$		0			0
$211$			8.57412i			0.590267i	0.955456	+	0.295133i	$0.0953641\pi$
$211$			8.57412i			0.590267i	−0.955456	+	0.295133i	$0.904636\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.14916i		−	0.555768i
$216$		0			0
$217$	−0.841553			−0.0571284
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.75311	+	0.0343913i	−0.319729	+	0.00231341i
$222$		0			0
$223$	−0.981401			−0.0657195			−0.0328597	−	0.999460i	$-0.510461\pi$
$223$	−0.981401			−0.0657195			−0.0328597	+	0.999460i	$0.510461\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			27.2626i			1.80949i	0.425958	+	0.904743i	$0.359937\pi$
$227$			27.2626i			1.80949i	−0.425958	+	0.904743i	$0.640063\pi$
$228$		0			0
$229$	11.8770			0.784854			0.392427	−	0.919783i	$-0.371635\pi$
$229$	11.8770			0.784854			0.392427	+	0.919783i	$0.371635\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.97017i		−	0.456631i	−0.973587	−	0.228315i	$-0.926678\pi$
$233$		−	6.97017i		−	0.456631i	0.973587	−	0.228315i	$-0.0733217\pi$
$234$		0			0
$235$		−	7.59850i		−	0.495672i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.777330			0.0502813			0.0251406	−	0.999684i	$-0.491997\pi$
$239$	0.777330			0.0502813			0.0251406	+	0.999684i	$0.491997\pi$
$240$		0			0
$241$		−	23.9098i		−	1.54016i	−0.637945	−	0.770082i	$-0.720215\pi$
$241$		−	23.9098i		−	1.54016i	0.637945	−	0.770082i	$-0.279785\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	1.15845i		−	0.0740104i
$246$		0			0
$247$	−0.753115			−0.0479195
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.9675			1.38658			0.693288	−	0.720661i	$-0.256161\pi$
$251$	21.9675			1.38658			0.693288	+	0.720661i	$0.256161\pi$
$252$		0			0
$253$	−16.2460			−1.02138
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.58238			0.161084			0.0805421	−	0.996751i	$-0.474335\pi$
$257$	2.58238			0.161084			0.0805421	+	0.996751i	$0.474335\pi$
$258$		0			0
$259$	−2.97017			−0.184557
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.08213			0.128390			0.0641949	−	0.997937i	$-0.479552\pi$
$263$	2.08213			0.128390			0.0641949	+	0.997937i	$0.479552\pi$
$264$		0			0
$265$		−	7.99894i		−	0.491371i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			19.3065i			1.17714i	0.808446	+	0.588571i	$0.200309\pi$
$269$			19.3065i			1.17714i	−0.808446	+	0.588571i	$0.799691\pi$
$270$		0			0
$271$	9.90961			0.601966			0.300983	−	0.953629i	$-0.402685\pi$
$271$	9.90961			0.601966			0.300983	+	0.953629i	$0.402685\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			7.76593i			0.468303i
$276$		0			0
$277$		−	20.7575i		−	1.24720i	−0.781745	−	0.623598i	$-0.785670\pi$
$277$		−	20.7575i		−	1.24720i	0.781745	−	0.623598i	$-0.214330\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.42235			−0.263815			−0.131908	−	0.991262i	$-0.542110\pi$
$281$	−4.42235			−0.263815			−0.131908	+	0.991262i	$0.542110\pi$
$282$		0			0
$283$			29.1567i			1.73318i	0.499017	+	0.866592i	$0.333694\pi$
$283$			29.1567i			1.73318i	−0.499017	+	0.866592i	$0.666306\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.81083			−0.579115
$288$		0			0
$289$	16.9982	−	0.245995i	0.999895	−	0.0144703i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.0194886			−0.00113853			−0.000569267	−	1.00000i	$-0.500181\pi$
$293$	−0.0194886			−0.00113853			−0.000569267		1.00000i	$0.500181\pi$
$294$		0			0
$295$			1.07735i			0.0627255i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	8.82190i		−	0.510183i
$300$		0			0
$301$			7.03456i			0.405465i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.36443			−0.364426
$306$		0			0
$307$	−12.3979			−0.707588			−0.353794	−	0.935323i	$-0.615109\pi$
$307$	−12.3979			−0.707588			−0.353794	+	0.935323i	$0.615109\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.02699i		−	0.455169i	−0.973758	−	0.227585i	$-0.926917\pi$
