LMFDB - Embedded newform 1004.2.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1004,2,Mod(1,1004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1004, 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("1004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1004 = 2^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1004.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:	$8.01698036294$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.844838$ of defining polynomial
Character	$\chi$	$=$	1004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.844838 q^{3} -2.47034 q^{5} -0.972158 q^{7} -2.28625 q^{9} +O(q^{10})$	$q+0.844838 q^{3} -2.47034 q^{5} -0.972158 q^{7} -2.28625 q^{9} +3.50942 q^{11} +2.87894 q^{13} -2.08704 q^{15} -2.27070 q^{17} -6.78729 q^{19} -0.821315 q^{21} +3.58090 q^{23} +1.10259 q^{25} -4.46602 q^{27} -7.60769 q^{29} -0.279626 q^{31} +2.96489 q^{33} +2.40156 q^{35} -9.37051 q^{37} +2.43223 q^{39} -10.4728 q^{41} -5.70457 q^{43} +5.64782 q^{45} -8.12711 q^{47} -6.05491 q^{49} -1.91837 q^{51} +1.01555 q^{53} -8.66946 q^{55} -5.73416 q^{57} +12.3081 q^{59} +9.95423 q^{61} +2.22260 q^{63} -7.11196 q^{65} -4.50345 q^{67} +3.02528 q^{69} +9.82624 q^{71} +6.37302 q^{73} +0.931509 q^{75} -3.41171 q^{77} +1.13734 q^{79} +3.08569 q^{81} -1.53106 q^{83} +5.60940 q^{85} -6.42726 q^{87} +15.3254 q^{89} -2.79878 q^{91} -0.236239 q^{93} +16.7669 q^{95} -0.254712 q^{97} -8.02340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * 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
$3$	0.844838	0.487767	0.243884	−	0.969805i	$-0.421579\pi$
$3$	0.844838	0.487767	0.243884	+	0.969805i	$0.421579\pi$
$4$	0	0
$5$	−2.47034	−1.10477	−0.552385	−	0.833589i	$-0.686282\pi$
$5$	−2.47034	−1.10477	−0.552385	+	0.833589i	$0.686282\pi$
$6$	0	0
$7$	−0.972158	−0.367441	−0.183721	−	0.982979i	$-0.558814\pi$
$7$	−0.972158	−0.367441	−0.183721	+	0.982979i	$0.558814\pi$
$8$	0	0
$9$	−2.28625	−0.762083
$10$	0	0
$11$	3.50942	1.05813	0.529064	−	0.848582i	$-0.322543\pi$
$11$	3.50942	1.05813	0.529064	+	0.848582i	$0.322543\pi$
$12$	0	0
$13$	2.87894	0.798473	0.399237	−	0.916848i	$-0.369275\pi$
$13$	2.87894	0.798473	0.399237	+	0.916848i	$0.369275\pi$
$14$	0	0
$15$	−2.08704	−0.538871
$16$	0	0
$17$	−2.27070	−0.550725	−0.275362	−	0.961340i	$-0.588798\pi$
$17$	−2.27070	−0.550725	−0.275362	+	0.961340i	$0.588798\pi$
$18$	0	0
$19$	−6.78729	−1.55711	−0.778556	−	0.627576i	$-0.784047\pi$
$19$	−6.78729	−1.55711	−0.778556	+	0.627576i	$0.784047\pi$
$20$	0	0
$21$	−0.821315	−0.179226
$22$	0	0
$23$	3.58090	0.746670	0.373335	−	0.927697i	$-0.378214\pi$
$23$	3.58090	0.746670	0.373335	+	0.927697i	$0.378214\pi$
$24$	0	0
$25$	1.10259	0.220518
$26$	0	0
$27$	−4.46602	−0.859486
$28$	0	0
$29$	−7.60769	−1.41271	−0.706356	−	0.707857i	$-0.749662\pi$
$29$	−7.60769	−1.41271	−0.706356	+	0.707857i	$0.749662\pi$
$30$	0	0
$31$	−0.279626	−0.0502224	−0.0251112	−	0.999685i	$-0.507994\pi$
$31$	−0.279626	−0.0502224	−0.0251112	+	0.999685i	$0.507994\pi$
$32$	0	0
$33$	2.96489	0.516121
$34$	0	0
$35$	2.40156	0.405938
$36$	0	0
$37$	−9.37051	−1.54050	−0.770251	−	0.637741i	$-0.779869\pi$
$37$	−9.37051	−1.54050	−0.770251	+	0.637741i	$0.779869\pi$
$38$	0	0
$39$	2.43223	0.389469
$40$	0	0
$41$	−10.4728	−1.63558	−0.817791	−	0.575516i	$-0.804801\pi$
$41$	−10.4728	−1.63558	−0.817791	+	0.575516i	$0.804801\pi$
$42$	0	0
$43$	−5.70457	−0.869939	−0.434970	−	0.900445i	$-0.643241\pi$
$43$	−5.70457	−0.869939	−0.434970	+	0.900445i	$0.643241\pi$
$44$	0	0
$45$	5.64782	0.841927
$46$	0	0
$47$	−8.12711	−1.18546	−0.592730	−	0.805401i	$-0.701950\pi$
$47$	−8.12711	−1.18546	−0.592730	+	0.805401i	$0.701950\pi$
$48$	0	0
$49$	−6.05491	−0.864987
$50$	0	0
$51$	−1.91837	−0.268626
$52$	0	0
$53$	1.01555	0.139497	0.0697484	−	0.997565i	$-0.477780\pi$
$53$	1.01555	0.139497	0.0697484	+	0.997565i	$0.477780\pi$
$54$	0	0
$55$	−8.66946	−1.16899
$56$	0	0
$57$	−5.73416	−0.759508
$58$	0	0
$59$	12.3081	1.60238	0.801188	−	0.598413i	$-0.204202\pi$
$59$	12.3081	1.60238	0.801188	+	0.598413i	$0.204202\pi$
$60$	0	0
$61$	9.95423	1.27451	0.637255	−	0.770653i	$-0.280070\pi$
$61$	9.95423	1.27451	0.637255	+	0.770653i	$0.280070\pi$
$62$	0	0
$63$	2.22260	0.280021
$64$	0	0
$65$	−7.11196	−0.882130
$66$	0	0
$67$	−4.50345	−0.550184	−0.275092	−	0.961418i	$-0.588708\pi$
$67$	−4.50345	−0.550184	−0.275092	+	0.961418i	$0.588708\pi$
$68$	0	0
$69$	3.02528	0.364201
$70$	0	0
$71$	9.82624	1.16616	0.583080	−	0.812415i	$-0.301847\pi$
$71$	9.82624	1.16616	0.583080	+	0.812415i	$0.301847\pi$
$72$	0	0
