LMFDB - Embedded newform 8001.2.a.z.1.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56027 q^{2} +0.434445 q^{4} -1.69832 q^{5} -1.00000 q^{7} -2.44269 q^{8} +O(q^{10})$	\(q+1.56027 q^{2} +0.434445 q^{4} -1.69832 q^{5} -1.00000 q^{7} -2.44269 q^{8} -2.64985 q^{10} -1.57750 q^{11} +5.37691 q^{13} -1.56027 q^{14} -4.68015 q^{16} +7.38180 q^{17} -0.683565 q^{19} -0.737828 q^{20} -2.46133 q^{22} -8.53565 q^{23} -2.11570 q^{25} +8.38943 q^{26} -0.434445 q^{28} +5.81934 q^{29} +7.97047 q^{31} -2.41692 q^{32} +11.5176 q^{34} +1.69832 q^{35} -11.4098 q^{37} -1.06655 q^{38} +4.14848 q^{40} +5.44263 q^{41} +4.36687 q^{43} -0.685336 q^{44} -13.3179 q^{46} -3.11843 q^{47} +1.00000 q^{49} -3.30106 q^{50} +2.33597 q^{52} +7.03105 q^{53} +2.67911 q^{55} +2.44269 q^{56} +9.07974 q^{58} -5.82826 q^{59} -6.57944 q^{61} +12.4361 q^{62} +5.58925 q^{64} -9.13173 q^{65} -6.68183 q^{67} +3.20698 q^{68} +2.64985 q^{70} +4.90031 q^{71} -7.91813 q^{73} -17.8024 q^{74} -0.296971 q^{76} +1.57750 q^{77} -15.7765 q^{79} +7.94841 q^{80} +8.49197 q^{82} +6.39375 q^{83} -12.5367 q^{85} +6.81350 q^{86} +3.85334 q^{88} +2.66271 q^{89} -5.37691 q^{91} -3.70827 q^{92} -4.86560 q^{94} +1.16092 q^{95} +5.37034 q^{97} +1.56027 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

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 $

$2$	1.56027	1.10328	0.551639	−	0.834083i	$-0.314003\pi$
$2$	1.56027	1.10328	0.551639	+	0.834083i	$0.314003\pi$
$3$	0	0
$3$	0	0
$4$	0.434445	0.217222
$5$	−1.69832	−0.759514	−0.379757	−	0.925086i	$-0.623992\pi$
$5$	−1.69832	−0.759514	−0.379757	+	0.925086i	$0.623992\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.44269	−0.863621
$9$	0	0
$10$	−2.64985	−0.837955
$11$	−1.57750	−0.475634	−0.237817	−	0.971310i	$-0.576432\pi$
$11$	−1.57750	−0.475634	−0.237817	+	0.971310i	$0.576432\pi$
$12$	0	0
$13$	5.37691	1.49129	0.745643	−	0.666346i	$-0.232143\pi$
$13$	5.37691	1.49129	0.745643	+	0.666346i	$0.232143\pi$
$14$	−1.56027	−0.417000
$15$	0	0
$16$	−4.68015	−1.17004
$17$	7.38180	1.79035	0.895174	−	0.445717i	$-0.147051\pi$
$17$	7.38180	1.79035	0.895174	+	0.445717i	$0.147051\pi$
$18$	0	0
$19$	−0.683565	−0.156821	−0.0784103	−	0.996921i	$-0.524984\pi$
$19$	−0.683565	−0.156821	−0.0784103	+	0.996921i	$0.524984\pi$
$20$	−0.737828	−0.164983
$21$	0	0
$22$	−2.46133	−0.524757
$23$	−8.53565	−1.77981	−0.889903	−	0.456149i	$-0.849228\pi$
$23$	−8.53565	−1.77981	−0.889903	+	0.456149i	$0.849228\pi$
$24$	0	0
$25$	−2.11570	−0.423139
$26$	8.38943	1.64530
$27$	0	0
$28$	−0.434445	−0.0821023
$29$	5.81934	1.08062	0.540312	−	0.841465i	$-0.318306\pi$
$29$	5.81934	1.08062	0.540312	+	0.841465i	$0.318306\pi$
$30$	0	0
$31$	7.97047	1.43154	0.715770	−	0.698336i	$-0.246076\pi$
$31$	7.97047	1.43154	0.715770	+	0.698336i	$0.246076\pi$
$32$	−2.41692	−0.427254
$33$	0	0
$34$	11.5176	1.97525
$35$	1.69832	0.287069
$36$	0	0
$37$	−11.4098	−1.87576	−0.937881	−	0.346956i	$-0.887215\pi$
$37$	−11.4098	−1.87576	−0.937881	+	0.346956i	$0.887215\pi$
$38$	−1.06655	−0.173017
$39$	0	0
$40$	4.14848	0.655932
$41$	5.44263	0.849995	0.424998	−	0.905194i	$-0.360275\pi$
$41$	5.44263	0.849995	0.424998	+	0.905194i	$0.360275\pi$
$42$	0	0
$43$	4.36687	0.665942	0.332971	−	0.942937i	$-0.391949\pi$
$43$	4.36687	0.665942	0.332971	+	0.942937i	$0.391949\pi$
$44$	−0.685336	−0.103318
$45$	0	0
$46$	−13.3179	−1.96362
$47$	−3.11843	−0.454870	−0.227435	−	0.973793i	$-0.573034\pi$
$47$	−3.11843	−0.454870	−0.227435	+	0.973793i	$0.573034\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.30106	−0.466840
$51$	0	0
$52$	2.33597	0.323941
$53$	7.03105	0.965789	0.482894	−	0.875679i	$-0.339585\pi$
$53$	7.03105	0.965789	0.482894	+	0.875679i	$0.339585\pi$
$54$	0	0
$55$	2.67911	0.361250
$56$	2.44269	0.326418
$57$	0	0
$58$	9.07974	1.19223
$59$	−5.82826	−0.758775	−0.379388	−	0.925238i	$-0.623865\pi$
$59$	−5.82826	−0.758775	−0.379388	+	0.925238i	$0.623865\pi$
$60$	0	0
$61$	−6.57944	−0.842411	−0.421206	−	0.906965i	$-0.638393\pi$
$61$	−6.57944	−0.842411	−0.421206	+	0.906965i	$0.638393\pi$
$62$	12.4361	1.57939
$63$	0	0
$64$	5.58925	0.698656
$65$	−9.13173	−1.13265
$66$	0	0
$67$	−6.68183	−0.816316	−0.408158	−	0.912911i	$-0.633829\pi$
$67$	−6.68183	−0.816316	−0.408158	+	0.912911i	$0.633829\pi$
$68$	3.20698	0.388904
$69$	0	0
$70$	2.64985	0.316717
$71$	4.90031	0.581559	0.290780	−	0.956790i	$-0.406085\pi$
$71$	4.90031	0.581559	0.290780	+	0.956790i	$0.406085\pi$
$72$	0	0
$73$	−7.91813	−0.926747	−0.463373	−	0.886163i	$-0.653361\pi$
$73$	−7.91813	−0.926747	−0.463373	+	0.886163i	$0.653361\pi$
$74$	−17.8024	−2.06949
$75$	0	0
$76$	−0.296971	−0.0340650
$77$	1.57750	0.179773
$78$	0	0
$79$	−15.7765	−1.77499	−0.887496	−	0.460815i	$-0.847557\pi$
$79$	−15.7765	−1.77499	−0.887496	+	0.460815i	$0.847557\pi$
$80$	7.94841	0.888659
$81$	0	0
$82$	8.49197	0.937781
