LMFDB - Embedded newform 1936.4.a.bv.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1936,4,Mod(1,1936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1936.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,8,0,-13,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$114.227697771$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 87x^{6} + 12x^{5} + 2157x^{4} + 2939x^{3} - 5906x^{2} - 3030x + 45 $ x^8 - 2x^7 - 87x^6 + 12x^5 + 2157x^4 + 2939x^3 - 5906x^2 - 3030*x + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 11 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.92777$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31763 q^{3} +6.10149 q^{5} +4.57313 q^{7} -21.6286 q^{9} +47.5050 q^{13} -14.1410 q^{15} -63.6175 q^{17} +18.4503 q^{19} -10.5988 q^{21} +51.9123 q^{23} -87.7719 q^{25} +112.703 q^{27} +77.1748 q^{29} -109.161 q^{31} +27.9029 q^{35} -139.985 q^{37} -110.099 q^{39} +158.382 q^{41} -26.9236 q^{43} -131.967 q^{45} +258.102 q^{47} -322.087 q^{49} +147.442 q^{51} -148.093 q^{53} -42.7609 q^{57} +725.622 q^{59} -482.462 q^{61} -98.9103 q^{63} +289.851 q^{65} +464.330 q^{67} -120.314 q^{69} -912.142 q^{71} -917.279 q^{73} +203.423 q^{75} -759.656 q^{79} +322.768 q^{81} +698.692 q^{83} -388.161 q^{85} -178.863 q^{87} -564.403 q^{89} +217.246 q^{91} +252.996 q^{93} +112.574 q^{95} +1167.45 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 13 q^{5} - 9 q^{7} + 36 q^{9} - 7 q^{13} + 66 q^{15} - 94 q^{17} - 92 q^{19} - 17 q^{21} + 46 q^{23} + 101 q^{25} - 124 q^{27} - 241 q^{29} + 265 q^{31} - 664 q^{35} - 469 q^{37} + 788 q^{39}+ \cdots - 4702 q^{97}+O(q^{100}) $ 8 * q + 8 * q^3 - 13 * q^5 - 9 * q^7 + 36 * q^9 - 7 * q^13 + 66 * q^15 - 94 * q^17 - 92 * q^19 - 17 * q^21 + 46 * q^23 + 101 * q^25 - 124 * q^27 - 241 * q^29 + 265 * q^31 - 664 * q^35 - 469 * q^37 + 788 * q^39 + 74 * q^41 + 35 * q^43 - 566 * q^45 - 197 * q^47 + 221 * q^49 - 608 * q^51 - 247 * q^53 + 979 * q^57 + 1208 * q^59 - 2061 * q^61 - 1617 * q^63 - 837 * q^65 + 829 * q^67 - 1860 * q^69 + 2005 * q^71 - 320 * q^73 + 2323 * q^75 - 1581 * q^79 - 3752 * q^81 + 692 * q^83 - 21 * q^85 - 131 * q^87 - 3205 * q^89 + 2724 * q^91 - 722 * q^93 - 432 * q^95 - 4702 * q^97

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
$3$	−2.31763	−0.446028	−0.223014	−	0.974815i	$-0.571590\pi$
$3$	−2.31763	−0.446028	−0.223014	+	0.974815i	$0.571590\pi$
$4$	0	0
$5$	6.10149	0.545734	0.272867	−	0.962052i	$-0.412028\pi$
$5$	6.10149	0.545734	0.272867	+	0.962052i	$0.412028\pi$
$6$	0	0
$7$	4.57313	0.246926	0.123463	−	0.992349i	$-0.460600\pi$
$7$	4.57313	0.246926	0.123463	+	0.992349i	$0.460600\pi$
$8$	0	0
$9$	−21.6286	−0.801059
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	47.5050	1.01350	0.506751	−	0.862093i	$-0.330847\pi$
$13$	47.5050	1.01350	0.506751	+	0.862093i	$0.330847\pi$
$14$	0	0
$15$	−14.1410	−0.243413
$16$	0	0
$17$	−63.6175	−0.907618	−0.453809	−	0.891099i	$-0.649935\pi$
$17$	−63.6175	−0.907618	−0.453809	+	0.891099i	$0.649935\pi$
$18$	0	0
$19$	18.4503	0.222778	0.111389	−	0.993777i	$-0.464470\pi$
$19$	18.4503	0.222778	0.111389	+	0.993777i	$0.464470\pi$
$20$	0	0
$21$	−10.5988	−0.110136
$22$	0	0
$23$	51.9123	0.470629	0.235315	−	0.971919i	$-0.424388\pi$
$23$	51.9123	0.470629	0.235315	+	0.971919i	$0.424388\pi$
$24$	0	0
$25$	−87.7719	−0.702175
$26$	0	0
$27$	112.703	0.803323
$28$	0	0
$29$	77.1748	0.494173	0.247086	−	0.968993i	$-0.420527\pi$
$29$	77.1748	0.494173	0.247086	+	0.968993i	$0.420527\pi$
$30$	0	0
$31$	−109.161	−0.632451	−0.316226	−	0.948684i	$-0.602416\pi$
$31$	−109.161	−0.632451	−0.316226	+	0.948684i	$0.602416\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	27.9029	0.134756
$36$	0	0
$37$	−139.985	−0.621983	−0.310992	−	0.950413i	$-0.600661\pi$
$37$	−139.985	−0.621983	−0.310992	+	0.950413i	$0.600661\pi$
$38$	0	0
$39$	−110.099	−0.452050
$40$	0	0
$41$	158.382	0.603297	0.301648	−	0.953419i	$-0.402463\pi$
$41$	158.382	0.603297	0.301648	+	0.953419i	$0.402463\pi$
$42$	0	0
$43$	−26.9236	−0.0954838	−0.0477419	−	0.998860i	$-0.515203\pi$
$43$	−26.9236	−0.0954838	−0.0477419	+	0.998860i	$0.515203\pi$
$44$	0	0
$45$	−131.967	−0.437165
$46$	0	0
$47$	258.102	0.801021	0.400510	−	0.916292i	$-0.368833\pi$
$47$	258.102	0.801021	0.400510	+	0.916292i	$0.368833\pi$
$48$	0	0
$49$	−322.087	−0.939028
$50$	0	0
$51$	147.442	0.404823
$52$	0	0
$53$	−148.093	−0.383815	−0.191907	−	0.981413i	$-0.561467\pi$
$53$	−148.093	−0.383815	−0.191907	+	0.981413i	$0.561467\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−42.7609	−0.0993653
$58$	0	0
$59$	725.622	1.60115	0.800575	−	0.599232i	$-0.204527\pi$
$59$	725.622	1.60115	0.800575	+	0.599232i	$0.204527\pi$
$60$	0	0
$61$	−482.462	−1.01267	−0.506336	−	0.862336i	$-0.669000\pi$