$73$	6.37302	0.745906	0.372953	−	0.927850i	$-0.378345\pi$
$73$	6.37302	0.745906	0.372953	+	0.927850i	$0.378345\pi$
$74$	0	0
$75$	0.931509	0.107561
$76$	0	0
$77$	−3.41171	−0.388800
$78$	0	0
$79$	1.13734	0.127961	0.0639806	−	0.997951i	$-0.479620\pi$
$79$	1.13734	0.127961	0.0639806	+	0.997951i	$0.479620\pi$
$80$	0	0
$81$	3.08569	0.342854
$82$	0	0
$83$	−1.53106	−0.168055	−0.0840277	−	0.996463i	$-0.526778\pi$
$83$	−1.53106	−0.168055	−0.0840277	+	0.996463i	$0.526778\pi$
$84$	0	0
$85$	5.60940	0.608425
$86$	0	0
$87$	−6.42726	−0.689074
$88$	0	0
$89$	15.3254	1.62449	0.812243	−	0.583320i	$-0.198246\pi$
$89$	15.3254	1.62449	0.812243	+	0.583320i	$0.198246\pi$
$90$	0	0
$91$	−2.79878	−0.293392
$92$	0	0
$93$	−0.236239	−0.0244968
$94$	0	0
$95$	16.7669	1.72025
$96$	0	0
$97$	−0.254712	−0.0258621	−0.0129310	−	0.999916i	$-0.504116\pi$
$97$	−0.254712	−0.0258621	−0.0129310	+	0.999916i	$0.504116\pi$
$98$	0	0
$99$	−8.02340	−0.806382
$100$	0	0
$101$	10.7612	1.07078	0.535391	−	0.844604i	$-0.320164\pi$
$101$	10.7612	1.07078	0.535391	+	0.844604i	$0.320164\pi$
$102$	0	0
$103$	−4.73867	−0.466915	−0.233458	−	0.972367i	$-0.575004\pi$
$103$	−4.73867	−0.466915	−0.233458	+	0.972367i	$0.575004\pi$
$104$	0	0
$105$	2.02893	0.198003
$106$	0	0
$107$	−5.02191	−0.485486	−0.242743	−	0.970091i	$-0.578047\pi$
$107$	−5.02191	−0.485486	−0.242743	+	0.970091i	$0.578047\pi$
$108$	0	0
$109$	0.0232276	0.00222480	0.00111240	−	0.999999i	$-0.499646\pi$
$109$	0.0232276	0.00222480	0.00111240	+	0.999999i	$0.499646\pi$
$110$	0	0
$111$	−7.91655	−0.751406
$112$	0	0
$113$	9.83680	0.925369	0.462684	−	0.886523i	$-0.346886\pi$
$113$	9.83680	0.925369	0.462684	+	0.886523i	$0.346886\pi$
$114$	0	0
$115$	−8.84605	−0.824899
$116$	0	0
$117$	−6.58197	−0.608503
$118$	0	0
$119$	2.20748	0.202359
$120$	0	0
$121$	1.31600	0.119637
$122$	0	0
$123$	−8.84784	−0.797783
$124$	0	0
$125$	9.62794	0.861149
$126$	0	0
$127$	−6.24438	−0.554099	−0.277049	−	0.960856i	$-0.589357\pi$
$127$	−6.24438	−0.554099	−0.277049	+	0.960856i	$0.589357\pi$
$128$	0	0
$129$	−4.81944	−0.424328
$130$	0	0
$131$	−21.8403	−1.90820	−0.954100	−	0.299489i	$-0.903184\pi$
$131$	−21.8403	−1.90820	−0.954100	+	0.299489i	$0.903184\pi$
$132$	0	0
$133$	6.59832	0.572147
$134$	0	0
$135$	11.0326	0.949535
$136$	0	0
$137$	−8.61906	−0.736376	−0.368188	−	0.929751i	$-0.620022\pi$
$137$	−8.61906	−0.736376	−0.368188	+	0.929751i	$0.620022\pi$
$138$	0	0
$139$	−5.86187	−0.497197	−0.248599	−	0.968607i	$-0.579970\pi$
$139$	−5.86187	−0.497197	−0.248599	+	0.968607i	$0.579970\pi$
$140$	0	0
$141$	−6.86608	−0.578229
$142$	0	0
$143$	10.1034	0.844888
$144$	0	0
$145$	18.7936	1.56072
$146$	0	0
$147$	−5.11541	−0.421912
$148$	0	0
$149$	−4.20332	−0.344349	−0.172175	−	0.985066i	$-0.555079\pi$
$149$	−4.20332	−0.344349	−0.172175	+	0.985066i	$0.555079\pi$
$150$	0	0
$151$	−11.9027	−0.968631	−0.484316	−	0.874893i	$-0.660931\pi$
$151$	−11.9027	−0.968631	−0.484316	+	0.874893i	$0.660931\pi$
$152$	0	0
$153$	5.19138	0.419698
$154$	0	0
$155$	0.690773	0.0554842
$156$	0	0
$157$	14.3438	1.14476	0.572378	−	0.819990i	$-0.306021\pi$
$157$	14.3438	1.14476	0.572378	+	0.819990i	$0.306021\pi$
$158$	0	0
$159$	0.857977	0.0680420
$160$	0	0
$161$	−3.48120	−0.274357
$162$	0	0
$163$	6.38647	0.500227	0.250114	−	0.968217i	$-0.419532\pi$
$163$	6.38647	0.500227	0.250114	+	0.968217i	$0.419532\pi$
$164$	0	0
$165$	−7.32428	−0.570195
$166$	0	0
$167$	20.7849	1.60838	0.804191	−	0.594371i	$-0.202599\pi$
$167$	20.7849	1.60838	0.804191	+	0.594371i	$0.202599\pi$
$168$	0	0
$169$	−4.71172	−0.362440
$170$	0	0
$171$	15.5174	1.18665
$172$	0	0
$173$	22.3065	1.69593	0.847964	−	0.530054i	$-0.177828\pi$
$173$	22.3065	1.69593	0.847964	+	0.530054i	$0.177828\pi$
$174$	0	0
$175$	−1.07189	−0.0810274
$176$	0	0
$177$	10.3983	0.781586
$178$	0	0
$179$	−4.57389	−0.341869	−0.170934	−	0.985282i	$-0.554679\pi$
$179$	−4.57389	−0.341869	−0.170934	+	0.985282i	$0.554679\pi$
$180$	0	0
$181$	−23.4328	−1.74175	−0.870874	−	0.491507i	$-0.836446\pi$
$181$	−23.4328	−1.74175	−0.870874	+	0.491507i	$0.836446\pi$
$182$	0	0
$183$	8.40971	0.621664
$184$	0	0
$185$	23.1484	1.70190
$186$	0	0
$187$	−7.96882	−0.582738
$188$	0	0
$189$	4.34168	0.315811
$190$	0	0
$191$	9.68497	0.700780	0.350390	−	0.936604i	$-0.386049\pi$
$191$	9.68497	0.700780	0.350390	+	0.936604i	$0.386049\pi$
$192$	0	0
$193$	−21.2162	−1.52718	−0.763588	−	0.645703i	$-0.776564\pi$
$193$	−21.2162	−1.52718	−0.763588	+	0.645703i	$0.776564\pi$
$194$	0	0
$195$	−6.00845	−0.430274
$196$	0	0
$197$	10.2301	0.728866	0.364433	−	0.931230i	$-0.381263\pi$
