LMFDB - Embedded newform 8044.2.a.a.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.94593 q^{3} +4.46459 q^{5} -3.96440 q^{7} +0.786636 q^{9} +O(q^{10})$	$q-1.94593 q^{3} +4.46459 q^{5} -3.96440 q^{7} +0.786636 q^{9} +1.06067 q^{11} -1.22694 q^{13} -8.68777 q^{15} -0.718680 q^{17} +0.102016 q^{19} +7.71444 q^{21} -5.81977 q^{23} +14.9326 q^{25} +4.30705 q^{27} +5.17729 q^{29} -6.83880 q^{31} -2.06398 q^{33} -17.6994 q^{35} +3.02403 q^{37} +2.38754 q^{39} -8.12621 q^{41} +3.66296 q^{43} +3.51201 q^{45} +5.79111 q^{47} +8.71647 q^{49} +1.39850 q^{51} -6.92217 q^{53} +4.73544 q^{55} -0.198516 q^{57} +0.466873 q^{59} +9.01937 q^{61} -3.11854 q^{63} -5.47778 q^{65} +5.64464 q^{67} +11.3249 q^{69} +4.12125 q^{71} +14.8516 q^{73} -29.0577 q^{75} -4.20491 q^{77} -13.7025 q^{79} -10.7411 q^{81} -8.05784 q^{83} -3.20861 q^{85} -10.0746 q^{87} +2.68682 q^{89} +4.86408 q^{91} +13.3078 q^{93} +0.455461 q^{95} +2.59116 q^{97} +0.834358 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.94593	−1.12348	−0.561741	−	0.827313i	$-0.689868\pi$
$3$	−1.94593	−1.12348	−0.561741	+	0.827313i	$0.689868\pi$
$4$	0	0
$5$	4.46459	1.99663	0.998313	−	0.0580588i	$-0.0184911\pi$
$5$	4.46459	1.99663	0.998313	+	0.0580588i	$0.0184911\pi$
$6$	0	0
$7$	−3.96440	−1.49840	−0.749201	−	0.662343i	$-0.769562\pi$
$7$	−3.96440	−1.49840	−0.749201	+	0.662343i	$0.769562\pi$
$8$	0	0
$9$	0.786636	0.262212
$10$	0	0
$11$	1.06067	0.319803	0.159902	−	0.987133i	$-0.448882\pi$
$11$	1.06067	0.319803	0.159902	+	0.987133i	$0.448882\pi$
$12$	0	0
$13$	−1.22694	−0.340292	−0.170146	−	0.985419i	$-0.554424\pi$
$13$	−1.22694	−0.340292	−0.170146	+	0.985419i	$0.554424\pi$
$14$	0	0
$15$	−8.68777	−2.24317
$16$	0	0
$17$	−0.718680	−0.174305	−0.0871527	−	0.996195i	$-0.527777\pi$
$17$	−0.718680	−0.174305	−0.0871527	+	0.996195i	$0.527777\pi$
$18$	0	0
$19$	0.102016	0.0234041	0.0117021	−	0.999932i	$-0.496275\pi$
$19$	0.102016	0.0234041	0.0117021	+	0.999932i	$0.496275\pi$
$20$	0	0
$21$	7.71444	1.68343
$22$	0	0
$23$	−5.81977	−1.21351	−0.606753	−	0.794890i	$-0.707528\pi$
$23$	−5.81977	−1.21351	−0.606753	+	0.794890i	$0.707528\pi$
$24$	0	0
$25$	14.9326	2.98652
$26$	0	0
$27$	4.30705	0.828892
$28$	0	0
$29$	5.17729	0.961398	0.480699	−	0.876886i	$-0.340383\pi$
$29$	5.17729	0.961398	0.480699	+	0.876886i	$0.340383\pi$
$30$	0	0
$31$	−6.83880	−1.22828	−0.614142	−	0.789195i	$-0.710498\pi$
$31$	−6.83880	−1.22828	−0.614142	+	0.789195i	$0.710498\pi$
$32$	0	0
$33$	−2.06398	−0.359293
$34$	0	0
$35$	−17.6994	−2.99175
$36$	0	0
$37$	3.02403	0.497148	0.248574	−	0.968613i	$-0.420038\pi$
$37$	3.02403	0.497148	0.248574	+	0.968613i	$0.420038\pi$
$38$	0	0
$39$	2.38754	0.382312
$40$	0	0
$41$	−8.12621	−1.26910	−0.634551	−	0.772881i	$-0.718815\pi$
$41$	−8.12621	−1.26910	−0.634551	+	0.772881i	$0.718815\pi$
$42$	0	0
$43$	3.66296	0.558596	0.279298	−	0.960204i	$-0.409898\pi$
$43$	3.66296	0.558596	0.279298	+	0.960204i	$0.409898\pi$
$44$	0	0
$45$	3.51201	0.523539
$46$	0	0
$47$	5.79111	0.844720	0.422360	−	0.906428i	$-0.361202\pi$
$47$	5.79111	0.844720	0.422360	+	0.906428i	$0.361202\pi$
$48$	0	0
$49$	8.71647	1.24521
$50$	0	0
$51$	1.39850	0.195829
$52$	0	0
$53$	−6.92217	−0.950833	−0.475416	−	0.879761i	$-0.657703\pi$
$53$	−6.92217	−0.950833	−0.475416	+	0.879761i	$0.657703\pi$
$54$	0	0
$55$	4.73544	0.638527
$56$	0	0
$57$	−0.198516	−0.0262941
$58$	0	0
$59$	0.466873	0.0607816	0.0303908	−	0.999538i	$-0.490325\pi$
$59$	0.466873	0.0607816	0.0303908	+	0.999538i	$0.490325\pi$
$60$	0	0
$61$	9.01937	1.15481	0.577406	−	0.816457i	$-0.304065\pi$
$61$	9.01937	1.15481	0.577406	+	0.816457i	$0.304065\pi$
$62$	0	0
$63$	−3.11854	−0.392899
$64$	0	0
$65$	−5.47778	−0.679436
$66$	0	0
$67$	5.64464	0.689602	0.344801	−	0.938676i	$-0.387946\pi$
$67$	5.64464	0.689602	0.344801	+	0.938676i	$0.387946\pi$
$68$	0	0
$69$	11.3249	1.36335
$70$	0	0
$71$	4.12125	0.489102	0.244551	−	0.969636i	$-0.421359\pi$
$71$	4.12125	0.489102	0.244551	+	0.969636i	$0.421359\pi$
$72$	0	0
$73$	14.8516	1.73824	0.869122	−	0.494599i	$-0.164685\pi$
$73$	14.8516	1.73824	0.869122	+	0.494599i	$0.164685\pi$
$74$	0	0
$75$	−29.0577	−3.35530
$76$	0	0
$77$	−4.20491	−0.479194
$78$	0	0
$79$	−13.7025	−1.54165	−0.770824	−	0.637048i	$-0.780155\pi$
$79$	−13.7025	−1.54165	−0.770824	+	0.637048i	$0.780155\pi$
$80$	0	0
$81$	−10.7411	−1.19346
$82$	0	0
$83$	−8.05784	−0.884463	−0.442231	−	0.896901i	$-0.645813\pi$
$83$	−8.05784	−0.884463	−0.442231	+	0.896901i	$0.645813\pi$
$84$	0	0
