LMFDB - Embedded newform 9800.2.a.dc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 22x^{8} - 16x^{7} + 146x^{6} + 200x^{5} - 206x^{4} - 440x^{3} - 124x^{2} + 72x + 28 $ x^10 - 22x^8 - 16x^7 + 146x^6 + 200x^5 - 206x^4 - 440x^3 - 124x^2 + 72x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.50994$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12212 q^{3} -1.74084 q^{9} +O(q^{10})$	$q-1.12212 q^{3} -1.74084 q^{9} -4.94252 q^{11} +4.64101 q^{13} -5.72505 q^{17} +7.53763 q^{19} -6.87122 q^{23} +5.31980 q^{27} +8.11636 q^{29} -3.87614 q^{31} +5.54611 q^{33} +5.18786 q^{37} -5.20778 q^{39} -9.45170 q^{41} -0.706904 q^{43} +2.20616 q^{47} +6.42420 q^{51} -1.32166 q^{53} -8.45814 q^{57} -3.94064 q^{59} -6.07542 q^{61} -8.24674 q^{67} +7.71035 q^{69} -4.50952 q^{71} +2.93470 q^{73} -11.0249 q^{79} -0.746943 q^{81} -5.27356 q^{83} -9.10755 q^{87} -9.49970 q^{89} +4.34950 q^{93} -1.74808 q^{97} +8.60415 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 14 q^{9}+O(q^{10}) $ 10 * q + 14 * q^9	$ 10 q + 14 q^{9} - 12 q^{11} + 16 q^{23} + 12 q^{29} + 36 q^{37} - 20 q^{39} + 24 q^{43} - 36 q^{51} + 8 q^{53} + 16 q^{57} + 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} + 48 q^{93} - 72 q^{99}+O(q^{100}) $ 10 * q + 14 * q^9 - 12 * q^11 + 16 * q^23 + 12 * q^29 + 36 * q^37 - 20 * q^39 + 24 * q^43 - 36 * q^51 + 8 * q^53 + 16 * q^57 + 40 * q^67 - 8 * q^71 - 4 * q^79 + 50 * q^81 + 48 * q^93 - 72 * 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
$3$	−1.12212	−0.647857	−0.323929	−	0.946081i	$-0.605004\pi$
$3$	−1.12212	−0.647857	−0.323929	+	0.946081i	$0.605004\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.74084	−0.580281
$10$	0	0
$11$	−4.94252	−1.49023	−0.745113	−	0.666938i	$-0.767604\pi$
$11$	−4.94252	−1.49023	−0.745113	+	0.666938i	$0.767604\pi$
$12$	0	0
$13$	4.64101	1.28718	0.643592	−	0.765369i	$-0.277443\pi$
$13$	4.64101	1.28718	0.643592	+	0.765369i	$0.277443\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.72505	−1.38853	−0.694264	−	0.719720i	$-0.744270\pi$
$17$	−5.72505	−1.38853	−0.694264	+	0.719720i	$0.744270\pi$
$18$	0	0
$19$	7.53763	1.72925	0.864625	−	0.502417i	$-0.167556\pi$
$19$	7.53763	1.72925	0.864625	+	0.502417i	$0.167556\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.87122	−1.43275	−0.716374	−	0.697716i	$-0.754200\pi$
$23$	−6.87122	−1.43275	−0.716374	+	0.697716i	$0.754200\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.31980	1.02380
$28$	0	0
$29$	8.11636	1.50717	0.753585	−	0.657350i	$-0.228323\pi$
$29$	8.11636	1.50717	0.753585	+	0.657350i	$0.228323\pi$
$30$	0	0
$31$	−3.87614	−0.696175	−0.348087	−	0.937462i	$-0.613169\pi$
$31$	−3.87614	−0.696175	−0.348087	+	0.937462i	$0.613169\pi$
$32$	0	0
$33$	5.54611	0.965454
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.18786	0.852879	0.426439	−	0.904516i	$-0.359768\pi$
$37$	5.18786	0.852879	0.426439	+	0.904516i	$0.359768\pi$
$38$	0	0
$39$	−5.20778	−0.833912
$40$	0	0
$41$	−9.45170	−1.47611	−0.738054	−	0.674742i	$-0.764255\pi$
$41$	−9.45170	−1.47611	−0.738054	+	0.674742i	$0.764255\pi$
$42$	0	0
$43$	−0.706904	−0.107802	−0.0539009	−	0.998546i	$-0.517166\pi$
$43$	−0.706904	−0.107802	−0.0539009	+	0.998546i	$0.517166\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.20616	0.321802	0.160901	−	0.986971i	$-0.448560\pi$
$47$	2.20616	0.321802	0.160901	+	0.986971i	$0.448560\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.42420	0.899568
$52$	0	0
$53$	−1.32166	−0.181544	−0.0907721	−	0.995872i	$-0.528933\pi$
$53$	−1.32166	−0.181544	−0.0907721	+	0.995872i	$0.528933\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.45814	−1.12031
$58$	0	0
$59$	−3.94064	−0.513027	−0.256514	−	0.966541i	$-0.582574\pi$
$59$	−3.94064	−0.513027	−0.256514	+	0.966541i	$0.582574\pi$
$60$	0	0
$61$	−6.07542	−0.777878	−0.388939	−	0.921264i	$-0.627158\pi$
$61$	−6.07542	−0.777878	−0.388939	+	0.921264i	$0.627158\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.24674	−1.00750	−0.503750	−	0.863850i	$-0.668047\pi$
$67$	−8.24674	−1.00750	−0.503750	+	0.863850i	$0.668047\pi$
$68$	0	0
$69$	7.71035	0.928217
$70$	0	0
$71$	−4.50952	−0.535182	−0.267591	−	0.963533i	$-0.586228\pi$
$71$	−4.50952	−0.535182	−0.267591	+	0.963533i	$0.586228\pi$
$72$	0	0
$73$	2.93470	0.343481	0.171741	−	0.985142i	$-0.445061\pi$
$73$	2.93470	0.343481	0.171741	+	0.985142i	$0.445061\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.0249	−1.24040	−0.620201	−	0.784443i	$-0.712949\pi$