$197$	10.2301	0.728866	0.364433	+	0.931230i	$0.381263\pi$
$198$	0	0
$199$	−2.03987	−0.144602	−0.0723011	−	0.997383i	$-0.523034\pi$
$199$	−2.03987	−0.144602	−0.0723011	+	0.997383i	$0.523034\pi$
$200$	0	0
$201$	−3.80468	−0.268362
$202$	0	0
$203$	7.39587	0.519088
$204$	0	0
$205$	25.8715	1.80694
$206$	0	0
$207$	−8.18684	−0.569024
$208$	0	0
$209$	−23.8194	−1.64762
$210$	0	0
$211$	−12.1422	−0.835901	−0.417951	−	0.908470i	$-0.637251\pi$
$211$	−12.1422	−0.835901	−0.417951	+	0.908470i	$0.637251\pi$
$212$	0	0
$213$	8.30158	0.568815
$214$	0	0
$215$	14.0922	0.961083
$216$	0	0
$217$	0.271841	0.0184538
$218$	0	0
$219$	5.38417	0.363828
$220$	0	0
$221$	−6.53719	−0.439739
$222$	0	0
$223$	−28.5639	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$223$	−28.5639	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$224$	0	0
$225$	−2.52080	−0.168053
$226$	0	0
$227$	−28.4032	−1.88519	−0.942593	−	0.333944i	$-0.891620\pi$
$227$	−28.4032	−1.88519	−0.942593	+	0.333944i	$0.891620\pi$
$228$	0	0
$229$	19.7016	1.30192	0.650958	−	0.759114i	$-0.274367\pi$
$229$	19.7016	1.30192	0.650958	+	0.759114i	$0.274367\pi$
$230$	0	0
$231$	−2.88234	−0.189644
$232$	0	0
$233$	−24.3686	−1.59644	−0.798221	−	0.602365i	$-0.794225\pi$
$233$	−24.3686	−1.59644	−0.798221	+	0.602365i	$0.794225\pi$
$234$	0	0
$235$	20.0767	1.30966
$236$	0	0
$237$	0.960871	0.0624153
$238$	0	0
$239$	−9.67986	−0.626138	−0.313069	−	0.949730i	$-0.601357\pi$
$239$	−9.67986	−0.626138	−0.313069	+	0.949730i	$0.601357\pi$
$240$	0	0
$241$	7.49541	0.482822	0.241411	−	0.970423i	$-0.422390\pi$
$241$	7.49541	0.482822	0.241411	+	0.970423i	$0.422390\pi$
$242$	0	0
$243$	16.0050	1.02672
$244$	0	0
$245$	14.9577	0.955612
$246$	0	0
$247$	−19.5402	−1.24331
$248$	0	0
$249$	−1.29349	−0.0819719
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	12.5669	0.790073
$254$	0	0
$255$	4.73903	0.296770
$256$	0	0
$257$	22.0330	1.37438	0.687189	−	0.726479i	$-0.258844\pi$
$257$	22.0330	1.37438	0.687189	+	0.726479i	$0.258844\pi$
$258$	0	0
$259$	9.10961	0.566044
$260$	0	0
$261$	17.3931	1.07660
$262$	0	0
$263$	−0.437637	−0.0269858	−0.0134929	−	0.999909i	$-0.504295\pi$
$263$	−0.437637	−0.0269858	−0.0134929	+	0.999909i	$0.504295\pi$
$264$	0	0
$265$	−2.50876	−0.154112
$266$	0	0
$267$	12.9474	0.792371
$268$	0	0
$269$	0.166462	0.0101494	0.00507468	−	0.999987i	$-0.498385\pi$
$269$	0.166462	0.0101494	0.00507468	+	0.999987i	$0.498385\pi$
$270$	0	0
$271$	27.1750	1.65076	0.825382	−	0.564574i	$-0.190960\pi$
$271$	27.1750	1.65076	0.825382	+	0.564574i	$0.190960\pi$
$272$	0	0
$273$	−2.36452	−0.143107
$274$	0	0
$275$	3.86945	0.233336
$276$	0	0
$277$	26.0074	1.56263	0.781315	−	0.624137i	$-0.214549\pi$
$277$	26.0074	1.56263	0.781315	+	0.624137i	$0.214549\pi$
$278$	0	0
$279$	0.639296	0.0382736
$280$	0	0
$281$	−14.1033	−0.841335	−0.420668	−	0.907215i	$-0.638204\pi$
$281$	−14.1033	−0.841335	−0.420668	+	0.907215i	$0.638204\pi$
$282$	0	0
$283$	−14.4293	−0.857734	−0.428867	−	0.903368i	$-0.641087\pi$
$283$	−14.4293	−0.857734	−0.428867	+	0.903368i	$0.641087\pi$
$284$	0	0
$285$	14.1653	0.839082
$286$	0	0
$287$	10.1812	0.600980
$288$	0	0
$289$	−11.8439	−0.696702
$290$	0	0
$291$	−0.215190	−0.0126147
$292$	0	0
$293$	28.0623	1.63942	0.819710	−	0.572779i	$-0.194135\pi$
$293$	28.0623	1.63942	0.819710	+	0.572779i	$0.194135\pi$
$294$	0	0
$295$	−30.4052	−1.77026
$296$	0	0
$297$	−15.6731	−0.909447
$298$	0	0
$299$	10.3092	0.596196
$300$	0	0
$301$	5.54574	0.319651
$302$	0	0
$303$	9.09149	0.522292
$304$	0	0
$305$	−24.5904	−1.40804
$306$	0	0
$307$	22.0019	1.25571	0.627857	−	0.778329i	$-0.283932\pi$
$307$	22.0019	1.25571	0.627857	+	0.778329i	$0.283932\pi$
$308$	0	0
$309$	−4.00341	−0.227746
$310$	0	0
$311$	−6.83660	−0.387668	−0.193834	−	0.981034i	$-0.562092\pi$
$311$	−6.83660	−0.387668	−0.193834	+	0.981034i	$0.562092\pi$
$312$	0	0
$313$	−22.1934	−1.25445	−0.627223	−	0.778839i	$-0.715809\pi$
$313$	−22.1934	−1.25445	−0.627223	+	0.778839i	$0.715809\pi$
$314$	0	0
$315$	−5.49057	−0.309359
$316$	0	0
$317$	14.4529	0.811755	0.405878	−	0.913927i	$-0.366966\pi$
$317$	14.4529	0.811755	0.405878	+	0.913927i	$0.366966\pi$
$318$	0	0
$319$	−26.6985	−1.49483
$320$	0	0
$321$	−4.24270	−0.236804
$322$	0	0
$323$	15.4119	0.857540
$324$	0	0
$325$	3.17429	0.176078
$326$	0	0
$327$	0.0196236	0.00108518
$328$	0	0
$329$	7.90083	0.435587
$330$	0	0
$331$	8.27651	0.454918	0.227459	−	0.973788i	$-0.426958\pi$
$331$	8.27651	0.454918	0.227459	+	0.973788i	$0.426958\pi$
$332$	0	0
$333$	21.4233	1.17399
$334$	0	0
$335$	11.1251	0.607827
$336$	0	0