$85$	−3.20861	−0.348023
$86$	0	0
$87$	−10.0746	−1.08011
$88$	0	0
$89$	2.68682	0.284803	0.142401	−	0.989809i	$-0.454518\pi$
$89$	2.68682	0.284803	0.142401	+	0.989809i	$0.454518\pi$
$90$	0	0
$91$	4.86408	0.509894
$92$	0	0
$93$	13.3078	1.37996
$94$	0	0
$95$	0.455461	0.0467293
$96$	0	0
$97$	2.59116	0.263092	0.131546	−	0.991310i	$-0.458006\pi$
$97$	2.59116	0.263092	0.131546	+	0.991310i	$0.458006\pi$
$98$	0	0
$99$	0.834358	0.0838561
$100$	0	0
$101$	−11.5198	−1.14627	−0.573134	−	0.819462i	$-0.694272\pi$
$101$	−11.5198	−1.14627	−0.573134	+	0.819462i	$0.694272\pi$
$102$	0	0
$103$	−15.5181	−1.52904	−0.764521	−	0.644599i	$-0.777024\pi$
$103$	−15.5181	−1.52904	−0.764521	+	0.644599i	$0.777024\pi$
$104$	0	0
$105$	34.4418	3.36118
$106$	0	0
$107$	−4.62192	−0.446818	−0.223409	−	0.974725i	$-0.571719\pi$
$107$	−4.62192	−0.446818	−0.223409	+	0.974725i	$0.571719\pi$
$108$	0	0
$109$	−17.7003	−1.69538	−0.847689	−	0.530494i	$-0.822007\pi$
$109$	−17.7003	−1.69538	−0.847689	+	0.530494i	$0.822007\pi$
$110$	0	0
$111$	−5.88455	−0.558537
$112$	0	0
$113$	−8.12471	−0.764309	−0.382154	−	0.924099i	$-0.624818\pi$
$113$	−8.12471	−0.764309	−0.382154	+	0.924099i	$0.624818\pi$
$114$	0	0
$115$	−25.9829	−2.42292
$116$	0	0
$117$	−0.965154	−0.0892285
$118$	0	0
$119$	2.84913	0.261180
$120$	0	0
$121$	−9.87499	−0.897726
$122$	0	0
$123$	15.8130	1.42581
$124$	0	0
$125$	44.3449	3.96633
$126$	0	0
$127$	18.3924	1.63206	0.816030	−	0.578009i	$-0.196170\pi$
$127$	18.3924	1.63206	0.816030	+	0.578009i	$0.196170\pi$
$128$	0	0
$129$	−7.12786	−0.627573
$130$	0	0
$131$	−5.68649	−0.496831	−0.248416	−	0.968654i	$-0.579910\pi$
$131$	−5.68649	−0.496831	−0.248416	+	0.968654i	$0.579910\pi$
$132$	0	0
$133$	−0.404433	−0.0350688
$134$	0	0
$135$	19.2292	1.65499
$136$	0	0
$137$	−0.756080	−0.0645963	−0.0322982	−	0.999478i	$-0.510283\pi$
$137$	−0.756080	−0.0645963	−0.0322982	+	0.999478i	$0.510283\pi$
$138$	0	0
$139$	−14.3790	−1.21961	−0.609805	−	0.792551i	$-0.708752\pi$
$139$	−14.3790	−1.21961	−0.609805	+	0.792551i	$0.708752\pi$
$140$	0	0
$141$	−11.2691	−0.949027
$142$	0	0
$143$	−1.30137	−0.108826
$144$	0	0
$145$	23.1145	1.91955
$146$	0	0
$147$	−16.9616	−1.39897
$148$	0	0
$149$	−9.06417	−0.742566	−0.371283	−	0.928520i	$-0.621082\pi$
$149$	−9.06417	−0.742566	−0.371283	+	0.928520i	$0.621082\pi$
$150$	0	0
$151$	6.70984	0.546039	0.273020	−	0.962008i	$-0.411978\pi$
$151$	6.70984	0.546039	0.273020	+	0.962008i	$0.411978\pi$
$152$	0	0
$153$	−0.565339	−0.0457050
$154$	0	0
$155$	−30.5324	−2.45242
$156$	0	0
$157$	12.5456	1.00125	0.500623	−	0.865665i	$-0.333104\pi$
$157$	12.5456	1.00125	0.500623	+	0.865665i	$0.333104\pi$
$158$	0	0
$159$	13.4700	1.06824
$160$	0	0
$161$	23.0719	1.81832
$162$	0	0
$163$	19.8179	1.55226	0.776130	−	0.630573i	$-0.217180\pi$
$163$	19.8179	1.55226	0.776130	+	0.630573i	$0.217180\pi$
$164$	0	0
$165$	−9.21483	−0.717374
$166$	0	0
$167$	−19.7266	−1.52649	−0.763246	−	0.646108i	$-0.776395\pi$
$167$	−19.7266	−1.52649	−0.763246	+	0.646108i	$0.776395\pi$
$168$	0	0
$169$	−11.4946	−0.884201
$170$	0	0
$171$	0.0802496	0.00613684
$172$	0	0
$173$	−21.8818	−1.66364	−0.831821	−	0.555044i	$-0.812702\pi$
$173$	−21.8818	−1.66364	−0.831821	+	0.555044i	$0.812702\pi$
$174$	0	0
$175$	−59.1987	−4.47500
$176$	0	0
$177$	−0.908500	−0.0682871
$178$	0	0
$179$	−3.78635	−0.283005	−0.141503	−	0.989938i	$-0.545193\pi$
$179$	−3.78635	−0.283005	−0.141503	+	0.989938i	$0.545193\pi$
$180$	0	0
$181$	−18.1905	−1.35209	−0.676045	−	0.736860i	$-0.736307\pi$
$181$	−18.1905	−1.35209	−0.676045	+	0.736860i	$0.736307\pi$
$182$	0	0
$183$	−17.5511	−1.29741
$184$	0	0
$185$	13.5011	0.992619
$186$	0	0
$187$	−0.762280	−0.0557434
$188$	0	0
$189$	−17.0749	−1.24201
$190$	0	0
$191$	−21.0834	−1.52554	−0.762771	−	0.646669i	$-0.776162\pi$
$191$	−21.0834	−1.52554	−0.762771	+	0.646669i	$0.776162\pi$
$192$	0	0
$193$	19.9969	1.43941	0.719705	−	0.694280i	$-0.244277\pi$
$193$	19.9969	1.43941	0.719705	+	0.694280i	$0.244277\pi$
$194$	0	0
$195$	10.6594	0.763334
$196$	0	0
$197$	20.9876	1.49531	0.747654	−	0.664089i	$-0.231180\pi$
$197$	20.9876	1.49531	0.747654	+	0.664089i	$0.231180\pi$
$198$	0	0
$199$	21.3697	1.51486	0.757429	−	0.652917i	$-0.226455\pi$
$199$	21.3697	1.51486	0.757429	+	0.652917i	$0.226455\pi$
$200$	0	0
$201$	−10.9841	−0.774756
$202$	0	0
$203$	−20.5248	−1.44056
$204$	0	0
$205$	−36.2802	−2.53392
$206$	0	0
$207$	−4.57804	−0.318196
$208$	0	0
$209$	0.108205	0.00748471
$210$	0	0
$211$	20.6796	1.42364	0.711820	−	0.702362i	$-0.247871\pi$