$79$	−11.0249	−1.24040	−0.620201	+	0.784443i	$0.712949\pi$
$80$	0	0
$81$	−0.746943	−0.0829937
$82$	0	0
$83$	−5.27356	−0.578848	−0.289424	−	0.957201i	$-0.593464\pi$
$83$	−5.27356	−0.578848	−0.289424	+	0.957201i	$0.593464\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.10755	−0.976432
$88$	0	0
$89$	−9.49970	−1.00697	−0.503483	−	0.864005i	$-0.667948\pi$
$89$	−9.49970	−1.00697	−0.503483	+	0.864005i	$0.667948\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.34950	0.451022
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.74808	−0.177491	−0.0887454	−	0.996054i	$-0.528286\pi$
$97$	−1.74808	−0.177491	−0.0887454	+	0.996054i	$0.528286\pi$
$98$	0	0
$99$	8.60415	0.864749
$100$	0	0
$101$	5.22585	0.519991	0.259996	−	0.965610i	$-0.416279\pi$
$101$	5.22585	0.519991	0.259996	+	0.965610i	$0.416279\pi$
$102$	0	0
$103$	−4.15144	−0.409053	−0.204527	−	0.978861i	$-0.565566\pi$
$103$	−4.15144	−0.409053	−0.204527	+	0.978861i	$0.565566\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.7234	1.61671	0.808356	−	0.588694i	$-0.200358\pi$
$107$	16.7234	1.61671	0.808356	+	0.588694i	$0.200358\pi$
$108$	0	0
$109$	11.3182	1.08409	0.542045	−	0.840349i	$-0.317650\pi$
$109$	11.3182	1.08409	0.542045	+	0.840349i	$0.317650\pi$
$110$	0	0
$111$	−5.82141	−0.552544
$112$	0	0
$113$	−0.614757	−0.0578315	−0.0289157	−	0.999582i	$-0.509205\pi$
$113$	−0.614757	−0.0578315	−0.0289157	+	0.999582i	$0.509205\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.07926	−0.746928
$118$	0	0
$119$	0	0
$120$	0	0
$121$	13.4285	1.22077
$122$	0	0
$123$	10.6060	0.956307
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.32166	0.472221	0.236111	−	0.971726i	$-0.424127\pi$
$127$	5.32166	0.472221	0.236111	+	0.971726i	$0.424127\pi$
$128$	0	0
$129$	0.793233	0.0698402
$130$	0	0
$131$	−7.30041	−0.637840	−0.318920	−	0.947782i	$-0.603320\pi$
$131$	−7.30041	−0.637840	−0.318920	+	0.947782i	$0.603320\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.9675	1.87681	0.938403	−	0.345542i	$-0.112305\pi$
$137$	21.9675	1.87681	0.938403	+	0.345542i	$0.112305\pi$
$138$	0	0
$139$	9.46849	0.803107	0.401553	−	0.915836i	$-0.368470\pi$
$139$	9.46849	0.803107	0.401553	+	0.915836i	$0.368470\pi$
$140$	0	0
$141$	−2.47558	−0.208482
$142$	0	0
$143$	−22.9383	−1.91820
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.7823	−1.29293	−0.646467	−	0.762942i	$-0.723754\pi$
$149$	−15.7823	−1.29293	−0.646467	+	0.762942i	$0.723754\pi$
$150$	0	0
$151$	11.2669	0.916884	0.458442	−	0.888724i	$-0.348408\pi$
$151$	11.2669	0.916884	0.458442	+	0.888724i	$0.348408\pi$
$152$	0	0
$153$	9.96640	0.805736
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.66981	−0.691926	−0.345963	−	0.938248i	$-0.612448\pi$
$157$	−8.66981	−0.691926	−0.345963	+	0.938248i	$0.612448\pi$
$158$	0	0
$159$	1.48307	0.117615
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.1275	1.88981	0.944905	−	0.327344i	$-0.106154\pi$
$163$	24.1275	1.88981	0.944905	+	0.327344i	$0.106154\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.9825	−1.31414	−0.657071	−	0.753829i	$-0.728205\pi$
$167$	−16.9825	−1.31414	−0.657071	+	0.753829i	$0.728205\pi$
$168$	0	0
$169$	8.53896	0.656843
$170$	0	0
$171$	−13.1218	−1.00345
$172$	0	0
$173$	19.7691	1.50302	0.751509	−	0.659723i	$-0.229327\pi$
$173$	19.7691	1.50302	0.751509	+	0.659723i	$0.229327\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.42188	0.332368
$178$	0	0
$179$	−2.11496	−0.158080	−0.0790398	−	0.996871i	$-0.525185\pi$
$179$	−2.11496	−0.158080	−0.0790398	+	0.996871i	$0.525185\pi$
$180$	0	0
$181$	9.23706	0.686585	0.343293	−	0.939229i	$-0.388458\pi$
$181$	9.23706	0.686585	0.343293	+	0.939229i	$0.388458\pi$
$182$	0	0
$183$	6.81736	0.503954
$184$	0	0
$185$	0	0
$186$	0	0
$187$	28.2962	2.06922
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.75734	−0.488944	−0.244472	−	0.969656i	$-0.578615\pi$
$191$	−6.75734	−0.488944	−0.244472	+	0.969656i	$0.578615\pi$
$192$	0	0
$193$	2.71570	0.195480	0.0977402	−	0.995212i	$-0.468839\pi$
$193$	2.71570	0.195480	0.0977402	+	0.995212i	$0.468839\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.2189	1.22680	0.613398	−	0.789774i	$-0.289802\pi$
$197$	17.2189	1.22680	0.613398	+	0.789774i	$0.289802\pi$
$198$	0	0
$199$	−14.0215	−0.993956	−0.496978	−	0.867763i	$-0.665557\pi$
$199$	−14.0215	−0.993956	−0.496978	+	0.867763i	$0.665557\pi$
$200$	0	0
$201$	9.25385	0.652716
