LMFDB - Embedded newform 7600.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.65109$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.65109 q^{3} -3.65109 q^{7} -0.273891 q^{9} +O(q^{10})$	$q+1.65109 q^{3} -3.65109 q^{7} -0.273891 q^{9} -2.65109 q^{11} +6.13161 q^{13} +2.34891 q^{17} -1.00000 q^{19} -6.02830 q^{21} -5.48052 q^{23} -5.40550 q^{27} +0.651093 q^{29} +6.67939 q^{31} -4.37720 q^{33} -8.70769 q^{37} +10.1239 q^{39} +1.93273 q^{41} +2.65884 q^{43} -3.71836 q^{47} +6.33048 q^{49} +3.87826 q^{51} +13.7544 q^{53} -1.65109 q^{57} +7.84997 q^{59} -1.92498 q^{61} +1.00000 q^{63} +4.44447 q^{67} -9.04884 q^{69} -3.54778 q^{71} -2.48052 q^{73} +9.67939 q^{77} +15.1599 q^{79} -8.10331 q^{81} +14.7282 q^{83} +1.07502 q^{87} -5.06727 q^{89} -22.3871 q^{91} +11.0283 q^{93} +3.22717 q^{97} +0.726109 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 4 q^{7} + q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 4 * q^7 + q^9	$ 3 q - 2 q^{3} - 4 q^{7} + q^{9} - q^{11} + 3 q^{13} + 14 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + q^{27} - 5 q^{29} + q^{31} - 8 q^{33} + 5 q^{37} + 11 q^{39} + q^{41} + 5 q^{43} - 9 q^{47} - 7 q^{49} - 18 q^{51} + 31 q^{53} + 2 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 13 q^{67} + q^{69} - 7 q^{71} + q^{73} + 10 q^{77} + 18 q^{79} - 21 q^{81} - 3 q^{83} + 12 q^{87} - 20 q^{89} - 17 q^{91} + 21 q^{93} - 13 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 4 * q^7 + q^9 - q^11 + 3 * q^13 + 14 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + q^27 - 5 * q^29 + q^31 - 8 * q^33 + 5 * q^37 + 11 * q^39 + q^41 + 5 * q^43 - 9 * q^47 - 7 * q^49 - 18 * q^51 + 31 * q^53 + 2 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 13 * q^67 + q^69 - 7 * q^71 + q^73 + 10 * q^77 + 18 * q^79 - 21 * q^81 - 3 * q^83 + 12 * q^87 - 20 * q^89 - 17 * q^91 + 21 * q^93 - 13 * q^97 + 4 * 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
$3$	1.65109	0.953259	0.476630	−	0.879104i	$-0.341858\pi$
$3$	1.65109	0.953259	0.476630	+	0.879104i	$0.341858\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.65109	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$7$	−3.65109	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$8$	0	0
$9$	−0.273891	−0.0912969
$10$	0	0
$11$	−2.65109	−0.799335	−0.399667	−	0.916660i	$-0.630874\pi$
$11$	−2.65109	−0.799335	−0.399667	+	0.916660i	$0.630874\pi$
$12$	0	0
$13$	6.13161	1.70060	0.850301	−	0.526297i	$-0.176420\pi$
$13$	6.13161	1.70060	0.850301	+	0.526297i	$0.176420\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.34891	0.569694	0.284847	−	0.958573i	$-0.408057\pi$
$17$	2.34891	0.569694	0.284847	+	0.958573i	$0.408057\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−6.02830	−1.31548
$22$	0	0
$23$	−5.48052	−1.14277	−0.571383	−	0.820683i	$-0.693593\pi$
$23$	−5.48052	−1.14277	−0.571383	+	0.820683i	$0.693593\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.40550	−1.04029
$28$	0	0
$29$	0.651093	0.120905	0.0604525	−	0.998171i	$-0.480746\pi$
$29$	0.651093	0.120905	0.0604525	+	0.998171i	$0.480746\pi$
$30$	0	0
$31$	6.67939	1.19965	0.599827	−	0.800130i	$-0.295236\pi$
$31$	6.67939	1.19965	0.599827	+	0.800130i	$0.295236\pi$
$32$	0	0
$33$	−4.37720	−0.761973
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.70769	−1.43153	−0.715767	−	0.698339i	$-0.753923\pi$
$37$	−8.70769	−1.43153	−0.715767	+	0.698339i	$0.753923\pi$
$38$	0	0
$39$	10.1239	1.62111
$40$	0	0
$41$	1.93273	0.301842	0.150921	−	0.988546i	$-0.451776\pi$
$41$	1.93273	0.301842	0.150921	+	0.988546i	$0.451776\pi$
$42$	0	0
$43$	2.65884	0.405470	0.202735	−	0.979234i	$-0.435017\pi$
$43$	2.65884	0.405470	0.202735	+	0.979234i	$0.435017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.71836	−0.542378	−0.271189	−	0.962526i	$-0.587417\pi$
$47$	−3.71836	−0.542378	−0.271189	+	0.962526i	$0.587417\pi$
$48$	0	0
$49$	6.33048	0.904355
$50$	0	0
$51$	3.87826	0.543066
$52$	0	0
$53$	13.7544	1.88931	0.944656	−	0.328061i	$-0.106395\pi$
$53$	13.7544	1.88931	0.944656	+	0.328061i	$0.106395\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.65109	−0.218693
$58$	0	0
$59$	7.84997	1.02198	0.510989	−	0.859587i	$-0.329279\pi$
$59$	7.84997	1.02198	0.510989	+	0.859587i	$0.329279\pi$
$60$	0	0
$61$	−1.92498	−0.246469	−0.123234	−	0.992378i	$-0.539327\pi$
$61$	−1.92498	−0.246469	−0.123234	+	0.992378i	$0.539327\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.44447	0.542978	0.271489	−	0.962442i	$-0.412484\pi$
$67$	4.44447	0.542978	0.271489	+	0.962442i	$0.412484\pi$
$68$	0	0
$69$	−9.04884	−1.08935
$70$	0	0
$71$	−3.54778	−0.421044	−0.210522	−	0.977589i	$-0.567516\pi$
$71$	−3.54778	−0.421044	−0.210522	+	0.977589i	$0.567516\pi$
$72$	0	0