$211$	20.6796	1.42364	0.711820	+	0.702362i	$0.247871\pi$
$212$	0	0
$213$	−8.01965	−0.549497
$214$	0	0
$215$	16.3536	1.11531
$216$	0	0
$217$	27.1117	1.84046
$218$	0	0
$219$	−28.9001	−1.95288
$220$	0	0
$221$	0.881777	0.0593147
$222$	0	0
$223$	0.0388488	0.00260151	0.00130075	−	0.999999i	$-0.499586\pi$
$223$	0.0388488	0.00260151	0.00130075	+	0.999999i	$0.499586\pi$
$224$	0	0
$225$	11.7465	0.783100
$226$	0	0
$227$	−17.6497	−1.17145	−0.585726	−	0.810509i	$-0.699190\pi$
$227$	−17.6497	−1.17145	−0.585726	+	0.810509i	$0.699190\pi$
$228$	0	0
$229$	−17.9370	−1.18531	−0.592654	−	0.805457i	$-0.701920\pi$
$229$	−17.9370	−1.18531	−0.592654	+	0.805457i	$0.701920\pi$
$230$	0	0
$231$	8.18245	0.538365
$232$	0	0
$233$	−25.2496	−1.65416	−0.827080	−	0.562085i	$-0.810000\pi$
$233$	−25.2496	−1.65416	−0.827080	+	0.562085i	$0.810000\pi$
$234$	0	0
$235$	25.8549	1.68659
$236$	0	0
$237$	26.6640	1.73201
$238$	0	0
$239$	−5.37929	−0.347957	−0.173979	−	0.984749i	$-0.555662\pi$
$239$	−5.37929	−0.347957	−0.173979	+	0.984749i	$0.555662\pi$
$240$	0	0
$241$	−24.8116	−1.59826	−0.799128	−	0.601161i	$-0.794705\pi$
$241$	−24.8116	−1.59826	−0.799128	+	0.601161i	$0.794705\pi$
$242$	0	0
$243$	7.98029	0.511936
$244$	0	0
$245$	38.9155	2.48622
$246$	0	0
$247$	−0.125168	−0.00796423
$248$	0	0
$249$	15.6800	0.993678
$250$	0	0
$251$	−30.8032	−1.94428	−0.972141	−	0.234396i	$-0.924689\pi$
$251$	−30.8032	−1.94428	−0.972141	+	0.234396i	$0.924689\pi$
$252$	0	0
$253$	−6.17284	−0.388083
$254$	0	0
$255$	6.24373	0.390997
$256$	0	0
$257$	9.59693	0.598640	0.299320	−	0.954153i	$-0.403240\pi$
$257$	9.59693	0.598640	0.299320	+	0.954153i	$0.403240\pi$
$258$	0	0
$259$	−11.9885	−0.744928
$260$	0	0
$261$	4.07264	0.252090
$262$	0	0
$263$	17.6939	1.09105	0.545527	−	0.838093i	$-0.316329\pi$
$263$	17.6939	1.09105	0.545527	+	0.838093i	$0.316329\pi$
$264$	0	0
$265$	−30.9046	−1.89846
$266$	0	0
$267$	−5.22836	−0.319971
$268$	0	0
$269$	11.6882	0.712641	0.356321	−	0.934364i	$-0.384031\pi$
$269$	11.6882	0.712641	0.356321	+	0.934364i	$0.384031\pi$
$270$	0	0
$271$	2.51081	0.152521	0.0762604	−	0.997088i	$-0.475702\pi$
$271$	2.51081	0.152521	0.0762604	+	0.997088i	$0.475702\pi$
$272$	0	0
$273$	−9.46515	−0.572857
$274$	0	0
$275$	15.8385	0.955097
$276$	0	0
$277$	−5.08842	−0.305733	−0.152867	−	0.988247i	$-0.548851\pi$
$277$	−5.08842	−0.305733	−0.152867	+	0.988247i	$0.548851\pi$
$278$	0	0
$279$	−5.37964	−0.322071
$280$	0	0
$281$	20.3461	1.21375	0.606875	−	0.794798i	$-0.292423\pi$
$281$	20.3461	1.21375	0.606875	+	0.794798i	$0.292423\pi$
$282$	0	0
$283$	5.68250	0.337789	0.168895	−	0.985634i	$-0.445980\pi$
$283$	5.68250	0.337789	0.168895	+	0.985634i	$0.445980\pi$
$284$	0	0
$285$	−0.886294	−0.0524995
$286$	0	0
$287$	32.2156	1.90162
$288$	0	0
$289$	−16.4835	−0.969618
$290$	0	0
$291$	−5.04221	−0.295579
$292$	0	0
$293$	7.45655	0.435616	0.217808	−	0.975992i	$-0.430109\pi$
$293$	7.45655	0.435616	0.217808	+	0.975992i	$0.430109\pi$
$294$	0	0
$295$	2.08440	0.121358
$296$	0	0
$297$	4.56834	0.265082
$298$	0	0
$299$	7.14051	0.412946
$300$	0	0
$301$	−14.5214	−0.837002
$302$	0	0
$303$	22.4168	1.28781
$304$	0	0
$305$	40.2678	2.30573
$306$	0	0
$307$	−30.0688	−1.71612	−0.858060	−	0.513550i	$-0.828330\pi$
$307$	−30.0688	−1.71612	−0.858060	+	0.513550i	$0.828330\pi$
$308$	0	0
$309$	30.1971	1.71785
$310$	0	0
$311$	22.1661	1.25692	0.628462	−	0.777841i	$-0.283685\pi$
$311$	22.1661	1.25692	0.628462	+	0.777841i	$0.283685\pi$
$312$	0	0
$313$	−4.58581	−0.259206	−0.129603	−	0.991566i	$-0.541370\pi$
$313$	−4.58581	−0.259206	−0.129603	+	0.991566i	$0.541370\pi$
$314$	0	0
$315$	−13.9230	−0.784472
$316$	0	0
$317$	23.0271	1.29333	0.646665	−	0.762774i	$-0.276163\pi$
$317$	23.0271	1.29333	0.646665	+	0.762774i	$0.276163\pi$
$318$	0	0
$319$	5.49137	0.307458
$320$	0	0
$321$	8.99392	0.501992
$322$	0	0
$323$	−0.0733170	−0.00407947
$324$	0	0
$325$	−18.3214	−1.01629
$326$	0	0
$327$	34.4434	1.90473
$328$	0	0
$329$	−22.9583	−1.26573
$330$	0	0
$331$	−20.6382	−1.13438	−0.567189	−	0.823588i	$-0.691969\pi$
$331$	−20.6382	−1.13438	−0.567189	+	0.823588i	$0.691969\pi$
$332$	0	0
$333$	2.37881	0.130358
$334$	0	0
$335$	25.2010	1.37688
$336$	0	0
$337$	−24.6776	−1.34428	−0.672138	−	0.740426i	$-0.734624\pi$
$337$	−24.6776	−1.34428	−0.672138	+	0.740426i	$0.734624\pi$
$338$	0	0
$339$	15.8101	0.858687
$340$	0	0
$341$	−7.25368	−0.392809
$342$	0	0
$343$	−6.80476	−0.367422
$344$	0	0
$345$	50.5609	2.72211
$346$	0	0