$311$		−	8.02699i		−	0.455169i	0.973758	−	0.227585i	$-0.0730828\pi$
$312$		0			0
$313$			31.7101i			1.79236i	0.443690	+	0.896180i	$0.353669\pi$
$313$			31.7101i			1.79236i	−0.443690	+	0.896180i	$0.646331\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			19.7678i			1.11027i	0.831760	+	0.555136i	$0.187334\pi$
$317$			19.7678i			1.11027i	−0.831760	+	0.555136i	$0.812666\pi$
$318$		0			0
$319$	−5.30655			−0.297110
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.69345	−	0.0194886i	0.149868	−	0.00108437i
$324$		0			0
$325$	−4.21705			−0.233920
$326$		0			0
$327$		0			0
$328$		0			0
$329$			6.55922i			0.361621i
$330$		0			0
$331$	−27.0224			−1.48529			−0.742644	−	0.669687i	$-0.766428\pi$
$331$	−27.0224			−1.48529			−0.742644	+	0.669687i	$0.766428\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			13.4230i			0.733379i
$336$		0			0
$337$		−	6.55343i		−	0.356988i	−0.983941	−	0.178494i	$-0.942877\pi$
$337$		−	6.55343i		−	0.356988i	0.983941	−	0.178494i	$-0.0571226\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.78662			0.0967507
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.0463i		−	0.968777i	−0.874853	−	0.484388i	$-0.839042\pi$
$347$		−	18.0463i		−	0.968777i	0.874853	−	0.484388i	$-0.160958\pi$
$348$		0			0
$349$	12.6439			0.676814			0.338407	−	0.941000i	$-0.390112\pi$
$349$	12.6439			0.676814			0.338407	+	0.941000i	$0.390112\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.25267			−0.173122			−0.0865611	−	0.996247i	$-0.527588\pi$
$353$	−3.25267			−0.173122			−0.0865611	+	0.996247i	$0.527588\pi$
$354$		0			0
$355$	−6.63012			−0.351890
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.2114			0.538939			0.269470	−	0.963009i	$-0.413152\pi$
$359$	10.2114			0.538939			0.269470	+	0.963009i	$0.413152\pi$
$360$		0			0
$361$	−18.5732			−0.977539
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.10687			−0.424333
$366$		0			0
$367$		−	3.18828i		−	0.166427i	−0.996532	−	0.0832134i	$-0.973482\pi$
$367$		−	3.18828i		−	0.166427i	0.996532	−	0.0832134i	$-0.0265183\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			6.90489i			0.358484i
$372$		0			0
$373$	2.23301			0.115621			0.0578105	−	0.998328i	$-0.481588\pi$
$373$	2.23301			0.115621			0.0578105	+	0.998328i	$0.481588\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.88156i		−	0.148408i
$378$		0			0
$379$			25.5657i			1.31322i	0.754230	+	0.656610i	$0.228010\pi$
$379$			25.5657i			1.31322i	−0.754230	+	0.656610i	$0.771990\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.9946			0.817283			0.408642	−	0.912695i	$-0.366003\pi$
$383$	15.9946			0.817283			0.408642	+	0.912695i	$0.366003\pi$
$384$		0			0
$385$			2.45938i			0.125342i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.04668			−0.357281			−0.178640	−	0.983914i	$-0.557170\pi$
$389$	−7.04668			−0.357281			−0.178640	+	0.983914i	$0.557170\pi$
$390$		0			0
$391$	0.228287	+	31.5508i	0.0115450	+	1.59559i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.4671			−0.727916
$396$		0			0
$397$		−	39.0840i		−	1.96157i	−0.195094	−	0.980785i	$-0.562501\pi$
$397$		−	39.0840i		−	1.96157i	0.195094	−	0.980785i	$-0.437499\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.5115i		−	0.524918i	−0.964943	−	0.262459i	$-0.915467\pi$
$401$		−	10.5115i		−	0.524918i	0.964943	−	0.262459i	$-0.0845335\pi$
$402$		0			0
$403$			0.970168i			0.0483275i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.30566			0.312560
$408$		0			0
$409$	−24.6208			−1.21742			−0.608709	−	0.793393i	$-0.708312\pi$
$409$	−24.6208			−1.21742			−0.608709	+	0.793393i	$0.708312\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	0.929992i		−	0.0457619i
$414$		0			0