$61$	−482.462	−1.01267	−0.506336	+	0.862336i	$0.669000\pi$
$62$	0	0
$63$	−98.9103	−0.197802
$64$	0	0
$65$	289.851	0.553102
$66$	0	0
$67$	464.330	0.846671	0.423335	−	0.905973i	$-0.360859\pi$
$67$	464.330	0.846671	0.423335	+	0.905973i	$0.360859\pi$
$68$	0	0
$69$	−120.314	−0.209914
$70$	0	0
$71$	−912.142	−1.52467	−0.762333	−	0.647185i	$-0.775946\pi$
$71$	−912.142	−1.52467	−0.762333	+	0.647185i	$0.775946\pi$
$72$	0	0
$73$	−917.279	−1.47068	−0.735338	−	0.677700i	$-0.762977\pi$
$73$	−917.279	−1.47068	−0.735338	+	0.677700i	$0.762977\pi$
$74$	0	0
$75$	203.423	0.313190
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−759.656	−1.08187	−0.540936	−	0.841064i	$-0.681930\pi$
$79$	−759.656	−1.08187	−0.540936	+	0.841064i	$0.681930\pi$
$80$	0	0
$81$	322.768	0.442754
$82$	0	0
$83$	698.692	0.923992	0.461996	−	0.886882i	$-0.347133\pi$
$83$	698.692	0.923992	0.461996	+	0.886882i	$0.347133\pi$
$84$	0	0
$85$	−388.161	−0.495318
$86$	0	0
$87$	−178.863	−0.220415
$88$	0	0
$89$	−564.403	−0.672209	−0.336104	−	0.941825i	$-0.609109\pi$
$89$	−564.403	−0.672209	−0.336104	+	0.941825i	$0.609109\pi$
$90$	0	0
$91$	217.246	0.250259
$92$	0	0
$93$	252.996	0.282091
$94$	0	0
$95$	112.574	0.121577
$96$	0	0
$97$	1167.45	1.22203	0.611013	−	0.791620i	$-0.290762\pi$
$97$	1167.45	1.22203	0.611013	+	0.791620i	$0.290762\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1484.91	1.46291	0.731455	−	0.681890i	$-0.238842\pi$
$101$	1484.91	1.46291	0.731455	+	0.681890i	$0.238842\pi$
$102$	0	0
$103$	977.753	0.935348	0.467674	−	0.883901i	$-0.345092\pi$
$103$	977.753	0.935348	0.467674	+	0.883901i	$0.345092\pi$
$104$	0	0
$105$	−64.6685	−0.0601048
$106$	0	0
$107$	−1222.15	−1.10421	−0.552103	−	0.833776i	$-0.686175\pi$
$107$	−1222.15	−1.10421	−0.552103	+	0.833776i	$0.686175\pi$
$108$	0	0
$109$	250.590	0.220203	0.110102	−	0.993920i	$-0.464882\pi$
$109$	250.590	0.220203	0.110102	+	0.993920i	$0.464882\pi$
$110$	0	0
$111$	324.433	0.277422
$112$	0	0
$113$	−2151.57	−1.79118	−0.895589	−	0.444883i	$-0.853245\pi$
$113$	−2151.57	−1.79118	−0.895589	+	0.444883i	$0.853245\pi$
$114$	0	0
$115$	316.742	0.256838
$116$	0	0
$117$	−1027.47	−0.811874
$118$	0	0
$119$	−290.931	−0.224114
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−367.072	−0.269087
$124$	0	0
$125$	−1298.22	−0.928934
$126$	0	0
$127$	−1785.31	−1.24741	−0.623703	−	0.781662i	$-0.714372\pi$
$127$	−1785.31	−1.24741	−0.623703	+	0.781662i	$0.714372\pi$
$128$	0	0
$129$	62.3989	0.0425885
$130$	0	0
$131$	127.974	0.0853522	0.0426761	−	0.999089i	$-0.486412\pi$
$131$	127.974	0.0853522	0.0426761	+	0.999089i	$0.486412\pi$
$132$	0	0
$133$	84.3754	0.0550096
$134$	0	0
$135$	687.656	0.438400
$136$	0	0
$137$	429.255	0.267691	0.133846	−	0.991002i	$-0.457267\pi$
$137$	429.255	0.267691	0.133846	+	0.991002i	$0.457267\pi$
$138$	0	0
$139$	2437.56	1.48742	0.743708	−	0.668505i	$-0.233065\pi$
$139$	2437.56	1.48742	0.743708	+	0.668505i	$0.233065\pi$
$140$	0	0
$141$	−598.184	−0.357278
$142$	0	0
$143$	0	0
$144$	0	0
$145$	470.881	0.269687
$146$	0	0
$147$	746.478	0.418833
$148$	0	0
$149$	−2787.37	−1.53255	−0.766277	−	0.642511i	$-0.777893\pi$
$149$	−2787.37	−1.53255	−0.766277	+	0.642511i	$0.777893\pi$
$150$	0	0
$151$	1273.46	0.686308	0.343154	−	0.939279i	$-0.388505\pi$
$151$	1273.46	0.686308	0.343154	+	0.939279i	$0.388505\pi$
$152$	0	0
$153$	1375.96	0.727055
$154$	0	0
$155$	−666.047	−0.345150
$156$	0	0
$157$	−3651.44	−1.85616	−0.928078	−	0.372385i	$-0.878540\pi$
$157$	−3651.44	−1.85616	−0.928078	+	0.372385i	$0.878540\pi$
$158$	0	0
$159$	343.225	0.171192
$160$	0	0
$161$	237.402	0.116210
$162$	0	0
$163$	1688.61	0.811422	0.405711	−	0.914001i	$-0.367024\pi$
$163$	1688.61	0.811422	0.405711	+	0.914001i	$0.367024\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1963.34	−0.909748	−0.454874	−	0.890556i	$-0.650316\pi$
$167$	−1963.34	−0.909748	−0.454874	+	0.890556i	$0.650316\pi$
$168$	0	0
$169$	59.7247	0.0271847
$170$	0	0
$171$	−399.053	−0.178458
$172$	0	0
$173$	−2482.07	−1.09080	−0.545399	−	0.838177i	$-0.683622\pi$
$173$	−2482.07	−1.09080	−0.545399	+	0.838177i	$0.683622\pi$
$174$	0	0
$175$	−401.392	−0.173385
$176$	0	0
$177$	−1681.72	−0.714158
$178$	0	0
$179$	−445.341	−0.185957	−0.0929787	−	0.995668i	$-0.529639\pi$
$179$	−445.341	−0.185957	−0.0929787	+	0.995668i	$0.529639\pi$
$180$	0	0
$181$	1387.76	0.569898	0.284949	−	0.958543i	$-0.408023\pi$
$181$	1387.76	0.569898	0.284949	+	0.958543i	$0.408023\pi$
$182$	0	0
$183$	1118.17	0.451680
$184$	0	0
$185$	−854.116	−0.339437
$186$	0	0
$187$	0	0
$188$	0	0
$189$	515.406	0.198361
$190$	0	0
$191$	4319.98	1.63656	0.818280	−	0.574819i	$-0.194928\pi$
$191$	4319.98	1.63656	0.818280	+	0.574819i	$0.194928\pi$
$192$	0	0
$193$	−1831.34	−0.683018	−0.341509	−	0.939878i	$-0.610938\pi$