$83$	6.39375	0.701806	0.350903	−	0.936412i	$-0.385875\pi$
$83$	6.39375	0.701806	0.350903	+	0.936412i	$0.385875\pi$
$84$	0	0
$85$	−12.5367	−1.35979
$86$	6.81350	0.734719
$87$	0	0
$88$	3.85334	0.410768
$89$	2.66271	0.282247	0.141123	−	0.989992i	$-0.454929\pi$
$89$	2.66271	0.282247	0.141123	+	0.989992i	$0.454929\pi$
$90$	0	0
$91$	−5.37691	−0.563653
$92$	−3.70827	−0.386614
$93$	0	0
$94$	−4.86560	−0.501848
$95$	1.16092	0.119107
$96$	0	0
$97$	5.37034	0.545276	0.272638	−	0.962117i	$-0.412104\pi$
$97$	5.37034	0.545276	0.272638	+	0.962117i	$0.412104\pi$
$98$	1.56027	0.157611
$99$	0	0
$100$	−0.919153	−0.0919153
$101$	−18.9810	−1.88868	−0.944342	−	0.328965i	$-0.893300\pi$
$101$	−18.9810	−1.88868	−0.944342	+	0.328965i	$0.893300\pi$
$102$	0	0
$103$	−6.36094	−0.626762	−0.313381	−	0.949627i	$-0.601462\pi$
$103$	−6.36094	−0.626762	−0.313381	+	0.949627i	$0.601462\pi$
$104$	−13.1341	−1.28791
$105$	0	0
$106$	10.9703	1.06553
$107$	1.79748	0.173769	0.0868844	−	0.996218i	$-0.472309\pi$
$107$	1.79748	0.173769	0.0868844	+	0.996218i	$0.472309\pi$
$108$	0	0
$109$	−16.5252	−1.58283	−0.791413	−	0.611281i	$-0.790654\pi$
$109$	−16.5252	−1.58283	−0.791413	+	0.611281i	$0.790654\pi$
$110$	4.18013	0.398560
$111$	0	0
$112$	4.68015	0.442232
$113$	−14.5665	−1.37030	−0.685151	−	0.728401i	$-0.740264\pi$
$113$	−14.5665	−1.37030	−0.685151	+	0.728401i	$0.740264\pi$
$114$	0	0
$115$	14.4963	1.35179
$116$	2.52818	0.234736
$117$	0	0
$118$	−9.09367	−0.837140
$119$	−7.38180	−0.676688
$120$	0	0
$121$	−8.51150	−0.773772
$122$	−10.2657	−0.929414
$123$	0	0
$124$	3.46273	0.310962
$125$	12.0848	1.08089
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	13.5546	1.19807
$129$	0	0
$130$	−14.2480	−1.24963
$131$	13.0083	1.13654	0.568271	−	0.822841i	$-0.307612\pi$
$131$	13.0083	1.13654	0.568271	+	0.822841i	$0.307612\pi$
$132$	0	0
$133$	0.683565	0.0592726
$134$	−10.4255	−0.900624
$135$	0	0
$136$	−18.0314	−1.54618
$137$	11.0957	0.947969	0.473984	−	0.880533i	$-0.342815\pi$
$137$	11.0957	0.947969	0.473984	+	0.880533i	$0.342815\pi$
$138$	0	0
$139$	−4.43688	−0.376331	−0.188165	−	0.982137i	$-0.560254\pi$
$139$	−4.43688	−0.376331	−0.188165	+	0.982137i	$0.560254\pi$
$140$	0.737828	0.0623578
$141$	0	0
$142$	7.64580	0.641622
$143$	−8.48207	−0.709306
$144$	0	0
$145$	−9.88312	−0.820748
$146$	−12.3544	−1.02246
$147$	0	0
$148$	−4.95694	−0.407458
$149$	0.577502	0.0473108	0.0236554	−	0.999720i	$-0.492470\pi$
$149$	0.577502	0.0473108	0.0236554	+	0.999720i	$0.492470\pi$
$150$	0	0
$151$	−7.20366	−0.586225	−0.293113	−	0.956078i	$-0.594691\pi$
$151$	−7.20366	−0.586225	−0.293113	+	0.956078i	$0.594691\pi$
$152$	1.66974	0.135434
$153$	0	0
$154$	2.46133	0.198339
$155$	−13.5364	−1.08727
$156$	0	0
$157$	−4.25760	−0.339793	−0.169897	−	0.985462i	$-0.554343\pi$
$157$	−4.25760	−0.339793	−0.169897	+	0.985462i	$0.554343\pi$
$158$	−24.6156	−1.95831
$159$	0	0
$160$	4.10471	0.324506
$161$	8.53565	0.672704
$162$	0	0
$163$	−13.2923	−1.04113	−0.520567	−	0.853821i	$-0.674279\pi$
$163$	−13.2923	−1.04113	−0.520567	+	0.853821i	$0.674279\pi$
$164$	2.36452	0.184638
$165$	0	0
$166$	9.97598	0.774287
$167$	−7.11696	−0.550727	−0.275363	−	0.961340i	$-0.588798\pi$
$167$	−7.11696	−0.550727	−0.275363	+	0.961340i	$0.588798\pi$
$168$	0	0
$169$	15.9111	1.22393
$170$	−19.5606	−1.50023
$171$	0	0
$172$	1.89716	0.144657
$173$	−13.8378	−1.05207	−0.526035	−	0.850463i	$-0.676322\pi$
$173$	−13.8378	−1.05207	−0.526035	+	0.850463i	$0.676322\pi$
$174$	0	0
$175$	2.11570	0.159932
$176$	7.38293	0.556509
$177$	0	0
$178$	4.15455	0.311397
$179$	−17.3579	−1.29739	−0.648697	−	0.761047i	$-0.724686\pi$
$179$	−17.3579	−1.29739	−0.648697	+	0.761047i	$0.724686\pi$
$180$	0	0
$181$	20.9700	1.55869	0.779345	−	0.626595i	$-0.215552\pi$
$181$	20.9700	1.55869	0.779345	+	0.626595i	$0.215552\pi$
$182$	−8.38943	−0.621866
$183$	0	0
$184$	20.8500	1.53708
$185$	19.3776	1.42467
$186$	0	0
$187$	−11.6448	−0.851551
$188$	−1.35479	−0.0988080
$189$	0	0
$190$	1.81134	0.131409
$191$	−12.2635	−0.887358	−0.443679	−	0.896186i	$-0.646327\pi$
$191$	−12.2635	−0.887358	−0.443679	+	0.896186i	$0.646327\pi$
$192$	0	0
$193$	−17.6311	−1.26912	−0.634558	−	0.772875i	$-0.718818\pi$
$193$	−17.6311	−1.26912	−0.634558	+	0.772875i	$0.718818\pi$
$194$	8.37919	0.601591
$195$	0	0
$196$	0.434445	0.0310318
$197$	−19.0024	−1.35387	−0.676933	−	0.736044i	$-0.736691\pi$
$197$	−19.0024	−1.35387	−0.676933	+	0.736044i	$0.736691\pi$
$198$	0	0
$199$	8.01293	0.568021	0.284011	−	0.958821i	$-0.408335\pi$
$199$	8.01293	0.568021	0.284011	+	0.958821i	$0.408335\pi$
$200$	5.16799	0.365432
$201$	0	0
$202$	−29.6156	−2.08374
$203$	−5.81934	−0.408437
$204$	0	0
$205$	−9.24334	−0.645583
$206$	−9.92479	−0.691493
$207$	0	0
$208$	−25.1647	−1.74486
$209$	1.07832	0.0745892
$210$	0	0
$211$	4.07668	0.280650	0.140325	−	0.990105i	$-0.455185\pi$