$415$			13.2049i			0.648205i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	27.9579i		−	1.36583i	−0.730497	−	0.682916i	$-0.760712\pi$
$419$		−	27.9579i		−	1.36583i	0.730497	−	0.682916i	$-0.239288\pi$
$420$		0			0
$421$	3.41290			0.166334			0.0831672	−	0.996536i	$-0.473496\pi$
$421$	3.41290			0.166334			0.0831672	+	0.996536i	$0.473496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.0819	−	0.109126i	0.731581	−	0.00529338i
$426$		0			0
$427$	5.49394			0.265870
$428$		0			0
$429$		0			0
$430$		0			0
$431$			7.27216i			0.350288i	0.984543	+	0.175144i	$0.0560390\pi$
$431$			7.27216i			0.350288i	−0.984543	+	0.175144i	$0.943961\pi$
$432$		0			0
$433$	29.4853			1.41697			0.708487	−	0.705724i	$-0.249378\pi$
$433$	29.4853			1.41697			0.708487	+	0.705724i	$0.249378\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			4.99911i			0.239140i
$438$		0			0
$439$			26.6681i			1.27280i	0.771359	+	0.636401i	$0.219577\pi$
$439$			26.6681i			1.27280i	−0.771359	+	0.636401i	$0.780423\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.7994			0.798165			0.399083	−	0.916915i	$-0.369329\pi$
$443$	16.7994			0.798165			0.399083	+	0.916915i	$0.369329\pi$
$444$		0			0
$445$		−	7.82737i		−	0.371053i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	30.8938i		−	1.45797i	−0.684530	−	0.728985i	$-0.739992\pi$
$449$		−	30.8938i		−	1.45797i	0.684530	−	0.728985i	$-0.260008\pi$
$450$		0			0
$451$	20.8284			0.980770
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.33549			−0.0626088
$456$		0			0
$457$	21.3186			0.997244			0.498622	−	0.866820i	$-0.333840\pi$
$457$	21.3186			0.997244			0.498622	+	0.866820i	$0.333840\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.63957			−0.309236			−0.154618	−	0.987974i	$-0.549415\pi$
$461$	−6.63957			−0.309236			−0.154618	+	0.987974i	$0.549415\pi$
$462$		0			0
$463$	−12.0316			−0.559154			−0.279577	−	0.960123i	$-0.590194\pi$
$463$	−12.0316			−0.559154			−0.279577	+	0.960123i	$0.590194\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.1230			−0.931181			−0.465591	−	0.885000i	$-0.654158\pi$
$467$	−20.1230			−0.931181			−0.465591	+	0.885000i	$0.654158\pi$
$468$		0			0
$469$		−	11.5871i		−	0.535043i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	14.9344i		−	0.686682i
$474$		0			0
$475$	2.38968			0.109646
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	7.79014i		−	0.355941i	−0.984036	−	0.177970i	$-0.943047\pi$
$479$		−	7.79014i		−	0.355941i	0.984036	−	0.177970i	$-0.0569531\pi$
$480$		0			0
$481$			3.42410i			0.156125i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.9646			−0.724913
$486$		0			0
$487$			19.5388i			0.885389i	0.896672	+	0.442695i	$0.145977\pi$
$487$			19.5388i			0.885389i	−0.896672	+	0.442695i	$0.854023\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.0903			−0.906664			−0.453332	−	0.891342i	$-0.649765\pi$
$491$	−20.0903			−0.906664			−0.453332	+	0.891342i	$0.649765\pi$
$492$		0			0
$493$	0.0745670	+	10.3057i	0.00335833	+	0.464144i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.72328			0.256724
$498$		0			0
$499$		−	4.29374i		−	0.192214i	−0.995371	−	0.0961070i	$-0.969361\pi$
$499$		−	4.29374i		−	0.192214i	0.995371	−	0.0961070i	$-0.0306391\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.2863i		−	1.35040i	−0.737634	−	0.675201i	$-0.764057\pi$
$503$		−	30.2863i		−	1.35040i	0.737634	−	0.675201i	$-0.235943\pi$
$504$		0			0
$505$			3.80788i			0.169449i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	41.6448			1.84587			0.922937	−	0.384951i	$-0.125782\pi$
$509$	41.6448			1.84587			0.922937	+	0.384951i	$0.125782\pi$
$510$		0			0
$511$	6.99805			0.309576
$512$		0			0
$513$		0			0
$514$		0			0
$515$			15.3836i			0.677882i
$516$		0			0
