LMFDB - Embedded newform 1100.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(1,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1100.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1761}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 440 $ x^2 - x - 440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-20.4821$ of defining polynomial
Character	$\chi$	$=$	1100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+30.4821 q^{3} +170.411 q^{7} +686.161 q^{9} +O(q^{10})$	$q+30.4821 q^{3} +170.411 q^{7} +686.161 q^{9} -121.000 q^{11} +427.322 q^{13} -1874.77 q^{17} +1572.63 q^{19} +5194.48 q^{21} -4198.36 q^{23} +13508.5 q^{27} +401.017 q^{29} -4602.48 q^{31} -3688.34 q^{33} +8366.41 q^{37} +13025.7 q^{39} +15037.6 q^{41} +13744.6 q^{43} +26465.3 q^{47} +12232.8 q^{49} -57146.9 q^{51} +12854.6 q^{53} +47937.0 q^{57} +37211.0 q^{59} -9897.88 q^{61} +116929. q^{63} +646.966 q^{67} -127975. q^{69} +18780.0 q^{71} +40997.5 q^{73} -20619.7 q^{77} -67294.7 q^{79} +245030. q^{81} -97038.0 q^{83} +12223.8 q^{87} -50878.6 q^{89} +72820.2 q^{91} -140293. q^{93} -53822.9 q^{97} -83025.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 19 q^{3} + 131 q^{7} + 575 q^{9}+O(q^{10}) $ 2 * q + 19 * q^3 + 131 * q^7 + 575 * q^9	$ 2 q + 19 q^{3} + 131 q^{7} + 575 q^{9} - 242 q^{11} + 1610 q^{13} + 321 q^{17} + 3439 q^{19} + 5647 q^{21} - 4536 q^{23} + 17575 q^{27} - 5031 q^{29} - 4463 q^{31} - 2299 q^{33} + 2423 q^{37} - 554 q^{39} + 19668 q^{41} + 14900 q^{43} + 23052 q^{47} - 3021 q^{49} - 82359 q^{51} + 7203 q^{53} + 26507 q^{57} + 50838 q^{59} - 44177 q^{61} + 121310 q^{63} + 10610 q^{67} - 124098 q^{69} - 1089 q^{71} + 92654 q^{73} - 15851 q^{77} - 16334 q^{79} + 225350 q^{81} - 88410 q^{83} + 74595 q^{87} - 80817 q^{89} + 26210 q^{91} - 141895 q^{93} - 102694 q^{97} - 69575 q^{99}+O(q^{100}) $ 2 * q + 19 * q^3 + 131 * q^7 + 575 * q^9 - 242 * q^11 + 1610 * q^13 + 321 * q^17 + 3439 * q^19 + 5647 * q^21 - 4536 * q^23 + 17575 * q^27 - 5031 * q^29 - 4463 * q^31 - 2299 * q^33 + 2423 * q^37 - 554 * q^39 + 19668 * q^41 + 14900 * q^43 + 23052 * q^47 - 3021 * q^49 - 82359 * q^51 + 7203 * q^53 + 26507 * q^57 + 50838 * q^59 - 44177 * q^61 + 121310 * q^63 + 10610 * q^67 - 124098 * q^69 - 1089 * q^71 + 92654 * q^73 - 15851 * q^77 - 16334 * q^79 + 225350 * q^81 - 88410 * q^83 + 74595 * q^87 - 80817 * q^89 + 26210 * q^91 - 141895 * q^93 - 102694 * q^97 - 69575 * 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
$3$	30.4821	1.95543	0.977715	−	0.209937i	$-0.0673258\pi$
$3$	30.4821	1.95543	0.977715	+	0.209937i	$0.0673258\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	170.411	1.31447	0.657237	−	0.753684i	$-0.271725\pi$
$7$	170.411	1.31447	0.657237	+	0.753684i	$0.271725\pi$
$8$	0	0
$9$	686.161	2.82371
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	427.322	0.701288	0.350644	−	0.936509i	$-0.385963\pi$
$13$	427.322	0.701288	0.350644	+	0.936509i	$0.385963\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1874.77	−1.57335	−0.786674	−	0.617368i	$-0.788199\pi$
$17$	−1874.77	−1.57335	−0.786674	+	0.617368i	$0.788199\pi$
$18$	0	0
$19$	1572.63	0.999404	0.499702	−	0.866197i	$-0.333443\pi$
$19$	1572.63	0.999404	0.499702	+	0.866197i	$0.333443\pi$
$20$	0	0
$21$	5194.48	2.57036
$22$	0	0
$23$	−4198.36	−1.65485	−0.827427	−	0.561573i	$-0.810196\pi$
$23$	−4198.36	−1.65485	−0.827427	+	0.561573i	$0.810196\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	13508.5	3.56613
$28$	0	0
$29$	401.017	0.0885457	0.0442729	−	0.999019i	$-0.485903\pi$
$29$	401.017	0.0885457	0.0442729	+	0.999019i	$0.485903\pi$
$30$	0	0
$31$	−4602.48	−0.860177	−0.430088	−	0.902787i	$-0.641518\pi$
$31$	−4602.48	−0.860177	−0.430088	+	0.902787i	$0.641518\pi$
$32$	0	0
$33$	−3688.34	−0.589584
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8366.41	1.00470	0.502348	−	0.864665i	$-0.332470\pi$
$37$	8366.41	1.00470	0.502348	+	0.864665i	$0.332470\pi$
$38$	0	0
$39$	13025.7	1.37132
$40$	0	0
$41$	15037.6	1.39707	0.698535	−	0.715576i	$-0.253836\pi$
$41$	15037.6	1.39707	0.698535	+	0.715576i	$0.253836\pi$
$42$	0	0
$43$	13744.6	1.13361	0.566803	−	0.823853i	$-0.308180\pi$
$43$	13744.6	1.13361	0.566803	+	0.823853i	$0.308180\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	26465.3	1.74756	0.873780	−	0.486322i	$-0.161662\pi$
$47$	26465.3	1.74756	0.873780	+	0.486322i	$0.161662\pi$
$48$	0	0
$49$	12232.8	0.727840
$50$	0	0
$51$	−57146.9	−3.07657
$52$	0	0
$53$	12854.6	0.628593	0.314297	−	0.949325i	$-0.398231\pi$
$53$	12854.6	0.628593	0.314297	+	0.949325i	$0.398231\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	47937.0	1.95426
$58$	0	0
$59$	37211.0	1.39168	0.695842	−	0.718195i	$-0.255031\pi$
$59$	37211.0	1.39168	0.695842	+	0.718195i	$0.255031\pi$
$60$	0	0
$61$	−9897.88	−0.340579	−0.170289	−	0.985394i	$-0.554470\pi$