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	11.9617	0.831396
$208$	0	0
$209$	−37.2549	−2.57697
$210$	0	0
$211$	−9.78518	−0.673639	−0.336820	−	0.941569i	$-0.609351\pi$
$211$	−9.78518	−0.673639	−0.336820	+	0.941569i	$0.609351\pi$
$212$	0	0
$213$	5.06023	0.346721
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.29310	−0.222527
$220$	0	0
$221$	−26.5700	−1.78729
$222$	0	0
$223$	7.13608	0.477867	0.238934	−	0.971036i	$-0.423202\pi$
$223$	7.13608	0.477867	0.238934	+	0.971036i	$0.423202\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.773180	−0.0513178	−0.0256589	−	0.999671i	$-0.508168\pi$
$227$	−0.773180	−0.0513178	−0.0256589	+	0.999671i	$0.508168\pi$
$228$	0	0
$229$	14.2336	0.940580	0.470290	−	0.882512i	$-0.344149\pi$
$229$	14.2336	0.940580	0.470290	+	0.882512i	$0.344149\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.7275	−0.768294	−0.384147	−	0.923272i	$-0.625504\pi$
$233$	−11.7275	−0.768294	−0.384147	+	0.923272i	$0.625504\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.3713	0.803604
$238$	0	0
$239$	−14.3330	−0.927124	−0.463562	−	0.886065i	$-0.653429\pi$
$239$	−14.3330	−0.927124	−0.463562	+	0.886065i	$0.653429\pi$
$240$	0	0
$241$	2.07456	0.133634	0.0668171	−	0.997765i	$-0.478716\pi$
$241$	2.07456	0.133634	0.0668171	+	0.997765i	$0.478716\pi$
$242$	0	0
$243$	−15.1212	−0.970029
$244$	0	0
$245$	0	0
$246$	0	0
$247$	34.9822	2.22586
$248$	0	0
$249$	5.91758	0.375011
$250$	0	0
$251$	−2.81192	−0.177487	−0.0887435	−	0.996055i	$-0.528285\pi$
$251$	−2.81192	−0.177487	−0.0887435	+	0.996055i	$0.528285\pi$
$252$	0	0
$253$	33.9611	2.13512
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.5852	1.34644	0.673222	−	0.739440i	$-0.264910\pi$
$257$	21.5852	1.34644	0.673222	+	0.739440i	$0.264910\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−14.1293	−0.874582
$262$	0	0
$263$	16.9163	1.04310	0.521551	−	0.853220i	$-0.325354\pi$
$263$	16.9163	1.04310	0.521551	+	0.853220i	$0.325354\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.6598	0.652370
$268$	0	0
$269$	−11.3325	−0.690953	−0.345477	−	0.938427i	$-0.612283\pi$
$269$	−11.3325	−0.690953	−0.345477	+	0.938427i	$0.612283\pi$
$270$	0	0
$271$	−3.11364	−0.189140	−0.0945701	−	0.995518i	$-0.530148\pi$
$271$	−3.11364	−0.189140	−0.0945701	+	0.995518i	$0.530148\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.6607	1.60189	0.800945	−	0.598739i	$-0.204331\pi$
$277$	26.6607	1.60189	0.800945	+	0.598739i	$0.204331\pi$
$278$	0	0
$279$	6.74774	0.403977
$280$	0	0
$281$	28.2579	1.68572	0.842861	−	0.538131i	$-0.180869\pi$
$281$	28.2579	1.68572	0.842861	+	0.538131i	$0.180869\pi$
$282$	0	0
$283$	13.9531	0.829428	0.414714	−	0.909952i	$-0.363882\pi$
$283$	13.9531	0.829428	0.414714	+	0.909952i	$0.363882\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.7762	0.928011
$290$	0	0
$291$	1.96156	0.114989
$292$	0	0
$293$	6.50405	0.379970	0.189985	−	0.981787i	$-0.439156\pi$
$293$	6.50405	0.379970	0.189985	+	0.981787i	$0.439156\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−26.2932	−1.52569
$298$	0	0
$299$	−31.8894	−1.84421
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−5.86404	−0.336880
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.4580	1.62418	0.812091	−	0.583531i	$-0.198329\pi$
$307$	28.4580	1.62418	0.812091	+	0.583531i	$0.198329\pi$
$308$	0	0
$309$	4.65842	0.265008
$310$	0	0
$311$	−6.10465	−0.346163	−0.173081	−	0.984908i	$-0.555372\pi$
$311$	−6.10465	−0.346163	−0.173081	+	0.984908i	$0.555372\pi$
$312$	0	0
$313$	4.42029	0.249850	0.124925	−	0.992166i	$-0.460131\pi$
$313$	4.42029	0.249850	0.124925	+	0.992166i	$0.460131\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.67611	−0.318802	−0.159401	−	0.987214i	$-0.550956\pi$
$317$	−5.67611	−0.318802	−0.159401	+	0.987214i	$0.550956\pi$
$318$	0	0
$319$	−40.1153	−2.24602
$320$	0	0
$321$	−18.7657	−1.04740
$322$	0	0
$323$	−43.1533	−2.40111
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.7004	−0.702336
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.4091	−1.34165	−0.670823	−	0.741618i	$-0.734059\pi$
$331$	−24.4091	−1.34165	−0.670823	+	0.741618i	$0.734059\pi$
$332$	0	0
$333$	−9.03124	−0.494909
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.84018	0.154714	0.0773572	−	0.997003i	$-0.475352\pi$
$337$	2.84018	0.154714	0.0773572	+	0.997003i	$0.475352\pi$
$338$	0	0
$339$	0.689832	0.0374666
$340$	0	0
$341$	19.1579	1.03746