$347$	−4.74048	−0.254482	−0.127241	−	0.991872i	$-0.540612\pi$
$347$	−4.74048	−0.254482	−0.127241	+	0.991872i	$0.540612\pi$
$348$	0	0
$349$	13.1359	0.703147	0.351574	−	0.936160i	$-0.385647\pi$
$349$	13.1359	0.703147	0.351574	+	0.936160i	$0.385647\pi$
$350$	0	0
$351$	−5.28449	−0.282065
$352$	0	0
$353$	3.58018	0.190554	0.0952769	−	0.995451i	$-0.469626\pi$
$353$	3.58018	0.190554	0.0952769	+	0.995451i	$0.469626\pi$
$354$	0	0
$355$	18.3997	0.976554
$356$	0	0
$357$	−5.54421	−0.293431
$358$	0	0
$359$	23.0232	1.21512	0.607558	−	0.794275i	$-0.292149\pi$
$359$	23.0232	1.21512	0.607558	+	0.794275i	$0.292149\pi$
$360$	0	0
$361$	−18.9896	−0.999452
$362$	0	0
$363$	19.2160	1.00858
$364$	0	0
$365$	66.3061	3.47062
$366$	0	0
$367$	24.3371	1.27039	0.635194	−	0.772353i	$-0.280920\pi$
$367$	24.3371	1.27039	0.635194	+	0.772353i	$0.280920\pi$
$368$	0	0
$369$	−6.39237	−0.332773
$370$	0	0
$371$	27.4422	1.42473
$372$	0	0
$373$	−25.1208	−1.30071	−0.650353	−	0.759632i	$-0.725379\pi$
$373$	−25.1208	−1.30071	−0.650353	+	0.759632i	$0.725379\pi$
$374$	0	0
$375$	−86.2920	−4.45610
$376$	0	0
$377$	−6.35222	−0.327156
$378$	0	0
$379$	3.06685	0.157533	0.0787667	−	0.996893i	$-0.474902\pi$
$379$	3.06685	0.157533	0.0787667	+	0.996893i	$0.474902\pi$
$380$	0	0
$381$	−35.7903	−1.83359
$382$	0	0
$383$	16.8309	0.860018	0.430009	−	0.902825i	$-0.358510\pi$
$383$	16.8309	0.860018	0.430009	+	0.902825i	$0.358510\pi$
$384$	0	0
$385$	−18.7732	−0.956771
$386$	0	0
$387$	2.88142	0.146471
$388$	0	0
$389$	−0.726005	−0.0368099	−0.0184050	−	0.999831i	$-0.505859\pi$
$389$	−0.726005	−0.0368099	−0.0184050	+	0.999831i	$0.505859\pi$
$390$	0	0
$391$	4.18255	0.211521
$392$	0	0
$393$	11.0655	0.558181
$394$	0	0
$395$	−61.1759	−3.07809
$396$	0	0
$397$	15.8994	0.797969	0.398985	−	0.916958i	$-0.369363\pi$
$397$	15.8994	0.797969	0.398985	+	0.916958i	$0.369363\pi$
$398$	0	0
$399$	0.786997	0.0393991
$400$	0	0
$401$	−14.7885	−0.738500	−0.369250	−	0.929330i	$-0.620385\pi$
$401$	−14.7885	−0.738500	−0.369250	+	0.929330i	$0.620385\pi$
$402$	0	0
$403$	8.39079	0.417975
$404$	0	0
$405$	−47.9547	−2.38289
$406$	0	0
$407$	3.20749	0.158989
$408$	0	0
$409$	−14.6826	−0.726007	−0.363003	−	0.931788i	$-0.618249\pi$
$409$	−14.6826	−0.726007	−0.363003	+	0.931788i	$0.618249\pi$
$410$	0	0
$411$	1.47128	0.0725728
$412$	0	0
$413$	−1.85087	−0.0910753
$414$	0	0
$415$	−35.9750	−1.76594
$416$	0	0
$417$	27.9805	1.37021
$418$	0	0
$419$	−8.46045	−0.413320	−0.206660	−	0.978413i	$-0.566259\pi$
$419$	−8.46045	−0.413320	−0.206660	+	0.978413i	$0.566259\pi$
$420$	0	0
$421$	−28.4553	−1.38683	−0.693414	−	0.720539i	$-0.743894\pi$
$421$	−28.4553	−1.38683	−0.693414	+	0.720539i	$0.743894\pi$
$422$	0	0
$423$	4.55549	0.221496
$424$	0	0
$425$	−10.7317	−0.520566
$426$	0	0
$427$	−35.7564	−1.73037
$428$	0	0
$429$	2.53238	0.122264
$430$	0	0
$431$	−2.23816	−0.107808	−0.0539042	−	0.998546i	$-0.517167\pi$
$431$	−2.23816	−0.107808	−0.0539042	+	0.998546i	$0.517167\pi$
$432$	0	0
$433$	12.0454	0.578866	0.289433	−	0.957198i	$-0.406533\pi$
$433$	12.0454	0.578866	0.289433	+	0.957198i	$0.406533\pi$
$434$	0	0
$435$	−44.9791	−2.15658
$436$	0	0
$437$	−0.593711	−0.0284010
$438$	0	0
$439$	−1.11922	−0.0534172	−0.0267086	−	0.999643i	$-0.508503\pi$
$439$	−1.11922	−0.0534172	−0.0267086	+	0.999643i	$0.508503\pi$
$440$	0	0
$441$	6.85668	0.326509
$442$	0	0
$443$	14.4310	0.685638	0.342819	−	0.939401i	$-0.388618\pi$
$443$	14.4310	0.685638	0.342819	+	0.939401i	$0.388618\pi$
$444$	0	0
$445$	11.9956	0.568644
$446$	0	0
$447$	17.6382	0.834259
$448$	0	0
$449$	−4.17196	−0.196887	−0.0984436	−	0.995143i	$-0.531386\pi$
$449$	−4.17196	−0.196887	−0.0984436	+	0.995143i	$0.531386\pi$
$450$	0	0
$451$	−8.61920	−0.405862
$452$	0	0
$453$	−13.0569	−0.613465
$454$	0	0
$455$	21.7161	1.01807
$456$	0	0
$457$	−18.7507	−0.877119	−0.438559	−	0.898702i	$-0.644511\pi$
$457$	−18.7507	−0.877119	−0.438559	+	0.898702i	$0.644511\pi$
$458$	0	0
$459$	−3.09539	−0.144480
$460$	0	0
$461$	−14.7122	−0.685217	−0.342608	−	0.939478i	$-0.611310\pi$
$461$	−14.7122	−0.685217	−0.342608	+	0.939478i	$0.611310\pi$
$462$	0	0
$463$	−32.9642	−1.53198	−0.765988	−	0.642855i	$-0.777750\pi$
$463$	−32.9642	−1.53198	−0.765988	+	0.642855i	$0.777750\pi$
$464$	0	0
$465$	59.4139	2.75526
$466$	0	0
$467$	22.4316	1.03801	0.519006	−	0.854771i	$-0.326302\pi$
$467$	22.4316	1.03801	0.519006	+	0.854771i	$0.326302\pi$
$468$	0	0
$469$	−22.3776	−1.03330
$470$	0	0
$471$	−24.4128	−1.12488
$472$	0	0
$473$	3.88518	0.178641