$61$	−9897.88	−0.340579	−0.170289	+	0.985394i	$0.554470\pi$
$62$	0	0
$63$	116929.	3.71169
$64$	0	0
$65$	0	0
$66$	0	0
$67$	646.966	0.0176074	0.00880368	−	0.999961i	$-0.497198\pi$
$67$	646.966	0.0176074	0.00880368	+	0.999961i	$0.497198\pi$
$68$	0	0
$69$	−127975.	−3.23595
$70$	0	0
$71$	18780.0	0.442131	0.221065	−	0.975259i	$-0.429047\pi$
$71$	18780.0	0.442131	0.221065	+	0.975259i	$0.429047\pi$
$72$	0	0
$73$	40997.5	0.900431	0.450216	−	0.892920i	$-0.351347\pi$
$73$	40997.5	0.900431	0.450216	+	0.892920i	$0.351347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20619.7	−0.396329
$78$	0	0
$79$	−67294.7	−1.21315	−0.606573	−	0.795028i	$-0.707456\pi$
$79$	−67294.7	−1.21315	−0.606573	+	0.795028i	$0.707456\pi$
$80$	0	0
$81$	245030.	4.14961
$82$	0	0
$83$	−97038.0	−1.54613	−0.773066	−	0.634325i	$-0.781278\pi$
$83$	−97038.0	−1.54613	−0.773066	+	0.634325i	$0.781278\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12223.8	0.173145
$88$	0	0
$89$	−50878.6	−0.680863	−0.340432	−	0.940269i	$-0.610573\pi$
$89$	−50878.6	−0.680863	−0.340432	+	0.940269i	$0.610573\pi$
$90$	0	0
$91$	72820.2	0.921824
$92$	0	0
$93$	−140293.	−1.68202
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−53822.9	−0.580815	−0.290407	−	0.956903i	$-0.593791\pi$
$97$	−53822.9	−0.580815	−0.290407	+	0.956903i	$0.593791\pi$
$98$	0	0
$99$	−83025.4	−0.851379
$100$	0	0
$101$	−119759.	−1.16816	−0.584082	−	0.811695i	$-0.698545\pi$
$101$	−119759.	−1.16816	−0.584082	+	0.811695i	$0.698545\pi$
$102$	0	0
$103$	105473.	0.979596	0.489798	−	0.871836i	$-0.337071\pi$
$103$	105473.	0.979596	0.489798	+	0.871836i	$0.337071\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19514.1	−0.164774	−0.0823871	−	0.996600i	$-0.526254\pi$
$107$	−19514.1	−0.164774	−0.0823871	+	0.996600i	$0.526254\pi$
$108$	0	0
$109$	−146588.	−1.18177	−0.590885	−	0.806756i	$-0.701221\pi$
$109$	−146588.	−1.18177	−0.590885	+	0.806756i	$0.701221\pi$
$110$	0	0
$111$	255026.	1.96461
$112$	0	0
$113$	−134319.	−0.989558	−0.494779	−	0.869019i	$-0.664751\pi$
$113$	−134319.	−0.989558	−0.494779	+	0.869019i	$0.664751\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	293211.	1.98023
$118$	0	0
$119$	−319480.	−2.06812
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	458377.	2.73187
$124$	0	0
$125$	0	0
$126$	0	0
$127$	204792.	1.12669	0.563345	−	0.826222i	$-0.309514\pi$
$127$	204792.	1.12669	0.563345	+	0.826222i	$0.309514\pi$
$128$	0	0
$129$	418966.	2.21669
$130$	0	0
$131$	158250.	0.805687	0.402843	−	0.915269i	$-0.368022\pi$
$131$	158250.	0.805687	0.402843	+	0.915269i	$0.368022\pi$
$132$	0	0
$133$	267992.	1.31369
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−193533.	−0.880955	−0.440477	−	0.897764i	$-0.645191\pi$
$137$	−193533.	−0.880955	−0.440477	+	0.897764i	$0.645191\pi$
$138$	0	0
$139$	63034.6	0.276721	0.138361	−	0.990382i	$-0.455817\pi$
$139$	63034.6	0.276721	0.138361	+	0.990382i	$0.455817\pi$
$140$	0	0
$141$	806718.	3.41723
$142$	0	0
$143$	−51705.9	−0.211446
$144$	0	0
$145$	0	0
$146$	0	0
$147$	372882.	1.42324
$148$	0	0
$149$	−49184.0	−0.181492	−0.0907461	−	0.995874i	$-0.528925\pi$
$149$	−49184.0	−0.181492	−0.0907461	+	0.995874i	$0.528925\pi$
$150$	0	0
$151$	61300.9	0.218788	0.109394	−	0.993998i	$-0.465109\pi$
$151$	61300.9	0.218788	0.109394	+	0.993998i	$0.465109\pi$
$152$	0	0
$153$	−1.28639e6	−4.44267
$154$	0	0
$155$	0	0
$156$	0	0
$157$	215550.	0.697910	0.348955	−	0.937139i	$-0.386537\pi$
$157$	215550.	0.697910	0.348955	+	0.937139i	$0.386537\pi$
$158$	0	0
$159$	391836.	1.22917
$160$	0	0
$161$	−715445.	−2.17526
$162$	0	0
$163$	363443.	1.07144	0.535719	−	0.844396i	$-0.320041\pi$
$163$	363443.	1.07144	0.535719	+	0.844396i	$0.320041\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−506400.	−1.40508	−0.702542	−	0.711642i	$-0.747952\pi$
$167$	−506400.	−1.40508	−0.702542	+	0.711642i	$0.747952\pi$
$168$	0	0
$169$	−188689.	−0.508195
$170$	0	0
$171$	1.07907e6	2.82202
$172$	0	0
$173$	654807.	1.66341	0.831703	−	0.555221i	$-0.187366\pi$
$173$	654807.	1.66341	0.831703	+	0.555221i	$0.187366\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.13427e6	2.72134
$178$	0	0
$179$	−654658.	−1.52715	−0.763576	−	0.645718i	$-0.776558\pi$
$179$	−654658.	−1.52715	−0.763576	+	0.645718i	$0.776558\pi$
$180$	0	0
$181$	333040.	0.755614	0.377807	−	0.925884i	$-0.376678\pi$
$181$	333040.	0.755614	0.377807	+	0.925884i	$0.376678\pi$
$182$	0	0
$183$	−301708.	−0.665978
$184$	0	0
$185$	0	0
$186$	0	0
$187$	226847.	0.474383
$188$	0	0
$189$	2.30199e6	4.68758
$190$	0	0
$191$	−26109.8	−0.0517869	−0.0258934	−	0.999665i	$-0.508243\pi$
$191$	−26109.8	−0.0517869	−0.0258934	+	0.999665i	$0.508243\pi$
$192$	0	0
$193$	459949.	0.888824	0.444412	−	0.895822i	$-0.353413\pi$
$193$	459949.	0.888824	0.444412	+	0.895822i	$0.353413\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	52356.8	0.0961185	0.0480593	−	0.998844i	$-0.484696\pi$