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.47921	0.294139	0.147070	−	0.989126i	$-0.453016\pi$
$347$	5.47921	0.294139	0.147070	+	0.989126i	$0.453016\pi$
$348$	0	0
$349$	−11.5197	−0.616637	−0.308318	−	0.951283i	$-0.599766\pi$
$349$	−11.5197	−0.616637	−0.308318	+	0.951283i	$0.599766\pi$
$350$	0	0
$351$	24.6893	1.31781
$352$	0	0
$353$	−11.1927	−0.595726	−0.297863	−	0.954609i	$-0.596274\pi$
$353$	−11.1927	−0.595726	−0.297863	+	0.954609i	$0.596274\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.2726	1.86161	0.930807	−	0.365511i	$-0.119106\pi$
$359$	35.2726	1.86161	0.930807	+	0.365511i	$0.119106\pi$
$360$	0	0
$361$	37.8159	1.99031
$362$	0	0
$363$	−15.0684	−0.790887
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−31.1383	−1.62541	−0.812704	−	0.582677i	$-0.802005\pi$
$367$	−31.1383	−1.62541	−0.812704	+	0.582677i	$0.802005\pi$
$368$	0	0
$369$	16.4539	0.856557
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.9531	0.567131	0.283565	−	0.958953i	$-0.408483\pi$
$373$	10.9531	0.567131	0.283565	+	0.958953i	$0.408483\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.6681	1.94001
$378$	0	0
$379$	7.34338	0.377204	0.188602	−	0.982054i	$-0.439604\pi$
$379$	7.34338	0.377204	0.188602	+	0.982054i	$0.439604\pi$
$380$	0	0
$381$	−5.97155	−0.305932
$382$	0	0
$383$	−3.73491	−0.190845	−0.0954224	−	0.995437i	$-0.530420\pi$
$383$	−3.73491	−0.190845	−0.0954224	+	0.995437i	$0.530420\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.23061	0.0625553
$388$	0	0
$389$	9.29745	0.471399	0.235700	−	0.971826i	$-0.424262\pi$
$389$	9.29745	0.471399	0.235700	+	0.971826i	$0.424262\pi$
$390$	0	0
$391$	39.3381	1.98941
$392$	0	0
$393$	8.19195	0.413229
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.2764	0.666325	0.333163	−	0.942869i	$-0.391884\pi$
$397$	13.2764	0.666325	0.333163	+	0.942869i	$0.391884\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.01019	−0.200260	−0.100130	−	0.994974i	$-0.531926\pi$
$401$	−4.01019	−0.200260	−0.100130	+	0.994974i	$0.531926\pi$
$402$	0	0
$403$	−17.9892	−0.896105
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.6411	−1.27098
$408$	0	0
$409$	29.2260	1.44513	0.722567	−	0.691301i	$-0.242962\pi$
$409$	29.2260	1.44513	0.722567	+	0.691301i	$0.242962\pi$
$410$	0	0
$411$	−24.6502	−1.21590
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.6248	−0.520299
$418$	0	0
$419$	29.8552	1.45852	0.729260	−	0.684237i	$-0.239864\pi$
$419$	29.8552	1.45852	0.729260	+	0.684237i	$0.239864\pi$
$420$	0	0
$421$	−9.81281	−0.478247	−0.239124	−	0.970989i	$-0.576860\pi$
$421$	−9.81281	−0.478247	−0.239124	+	0.970989i	$0.576860\pi$
$422$	0	0
$423$	−3.84058	−0.186735
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	25.7395	1.24272
$430$	0	0
$431$	−17.0840	−0.822908	−0.411454	−	0.911430i	$-0.634979\pi$
$431$	−17.0840	−0.822908	−0.411454	+	0.911430i	$0.634979\pi$
$432$	0	0
$433$	−9.70709	−0.466493	−0.233246	−	0.972418i	$-0.574935\pi$
$433$	−9.70709	−0.466493	−0.233246	+	0.972418i	$0.574935\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−51.7927	−2.47758
$438$	0	0
$439$	−10.4222	−0.497425	−0.248713	−	0.968577i	$-0.580007\pi$
$439$	−10.4222	−0.497425	−0.248713	+	0.968577i	$0.580007\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.8412	1.65535	0.827677	−	0.561205i	$-0.189662\pi$
$443$	34.8412	1.65535	0.827677	+	0.561205i	$0.189662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.7096	0.837637
$448$	0	0
$449$	−8.64043	−0.407767	−0.203883	−	0.978995i	$-0.565356\pi$
$449$	−8.64043	−0.407767	−0.203883	+	0.978995i	$0.565356\pi$
$450$	0	0
$451$	46.7152	2.19973
$452$	0	0
$453$	−12.6428	−0.594010
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.6058	1.19779	0.598893	−	0.800829i	$-0.295607\pi$
$457$	25.6058	1.19779	0.598893	+	0.800829i	$0.295607\pi$
$458$	0	0
$459$	−30.4561	−1.42157
$460$	0	0
$461$	23.8105	1.10896	0.554482	−	0.832196i	$-0.312917\pi$
$461$	23.8105	1.10896	0.554482	+	0.832196i	$0.312917\pi$
$462$	0	0
$463$	3.27660	0.152276	0.0761382	−	0.997097i	$-0.475741\pi$
$463$	3.27660	0.152276	0.0761382	+	0.997097i	$0.475741\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.1321	1.34807	0.674036	−	0.738699i	$-0.264559\pi$
$467$	29.1321	1.34807	0.674036	+	0.738699i	$0.264559\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.72859	0.448270
$472$	0	0
$473$	3.49389	0.160649
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.30080	0.105347
$478$	0	0
$479$	−7.58771	−0.346692	−0.173346	−	0.984861i	$-0.555458\pi$