$193$	−1831.34	−0.683018	−0.341509	+	0.939878i	$0.610938\pi$
$194$	0	0
$195$	−671.768	−0.246699
$196$	0	0
$197$	720.368	0.260529	0.130264	−	0.991479i	$-0.458417\pi$
$197$	720.368	0.260529	0.130264	+	0.991479i	$0.458417\pi$
$198$	0	0
$199$	1848.51	0.658478	0.329239	−	0.944247i	$-0.393208\pi$
$199$	1848.51	0.658478	0.329239	+	0.944247i	$0.393208\pi$
$200$	0	0
$201$	−1076.15	−0.377639
$202$	0	0
$203$	352.930	0.122024
$204$	0	0
$205$	966.368	0.329239
$206$	0	0
$207$	−1122.79	−0.377002
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−5819.77	−1.89881	−0.949407	−	0.314048i	$-0.898315\pi$
$211$	−5819.77	−1.89881	−0.949407	+	0.314048i	$0.898315\pi$
$212$	0	0
$213$	2114.01	0.680044
$214$	0	0
$215$	−164.274	−0.0521087
$216$	0	0
$217$	−499.209	−0.156168
$218$	0	0
$219$	2125.91	0.655963
$220$	0	0
$221$	−3022.15	−0.919872
$222$	0	0
$223$	−955.771	−0.287010	−0.143505	−	0.989650i	$-0.545837\pi$
$223$	−955.771	−0.287010	−0.143505	+	0.989650i	$0.545837\pi$
$224$	0	0
$225$	1898.38	0.562483
$226$	0	0
$227$	2061.22	0.602677	0.301339	−	0.953517i	$-0.402567\pi$
$227$	2061.22	0.602677	0.301339	+	0.953517i	$0.402567\pi$
$228$	0	0
$229$	4092.88	1.18107	0.590535	−	0.807012i	$-0.298917\pi$
$229$	4092.88	1.18107	0.590535	+	0.807012i	$0.298917\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2635.96	−0.741149	−0.370574	−	0.928803i	$-0.620839\pi$
$233$	−2635.96	−0.741149	−0.370574	+	0.928803i	$0.620839\pi$
$234$	0	0
$235$	1574.80	0.437144
$236$	0	0
$237$	1760.60	0.482546
$238$	0	0
$239$	−6625.08	−1.79306	−0.896529	−	0.442986i	$-0.853919\pi$
$239$	−6625.08	−1.79306	−0.896529	+	0.442986i	$0.853919\pi$
$240$	0	0
$241$	−19.3078	−0.00516069	−0.00258034	−	0.999997i	$-0.500821\pi$
$241$	−19.3078	−0.00516069	−0.00258034	+	0.999997i	$0.500821\pi$
$242$	0	0
$243$	−3791.04	−1.00080
$244$	0	0
$245$	−1965.21	−0.512459
$246$	0	0
$247$	876.480	0.225786
$248$	0	0
$249$	−1619.31	−0.412127
$250$	0	0
$251$	−913.662	−0.229760	−0.114880	−	0.993379i	$-0.536648\pi$
$251$	−913.662	−0.229760	−0.114880	+	0.993379i	$0.536648\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	899.614	0.220926
$256$	0	0
$257$	3024.41	0.734077	0.367038	−	0.930206i	$-0.380372\pi$
$257$	3024.41	0.734077	0.367038	+	0.930206i	$0.380372\pi$
$258$	0	0
$259$	−640.169	−0.153584
$260$	0	0
$261$	−1669.18	−0.395861
$262$	0	0
$263$	302.593	0.0709456	0.0354728	−	0.999371i	$-0.488706\pi$
$263$	302.593	0.0709456	0.0354728	+	0.999371i	$0.488706\pi$
$264$	0	0
$265$	−903.589	−0.209460
$266$	0	0
$267$	1308.08	0.299824
$268$	0	0
$269$	−5429.36	−1.23061	−0.615305	−	0.788289i	$-0.710967\pi$
$269$	−5429.36	−1.23061	−0.615305	+	0.788289i	$0.710967\pi$
$270$	0	0
$271$	−4003.29	−0.897353	−0.448677	−	0.893694i	$-0.648104\pi$
$271$	−4003.29	−0.897353	−0.448677	+	0.893694i	$0.648104\pi$
$272$	0	0
$273$	−503.497	−0.111623
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7343.17	−1.59281	−0.796405	−	0.604764i	$-0.793267\pi$
$277$	−7343.17	−1.59281	−0.796405	+	0.604764i	$0.793267\pi$
$278$	0	0
$279$	2361.01	0.506630
$280$	0	0
$281$	−6982.61	−1.48238	−0.741188	−	0.671298i	$-0.765737\pi$
$281$	−6982.61	−1.48238	−0.741188	+	0.671298i	$0.765737\pi$
$282$	0	0
$283$	−4276.66	−0.898307	−0.449154	−	0.893454i	$-0.648274\pi$
$283$	−4276.66	−0.898307	−0.449154	+	0.893454i	$0.648274\pi$
$284$	0	0
$285$	−260.905	−0.0542270
$286$	0	0
$287$	724.303	0.148969
$288$	0	0
$289$	−865.816	−0.176229
$290$	0	0
$291$	−2705.72	−0.545058
$292$	0	0
$293$	−5548.43	−1.10629	−0.553144	−	0.833085i	$-0.686572\pi$
$293$	−5548.43	−1.10629	−0.553144	+	0.833085i	$0.686572\pi$
$294$	0	0
$295$	4427.37	0.873802
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2466.10	0.476983
$300$	0	0
$301$	−123.125	−0.0235774
$302$	0	0
$303$	−3441.47	−0.652499
$304$	0	0
$305$	−2943.74	−0.552649
$306$	0	0
$307$	−7074.54	−1.31520	−0.657598	−	0.753369i	$-0.728428\pi$
$307$	−7074.54	−1.31520	−0.657598	+	0.753369i	$0.728428\pi$
$308$	0	0
$309$	−2266.07	−0.417192
$310$	0	0
$311$	−10632.8	−1.93868	−0.969342	−	0.245716i	$-0.920977\pi$
$311$	−10632.8	−1.93868	−0.969342	+	0.245716i	$0.920977\pi$
$312$	0	0
$313$	2550.43	0.460571	0.230286	−	0.973123i	$-0.426034\pi$
$313$	2550.43	0.460571	0.230286	+	0.973123i	$0.426034\pi$
$314$	0	0
$315$	−603.500	−0.107947
$316$	0	0
$317$	3921.78	0.694855	0.347428	−	0.937707i	$-0.387055\pi$
$317$	3921.78	0.694855	0.347428	+	0.937707i	$0.387055\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2832.50	0.492507
$322$	0	0
$323$	−1173.76	−0.202197
$324$	0	0
$325$	−4169.60	−0.711655
$326$	0	0
$327$	−580.775	−0.0982169
$328$	0	0
$329$	1180.33	0.197793
$330$	0	0
$331$	−7694.76	−1.27777	−0.638886	−	0.769302i	$-0.720604\pi$
$331$	−7694.76	−1.27777	−0.638886	+	0.769302i	$0.720604\pi$