$517$		−	13.9252i		−	0.612430i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	7.74171i		−	0.339171i	−0.985516	−	0.169585i	$-0.945757\pi$
$521$		−	7.74171i		−	0.339171i	0.985516	−	0.169585i	$-0.0542428\pi$
$522$		0			0
$523$	21.6134			0.945088			0.472544	−	0.881307i	$-0.343336\pi$
$523$	21.6134			0.945088			0.472544	+	0.881307i	$0.343336\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.0251053	−	3.46972i	−0.00109360	−	0.151144i
$528$		0			0
$529$	−35.5590			−1.54604
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.3102i			0.489900i
$534$		0			0
$535$	−8.72239			−0.377102
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2.12300i		−	0.0914440i
$540$		0			0
$541$		−	34.9666i		−	1.50333i	−0.659544	−	0.751666i	$-0.729251\pi$
$541$		−	34.9666i		−	1.50333i	0.659544	−	0.751666i	$-0.270749\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−18.4530			−0.790440
$546$		0			0
$547$		−	26.8856i		−	1.14954i	−0.818314	−	0.574772i	$-0.805091\pi$
$547$		−	26.8856i		−	1.14954i	0.818314	−	0.574772i	$-0.194909\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1.63290i			0.0695637i
$552$		0			0
$553$	12.4883			0.531057
$554$		0			0
$555$		0			0
$556$		0			0
$557$	39.4297			1.67069			0.835344	−	0.549727i	$-0.185268\pi$
$557$	39.4297			1.67069			0.835344	+	0.549727i	$0.185268\pi$
$558$		0			0
$559$	8.10965			0.343002
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.3197			−0.561357			−0.280679	−	0.959802i	$-0.590560\pi$
$563$	−13.3197			−0.561357			−0.280679	+	0.959802i	$0.590560\pi$
$564$		0			0
$565$	20.2022			0.849911
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.88420			−0.0789898			−0.0394949	−	0.999220i	$-0.512575\pi$
$569$	−1.88420			−0.0789898			−0.0394949	+	0.999220i	$0.512575\pi$
$570$		0			0
$571$			32.6866i			1.36789i	0.729533	+	0.683945i	$0.239737\pi$
$571$			32.6866i			1.36789i	−0.729533	+	0.683945i	$0.760263\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			27.9924i			1.16737i
$576$		0			0
$577$	7.84840			0.326733			0.163366	−	0.986565i	$-0.447765\pi$
$577$	7.84840			0.326733			0.163366	+	0.986565i	$0.447765\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	11.3988i		−	0.472903i
$582$		0			0
$583$		−	14.6591i		−	0.607116i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.68500			0.0695472			0.0347736	−	0.999395i	$-0.488929\pi$
$587$	1.68500			0.0695472			0.0347736	+	0.999395i	$0.488929\pi$
$588$		0			0
$589$		−	0.549766i		−	0.0226527i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	40.0065			1.64287			0.821435	−	0.570303i	$-0.193174\pi$
$593$	40.0065			1.64287			0.821435	+	0.570303i	$0.193174\pi$
$594$		0			0
$595$	4.77627	−	0.0345589i	0.195808	−	0.00141678i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	35.3774			1.44548			0.722741	−	0.691119i	$-0.242882\pi$
$599$	35.3774			1.44548			0.722741	+	0.691119i	$0.242882\pi$
$600$		0			0
$601$		−	20.8664i		−	0.851160i	−0.904921	−	0.425580i	$-0.860070\pi$
$601$		−	20.8664i		−	0.851160i	0.904921	−	0.425580i	$-0.139930\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			7.52166i			0.305799i
$606$		0			0
$607$		−	13.3223i		−	0.540736i	−0.962757	−	0.270368i	$-0.912855\pi$
$607$		−	13.3223i		−	0.540736i	0.962757	−	0.270368i	$-0.0871454\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.56166			0.305912
$612$		0			0
$613$	−21.3299			−0.861506			−0.430753	−	0.902470i	$-0.641752\pi$
$613$	−21.3299			−0.861506			−0.430753	+	0.902470i	$0.641752\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10.8277i		−	0.435906i	−0.975959	−	0.217953i	$-0.930062\pi$
$617$		−	10.8277i		−	0.435906i	0.975959	−	0.217953i	$-0.0699380\pi$
$618$		0			0
$619$		−	16.5534i		−	0.665339i	−0.943044	−	0.332669i	$-0.892051\pi$
$619$		−	16.5534i		−	0.665339i	0.943044	−	0.332669i	$-0.107949\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			6.75678i			0.270705i