$474$	0	0
$475$	1.52337	0.0698968
$476$	0	0
$477$	−5.44522	−0.249320
$478$	0	0
$479$	−21.1930	−0.968332	−0.484166	−	0.874976i	$-0.660877\pi$
$479$	−21.1930	−0.968332	−0.484166	+	0.874976i	$0.660877\pi$
$480$	0	0
$481$	−3.71031	−0.169175
$482$	0	0
$483$	−44.8963	−2.04285
$484$	0	0
$485$	11.5685	0.525297
$486$	0	0
$487$	−13.6401	−0.618091	−0.309045	−	0.951047i	$-0.600010\pi$
$487$	−13.6401	−0.618091	−0.309045	+	0.951047i	$0.600010\pi$
$488$	0	0
$489$	−38.5642	−1.74394
$490$	0	0
$491$	−15.8262	−0.714228	−0.357114	−	0.934061i	$-0.616239\pi$
$491$	−15.8262	−0.714228	−0.357114	+	0.934061i	$0.616239\pi$
$492$	0	0
$493$	−3.72081	−0.167577
$494$	0	0
$495$	3.72507	0.167429
$496$	0	0
$497$	−16.3383	−0.732872
$498$	0	0
$499$	−24.8284	−1.11147	−0.555735	−	0.831359i	$-0.687563\pi$
$499$	−24.8284	−1.11147	−0.555735	+	0.831359i	$0.687563\pi$
$500$	0	0
$501$	38.3866	1.71499
$502$	0	0
$503$	10.1107	0.450812	0.225406	−	0.974265i	$-0.427629\pi$
$503$	10.1107	0.450812	0.225406	+	0.974265i	$0.427629\pi$
$504$	0	0
$505$	−51.4314	−2.28867
$506$	0	0
$507$	22.3677	0.993384
$508$	0	0
$509$	−4.40769	−0.195367	−0.0976837	−	0.995218i	$-0.531143\pi$
$509$	−4.40769	−0.195367	−0.0976837	+	0.995218i	$0.531143\pi$
$510$	0	0
$511$	−58.8775	−2.60459
$512$	0	0
$513$	0.439389	0.0193995
$514$	0	0
$515$	−69.2819	−3.05292
$516$	0	0
$517$	6.14243	0.270144
$518$	0	0
$519$	42.5804	1.86907
$520$	0	0
$521$	−30.0028	−1.31444	−0.657222	−	0.753697i	$-0.728269\pi$
$521$	−30.0028	−1.31444	−0.657222	+	0.753697i	$0.728269\pi$
$522$	0	0
$523$	−19.2873	−0.843375	−0.421688	−	0.906741i	$-0.638562\pi$
$523$	−19.2873	−0.843375	−0.421688	+	0.906741i	$0.638562\pi$
$524$	0	0
$525$	115.196	5.02759
$526$	0	0
$527$	4.91491	0.214097
$528$	0	0
$529$	10.8697	0.472597
$530$	0	0
$531$	0.367259	0.0159377
$532$	0	0
$533$	9.97037	0.431865
$534$	0	0
$535$	−20.6350	−0.892128
$536$	0	0
$537$	7.36797	0.317951
$538$	0	0
$539$	9.24527	0.398222
$540$	0	0
$541$	22.0003	0.945868	0.472934	−	0.881098i	$-0.343195\pi$
$541$	22.0003	0.945868	0.472934	+	0.881098i	$0.343195\pi$
$542$	0	0
$543$	35.3974	1.51905
$544$	0	0
$545$	−79.0245	−3.38504
$546$	0	0
$547$	7.46411	0.319142	0.159571	−	0.987186i	$-0.448989\pi$
$547$	7.46411	0.319142	0.159571	+	0.987186i	$0.448989\pi$
$548$	0	0
$549$	7.09496	0.302806
$550$	0	0
$551$	0.528167	0.0225007
$552$	0	0
$553$	54.3221	2.31001
$554$	0	0
$555$	−26.2721	−1.11519
$556$	0	0
$557$	−15.0261	−0.636676	−0.318338	−	0.947977i	$-0.603125\pi$
$557$	−15.0261	−0.636676	−0.318338	+	0.947977i	$0.603125\pi$
$558$	0	0
$559$	−4.49423	−0.190086
$560$	0	0
$561$	1.48334	0.0626267
$562$	0	0
$563$	−41.6787	−1.75655	−0.878274	−	0.478158i	$-0.841305\pi$
$563$	−41.6787	−1.75655	−0.878274	+	0.478158i	$0.841305\pi$
$564$	0	0
$565$	−36.2735	−1.52604
$566$	0	0
$567$	42.5821	1.78828
$568$	0	0
$569$	23.6609	0.991917	0.495959	−	0.868346i	$-0.334817\pi$
$569$	23.6609	0.991917	0.495959	+	0.868346i	$0.334817\pi$
$570$	0	0
$571$	−30.2829	−1.26730	−0.633650	−	0.773620i	$-0.718444\pi$
$571$	−30.2829	−1.26730	−0.633650	+	0.773620i	$0.718444\pi$
$572$	0	0
$573$	41.0268	1.71392
$574$	0	0
$575$	−86.9042	−3.62416
$576$	0	0
$577$	−11.2004	−0.466277	−0.233139	−	0.972443i	$-0.574900\pi$
$577$	−11.2004	−0.466277	−0.233139	+	0.972443i	$0.574900\pi$
$578$	0	0
$579$	−38.9126	−1.61715
$580$	0	0
$581$	31.9445	1.32528
$582$	0	0
$583$	−7.34211	−0.304079
$584$	0	0
$585$	−4.30902	−0.178156
$586$	0	0
$587$	−29.5742	−1.22066	−0.610330	−	0.792147i	$-0.708963\pi$
$587$	−29.5742	−1.22066	−0.610330	+	0.792147i	$0.708963\pi$
$588$	0	0
$589$	−0.697668	−0.0287469
$590$	0	0
$591$	−40.8404	−1.67995
$592$	0	0
$593$	0.941603	0.0386670	0.0193335	−	0.999813i	$-0.493846\pi$
$593$	0.941603	0.0386670	0.0193335	+	0.999813i	$0.493846\pi$
$594$	0	0
$595$	12.7202	0.521478
$596$	0	0
$597$	−41.5839	−1.70192
$598$	0	0
$599$	−21.7115	−0.887109	−0.443555	−	0.896247i	$-0.646283\pi$
$599$	−21.7115	−0.887109	−0.443555	+	0.896247i	$0.646283\pi$
$600$	0	0
$601$	32.1848	1.31285	0.656423	−	0.754393i	$-0.272069\pi$
$601$	32.1848	1.31285	0.656423	+	0.754393i	$0.272069\pi$
$602$	0	0
$603$	4.44027	0.180822
$604$	0	0
$605$	−44.0878	−1.79242
$606$	0	0
$607$	−26.4532	−1.07370	−0.536850	−	0.843677i	$-0.680386\pi$
$607$	−26.4532	−1.07370	−0.536850	+	0.843677i	$0.680386\pi$
$608$	0	0
$609$	39.9398	1.61844
$610$	0	0
$611$	−7.10534	−0.287451
$612$	0	0
$613$	21.0471	0.850084	0.425042	−	0.905174i	$-0.360259\pi$