$332$	0	0
$333$	3027.68	0.498245
$334$	0	0
$335$	2833.10	0.462057
$336$	0	0
$337$	11047.7	1.78578	0.892890	−	0.450275i	$-0.148674\pi$
$337$	11047.7	1.78578	0.892890	+	0.450275i	$0.148674\pi$
$338$	0	0
$339$	4986.55	0.798916
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3041.52	−0.478796
$344$	0	0
$345$	−734.092	−0.114557
$346$	0	0
$347$	−12119.6	−1.87497	−0.937487	−	0.348021i	$-0.886854\pi$
$347$	−12119.6	−1.87497	−0.937487	+	0.348021i	$0.886854\pi$
$348$	0	0
$349$	−6056.42	−0.928919	−0.464460	−	0.885594i	$-0.653751\pi$
$349$	−6056.42	−0.928919	−0.464460	+	0.885594i	$0.653751\pi$
$350$	0	0
$351$	5353.96	0.814169
$352$	0	0
$353$	−1840.84	−0.277558	−0.138779	−	0.990323i	$-0.544318\pi$
$353$	−1840.84	−0.277558	−0.138779	+	0.990323i	$0.544318\pi$
$354$	0	0
$355$	−5565.42	−0.832062
$356$	0	0
$357$	674.270	0.0999612
$358$	0	0
$359$	−3441.99	−0.506021	−0.253010	−	0.967464i	$-0.581421\pi$
$359$	−3441.99	−0.506021	−0.253010	+	0.967464i	$0.581421\pi$
$360$	0	0
$361$	−6518.59	−0.950370
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5596.77	−0.802597
$366$	0	0
$367$	−4887.65	−0.695186	−0.347593	−	0.937645i	$-0.613001\pi$
$367$	−4887.65	−0.695186	−0.347593	+	0.937645i	$0.613001\pi$
$368$	0	0
$369$	−3425.59	−0.483276
$370$	0	0
$371$	−677.249	−0.0947736
$372$	0	0
$373$	−9784.78	−1.35828	−0.679138	−	0.734011i	$-0.737646\pi$
$373$	−9784.78	−1.35828	−0.679138	+	0.734011i	$0.737646\pi$
$374$	0	0
$375$	3008.81	0.414331
$376$	0	0
$377$	3666.19	0.500845
$378$	0	0
$379$	8259.50	1.11942	0.559712	−	0.828687i	$-0.310912\pi$
$379$	8259.50	1.11942	0.559712	+	0.828687i	$0.310912\pi$
$380$	0	0
$381$	4137.69	0.556378
$382$	0	0
$383$	−7248.34	−0.967031	−0.483515	−	0.875336i	$-0.660640\pi$
$383$	−7248.34	−0.967031	−0.483515	+	0.875336i	$0.660640\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	582.319	0.0764882
$388$	0	0
$389$	11005.2	1.43441	0.717203	−	0.696864i	$-0.245422\pi$
$389$	11005.2	1.43441	0.717203	+	0.696864i	$0.245422\pi$
$390$	0	0
$391$	−3302.53	−0.427151
$392$	0	0
$393$	−296.596	−0.0380695
$394$	0	0
$395$	−4635.03	−0.590414
$396$	0	0
$397$	7715.18	0.975349	0.487675	−	0.873025i	$-0.337845\pi$
$397$	7715.18	0.975349	0.487675	+	0.873025i	$0.337845\pi$
$398$	0	0
$399$	−195.551	−0.0245358
$400$	0	0
$401$	2467.05	0.307228	0.153614	−	0.988131i	$-0.450909\pi$
$401$	2467.05	0.307228	0.153614	+	0.988131i	$0.450909\pi$
$402$	0	0
$403$	−5185.72	−0.640990
$404$	0	0
$405$	1969.36	0.241626
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8171.50	−0.987909	−0.493955	−	0.869488i	$-0.664449\pi$
$409$	−8171.50	−0.987909	−0.493955	+	0.869488i	$0.664449\pi$
$410$	0	0
$411$	−994.854	−0.119398
$412$	0	0
$413$	3318.36	0.395365
$414$	0	0
$415$	4263.06	0.504254
$416$	0	0
$417$	−5649.35	−0.663429
$418$	0	0
$419$	−3563.75	−0.415514	−0.207757	−	0.978180i	$-0.566616\pi$
$419$	−3563.75	−0.415514	−0.207757	+	0.978180i	$0.566616\pi$
$420$	0	0
$421$	−2252.68	−0.260781	−0.130391	−	0.991463i	$-0.541623\pi$
$421$	−2252.68	−0.260781	−0.130391	+	0.991463i	$0.541623\pi$
$422$	0	0
$423$	−5582.37	−0.641665
$424$	0	0
$425$	5583.83	0.637307
$426$	0	0
$427$	−2206.36	−0.250055
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10621.6	1.18706	0.593530	−	0.804811i	$-0.297734\pi$
$431$	10621.6	1.18706	0.593530	+	0.804811i	$0.297734\pi$
$432$	0	0
$433$	17520.7	1.94455	0.972276	−	0.233835i	$-0.0751276\pi$
$433$	17520.7	1.94455	0.972276	+	0.233835i	$0.0751276\pi$
$434$	0	0
$435$	−1091.33	−0.120288
$436$	0	0
$437$	957.796	0.104846
$438$	0	0
$439$	13581.6	1.47657	0.738283	−	0.674491i	$-0.235637\pi$
$439$	13581.6	1.47657	0.738283	+	0.674491i	$0.235637\pi$
$440$	0	0
$441$	6966.28	0.752216
$442$	0	0
$443$	502.605	0.0539040	0.0269520	−	0.999637i	$-0.491420\pi$
$443$	502.605	0.0539040	0.0269520	+	0.999637i	$0.491420\pi$
$444$	0	0
$445$	−3443.70	−0.366847
$446$	0	0
$447$	6460.10	0.683562
$448$	0	0
$449$	2526.93	0.265598	0.132799	−	0.991143i	$-0.457604\pi$
$449$	2526.93	0.265598	0.132799	+	0.991143i	$0.457604\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2951.41	−0.306113
$454$	0	0
$455$	1325.53	0.136575
$456$	0	0
$457$	6870.17	0.703223	0.351611	−	0.936146i	$-0.385634\pi$
$457$	6870.17	0.703223	0.351611	+	0.936146i	$0.385634\pi$
$458$	0	0
$459$	−7169.89	−0.729110
$460$	0	0
$461$	6760.39	0.683000	0.341500	−	0.939882i	$-0.389065\pi$
$461$	6760.39	0.683000	0.341500	+	0.939882i	$0.389065\pi$
$462$	0	0
$463$	−12796.6	−1.28447	−0.642236	−	0.766507i	$-0.721993\pi$
$463$	−12796.6	−1.28447	−0.642236	+	0.766507i	$0.721993\pi$
$464$	0	0
$465$	1543.65	0.153947
$466$	0	0
$467$	853.768	0.0845989	0.0422994	−	0.999105i	$-0.486532\pi$
$467$	853.768	0.0845989	0.0422994	+	0.999105i	$0.486532\pi$
$468$	0	0
$469$	2123.44	0.209065