$211$	4.07668	0.280650	0.140325	+	0.990105i	$0.455185\pi$
$212$	3.05460	0.209791
$213$	0	0
$214$	2.80455	0.191715
$215$	−7.41636	−0.505792
$216$	0	0
$217$	−7.97047	−0.541071
$218$	−25.7838	−1.74630
$219$	0	0
$220$	1.16392	0.0784717
$221$	39.6912	2.66992
$222$	0	0
$223$	5.02212	0.336306	0.168153	−	0.985761i	$-0.446220\pi$
$223$	5.02212	0.336306	0.168153	+	0.985761i	$0.446220\pi$
$224$	2.41692	0.161487
$225$	0	0
$226$	−22.7277	−1.51183
$227$	5.67133	0.376420	0.188210	−	0.982129i	$-0.439732\pi$
$227$	5.67133	0.376420	0.188210	+	0.982129i	$0.439732\pi$
$228$	0	0
$229$	11.8322	0.781892	0.390946	−	0.920414i	$-0.372148\pi$
$229$	11.8322	0.781892	0.390946	+	0.920414i	$0.372148\pi$
$230$	22.6182	1.49140
$231$	0	0
$232$	−14.2148	−0.933250
$233$	22.7974	1.49351	0.746755	−	0.665099i	$-0.231611\pi$
$233$	22.7974	1.49351	0.746755	+	0.665099i	$0.231611\pi$
$234$	0	0
$235$	5.29611	0.345480
$236$	−2.53206	−0.164823
$237$	0	0
$238$	−11.5176	−0.746575
$239$	−10.1782	−0.658374	−0.329187	−	0.944265i	$-0.606775\pi$
$239$	−10.1782	−0.658374	−0.329187	+	0.944265i	$0.606775\pi$
$240$	0	0
$241$	−26.7920	−1.72583	−0.862913	−	0.505353i	$-0.831362\pi$
$241$	−26.7920	−1.72583	−0.862913	+	0.505353i	$0.831362\pi$
$242$	−13.2802	−0.853686
$243$	0	0
$244$	−2.85840	−0.182991
$245$	−1.69832	−0.108502
$246$	0	0
$247$	−3.67547	−0.233864
$248$	−19.4694	−1.23631
$249$	0	0
$250$	18.8555	1.19253
$251$	−20.6400	−1.30279	−0.651393	−	0.758741i	$-0.725815\pi$
$251$	−20.6400	−1.30279	−0.651393	+	0.758741i	$0.725815\pi$
$252$	0	0
$253$	13.4650	0.846537
$254$	−1.56027	−0.0979001
$255$	0	0
$256$	9.97031	0.623144
$257$	−12.6267	−0.787632	−0.393816	−	0.919189i	$-0.628845\pi$
$257$	−12.6267	−0.787632	−0.393816	+	0.919189i	$0.628845\pi$
$258$	0	0
$259$	11.4098	0.708972
$260$	−3.96723	−0.246037
$261$	0	0
$262$	20.2965	1.25392
$263$	1.56670	0.0966067	0.0483034	−	0.998833i	$-0.484619\pi$
$263$	1.56670	0.0966067	0.0483034	+	0.998833i	$0.484619\pi$
$264$	0	0
$265$	−11.9410	−0.733530
$266$	1.06655	0.0653942
$267$	0	0
$268$	−2.90289	−0.177322
$269$	21.7358	1.32526	0.662629	−	0.748948i	$-0.269441\pi$
$269$	21.7358	1.32526	0.662629	+	0.748948i	$0.269441\pi$
$270$	0	0
$271$	−15.5957	−0.947370	−0.473685	−	0.880694i	$-0.657076\pi$
$271$	−15.5957	−0.947370	−0.473685	+	0.880694i	$0.657076\pi$
$272$	−34.5479	−2.09477
$273$	0	0
$274$	17.3123	1.04587
$275$	3.33751	0.201259
$276$	0	0
$277$	9.01303	0.541541	0.270770	−	0.962644i	$-0.412722\pi$
$277$	9.01303	0.541541	0.270770	+	0.962644i	$0.412722\pi$
$278$	−6.92273	−0.415198
$279$	0	0
$280$	−4.14848	−0.247919
$281$	5.00890	0.298806	0.149403	−	0.988776i	$-0.452265\pi$
$281$	5.00890	0.298806	0.149403	+	0.988776i	$0.452265\pi$
$282$	0	0
$283$	1.98588	0.118048	0.0590242	−	0.998257i	$-0.481201\pi$
$283$	1.98588	0.118048	0.0590242	+	0.998257i	$0.481201\pi$
$284$	2.12891	0.126328
$285$	0	0
$286$	−13.2343	−0.782562
$287$	−5.44263	−0.321268
$288$	0	0
$289$	37.4909	2.20535
$290$	−15.4203	−0.905514
$291$	0	0
$292$	−3.43999	−0.201310
$293$	−13.9204	−0.813239	−0.406620	−	0.913598i	$-0.633293\pi$
$293$	−13.9204	−0.813239	−0.406620	+	0.913598i	$0.633293\pi$
$294$	0	0
$295$	9.89828	0.576300
$296$	27.8707	1.61995
$297$	0	0
$298$	0.901059	0.0521970
$299$	−45.8954	−2.65420
$300$	0	0
$301$	−4.36687	−0.251702
$302$	−11.2397	−0.646770
$303$	0	0
$304$	3.19919	0.183486
$305$	11.1740	0.639823
$306$	0	0
$307$	−28.6836	−1.63706	−0.818529	−	0.574465i	$-0.805210\pi$
$307$	−28.6836	−1.63706	−0.818529	+	0.574465i	$0.805210\pi$
$308$	0.685336	0.0390507
$309$	0	0
$310$	−21.1205	−1.19957
$311$	10.1277	0.574288	0.287144	−	0.957887i	$-0.407294\pi$
$311$	10.1277	0.574288	0.287144	+	0.957887i	$0.407294\pi$
$312$	0	0
$313$	−4.26249	−0.240930	−0.120465	−	0.992718i	$-0.538439\pi$
$313$	−4.26249	−0.240930	−0.120465	+	0.992718i	$0.538439\pi$
$314$	−6.64300	−0.374886
$315$	0	0
$316$	−6.85401	−0.385568
$317$	10.8781	0.610973	0.305487	−	0.952196i	$-0.401181\pi$
$317$	10.8781	0.610973	0.305487	+	0.952196i	$0.401181\pi$
$318$	0	0
$319$	−9.18000	−0.513981
$320$	−9.49236	−0.530639
$321$	0	0
$322$	13.3179	0.742179
$323$	−5.04594	−0.280764
$324$	0	0
$325$	−11.3759	−0.631021
$326$	−20.7396	−1.14866
$327$	0	0
$328$	−13.2946	−0.734074
$329$	3.11843	0.171925
$330$	0	0
$331$	−14.9176	−0.819946	−0.409973	−	0.912098i	$-0.634462\pi$
$331$	−14.9176	−0.819946	−0.409973	+	0.912098i	$0.634462\pi$
$332$	2.77773	0.152448
$333$	0	0
$334$	−11.1044	−0.607605
$335$	11.3479	0.620003
$336$	0	0
$337$	9.00961	0.490785	0.245392	−	0.969424i	$-0.421083\pi$
$337$	9.00961	0.490785	0.245392	+	0.969424i	$0.421083\pi$
$338$	24.8257	1.35034
$339$	0	0
$340$	−5.44649	−0.295378
$341$	−12.5734	−0.680889
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−10.6669	−0.575122
$345$	0	0
$346$	−21.5907	−1.16073
$347$	−2.34030	−0.125634	−0.0628169	−	0.998025i	$-0.520008\pi$