$73$	−2.48052	−0.290322	−0.145161	−	0.989408i	$-0.546370\pi$
$73$	−2.48052	−0.290322	−0.145161	+	0.989408i	$0.546370\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.67939	1.10307
$78$	0	0
$79$	15.1599	1.70562	0.852811	−	0.522219i	$-0.174896\pi$
$79$	15.1599	1.70562	0.852811	+	0.522219i	$0.174896\pi$
$80$	0	0
$81$	−8.10331	−0.900368
$82$	0	0
$83$	14.7282	1.61663	0.808317	−	0.588748i	$-0.200379\pi$
$83$	14.7282	1.61663	0.808317	+	0.588748i	$0.200379\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.07502	0.115254
$88$	0	0
$89$	−5.06727	−0.537129	−0.268565	−	0.963262i	$-0.586549\pi$
$89$	−5.06727	−0.537129	−0.268565	+	0.963262i	$0.586549\pi$
$90$	0	0
$91$	−22.3871	−2.34680
$92$	0	0
$93$	11.0283	1.14358
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.22717	0.327670	0.163835	−	0.986488i	$-0.447614\pi$
$97$	3.22717	0.327670	0.163835	+	0.986488i	$0.447614\pi$
$98$	0	0
$99$	0.726109	0.0729767
$100$	0	0
$101$	16.8032	1.67199	0.835993	−	0.548740i	$-0.184892\pi$
$101$	16.8032	1.67199	0.835993	+	0.548740i	$0.184892\pi$
$102$	0	0
$103$	9.85772	0.971310	0.485655	−	0.874151i	$-0.338581\pi$
$103$	9.85772	0.971310	0.485655	+	0.874151i	$0.338581\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.13936	0.496841	0.248420	−	0.968652i	$-0.420089\pi$
$107$	5.13936	0.496841	0.248420	+	0.968652i	$0.420089\pi$
$108$	0	0
$109$	13.9738	1.33845	0.669225	−	0.743060i	$-0.266626\pi$
$109$	13.9738	1.33845	0.669225	+	0.743060i	$0.266626\pi$
$110$	0	0
$111$	−14.3772	−1.36462
$112$	0	0
$113$	−0.527235	−0.0495981	−0.0247990	−	0.999692i	$-0.507895\pi$
$113$	−0.527235	−0.0495981	−0.0247990	+	0.999692i	$0.507895\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.67939	−0.155260
$118$	0	0
$119$	−8.57608	−0.786168
$120$	0	0
$121$	−3.97170	−0.361064
$122$	0	0
$123$	3.19112	0.287734
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.8217	1.04900	0.524502	−	0.851409i	$-0.324252\pi$
$127$	11.8217	1.04900	0.524502	+	0.851409i	$0.324252\pi$
$128$	0	0
$129$	4.39000	0.386518
$130$	0	0
$131$	−8.12386	−0.709785	−0.354892	−	0.934907i	$-0.615483\pi$
$131$	−8.12386	−0.709785	−0.354892	+	0.934907i	$0.615483\pi$
$132$	0	0
$133$	3.65109	0.316590
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.5761	1.50163	0.750813	−	0.660515i	$-0.229662\pi$
$137$	17.5761	1.50163	0.750813	+	0.660515i	$0.229662\pi$
$138$	0	0
$139$	−18.4154	−1.56197	−0.780986	−	0.624549i	$-0.785283\pi$
$139$	−18.4154	−1.56197	−0.780986	+	0.624549i	$0.785283\pi$
$140$	0	0
$141$	−6.13936	−0.517027
$142$	0	0
$143$	−16.2555	−1.35935
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.4522	0.862084
$148$	0	0
$149$	−17.2915	−1.41658	−0.708288	−	0.705924i	$-0.750532\pi$
$149$	−17.2915	−1.41658	−0.708288	+	0.705924i	$0.750532\pi$
$150$	0	0
$151$	−2.58383	−0.210269	−0.105134	−	0.994458i	$-0.533527\pi$
$151$	−2.58383	−0.210269	−0.105134	+	0.994458i	$0.533527\pi$
$152$	0	0
$153$	−0.643343	−0.0520112
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.9738	−0.875807	−0.437903	−	0.899022i	$-0.644279\pi$
$157$	−10.9738	−0.875807	−0.437903	+	0.899022i	$0.644279\pi$
$158$	0	0
$159$	22.7098	1.80100
$160$	0	0
$161$	20.0099	1.57700
$162$	0	0
$163$	−22.6794	−1.77639	−0.888193	−	0.459470i	$-0.848039\pi$
$163$	−22.6794	−1.77639	−0.888193	+	0.459470i	$0.848039\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.16283	0.399512	0.199756	−	0.979846i	$-0.435985\pi$
$167$	5.16283	0.399512	0.199756	+	0.979846i	$0.435985\pi$
$168$	0	0
$169$	24.5966	1.89205
$170$	0	0
$171$	0.273891	0.0209449
$172$	0	0
$173$	−17.2165	−1.30895	−0.654473	−	0.756085i	$-0.727109\pi$
$173$	−17.2165	−1.30895	−0.654473	+	0.756085i	$0.727109\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.9610	0.974211
$178$	0	0
$179$	−10.6999	−0.799751	−0.399875	−	0.916570i	$-0.630947\pi$
$179$	−10.6999	−0.799751	−0.399875	+	0.916570i	$0.630947\pi$
$180$	0	0
$181$	−16.7720	−1.24666	−0.623328	−	0.781961i	$-0.714220\pi$
$181$	−16.7720	−1.24666	−0.623328	+	0.781961i	$0.714220\pi$
$182$	0	0
$183$	−3.17833	−0.234949
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.22717	−0.455376
$188$	0	0
$189$	19.7360	1.43558
$190$	0	0
$191$	13.8598	1.00286	0.501431	−	0.865197i	$-0.332807\pi$
$191$	13.8598	1.00286	0.501431	+	0.865197i	$0.332807\pi$
$192$	0	0
$193$	21.0694	1.51661	0.758304	−	0.651901i	$-0.226028\pi$
$193$	21.0694	1.51661	0.758304	+	0.651901i	$0.226028\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.2555	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$197$	21.2555	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$198$	0	0
$199$	7.69006	0.545134	0.272567	−	0.962137i	$-0.412127\pi$
$199$	7.69006	0.545134	0.272567	+	0.962137i	$0.412127\pi$
$200$	0	0