$470$	0	0
$471$	8462.69	0.827898
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1619.41	−0.156429
$476$	0	0
$477$	3203.05	0.307458
$478$	0	0
$479$	−4347.00	−0.414654	−0.207327	−	0.978272i	$-0.566476\pi$
$479$	−4347.00	−0.414654	−0.207327	+	0.978272i	$0.566476\pi$
$480$	0	0
$481$	−6649.98	−0.630381
$482$	0	0
$483$	−550.209	−0.0518331
$484$	0	0
$485$	7123.18	0.666901
$486$	0	0
$487$	−12353.4	−1.14946	−0.574730	−	0.818343i	$-0.694893\pi$
$487$	−12353.4	−1.14946	−0.574730	+	0.818343i	$0.694893\pi$
$488$	0	0
$489$	−3913.56	−0.361917
$490$	0	0
$491$	−9316.08	−0.856271	−0.428135	−	0.903715i	$-0.640829\pi$
$491$	−9316.08	−0.856271	−0.428135	+	0.903715i	$0.640829\pi$
$492$	0	0
$493$	−4909.67	−0.448520
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4171.34	−0.376479
$498$	0	0
$499$	2926.15	0.262510	0.131255	−	0.991349i	$-0.458099\pi$
$499$	2926.15	0.262510	0.131255	+	0.991349i	$0.458099\pi$
$500$	0	0
$501$	4550.30	0.405773
$502$	0	0
$503$	11343.0	1.00548	0.502742	−	0.864436i	$-0.332324\pi$
$503$	11343.0	1.00548	0.502742	+	0.864436i	$0.332324\pi$
$504$	0	0
$505$	9060.15	0.798359
$506$	0	0
$507$	−138.420	−0.0121251
$508$	0	0
$509$	14351.9	1.24977	0.624887	−	0.780715i	$-0.285145\pi$
$509$	14351.9	1.24977	0.624887	+	0.780715i	$0.285145\pi$
$510$	0	0
$511$	−4194.83	−0.363148
$512$	0	0
$513$	2079.40	0.178963
$514$	0	0
$515$	5965.75	0.510451
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5752.51	0.486526
$520$	0	0
$521$	2623.27	0.220590	0.110295	−	0.993899i	$-0.464820\pi$
$521$	2623.27	0.220590	0.110295	+	0.993899i	$0.464820\pi$
$522$	0	0
$523$	3193.11	0.266969	0.133485	−	0.991051i	$-0.457383\pi$
$523$	3193.11	0.266969	0.133485	+	0.991051i	$0.457383\pi$
$524$	0	0
$525$	930.278	0.0773346
$526$	0	0
$527$	6944.58	0.574024
$528$	0	0
$529$	−9472.11	−0.778508
$530$	0	0
$531$	−15694.2	−1.28262
$532$	0	0
$533$	7523.95	0.611442
$534$	0	0
$535$	−7456.96	−0.602603
$536$	0	0
$537$	1032.14	0.0829423
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8057.19	0.640306	0.320153	−	0.947366i	$-0.396266\pi$
$541$	8057.19	0.640306	0.320153	+	0.947366i	$0.396266\pi$
$542$	0	0
$543$	−3216.32	−0.254191
$544$	0	0
$545$	1528.97	0.120172
$546$	0	0
$547$	1802.34	0.140882	0.0704410	−	0.997516i	$-0.477559\pi$
$547$	1802.34	0.140882	0.0704410	+	0.997516i	$0.477559\pi$
$548$	0	0
$549$	10435.0	0.811209
$550$	0	0
$551$	1423.90	0.110091
$552$	0	0
$553$	−3474.00	−0.267142
$554$	0	0
$555$	1979.53	0.151398
$556$	0	0
$557$	−858.069	−0.0652739	−0.0326369	−	0.999467i	$-0.510391\pi$
$557$	−858.069	−0.0652739	−0.0326369	+	0.999467i	$0.510391\pi$
$558$	0	0
$559$	−1279.00	−0.0967730
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11092.2	−0.830342	−0.415171	−	0.909743i	$-0.636278\pi$
$563$	−11092.2	−0.830342	−0.415171	+	0.909743i	$0.636278\pi$
$564$	0	0
$565$	−13127.8	−0.977506
$566$	0	0
$567$	1476.06	0.109327
$568$	0	0
$569$	13685.6	1.00832	0.504158	−	0.863611i	$-0.331803\pi$
$569$	13685.6	1.00832	0.504158	+	0.863611i	$0.331803\pi$
$570$	0	0
$571$	1193.39	0.0874637	0.0437319	−	0.999043i	$-0.486075\pi$
$571$	1193.39	0.0874637	0.0437319	+	0.999043i	$0.486075\pi$
$572$	0	0
$573$	−10012.1	−0.729952
$574$	0	0
$575$	−4556.44	−0.330464
$576$	0	0
$577$	−12284.3	−0.886313	−0.443157	−	0.896444i	$-0.646141\pi$
$577$	−12284.3	−0.886313	−0.443157	+	0.896444i	$0.646141\pi$
$578$	0	0
$579$	4244.36	0.304645
$580$	0	0
$581$	3195.20	0.228157
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6269.07	−0.443067
$586$	0	0
$587$	−17013.3	−1.19628	−0.598139	−	0.801392i	$-0.704093\pi$
$587$	−17013.3	−1.19628	−0.598139	+	0.801392i	$0.704093\pi$
$588$	0	0
$589$	−2014.06	−0.140896
$590$	0	0
$591$	−1669.55	−0.116203
$592$	0	0
$593$	−898.618	−0.0622291	−0.0311145	−	0.999516i	$-0.509906\pi$
$593$	−898.618	−0.0622291	−0.0311145	+	0.999516i	$0.509906\pi$
$594$	0	0
$595$	−1775.11	−0.122307
$596$	0	0
$597$	−4284.15	−0.293700
$598$	0	0
$599$	−21083.4	−1.43814	−0.719068	−	0.694940i	$-0.755431\pi$
$599$	−21083.4	−1.43814	−0.719068	+	0.694940i	$0.755431\pi$
$600$	0	0
$601$	−9414.85	−0.639001	−0.319500	−	0.947586i	$-0.603515\pi$
$601$	−9414.85	−0.639001	−0.319500	+	0.947586i	$0.603515\pi$
$602$	0	0
$603$	−10042.8	−0.678233
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23396.4	1.56447	0.782233	−	0.622987i	$-0.214081\pi$
$607$	23396.4	1.56447	0.782233	+	0.622987i	$0.214081\pi$
$608$	0	0
$609$	−817.962	−0.0544261
$610$	0	0
$611$	12261.1	0.811836
$612$	0	0
$613$	−11447.9	−0.754288	−0.377144	−	0.926155i	$-0.623094\pi$
$613$	−11447.9	−0.754288	−0.377144	+	0.926155i	$0.623094\pi$
$614$	0	0
$615$	−2239.68	−0.146850
$616$	0	0
$617$	18827.4	1.22847	0.614233	−	0.789125i	$-0.289465\pi$
$617$	18827.4	1.22847	0.614233	+	0.789125i	$0.289465\pi$