$624$		0			0
$625$	6.67098			0.266839
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.0886064	−	12.2460i	−0.00353297	−	0.488280i
$630$		0			0
$631$	−31.6334			−1.25931			−0.629653	−	0.776876i	$-0.716803\pi$
$631$	−31.6334			−1.25931			−0.629653	+	0.776876i	$0.716803\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	9.86660i		−	0.391544i
$636$		0			0
$637$	1.15283			0.0456768
$638$		0			0
$639$		0			0
$640$		0			0
$641$			9.84628i			0.388905i	0.980912	+	0.194452i	$0.0622930\pi$
$641$			9.84628i			0.388905i	−0.980912	+	0.194452i	$0.937707\pi$
$642$		0			0
$643$			2.31795i			0.0914110i	0.998955	+	0.0457055i	$0.0145536\pi$
$643$			2.31795i			0.0914110i	−0.998955	+	0.0457055i	$0.985446\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.7231			−0.736080			−0.368040	−	0.929810i	$-0.619971\pi$
$647$	−18.7231			−0.736080			−0.368040	+	0.929810i	$0.619971\pi$
$648$		0			0
$649$			1.97437i			0.0775009i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	38.9282i		−	1.52338i	−0.647942	−	0.761689i	$-0.724370\pi$
$653$		−	38.9282i		−	1.52338i	0.647942	−	0.761689i	$-0.275630\pi$
$654$		0			0
$655$	3.12022			0.121917
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.6904			−1.27344			−0.636719	−	0.771096i	$-0.719709\pi$
$659$	−32.6904			−1.27344			−0.636719	+	0.771096i	$0.719709\pi$
$660$		0			0
$661$	17.9507			0.698202			0.349101	−	0.937085i	$-0.386487\pi$
$661$	17.9507			0.698202			0.349101	+	0.937085i	$0.386487\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.756784			0.0293468
$666$		0			0
$667$	−19.1276			−0.740622
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11.6636			−0.450269
$672$		0			0
$673$		−	42.8631i		−	1.65225i	−0.563486	−	0.826126i	$-0.690540\pi$
$673$		−	42.8631i		−	1.65225i	0.563486	−	0.826126i	$-0.309460\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			22.8110i			0.876698i	0.898805	+	0.438349i	$0.144437\pi$
$677$			22.8110i			0.876698i	−0.898805	+	0.438349i	$0.855563\pi$
$678$		0			0
$679$	13.7810			0.528866
$680$		0			0
$681$		0			0
$682$		0			0
$683$			37.2068i			1.42368i	0.702342	+	0.711840i	$0.252138\pi$
$683$			37.2068i			1.42368i	−0.702342	+	0.711840i	$0.747862\pi$
$684$		0			0
$685$		−	5.67007i		−	0.216642i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.96016			0.303258
$690$		0			0
$691$		−	15.9263i		−	0.605864i	−0.953012	−	0.302932i	$-0.902035\pi$
$691$		−	15.9263i		−	0.605864i	0.953012	−	0.302932i	$-0.0979655\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.55785			−0.0590927
$696$		0			0
$697$	−0.292678	−	40.4500i	−0.0110860	−	1.53215i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.1249			−1.36442			−0.682209	−	0.731157i	$-0.738981\pi$
$701$	−36.1249			−1.36442			−0.682209	+	0.731157i	$0.738981\pi$
$702$		0			0
$703$		−	1.94034i		−	0.0731812i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.28706i		−	0.123623i
$708$		0			0
$709$			24.3906i			0.916008i	0.888950	+	0.458004i	$0.151435\pi$
$709$			24.3906i			0.916008i	−0.888950	+	0.458004i	$0.848565\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.43989			0.241176
$714$		0			0
$715$	2.83525			0.106032
$716$		0			0
$717$		0			0
$718$		0			0
$719$			26.6103i			0.992395i	0.868210	+	0.496198i	$0.165271\pi$
$719$			26.6103i			0.992395i	−0.868210	+	0.496198i	$0.834729\pi$
$720$		0			0
$721$		−	13.2795i		−	0.494554i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			9.14338i			0.339576i
$726$		0			0
$727$	45.7943			1.69842			0.849208	−	0.528058i	$-0.177080\pi$
$727$	45.7943			1.69842			0.849208	+	0.528058i	$0.177080\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−29.0035	+	0.209856i	−1.07273	+	0.00776179i
$732$		0			0