$201$	7.33823	0.517599
$202$	0	0
$203$	−2.37720	−0.166847
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.50106	0.104331
$208$	0	0
$209$	2.65109	0.183380
$210$	0	0
$211$	1.69781	0.116882	0.0584411	−	0.998291i	$-0.481387\pi$
$211$	1.69781	0.116882	0.0584411	+	0.998291i	$0.481387\pi$
$212$	0	0
$213$	−5.85772	−0.401364
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.3871	−1.65550
$218$	0	0
$219$	−4.09556	−0.276752
$220$	0	0
$221$	14.4026	0.968822
$222$	0	0
$223$	25.2632	1.69175	0.845875	−	0.533381i	$-0.179079\pi$
$223$	25.2632	1.69175	0.845875	+	0.533381i	$0.179079\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.8217	1.11649	0.558247	−	0.829675i	$-0.311474\pi$
$227$	16.8217	1.11649	0.558247	+	0.829675i	$0.311474\pi$
$228$	0	0
$229$	−14.6249	−0.966442	−0.483221	−	0.875498i	$-0.660533\pi$
$229$	−14.6249	−0.966442	−0.483221	+	0.875498i	$0.660533\pi$
$230$	0	0
$231$	15.9816	1.05151
$232$	0	0
$233$	8.91431	0.583996	0.291998	−	0.956419i	$-0.405680\pi$
$233$	8.91431	0.583996	0.291998	+	0.956419i	$0.405680\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	25.0304	1.62590
$238$	0	0
$239$	24.6015	1.59134	0.795668	−	0.605733i	$-0.207120\pi$
$239$	24.6015	1.59134	0.795668	+	0.605733i	$0.207120\pi$
$240$	0	0
$241$	−14.6150	−0.941438	−0.470719	−	0.882283i	$-0.656005\pi$
$241$	−14.6150	−0.941438	−0.470719	+	0.882283i	$0.656005\pi$
$242$	0	0
$243$	2.83717	0.182005
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.13161	−0.390145
$248$	0	0
$249$	24.3177	1.54107
$250$	0	0
$251$	13.3382	0.841902	0.420951	−	0.907083i	$-0.361696\pi$
$251$	13.3382	0.841902	0.420951	+	0.907083i	$0.361696\pi$
$252$	0	0
$253$	14.5294	0.913453
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.35103	−0.209031	−0.104516	−	0.994523i	$-0.533329\pi$
$257$	−3.35103	−0.209031	−0.104516	+	0.994523i	$0.533329\pi$
$258$	0	0
$259$	31.7926	1.97549
$260$	0	0
$261$	−0.178328	−0.0110382
$262$	0	0
$263$	−10.8860	−0.671260	−0.335630	−	0.941994i	$-0.608949\pi$
$263$	−10.8860	−0.671260	−0.335630	+	0.941994i	$0.608949\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.36653	−0.512023
$268$	0	0
$269$	18.4338	1.12393	0.561964	−	0.827162i	$-0.310046\pi$
$269$	18.4338	1.12393	0.561964	+	0.827162i	$0.310046\pi$
$270$	0	0
$271$	20.4076	1.23967	0.619837	−	0.784730i	$-0.287199\pi$
$271$	20.4076	1.23967	0.619837	+	0.784730i	$0.287199\pi$
$272$	0	0
$273$	−36.9632	−2.23711
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.69994	−0.162223	−0.0811117	−	0.996705i	$-0.525847\pi$
$277$	−2.69994	−0.162223	−0.0811117	+	0.996705i	$0.525847\pi$
$278$	0	0
$279$	−1.82942	−0.109525
$280$	0	0
$281$	−15.2242	−0.908202	−0.454101	−	0.890950i	$-0.650040\pi$
$281$	−15.2242	−0.908202	−0.454101	+	0.890950i	$0.650040\pi$
$282$	0	0
$283$	−6.18045	−0.367390	−0.183695	−	0.982983i	$-0.558806\pi$
$283$	−6.18045	−0.367390	−0.183695	+	0.982983i	$0.558806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.05659	−0.416537
$288$	0	0
$289$	−11.4826	−0.675449
$290$	0	0
$291$	5.32836	0.312354
$292$	0	0
$293$	30.6893	1.79289	0.896443	−	0.443159i	$-0.146142\pi$
$293$	30.6893	1.79289	0.896443	+	0.443159i	$0.146142\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.3305	0.831539
$298$	0	0
$299$	−33.6044	−1.94339
$300$	0	0
$301$	−9.70769	−0.559542
$302$	0	0
$303$	27.7437	1.59384
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.7827	0.843693	0.421847	−	0.906667i	$-0.361382\pi$
$307$	14.7827	0.843693	0.421847	+	0.906667i	$0.361382\pi$
$308$	0	0
$309$	16.2760	0.925910
$310$	0	0
$311$	26.9992	1.53098	0.765492	−	0.643445i	$-0.222496\pi$
$311$	26.9992	1.53098	0.765492	+	0.643445i	$0.222496\pi$
$312$	0	0
$313$	−9.74666	−0.550914	−0.275457	−	0.961313i	$-0.588829\pi$
$313$	−9.74666	−0.550914	−0.275457	+	0.961313i	$0.588829\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.7261	1.10793	0.553964	−	0.832540i	$-0.313114\pi$
$317$	19.7261	1.10793	0.553964	+	0.832540i	$0.313114\pi$
$318$	0	0
$319$	−1.72611	−0.0966436
$320$	0	0
$321$	8.48556	0.473618
$322$	0	0
$323$	−2.34891	−0.130697
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.0721	1.27589
$328$	0	0
$329$	13.5761	0.748473
$330$	0	0
$331$	−19.9426	−1.09614	−0.548072	−	0.836431i	$-0.684638\pi$
$331$	−19.9426	−1.09614	−0.548072	+	0.836431i	$0.684638\pi$
$332$	0	0
$333$	2.38495	0.130695
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.05952	0.112189	0.0560945	−	0.998425i	$-0.482135\pi$
$337$	2.05952	0.112189	0.0560945	+	0.998425i	$0.482135\pi$
$338$	0	0
$339$	−0.870514	−0.0472798
$340$	0	0
$341$	−17.7077	−0.958925
$342$	0	0
$343$	2.44447	0.131989
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.5011	0.563727	0.281863	−	0.959455i	$-0.409048\pi$