$197$	52356.8	0.0961185	0.0480593	+	0.998844i	$0.484696\pi$
$198$	0	0
$199$	139469.	0.249658	0.124829	−	0.992178i	$-0.460162\pi$
$199$	139469.	0.249658	0.124829	+	0.992178i	$0.460162\pi$
$200$	0	0
$201$	19720.9	0.0344300
$202$	0	0
$203$	68337.5	0.116391
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.88075e6	−4.67282
$208$	0	0
$209$	−190288.	−0.301332
$210$	0	0
$211$	36639.5	0.0566556	0.0283278	−	0.999599i	$-0.490982\pi$
$211$	36639.5	0.0566556	0.0283278	+	0.999599i	$0.490982\pi$
$212$	0	0
$213$	572456.	0.864556
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−784312.	−1.13068
$218$	0	0
$219$	1.24969e6	1.76073
$220$	0	0
$221$	−801128.	−1.10337
$222$	0	0
$223$	−567182.	−0.763765	−0.381883	−	0.924211i	$-0.624724\pi$
$223$	−567182.	−0.763765	−0.381883	+	0.924211i	$0.624724\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−423993.	−0.546128	−0.273064	−	0.961996i	$-0.588037\pi$
$227$	−423993.	−0.546128	−0.273064	+	0.961996i	$0.588037\pi$
$228$	0	0
$229$	−960980.	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$229$	−960980.	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$230$	0	0
$231$	−628532.	−0.774993
$232$	0	0
$233$	57215.6	0.0690438	0.0345219	−	0.999404i	$-0.489009\pi$
$233$	57215.6	0.0690438	0.0345219	+	0.999404i	$0.489009\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.05128e6	−2.37222
$238$	0	0
$239$	−334253.	−0.378513	−0.189256	−	0.981928i	$-0.560608\pi$
$239$	−334253.	−0.378513	−0.189256	+	0.981928i	$0.560608\pi$
$240$	0	0
$241$	711829.	0.789465	0.394733	−	0.918796i	$-0.370837\pi$
$241$	711829.	0.789465	0.394733	+	0.918796i	$0.370837\pi$
$242$	0	0
$243$	4.18649e6	4.54814
$244$	0	0
$245$	0	0
$246$	0	0
$247$	672017.	0.700870
$248$	0	0
$249$	−2.95793e6	−3.02335
$250$	0	0
$251$	−763017.	−0.764451	−0.382226	−	0.924069i	$-0.624842\pi$
$251$	−763017.	−0.764451	−0.382226	+	0.924069i	$0.624842\pi$
$252$	0	0
$253$	508001.	0.498957
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−49015.6	−0.0462915	−0.0231458	−	0.999732i	$-0.507368\pi$
$257$	−49015.6	−0.0462915	−0.0231458	+	0.999732i	$0.507368\pi$
$258$	0	0
$259$	1.42573e6	1.32065
$260$	0	0
$261$	275162.	0.250027
$262$	0	0
$263$	74875.7	0.0667500	0.0333750	−	0.999443i	$-0.489374\pi$
$263$	74875.7	0.0667500	0.0333750	+	0.999443i	$0.489374\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.55089e6	−1.33138
$268$	0	0
$269$	514243.	0.433299	0.216649	−	0.976249i	$-0.430487\pi$
$269$	514243.	0.433299	0.216649	+	0.976249i	$0.430487\pi$
$270$	0	0
$271$	−1.20920e6	−1.00017	−0.500086	−	0.865976i	$-0.666698\pi$
$271$	−1.20920e6	−1.00017	−0.500086	+	0.865976i	$0.666698\pi$
$272$	0	0
$273$	2.21971e6	1.80256
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−473442.	−0.370738	−0.185369	−	0.982669i	$-0.559348\pi$
$277$	−473442.	−0.370738	−0.185369	+	0.982669i	$0.559348\pi$
$278$	0	0
$279$	−3.15804e6	−2.42889
$280$	0	0
$281$	−1.28353e6	−0.969709	−0.484855	−	0.874595i	$-0.661127\pi$
$281$	−1.28353e6	−0.969709	−0.484855	+	0.874595i	$0.661127\pi$
$282$	0	0
$283$	−908348.	−0.674196	−0.337098	−	0.941469i	$-0.609445\pi$
$283$	−908348.	−0.674196	−0.337098	+	0.941469i	$0.609445\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.56256e6	1.83641
$288$	0	0
$289$	2.09489e6	1.47543
$290$	0	0
$291$	−1.64064e6	−1.13574
$292$	0	0
$293$	2.80346e6	1.90777	0.953885	−	0.300173i	$-0.0970445\pi$
$293$	2.80346e6	1.90777	0.953885	+	0.300173i	$0.0970445\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.63453e6	−1.07523
$298$	0	0
$299$	−1.79405e6	−1.16053
$300$	0	0
$301$	2.34223e6	1.49009
$302$	0	0
$303$	−3.65050e6	−2.28426
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.70355e6	1.63715	0.818576	−	0.574399i	$-0.194764\pi$
$307$	2.70355e6	1.63715	0.818576	+	0.574399i	$0.194764\pi$
$308$	0	0
$309$	3.21503e6	1.91553
$310$	0	0
$311$	−1.48422e6	−0.870154	−0.435077	−	0.900393i	$-0.643279\pi$
$311$	−1.48422e6	−0.870154	−0.435077	+	0.900393i	$0.643279\pi$
$312$	0	0
$313$	−1.26860e6	−0.731920	−0.365960	−	0.930631i	$-0.619259\pi$
$313$	−1.26860e6	−0.731920	−0.365960	+	0.930631i	$0.619259\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.55654e6	−0.869987	−0.434994	−	0.900434i	$-0.643249\pi$
$317$	−1.55654e6	−0.869987	−0.434994	+	0.900434i	$0.643249\pi$
$318$	0	0
$319$	−48523.0	−0.0266975
$320$	0	0
$321$	−594832.	−0.322205
$322$	0	0
$323$	−2.94831e6	−1.57241
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.46832e6	−2.31087
$328$	0	0
$329$	4.50997e6	2.29712
$330$	0	0
$331$	−1.95192e6	−0.979245	−0.489623	−	0.871934i	$-0.662865\pi$
$331$	−1.95192e6	−0.979245	−0.489623	+	0.871934i	$0.662865\pi$
$332$	0	0
$333$	5.74070e6	2.83697
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.70471e6	1.77697	0.888483	−	0.458910i	$-0.151760\pi$
$337$	3.70471e6	1.77697	0.888483	+	0.458910i	$0.151760\pi$
$338$	0	0