$337$	6.45813	0.351797	0.175898	−	0.984408i	$-0.443717\pi$
$337$	6.45813	0.351797	0.175898	+	0.984408i	$0.443717\pi$
$338$	0	0
$339$	8.31050	0.451364
$340$	0	0
$341$	−0.981326	−0.0531418
$342$	0	0
$343$	12.6914	0.685273
$344$	0	0
$345$	−7.47348	−0.402358
$346$	0	0
$347$	15.2559	0.818979	0.409490	−	0.912315i	$-0.365707\pi$
$347$	15.2559	0.818979	0.409490	+	0.912315i	$0.365707\pi$
$348$	0	0
$349$	1.41721	0.0758616	0.0379308	−	0.999280i	$-0.487923\pi$
$349$	1.41721	0.0758616	0.0379308	+	0.999280i	$0.487923\pi$
$350$	0	0
$351$	−12.8574	−0.686277
$352$	0	0
$353$	13.5027	0.718674	0.359337	−	0.933208i	$-0.383003\pi$
$353$	13.5027	0.718674	0.359337	+	0.933208i	$0.383003\pi$
$354$	0	0
$355$	−24.2742	−1.28834
$356$	0	0
$357$	1.86496	0.0987041
$358$	0	0
$359$	−5.44741	−0.287503	−0.143752	−	0.989614i	$-0.545917\pi$
$359$	−5.44741	−0.287503	−0.143752	+	0.989614i	$0.545917\pi$
$360$	0	0
$361$	27.0673	1.42460
$362$	0	0
$363$	1.11181	0.0583549
$364$	0	0
$365$	−15.7435	−0.824055
$366$	0	0
$367$	6.36536	0.332269	0.166135	−	0.986103i	$-0.446871\pi$
$367$	6.36536	0.332269	0.166135	+	0.986103i	$0.446871\pi$
$368$	0	0
$369$	23.9435	1.24645
$370$	0	0
$371$	−0.987277	−0.0512569
$372$	0	0
$373$	−10.0940	−0.522646	−0.261323	−	0.965251i	$-0.584159\pi$
$373$	−10.0940	−0.522646	−0.261323	+	0.965251i	$0.584159\pi$
$374$	0	0
$375$	8.13404	0.420040
$376$	0	0
$377$	−21.9020	−1.12801
$378$	0	0
$379$	−23.6557	−1.21511	−0.607557	−	0.794276i	$-0.707850\pi$
$379$	−23.6557	−1.21511	−0.607557	+	0.794276i	$0.707850\pi$
$380$	0	0
$381$	−5.27548	−0.270271
$382$	0	0
$383$	−29.3212	−1.49825	−0.749123	−	0.662431i	$-0.769525\pi$
$383$	−29.3212	−1.49825	−0.749123	+	0.662431i	$0.769525\pi$
$384$	0	0
$385$	8.42808	0.429535
$386$	0	0
$387$	13.0421	0.662966
$388$	0	0
$389$	24.2891	1.23151	0.615753	−	0.787940i	$-0.288852\pi$
$389$	24.2891	1.23151	0.615753	+	0.787940i	$0.288852\pi$
$390$	0	0
$391$	−8.13114	−0.411210
$392$	0	0
$393$	−18.4515	−0.930757
$394$	0	0
$395$	−2.80963	−0.141368
$396$	0	0
$397$	−10.4398	−0.523961	−0.261980	−	0.965073i	$-0.584376\pi$
$397$	−10.4398	−0.523961	−0.261980	+	0.965073i	$0.584376\pi$
$398$	0	0
$399$	5.57451	0.279074
$400$	0	0
$401$	27.6476	1.38066	0.690328	−	0.723496i	$-0.257466\pi$
$401$	27.6476	1.38066	0.690328	+	0.723496i	$0.257466\pi$
$402$	0	0
$403$	−0.805027	−0.0401012
$404$	0	0
$405$	−7.62270	−0.378775
$406$	0	0
$407$	−32.8850	−1.63005
$408$	0	0
$409$	−20.0738	−0.992587	−0.496294	−	0.868155i	$-0.665306\pi$
$409$	−20.0738	−0.992587	−0.496294	+	0.868155i	$0.665306\pi$
$410$	0	0
$411$	−7.28170	−0.359180
$412$	0	0
$413$	−11.9654	−0.588779
$414$	0	0
$415$	3.78224	0.185663
$416$	0	0
$417$	−4.95233	−0.242516
$418$	0	0
$419$	5.41538	0.264558	0.132279	−	0.991212i	$-0.457770\pi$
$419$	5.41538	0.264558	0.132279	+	0.991212i	$0.457770\pi$
$420$	0	0
$421$	−23.1966	−1.13053	−0.565266	−	0.824909i	$-0.691226\pi$
$421$	−23.1966	−1.13053	−0.565266	+	0.824909i	$0.691226\pi$
$422$	0	0
$423$	18.5806	0.903419
$424$	0	0
$425$	−2.50365	−0.121445
$426$	0	0
$427$	−9.67709	−0.468307
$428$	0	0
$429$	8.53572	0.412109
$430$	0	0
$431$	−31.4498	−1.51488	−0.757441	−	0.652904i	$-0.773551\pi$
$431$	−31.4498	−1.51488	−0.757441	+	0.652904i	$0.773551\pi$
$432$	0	0
$433$	−9.30587	−0.447211	−0.223606	−	0.974680i	$-0.571783\pi$
$433$	−9.30587	−0.447211	−0.223606	+	0.974680i	$0.571783\pi$
$434$	0	0
$435$	15.8775	0.761269
$436$	0	0
$437$	−24.3046	−1.16265
$438$	0	0
$439$	−31.9914	−1.52687	−0.763434	−	0.645886i	$-0.776488\pi$
$439$	−31.9914	−1.52687	−0.763434	+	0.645886i	$0.776488\pi$
$440$	0	0
$441$	13.8430	0.659192
$442$	0	0
$443$	−9.46628	−0.449757	−0.224878	−	0.974387i	$-0.572198\pi$
$443$	−9.46628	−0.449757	−0.224878	+	0.974387i	$0.572198\pi$
$444$	0	0
$445$	−37.8589	−1.79468
$446$	0	0
$447$	−3.55112	−0.167962
$448$	0	0
$449$	−20.4021	−0.962835	−0.481418	−	0.876491i	$-0.659878\pi$
$449$	−20.4021	−0.962835	−0.481418	+	0.876491i	$0.659878\pi$
$450$	0	0
$451$	−36.7535	−1.73066
$452$	0	0
$453$	−10.0559	−0.472467
$454$	0	0
$455$	6.91395	0.324131
$456$	0	0
$457$	5.97184	0.279351	0.139675	−	0.990197i	$-0.455394\pi$
$457$	5.97184	0.279351	0.139675	+	0.990197i	$0.455394\pi$
$458$	0	0
$459$	10.1410	0.473341
$460$	0	0
$461$	6.01449	0.280123	0.140061	−	0.990143i	$-0.455270\pi$
$461$	6.01449	0.280123	0.140061	+	0.990143i	$0.455270\pi$
$462$	0	0
$463$	11.4730	0.533197	0.266598	−	0.963808i	$-0.414100\pi$
$463$	11.4730	0.533197	0.266598	+	0.963808i	$0.414100\pi$
$464$	0	0
$465$	0.583591	0.0270634
$466$	0	0
$467$	−26.1886	−1.21187	−0.605933	−	0.795516i	$-0.707200\pi$