$733$	0.380227			0.0140440			0.00702199	−	0.999975i	$-0.497765\pi$
$733$	0.380227			0.0140440			0.00702199	+	0.999975i	$0.497765\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			24.5994i			0.906130i
$738$		0			0
$739$	−29.7613			−1.09479			−0.547394	−	0.836875i	$-0.684380\pi$
$739$	−29.7613			−1.09479			−0.547394	+	0.836875i	$0.684380\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.09336i			0.296917i	0.988919	+	0.148458i	$0.0474311\pi$
$743$			8.09336i			0.296917i	−0.988919	+	0.148458i	$0.952569\pi$
$744$		0			0
$745$		−	14.8665i		−	0.544665i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.52939			0.275118
$750$		0			0
$751$		−	32.2728i		−	1.17765i	−0.808260	−	0.588826i	$-0.799590\pi$
$751$		−	32.2728i		−	1.17765i	0.808260	−	0.588826i	$-0.200410\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			14.0761i			0.512283i
$756$		0			0
$757$	−31.1707			−1.13292			−0.566459	−	0.824090i	$-0.691687\pi$
$757$	−31.1707			−1.13292			−0.566459	+	0.824090i	$0.691687\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.2779			−1.09757			−0.548787	−	0.835962i	$-0.684910\pi$
$761$	−30.2779			−1.09757			−0.548787	+	0.835962i	$0.684910\pi$
$762$		0			0
$763$	15.9291			0.576672
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.07212			−0.0387121
$768$		0			0
$769$	44.7810			1.61484			0.807421	−	0.589975i	$-0.200862\pi$
$769$	44.7810			1.61484			0.807421	+	0.589975i	$0.200862\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21.2600			−0.764669			−0.382335	−	0.924024i	$-0.624880\pi$
$773$	−21.2600			−0.764669			−0.382335	+	0.924024i	$0.624880\pi$
$774$		0			0
$775$		−	3.07840i		−	0.110580i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	6.40917i		−	0.229632i
$780$		0			0
$781$	−12.1505			−0.434780
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	17.8161i		−	0.635885i
$786$		0			0
$787$		−	30.1828i		−	1.07590i	−0.842977	−	0.537950i	$-0.819199\pi$
$787$		−	30.1828i		−	1.07590i	0.842977	−	0.537950i	$-0.180801\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.4390			−0.620059
$792$		0			0
$793$		−	6.33358i		−	0.224912i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.8320			−0.454534			−0.227267	−	0.973832i	$-0.572979\pi$
$797$	−12.8320			−0.454534			−0.227267	+	0.973832i	$0.572979\pi$
$798$		0			0
$799$	−27.0436	+	0.195675i	−0.956735	+	0.00692250i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.8568			−0.524287
$804$		0			0
$805$			8.86488i			0.312446i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	10.4667i		−	0.367990i	−0.982927	−	0.183995i	$-0.941097\pi$
$809$		−	10.4667i		−	0.367990i	0.982927	−	0.183995i	$-0.0589031\pi$
$810$		0			0
$811$			1.42304i			0.0499697i	0.999688	+	0.0249849i	$0.00795375\pi$
$811$			1.42304i			0.0499697i	−0.999688	+	0.0249849i	$0.992046\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.58416			−0.300690
$816$		0			0
$817$	−4.59550			−0.160776
$818$		0			0
$819$		0			0
$820$		0			0
$821$			8.39215i			0.292888i	0.989219	+	0.146444i	$0.0467828\pi$
$821$			8.39215i			0.292888i	−0.989219	+	0.146444i	$0.953217\pi$
$822$		0			0
$823$		−	9.64624i		−	0.336247i	−0.985766	−	0.168123i	$-0.946229\pi$
$823$		−	9.64624i		−	0.336247i	0.985766	−	0.168123i	$-0.0537707\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	30.0944i		−	1.04648i	−0.852184	−	0.523242i	$-0.824722\pi$
$827$		−	30.0944i		−	1.04648i	0.852184	−	0.523242i	$-0.175278\pi$
$828$		0			0
$829$	−10.1537			−0.352653			−0.176327	−	0.984332i	$-0.556421\pi$
$829$	−10.1537			−0.352653			−0.176327	+	0.984332i	$0.556421\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.12300	+	0.0298321i	−0.142853	+	0.00103362i
$834$		0			0
$835$	−17.5127			−0.606051
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.57501i			0.0543755i	0.999630	+	0.0271877i	$0.00865519\pi$