$613$	21.0471	0.850084	0.425042	+	0.905174i	$0.360259\pi$
$614$	0	0
$615$	70.5987	2.84681
$616$	0	0
$617$	−16.1187	−0.648915	−0.324457	−	0.945900i	$-0.605182\pi$
$617$	−16.1187	−0.648915	−0.324457	+	0.945900i	$0.605182\pi$
$618$	0	0
$619$	−33.1197	−1.33119	−0.665597	−	0.746311i	$-0.731823\pi$
$619$	−33.1197	−1.33119	−0.665597	+	0.746311i	$0.731823\pi$
$620$	0	0
$621$	−25.0660	−1.00587
$622$	0	0
$623$	−10.6516	−0.426749
$624$	0	0
$625$	123.319	4.93277
$626$	0	0
$627$	−0.210559	−0.00840894
$628$	0	0
$629$	−2.17331	−0.0866556
$630$	0	0
$631$	−16.6581	−0.663150	−0.331575	−	0.943429i	$-0.607580\pi$
$631$	−16.6581	−0.663150	−0.331575	+	0.943429i	$0.607580\pi$
$632$	0	0
$633$	−40.2409	−1.59943
$634$	0	0
$635$	82.1145	3.25862
$636$	0	0
$637$	−10.6946	−0.423735
$638$	0	0
$639$	3.24192	0.128248
$640$	0	0
$641$	38.0041	1.50107	0.750535	−	0.660831i	$-0.229796\pi$
$641$	38.0041	1.50107	0.750535	+	0.660831i	$0.229796\pi$
$642$	0	0
$643$	−1.06815	−0.0421238	−0.0210619	−	0.999778i	$-0.506705\pi$
$643$	−1.06815	−0.0421238	−0.0210619	+	0.999778i	$0.506705\pi$
$644$	0	0
$645$	−31.8230	−1.25303
$646$	0	0
$647$	−19.5326	−0.767907	−0.383954	−	0.923352i	$-0.625438\pi$
$647$	−19.5326	−0.767907	−0.383954	+	0.923352i	$0.625438\pi$
$648$	0	0
$649$	0.495196	0.0194381
$650$	0	0
$651$	−52.7575	−2.06773
$652$	0	0
$653$	47.6553	1.86489	0.932447	−	0.361306i	$-0.117669\pi$
$653$	47.6553	1.86489	0.932447	+	0.361306i	$0.117669\pi$
$654$	0	0
$655$	−25.3879	−0.991986
$656$	0	0
$657$	11.6828	0.455788
$658$	0	0
$659$	35.3280	1.37618	0.688092	−	0.725623i	$-0.258448\pi$
$659$	35.3280	1.37618	0.688092	+	0.725623i	$0.258448\pi$
$660$	0	0
$661$	−5.71535	−0.222302	−0.111151	−	0.993804i	$-0.535454\pi$
$661$	−5.71535	−0.222302	−0.111151	+	0.993804i	$0.535454\pi$
$662$	0	0
$663$	−1.71587	−0.0666390
$664$	0	0
$665$	−1.80563	−0.0700193
$666$	0	0
$667$	−30.1306	−1.16666
$668$	0	0
$669$	−0.0755970	−0.00292275
$670$	0	0
$671$	9.56655	0.369313
$672$	0	0
$673$	3.25033	0.125291	0.0626456	−	0.998036i	$-0.480046\pi$
$673$	3.25033	0.125291	0.0626456	+	0.998036i	$0.480046\pi$
$674$	0	0
$675$	64.3153	2.47550
$676$	0	0
$677$	−21.5735	−0.829137	−0.414568	−	0.910018i	$-0.636067\pi$
$677$	−21.5735	−0.829137	−0.414568	+	0.910018i	$0.636067\pi$
$678$	0	0
$679$	−10.2724	−0.394218
$680$	0	0
$681$	34.3450	1.31610
$682$	0	0
$683$	−32.6630	−1.24981	−0.624907	−	0.780699i	$-0.714863\pi$
$683$	−32.6630	−1.24981	−0.624907	+	0.780699i	$0.714863\pi$
$684$	0	0
$685$	−3.37559	−0.128975
$686$	0	0
$687$	34.9040	1.33167
$688$	0	0
$689$	8.49308	0.323561
$690$	0	0
$691$	−7.10732	−0.270375	−0.135188	−	0.990820i	$-0.543164\pi$
$691$	−7.10732	−0.270375	−0.135188	+	0.990820i	$0.543164\pi$
$692$	0	0
$693$	−3.30773	−0.125650
$694$	0	0
$695$	−64.1963	−2.43511
$696$	0	0
$697$	5.84014	0.221211
$698$	0	0
$699$	49.1340	1.85842
$700$	0	0
$701$	−29.4279	−1.11148	−0.555738	−	0.831358i	$-0.687564\pi$
$701$	−29.4279	−1.11148	−0.555738	+	0.831358i	$0.687564\pi$
$702$	0	0
$703$	0.308500	0.0116353
$704$	0	0
$705$	−50.3118	−1.89485
$706$	0	0
$707$	45.6693	1.71757
$708$	0	0
$709$	−31.1327	−1.16921	−0.584607	−	0.811316i	$-0.698751\pi$
$709$	−31.1327	−1.16921	−0.584607	+	0.811316i	$0.698751\pi$
$710$	0	0
$711$	−10.7788	−0.404238
$712$	0	0
$713$	39.8002	1.49053
$714$	0	0
$715$	−5.81010	−0.217286
$716$	0	0
$717$	10.4677	0.390924
$718$	0	0
$719$	46.2503	1.72485	0.862423	−	0.506189i	$-0.168946\pi$
$719$	46.2503	1.72485	0.862423	+	0.506189i	$0.168946\pi$
$720$	0	0
$721$	61.5198	2.29112
$722$	0	0
$723$	48.2816	1.79561
$724$	0	0
$725$	77.3103	2.87123
$726$	0	0
$727$	35.0459	1.29978	0.649889	−	0.760029i	$-0.274815\pi$
$727$	35.0459	1.29978	0.649889	+	0.760029i	$0.274815\pi$
$728$	0	0
$729$	16.6943	0.618306
$730$	0	0
$731$	−2.63250	−0.0973664
$732$	0	0
$733$	−25.4379	−0.939570	−0.469785	−	0.882781i	$-0.655669\pi$
$733$	−25.4379	−0.939570	−0.469785	+	0.882781i	$0.655669\pi$
$734$	0	0
$735$	−75.7267	−2.79322
$736$	0	0
$737$	5.98708	0.220537
$738$	0	0
$739$	−15.9683	−0.587404	−0.293702	−	0.955897i	$-0.594887\pi$
$739$	−15.9683	−0.587404	−0.293702	+	0.955897i	$0.594887\pi$
$740$	0	0
$741$	0.243567	0.00894767
$742$	0	0
$743$	−20.2327	−0.742266	−0.371133	−	0.928580i	$-0.621031\pi$
$743$	−20.2327	−0.742266	−0.371133	+	0.928580i	$0.621031\pi$
$744$	0	0
$745$	−40.4678	−1.48263
$746$	0	0
$747$	−6.33858	−0.231917
$748$	0	0
$749$	18.3231	0.669513
$750$	0	0
$751$	8.36203	0.305135	0.152567	−	0.988293i	$-0.451246\pi$