$347$	−2.34030	−0.125634	−0.0628169	+	0.998025i	$0.520008\pi$
$348$	0	0
$349$	−16.1455	−0.864249	−0.432125	−	0.901814i	$-0.642236\pi$
$349$	−16.1455	−0.864249	−0.432125	+	0.901814i	$0.642236\pi$
$350$	3.30106	0.176449
$351$	0	0
$352$	3.81268	0.203217
$353$	15.4517	0.822412	0.411206	−	0.911542i	$-0.365108\pi$
$353$	15.4517	0.822412	0.411206	+	0.911542i	$0.365108\pi$
$354$	0	0
$355$	−8.32231	−0.441702
$356$	1.15680	0.0613103
$357$	0	0
$358$	−27.0831	−1.43139
$359$	−22.3811	−1.18123	−0.590616	−	0.806953i	$-0.701115\pi$
$359$	−22.3811	−1.18123	−0.590616	+	0.806953i	$0.701115\pi$
$360$	0	0
$361$	−18.5327	−0.975407
$362$	32.7189	1.71967
$363$	0	0
$364$	−2.33597	−0.122438
$365$	13.4475	0.703877
$366$	0	0
$367$	−23.4592	−1.22456	−0.612280	−	0.790641i	$-0.709747\pi$
$367$	−23.4592	−1.22456	−0.612280	+	0.790641i	$0.709747\pi$
$368$	39.9481	2.08244
$369$	0	0
$370$	30.2343	1.57180
$371$	−7.03105	−0.365034
$372$	0	0
$373$	26.8125	1.38830	0.694150	−	0.719830i	$-0.255780\pi$
$373$	26.8125	1.38830	0.694150	+	0.719830i	$0.255780\pi$
$374$	−18.1690	−0.939497
$375$	0	0
$376$	7.61737	0.392836
$377$	31.2900	1.61152
$378$	0	0
$379$	19.2840	0.990551	0.495275	−	0.868736i	$-0.335067\pi$
$379$	19.2840	0.990551	0.495275	+	0.868736i	$0.335067\pi$
$380$	0.504354	0.0258728
$381$	0	0
$382$	−19.1344	−0.979002
$383$	−28.5258	−1.45760	−0.728799	−	0.684727i	$-0.759921\pi$
$383$	−28.5258	−1.45760	−0.728799	+	0.684727i	$0.759921\pi$
$384$	0	0
$385$	−2.67911	−0.136540
$386$	−27.5093	−1.40019
$387$	0	0
$388$	2.33312	0.118446
$389$	−27.3537	−1.38689	−0.693443	−	0.720511i	$-0.743907\pi$
$389$	−27.3537	−1.38689	−0.693443	+	0.720511i	$0.743907\pi$
$390$	0	0
$391$	−63.0084	−3.18647
$392$	−2.44269	−0.123374
$393$	0	0
$394$	−29.6489	−1.49369
$395$	26.7936	1.34813
$396$	0	0
$397$	−16.9610	−0.851250	−0.425625	−	0.904900i	$-0.639946\pi$
$397$	−16.9610	−0.851250	−0.425625	+	0.904900i	$0.639946\pi$
$398$	12.5023	0.626685
$399$	0	0
$400$	9.90177	0.495088
$401$	17.7166	0.884723	0.442362	−	0.896837i	$-0.354141\pi$
$401$	17.7166	0.884723	0.442362	+	0.896837i	$0.354141\pi$
$402$	0	0
$403$	42.8565	2.13483
$404$	−8.24621	−0.410264
$405$	0	0
$406$	−9.07974	−0.450620
$407$	17.9990	0.892177
$408$	0	0
$409$	−1.61511	−0.0798619	−0.0399310	−	0.999202i	$-0.512714\pi$
$409$	−1.61511	−0.0798619	−0.0399310	+	0.999202i	$0.512714\pi$
$410$	−14.4221	−0.712257
$411$	0	0
$412$	−2.76348	−0.136147
$413$	5.82826	0.286790
$414$	0	0
$415$	−10.8587	−0.533031
$416$	−12.9955	−0.637158
$417$	0	0
$418$	1.68248	0.0822927
$419$	15.8383	0.773753	0.386876	−	0.922132i	$-0.373554\pi$
$419$	15.8383	0.773753	0.386876	+	0.922132i	$0.373554\pi$
$420$	0	0
$421$	−15.1541	−0.738564	−0.369282	−	0.929317i	$-0.620396\pi$
$421$	−15.1541	−0.738564	−0.369282	+	0.929317i	$0.620396\pi$
$422$	6.36072	0.309635
$423$	0	0
$424$	−17.1747	−0.834076
$425$	−15.6176	−0.757567
$426$	0	0
$427$	6.57944	0.318401
$428$	0.780905	0.0377465
$429$	0	0
$430$	−11.5715	−0.558029
$431$	−28.5503	−1.37522	−0.687610	−	0.726080i	$-0.741340\pi$
$431$	−28.5503	−1.37522	−0.687610	+	0.726080i	$0.741340\pi$
$432$	0	0
$433$	32.9912	1.58546	0.792729	−	0.609575i	$-0.208660\pi$
$433$	32.9912	1.58546	0.792729	+	0.609575i	$0.208660\pi$
$434$	−12.4361	−0.596952
$435$	0	0
$436$	−7.17928	−0.343825
$437$	5.83468	0.279110
$438$	0	0
$439$	−20.1904	−0.963636	−0.481818	−	0.876271i	$-0.660023\pi$
$439$	−20.1904	−0.963636	−0.481818	+	0.876271i	$0.660023\pi$
$440$	−6.54422	−0.311984
$441$	0	0
$442$	61.9290	2.94566
$443$	40.8990	1.94317	0.971584	−	0.236694i	$-0.0760640\pi$
$443$	40.8990	1.94317	0.971584	+	0.236694i	$0.0760640\pi$
$444$	0	0
$445$	−4.52215	−0.214370
$446$	7.83587	0.371039
$447$	0	0
$448$	−5.58925	−0.264067
$449$	−18.4836	−0.872293	−0.436146	−	0.899876i	$-0.643657\pi$
$449$	−18.4836	−0.872293	−0.436146	+	0.899876i	$0.643657\pi$
$450$	0	0
$451$	−8.58574	−0.404287
$452$	−6.32835	−0.297660
$453$	0	0
$454$	8.84882	0.415296
$455$	9.13173	0.428102
$456$	0	0
$457$	4.17547	0.195320	0.0976601	−	0.995220i	$-0.468864\pi$
$457$	4.17547	0.195320	0.0976601	+	0.995220i	$0.468864\pi$
$458$	18.4614	0.862645
$459$	0	0
$460$	6.29784	0.293638
$461$	27.4411	1.27806	0.639029	−	0.769182i	$-0.279336\pi$
$461$	27.4411	1.27806	0.639029	+	0.769182i	$0.279336\pi$
$462$	0	0
$463$	−28.4925	−1.32416	−0.662078	−	0.749435i	$-0.730325\pi$
$463$	−28.4925	−1.32416	−0.662078	+	0.749435i	$0.730325\pi$
$464$	−27.2354	−1.26437
$465$	0	0
$466$	35.5702	1.64776
$467$	−31.9965	−1.48062	−0.740311	−	0.672265i	$-0.765322\pi$
$467$	−31.9965	−1.48062	−0.740311	+	0.672265i	$0.765322\pi$
$468$	0	0
$469$	6.68183	0.308538
$470$	8.26336	0.381161
$471$	0	0
$472$	14.2366	0.655295
$473$	−6.88874	−0.316745
$474$	0	0
$475$	1.44622	0.0663570
$476$	−3.20698	−0.146992
$477$	0	0
$478$	−15.8808	−0.726369