$839$			1.57501i			0.0543755i	−0.999630	+	0.0271877i	$0.991345\pi$
$840$		0			0
$841$	22.7522			0.784559
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	13.5202i		−	0.465109i
$846$		0			0
$847$		−	6.49288i		−	0.223098i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	22.7289			0.779136
$852$		0			0
$853$			1.14407i			0.0391721i	0.999808	+	0.0195861i	$0.00623483\pi$
$853$			1.14407i			0.0391721i	−0.999808	+	0.0195861i	$0.993765\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	15.0875i		−	0.515381i	−0.966228	−	0.257690i	$-0.917039\pi$
$857$		−	15.0875i		−	0.515381i	0.966228	−	0.257690i	$-0.0829615\pi$
$858$		0			0
$859$	−19.5808			−0.668088			−0.334044	−	0.942557i	$-0.608413\pi$
$859$	−19.5808			−0.668088			−0.334044	+	0.942557i	$0.608413\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	54.6083			1.85889			0.929444	−	0.368964i	$-0.120287\pi$
$863$	54.6083			1.85889			0.929444	+	0.368964i	$0.120287\pi$
$864$		0			0
$865$	−11.6393			−0.395750
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−26.5127			−0.899381
$870$		0			0
$871$	−13.3580			−0.452617
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.0298			0.339070
$876$		0			0
$877$		−	0.753805i		−	0.0254542i	−0.999919	−	0.0127271i	$-0.995949\pi$
$877$		−	0.753805i		−	0.0254542i	0.999919	−	0.0127271i	$-0.00405127\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17.7585i			0.598300i	0.954206	+	0.299150i	$0.0967031\pi$
$881$			17.7585i			0.598300i	−0.954206	+	0.299150i	$0.903297\pi$
$882$		0			0
$883$	−6.97578			−0.234754			−0.117377	−	0.993087i	$-0.537449\pi$
$883$	−6.97578			−0.234754			−0.117377	+	0.993087i	$0.537449\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	19.6504i		−	0.659797i	−0.944016	−	0.329899i	$-0.892985\pi$
$887$		−	19.6504i		−	0.659797i	0.944016	−	0.329899i	$-0.107015\pi$
$888$		0			0
$889$			8.51710i			0.285654i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.28497			−0.143391
$894$		0			0
$895$		−	18.4811i		−	0.617754i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.10351			0.0701560
$900$		0			0
$901$	−28.4688	+	0.205987i	−0.948434	+	0.00686243i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.59453			0.219209
$906$		0			0
$907$			17.8828i			0.593788i	0.954910	+	0.296894i	$0.0959508\pi$
$907$			17.8828i			0.593788i	−0.954910	+	0.296894i	$0.904049\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	38.2247i		−	1.26644i	−0.773971	−	0.633221i	$-0.781732\pi$
$911$		−	38.2247i		−	1.26644i	0.773971	−	0.633221i	$-0.218268\pi$
$912$		0			0
$913$			24.1997i			0.800893i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.69345			−0.0889456
$918$		0			0
$919$	28.6429			0.944841			0.472421	−	0.881373i	$-0.343380\pi$
$919$	28.6429			0.944841			0.472421	+	0.881373i	$0.343380\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.59797i		−	0.217175i
$924$		0			0
$925$		−	10.8649i		−	0.357235i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	8.78923i		−	0.288365i	−0.989551	−	0.144183i	$-0.953945\pi$
$929$		−	8.78923i		−	0.288365i	0.989551	−	0.144183i	$-0.0460553\pi$
$930$		0			0
$931$	−0.653275			−0.0214102
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.1400	+	0.0733685i	−0.331614	+	0.00239941i
$936$		0			0
$937$	−18.4664			−0.603270			−0.301635	−	0.953424i	$-0.597532\pi$
$937$	−18.4664			−0.603270			−0.301635	+	0.953424i	$0.597532\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	45.0296i		−	1.46792i	−0.679192	−	0.733961i	$-0.737669\pi$
$941$		−	45.0296i		−	1.46792i	0.679192	−	0.733961i	$-0.262331\pi$
$942$		0			0
$943$	75.0763			2.44482
$944$		0			0
$945$		0			0
$946$		0			0
$947$			34.7361i			1.12877i	0.825511	+	0.564386i	$0.190887\pi$
$947$			34.7361i			1.12877i	−0.825511	+	0.564386i	$0.809113\pi$