$467$	−26.1886	−1.21187	−0.605933	+	0.795516i	$0.707200\pi$
$468$	0	0
$469$	4.37806	0.202160
$470$	0	0
$471$	12.1181	0.558374
$472$	0	0
$473$	−20.0197	−0.920508
$474$	0	0
$475$	−7.48360	−0.343371
$476$	0	0
$477$	−2.32181	−0.106308
$478$	0	0
$479$	−5.68587	−0.259794	−0.129897	−	0.991527i	$-0.541465\pi$
$479$	−5.68587	−0.259794	−0.129897	+	0.991527i	$0.541465\pi$
$480$	0	0
$481$	−26.9771	−1.23005
$482$	0	0
$483$	−2.94105	−0.133822
$484$	0	0
$485$	0.629226	0.0285717
$486$	0	0
$487$	16.9258	0.766983	0.383491	−	0.923544i	$-0.374722\pi$
$487$	16.9258	0.766983	0.383491	+	0.923544i	$0.374722\pi$
$488$	0	0
$489$	5.39553	0.243994
$490$	0	0
$491$	0.457098	0.0206285	0.0103143	−	0.999947i	$-0.496717\pi$
$491$	0.457098	0.0206285	0.0103143	+	0.999947i	$0.496717\pi$
$492$	0	0
$493$	17.2748	0.778016
$494$	0	0
$495$	19.8205	0.890867
$496$	0	0
$497$	−9.55266	−0.428495
$498$	0	0
$499$	20.9818	0.939274	0.469637	−	0.882860i	$-0.344385\pi$
$499$	20.9818	0.939274	0.469637	+	0.882860i	$0.344385\pi$
$500$	0	0
$501$	17.5598	0.784516
$502$	0	0
$503$	7.22680	0.322227	0.161113	−	0.986936i	$-0.448491\pi$
$503$	7.22680	0.322227	0.161113	+	0.986936i	$0.448491\pi$
$504$	0	0
$505$	−26.5839	−1.18297
$506$	0	0
$507$	−3.98064	−0.176786
$508$	0	0
$509$	−27.1215	−1.20214	−0.601069	−	0.799197i	$-0.705258\pi$
$509$	−27.1215	−1.20214	−0.601069	+	0.799197i	$0.705258\pi$
$510$	0	0
$511$	−6.19558	−0.274077
$512$	0	0
$513$	30.3122	1.33832
$514$	0	0
$515$	11.7061	0.515834
$516$	0	0
$517$	−28.5214	−1.25437
$518$	0	0
$519$	18.8453	0.827218
$520$	0	0
$521$	8.03645	0.352083	0.176042	−	0.984383i	$-0.443671\pi$
$521$	8.03645	0.352083	0.176042	+	0.984383i	$0.443671\pi$
$522$	0	0
$523$	35.6907	1.56064	0.780322	−	0.625378i	$-0.215055\pi$
$523$	35.6907	1.56064	0.780322	+	0.625378i	$0.215055\pi$
$524$	0	0
$525$	−0.905574	−0.0395225
$526$	0	0
$527$	0.634947	0.0276587
$528$	0	0
$529$	−10.1771	−0.442484
$530$	0	0
$531$	−28.1393	−1.22114
$532$	0	0
$533$	−30.1506	−1.30597
$534$	0	0
$535$	12.4058	0.536351
$536$	0	0
$537$	−3.86420	−0.166752
$538$	0	0
$539$	−21.2492	−0.915268
$540$	0	0
$541$	−23.7224	−1.01991	−0.509953	−	0.860202i	$-0.670337\pi$
$541$	−23.7224	−1.01991	−0.509953	+	0.860202i	$0.670337\pi$
$542$	0	0
$543$	−19.7969	−0.849567
$544$	0	0
$545$	−0.0573801	−0.00245789
$546$	0	0
$547$	0.652965	0.0279188	0.0139594	−	0.999903i	$-0.495556\pi$
$547$	0.652965	0.0279188	0.0139594	+	0.999903i	$0.495556\pi$
$548$	0	0
$549$	−22.7579	−0.971282
$550$	0	0
$551$	51.6356	2.19975
$552$	0	0
$553$	−1.10568	−0.0470182
$554$	0	0
$555$	19.5566	0.830131
$556$	0	0
$557$	10.7803	0.456776	0.228388	−	0.973570i	$-0.426655\pi$
$557$	10.7803	0.456776	0.228388	+	0.973570i	$0.426655\pi$
$558$	0	0
$559$	−16.4231	−0.694623
$560$	0	0
$561$	−6.73236	−0.284240
$562$	0	0
$563$	−17.4873	−0.737000	−0.368500	−	0.929628i	$-0.620129\pi$
$563$	−17.4873	−0.737000	−0.368500	+	0.929628i	$0.620129\pi$
$564$	0	0
$565$	−24.3003	−1.02232
$566$	0	0
$567$	−2.99977	−0.125979
$568$	0	0
$569$	3.99957	0.167671	0.0838353	−	0.996480i	$-0.473283\pi$
$569$	3.99957	0.167671	0.0838353	+	0.996480i	$0.473283\pi$
$570$	0	0
$571$	39.9033	1.66990	0.834950	−	0.550325i	$-0.185496\pi$
$571$	39.9033	1.66990	0.834950	+	0.550325i	$0.185496\pi$
$572$	0	0
$573$	8.18223	0.341817
$574$	0	0
$575$	3.94827	0.164654
$576$	0	0
$577$	−8.27664	−0.344561	−0.172280	−	0.985048i	$-0.555114\pi$
$577$	−8.27664	−0.344561	−0.172280	+	0.985048i	$0.555114\pi$
$578$	0	0
$579$	−17.9243	−0.744907
$580$	0	0
$581$	1.48843	0.0617505
$582$	0	0
$583$	3.56400	0.147606
$584$	0	0
$585$	16.2597	0.672256
$586$	0	0
$587$	39.2614	1.62049	0.810246	−	0.586090i	$-0.199333\pi$
$587$	39.2614	1.62049	0.810246	+	0.586090i	$0.199333\pi$
$588$	0	0
$589$	1.89791	0.0782018
$590$	0	0
$591$	8.64280	0.355517
$592$	0	0
$593$	24.9457	1.02440	0.512198	−	0.858868i	$-0.328831\pi$
$593$	24.9457	1.02440	0.512198	+	0.858868i	$0.328831\pi$
$594$	0	0
$595$	−5.45322	−0.223560
$596$	0	0
$597$	−1.72335	−0.0705322
$598$	0	0
$599$	−12.0175	−0.491020	−0.245510	−	0.969394i	$-0.578955\pi$
$599$	−12.0175	−0.491020	−0.245510	+	0.969394i	$0.578955\pi$
$600$	0	0
$601$	−11.9326	−0.486742	−0.243371	−	0.969933i	$-0.578253\pi$
$601$	−11.9326	−0.486742	−0.243371	+	0.969933i	$0.578253\pi$
$602$	0	0
$603$	10.2960	0.419286
$604$	0	0
$605$	−3.25098	−0.132171
$606$	0	0
$607$	−38.2623	−1.55302	−0.776509	−	0.630106i	$-0.783011\pi$
$607$	−38.2623	−1.55302	−0.776509	+	0.630106i	$0.783011\pi$
$608$	0	0
$609$	6.24831	0.253194
$610$	0	0
$611$	−23.3974	−0.946558
$612$	0	0