$479$	0.125524	0.00573535	0.00286767	−	0.999996i	$-0.499087\pi$
$479$	0.125524	0.00573535	0.00286767	+	0.999996i	$0.499087\pi$
$480$	0	0
$481$	−61.3495	−2.79730
$482$	−41.8028	−1.90406
$483$	0	0
$484$	−3.69777	−0.168081
$485$	−9.12058	−0.414144
$486$	0	0
$487$	23.4115	1.06088	0.530439	−	0.847723i	$-0.322027\pi$
$487$	23.4115	1.06088	0.530439	+	0.847723i	$0.322027\pi$
$488$	16.0715	0.727524
$489$	0	0
$490$	−2.64985	−0.119708
$491$	20.9037	0.943369	0.471685	−	0.881767i	$-0.343646\pi$
$491$	20.9037	0.943369	0.471685	+	0.881767i	$0.343646\pi$
$492$	0	0
$493$	42.9572	1.93469
$494$	−5.73472	−0.258017
$495$	0	0
$496$	−37.3030	−1.67495
$497$	−4.90031	−0.219809
$498$	0	0
$499$	5.86107	0.262377	0.131189	−	0.991357i	$-0.458121\pi$
$499$	5.86107	0.262377	0.131189	+	0.991357i	$0.458121\pi$
$500$	5.25016	0.234794
$501$	0	0
$502$	−32.2040	−1.43733
$503$	30.4405	1.35728	0.678638	−	0.734473i	$-0.262571\pi$
$503$	30.4405	1.35728	0.678638	+	0.734473i	$0.262571\pi$
$504$	0	0
$505$	32.2360	1.43448
$506$	21.0090	0.933965
$507$	0	0
$508$	−0.434445	−0.0192754
$509$	−10.0065	−0.443531	−0.221766	−	0.975100i	$-0.571182\pi$
$509$	−10.0065	−0.443531	−0.221766	+	0.975100i	$0.571182\pi$
$510$	0	0
$511$	7.91813	0.350277
$512$	−11.5528	−0.510565
$513$	0	0
$514$	−19.7011	−0.868977
$515$	10.8029	0.476034
$516$	0	0
$517$	4.91933	0.216352
$518$	17.8024	0.782193
$519$	0	0
$520$	22.3060	0.978182
$521$	9.37377	0.410672	0.205336	−	0.978692i	$-0.434171\pi$
$521$	9.37377	0.410672	0.205336	+	0.978692i	$0.434171\pi$
$522$	0	0
$523$	42.7780	1.87055	0.935277	−	0.353918i	$-0.115151\pi$
$523$	42.7780	1.87055	0.935277	+	0.353918i	$0.115151\pi$
$524$	5.65140	0.246882
$525$	0	0
$526$	2.44447	0.106584
$527$	58.8364	2.56295
$528$	0	0
$529$	49.8574	2.16771
$530$	−18.6312	−0.809287
$531$	0	0
$532$	0.296971	0.0128753
$533$	29.2645	1.26759
$534$	0	0
$535$	−3.05270	−0.131980
$536$	16.3217	0.704988
$537$	0	0
$538$	33.9138	1.46213
$539$	−1.57750	−0.0679477
$540$	0	0
$541$	31.6416	1.36038	0.680189	−	0.733037i	$-0.261898\pi$
$541$	31.6416	1.36038	0.680189	+	0.733037i	$0.261898\pi$
$542$	−24.3335	−1.04521
$543$	0	0
$544$	−17.8412	−0.764934
$545$	28.0651	1.20218
$546$	0	0
$547$	−23.6839	−1.01265	−0.506326	−	0.862342i	$-0.668997\pi$
$547$	−23.6839	−1.01265	−0.506326	+	0.862342i	$0.668997\pi$
$548$	4.82046	0.205920
$549$	0	0
$550$	5.20742	0.222045
$551$	−3.97790	−0.169464
$552$	0	0
$553$	15.7765	0.670884
$554$	14.0628	0.597470
$555$	0	0
$556$	−1.92758	−0.0817475
$557$	20.7846	0.880671	0.440336	−	0.897833i	$-0.354859\pi$
$557$	20.7846	0.880671	0.440336	+	0.897833i	$0.354859\pi$
$558$	0	0
$559$	23.4803	0.993109
$560$	−7.94841	−0.335881
$561$	0	0
$562$	7.81523	0.329666
$563$	12.9904	0.547479	0.273739	−	0.961804i	$-0.411739\pi$
$563$	12.9904	0.547479	0.273739	+	0.961804i	$0.411739\pi$
$564$	0	0
$565$	24.7387	1.04076
$566$	3.09851	0.130240
$567$	0	0
$568$	−11.9699	−0.502247
$569$	29.1174	1.22067	0.610333	−	0.792145i	$-0.291036\pi$
$569$	29.1174	1.22067	0.610333	+	0.792145i	$0.291036\pi$
$570$	0	0
$571$	−21.9240	−0.917491	−0.458745	−	0.888568i	$-0.651701\pi$
$571$	−21.9240	−0.917491	−0.458745	+	0.888568i	$0.651701\pi$
$572$	−3.68499	−0.154077
$573$	0	0
$574$	−8.49197	−0.354448
$575$	18.0588	0.753106
$576$	0	0
$577$	15.9248	0.662958	0.331479	−	0.943463i	$-0.392452\pi$
$577$	15.9248	0.662958	0.331479	+	0.943463i	$0.392452\pi$
$578$	58.4960	2.43311
$579$	0	0
$580$	−4.29367	−0.178285
$581$	−6.39375	−0.265258
$582$	0	0
$583$	−11.0915	−0.459362
$584$	19.3415	0.800358
$585$	0	0
$586$	−21.7196	−0.897229
$587$	1.21000	0.0499420	0.0249710	−	0.999688i	$-0.492051\pi$
$587$	1.21000	0.0499420	0.0249710	+	0.999688i	$0.492051\pi$
$588$	0	0
$589$	−5.44834	−0.224495
$590$	15.4440	0.635819
$591$	0	0
$592$	53.3996	2.19471
$593$	−18.1805	−0.746585	−0.373292	−	0.927714i	$-0.621771\pi$
$593$	−18.1805	−0.746585	−0.373292	+	0.927714i	$0.621771\pi$
$594$	0	0
$595$	12.5367	0.513954
$596$	0.250893	0.0102770
$597$	0	0
$598$	−71.6093	−2.92832
$599$	−3.41903	−0.139698	−0.0698489	−	0.997558i	$-0.522252\pi$
$599$	−3.41903	−0.139698	−0.0698489	+	0.997558i	$0.522252\pi$
$600$	0	0
$601$	20.4048	0.832328	0.416164	−	0.909290i	$-0.363374\pi$
$601$	20.4048	0.832328	0.416164	+	0.909290i	$0.363374\pi$
$602$	−6.81350	−0.277698
$603$	0	0
$604$	−3.12959	−0.127341
$605$	14.4553	0.587691
$606$	0	0
$607$	10.0628	0.408435	0.204217	−	0.978926i	$-0.434535\pi$
$607$	10.0628	0.408435	0.204217	+	0.978926i	$0.434535\pi$
$608$	1.65212	0.0670023
$609$	0	0
$610$	17.4345	0.705902
$611$	−16.7675	−0.678341
$612$	0	0
$613$	0.173649	0.00701361	0.00350680	−	0.999994i	$-0.498884\pi$
$613$	0.173649	0.00701361	0.00350680	+	0.999994i	$0.498884\pi$
$614$	−44.7542	−1.80613
$615$	0	0
$616$	−3.85334	−0.155256
$617$	−19.8820	−0.800421	−0.400210	−	0.916423i	$-0.631063\pi$