$339$	−4.09433e6	−1.93501
$340$	0	0
$341$	556900.	0.259353
$342$	0	0
$343$	−779493.	−0.357748
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.52580e6	1.57193	0.785966	−	0.618270i	$-0.212166\pi$
$347$	3.52580e6	1.57193	0.785966	+	0.618270i	$0.212166\pi$
$348$	0	0
$349$	−2.44140e6	−1.07294	−0.536469	−	0.843920i	$-0.680242\pi$
$349$	−2.44140e6	−1.07294	−0.536469	+	0.843920i	$0.680242\pi$
$350$	0	0
$351$	5.77247e6	2.50088
$352$	0	0
$353$	−2.38265e6	−1.01771	−0.508855	−	0.860852i	$-0.669931\pi$
$353$	−2.38265e6	−1.01771	−0.508855	+	0.860852i	$0.669931\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.73844e6	−4.04407
$358$	0	0
$359$	−2.84069e6	−1.16329	−0.581645	−	0.813443i	$-0.697591\pi$
$359$	−2.84069e6	−1.16329	−0.581645	+	0.813443i	$0.697591\pi$
$360$	0	0
$361$	−2949.44	−0.00119116
$362$	0	0
$363$	446289.	0.177766
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.96602e6	−1.53706	−0.768529	−	0.639815i	$-0.779011\pi$
$367$	−3.96602e6	−1.53706	−0.768529	+	0.639815i	$0.779011\pi$
$368$	0	0
$369$	1.03182e7	3.94491
$370$	0	0
$371$	2.19056e6	0.826269
$372$	0	0
$373$	2.96949e6	1.10512	0.552560	−	0.833473i	$-0.313651\pi$
$373$	2.96949e6	1.10512	0.552560	+	0.833473i	$0.313651\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	171363.	0.0620961
$378$	0	0
$379$	−2.50239e6	−0.894865	−0.447433	−	0.894318i	$-0.647662\pi$
$379$	−2.50239e6	−0.894865	−0.447433	+	0.894318i	$0.647662\pi$
$380$	0	0
$381$	6.24251e6	2.20316
$382$	0	0
$383$	−500750.	−0.174431	−0.0872155	−	0.996189i	$-0.527797\pi$
$383$	−500750.	−0.174431	−0.0872155	+	0.996189i	$0.527797\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.43103e6	3.20097
$388$	0	0
$389$	2.52051e6	0.844530	0.422265	−	0.906473i	$-0.361235\pi$
$389$	2.52051e6	0.844530	0.422265	+	0.906473i	$0.361235\pi$
$390$	0	0
$391$	7.87094e6	2.60366
$392$	0	0
$393$	4.82381e6	1.57546
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.44499e6	1.41545	0.707726	−	0.706487i	$-0.249721\pi$
$397$	4.44499e6	1.41545	0.707726	+	0.706487i	$0.249721\pi$
$398$	0	0
$399$	8.16897e6	2.56883
$400$	0	0
$401$	1.49785e6	0.465165	0.232583	−	0.972577i	$-0.425282\pi$
$401$	1.49785e6	0.465165	0.232583	+	0.972577i	$0.425282\pi$
$402$	0	0
$403$	−1.96674e6	−0.603232
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.01234e6	−0.302927
$408$	0	0
$409$	−1.37262e6	−0.405733	−0.202867	−	0.979206i	$-0.565026\pi$
$409$	−1.37262e6	−0.405733	−0.202867	+	0.979206i	$0.565026\pi$
$410$	0	0
$411$	−5.89930e6	−1.72264
$412$	0	0
$413$	6.34114e6	1.82933
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.92143e6	0.541109
$418$	0	0
$419$	676711.	0.188308	0.0941539	−	0.995558i	$-0.469985\pi$
$419$	676711.	0.188308	0.0941539	+	0.995558i	$0.469985\pi$
$420$	0	0
$421$	1.31227e6	0.360842	0.180421	−	0.983590i	$-0.442254\pi$
$421$	1.31227e6	0.360842	0.180421	+	0.983590i	$0.442254\pi$
$422$	0	0
$423$	1.81594e7	4.93459
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.68670e6	−0.447682
$428$	0	0
$429$	−1.57611e6	−0.413468
$430$	0	0
$431$	−3.43100e6	−0.889666	−0.444833	−	0.895613i	$-0.646737\pi$
$431$	−3.43100e6	−0.889666	−0.444833	+	0.895613i	$0.646737\pi$
$432$	0	0
$433$	−4.34873e6	−1.11466	−0.557330	−	0.830291i	$-0.688174\pi$
$433$	−4.34873e6	−1.11466	−0.557330	+	0.830291i	$0.688174\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.60244e6	−1.65387
$438$	0	0
$439$	1.40000e6	0.346709	0.173355	−	0.984859i	$-0.444539\pi$
$439$	1.40000e6	0.346709	0.173355	+	0.984859i	$0.444539\pi$
$440$	0	0
$441$	8.39366e6	2.05520
$442$	0	0
$443$	3.05910e6	0.740600	0.370300	−	0.928912i	$-0.379255\pi$
$443$	3.05910e6	0.740600	0.370300	+	0.928912i	$0.379255\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.49923e6	−0.354895
$448$	0	0
$449$	−3.37359e6	−0.789726	−0.394863	−	0.918740i	$-0.629208\pi$
$449$	−3.37359e6	−0.789726	−0.394863	+	0.918740i	$0.629208\pi$
$450$	0	0
$451$	−1.81955e6	−0.421232
$452$	0	0
$453$	1.86858e6	0.427825
$454$	0	0
$455$	0	0
$456$	0	0
$457$	469470.	0.105152	0.0525760	−	0.998617i	$-0.483257\pi$
$457$	469470.	0.105152	0.0525760	+	0.998617i	$0.483257\pi$
$458$	0	0
$459$	−2.53253e7	−5.61077
$460$	0	0
$461$	4.29980e6	0.942315	0.471158	−	0.882049i	$-0.343836\pi$
$461$	4.29980e6	0.942315	0.471158	+	0.882049i	$0.343836\pi$
$462$	0	0
$463$	−8.43399e6	−1.82844	−0.914220	−	0.405218i	$-0.867196\pi$
$463$	−8.43399e6	−1.82844	−0.914220	+	0.405218i	$0.867196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.72944e6	−0.366955	−0.183477	−	0.983024i	$-0.558735\pi$
$467$	−1.72944e6	−0.366955	−0.183477	+	0.983024i	$0.558735\pi$
$468$	0	0
$469$	110250.	0.0231444
$470$	0	0
$471$	6.57044e6	1.36471
$472$	0	0
$473$	−1.66310e6	−0.341795
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.82033e6	1.77496
$478$	0	0
$479$	−1.64804e6	−0.328193	−0.164096	−	0.986444i	$-0.552471\pi$