$613$	32.5152	1.31328	0.656639	−	0.754205i	$-0.271977\pi$
$613$	32.5152	1.31328	0.656639	+	0.754205i	$0.271977\pi$
$614$	0	0
$615$	21.8572	0.881367
$616$	0	0
$617$	−16.0903	−0.647770	−0.323885	−	0.946096i	$-0.604989\pi$
$617$	−16.0903	−0.647770	−0.323885	+	0.946096i	$0.604989\pi$
$618$	0	0
$619$	44.4841	1.78797	0.893984	−	0.448099i	$-0.147899\pi$
$619$	44.4841	1.78797	0.893984	+	0.448099i	$0.147899\pi$
$620$	0	0
$621$	−15.9924	−0.641752
$622$	0	0
$623$	−14.8987	−0.596903
$624$	0	0
$625$	−29.2972	−1.17189
$626$	0	0
$627$	−20.1235	−0.803657
$628$	0	0
$629$	21.2776	0.848393
$630$	0	0
$631$	−42.6044	−1.69606	−0.848028	−	0.529952i	$-0.822210\pi$
$631$	−42.6044	−1.69606	−0.848028	+	0.529952i	$0.822210\pi$
$632$	0	0
$633$	−10.2582	−0.407725
$634$	0	0
$635$	15.4257	0.612152
$636$	0	0
$637$	−17.4317	−0.690669
$638$	0	0
$639$	−22.4652	−0.888711
$640$	0	0
$641$	−26.8680	−1.06122	−0.530612	−	0.847615i	$-0.678038\pi$
$641$	−26.8680	−1.06122	−0.530612	+	0.847615i	$0.678038\pi$
$642$	0	0
$643$	4.23302	0.166934	0.0834670	−	0.996511i	$-0.473401\pi$
$643$	4.23302	0.166934	0.0834670	+	0.996511i	$0.473401\pi$
$644$	0	0
$645$	11.9057	0.468785
$646$	0	0
$647$	−6.04546	−0.237672	−0.118836	−	0.992914i	$-0.537916\pi$
$647$	−6.04546	−0.237672	−0.118836	+	0.992914i	$0.537916\pi$
$648$	0	0
$649$	43.1942	1.69552
$650$	0	0
$651$	0.229662	0.00900114
$652$	0	0
$653$	−1.66383	−0.0651106	−0.0325553	−	0.999470i	$-0.510365\pi$
$653$	−1.66383	−0.0651106	−0.0325553	+	0.999470i	$0.510365\pi$
$654$	0	0
$655$	53.9531	2.10812
$656$	0	0
$657$	−14.5703	−0.568442
$658$	0	0
$659$	−7.82120	−0.304671	−0.152335	−	0.988329i	$-0.548679\pi$
$659$	−7.82120	−0.304671	−0.152335	+	0.988329i	$0.548679\pi$
$660$	0	0
$661$	22.4534	0.873336	0.436668	−	0.899623i	$-0.356159\pi$
$661$	22.4534	0.873336	0.436668	+	0.899623i	$0.356159\pi$
$662$	0	0
$663$	−5.52287	−0.214490
$664$	0	0
$665$	−16.3001	−0.632091
$666$	0	0
$667$	−27.2424	−1.05483
$668$	0	0
$669$	−24.1318	−0.932990
$670$	0	0
$671$	34.9336	1.34859
$672$	0	0
$673$	−24.6777	−0.951256	−0.475628	−	0.879647i	$-0.657779\pi$
$673$	−24.6777	−0.951256	−0.475628	+	0.879647i	$0.657779\pi$
$674$	0	0
$675$	−4.92419	−0.189532
$676$	0	0
$677$	38.6413	1.48510	0.742552	−	0.669788i	$-0.233615\pi$
$677$	38.6413	1.48510	0.742552	+	0.669788i	$0.233615\pi$
$678$	0	0
$679$	0.247620	0.00950279
$680$	0	0
$681$	−23.9961	−0.919532
$682$	0	0
$683$	13.3316	0.510120	0.255060	−	0.966925i	$-0.417905\pi$
$683$	13.3316	0.510120	0.255060	+	0.966925i	$0.417905\pi$
$684$	0	0
$685$	21.2920	0.813526
$686$	0	0
$687$	16.6446	0.635032
$688$	0	0
$689$	2.92371	0.111385
$690$	0	0
$691$	−21.1199	−0.803439	−0.401720	−	0.915763i	$-0.631587\pi$
$691$	−21.1199	−0.803439	−0.401720	+	0.915763i	$0.631587\pi$
$692$	0	0
$693$	7.80001	0.296298
$694$	0	0
$695$	14.4808	0.549289
$696$	0	0
$697$	23.7806	0.900755
$698$	0	0
$699$	−20.5875	−0.778692
$700$	0	0
$701$	−20.8009	−0.785638	−0.392819	−	0.919616i	$-0.628500\pi$
$701$	−20.8009	−0.785638	−0.392819	+	0.919616i	$0.628500\pi$
$702$	0	0
$703$	63.6003	2.39873
$704$	0	0
$705$	16.9616	0.638810
$706$	0	0
$707$	−10.4616	−0.393449
$708$	0	0
$709$	28.0183	1.05225	0.526125	−	0.850407i	$-0.323645\pi$
$709$	28.0183	1.05225	0.526125	+	0.850407i	$0.323645\pi$
$710$	0	0
$711$	−2.60025	−0.0975171
$712$	0	0
$713$	−1.00131	−0.0374995
$714$	0	0
$715$	−24.9588	−0.933407
$716$	0	0
$717$	−8.17791	−0.305410
$718$	0	0
$719$	−23.5265	−0.877391	−0.438695	−	0.898636i	$-0.644559\pi$
$719$	−23.5265	−0.877391	−0.438695	+	0.898636i	$0.644559\pi$
$720$	0	0
$721$	4.60674	0.171564
$722$	0	0
$723$	6.33240	0.235505
$724$	0	0
$725$	−8.38816	−0.311528
$726$	0	0
$727$	13.6375	0.505786	0.252893	−	0.967494i	$-0.418618\pi$
$727$	13.6375	0.505786	0.252893	+	0.967494i	$0.418618\pi$
$728$	0	0
$729$	4.26454	0.157946
$730$	0	0
$731$	12.9534	0.479097
$732$	0	0
$733$	−45.2947	−1.67300	−0.836498	−	0.547970i	$-0.815401\pi$
$733$	−45.2947	−1.67300	−0.836498	+	0.547970i	$0.815401\pi$
$734$	0	0
$735$	12.6368	0.466116
$736$	0	0
$737$	−15.8045	−0.582165
$738$	0	0
$739$	−38.2582	−1.40735	−0.703676	−	0.710521i	$-0.748459\pi$
$739$	−38.2582	−1.40735	−0.703676	+	0.710521i	$0.748459\pi$
$740$	0	0
$741$	−16.5083	−0.606447
$742$	0	0
$743$	−21.5410	−0.790264	−0.395132	−	0.918624i	$-0.629301\pi$
$743$	−21.5410	−0.790264	−0.395132	+	0.918624i	$0.629301\pi$
$744$	0	0
$745$	10.3836	0.380427
$746$	0	0
$747$	3.50038	0.128072
$748$	0	0
$749$	4.88209	0.178388
$750$	0	0
$751$	−2.05259	−0.0748999	−0.0374500	−	0.999299i	$-0.511923\pi$
$751$	−2.05259	−0.0748999	−0.0374500	+	0.999299i	$0.511923\pi$