$618$	0	0
$619$	22405.3	1.45484	0.727420	−	0.686192i	$-0.240719\pi$
$619$	22405.3	1.45484	0.727420	+	0.686192i	$0.240719\pi$
$620$	0	0
$621$	5850.68	0.378067
$622$	0	0
$623$	−2581.09	−0.165986
$624$	0	0
$625$	3050.38	0.195224
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8905.49	0.564523
$630$	0	0
$631$	−2506.14	−0.158111	−0.0790554	−	0.996870i	$-0.525190\pi$
$631$	−2506.14	−0.158111	−0.0790554	+	0.996870i	$0.525190\pi$
$632$	0	0
$633$	13488.1	0.846925
$634$	0	0
$635$	−10893.0	−0.680751
$636$	0	0
$637$	−15300.7	−0.951706
$638$	0	0
$639$	19728.3	1.22135
$640$	0	0
$641$	−1636.46	−0.100837	−0.0504185	−	0.998728i	$-0.516055\pi$
$641$	−1636.46	−0.100837	−0.0504185	+	0.998728i	$0.516055\pi$
$642$	0	0
$643$	30330.6	1.86022	0.930110	−	0.367282i	$-0.119712\pi$
$643$	30330.6	1.86022	0.930110	+	0.367282i	$0.119712\pi$
$644$	0	0
$645$	380.726	0.0232420
$646$	0	0
$647$	1203.77	0.0731455	0.0365727	−	0.999331i	$-0.488356\pi$
$647$	1203.77	0.0731455	0.0365727	+	0.999331i	$0.488356\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1156.98	0.0696555
$652$	0	0
$653$	−19170.8	−1.14887	−0.574435	−	0.818550i	$-0.694778\pi$
$653$	−19170.8	−1.14887	−0.574435	+	0.818550i	$0.694778\pi$
$654$	0	0
$655$	780.832	0.0465796
$656$	0	0
$657$	19839.4	1.17810
$658$	0	0
$659$	−2265.97	−0.133945	−0.0669723	−	0.997755i	$-0.521334\pi$
$659$	−2265.97	−0.133945	−0.0669723	+	0.997755i	$0.521334\pi$
$660$	0	0
$661$	9959.70	0.586063	0.293031	−	0.956103i	$-0.405336\pi$
$661$	9959.70	0.586063	0.293031	+	0.956103i	$0.405336\pi$
$662$	0	0
$663$	7004.22	0.410289
$664$	0	0
$665$	514.815	0.0300206
$666$	0	0
$667$	4006.33	0.232572
$668$	0	0
$669$	2215.12	0.128014
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6240.28	0.357422	0.178711	−	0.983902i	$-0.442807\pi$
$673$	6240.28	0.357422	0.178711	+	0.983902i	$0.442807\pi$
$674$	0	0
$675$	−9892.16	−0.564073
$676$	0	0
$677$	−15884.6	−0.901765	−0.450882	−	0.892583i	$-0.648891\pi$
$677$	−15884.6	−0.901765	−0.450882	+	0.892583i	$0.648891\pi$
$678$	0	0
$679$	5338.90	0.301750
$680$	0	0
$681$	−4777.14	−0.268811
$682$	0	0
$683$	8537.29	0.478288	0.239144	−	0.970984i	$-0.423133\pi$
$683$	8537.29	0.478288	0.239144	+	0.970984i	$0.423133\pi$
$684$	0	0
$685$	2619.09	0.146088
$686$	0	0
$687$	−9485.78	−0.526791
$688$	0	0
$689$	−7035.17	−0.388996
$690$	0	0
$691$	−14166.2	−0.779897	−0.389949	−	0.920837i	$-0.627507\pi$
$691$	−14166.2	−0.779897	−0.389949	+	0.920837i	$0.627507\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14872.7	0.811732
$696$	0	0
$697$	−10075.9	−0.547563
$698$	0	0
$699$	6109.19	0.330573
$700$	0	0
$701$	17384.2	0.936648	0.468324	−	0.883557i	$-0.344858\pi$
$701$	17384.2	0.936648	0.468324	+	0.883557i	$0.344858\pi$
$702$	0	0
$703$	−2582.76	−0.138564
$704$	0	0
$705$	−3649.81	−0.194979
$706$	0	0
$707$	6790.67	0.361230
$708$	0	0
$709$	−5210.69	−0.276011	−0.138005	−	0.990431i	$-0.544069\pi$
$709$	−5210.69	−0.276011	−0.138005	+	0.990431i	$0.544069\pi$
$710$	0	0
$711$	16430.3	0.866644
$712$	0	0
$713$	−5666.83	−0.297650
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15354.5	0.799754
$718$	0	0
$719$	25865.1	1.34159	0.670795	−	0.741643i	$-0.265953\pi$
$719$	25865.1	1.34159	0.670795	+	0.741643i	$0.265953\pi$
$720$	0	0
$721$	4471.39	0.230961
$722$	0	0
$723$	44.7484	0.00230181
$724$	0	0
$725$	−6773.78	−0.346996
$726$	0	0
$727$	29046.8	1.48182	0.740911	−	0.671603i	$-0.234394\pi$
$727$	29046.8	1.48182	0.740911	+	0.671603i	$0.234394\pi$
$728$	0	0
$729$	71.5012	0.00363264
$730$	0	0
$731$	1712.81	0.0866628
$732$	0	0
$733$	15552.9	0.783712	0.391856	−	0.920027i	$-0.371833\pi$
$733$	15552.9	0.783712	0.391856	+	0.920027i	$0.371833\pi$
$734$	0	0
$735$	4554.62	0.228571
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−38184.8	−1.90074	−0.950372	−	0.311114i	$-0.899298\pi$
$739$	−38184.8	−1.90074	−0.950372	+	0.311114i	$0.899298\pi$
$740$	0	0
$741$	−2031.36	−0.100707
$742$	0	0
$743$	17912.9	0.884470	0.442235	−	0.896899i	$-0.354186\pi$
$743$	17912.9	0.884470	0.442235	+	0.896899i	$0.354186\pi$
$744$	0	0
$745$	−17007.1	−0.836366
$746$	0	0
$747$	−15111.7	−0.740172
$748$	0	0
$749$	−5589.07	−0.272657
$750$	0	0
$751$	40384.2	1.96224	0.981119	−	0.193407i	$-0.0619537\pi$
$751$	40384.2	1.96224	0.981119	+	0.193407i	$0.0619537\pi$
$752$	0	0
$753$	2117.53	0.102480
$754$	0	0
$755$	7769.99	0.374542
$756$	0	0
$757$	−27904.1	−1.33975	−0.669876	−	0.742473i	$-0.733653\pi$
$757$	−27904.1	−1.33975	−0.669876	+	0.742473i	$0.733653\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20312.6	0.967583	0.483791	−	0.875183i	$-0.339259\pi$
$761$	20312.6	0.967583	0.483791	+	0.875183i	$0.339259\pi$
$762$	0	0
$763$	1145.98	0.0543739
$764$	0	0
$765$	8395.38	0.396779
$766$	0	0
$767$	34470.7	1.62277