$751$	8.36203	0.305135	0.152567	+	0.988293i	$0.451246\pi$
$752$	0	0
$753$	59.9409	2.18437
$754$	0	0
$755$	29.9567	1.09024
$756$	0	0
$757$	3.98313	0.144769	0.0723847	−	0.997377i	$-0.476939\pi$
$757$	3.98313	0.144769	0.0723847	+	0.997377i	$0.476939\pi$
$758$	0	0
$759$	12.0119	0.436004
$760$	0	0
$761$	24.6177	0.892391	0.446195	−	0.894936i	$-0.352779\pi$
$761$	24.6177	0.892391	0.446195	+	0.894936i	$0.352779\pi$
$762$	0	0
$763$	70.1709	2.54036
$764$	0	0
$765$	−2.52401	−0.0912557
$766$	0	0
$767$	−0.572824	−0.0206835
$768$	0	0
$769$	−4.75269	−0.171386	−0.0856932	−	0.996322i	$-0.527310\pi$
$769$	−4.75269	−0.171386	−0.0856932	+	0.996322i	$0.527310\pi$
$770$	0	0
$771$	−18.6749	−0.672561
$772$	0	0
$773$	−37.4185	−1.34585	−0.672925	−	0.739711i	$-0.734962\pi$
$773$	−37.4185	−1.34585	−0.672925	+	0.739711i	$0.734962\pi$
$774$	0	0
$775$	−102.121	−3.66829
$776$	0	0
$777$	23.3287	0.836913
$778$	0	0
$779$	−0.829005	−0.0297022
$780$	0	0
$781$	4.37127	0.156416
$782$	0	0
$783$	22.2988	0.796895
$784$	0	0
$785$	56.0109	1.99912
$786$	0	0
$787$	−12.8790	−0.459087	−0.229543	−	0.973298i	$-0.573723\pi$
$787$	−12.8790	−0.459087	−0.229543	+	0.973298i	$0.573723\pi$
$788$	0	0
$789$	−34.4311	−1.22578
$790$	0	0
$791$	32.2096	1.14524
$792$	0	0
$793$	−11.0662	−0.392973
$794$	0	0
$795$	60.1382	2.13288
$796$	0	0
$797$	−19.6454	−0.695876	−0.347938	−	0.937518i	$-0.613118\pi$
$797$	−19.6454	−0.695876	−0.347938	+	0.937518i	$0.613118\pi$
$798$	0	0
$799$	−4.16195	−0.147239
$800$	0	0
$801$	2.11355	0.0746786
$802$	0	0
$803$	15.7526	0.555895
$804$	0	0
$805$	103.007	3.63051
$806$	0	0
$807$	−22.7444	−0.800639
$808$	0	0
$809$	−35.2645	−1.23983	−0.619917	−	0.784668i	$-0.712834\pi$
$809$	−35.2645	−1.23983	−0.619917	+	0.784668i	$0.712834\pi$
$810$	0	0
$811$	−24.1766	−0.848957	−0.424478	−	0.905438i	$-0.639543\pi$
$811$	−24.1766	−0.848957	−0.424478	+	0.905438i	$0.639543\pi$
$812$	0	0
$813$	−4.88585	−0.171354
$814$	0	0
$815$	88.4789	3.09928
$816$	0	0
$817$	0.373681	0.0130735
$818$	0	0
$819$	3.82626	0.133700
$820$	0	0
$821$	36.9365	1.28909	0.644546	−	0.764566i	$-0.277047\pi$
$821$	36.9365	1.28909	0.644546	+	0.764566i	$0.277047\pi$
$822$	0	0
$823$	42.2132	1.47146	0.735730	−	0.677275i	$-0.236839\pi$
$823$	42.2132	1.47146	0.735730	+	0.677275i	$0.236839\pi$
$824$	0	0
$825$	−30.8206	−1.07303
$826$	0	0
$827$	−40.1419	−1.39587	−0.697936	−	0.716160i	$-0.745898\pi$
$827$	−40.1419	−1.39587	−0.697936	+	0.716160i	$0.745898\pi$
$828$	0	0
$829$	−13.1004	−0.454994	−0.227497	−	0.973779i	$-0.573054\pi$
$829$	−13.1004	−0.454994	−0.227497	+	0.973779i	$0.573054\pi$
$830$	0	0
$831$	9.90169	0.343486
$832$	0	0
$833$	−6.26435	−0.217047
$834$	0	0
$835$	−88.0713	−3.04783
$836$	0	0
$837$	−29.4550	−1.01811
$838$	0	0
$839$	2.95822	0.102129	0.0510646	−	0.998695i	$-0.483739\pi$
$839$	2.95822	0.102129	0.0510646	+	0.998695i	$0.483739\pi$
$840$	0	0
$841$	−2.19571	−0.0757141
$842$	0	0
$843$	−39.5921	−1.36363
$844$	0	0
$845$	−51.3188	−1.76542
$846$	0	0
$847$	39.1484	1.34515
$848$	0	0
$849$	−11.0577	−0.379500
$850$	0	0
$851$	−17.5992	−0.603292
$852$	0	0
$853$	−13.4257	−0.459688	−0.229844	−	0.973227i	$-0.573822\pi$
$853$	−13.4257	−0.459688	−0.229844	+	0.973227i	$0.573822\pi$
$854$	0	0
$855$	0.358282	0.0122530
$856$	0	0
$857$	−23.1200	−0.789765	−0.394882	−	0.918732i	$-0.629215\pi$
$857$	−23.1200	−0.789765	−0.394882	+	0.918732i	$0.629215\pi$
$858$	0	0
$859$	13.8513	0.472600	0.236300	−	0.971680i	$-0.424065\pi$
$859$	13.8513	0.472600	0.236300	+	0.971680i	$0.424065\pi$
$860$	0	0
$861$	−62.6892	−2.13644
$862$	0	0
$863$	−22.8854	−0.779027	−0.389513	−	0.921021i	$-0.627357\pi$
$863$	−22.8854	−0.779027	−0.389513	+	0.921021i	$0.627357\pi$
$864$	0	0
$865$	−97.6933	−3.32167
$866$	0	0
$867$	32.0757	1.08935
$868$	0	0
$869$	−14.5338	−0.493024
$870$	0	0
$871$	−6.92563	−0.234666
$872$	0	0
$873$	2.03830	0.0689859
$874$	0	0
$875$	−175.801	−5.94316
$876$	0	0
$877$	−17.3848	−0.587042	−0.293521	−	0.955953i	$-0.594827\pi$
$877$	−17.3848	−0.587042	−0.293521	+	0.955953i	$0.594827\pi$
$878$	0	0
$879$	−14.5099	−0.489407
$880$	0	0
$881$	−24.5500	−0.827110	−0.413555	−	0.910479i	$-0.635713\pi$
$881$	−24.5500	−0.827110	−0.413555	+	0.910479i	$0.635713\pi$
$882$	0	0
$883$	51.5456	1.73465	0.867324	−	0.497744i	$-0.165838\pi$
$883$	51.5456	1.73465	0.867324	+	0.497744i	$0.165838\pi$
$884$	0	0
$885$	−4.05608	−0.136344
$886$	0	0
$887$	45.8783	1.54044	0.770221	−	0.637778i	$-0.220146\pi$
$887$	45.8783	1.54044	0.770221	+	0.637778i	$0.220146\pi$
$888$	0	0