$347$	10.5011	0.563727	0.281863	+	0.959455i	$0.409048\pi$
$348$	0	0
$349$	16.5059	0.883540	0.441770	−	0.897128i	$-0.354351\pi$
$349$	16.5059	0.883540	0.441770	+	0.897128i	$0.354351\pi$
$350$	0	0
$351$	−33.1444	−1.76912
$352$	0	0
$353$	15.8860	0.845527	0.422764	−	0.906240i	$-0.361060\pi$
$353$	15.8860	0.845527	0.422764	+	0.906240i	$0.361060\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14.1599	−0.749422
$358$	0	0
$359$	−1.28376	−0.0677544	−0.0338772	−	0.999426i	$-0.510786\pi$
$359$	−1.28376	−0.0677544	−0.0338772	+	0.999426i	$0.510786\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−6.55765	−0.344188
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.8804	1.03775	0.518874	−	0.854851i	$-0.326351\pi$
$367$	19.8804	1.03775	0.518874	+	0.854851i	$0.326351\pi$
$368$	0	0
$369$	−0.529358	−0.0275573
$370$	0	0
$371$	−50.2186	−2.60722
$372$	0	0
$373$	−16.4359	−0.851020	−0.425510	−	0.904954i	$-0.639905\pi$
$373$	−16.4359	−0.851020	−0.425510	+	0.904954i	$0.639905\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.99225	0.205611
$378$	0	0
$379$	−0.338233	−0.0173739	−0.00868694	−	0.999962i	$-0.502765\pi$
$379$	−0.338233	−0.0173739	−0.00868694	+	0.999962i	$0.502765\pi$
$380$	0	0
$381$	19.5187	0.999972
$382$	0	0
$383$	13.7339	0.701767	0.350884	−	0.936419i	$-0.385881\pi$
$383$	13.7339	0.701767	0.350884	+	0.936419i	$0.385881\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.728232	−0.0370181
$388$	0	0
$389$	−2.26109	−0.114642	−0.0573210	−	0.998356i	$-0.518256\pi$
$389$	−2.26109	−0.114642	−0.0573210	+	0.998356i	$0.518256\pi$
$390$	0	0
$391$	−12.8732	−0.651027
$392$	0	0
$393$	−13.4132	−0.676609
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.82942	−0.392947	−0.196474	−	0.980509i	$-0.562949\pi$
$397$	−7.82942	−0.392947	−0.196474	+	0.980509i	$0.562949\pi$
$398$	0	0
$399$	6.02830	0.301792
$400$	0	0
$401$	−35.0510	−1.75036	−0.875181	−	0.483796i	$-0.839258\pi$
$401$	−35.0510	−1.75036	−0.875181	+	0.483796i	$0.839258\pi$
$402$	0	0
$403$	40.9554	2.04013
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.0849	1.14428
$408$	0	0
$409$	−8.34811	−0.412787	−0.206394	−	0.978469i	$-0.566173\pi$
$409$	−8.34811	−0.412787	−0.206394	+	0.978469i	$0.566173\pi$
$410$	0	0
$411$	29.0197	1.43144
$412$	0	0
$413$	−28.6610	−1.41031
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−30.4055	−1.48896
$418$	0	0
$419$	19.8705	0.970738	0.485369	−	0.874309i	$-0.338685\pi$
$419$	19.8705	0.970738	0.485369	+	0.874309i	$0.338685\pi$
$420$	0	0
$421$	−0.400672	−0.0195276	−0.00976379	−	0.999952i	$-0.503108\pi$
$421$	−0.400672	−0.0195276	−0.00976379	+	0.999952i	$0.503108\pi$
$422$	0	0
$423$	1.01842	0.0495174
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.02830	0.340123
$428$	0	0
$429$	−26.8393	−1.29581
$430$	0	0
$431$	14.2533	0.686559	0.343280	−	0.939233i	$-0.388462\pi$
$431$	14.2533	0.686559	0.343280	+	0.939233i	$0.388462\pi$
$432$	0	0
$433$	12.8393	0.617017	0.308509	−	0.951222i	$-0.400170\pi$
$433$	12.8393	0.617017	0.308509	+	0.951222i	$0.400170\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.48052	0.262169
$438$	0	0
$439$	13.2555	0.632649	0.316324	−	0.948651i	$-0.397551\pi$
$439$	13.2555	0.632649	0.316324	+	0.948651i	$0.397551\pi$
$440$	0	0
$441$	−1.73386	−0.0825647
$442$	0	0
$443$	−5.72611	−0.272056	−0.136028	−	0.990705i	$-0.543434\pi$
$443$	−5.72611	−0.272056	−0.136028	+	0.990705i	$0.543434\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−28.5499	−1.35036
$448$	0	0
$449$	3.11399	0.146958	0.0734790	−	0.997297i	$-0.476590\pi$
$449$	3.11399	0.146958	0.0734790	+	0.997297i	$0.476590\pi$
$450$	0	0
$451$	−5.12386	−0.241273
$452$	0	0
$453$	−4.26614	−0.200441
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.4535	1.28422	0.642111	−	0.766611i	$-0.278059\pi$
$457$	27.4535	1.28422	0.642111	+	0.766611i	$0.278059\pi$
$458$	0	0
$459$	−12.6970	−0.592646
$460$	0	0
$461$	13.9837	0.651286	0.325643	−	0.945493i	$-0.394419\pi$
$461$	13.9837	0.651286	0.325643	+	0.945493i	$0.394419\pi$
$462$	0	0
$463$	−17.9992	−0.836494	−0.418247	−	0.908333i	$-0.637355\pi$
$463$	−17.9992	−0.836494	−0.418247	+	0.908333i	$0.637355\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.6871	−0.587091	−0.293545	−	0.955945i	$-0.594835\pi$
$467$	−12.6871	−0.587091	−0.293545	+	0.955945i	$0.594835\pi$
$468$	0	0
$469$	−16.2272	−0.749301
$470$	0	0
$471$	−18.1188	−0.834871
$472$	0	0
$473$	−7.04884	−0.324106
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.76720	−0.172488
$478$	0	0
$479$	−24.7544	−1.13106	−0.565529	−	0.824729i	$-0.691328\pi$
$479$	−24.7544	−1.13106	−0.565529	+	0.824729i	$0.691328\pi$
$480$	0	0
$481$	−53.3921	−2.43447
$482$	0	0
$483$	33.0382	1.50329