$768$	0	0
$769$	19314.1	0.905701	0.452850	−	0.891587i	$-0.350407\pi$
$769$	19314.1	0.905701	0.452850	+	0.891587i	$0.350407\pi$
$770$	0	0
$771$	−7009.47	−0.327419
$772$	0	0
$773$	4864.29	0.226334	0.113167	−	0.993576i	$-0.463900\pi$
$773$	4864.29	0.226334	0.113167	+	0.993576i	$0.463900\pi$
$774$	0	0
$775$	9581.31	0.444091
$776$	0	0
$777$	1483.67	0.0685026
$778$	0	0
$779$	2922.20	0.134401
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8697.84	0.396980
$784$	0	0
$785$	−22279.2	−1.01297
$786$	0	0
$787$	16349.1	0.740511	0.370255	−	0.928930i	$-0.379270\pi$
$787$	16349.1	0.740511	0.370255	+	0.928930i	$0.379270\pi$
$788$	0	0
$789$	−701.299	−0.0316437
$790$	0	0
$791$	−9839.42	−0.442288
$792$	0	0
$793$	−22919.4	−1.02634
$794$	0	0
$795$	2094.19	0.0934253
$796$	0	0
$797$	4999.44	0.222195	0.111097	−	0.993810i	$-0.464563\pi$
$797$	4999.44	0.222195	0.111097	+	0.993810i	$0.464563\pi$
$798$	0	0
$799$	−16419.8	−0.727021
$800$	0	0
$801$	12207.2	0.538479
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1448.50	0.0634199
$806$	0	0
$807$	12583.3	0.548887
$808$	0	0
$809$	3239.69	0.140793	0.0703964	−	0.997519i	$-0.477574\pi$
$809$	3239.69	0.140793	0.0703964	+	0.997519i	$0.477574\pi$
$810$	0	0
$811$	16989.9	0.735629	0.367814	−	0.929899i	$-0.380106\pi$
$811$	16989.9	0.735629	0.367814	+	0.929899i	$0.380106\pi$
$812$	0	0
$813$	9278.15	0.400245
$814$	0	0
$815$	10303.0	0.442820
$816$	0	0
$817$	−496.747	−0.0212717
$818$	0	0
$819$	−4698.73	−0.200473
$820$	0	0
$821$	−34839.6	−1.48101	−0.740506	−	0.672049i	$-0.765414\pi$
$821$	−34839.6	−1.48101	−0.740506	+	0.672049i	$0.765414\pi$
$822$	0	0
$823$	2766.92	0.117192	0.0585958	−	0.998282i	$-0.481338\pi$
$823$	2766.92	0.117192	0.0585958	+	0.998282i	$0.481338\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33074.5	−1.39070	−0.695352	−	0.718670i	$-0.744751\pi$
$827$	−33074.5	−1.39070	−0.695352	+	0.718670i	$0.744751\pi$
$828$	0	0
$829$	−6438.40	−0.269740	−0.134870	−	0.990863i	$-0.543062\pi$
$829$	−6438.40	−0.269740	−0.134870	+	0.990863i	$0.543062\pi$
$830$	0	0
$831$	17018.8	0.710438
$832$	0	0
$833$	20490.3	0.852279
$834$	0	0
$835$	−11979.3	−0.496480
$836$	0	0
$837$	−12302.8	−0.508062
$838$	0	0
$839$	27754.6	1.14207	0.571034	−	0.820926i	$-0.306542\pi$
$839$	27754.6	1.14207	0.571034	+	0.820926i	$0.306542\pi$
$840$	0	0
$841$	−18433.0	−0.755793
$842$	0	0
$843$	16183.1	0.661181
$844$	0	0
$845$	364.410	0.0148356
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9911.72	0.400670
$850$	0	0
$851$	−7266.94	−0.292723
$852$	0	0
$853$	10427.2	0.418547	0.209274	−	0.977857i	$-0.432890\pi$
$853$	10427.2	0.418547	0.209274	+	0.977857i	$0.432890\pi$
$854$	0	0
$855$	−2434.82	−0.0973907
$856$	0	0
$857$	−26763.2	−1.06676	−0.533379	−	0.845876i	$-0.679078\pi$
$857$	−26763.2	−1.06676	−0.533379	+	0.845876i	$0.679078\pi$
$858$	0	0
$859$	−29313.2	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$859$	−29313.2	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$860$	0	0
$861$	−1678.67	−0.0664446
$862$	0	0
$863$	−46853.5	−1.84810	−0.924052	−	0.382267i	$-0.875143\pi$
$863$	−46853.5	−1.84810	−0.924052	+	0.382267i	$0.875143\pi$
$864$	0	0
$865$	−15144.3	−0.595285
$866$	0	0
$867$	2006.64	0.0786033
$868$	0	0
$869$	0	0
$870$	0	0
$871$	22058.0	0.858102
$872$	0	0
$873$	−25250.3	−0.978915
$874$	0	0
$875$	−5936.95	−0.229378
$876$	0	0
$877$	36758.6	1.41534	0.707669	−	0.706545i	$-0.249747\pi$
$877$	36758.6	1.41534	0.707669	+	0.706545i	$0.249747\pi$
$878$	0	0
$879$	12859.2	0.493436
$880$	0	0
$881$	8902.03	0.340428	0.170214	−	0.985407i	$-0.445554\pi$
$881$	8902.03	0.340428	0.170214	+	0.985407i	$0.445554\pi$
$882$	0	0
$883$	46557.8	1.77440	0.887199	−	0.461387i	$-0.152648\pi$
$883$	46557.8	1.77440	0.887199	+	0.461387i	$0.152648\pi$
$884$	0	0
$885$	−10261.0	−0.389740
$886$	0	0
$887$	17155.9	0.649422	0.324711	−	0.945813i	$-0.394733\pi$
$887$	17155.9	0.649422	0.324711	+	0.945813i	$0.394733\pi$
$888$	0	0
$889$	−8164.44	−0.308016
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4762.04	0.178450
$894$	0	0
$895$	−2717.24	−0.101483
$896$	0	0
$897$	−5715.50	−0.212748
$898$	0	0
$899$	−8424.52	−0.312540
$900$	0	0
$901$	9421.32	0.348357
$902$	0	0
$903$	285.358	0.0105162
$904$	0	0
$905$	8467.41	0.311013
$906$	0	0
$907$	19481.7	0.713207	0.356604	−	0.934256i	$-0.383935\pi$
$907$	19481.7	0.713207	0.356604	+	0.934256i	$0.383935\pi$
$908$	0	0
$909$	−32116.5	−1.17188
$910$	0	0
$911$	−53205.0	−1.93497	−0.967486	−	0.252925i	$-0.918607\pi$
$911$	−53205.0	−1.93497	−0.967486	+	0.252925i	$0.918607\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6822.50	0.246497
$916$	0	0
$917$	585.241	0.0210756
$918$	0	0
$919$	12297.6	0.441413	0.220707	−	0.975340i	$-0.429164\pi$