$889$	−72.9148	−2.44548
$890$	0	0
$891$	−11.3927	−0.381671
$892$	0	0
$893$	0.590787	0.0197699
$894$	0	0
$895$	−16.9045	−0.565056
$896$	0	0
$897$	−13.8949	−0.463938
$898$	0	0
$899$	−35.4064	−1.18087
$900$	0	0
$901$	4.97482	0.165735
$902$	0	0
$903$	28.2577	0.940357
$904$	0	0
$905$	−81.2132	−2.69962
$906$	0	0
$907$	−29.6926	−0.985926	−0.492963	−	0.870050i	$-0.664086\pi$
$907$	−29.6926	−0.985926	−0.492963	+	0.870050i	$0.664086\pi$
$908$	0	0
$909$	−9.06192	−0.300565
$910$	0	0
$911$	−27.9664	−0.926569	−0.463285	−	0.886210i	$-0.653329\pi$
$911$	−27.9664	−0.926569	−0.463285	+	0.886210i	$0.653329\pi$
$912$	0	0
$913$	−8.54668	−0.282854
$914$	0	0
$915$	−78.3583	−2.59045
$916$	0	0
$917$	22.5435	0.744453
$918$	0	0
$919$	−4.64591	−0.153254	−0.0766271	−	0.997060i	$-0.524415\pi$
$919$	−4.64591	−0.153254	−0.0766271	+	0.997060i	$0.524415\pi$
$920$	0	0
$921$	58.5118	1.92803
$922$	0	0
$923$	−5.05652	−0.166437
$924$	0	0
$925$	45.1566	1.48474
$926$	0	0
$927$	−12.2071	−0.400933
$928$	0	0
$929$	−28.0845	−0.921422	−0.460711	−	0.887550i	$-0.652406\pi$
$929$	−28.0845	−0.921422	−0.460711	+	0.887550i	$0.652406\pi$
$930$	0	0
$931$	0.889221	0.0291430
$932$	0	0
$933$	−43.1336	−1.41213
$934$	0	0
$935$	−3.40327	−0.111299
$936$	0	0
$937$	5.92236	0.193475	0.0967376	−	0.995310i	$-0.469159\pi$
$937$	5.92236	0.193475	0.0967376	+	0.995310i	$0.469159\pi$
$938$	0	0
$939$	8.92367	0.291213
$940$	0	0
$941$	30.7233	1.00155	0.500776	−	0.865577i	$-0.333048\pi$
$941$	30.7233	1.00155	0.500776	+	0.865577i	$0.333048\pi$
$942$	0	0
$943$	47.2927	1.54006
$944$	0	0
$945$	−76.2323	−2.47984
$946$	0	0
$947$	0.0661907	0.00215091	0.00107545	−	0.999999i	$-0.499658\pi$
$947$	0.0661907	0.00215091	0.00107545	+	0.999999i	$0.499658\pi$
$948$	0	0
$949$	−18.2220	−0.591510
$950$	0	0
$951$	−44.8090	−1.45303
$952$	0	0
$953$	−19.5857	−0.634444	−0.317222	−	0.948351i	$-0.602750\pi$
$953$	−19.5857	−0.634444	−0.317222	+	0.948351i	$0.602750\pi$
$954$	0	0
$955$	−94.1288	−3.04594
$956$	0	0
$957$	−10.6858	−0.345423
$958$	0	0
$959$	2.99741	0.0967913
$960$	0	0
$961$	15.7692	0.508682
$962$	0	0
$963$	−3.63577	−0.117161
$964$	0	0
$965$	89.2781	2.87396
$966$	0	0
$967$	37.1440	1.19447	0.597235	−	0.802067i	$-0.296266\pi$
$967$	37.1440	1.19447	0.597235	+	0.802067i	$0.296266\pi$
$968$	0	0
$969$	0.142670	0.00458321
$970$	0	0
$971$	21.8701	0.701845	0.350922	−	0.936405i	$-0.385868\pi$
$971$	21.8701	0.701845	0.350922	+	0.936405i	$0.385868\pi$
$972$	0	0
$973$	57.0041	1.82747
$974$	0	0
$975$	35.6521	1.14178
$976$	0	0
$977$	40.1084	1.28318	0.641591	−	0.767047i	$-0.278275\pi$
$977$	40.1084	1.28318	0.641591	+	0.767047i	$0.278275\pi$
$978$	0	0
$979$	2.84982	0.0910807
$980$	0	0
$981$	−13.9237	−0.444548
$982$	0	0
$983$	−8.30777	−0.264977	−0.132488	−	0.991185i	$-0.542297\pi$
$983$	−8.30777	−0.264977	−0.132488	+	0.991185i	$0.542297\pi$
$984$	0	0
$985$	93.7012	2.98557
$986$	0	0
$987$	44.6751	1.42202
$988$	0	0
$989$	−21.3176	−0.677860
$990$	0	0
$991$	31.8595	1.01205	0.506026	−	0.862518i	$-0.331114\pi$
$991$	31.8595	1.01205	0.506026	+	0.862518i	$0.331114\pi$
$992$	0	0
$993$	40.1605	1.27445
$994$	0	0
$995$	95.4071	3.02461
$996$	0	0
$997$	−48.0219	−1.52087	−0.760434	−	0.649415i	$-0.775014\pi$
$997$	−48.0219	−1.52087	−0.760434	+	0.649415i	$0.775014\pi$
$998$	0	0
$999$	13.0247	0.412082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.18	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.18	✓	80	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.94593 q^{3} +4.46459 q^{5} -3.96440 q^{7} +0.786636 q^{9} +O(q^{10})\)	\(q-1.94593 q^{3} +4.46459 q^{5} -3.96440 q^{7} +0.786636 q^{9} +1.06067 q^{11} -1.22694 q^{13} -8.68777 q^{15} -0.718680 q^{17} +0.102016 q^{19} +7.71444 q^{21} -5.81977 q^{23} +14.9326 q^{25} +4.30705 q^{27} +5.17729 q^{29} -6.83880 q^{31} -2.06398 q^{33} -17.6994 q^{35} +3.02403 q^{37} +2.38754 q^{39} -8.12621 q^{41} +3.66296 q^{43} +3.51201 q^{45} +5.79111 q^{47} +8.71647 q^{49} +1.39850 q^{51} -6.92217 q^{53} +4.73544 q^{55} -0.198516 q^{57} +0.466873 q^{59} +9.01937 q^{61} -3.11854 q^{63} -5.47778 q^{65} +5.64464 q^{67} +11.3249 q^{69} +4.12125 q^{71} +14.8516 q^{73} -29.0577 q^{75} -4.20491 q^{77} -13.7025 q^{79} -10.7411 q^{81} -8.05784 q^{83} -3.20861 q^{85} -10.0746 q^{87} +2.68682 q^{89} +4.86408 q^{91} +13.3078 q^{93} +0.455461 q^{95} +2.59116 q^{97} +0.834358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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