$479$	−7.58771	−0.346692	−0.173346	+	0.984861i	$0.555458\pi$
$480$	0	0
$481$	24.0769	1.09781
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.86118	0.220281	0.110140	−	0.993916i	$-0.464870\pi$
$487$	4.86118	0.220281	0.110140	+	0.993916i	$0.464870\pi$
$488$	0	0
$489$	−27.0740	−1.22433
$490$	0	0
$491$	5.95002	0.268521	0.134260	−	0.990946i	$-0.457134\pi$
$491$	5.95002	0.268521	0.134260	+	0.990946i	$0.457134\pi$
$492$	0	0
$493$	−46.4666	−2.09275
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.08297	−0.0932467	−0.0466234	−	0.998913i	$-0.514846\pi$
$499$	−2.08297	−0.0932467	−0.0466234	+	0.998913i	$0.514846\pi$
$500$	0	0
$501$	19.0564	0.851377
$502$	0	0
$503$	−2.22949	−0.0994081	−0.0497041	−	0.998764i	$-0.515828\pi$
$503$	−2.22949	−0.0994081	−0.0497041	+	0.998764i	$0.515828\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.58176	−0.425541
$508$	0	0
$509$	−1.75149	−0.0776334	−0.0388167	−	0.999246i	$-0.512359\pi$
$509$	−1.75149	−0.0776334	−0.0388167	+	0.999246i	$0.512359\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	40.0987	1.77040
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10.9040	−0.479557
$518$	0	0
$519$	−22.1833	−0.973741
$520$	0	0
$521$	16.7933	0.735727	0.367864	−	0.929880i	$-0.380089\pi$
$521$	16.7933	0.735727	0.367864	+	0.929880i	$0.380089\pi$
$522$	0	0
$523$	20.3763	0.890995	0.445497	−	0.895283i	$-0.353027\pi$
$523$	20.3763	0.890995	0.445497	+	0.895283i	$0.353027\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.1911	0.966658
$528$	0	0
$529$	24.2137	1.05277
$530$	0	0
$531$	6.86002	0.297700
$532$	0	0
$533$	−43.8654	−1.90002
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.37324	0.102413
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.4407	−0.534867	−0.267434	−	0.963576i	$-0.586176\pi$
$541$	−12.4407	−0.534867	−0.267434	+	0.963576i	$0.586176\pi$
$542$	0	0
$543$	−10.3651	−0.444809
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.2004	1.41955	0.709773	−	0.704431i	$-0.248798\pi$
$547$	33.2004	1.41955	0.709773	+	0.704431i	$0.248798\pi$
$548$	0	0
$549$	10.5763	0.451388
$550$	0	0
$551$	61.1781	2.60628
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.0813	−1.10510	−0.552551	−	0.833479i	$-0.686345\pi$
$557$	−26.0813	−1.10510	−0.552551	+	0.833479i	$0.686345\pi$
$558$	0	0
$559$	−3.28075	−0.138761
$560$	0	0
$561$	−31.7517	−1.34056
$562$	0	0
$563$	−39.4346	−1.66197	−0.830986	−	0.556294i	$-0.812223\pi$
$563$	−39.4346	−1.66197	−0.830986	+	0.556294i	$0.812223\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.09101	−0.255348	−0.127674	−	0.991816i	$-0.540751\pi$
$569$	−6.09101	−0.255348	−0.127674	+	0.991816i	$0.540751\pi$
$570$	0	0
$571$	−24.9957	−1.04604	−0.523019	−	0.852321i	$-0.675194\pi$
$571$	−24.9957	−1.04604	−0.523019	+	0.852321i	$0.675194\pi$
$572$	0	0
$573$	7.58256	0.316766
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.4463	0.976083	0.488042	−	0.872820i	$-0.337711\pi$
$577$	23.4463	0.976083	0.488042	+	0.872820i	$0.337711\pi$
$578$	0	0
$579$	−3.04735	−0.126643
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.53234	0.270542
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.6033	1.13931	0.569656	−	0.821884i	$-0.307077\pi$
$587$	27.6033	1.13931	0.569656	+	0.821884i	$0.307077\pi$
$588$	0	0
$589$	−29.2169	−1.20386
$590$	0	0
$591$	−19.3217	−0.794789
$592$	0	0
$593$	5.40275	0.221864	0.110932	−	0.993828i	$-0.464616\pi$
$593$	5.40275	0.221864	0.110932	+	0.993828i	$0.464616\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.7338	0.643942
$598$	0	0
$599$	42.1269	1.72126	0.860628	−	0.509234i	$-0.170071\pi$
$599$	42.1269	1.72126	0.860628	+	0.509234i	$0.170071\pi$
$600$	0	0
$601$	1.70777	0.0696615	0.0348307	−	0.999393i	$-0.488911\pi$
$601$	1.70777	0.0696615	0.0348307	+	0.999393i	$0.488911\pi$
$602$	0	0
$603$	14.3563	0.584632
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13.1419	−0.533412	−0.266706	−	0.963778i	$-0.585935\pi$
$607$	−13.1419	−0.533412	−0.266706	+	0.963778i	$0.585935\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.2388	0.414218
$612$	0	0
$613$	38.3179	1.54764	0.773822	−	0.633403i	$-0.218342\pi$
$613$	38.3179	1.54764	0.773822	+	0.633403i	$0.218342\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.55459	−0.384653	−0.192327	−	0.981331i	$-0.561603\pi$
$617$	−9.55459	−0.384653	−0.192327	+	0.981331i	$0.561603\pi$
$618$	0	0
$619$	30.6354	1.23134	0.615670	−	0.788004i	$-0.288885\pi$