$617$	−19.8820	−0.800421	−0.400210	+	0.916423i	$0.631063\pi$
$618$	0	0
$619$	−41.5677	−1.67075	−0.835373	−	0.549684i	$-0.814748\pi$
$619$	−41.5677	−1.67075	−0.835373	+	0.549684i	$0.814748\pi$
$620$	−5.88084	−0.236180
$621$	0	0
$622$	15.8019	0.633599
$623$	−2.66271	−0.106679
$624$	0	0
$625$	−9.94535	−0.397814
$626$	−6.65064	−0.265813
$627$	0	0
$628$	−1.84969	−0.0738107
$629$	−84.2250	−3.35827
$630$	0	0
$631$	21.7909	0.867483	0.433742	−	0.901037i	$-0.357193\pi$
$631$	21.7909	0.867483	0.433742	+	0.901037i	$0.357193\pi$
$632$	38.5370	1.53292
$633$	0	0
$634$	16.9727	0.674073
$635$	1.69832	0.0673959
$636$	0	0
$637$	5.37691	0.213041
$638$	−14.3233	−0.567064
$639$	0	0
$640$	−23.0201	−0.909948
$641$	14.0620	0.555415	0.277707	−	0.960666i	$-0.410425\pi$
$641$	14.0620	0.555415	0.277707	+	0.960666i	$0.410425\pi$
$642$	0	0
$643$	−3.13529	−0.123644	−0.0618219	−	0.998087i	$-0.519691\pi$
$643$	−3.13529	−0.123644	−0.0618219	+	0.998087i	$0.519691\pi$
$644$	3.70827	0.146126
$645$	0	0
$646$	−7.87303	−0.309760
$647$	31.1920	1.22628	0.613142	−	0.789973i	$-0.289906\pi$
$647$	31.1920	1.22628	0.613142	+	0.789973i	$0.289906\pi$
$648$	0	0
$649$	9.19408	0.360899
$650$	−17.7495	−0.696192
$651$	0	0
$652$	−5.77477	−0.226157
$653$	1.25769	0.0492171	0.0246085	−	0.999697i	$-0.492166\pi$
$653$	1.25769	0.0492171	0.0246085	+	0.999697i	$0.492166\pi$
$654$	0	0
$655$	−22.0923	−0.863219
$656$	−25.4723	−0.994526
$657$	0	0
$658$	4.86560	0.189681
$659$	25.2125	0.982140	0.491070	−	0.871120i	$-0.336606\pi$
$659$	25.2125	0.982140	0.491070	+	0.871120i	$0.336606\pi$
$660$	0	0
$661$	42.0623	1.63603	0.818017	−	0.575193i	$-0.195073\pi$
$661$	42.0623	1.63603	0.818017	+	0.575193i	$0.195073\pi$
$662$	−23.2755	−0.904628
$663$	0	0
$664$	−15.6180	−0.606094
$665$	−1.16092	−0.0450184
$666$	0	0
$667$	−49.6719	−1.92330
$668$	−3.09193	−0.119630
$669$	0	0
$670$	17.7058	0.684036
$671$	10.3791	0.400679
$672$	0	0
$673$	−25.1327	−0.968793	−0.484396	−	0.874849i	$-0.660961\pi$
$673$	−25.1327	−0.968793	−0.484396	+	0.874849i	$0.660961\pi$
$674$	14.0574	0.541472
$675$	0	0
$676$	6.91250	0.265865
$677$	−18.1849	−0.698903	−0.349452	−	0.936954i	$-0.613632\pi$
$677$	−18.1849	−0.698903	−0.349452	+	0.936954i	$0.613632\pi$
$678$	0	0
$679$	−5.37034	−0.206095
$680$	30.6232	1.17435
$681$	0	0
$682$	−19.6179	−0.751210
$683$	−1.19845	−0.0458573	−0.0229286	−	0.999737i	$-0.507299\pi$
$683$	−1.19845	−0.0458573	−0.0229286	+	0.999737i	$0.507299\pi$
$684$	0	0
$685$	−18.8441	−0.719995
$686$	−1.56027	−0.0595714
$687$	0	0
$688$	−20.4376	−0.779176
$689$	37.8053	1.44027
$690$	0	0
$691$	−35.3781	−1.34585	−0.672924	−	0.739712i	$-0.734962\pi$
$691$	−35.3781	−1.34585	−0.672924	+	0.739712i	$0.734962\pi$
$692$	−6.01177	−0.228533
$693$	0	0
$694$	−3.65150	−0.138609
$695$	7.53525	0.285828
$696$	0	0
$697$	40.1763	1.52179
$698$	−25.1914	−0.953507
$699$	0	0
$700$	0.919153	0.0347407
$701$	21.6244	0.816740	0.408370	−	0.912816i	$-0.366097\pi$
$701$	21.6244	0.816740	0.408370	+	0.912816i	$0.366097\pi$
$702$	0	0
$703$	7.79936	0.294158
$704$	−8.81704	−0.332305
$705$	0	0
$706$	24.1089	0.907349
$707$	18.9810	0.713855
$708$	0	0
$709$	−20.0892	−0.754467	−0.377233	−	0.926118i	$-0.623125\pi$
$709$	−20.0892	−0.754467	−0.377233	+	0.926118i	$0.623125\pi$
$710$	−12.9850	−0.487320
$711$	0	0
$712$	−6.50418	−0.243754
$713$	−68.0332	−2.54786
$714$	0	0
$715$	14.4053	0.538728
$716$	−7.54107	−0.281823
$717$	0	0
$718$	−34.9206	−1.30323
$719$	1.69180	0.0630936	0.0315468	−	0.999502i	$-0.489957\pi$
$719$	1.69180	0.0630936	0.0315468	+	0.999502i	$0.489957\pi$
$720$	0	0
$721$	6.36094	0.236894
$722$	−28.9161	−1.07615
$723$	0	0
$724$	9.11032	0.338582
$725$	−12.3119	−0.457254
$726$	0	0
$727$	−19.4327	−0.720718	−0.360359	−	0.932814i	$-0.617346\pi$
$727$	−19.4327	−0.720718	−0.360359	+	0.932814i	$0.617346\pi$
$728$	13.1341	0.486783
$729$	0	0
$730$	20.9818	0.776572
$731$	32.2354	1.19227
$732$	0	0
$733$	1.18340	0.0437098	0.0218549	−	0.999761i	$-0.493043\pi$
$733$	1.18340	0.0437098	0.0218549	+	0.999761i	$0.493043\pi$
$734$	−36.6027	−1.35103
$735$	0	0
$736$	20.6300	0.760430
$737$	10.5406	0.388268
$738$	0	0
$739$	−13.1398	−0.483355	−0.241677	−	0.970357i	$-0.577698\pi$
$739$	−13.1398	−0.483355	−0.241677	+	0.970357i	$0.577698\pi$
$740$	8.41848	0.309470
$741$	0	0
$742$	−10.9703	−0.402734
$743$	49.1903	1.80462	0.902308	−	0.431091i	$-0.141871\pi$
$743$	49.1903	1.80462	0.902308	+	0.431091i	$0.141871\pi$
$744$	0	0
$745$	−0.980785	−0.0359332
$746$	41.8348	1.53168
$747$	0	0
$748$	−5.05901	−0.184976
$749$	−1.79748	−0.0656785
$750$	0	0
$751$	3.63243	0.132549	0.0662746	−	0.997801i	$-0.478889\pi$
$751$	3.63243	0.132549	0.0662746	+	0.997801i	$0.478889\pi$
$752$	14.5947	0.532215
$753$	0	0
$754$	48.8209	1.77795
$755$	12.2341	0.445246
$756$	0	0
$757$	−46.0208	−1.67265	−0.836327	−	0.548231i	$-0.815301\pi$