$752$	0	0
$753$	0.844838	0.0307876
$754$	0	0
$755$	29.4038	1.07012
$756$	0	0
$757$	−24.1139	−0.876434	−0.438217	−	0.898869i	$-0.644390\pi$
$757$	−24.1139	−0.876434	−0.438217	+	0.898869i	$0.644390\pi$
$758$	0	0
$759$	10.6170	0.385372
$760$	0	0
$761$	−23.7308	−0.860239	−0.430120	−	0.902772i	$-0.641529\pi$
$761$	−23.7308	−0.860239	−0.430120	+	0.902772i	$0.641529\pi$
$762$	0	0
$763$	−0.0225809	−0.000817483 0
$764$	0	0
$765$	−12.8245	−0.463670
$766$	0	0
$767$	35.4342	1.27945
$768$	0	0
$769$	37.2832	1.34447	0.672233	−	0.740340i	$-0.265335\pi$
$769$	37.2832	1.34447	0.672233	+	0.740340i	$0.265335\pi$
$770$	0	0
$771$	18.6143	0.670376
$772$	0	0
$773$	0.720896	0.0259288	0.0129644	−	0.999916i	$-0.495873\pi$
$773$	0.720896	0.0259288	0.0129644	+	0.999916i	$0.495873\pi$
$774$	0	0
$775$	−0.308313	−0.0110749
$776$	0	0
$777$	7.69614	0.276097
$778$	0	0
$779$	71.0821	2.54678
$780$	0	0
$781$	34.4844	1.23395
$782$	0	0
$783$	33.9761	1.21421
$784$	0	0
$785$	−35.4340	−1.26469
$786$	0	0
$787$	−50.8920	−1.81410	−0.907051	−	0.421020i	$-0.861672\pi$
$787$	−50.8920	−1.81410	−0.907051	+	0.421020i	$0.861672\pi$
$788$	0	0
$789$	−0.369732	−0.0131628
$790$	0	0
$791$	−9.56293	−0.340018
$792$	0	0
$793$	28.6576	1.01766
$794$	0	0
$795$	−2.11950	−0.0751708
$796$	0	0
$797$	−14.0627	−0.498126	−0.249063	−	0.968487i	$-0.580123\pi$
$797$	−14.0627	−0.498126	−0.249063	+	0.968487i	$0.580123\pi$
$798$	0	0
$799$	18.4542	0.652863
$800$	0	0
$801$	−35.0376	−1.23799
$802$	0	0
$803$	22.3656	0.789265
$804$	0	0
$805$	8.59976	0.303102
$806$	0	0
$807$	0.140633	0.00495053
$808$	0	0
$809$	−44.6840	−1.57100	−0.785502	−	0.618859i	$-0.787595\pi$
$809$	−44.6840	−1.57100	−0.785502	+	0.618859i	$0.787595\pi$
$810$	0	0
$811$	9.73808	0.341950	0.170975	−	0.985275i	$-0.445308\pi$
$811$	9.73808	0.341950	0.170975	+	0.985275i	$0.445308\pi$
$812$	0	0
$813$	22.9585	0.805189
$814$	0	0
$815$	−15.7768	−0.552636
$816$	0	0
$817$	38.7186	1.35459
$818$	0	0
$819$	6.39871	0.223589
$820$	0	0
$821$	37.0650	1.29358	0.646788	−	0.762670i	$-0.276112\pi$
$821$	37.0650	1.29358	0.646788	+	0.762670i	$0.276112\pi$
$822$	0	0
$823$	−53.7145	−1.87237	−0.936185	−	0.351508i	$-0.885669\pi$
$823$	−53.7145	−1.87237	−0.936185	+	0.351508i	$0.885669\pi$
$824$	0	0
$825$	3.26905	0.113814
$826$	0	0
$827$	33.6240	1.16922	0.584611	−	0.811314i	$-0.301247\pi$
$827$	33.6240	1.16922	0.584611	+	0.811314i	$0.301247\pi$
$828$	0	0
$829$	−4.34280	−0.150832	−0.0754159	−	0.997152i	$-0.524028\pi$
$829$	−4.34280	−0.150832	−0.0754159	+	0.997152i	$0.524028\pi$
$830$	0	0
$831$	21.9720	0.762200
$832$	0	0
$833$	13.7489	0.476370
$834$	0	0
$835$	−51.3458	−1.77689
$836$	0	0
$837$	1.24882	0.0431655
$838$	0	0
$839$	20.3087	0.701135	0.350567	−	0.936538i	$-0.385989\pi$
$839$	20.3087	0.701135	0.350567	+	0.936538i	$0.385989\pi$
$840$	0	0
$841$	28.8769	0.995755
$842$	0	0
$843$	−11.9150	−0.410376
$844$	0	0
$845$	11.6396	0.400413
$846$	0	0
$847$	−1.27936	−0.0439595
$848$	0	0
$849$	−12.1904	−0.418375
$850$	0	0
$851$	−33.5549	−1.15025
$852$	0	0
$853$	4.46704	0.152948	0.0764742	−	0.997072i	$-0.475634\pi$
$853$	4.46704	0.152948	0.0764742	+	0.997072i	$0.475634\pi$
$854$	0	0
$855$	−38.3334	−1.31097
$856$	0	0
$857$	28.2401	0.964664	0.482332	−	0.875989i	$-0.339790\pi$
$857$	28.2401	0.964664	0.482332	+	0.875989i	$0.339790\pi$
$858$	0	0
$859$	−44.2974	−1.51141	−0.755703	−	0.654914i	$-0.772705\pi$
$859$	−44.2974	−1.51141	−0.755703	+	0.654914i	$0.772705\pi$
$860$	0	0
$861$	8.60150	0.293138
$862$	0	0
$863$	−8.12320	−0.276517	−0.138259	−	0.990396i	$-0.544150\pi$
$863$	−8.12320	−0.276517	−0.138259	+	0.990396i	$0.544150\pi$
$864$	0	0
$865$	−55.1046	−1.87361
$866$	0	0
$867$	−10.0062	−0.339828
$868$	0	0
$869$	3.99142	0.135399
$870$	0	0
$871$	−12.9651	−0.439307
$872$	0	0
$873$	0.582335	0.0197091
$874$	0	0
$875$	−9.35987	−0.316421
$876$	0	0
$877$	9.21603	0.311203	0.155602	−	0.987820i	$-0.450268\pi$
$877$	9.21603	0.311203	0.155602	+	0.987820i	$0.450268\pi$
$878$	0	0
$879$	23.7081	0.799655
$880$	0	0
$881$	−27.3672	−0.922025	−0.461013	−	0.887394i	$-0.652514\pi$
$881$	−27.3672	−0.922025	−0.461013	+	0.887394i	$0.652514\pi$
$882$	0	0
$883$	28.6386	0.963765	0.481883	−	0.876236i	$-0.339953\pi$
$883$	28.6386	0.963765	0.481883	+	0.876236i	$0.339953\pi$
$884$	0	0
$885$	−25.6874	−0.863474
$886$	0	0
$887$	31.8728	1.07018	0.535092	−	0.844794i	$-0.320277\pi$
$887$	31.8728	1.07018	0.535092	+	0.844794i	$0.320277\pi$
$888$	0	0
$889$	6.07052	0.203599
$890$	0	0
$891$	10.8290	0.362784
$892$	0	0
$893$	55.1610	1.84589
$894$	0	0