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.08277	0.185008	0.0925039	−	0.995712i	$-0.470513\pi$
$487$	4.08277	0.185008	0.0925039	+	0.995712i	$0.470513\pi$
$488$	0	0
$489$	−37.4458	−1.69336
$490$	0	0
$491$	−29.2547	−1.32024	−0.660122	−	0.751158i	$-0.729496\pi$
$491$	−29.2547	−1.32024	−0.660122	+	0.751158i	$0.729496\pi$
$492$	0	0
$493$	1.52936	0.0688788
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.9533	0.581034
$498$	0	0
$499$	−1.57315	−0.0704240	−0.0352120	−	0.999380i	$-0.511211\pi$
$499$	−1.57315	−0.0704240	−0.0352120	+	0.999380i	$0.511211\pi$
$500$	0	0
$501$	8.52431	0.380838
$502$	0	0
$503$	−20.7819	−0.926619	−0.463310	−	0.886196i	$-0.653338\pi$
$503$	−20.7819	−0.926619	−0.463310	+	0.886196i	$0.653338\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	40.6113	1.80361
$508$	0	0
$509$	−31.8238	−1.41056	−0.705282	−	0.708926i	$-0.749180\pi$
$509$	−31.8238	−1.41056	−0.705282	+	0.708926i	$0.749180\pi$
$510$	0	0
$511$	9.05659	0.400640
$512$	0	0
$513$	5.40550	0.238659
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.85772	0.433542
$518$	0	0
$519$	−28.4260	−1.24776
$520$	0	0
$521$	−27.4720	−1.20357	−0.601784	−	0.798659i	$-0.705543\pi$
$521$	−27.4720	−1.20357	−0.601784	+	0.798659i	$0.705543\pi$
$522$	0	0
$523$	10.4466	0.456798	0.228399	−	0.973568i	$-0.426651\pi$
$523$	10.4466	0.456798	0.228399	+	0.973568i	$0.426651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.6893	0.683435
$528$	0	0
$529$	7.03605	0.305915
$530$	0	0
$531$	−2.15003	−0.0933034
$532$	0	0
$533$	11.8508	0.513314
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17.6666	−0.762370
$538$	0	0
$539$	−16.7827	−0.722882
$540$	0	0
$541$	13.4989	0.580365	0.290182	−	0.956971i	$-0.406284\pi$
$541$	13.4989	0.580365	0.290182	+	0.956971i	$0.406284\pi$
$542$	0	0
$543$	−27.6922	−1.18839
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.54698	−0.365442	−0.182721	−	0.983165i	$-0.558491\pi$
$547$	−8.54698	−0.365442	−0.182721	+	0.983165i	$0.558491\pi$
$548$	0	0
$549$	0.527235	0.0225018
$550$	0	0
$551$	−0.651093	−0.0277375
$552$	0	0
$553$	−55.3502	−2.35373
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.2378	1.15410	0.577052	−	0.816707i	$-0.304203\pi$
$557$	27.2378	1.15410	0.577052	+	0.816707i	$0.304203\pi$
$558$	0	0
$559$	16.3030	0.689543
$560$	0	0
$561$	−10.2816	−0.434091
$562$	0	0
$563$	1.44235	0.0607876	0.0303938	−	0.999538i	$-0.490324\pi$
$563$	1.44235	0.0607876	0.0303938	+	0.999538i	$0.490324\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	29.5860	1.24249
$568$	0	0
$569$	−14.2632	−0.597945	−0.298973	−	0.954262i	$-0.596644\pi$
$569$	−14.2632	−0.597945	−0.298973	+	0.954262i	$0.596644\pi$
$570$	0	0
$571$	−9.91723	−0.415023	−0.207512	−	0.978233i	$-0.566536\pi$
$571$	−9.91723	−0.415023	−0.207512	+	0.978233i	$0.566536\pi$
$572$	0	0
$573$	22.8839	0.955988
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.23997	−0.343034	−0.171517	−	0.985181i	$-0.554867\pi$
$577$	−8.23997	−0.343034	−0.171517	+	0.985181i	$0.554867\pi$
$578$	0	0
$579$	34.7875	1.44572
$580$	0	0
$581$	−53.7742	−2.23093
$582$	0	0
$583$	−36.4642	−1.51019
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−36.9554	−1.52531	−0.762656	−	0.646804i	$-0.776105\pi$
$587$	−36.9554	−1.52531	−0.762656	+	0.646804i	$0.776105\pi$
$588$	0	0
$589$	−6.67939	−0.275219
$590$	0	0
$591$	35.0948	1.44361
$592$	0	0
$593$	−1.36170	−0.0559184	−0.0279592	−	0.999609i	$-0.508901\pi$
$593$	−1.36170	−0.0559184	−0.0279592	+	0.999609i	$0.508901\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.6970	0.519654
$598$	0	0
$599$	−33.5753	−1.37185	−0.685924	−	0.727673i	$-0.740602\pi$
$599$	−33.5753	−1.37185	−0.685924	+	0.727673i	$0.740602\pi$
$600$	0	0
$601$	12.6561	0.516255	0.258127	−	0.966111i	$-0.416895\pi$
$601$	12.6561	0.516255	0.258127	+	0.966111i	$0.416895\pi$
$602$	0	0
$603$	−1.21730	−0.0495722
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.4047	−0.706435	−0.353217	−	0.935541i	$-0.614912\pi$
$607$	−17.4047	−0.706435	−0.353217	+	0.935541i	$0.614912\pi$
$608$	0	0
$609$	−3.92498	−0.159048
$610$	0	0
$611$	−22.7995	−0.922370
$612$	0	0
$613$	−37.2603	−1.50493	−0.752465	−	0.658633i	$-0.771135\pi$
$613$	−37.2603	−1.50493	−0.752465	+	0.658633i	$0.771135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.3764	−0.739806	−0.369903	−	0.929070i	$-0.620609\pi$
$617$	−18.3764	−0.739806	−0.369903	+	0.929070i	$0.620609\pi$
$618$	0	0
$619$	−14.5526	−0.584919	−0.292459	−	0.956278i	$-0.594474\pi$
$619$	−14.5526	−0.584919	−0.292459	+	0.956278i	$0.594474\pi$
$620$	0	0
$621$	29.6249	1.18881
$622$	0	0
$623$	18.5011	0.741229
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.37720	0.174809
$628$	0	0
$629$	−20.4535	−0.815536
$630$	0	0
$631$	−22.0304	−0.877017	−0.438509	−	0.898727i	$-0.644493\pi$