$619$	30.6354	1.23134	0.615670	+	0.788004i	$0.288885\pi$
$620$	0	0
$621$	−36.5535	−1.46684
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	41.8045	1.66951
$628$	0	0
$629$	−29.7007	−1.18425
$630$	0	0
$631$	−31.4786	−1.25314	−0.626571	−	0.779364i	$-0.715542\pi$
$631$	−31.4786	−1.25314	−0.626571	+	0.779364i	$0.715542\pi$
$632$	0	0
$633$	10.9802	0.436422
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.85036	0.310556
$640$	0	0
$641$	21.4201	0.846041	0.423021	−	0.906120i	$-0.360970\pi$
$641$	21.4201	0.846041	0.423021	+	0.906120i	$0.360970\pi$
$642$	0	0
$643$	−5.83210	−0.229996	−0.114998	−	0.993366i	$-0.536686\pi$
$643$	−5.83210	−0.229996	−0.114998	+	0.993366i	$0.536686\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.1428	−1.02778	−0.513890	−	0.857856i	$-0.671796\pi$
$647$	−26.1428	−1.02778	−0.513890	+	0.857856i	$0.671796\pi$
$648$	0	0
$649$	19.4767	0.764526
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.1019	−1.45191	−0.725955	−	0.687742i	$-0.758602\pi$
$653$	−37.1019	−1.45191	−0.725955	+	0.687742i	$0.758602\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.10886	−0.199315
$658$	0	0
$659$	−49.4758	−1.92730	−0.963652	−	0.267159i	$-0.913915\pi$
$659$	−49.4758	−1.92730	−0.963652	+	0.267159i	$0.913915\pi$
$660$	0	0
$661$	−28.1885	−1.09640	−0.548202	−	0.836346i	$-0.684687\pi$
$661$	−28.1885	−1.09640	−0.548202	+	0.836346i	$0.684687\pi$
$662$	0	0
$663$	29.8148	1.15791
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−55.7693	−2.15940
$668$	0	0
$669$	−8.00755	−0.309590
$670$	0	0
$671$	30.0279	1.15921
$672$	0	0
$673$	−14.0752	−0.542558	−0.271279	−	0.962501i	$-0.587447\pi$
$673$	−14.0752	−0.542558	−0.271279	+	0.962501i	$0.587447\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.9999	1.07612	0.538062	−	0.842906i	$-0.319157\pi$
$677$	27.9999	1.07612	0.538062	+	0.842906i	$0.319157\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.867603	0.0332466
$682$	0	0
$683$	−25.1558	−0.962562	−0.481281	−	0.876566i	$-0.659828\pi$
$683$	−25.1558	−0.962562	−0.481281	+	0.876566i	$0.659828\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15.9718	−0.609362
$688$	0	0
$689$	−6.13384	−0.233681
$690$	0	0
$691$	−18.6990	−0.711344	−0.355672	−	0.934611i	$-0.615748\pi$
$691$	−18.6990	−0.711344	−0.355672	+	0.934611i	$0.615748\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	54.1114	2.04962
$698$	0	0
$699$	13.1597	0.497745
$700$	0	0
$701$	25.1354	0.949351	0.474676	−	0.880161i	$-0.342565\pi$
$701$	25.1354	0.949351	0.474676	+	0.880161i	$0.342565\pi$
$702$	0	0
$703$	39.1042	1.47484
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.6659	0.663458	0.331729	−	0.943375i	$-0.392368\pi$
$709$	17.6659	0.663458	0.331729	+	0.943375i	$0.392368\pi$
$710$	0	0
$711$	19.1927	0.719782
$712$	0	0
$713$	26.6338	0.997443
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0834	0.600644
$718$	0	0
$719$	−4.06812	−0.151715	−0.0758576	−	0.997119i	$-0.524169\pi$
$719$	−4.06812	−0.151715	−0.0758576	+	0.997119i	$0.524169\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.32791	−0.0865760
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35.5626	−1.31894	−0.659471	−	0.751730i	$-0.729220\pi$
$727$	−35.5626	−1.31894	−0.659471	+	0.751730i	$0.729220\pi$
$728$	0	0
$729$	19.2087	0.711434
$730$	0	0
$731$	4.04706	0.149686
$732$	0	0
$733$	−1.90581	−0.0703929	−0.0351964	−	0.999380i	$-0.511206\pi$
$733$	−1.90581	−0.0703929	−0.0351964	+	0.999380i	$0.511206\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.7597	1.50140
$738$	0	0
$739$	−5.44004	−0.200115	−0.100058	−	0.994982i	$-0.531903\pi$
$739$	−5.44004	−0.200115	−0.100058	+	0.994982i	$0.531903\pi$
$740$	0	0
$741$	−39.2543	−1.44204
$742$	0	0
$743$	42.7120	1.56695	0.783475	−	0.621423i	$-0.213445\pi$
$743$	42.7120	1.56695	0.783475	+	0.621423i	$0.213445\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.18043	0.335895
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15.9029	−0.580306	−0.290153	−	0.956980i	$-0.593706\pi$
$751$	−15.9029	−0.580306	−0.290153	+	0.956980i	$0.593706\pi$
$752$	0	0
$753$	3.15532	0.114986
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.28840	0.0468277	0.0234138	−	0.999726i	$-0.492546\pi$
$757$	1.28840	0.0468277	0.0234138	+	0.999726i	$0.492546\pi$
$758$	0	0
$759$	−38.1085	−1.38325
$760$	0	0
$761$	−18.7660	−0.680266	−0.340133	−	0.940377i	$-0.610472\pi$