$479$	−1.64804e6	−0.328193	−0.164096	+	0.986444i	$0.552471\pi$
$480$	0	0
$481$	3.57515e6	0.704581
$482$	0	0
$483$	−2.18083e7	−4.25357
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.41742e6	−1.41720	−0.708599	−	0.705612i	$-0.750672\pi$
$487$	−7.41742e6	−1.41720	−0.708599	+	0.705612i	$0.750672\pi$
$488$	0	0
$489$	1.10785e7	2.09512
$490$	0	0
$491$	8.33455e6	1.56019	0.780097	−	0.625658i	$-0.215170\pi$
$491$	8.33455e6	1.56019	0.780097	+	0.625658i	$0.215170\pi$
$492$	0	0
$493$	−751813.	−0.139313
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.20032e6	0.581169
$498$	0	0
$499$	−7.22844e6	−1.29955	−0.649775	−	0.760127i	$-0.725137\pi$
$499$	−7.22844e6	−1.29955	−0.649775	+	0.760127i	$0.725137\pi$
$500$	0	0
$501$	−1.54361e7	−2.74754
$502$	0	0
$503$	899953.	0.158599	0.0792994	−	0.996851i	$-0.474732\pi$
$503$	899953.	0.158599	0.0792994	+	0.996851i	$0.474732\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.75165e6	−0.993740
$508$	0	0
$509$	−8.32180e6	−1.42371	−0.711857	−	0.702325i	$-0.752146\pi$
$509$	−8.32180e6	−1.42371	−0.711857	+	0.702325i	$0.752146\pi$
$510$	0	0
$511$	6.98642e6	1.18359
$512$	0	0
$513$	2.12438e7	3.56400
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.20230e6	−0.526909
$518$	0	0
$519$	1.99599e7	3.25267
$520$	0	0
$521$	−6.88886e6	−1.11187	−0.555934	−	0.831227i	$-0.687639\pi$
$521$	−6.88886e6	−1.11187	−0.555934	+	0.831227i	$0.687639\pi$
$522$	0	0
$523$	6.43946e6	1.02943	0.514713	−	0.857363i	$-0.327899\pi$
$523$	6.43946e6	1.02943	0.514713	+	0.857363i	$0.327899\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.62858e6	1.35336
$528$	0	0
$529$	1.11899e7	1.73854
$530$	0	0
$531$	2.55327e7	3.92971
$532$	0	0
$533$	6.42588e6	0.979748
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.99554e7	−2.98624
$538$	0	0
$539$	−1.48017e6	−0.219452
$540$	0	0
$541$	2.04534e6	0.300451	0.150225	−	0.988652i	$-0.452000\pi$
$541$	2.04534e6	0.300451	0.150225	+	0.988652i	$0.452000\pi$
$542$	0	0
$543$	1.01518e7	1.47755
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.46715e6	1.06705	0.533527	−	0.845783i	$-0.320866\pi$
$547$	7.46715e6	1.06705	0.533527	+	0.845783i	$0.320866\pi$
$548$	0	0
$549$	−6.79153e6	−0.961694
$550$	0	0
$551$	630649.	0.0884930
$552$	0	0
$553$	−1.14677e7	−1.59465
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.14446e6	0.566018	0.283009	−	0.959117i	$-0.408667\pi$
$557$	4.14446e6	0.566018	0.283009	+	0.959117i	$0.408667\pi$
$558$	0	0
$559$	5.87338e6	0.794985
$560$	0	0
$561$	6.91478e6	0.927622
$562$	0	0
$563$	−9.56825e6	−1.27222	−0.636109	−	0.771599i	$-0.719457\pi$
$563$	−9.56825e6	−1.27222	−0.636109	+	0.771599i	$0.719457\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.17558e7	5.45455
$568$	0	0
$569$	−4.51090e6	−0.584093	−0.292047	−	0.956404i	$-0.594336\pi$
$569$	−4.51090e6	−0.584093	−0.292047	+	0.956404i	$0.594336\pi$
$570$	0	0
$571$	1.13956e7	1.46268	0.731338	−	0.682016i	$-0.238896\pi$
$571$	1.13956e7	1.46268	0.731338	+	0.682016i	$0.238896\pi$
$572$	0	0
$573$	−795882.	−0.101266
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.19639e6	0.899860	0.449930	−	0.893064i	$-0.351449\pi$
$577$	7.19639e6	0.899860	0.449930	+	0.893064i	$0.351449\pi$
$578$	0	0
$579$	1.40202e7	1.73803
$580$	0	0
$581$	−1.65363e7	−2.03235
$582$	0	0
$583$	−1.55541e6	−0.189528
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.78491e6	0.573163	0.286582	−	0.958056i	$-0.407481\pi$
$587$	4.78491e6	0.573163	0.286582	+	0.958056i	$0.407481\pi$
$588$	0	0
$589$	−7.23798e6	−0.859664
$590$	0	0
$591$	1.59595e6	0.187953
$592$	0	0
$593$	−1.25901e7	−1.47025	−0.735126	−	0.677931i	$-0.762877\pi$
$593$	−1.25901e7	−1.47025	−0.735126	+	0.677931i	$0.762877\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.25132e6	0.488188
$598$	0	0
$599$	−7.28664e6	−0.829775	−0.414887	−	0.909873i	$-0.636179\pi$
$599$	−7.28664e6	−0.829775	−0.414887	+	0.909873i	$0.636179\pi$
$600$	0	0
$601$	−1.37664e6	−0.155465	−0.0777326	−	0.996974i	$-0.524768\pi$
$601$	−1.37664e6	−0.155465	−0.0777326	+	0.996974i	$0.524768\pi$
$602$	0	0
$603$	443923.	0.0497180
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.37512e6	−1.03277	−0.516387	−	0.856356i	$-0.672723\pi$
$607$	−9.37512e6	−1.03277	−0.516387	+	0.856356i	$0.672723\pi$
$608$	0	0
$609$	2.08307e6	0.227594
$610$	0	0
$611$	1.13092e7	1.22554
$612$	0	0
$613$	1.65526e6	0.177916	0.0889580	−	0.996035i	$-0.471646\pi$
$613$	1.65526e6	0.177916	0.0889580	+	0.996035i	$0.471646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.39771e6	0.993823	0.496912	−	0.867801i	$-0.334467\pi$
$617$	9.39771e6	0.993823	0.496912	+	0.867801i	$0.334467\pi$
$618$	0	0
$619$	−1.02554e7	−1.07579	−0.537894	−	0.843013i	$-0.680780\pi$
$619$	−1.02554e7	−1.07579	−0.537894	+	0.843013i	$0.680780\pi$
$620$	0	0
$621$	−5.67134e7	−5.90142
$622$	0	0
$623$	−8.67025e6	−0.894977
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.80037e6	−0.589233