$631$	−22.0304	−0.877017	−0.438509	+	0.898727i	$0.644493\pi$
$632$	0	0
$633$	2.80325	0.111419
$634$	0	0
$635$	0	0
$636$	0	0
$637$	38.8160	1.53795
$638$	0	0
$639$	0.971704	0.0384400
$640$	0	0
$641$	−43.6765	−1.72512	−0.862558	−	0.505958i	$-0.831139\pi$
$641$	−43.6765	−1.72512	−0.862558	+	0.505958i	$0.831139\pi$
$642$	0	0
$643$	25.9263	1.02243	0.511217	−	0.859452i	$-0.329195\pi$
$643$	25.9263	1.02243	0.511217	+	0.859452i	$0.329195\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.1046	1.65530	0.827652	−	0.561242i	$-0.189676\pi$
$647$	42.1046	1.65530	0.827652	+	0.561242i	$0.189676\pi$
$648$	0	0
$649$	−20.8110	−0.816903
$650$	0	0
$651$	−40.2653	−1.57812
$652$	0	0
$653$	40.4671	1.58360	0.791801	−	0.610779i	$-0.209144\pi$
$653$	40.4671	1.58360	0.791801	+	0.610779i	$0.209144\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.679390	0.0265055
$658$	0	0
$659$	−33.4204	−1.30187	−0.650937	−	0.759131i	$-0.725624\pi$
$659$	−33.4204	−1.30187	−0.650937	+	0.759131i	$0.725624\pi$
$660$	0	0
$661$	19.4883	0.758006	0.379003	−	0.925396i	$-0.376267\pi$
$661$	19.4883	0.758006	0.379003	+	0.925396i	$0.376267\pi$
$662$	0	0
$663$	23.7800	0.923539
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.56833	−0.138166
$668$	0	0
$669$	41.7119	1.61268
$670$	0	0
$671$	5.10331	0.197011
$672$	0	0
$673$	−0.747456	−0.0288123	−0.0144062	−	0.999896i	$-0.504586\pi$
$673$	−0.747456	−0.0288123	−0.0144062	+	0.999896i	$0.504586\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.0275	0.961885	0.480942	−	0.876752i	$-0.340295\pi$
$677$	25.0275	0.961885	0.480942	+	0.876752i	$0.340295\pi$
$678$	0	0
$679$	−11.7827	−0.452179
$680$	0	0
$681$	27.7742	1.06431
$682$	0	0
$683$	36.7253	1.40525	0.702627	−	0.711558i	$-0.252010\pi$
$683$	36.7253	1.40525	0.702627	+	0.711558i	$0.252010\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.1471	−0.921270
$688$	0	0
$689$	84.3366	3.21297
$690$	0	0
$691$	21.3177	0.810963	0.405482	−	0.914103i	$-0.367104\pi$
$691$	21.3177	0.810963	0.405482	+	0.914103i	$0.367104\pi$
$692$	0	0
$693$	−2.65109	−0.100707
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.53981	0.171958
$698$	0	0
$699$	14.7184	0.556699
$700$	0	0
$701$	27.1161	1.02416	0.512081	−	0.858937i	$-0.328875\pi$
$701$	27.1161	1.02416	0.512081	+	0.858937i	$0.328875\pi$
$702$	0	0
$703$	8.70769	0.328417
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−61.3502	−2.30731
$708$	0	0
$709$	−10.1161	−0.379918	−0.189959	−	0.981792i	$-0.560836\pi$
$709$	−10.1161	−0.379918	−0.189959	+	0.981792i	$0.560836\pi$
$710$	0	0
$711$	−4.15215	−0.155718
$712$	0	0
$713$	−36.6065	−1.37092
$714$	0	0
$715$	0	0
$716$	0	0
$717$	40.6193	1.51696
$718$	0	0
$719$	−24.1471	−0.900535	−0.450268	−	0.892894i	$-0.648671\pi$
$719$	−24.1471	−0.900535	−0.450268	+	0.892894i	$0.648671\pi$
$720$	0	0
$721$	−35.9914	−1.34039
$722$	0	0
$723$	−24.1308	−0.897434
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.8988	0.738006	0.369003	−	0.929428i	$-0.379699\pi$
$727$	19.8988	0.738006	0.369003	+	0.929428i	$0.379699\pi$
$728$	0	0
$729$	28.9944	1.07387
$730$	0	0
$731$	6.24537	0.230994
$732$	0	0
$733$	19.9455	0.736705	0.368352	−	0.929686i	$-0.379922\pi$
$733$	19.9455	0.736705	0.368352	+	0.929686i	$0.379922\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.7827	−0.434021
$738$	0	0
$739$	38.2624	1.40751	0.703753	−	0.710445i	$-0.251506\pi$
$739$	38.2624	1.40751	0.703753	+	0.710445i	$0.251506\pi$
$740$	0	0
$741$	−10.1239	−0.371909
$742$	0	0
$743$	12.5547	0.460588	0.230294	−	0.973121i	$-0.426031\pi$
$743$	12.5547	0.460588	0.230294	+	0.973121i	$0.426031\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.03392	−0.147594
$748$	0	0
$749$	−18.7643	−0.685632
$750$	0	0
$751$	−23.2215	−0.847366	−0.423683	−	0.905810i	$-0.639263\pi$
$751$	−23.2215	−0.847366	−0.423683	+	0.905810i	$0.639263\pi$
$752$	0	0
$753$	22.0227	0.802551
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.4105	0.669143	0.334571	−	0.942370i	$-0.391408\pi$
$757$	18.4105	0.669143	0.334571	+	0.942370i	$0.391408\pi$
$758$	0	0
$759$	23.9893	0.870757
$760$	0	0
$761$	11.7982	0.427684	0.213842	−	0.976868i	$-0.431402\pi$
$761$	11.7982	0.427684	0.213842	+	0.976868i	$0.431402\pi$
$762$	0	0
$763$	−51.0197	−1.84704
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48.1329	1.73798
$768$	0	0
$769$	41.2653	1.48807	0.744033	−	0.668143i	$-0.232910\pi$
$769$	41.2653	1.48807	0.744033	+	0.668143i	$0.232910\pi$
$770$	0	0
$771$	−5.53286	−0.199261
$772$	0	0
$773$	5.51736	0.198446	0.0992229	−	0.995065i	$-0.468364\pi$
$773$	5.51736	0.198446	0.0992229	+	0.995065i	$0.468364\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	52.4925	1.88316
$778$	0	0
$779$	−1.93273	−0.0692474
$780$	0	0