$761$	−18.7660	−0.680266	−0.340133	+	0.940377i	$0.610472\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.2885	−0.660360
$768$	0	0
$769$	39.3190	1.41788	0.708939	−	0.705269i	$-0.249174\pi$
$769$	39.3190	1.41788	0.708939	+	0.705269i	$0.249174\pi$
$770$	0	0
$771$	−24.2212	−0.872304
$772$	0	0
$773$	11.9516	0.429868	0.214934	−	0.976629i	$-0.431046\pi$
$773$	11.9516	0.429868	0.214934	+	0.976629i	$0.431046\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−71.2434	−2.55256
$780$	0	0
$781$	22.2884	0.797541
$782$	0	0
$783$	43.1774	1.54304
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−36.6318	−1.30578	−0.652892	−	0.757451i	$-0.726445\pi$
$787$	−36.6318	−1.30578	−0.652892	+	0.757451i	$0.726445\pi$
$788$	0	0
$789$	−18.9821	−0.675782
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−28.1961	−1.00127
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.35871	−0.0481281	−0.0240641	−	0.999710i	$-0.507661\pi$
$797$	−1.35871	−0.0481281	−0.0240641	+	0.999710i	$0.507661\pi$
$798$	0	0
$799$	−12.6304	−0.446831
$800$	0	0
$801$	16.5375	0.584323
$802$	0	0
$803$	−14.5048	−0.511864
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.7164	0.447639
$808$	0	0
$809$	55.6391	1.95617	0.978084	−	0.208211i	$-0.0667642\pi$
$809$	55.6391	1.95617	0.978084	+	0.208211i	$0.0667642\pi$
$810$	0	0
$811$	−54.1690	−1.90213	−0.951065	−	0.308990i	$-0.900009\pi$
$811$	−54.1690	−1.90213	−0.951065	+	0.308990i	$0.900009\pi$
$812$	0	0
$813$	3.49389	0.122536
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.32838	−0.186416
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46.2489	1.61410	0.807048	−	0.590486i	$-0.201064\pi$
$821$	46.2489	1.61410	0.807048	+	0.590486i	$0.201064\pi$
$822$	0	0
$823$	54.1540	1.88769	0.943845	−	0.330388i	$-0.107179\pi$
$823$	54.1540	1.88769	0.943845	+	0.330388i	$0.107179\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.0834	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$827$	10.0834	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$828$	0	0
$829$	39.0572	1.35651	0.678256	−	0.734825i	$-0.262736\pi$
$829$	39.0572	1.35651	0.678256	+	0.734825i	$0.262736\pi$
$830$	0	0
$831$	−29.9166	−1.03780
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−20.6203	−0.712741
$838$	0	0
$839$	44.4278	1.53382	0.766909	−	0.641756i	$-0.221794\pi$
$839$	44.4278	1.53382	0.766909	+	0.641756i	$0.221794\pi$
$840$	0	0
$841$	36.8753	1.27156
$842$	0	0
$843$	−31.7088	−1.09211
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−15.6571	−0.537351
$850$	0	0
$851$	−35.6469	−1.22196
$852$	0	0
$853$	−49.0774	−1.68038	−0.840189	−	0.542293i	$-0.817556\pi$
$853$	−49.0774	−1.68038	−0.840189	+	0.542293i	$0.817556\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.7257	1.15205	0.576025	−	0.817432i	$-0.304603\pi$
$857$	33.7257	1.15205	0.576025	+	0.817432i	$0.304603\pi$
$858$	0	0
$859$	−27.8680	−0.950843	−0.475421	−	0.879758i	$-0.657704\pi$
$859$	−27.8680	−0.950843	−0.475421	+	0.879758i	$0.657704\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.3459	−0.794702	−0.397351	−	0.917667i	$-0.630070\pi$
$863$	−23.3459	−0.794702	−0.397351	+	0.917667i	$0.630070\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−17.7028	−0.601219
$868$	0	0
$869$	54.4910	1.84848
$870$	0	0
$871$	−38.2732	−1.29684
$872$	0	0
$873$	3.04313	0.102994
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.4536	0.758206	0.379103	−	0.925355i	$-0.376233\pi$
$877$	22.4536	0.758206	0.379103	+	0.925355i	$0.376233\pi$
$878$	0	0
$879$	−7.29833	−0.246167
$880$	0	0
$881$	−2.44467	−0.0823629	−0.0411814	−	0.999152i	$-0.513112\pi$
$881$	−2.44467	−0.0823629	−0.0411814	+	0.999152i	$0.513112\pi$
$882$	0	0
$883$	42.2730	1.42260	0.711299	−	0.702889i	$-0.248107\pi$
$883$	42.2730	1.42260	0.711299	+	0.702889i	$0.248107\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.29839	−0.0771725	−0.0385863	−	0.999255i	$-0.512285\pi$
$887$	−2.29839	−0.0771725	−0.0385863	+	0.999255i	$0.512285\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.69178	0.123679
$892$	0	0
$893$	16.6292	0.556476
$894$	0	0
$895$	0	0
$896$	0	0
$897$	35.7838	1.19479
$898$	0	0
$899$	−31.4601	−1.04925
$900$	0	0
$901$	7.56657	0.252079
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	54.1626	1.79844	0.899220	−	0.437497i	$-0.144135\pi$
$907$	54.1626	1.79844	0.899220	+	0.437497i	$0.144135\pi$
$908$	0	0
$909$	−9.09738	−0.301741
$910$	0	0
$911$	7.71481	0.255603	0.127801	−	0.991800i	$-0.459208\pi$