$628$	0	0
$629$	−1.56851e7	−1.58074
$630$	0	0
$631$	−4.45162e6	−0.445087	−0.222543	−	0.974923i	$-0.571436\pi$
$631$	−4.45162e6	−0.445087	−0.222543	+	0.974923i	$0.571436\pi$
$632$	0	0
$633$	1.11685e6	0.110786
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.22734e6	0.510425
$638$	0	0
$639$	1.28861e7	1.24845
$640$	0	0
$641$	−1.35274e6	−0.130038	−0.0650189	−	0.997884i	$-0.520711\pi$
$641$	−1.35274e6	−0.130038	−0.0650189	+	0.997884i	$0.520711\pi$
$642$	0	0
$643$	−7.81676e6	−0.745589	−0.372795	−	0.927914i	$-0.621600\pi$
$643$	−7.81676e6	−0.745589	−0.372795	+	0.927914i	$0.621600\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.84310e6	−0.548760	−0.274380	−	0.961621i	$-0.588473\pi$
$647$	−5.84310e6	−0.548760	−0.274380	+	0.961621i	$0.588473\pi$
$648$	0	0
$649$	−4.50253e6	−0.419609
$650$	0	0
$651$	−2.39075e7	−2.21096
$652$	0	0
$653$	9.90242e6	0.908779	0.454390	−	0.890803i	$-0.349857\pi$
$653$	9.90242e6	0.908779	0.454390	+	0.890803i	$0.349857\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.81309e7	2.54255
$658$	0	0
$659$	−1.62524e7	−1.45782	−0.728910	−	0.684610i	$-0.759973\pi$
$659$	−1.62524e7	−1.45782	−0.728910	+	0.684610i	$0.759973\pi$
$660$	0	0
$661$	957911.	0.0852749	0.0426375	−	0.999091i	$-0.486424\pi$
$661$	957911.	0.0852749	0.0426375	+	0.999091i	$0.486424\pi$
$662$	0	0
$663$	−2.44201e7	−2.15756
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.68361e6	−0.146530
$668$	0	0
$669$	−1.72889e7	−1.49349
$670$	0	0
$671$	1.19764e6	0.102688
$672$	0	0
$673$	1.57601e7	1.34128	0.670642	−	0.741781i	$-0.266019\pi$
$673$	1.57601e7	1.34128	0.670642	+	0.741781i	$0.266019\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.91072e7	−1.60223	−0.801116	−	0.598510i	$-0.795760\pi$
$677$	−1.91072e7	−1.60223	−0.801116	+	0.598510i	$0.795760\pi$
$678$	0	0
$679$	−9.17200e6	−0.763465
$680$	0	0
$681$	−1.29242e7	−1.06791
$682$	0	0
$683$	−6.87828e6	−0.564193	−0.282097	−	0.959386i	$-0.591030\pi$
$683$	−6.87828e6	−0.564193	−0.282097	+	0.959386i	$0.591030\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.92927e7	−2.36792
$688$	0	0
$689$	5.49306e6	0.440825
$690$	0	0
$691$	1.77961e7	1.41785	0.708925	−	0.705284i	$-0.249181\pi$
$691$	1.77961e7	1.41785	0.708925	+	0.705284i	$0.249181\pi$
$692$	0	0
$693$	−1.41484e7	−1.11912
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.81919e7	−2.19808
$698$	0	0
$699$	1.74405e6	0.135010
$700$	0	0
$701$	−7.13494e6	−0.548398	−0.274199	−	0.961673i	$-0.588413\pi$
$701$	−7.13494e6	−0.548398	−0.274199	+	0.961673i	$0.588413\pi$
$702$	0	0
$703$	1.31572e7	1.00410
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.04082e7	−1.53552
$708$	0	0
$709$	1.55820e6	0.116414	0.0582072	−	0.998305i	$-0.481462\pi$
$709$	1.55820e6	0.116414	0.0582072	+	0.998305i	$0.481462\pi$
$710$	0	0
$711$	−4.61749e7	−3.42557
$712$	0	0
$713$	1.93229e7	1.42347
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.01887e7	−0.740155
$718$	0	0
$719$	2.14780e6	0.154943	0.0774714	−	0.996995i	$-0.475315\pi$
$719$	2.14780e6	0.154943	0.0774714	+	0.996995i	$0.475315\pi$
$720$	0	0
$721$	1.79737e7	1.28765
$722$	0	0
$723$	2.16981e7	1.54374
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−7.24128e6	−0.508135	−0.254068	−	0.967186i	$-0.581769\pi$
$727$	−7.24128e6	−0.508135	−0.254068	+	0.967186i	$0.581769\pi$
$728$	0	0
$729$	6.80707e7	4.74396
$730$	0	0
$731$	−2.57680e7	−1.78356
$732$	0	0
$733$	−9.07708e6	−0.624003	−0.312001	−	0.950082i	$-0.600999\pi$
$733$	−9.07708e6	−0.624003	−0.312001	+	0.950082i	$0.600999\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−78282.9	−0.00530882
$738$	0	0
$739$	2.08886e7	1.40701	0.703507	−	0.710688i	$-0.251616\pi$
$739$	2.08886e7	1.40701	0.703507	+	0.710688i	$0.251616\pi$
$740$	0	0
$741$	2.04845e7	1.37050
$742$	0	0
$743$	3.97449e6	0.264125	0.132063	−	0.991241i	$-0.457840\pi$
$743$	3.97449e6	0.264125	0.132063	+	0.991241i	$0.457840\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6.65837e7	−4.36582
$748$	0	0
$749$	−3.32541e6	−0.216591
$750$	0	0
$751$	−1.07786e7	−0.697371	−0.348686	−	0.937240i	$-0.613372\pi$
$751$	−1.07786e7	−0.697371	−0.348686	+	0.937240i	$0.613372\pi$
$752$	0	0
$753$	−2.32584e7	−1.49483
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.04815e7	−0.664790	−0.332395	−	0.943140i	$-0.607857\pi$
$757$	−1.04815e7	−0.664790	−0.332395	+	0.943140i	$0.607857\pi$
$758$	0	0
$759$	1.54850e7	0.975676
$760$	0	0
$761$	−1.94293e7	−1.21617	−0.608085	−	0.793872i	$-0.708062\pi$
$761$	−1.94293e7	−1.21617	−0.608085	+	0.793872i	$0.708062\pi$
$762$	0	0
$763$	−2.49802e7	−1.55340
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.59010e7	0.975972
$768$	0	0
$769$	−1.21703e7	−0.742138	−0.371069	−	0.928605i	$-0.621009\pi$
$769$	−1.21703e7	−0.742138	−0.371069	+	0.928605i	$0.621009\pi$
$770$	0	0
$771$	−1.49410e6	−0.0905199
$772$	0	0