$781$	9.40550	0.336555
$782$	0	0
$783$	−3.51948	−0.125776
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.7771	−0.598038	−0.299019	−	0.954247i	$-0.596659\pi$
$787$	−16.7771	−0.598038	−0.299019	+	0.954247i	$0.596659\pi$
$788$	0	0
$789$	−17.9738	−0.639885
$790$	0	0
$791$	1.92498	0.0684446
$792$	0	0
$793$	−11.8032	−0.419146
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.4260	−1.04232	−0.521162	−	0.853458i	$-0.674501\pi$
$797$	−29.4260	−1.04232	−0.521162	+	0.853458i	$0.674501\pi$
$798$	0	0
$799$	−8.73408	−0.308989
$800$	0	0
$801$	1.38788	0.0490382
$802$	0	0
$803$	6.57608	0.232065
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.4359	1.07140
$808$	0	0
$809$	5.48614	0.192883	0.0964413	−	0.995339i	$-0.469254\pi$
$809$	5.48614	0.192883	0.0964413	+	0.995339i	$0.469254\pi$
$810$	0	0
$811$	−16.3927	−0.575626	−0.287813	−	0.957687i	$-0.592928\pi$
$811$	−16.3927	−0.575626	−0.287813	+	0.957687i	$0.592928\pi$
$812$	0	0
$813$	33.6949	1.18173
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.65884	−0.0930212
$818$	0	0
$819$	6.13161	0.214256
$820$	0	0
$821$	15.1628	0.529186	0.264593	−	0.964360i	$-0.414762\pi$
$821$	15.1628	0.529186	0.264593	+	0.964360i	$0.414762\pi$
$822$	0	0
$823$	−16.6094	−0.578968	−0.289484	−	0.957183i	$-0.593484\pi$
$823$	−16.6094	−0.578968	−0.289484	+	0.957183i	$0.593484\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.2293	−0.529574	−0.264787	−	0.964307i	$-0.585302\pi$
$827$	−15.2293	−0.529574	−0.264787	+	0.964307i	$0.585302\pi$
$828$	0	0
$829$	47.4565	1.64823	0.824116	−	0.566422i	$-0.191673\pi$
$829$	47.4565	1.64823	0.824116	+	0.566422i	$0.191673\pi$
$830$	0	0
$831$	−4.45785	−0.154641
$832$	0	0
$833$	14.8697	0.515205
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−36.1054	−1.24799
$838$	0	0
$839$	−2.26614	−0.0782359	−0.0391179	−	0.999235i	$-0.512455\pi$
$839$	−2.26614	−0.0782359	−0.0391179	+	0.999235i	$0.512455\pi$
$840$	0	0
$841$	−28.5761	−0.985382
$842$	0	0
$843$	−25.1367	−0.865752
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.5011	0.498262
$848$	0	0
$849$	−10.2045	−0.350218
$850$	0	0
$851$	47.7226	1.63591
$852$	0	0
$853$	51.8753	1.77618	0.888089	−	0.459672i	$-0.152033\pi$
$853$	51.8753	1.77618	0.888089	+	0.459672i	$0.152033\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.17058	−0.210783	−0.105391	−	0.994431i	$-0.533610\pi$
$857$	−6.17058	−0.210783	−0.105391	+	0.994431i	$0.533610\pi$
$858$	0	0
$859$	31.0635	1.05987	0.529937	−	0.848037i	$-0.322215\pi$
$859$	31.0635	1.05987	0.529937	+	0.848037i	$0.322215\pi$
$860$	0	0
$861$	−11.6511	−0.397068
$862$	0	0
$863$	3.24772	0.110554	0.0552768	−	0.998471i	$-0.482396\pi$
$863$	3.24772	0.110554	0.0552768	+	0.998471i	$0.482396\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−18.9589	−0.643878
$868$	0	0
$869$	−40.1903	−1.36336
$870$	0	0
$871$	27.2517	0.923390
$872$	0	0
$873$	−0.883892	−0.0299152
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.7042	−1.40825	−0.704125	−	0.710076i	$-0.748661\pi$
$877$	−41.7042	−1.40825	−0.704125	+	0.710076i	$0.748661\pi$
$878$	0	0
$879$	50.6708	1.70908
$880$	0	0
$881$	7.42392	0.250118	0.125059	−	0.992149i	$-0.460088\pi$
$881$	7.42392	0.250118	0.125059	+	0.992149i	$0.460088\pi$
$882$	0	0
$883$	−32.0333	−1.07801	−0.539004	−	0.842303i	$-0.681199\pi$
$883$	−32.0333	−1.07801	−0.539004	+	0.842303i	$0.681199\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.0275	−0.672457	−0.336229	−	0.941780i	$-0.609152\pi$
$887$	−20.0275	−0.672457	−0.336229	+	0.941780i	$0.609152\pi$
$888$	0	0
$889$	−43.1620	−1.44761
$890$	0	0
$891$	21.4826	0.719695
$892$	0	0
$893$	3.71836	0.124430
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−55.4840	−1.85256
$898$	0	0
$899$	4.34891	0.145044
$900$	0	0
$901$	32.3078	1.07633
$902$	0	0
$903$	−16.0283	−0.533388
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.94823	−0.164303	−0.0821517	−	0.996620i	$-0.526179\pi$
$907$	−4.94823	−0.164303	−0.0821517	+	0.996620i	$0.526179\pi$
$908$	0	0
$909$	−4.60225	−0.152647
$910$	0	0
$911$	−32.3014	−1.07019	−0.535096	−	0.844791i	$-0.679725\pi$
$911$	−32.3014	−1.07019	−0.535096	+	0.844791i	$0.679725\pi$
$912$	0	0
$913$	−39.0459	−1.29223
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29.6610	0.979491
$918$	0	0
$919$	−21.3072	−0.702861	−0.351430	−	0.936214i	$-0.614305\pi$
$919$	−21.3072	−0.702861	−0.351430	+	0.936214i	$0.614305\pi$
$920$	0	0
$921$	24.4076	0.804258
$922$	0	0
$923$	−21.7536	−0.716029
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.69994	−0.0886775
$928$	0	0
$929$	24.7848	0.813164	0.406582	−	0.913614i	$-0.366721\pi$
$929$	24.7848	0.813164	0.406582	+	0.913614i	$0.366721\pi$
$930$	0	0
$931$	−6.33048	−0.207473
$932$	0	0
$933$	44.5782	1.45942