$919$	12297.6	0.441413	0.220707	+	0.975340i	$0.429164\pi$
$920$	0	0
$921$	16396.2	0.586615
$922$	0	0
$923$	−43331.3	−1.54525
$924$	0	0
$925$	12286.7	0.436741
$926$	0	0
$927$	−21147.4	−0.749269
$928$	0	0
$929$	38058.5	1.34409	0.672045	−	0.740510i	$-0.265416\pi$
$929$	38058.5	1.34409	0.672045	+	0.740510i	$0.265416\pi$
$930$	0	0
$931$	−5942.58	−0.209195
$932$	0	0
$933$	24642.9	0.864708
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41777.3	1.45657	0.728285	−	0.685275i	$-0.240318\pi$
$937$	41777.3	1.45657	0.728285	+	0.685275i	$0.240318\pi$
$938$	0	0
$939$	−5910.95	−0.205428
$940$	0	0
$941$	32298.5	1.11892	0.559459	−	0.828858i	$-0.311009\pi$
$941$	32298.5	1.11892	0.559459	+	0.828858i	$0.311009\pi$
$942$	0	0
$943$	8222.00	0.283929
$944$	0	0
$945$	3144.74	0.108252
$946$	0	0
$947$	−1027.50	−0.0352579	−0.0176290	−	0.999845i	$-0.505612\pi$
$947$	−1027.50	−0.0352579	−0.0176290	+	0.999845i	$0.505612\pi$
$948$	0	0
$949$	−43575.3	−1.49053
$950$	0	0
$951$	−9089.24	−0.309925
$952$	0	0
$953$	−36866.8	−1.25313	−0.626565	−	0.779369i	$-0.715540\pi$
$953$	−36866.8	−1.25313	−0.626565	+	0.779369i	$0.715540\pi$
$954$	0	0
$955$	26358.3	0.893126
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1963.04	0.0660998
$960$	0	0
$961$	−17874.8	−0.600006
$962$	0	0
$963$	26433.5	0.884535
$964$	0	0
$965$	−11173.9	−0.372746
$966$	0	0
$967$	58557.8	1.94735	0.973677	−	0.227931i	$-0.0731960\pi$
$967$	58557.8	1.94735	0.973677	+	0.227931i	$0.0731960\pi$
$968$	0	0
$969$	2720.34	0.0901857
$970$	0	0
$971$	17765.0	0.587131	0.293566	−	0.955939i	$-0.405158\pi$
$971$	17765.0	0.587131	0.293566	+	0.955939i	$0.405158\pi$
$972$	0	0
$973$	11147.2	0.367281
$974$	0	0
$975$	9663.60	0.317418
$976$	0	0
$977$	−18400.0	−0.602525	−0.301263	−	0.953541i	$-0.597408\pi$
$977$	−18400.0	−0.602525	−0.301263	+	0.953541i	$0.597408\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5419.91	−0.176396
$982$	0	0
$983$	−12366.3	−0.401246	−0.200623	−	0.979668i	$-0.564297\pi$
$983$	−12366.3	−0.401246	−0.200623	+	0.979668i	$0.564297\pi$
$984$	0	0
$985$	4395.32	0.142179
$986$	0	0
$987$	−2735.57	−0.0882211
$988$	0	0
$989$	−1397.66	−0.0449375
$990$	0	0
$991$	54570.4	1.74923	0.874615	−	0.484818i	$-0.161114\pi$
$991$	54570.4	1.74923	0.874615	+	0.484818i	$0.161114\pi$
$992$	0	0
$993$	17833.6	0.569922
$994$	0	0
$995$	11278.6	0.359354
$996$	0	0
$997$	40839.2	1.29728	0.648641	−	0.761094i	$-0.275338\pi$
$997$	40839.2	1.29728	0.648641	+	0.761094i	$0.275338\pi$
$998$	0	0
$999$	−15776.7	−0.499653

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bv.1.3		8
4.3	odd	2		968.4.a.o.1.6		8
11.7	odd	10		176.4.m.e.49.3		16
11.8	odd	10		176.4.m.e.97.3		16
11.10	odd	2		1936.4.a.bw.1.3		8
44.7	even	10		88.4.i.a.49.2	yes	16
44.19	even	10		88.4.i.a.9.2	✓	16
44.43	even	2		968.4.a.n.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.a.9.2	✓	16	44.19	even	10
88.4.i.a.49.2	yes	16	44.7	even	10
176.4.m.e.49.3		16	11.7	odd	10
176.4.m.e.97.3		16	11.8	odd	10
968.4.a.n.1.6		8	44.43	even	2
968.4.a.o.1.6		8	4.3	odd	2
1936.4.a.bv.1.3		8	1.1	even	1	trivial
1936.4.a.bw.1.3		8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31763 q^{3} +6.10149 q^{5} +4.57313 q^{7} -21.6286 q^{9} +47.5050 q^{13} -14.1410 q^{15} -63.6175 q^{17} +18.4503 q^{19} -10.5988 q^{21} +51.9123 q^{23} -87.7719 q^{25} +112.703 q^{27} +77.1748 q^{29} -109.161 q^{31} +27.9029 q^{35} -139.985 q^{37} -110.099 q^{39} +158.382 q^{41} -26.9236 q^{43} -131.967 q^{45} +258.102 q^{47} -322.087 q^{49} +147.442 q^{51} -148.093 q^{53} -42.7609 q^{57} +725.622 q^{59} -482.462 q^{61} -98.9103 q^{63} +289.851 q^{65} +464.330 q^{67} -120.314 q^{69} -912.142 q^{71} -917.279 q^{73} +203.423 q^{75} -759.656 q^{79} +322.768 q^{81} +698.692 q^{83} -388.161 q^{85} -178.863 q^{87} -564.403 q^{89} +217.246 q^{91} +252.996 q^{93} +112.574 q^{95} +1167.45 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 13 q^{5} - 9 q^{7} + 36 q^{9} - 7 q^{13} + 66 q^{15} - 94 q^{17} - 92 q^{19} - 17 q^{21} + 46 q^{23} + 101 q^{25} - 124 q^{27} - 241 q^{29} + 265 q^{31} - 664 q^{35} - 469 q^{37} + 788 q^{39}+ \cdots - 4702 q^{97}+O(q^{100}) \) 8 * q + 8 * q^3 - 13 * q^5 - 9 * q^7 + 36 * q^9 - 7 * q^13 + 66 * q^15 - 94 * q^17 - 92 * q^19 - 17 * q^21 + 46 * q^23 + 101 * q^25 - 124 * q^27 - 241 * q^29 + 265 * q^31 - 664 * q^35 - 469 * q^37 + 788 * q^39 + 74 * q^41 + 35 * q^43 - 566 * q^45 - 197 * q^47 + 221 * q^49 - 608 * q^51 - 247 * q^53 + 979 * q^57 + 1208 * q^59 - 2061 * q^61 - 1617 * q^63 - 837 * q^65 + 829 * q^67 - 1860 * q^69 + 2005 * q^71 - 320 * q^73 + 2323 * q^75 - 1581 * q^79 - 3752 * q^81 + 692 * q^83 - 21 * q^85 - 131 * q^87 - 3205 * q^89 + 2724 * q^91 - 722 * q^93 - 432 * q^95 - 4702 * q^97

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