$948$		0			0
$949$		−	8.06756i		−	0.261884i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.2590			−0.364714			−0.182357	−	0.983232i	$-0.558373\pi$
$953$	−11.2590			−0.364714			−0.182357	+	0.983232i	$0.558373\pi$
$954$		0			0
$955$			9.44339i			0.305581i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			4.89454i			0.158053i
$960$		0			0
$961$	30.2918			0.977154
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.38043			0.108820
$966$		0			0
$967$	−47.7038			−1.53405			−0.767025	−	0.641617i	$-0.778264\pi$
$967$	−47.7038			−1.53405			−0.767025	+	0.641617i	$0.778264\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.7589			0.602002			0.301001	−	0.953624i	$-0.402679\pi$
$971$	18.7589			0.602002			0.301001	+	0.953624i	$0.402679\pi$
$972$		0			0
$973$	1.34478			0.0431116
$974$		0			0
$975$		0			0
$976$		0			0
$977$	42.4708			1.35876			0.679380	−	0.733786i	$-0.262249\pi$
$977$	42.4708			1.35876			0.679380	+	0.733786i	$0.262249\pi$
$978$		0			0
$979$		−	14.3446i		−	0.458457i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			16.4611i			0.525028i	0.964928	+	0.262514i	$0.0845515\pi$
$983$			16.4611i			0.525028i	−0.964928	+	0.262514i	$0.915448\pi$
$984$		0			0
$985$	−10.8189			−0.344719
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	53.8312i		−	1.71173i
$990$		0			0
$991$		−	38.2811i		−	1.21604i	−0.793922	−	0.608019i	$-0.791964\pi$
$991$		−	38.2811i		−	1.21604i	0.793922	−	0.608019i	$-0.208036\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−15.0445			−0.476944
$996$		0			0
$997$		−	32.7205i		−	1.03627i	−0.855300	−	0.518134i	$-0.826627\pi$
$997$		−	32.7205i		−	1.03627i	0.855300	−	0.518134i	$-0.173373\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4284.2.d.e.3025.5		8
3.2	odd	2		476.2.b.a.169.4	✓	8
12.11	even	2		1904.2.c.f.1121.5		8
17.16	even	2	inner	4284.2.d.e.3025.4		8
21.20	even	2		3332.2.b.b.2549.5		8
51.38	odd	4		8092.2.a.q.1.3		4
51.47	odd	4		8092.2.a.r.1.2		4
51.50	odd	2		476.2.b.a.169.5	yes	8
204.203	even	2		1904.2.c.f.1121.4		8
357.356	even	2		3332.2.b.b.2549.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.b.a.169.4	✓	8	3.2	odd	2
476.2.b.a.169.5	yes	8	51.50	odd	2
1904.2.c.f.1121.4		8	204.203	even	2
1904.2.c.f.1121.5		8	12.11	even	2
3332.2.b.b.2549.4		8	357.356	even	2
3332.2.b.b.2549.5		8	21.20	even	2
4284.2.d.e.3025.4		8	17.16	even	2	inner
4284.2.d.e.3025.5		8	1.1	even	1	trivial
8092.2.a.q.1.3		4	51.38	odd	4
8092.2.a.r.1.2		4	51.47	odd	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:	\( N \)	\(=\)	\( 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4284.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.15845i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+1.15845i q^{5} -1.00000i q^{7} +2.12300i q^{11} -1.15283 q^{13} +(4.12300 - 0.0298321i) q^{17} +0.653275 q^{19} +7.65238i q^{23} +3.65800 q^{25} +2.49956i q^{29} -0.841553i q^{31} +1.15845 q^{35} -2.97017i q^{37} -9.81083i q^{41} -7.03456 q^{43} -6.55922 q^{47} -1.00000 q^{49} -6.90489 q^{53} -2.45938 q^{55} +0.929992 q^{59} +5.49394i q^{61} -1.33549i q^{65} +11.5871 q^{67} +5.72328i q^{71} +6.99805i q^{73} +2.12300 q^{77} +12.4883i q^{79} +11.3988 q^{83} +(0.0345589 + 4.77627i) q^{85} -6.75678 q^{89} +1.15283i q^{91} +0.756784i q^{95} +13.7810i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 20 q^{13} + 12 q^{19} - 12 q^{25} + 4 q^{35} - 12 q^{43} - 24 q^{47} - 8 q^{49} - 20 q^{53} - 4 q^{55} + 24 q^{59} - 4 q^{67} - 16 q^{77} - 4 q^{83} - 44 q^{85}+O(q^{100}) \) 8 * q + 20 * q^13 + 12 * q^19 - 12 * q^25 + 4 * q^35 - 12 * q^43 - 24 * q^47 - 8 * q^49 - 20 * q^53 - 4 * q^55 + 24 * q^59 - 4 * q^67 - 16 * q^77 - 4 * q^83 - 44 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more