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.3687	−1.05744	−0.528719	−	0.848797i	$-0.677327\pi$
$937$	−32.3687	−1.05744	−0.528719	+	0.848797i	$0.677327\pi$
$938$	0	0
$939$	−16.0926	−0.525163
$940$	0	0
$941$	−1.48264	−0.0483326	−0.0241663	−	0.999708i	$-0.507693\pi$
$941$	−1.48264	−0.0483326	−0.0241663	+	0.999708i	$0.507693\pi$
$942$	0	0
$943$	−10.5924	−0.344935
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.86064	0.287932	0.143966	−	0.989583i	$-0.454014\pi$
$947$	8.86064	0.287932	0.143966	+	0.989583i	$0.454014\pi$
$948$	0	0
$949$	−15.2095	−0.493723
$950$	0	0
$951$	32.5696	1.05614
$952$	0	0
$953$	10.1239	0.327944	0.163972	−	0.986465i	$-0.447569\pi$
$953$	10.1239	0.327944	0.163972	+	0.986465i	$0.447569\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.84997	−0.0921264
$958$	0	0
$959$	−64.1719	−2.07222
$960$	0	0
$961$	13.6142	0.439169
$962$	0	0
$963$	−1.40762	−0.0453600
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41.8139	−1.34465	−0.672323	−	0.740258i	$-0.734703\pi$
$967$	−41.8139	−1.34465	−0.672323	+	0.740258i	$0.734703\pi$
$968$	0	0
$969$	−3.87826	−0.124588
$970$	0	0
$971$	5.53016	0.177471	0.0887356	−	0.996055i	$-0.471717\pi$
$971$	5.53016	0.177471	0.0887356	+	0.996055i	$0.471717\pi$
$972$	0	0
$973$	67.2362	2.15549
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.9447	−1.02200	−0.511001	−	0.859580i	$-0.670725\pi$
$977$	−31.9447	−1.02200	−0.511001	+	0.859580i	$0.670725\pi$
$978$	0	0
$979$	13.4338	0.429346
$980$	0	0
$981$	−3.82730	−0.122196
$982$	0	0
$983$	−8.82862	−0.281589	−0.140795	−	0.990039i	$-0.544966\pi$
$983$	−8.82862	−0.281589	−0.140795	+	0.990039i	$0.544966\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	22.4154	0.713489
$988$	0	0
$989$	−14.5718	−0.463357
$990$	0	0
$991$	43.7304	1.38914	0.694570	−	0.719425i	$-0.255595\pi$
$991$	43.7304	1.38914	0.694570	+	0.719425i	$0.255595\pi$
$992$	0	0
$993$	−32.9271	−1.04491
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−46.6738	−1.47817	−0.739086	−	0.673611i	$-0.764743\pi$
$997$	−46.6738	−1.47817	−0.739086	+	0.673611i	$0.764743\pi$
$998$	0	0
$999$	47.0694	1.48921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bh.1.3		3
4.3	odd	2		475.2.a.g.1.2	yes	3
5.4	even	2		7600.2.a.cc.1.1		3
12.11	even	2		4275.2.a.ba.1.2		3
20.3	even	4		475.2.b.b.324.3		6
20.7	even	4		475.2.b.b.324.4		6
20.19	odd	2		475.2.a.e.1.2	✓	3
60.59	even	2		4275.2.a.bm.1.2		3
76.75	even	2		9025.2.a.y.1.2		3
380.379	even	2		9025.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.2	✓	3	20.19	odd	2
475.2.a.g.1.2	yes	3	4.3	odd	2
475.2.b.b.324.3		6	20.3	even	4
475.2.b.b.324.4		6	20.7	even	4
4275.2.a.ba.1.2		3	12.11	even	2
4275.2.a.bm.1.2		3	60.59	even	2
7600.2.a.bh.1.3		3	1.1	even	1	trivial
7600.2.a.cc.1.1		3	5.4	even	2
9025.2.a.y.1.2		3	76.75	even	2
9025.2.a.bc.1.2		3	380.379	even	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.65109 q^{3} -3.65109 q^{7} -0.273891 q^{9} +O(q^{10})\)	\(q+1.65109 q^{3} -3.65109 q^{7} -0.273891 q^{9} -2.65109 q^{11} +6.13161 q^{13} +2.34891 q^{17} -1.00000 q^{19} -6.02830 q^{21} -5.48052 q^{23} -5.40550 q^{27} +0.651093 q^{29} +6.67939 q^{31} -4.37720 q^{33} -8.70769 q^{37} +10.1239 q^{39} +1.93273 q^{41} +2.65884 q^{43} -3.71836 q^{47} +6.33048 q^{49} +3.87826 q^{51} +13.7544 q^{53} -1.65109 q^{57} +7.84997 q^{59} -1.92498 q^{61} +1.00000 q^{63} +4.44447 q^{67} -9.04884 q^{69} -3.54778 q^{71} -2.48052 q^{73} +9.67939 q^{77} +15.1599 q^{79} -8.10331 q^{81} +14.7282 q^{83} +1.07502 q^{87} -5.06727 q^{89} -22.3871 q^{91} +11.0283 q^{93} +3.22717 q^{97} +0.726109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 4 q^{7} + q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 4 * q^7 + q^9	\( 3 q - 2 q^{3} - 4 q^{7} + q^{9} - q^{11} + 3 q^{13} + 14 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + q^{27} - 5 q^{29} + q^{31} - 8 q^{33} + 5 q^{37} + 11 q^{39} + q^{41} + 5 q^{43} - 9 q^{47} - 7 q^{49} - 18 q^{51} + 31 q^{53} + 2 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 13 q^{67} + q^{69} - 7 q^{71} + q^{73} + 10 q^{77} + 18 q^{79} - 21 q^{81} - 3 q^{83} + 12 q^{87} - 20 q^{89} - 17 q^{91} + 21 q^{93} - 13 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 4 * q^7 + q^9 - q^11 + 3 * q^13 + 14 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + q^27 - 5 * q^29 + q^31 - 8 * q^33 + 5 * q^37 + 11 * q^39 + q^41 + 5 * q^43 - 9 * q^47 - 7 * q^49 - 18 * q^51 + 31 * q^53 + 2 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 13 * q^67 + q^69 - 7 * q^71 + q^73 + 10 * q^77 + 18 * q^79 - 21 * q^81 - 3 * q^83 + 12 * q^87 - 20 * q^89 - 17 * q^91 + 21 * q^93 - 13 * q^97 + 4 * q^99

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