$911$	7.71481	0.255603	0.127801	+	0.991800i	$0.459208\pi$
$912$	0	0
$913$	26.0647	0.862615
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.86010	0.226294	0.113147	−	0.993578i	$-0.463907\pi$
$919$	6.86010	0.226294	0.113147	+	0.993578i	$0.463907\pi$
$920$	0	0
$921$	−31.9333	−1.05224
$922$	0	0
$923$	−20.9287	−0.688877
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.22700	0.237366
$928$	0	0
$929$	19.2013	0.629975	0.314988	−	0.949096i	$-0.398000\pi$
$929$	19.2013	0.629975	0.314988	+	0.949096i	$0.398000\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	6.85016	0.224264
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.3868	0.666009	0.333004	−	0.942925i	$-0.391938\pi$
$937$	20.3868	0.666009	0.333004	+	0.942925i	$0.391938\pi$
$938$	0	0
$939$	−4.96011	−0.161867
$940$	0	0
$941$	44.2362	1.44206	0.721030	−	0.692904i	$-0.243669\pi$
$941$	44.2362	1.44206	0.721030	+	0.692904i	$0.243669\pi$
$942$	0	0
$943$	64.9447	2.11489
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.4842	1.18558	0.592788	−	0.805359i	$-0.298027\pi$
$947$	36.4842	1.18558	0.592788	+	0.805359i	$0.298027\pi$
$948$	0	0
$949$	13.6200	0.442123
$950$	0	0
$951$	6.36929	0.206538
$952$	0	0
$953$	2.51474	0.0814606	0.0407303	−	0.999170i	$-0.487032\pi$
$953$	2.51474	0.0814606	0.0407303	+	0.999170i	$0.487032\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	45.0142	1.45510
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.9756	−0.515341
$962$	0	0
$963$	−29.1128	−0.938147
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.49369	0.0801915	0.0400958	−	0.999196i	$-0.487234\pi$
$967$	2.49369	0.0801915	0.0400958	+	0.999196i	$0.487234\pi$
$968$	0	0
$969$	48.4233	1.55558
$970$	0	0
$971$	15.3021	0.491069	0.245534	−	0.969388i	$-0.421037\pi$
$971$	15.3021	0.491069	0.245534	+	0.969388i	$0.421037\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.5170	−1.23227	−0.616134	−	0.787641i	$-0.711302\pi$
$977$	−38.5170	−1.23227	−0.616134	+	0.787641i	$0.711302\pi$
$978$	0	0
$979$	46.9524	1.50061
$980$	0	0
$981$	−19.7033	−0.629077
$982$	0	0
$983$	19.9438	0.636107	0.318054	−	0.948073i	$-0.396971\pi$
$983$	19.9438	0.636107	0.318054	+	0.948073i	$0.396971\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.85729	0.154453
$990$	0	0
$991$	−1.09531	−0.0347937	−0.0173968	−	0.999849i	$-0.505538\pi$
$991$	−1.09531	−0.0347937	−0.0173968	+	0.999849i	$0.505538\pi$
$992$	0	0
$993$	27.3900	0.869195
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−43.8274	−1.38803	−0.694014	−	0.719961i	$-0.744159\pi$
$997$	−43.8274	−1.38803	−0.694014	+	0.719961i	$0.744159\pi$
$998$	0	0
$999$	27.5984	0.873175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.dc.1.4		10
5.2	odd	4		1960.2.g.g.1569.14	yes	20
5.3	odd	4		1960.2.g.g.1569.8	yes	20
5.4	even	2		9800.2.a.db.1.7		10
7.6	odd	2	inner	9800.2.a.dc.1.7		10
35.13	even	4		1960.2.g.g.1569.13	yes	20
35.27	even	4		1960.2.g.g.1569.7	✓	20
35.34	odd	2		9800.2.a.db.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.g.g.1569.7	✓	20	35.27	even	4
1960.2.g.g.1569.8	yes	20	5.3	odd	4
1960.2.g.g.1569.13	yes	20	35.13	even	4
1960.2.g.g.1569.14	yes	20	5.2	odd	4
9800.2.a.db.1.4		10	35.34	odd	2
9800.2.a.db.1.7		10	5.4	even	2
9800.2.a.dc.1.4		10	1.1	even	1	trivial
9800.2.a.dc.1.7		10	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.12212 q^{3} -1.74084 q^{9} +O(q^{10})\)	\(q-1.12212 q^{3} -1.74084 q^{9} -4.94252 q^{11} +4.64101 q^{13} -5.72505 q^{17} +7.53763 q^{19} -6.87122 q^{23} +5.31980 q^{27} +8.11636 q^{29} -3.87614 q^{31} +5.54611 q^{33} +5.18786 q^{37} -5.20778 q^{39} -9.45170 q^{41} -0.706904 q^{43} +2.20616 q^{47} +6.42420 q^{51} -1.32166 q^{53} -8.45814 q^{57} -3.94064 q^{59} -6.07542 q^{61} -8.24674 q^{67} +7.71035 q^{69} -4.50952 q^{71} +2.93470 q^{73} -11.0249 q^{79} -0.746943 q^{81} -5.27356 q^{83} -9.10755 q^{87} -9.49970 q^{89} +4.34950 q^{93} -1.74808 q^{97} +8.60415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 14 q^{9}+O(q^{10}) \) 10 * q + 14 * q^9	\( 10 q + 14 q^{9} - 12 q^{11} + 16 q^{23} + 12 q^{29} + 36 q^{37} - 20 q^{39} + 24 q^{43} - 36 q^{51} + 8 q^{53} + 16 q^{57} + 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} + 48 q^{93} - 72 q^{99}+O(q^{100}) \) 10 * q + 14 * q^9 - 12 * q^11 + 16 * q^23 + 12 * q^29 + 36 * q^37 - 20 * q^39 + 24 * q^43 - 36 * q^51 + 8 * q^53 + 16 * q^57 + 40 * q^67 - 8 * q^71 - 4 * q^79 + 50 * q^81 + 48 * q^93 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more