$757$	−46.0208	−1.67265	−0.836327	+	0.548231i	$0.815301\pi$
$758$	30.0882	1.09285
$759$	0	0
$760$	−2.83576	−0.102864
$761$	6.59912	0.239218	0.119609	−	0.992821i	$-0.461836\pi$
$761$	6.59912	0.239218	0.119609	+	0.992821i	$0.461836\pi$
$762$	0	0
$763$	16.5252	0.598252
$764$	−5.32782	−0.192754
$765$	0	0
$766$	−44.5079	−1.60814
$767$	−31.3380	−1.13155
$768$	0	0
$769$	18.9717	0.684138	0.342069	−	0.939675i	$-0.388872\pi$
$769$	18.9717	0.684138	0.342069	+	0.939675i	$0.388872\pi$
$770$	−4.18013	−0.150641
$771$	0	0
$772$	−7.65975	−0.275681
$773$	21.5176	0.773934	0.386967	−	0.922094i	$-0.373523\pi$
$773$	21.5176	0.773934	0.386967	+	0.922094i	$0.373523\pi$
$774$	0	0
$775$	−16.8631	−0.605741
$776$	−13.1181	−0.470912
$777$	0	0
$778$	−42.6792	−1.53012
$779$	−3.72039	−0.133297
$780$	0	0
$781$	−7.73023	−0.276609
$782$	−98.3102	−3.51557
$783$	0	0
$784$	−4.68015	−0.167148
$785$	7.23078	0.258078
$786$	0	0
$787$	−19.9237	−0.710205	−0.355102	−	0.934827i	$-0.615554\pi$
$787$	−19.9237	−0.710205	−0.355102	+	0.934827i	$0.615554\pi$
$788$	−8.25550	−0.294090
$789$	0	0
$790$	41.8052	1.48736
$791$	14.5665	0.517926
$792$	0	0
$793$	−35.3770	−1.25628
$794$	−26.4638	−0.939166
$795$	0	0
$796$	3.48117	0.123387
$797$	8.77923	0.310976	0.155488	−	0.987838i	$-0.450305\pi$
$797$	8.77923	0.310976	0.155488	+	0.987838i	$0.450305\pi$
$798$	0	0
$799$	−23.0196	−0.814376
$800$	5.11346	0.180788
$801$	0	0
$802$	27.6426	0.976096
$803$	12.4908	0.440792
$804$	0	0
$805$	−14.4963	−0.510928
$806$	66.8677	2.35532
$807$	0	0
$808$	46.3648	1.63111
$809$	−10.0995	−0.355080	−0.177540	−	0.984114i	$-0.556814\pi$
$809$	−10.0995	−0.355080	−0.177540	+	0.984114i	$0.556814\pi$
$810$	0	0
$811$	−17.2459	−0.605584	−0.302792	−	0.953057i	$-0.597919\pi$
$811$	−17.2459	−0.605584	−0.302792	+	0.953057i	$0.597919\pi$
$812$	−2.52818	−0.0887217
$813$	0	0
$814$	28.0833	0.984319
$815$	22.5746	0.790755
$816$	0	0
$817$	−2.98504	−0.104433
$818$	−2.52000	−0.0881099
$819$	0	0
$820$	−4.01572	−0.140235
$821$	−2.22336	−0.0775958	−0.0387979	−	0.999247i	$-0.512353\pi$
$821$	−2.22336	−0.0775958	−0.0387979	+	0.999247i	$0.512353\pi$
$822$	0	0
$823$	−49.5127	−1.72590	−0.862951	−	0.505287i	$-0.831387\pi$
$823$	−49.5127	−1.72590	−0.862951	+	0.505287i	$0.831387\pi$
$824$	15.5378	0.541285
$825$	0	0
$826$	9.09367	0.316409
$827$	−33.2176	−1.15509	−0.577544	−	0.816360i	$-0.695989\pi$
$827$	−33.2176	−1.15509	−0.577544	+	0.816360i	$0.695989\pi$
$828$	0	0
$829$	−56.8197	−1.97343	−0.986715	−	0.162464i	$-0.948056\pi$
$829$	−56.8197	−1.97343	−0.986715	+	0.162464i	$0.948056\pi$
$830$	−16.9425	−0.588081
$831$	0	0
$832$	30.0529	1.04190
$833$	7.38180	0.255764
$834$	0	0
$835$	12.0869	0.418285
$836$	0.468472	0.0162025
$837$	0	0
$838$	24.7121	0.853664
$839$	−11.2187	−0.387312	−0.193656	−	0.981070i	$-0.562035\pi$
$839$	−11.2187	−0.387312	−0.193656	+	0.981070i	$0.562035\pi$
$840$	0	0
$841$	4.86469	0.167748
$842$	−23.6445	−0.814842
$843$	0	0
$844$	1.77109	0.0609635
$845$	−27.0222	−0.929593
$846$	0	0
$847$	8.51150	0.292458
$848$	−32.9063	−1.13001
$849$	0	0
$850$	−24.3677	−0.835807
$851$	97.3903	3.33850
$852$	0	0
$853$	−15.1859	−0.519956	−0.259978	−	0.965614i	$-0.583715\pi$
$853$	−15.1859	−0.519956	−0.259978	+	0.965614i	$0.583715\pi$
$854$	10.2657	0.351285
$855$	0	0
$856$	−4.39068	−0.150071
$857$	−7.84113	−0.267848	−0.133924	−	0.990992i	$-0.542758\pi$
$857$	−7.84113	−0.267848	−0.133924	+	0.990992i	$0.542758\pi$
$858$	0	0
$859$	5.04032	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$859$	5.04032	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$860$	−3.22200	−0.109869
$861$	0	0
$862$	−44.5462	−1.51725
$863$	22.2053	0.755876	0.377938	−	0.925831i	$-0.376633\pi$
$863$	22.2053	0.755876	0.377938	+	0.925831i	$0.376633\pi$
$864$	0	0
$865$	23.5011	0.799061
$866$	51.4752	1.74920
$867$	0	0
$868$	−3.46273	−0.117533
$869$	24.8874	0.844247
$870$	0	0
$871$	−35.9276	−1.21736
$872$	40.3659	1.36696
$873$	0	0
$874$	9.10368	0.307936
$875$	−12.0848	−0.408539
$876$	0	0
$877$	41.0971	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$877$	41.0971	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$878$	−31.5025	−1.06316
$879$	0	0
$880$	−12.5386	−0.422676
$881$	41.5727	1.40062	0.700309	−	0.713840i	$-0.253046\pi$
$881$	41.5727	1.40062	0.700309	+	0.713840i	$0.253046\pi$
$882$	0	0
$883$	29.2046	0.982812	0.491406	−	0.870931i	$-0.336483\pi$
$883$	29.2046	0.982812	0.491406	+	0.870931i	$0.336483\pi$
$884$	17.2436	0.579966
$885$	0	0
$886$	63.8135	2.14385
$887$	24.1854	0.812067	0.406033	−	0.913858i	$-0.366912\pi$
$887$	24.1854	0.812067	0.406033	+	0.913858i	$0.366912\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−7.05577	−0.236510
$891$	0	0
$892$	2.18183	0.0730532
$893$	2.13165	0.0713330
$894$	0	0
$895$	29.4794	0.985388
$896$	−13.5546	−0.452827
$897$	0	0
$898$	−28.8393	−0.962381
$899$	46.3829	1.54696