$895$	11.2991	0.377687
$896$	0	0
$897$	8.70959	0.290805
$898$	0	0
$899$	2.12731	0.0709498
$900$	0	0
$901$	−2.30601	−0.0768244
$902$	0	0
$903$	4.68525	0.155915
$904$	0	0
$905$	57.8871	1.92423
$906$	0	0
$907$	50.8857	1.68963	0.844816	−	0.535057i	$-0.179710\pi$
$907$	50.8857	1.68963	0.844816	+	0.535057i	$0.179710\pi$
$908$	0	0
$909$	−24.6028	−0.816025
$910$	0	0
$911$	18.1844	0.602477	0.301239	−	0.953549i	$-0.402600\pi$
$911$	18.1844	0.602477	0.301239	+	0.953549i	$0.402600\pi$
$912$	0	0
$913$	−5.37312	−0.177824
$914$	0	0
$915$	−20.7749	−0.686796
$916$	0	0
$917$	21.2323	0.701151
$918$	0	0
$919$	2.57600	0.0849745	0.0424872	−	0.999097i	$-0.486472\pi$
$919$	2.57600	0.0849745	0.0424872	+	0.999097i	$0.486472\pi$
$920$	0	0
$921$	18.5880	0.612496
$922$	0	0
$923$	28.2891	0.931148
$924$	0	0
$925$	−10.3318	−0.339708
$926$	0	0
$927$	10.8338	0.355828
$928$	0	0
$929$	23.1992	0.761142	0.380571	−	0.924752i	$-0.375727\pi$
$929$	23.1992	0.761142	0.380571	+	0.924752i	$0.375727\pi$
$930$	0	0
$931$	41.0964	1.34688
$932$	0	0
$933$	−5.77582	−0.189092
$934$	0	0
$935$	19.6857	0.643792
$936$	0	0
$937$	−15.9033	−0.519539	−0.259769	−	0.965671i	$-0.583647\pi$
$937$	−15.9033	−0.519539	−0.259769	+	0.965671i	$0.583647\pi$
$938$	0	0
$939$	−18.7498	−0.611878
$940$	0	0
$941$	−4.26329	−0.138979	−0.0694896	−	0.997583i	$-0.522137\pi$
$941$	−4.26329	−0.138979	−0.0694896	+	0.997583i	$0.522137\pi$
$942$	0	0
$943$	−37.5022	−1.22124
$944$	0	0
$945$	−10.7254	−0.348898
$946$	0	0
$947$	−7.78149	−0.252864	−0.126432	−	0.991975i	$-0.540353\pi$
$947$	−7.78149	−0.252864	−0.126432	+	0.991975i	$0.540353\pi$
$948$	0	0
$949$	18.3475	0.595586
$950$	0	0
$951$	12.2103	0.395947
$952$	0	0
$953$	18.4808	0.598650	0.299325	−	0.954151i	$-0.403238\pi$
$953$	18.4808	0.598650	0.299325	+	0.954151i	$0.403238\pi$
$954$	0	0
$955$	−23.9252	−0.774201
$956$	0	0
$957$	−22.5559	−0.729130
$958$	0	0
$959$	8.37908	0.270575
$960$	0	0
$961$	−30.9218	−0.997478
$962$	0	0
$963$	11.4813	0.369981
$964$	0	0
$965$	52.4113	1.68718
$966$	0	0
$967$	−8.95512	−0.287977	−0.143989	−	0.989579i	$-0.545993\pi$
$967$	−8.95512	−0.287977	−0.143989	+	0.989579i	$0.545993\pi$
$968$	0	0
$969$	13.0205	0.418280
$970$	0	0
$971$	−2.19052	−0.0702972	−0.0351486	−	0.999382i	$-0.511190\pi$
$971$	−2.19052	−0.0702972	−0.0351486	+	0.999382i	$0.511190\pi$
$972$	0	0
$973$	5.69866	0.182691
$974$	0	0
$975$	2.68176	0.0858850
$976$	0	0
$977$	−25.8102	−0.825741	−0.412871	−	0.910790i	$-0.635474\pi$
$977$	−25.8102	−0.825741	−0.412871	+	0.910790i	$0.635474\pi$
$978$	0	0
$979$	53.7831	1.71891
$980$	0	0
$981$	−0.0531041	−0.00169548
$982$	0	0
$983$	21.6394	0.690188	0.345094	−	0.938568i	$-0.387847\pi$
$983$	21.6394	0.690188	0.345094	+	0.938568i	$0.387847\pi$
$984$	0	0
$985$	−25.2719	−0.805230
$986$	0	0
$987$	6.67492	0.212465
$988$	0	0
$989$	−20.4275	−0.649557
$990$	0	0
$991$	−46.4242	−1.47471	−0.737357	−	0.675503i	$-0.763927\pi$
$991$	−46.4242	−1.47471	−0.737357	+	0.675503i	$0.763927\pi$
$992$	0	0
$993$	6.99231	0.221894
$994$	0	0
$995$	5.03916	0.159752
$996$	0	0
$997$	7.29268	0.230961	0.115481	−	0.993310i	$-0.463159\pi$
$997$	7.29268	0.230961	0.115481	+	0.993310i	$0.463159\pi$
$998$	0	0
$999$	41.8489	1.32404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1004.2.a.a.1.6	✓	7
3.2	odd	2		9036.2.a.i.1.7		7
4.3	odd	2		4016.2.a.g.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.6	✓	7	1.1	even	1	trivial
4016.2.a.g.1.2		7	4.3	odd	2
9036.2.a.i.1.7		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1004 = 2^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.844838 q^{3} -2.47034 q^{5} -0.972158 q^{7} -2.28625 q^{9} +O(q^{10})\)	\(q+0.844838 q^{3} -2.47034 q^{5} -0.972158 q^{7} -2.28625 q^{9} +3.50942 q^{11} +2.87894 q^{13} -2.08704 q^{15} -2.27070 q^{17} -6.78729 q^{19} -0.821315 q^{21} +3.58090 q^{23} +1.10259 q^{25} -4.46602 q^{27} -7.60769 q^{29} -0.279626 q^{31} +2.96489 q^{33} +2.40156 q^{35} -9.37051 q^{37} +2.43223 q^{39} -10.4728 q^{41} -5.70457 q^{43} +5.64782 q^{45} -8.12711 q^{47} -6.05491 q^{49} -1.91837 q^{51} +1.01555 q^{53} -8.66946 q^{55} -5.73416 q^{57} +12.3081 q^{59} +9.95423 q^{61} +2.22260 q^{63} -7.11196 q^{65} -4.50345 q^{67} +3.02528 q^{69} +9.82624 q^{71} +6.37302 q^{73} +0.931509 q^{75} -3.41171 q^{77} +1.13734 q^{79} +3.08569 q^{81} -1.53106 q^{83} +5.60940 q^{85} -6.42726 q^{87} +15.3254 q^{89} -2.79878 q^{91} -0.236239 q^{93} +16.7669 q^{95} -0.254712 q^{97} -8.02340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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