$773$	2.58790e6	0.155776	0.0778878	−	0.996962i	$-0.475182\pi$
$773$	2.58790e6	0.155776	0.0778878	+	0.996962i	$0.475182\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.34592e7	2.58243
$778$	0	0
$779$	2.36485e7	1.39624
$780$	0	0
$781$	−2.27239e6	−0.133307
$782$	0	0
$783$	5.41713e6	0.315765
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.94609e7	1.12002	0.560011	−	0.828485i	$-0.310797\pi$
$787$	1.94609e7	1.12002	0.560011	+	0.828485i	$0.310797\pi$
$788$	0	0
$789$	2.28237e6	0.130525
$790$	0	0
$791$	−2.28894e7	−1.30075
$792$	0	0
$793$	−4.22958e6	−0.238844
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.01556e6	0.0566318	0.0283159	−	0.999599i	$-0.490986\pi$
$797$	1.01556e6	0.0566318	0.0283159	+	0.999599i	$0.490986\pi$
$798$	0	0
$799$	−4.96162e7	−2.74952
$800$	0	0
$801$	−3.49109e7	−1.92256
$802$	0	0
$803$	−4.96070e6	−0.271490
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.56752e7	0.847285
$808$	0	0
$809$	−1.34100e7	−0.720371	−0.360185	−	0.932881i	$-0.617287\pi$
$809$	−1.34100e7	−0.720371	−0.360185	+	0.932881i	$0.617287\pi$
$810$	0	0
$811$	5.33950e6	0.285068	0.142534	−	0.989790i	$-0.454475\pi$
$811$	5.33950e6	0.285068	0.142534	+	0.989790i	$0.454475\pi$
$812$	0	0
$813$	−3.68590e7	−1.95577
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.16152e7	1.13293
$818$	0	0
$819$	4.99663e7	2.60296
$820$	0	0
$821$	9.69015e6	0.501733	0.250867	−	0.968022i	$-0.419284\pi$
$821$	9.69015e6	0.501733	0.250867	+	0.968022i	$0.419284\pi$
$822$	0	0
$823$	−3.31095e7	−1.70394	−0.851968	−	0.523594i	$-0.824591\pi$
$823$	−3.31095e7	−1.70394	−0.851968	+	0.523594i	$0.824591\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.63323e7	−1.33883	−0.669415	−	0.742889i	$-0.733455\pi$
$827$	−2.63323e7	−1.33883	−0.669415	+	0.742889i	$0.733455\pi$
$828$	0	0
$829$	−2.12119e7	−1.07200	−0.535998	−	0.844219i	$-0.680065\pi$
$829$	−2.12119e7	−1.07200	−0.535998	+	0.844219i	$0.680065\pi$
$830$	0	0
$831$	−1.44315e7	−0.724952
$832$	0	0
$833$	−2.29336e7	−1.14515
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.21725e7	−3.06750
$838$	0	0
$839$	−1.62277e7	−0.795889	−0.397944	−	0.917410i	$-0.630276\pi$
$839$	−1.62277e7	−0.795889	−0.397944	+	0.917410i	$0.630276\pi$
$840$	0	0
$841$	−2.03503e7	−0.992160
$842$	0	0
$843$	−3.91249e7	−1.89620
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.49498e6	0.119498
$848$	0	0
$849$	−2.76884e7	−1.31834
$850$	0	0
$851$	−3.51252e7	−1.66263
$852$	0	0
$853$	−1.28100e7	−0.602803	−0.301401	−	0.953497i	$-0.597454\pi$
$853$	−1.28100e7	−0.602803	−0.301401	+	0.953497i	$0.597454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.98494e7	1.38830	0.694151	−	0.719829i	$-0.255780\pi$
$857$	2.98494e7	1.38830	0.694151	+	0.719829i	$0.255780\pi$
$858$	0	0
$859$	1.86550e7	0.862604	0.431302	−	0.902208i	$-0.358054\pi$
$859$	1.86550e7	0.862604	0.431302	+	0.902208i	$0.358054\pi$
$860$	0	0
$861$	7.81124e7	3.59097
$862$	0	0
$863$	−2.41996e7	−1.10607	−0.553034	−	0.833159i	$-0.686530\pi$
$863$	−2.41996e7	−1.10607	−0.553034	+	0.833159i	$0.686530\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.38569e7	2.88509
$868$	0	0
$869$	8.14265e6	0.365777
$870$	0	0
$871$	276463.	0.0123478
$872$	0	0
$873$	−3.69311e7	−1.64005
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.82831e7	1.24173	0.620866	−	0.783916i	$-0.286781\pi$
$877$	2.82831e7	1.24173	0.620866	+	0.783916i	$0.286781\pi$
$878$	0	0
$879$	8.54556e7	3.73051
$880$	0	0
$881$	2.02090e7	0.877213	0.438607	−	0.898679i	$-0.355472\pi$
$881$	2.02090e7	0.877213	0.438607	+	0.898679i	$0.355472\pi$
$882$	0	0
$883$	3.54079e7	1.52827	0.764133	−	0.645059i	$-0.223167\pi$
$883$	3.54079e7	1.52827	0.764133	+	0.645059i	$0.223167\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.52470e6	0.193099	0.0965496	−	0.995328i	$-0.469219\pi$
$887$	4.52470e6	0.193099	0.0965496	+	0.995328i	$0.469219\pi$
$888$	0	0
$889$	3.48988e7	1.48100
$890$	0	0
$891$	−2.96487e7	−1.25115
$892$	0	0
$893$	4.16200e7	1.74652
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.46864e7	−2.26933
$898$	0	0
$899$	−1.84567e6	−0.0761650
$900$	0	0
$901$	−2.40994e7	−0.988996
$902$	0	0
$903$	7.13963e7	2.91378
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.64770e7	1.47232	0.736158	−	0.676810i	$-0.236638\pi$
$907$	3.64770e7	1.47232	0.736158	+	0.676810i	$0.236638\pi$
$908$	0	0
$909$	−8.21737e7	−3.29855
$910$	0	0
$911$	2.97945e7	1.18943	0.594717	−	0.803935i	$-0.297264\pi$
$911$	2.97945e7	1.18943	0.594717	+	0.803935i	$0.297264\pi$
$912$	0	0
$913$	1.17416e7	0.466176
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.69675e7	1.05905
$918$	0	0
$919$	−3.37510e7	−1.31825	−0.659126	−	0.752033i	$-0.729074\pi$
$919$	−3.37510e7	−1.31825	−0.659126	+	0.752033i	$0.729074\pi$
$920$	0	0
$921$	8.24101e7	3.20134
$922$	0	0
$923$	8.02512e6	0.310061
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.23712e7	2.76609