$900$	0	0
$901$	51.9018	1.72910
$902$	−13.3961	−0.446041
$903$	0	0
$904$	35.5815	1.18342
$905$	−35.6139	−1.18385
$906$	0	0
$907$	−49.8040	−1.65371	−0.826857	−	0.562412i	$-0.809874\pi$
$907$	−49.8040	−1.65371	−0.826857	+	0.562412i	$0.809874\pi$
$908$	2.46388	0.0817668
$909$	0	0
$910$	14.2480	0.472316
$911$	−54.6229	−1.80974	−0.904869	−	0.425690i	$-0.860031\pi$
$911$	−54.6229	−1.80974	−0.904869	+	0.425690i	$0.860031\pi$
$912$	0	0
$913$	−10.0861	−0.333803
$914$	6.51486	0.215492
$915$	0	0
$916$	5.14043	0.169845
$917$	−13.0083	−0.429573
$918$	0	0
$919$	−27.0540	−0.892428	−0.446214	−	0.894926i	$-0.647228\pi$
$919$	−27.0540	−0.892428	−0.446214	+	0.894926i	$0.647228\pi$
$920$	−35.4100	−1.16743
$921$	0	0
$922$	42.8155	1.41005
$923$	26.3485	0.867271
$924$	0	0
$925$	24.1397	0.793709
$926$	−44.4559	−1.46091
$927$	0	0
$928$	−14.0649	−0.461701
$929$	−17.5180	−0.574748	−0.287374	−	0.957818i	$-0.592782\pi$
$929$	−17.5180	−0.574748	−0.287374	+	0.957818i	$0.592782\pi$
$930$	0	0
$931$	−0.683565	−0.0224030
$932$	9.90423	0.324424
$933$	0	0
$934$	−49.9232	−1.63354
$935$	19.7766	0.646764
$936$	0	0
$937$	−52.0412	−1.70011	−0.850056	−	0.526692i	$-0.823432\pi$
$937$	−52.0412	−1.70011	−0.850056	+	0.526692i	$0.823432\pi$
$938$	10.4255	0.340404
$939$	0	0
$940$	2.30087	0.0750460
$941$	10.3014	0.335816	0.167908	−	0.985803i	$-0.446299\pi$
$941$	10.3014	0.335816	0.167908	+	0.985803i	$0.446299\pi$
$942$	0	0
$943$	−46.4564	−1.51283
$944$	27.2771	0.887795
$945$	0	0
$946$	−10.7483	−0.349457
$947$	31.0642	1.00945	0.504726	−	0.863280i	$-0.331594\pi$
$947$	31.0642	1.00945	0.504726	+	0.863280i	$0.331594\pi$
$948$	0	0
$949$	−42.5750	−1.38204
$950$	2.25649	0.0732102
$951$	0	0
$952$	18.0314	0.584402
$953$	−28.8878	−0.935768	−0.467884	−	0.883790i	$-0.654984\pi$
$953$	−28.8878	−0.935768	−0.467884	+	0.883790i	$0.654984\pi$
$954$	0	0
$955$	20.8274	0.673960
$956$	−4.42187	−0.143013
$957$	0	0
$958$	0.195852	0.00632768
$959$	−11.0957	−0.358299
$960$	0	0
$961$	32.5285	1.04931
$962$	−95.7219	−3.08620
$963$	0	0
$964$	−11.6396	−0.374888
$965$	29.9434	0.963911
$966$	0	0
$967$	33.6661	1.08263	0.541315	−	0.840820i	$-0.317927\pi$
$967$	33.6661	1.08263	0.541315	+	0.840820i	$0.317927\pi$
$968$	20.7909	0.668246
$969$	0	0
$970$	−14.2306	−0.456916
$971$	4.51372	0.144852	0.0724260	−	0.997374i	$-0.476926\pi$
$971$	4.51372	0.144852	0.0724260	+	0.997374i	$0.476926\pi$
$972$	0	0
$973$	4.43688	0.142240
$974$	36.5283	1.17044
$975$	0	0
$976$	30.7928	0.985652
$977$	49.7681	1.59222	0.796111	−	0.605151i	$-0.206887\pi$
$977$	49.7681	1.59222	0.796111	+	0.605151i	$0.206887\pi$
$978$	0	0
$979$	−4.20043	−0.134246
$980$	−0.737828	−0.0235690
$981$	0	0
$982$	32.6154	1.04080
$983$	−28.8892	−0.921422	−0.460711	−	0.887550i	$-0.652406\pi$
$983$	−28.8892	−0.921422	−0.460711	+	0.887550i	$0.652406\pi$
$984$	0	0
$985$	32.2723	1.02828
$986$	67.0248	2.13450
$987$	0	0
$988$	−1.59679	−0.0508006
$989$	−37.2741	−1.18525
$990$	0	0
$991$	−16.6458	−0.528770	−0.264385	−	0.964417i	$-0.585169\pi$
$991$	−16.6458	−0.528770	−0.264385	+	0.964417i	$0.585169\pi$
$992$	−19.2640	−0.611632
$993$	0	0
$994$	−7.64580	−0.242510
$995$	−13.6085	−0.431420
$996$	0	0
$997$	9.55174	0.302507	0.151253	−	0.988495i	$-0.451669\pi$
$997$	9.55174	0.302507	0.151253	+	0.988495i	$0.451669\pi$
$998$	9.14485	0.289475
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.24	yes	32
3.2	odd	2	inner	8001.2.a.z.1.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.9	✓	32	3.2	odd	2	inner
8001.2.a.z.1.24	yes	32	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56027 q^{2} +0.434445 q^{4} -1.69832 q^{5} -1.00000 q^{7} -2.44269 q^{8} +O(q^{10})\)	\(q+1.56027 q^{2} +0.434445 q^{4} -1.69832 q^{5} -1.00000 q^{7} -2.44269 q^{8} -2.64985 q^{10} -1.57750 q^{11} +5.37691 q^{13} -1.56027 q^{14} -4.68015 q^{16} +7.38180 q^{17} -0.683565 q^{19} -0.737828 q^{20} -2.46133 q^{22} -8.53565 q^{23} -2.11570 q^{25} +8.38943 q^{26} -0.434445 q^{28} +5.81934 q^{29} +7.97047 q^{31} -2.41692 q^{32} +11.5176 q^{34} +1.69832 q^{35} -11.4098 q^{37} -1.06655 q^{38} +4.14848 q^{40} +5.44263 q^{41} +4.36687 q^{43} -0.685336 q^{44} -13.3179 q^{46} -3.11843 q^{47} +1.00000 q^{49} -3.30106 q^{50} +2.33597 q^{52} +7.03105 q^{53} +2.67911 q^{55} +2.44269 q^{56} +9.07974 q^{58} -5.82826 q^{59} -6.57944 q^{61} +12.4361 q^{62} +5.58925 q^{64} -9.13173 q^{65} -6.68183 q^{67} +3.20698 q^{68} +2.64985 q^{70} +4.90031 q^{71} -7.91813 q^{73} -17.8024 q^{74} -0.296971 q^{76} +1.57750 q^{77} -15.7765 q^{79} +7.94841 q^{80} +8.49197 q^{82} +6.39375 q^{83} -12.5367 q^{85} +6.81350 q^{86} +3.85334 q^{88} +2.66271 q^{89} -5.37691 q^{91} -3.70827 q^{92} -4.86560 q^{94} +1.16092 q^{95} +5.37034 q^{97} +1.56027 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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