$928$	0	0
$929$	1.48619e7	0.564983	0.282492	−	0.959270i	$-0.408839\pi$
$929$	1.48619e7	0.564983	0.282492	+	0.959270i	$0.408839\pi$
$930$	0	0
$931$	1.92376e7	0.727406
$932$	0	0
$933$	−4.52421e7	−1.70153
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.33845e7	−0.870120	−0.435060	−	0.900402i	$-0.643273\pi$
$937$	−2.33845e7	−0.870120	−0.435060	+	0.900402i	$0.643273\pi$
$938$	0	0
$939$	−3.86696e7	−1.43122
$940$	0	0
$941$	9.00073e6	0.331363	0.165681	−	0.986179i	$-0.447018\pi$
$941$	9.00073e6	0.331363	0.165681	+	0.986179i	$0.447018\pi$
$942$	0	0
$943$	−6.31331e7	−2.31195
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.81160e7	1.74347	0.871734	−	0.489979i	$-0.162996\pi$
$947$	4.81160e7	1.74347	0.871734	+	0.489979i	$0.162996\pi$
$948$	0	0
$949$	1.75191e7	0.631462
$950$	0	0
$951$	−4.74467e7	−1.70120
$952$	0	0
$953$	7.30065e6	0.260393	0.130196	−	0.991488i	$-0.458439\pi$
$953$	7.30065e6	0.260393	0.130196	+	0.991488i	$0.458439\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.47909e6	−0.0522052
$958$	0	0
$959$	−3.29801e7	−1.15799
$960$	0	0
$961$	−7.44632e6	−0.260096
$962$	0	0
$963$	−1.33898e7	−0.465274
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.11094e7	−1.75766	−0.878829	−	0.477138i	$-0.841674\pi$
$967$	−5.11094e7	−1.75766	−0.878829	+	0.477138i	$0.841674\pi$
$968$	0	0
$969$	−8.98707e7	−3.07474
$970$	0	0
$971$	5.67267e7	1.93081	0.965405	−	0.260754i	$-0.0839712\pi$
$971$	5.67267e7	1.93081	0.965405	+	0.260754i	$0.0839712\pi$
$972$	0	0
$973$	1.07418e7	0.363742
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.01264e7	−0.674573	−0.337286	−	0.941402i	$-0.609509\pi$
$977$	−2.01264e7	−0.674573	−0.337286	+	0.941402i	$0.609509\pi$
$978$	0	0
$979$	6.15631e6	0.205288
$980$	0	0
$981$	−1.00583e8	−3.33697
$982$	0	0
$983$	1.07550e7	0.355000	0.177500	−	0.984121i	$-0.443199\pi$
$983$	1.07550e7	0.355000	0.177500	+	0.984121i	$0.443199\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.37473e8	4.49186
$988$	0	0
$989$	−5.77049e7	−1.87595
$990$	0	0
$991$	−4.32009e7	−1.39736	−0.698680	−	0.715434i	$-0.746229\pi$
$991$	−4.32009e7	−1.39736	−0.698680	+	0.715434i	$0.746229\pi$
$992$	0	0
$993$	−5.94986e7	−1.91485
$994$	0	0
$995$	0	0
$996$	0	0
$997$	575481.	0.0183355	0.00916776	−	0.999958i	$-0.497082\pi$
$997$	575481.	0.0183355	0.00916776	+	0.999958i	$0.497082\pi$
$998$	0	0
$999$	1.13017e8	3.58288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.c.1.2		2
5.2	odd	4		1100.6.b.b.749.1		4
5.3	odd	4		1100.6.b.b.749.4		4
5.4	even	2		220.6.a.a.1.1	✓	2
20.19	odd	2		880.6.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.a.a.1.1	✓	2	5.4	even	2
880.6.a.h.1.2		2	20.19	odd	2
1100.6.a.c.1.2		2	1.1	even	1	trivial
1100.6.b.b.749.1		4	5.2	odd	4
1100.6.b.b.749.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q+30.4821 q^{3} +170.411 q^{7} +686.161 q^{9} +O(q^{10})\)	\(q+30.4821 q^{3} +170.411 q^{7} +686.161 q^{9} -121.000 q^{11} +427.322 q^{13} -1874.77 q^{17} +1572.63 q^{19} +5194.48 q^{21} -4198.36 q^{23} +13508.5 q^{27} +401.017 q^{29} -4602.48 q^{31} -3688.34 q^{33} +8366.41 q^{37} +13025.7 q^{39} +15037.6 q^{41} +13744.6 q^{43} +26465.3 q^{47} +12232.8 q^{49} -57146.9 q^{51} +12854.6 q^{53} +47937.0 q^{57} +37211.0 q^{59} -9897.88 q^{61} +116929. q^{63} +646.966 q^{67} -127975. q^{69} +18780.0 q^{71} +40997.5 q^{73} -20619.7 q^{77} -67294.7 q^{79} +245030. q^{81} -97038.0 q^{83} +12223.8 q^{87} -50878.6 q^{89} +72820.2 q^{91} -140293. q^{93} -53822.9 q^{97} -83025.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 19 q^{3} + 131 q^{7} + 575 q^{9}+O(q^{10}) \) 2 * q + 19 * q^3 + 131 * q^7 + 575 * q^9	\( 2 q + 19 q^{3} + 131 q^{7} + 575 q^{9} - 242 q^{11} + 1610 q^{13} + 321 q^{17} + 3439 q^{19} + 5647 q^{21} - 4536 q^{23} + 17575 q^{27} - 5031 q^{29} - 4463 q^{31} - 2299 q^{33} + 2423 q^{37} - 554 q^{39} + 19668 q^{41} + 14900 q^{43} + 23052 q^{47} - 3021 q^{49} - 82359 q^{51} + 7203 q^{53} + 26507 q^{57} + 50838 q^{59} - 44177 q^{61} + 121310 q^{63} + 10610 q^{67} - 124098 q^{69} - 1089 q^{71} + 92654 q^{73} - 15851 q^{77} - 16334 q^{79} + 225350 q^{81} - 88410 q^{83} + 74595 q^{87} - 80817 q^{89} + 26210 q^{91} - 141895 q^{93} - 102694 q^{97} - 69575 q^{99}+O(q^{100}) \) 2 * q + 19 * q^3 + 131 * q^7 + 575 * q^9 - 242 * q^11 + 1610 * q^13 + 321 * q^17 + 3439 * q^19 + 5647 * q^21 - 4536 * q^23 + 17575 * q^27 - 5031 * q^29 - 4463 * q^31 - 2299 * q^33 + 2423 * q^37 - 554 * q^39 + 19668 * q^41 + 14900 * q^43 + 23052 * q^47 - 3021 * q^49 - 82359 * q^51 + 7203 * q^53 + 26507 * q^57 + 50838 * q^59 - 44177 * q^61 + 121310 * q^63 + 10610 * q^67 - 124098 * q^69 - 1089 * q^71 + 92654 * q^73 - 15851 * q^77 - 16334 * q^79 + 225350 * q^81 - 88410 * q^83 + 74595 * q^87 - 80817 * q^89 + 26210 * q^91 - 141895 * q^93 - 102694 * q^97 - 69575 * q^99

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