LMFDB - Embedded newform 1936.4.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.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:	$114.227697771$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{37})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 21x^{2} + 64 $ x^4 - 21*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.15942$ 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)

$f(q)$	$=$	$q-4.54138 q^{3} +6.17671 q^{5} -32.1271 q^{7} -6.37586 q^{9} +O(q^{10})$	$q-4.54138 q^{3} +6.17671 q^{5} -32.1271 q^{7} -6.37586 q^{9} -4.49714 q^{13} -28.0508 q^{15} +59.4763 q^{17} -28.9929 q^{19} +145.901 q^{21} +38.3004 q^{23} -86.8483 q^{25} +151.572 q^{27} +39.4345 q^{29} +266.057 q^{31} -198.440 q^{35} +112.232 q^{37} +20.4232 q^{39} -134.223 q^{41} +252.470 q^{43} -39.3818 q^{45} +182.276 q^{47} +689.150 q^{49} -270.104 q^{51} +42.7271 q^{53} +131.668 q^{57} +180.626 q^{59} -559.433 q^{61} +204.838 q^{63} -27.7775 q^{65} -770.935 q^{67} -173.937 q^{69} +26.5440 q^{71} +372.155 q^{73} +394.411 q^{75} +252.776 q^{79} -516.200 q^{81} -1055.18 q^{83} +367.368 q^{85} -179.087 q^{87} +58.5392 q^{89} +144.480 q^{91} -1208.27 q^{93} -179.081 q^{95} -597.668 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} - 11 q^{5} - 25 q^{7} - 62 q^{9}+O(q^{10}) $ 4 * q - 6 * q^3 - 11 * q^5 - 25 * q^7 - 62 * q^9	$ 4 q - 6 q^{3} - 11 q^{5} - 25 q^{7} - 62 q^{9} + 25 q^{13} - 2 q^{15} + 232 q^{17} - 154 q^{19} + 167 q^{21} + 6 q^{23} - 13 q^{25} + 144 q^{27} + 363 q^{29} - 37 q^{31} - 356 q^{35} + 93 q^{37} + 240 q^{39} + 152 q^{41} + 325 q^{43} + 226 q^{45} + 869 q^{47} - 245 q^{49} - 52 q^{51} + 811 q^{53} + 231 q^{57} - 178 q^{59} - 105 q^{61} - q^{63} + 895 q^{65} - 43 q^{67} - 1156 q^{69} - 629 q^{71} - 270 q^{73} + 815 q^{75} - 977 q^{79} + 52 q^{81} - 1686 q^{83} - 721 q^{85} - 1155 q^{87} + 1891 q^{89} + 80 q^{91} - 666 q^{93} - 1804 q^{95} - 1772 q^{97}+O(q^{100}) $ 4 * q - 6 * q^3 - 11 * q^5 - 25 * q^7 - 62 * q^9 + 25 * q^13 - 2 * q^15 + 232 * q^17 - 154 * q^19 + 167 * q^21 + 6 * q^23 - 13 * q^25 + 144 * q^27 + 363 * q^29 - 37 * q^31 - 356 * q^35 + 93 * q^37 + 240 * q^39 + 152 * q^41 + 325 * q^43 + 226 * q^45 + 869 * q^47 - 245 * q^49 - 52 * q^51 + 811 * q^53 + 231 * q^57 - 178 * q^59 - 105 * q^61 - q^63 + 895 * q^65 - 43 * q^67 - 1156 * q^69 - 629 * q^71 - 270 * q^73 + 815 * q^75 - 977 * q^79 + 52 * q^81 - 1686 * q^83 - 721 * q^85 - 1155 * q^87 + 1891 * q^89 + 80 * q^91 - 666 * q^93 - 1804 * q^95 - 1772 * q^97

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
$3$	−4.54138	−0.873989	−0.436995	−	0.899464i	$-0.643957\pi$
$3$	−4.54138	−0.873989	−0.436995	+	0.899464i	$0.643957\pi$
$4$	0	0
$5$	6.17671	0.552462	0.276231	−	0.961091i	$-0.410915\pi$
$5$	6.17671	0.552462	0.276231	+	0.961091i	$0.410915\pi$
$6$	0	0
$7$	−32.1271	−1.73470	−0.867350	−	0.497699i	$-0.834178\pi$
$7$	−32.1271	−1.73470	−0.867350	+	0.497699i	$0.834178\pi$
$8$	0	0
$9$	−6.37586	−0.236143
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.49714	−0.0959448	−0.0479724	−	0.998849i	$-0.515276\pi$
$13$	−4.49714	−0.0959448	−0.0479724	+	0.998849i	$0.515276\pi$
$14$	0	0
$15$	−28.0508	−0.482845
$16$	0	0
$17$	59.4763	0.848536	0.424268	−	0.905537i	$-0.360531\pi$
$17$	59.4763	0.848536	0.424268	+	0.905537i	$0.360531\pi$
$18$	0	0
$19$	−28.9929	−0.350075	−0.175037	−	0.984562i	$-0.556005\pi$
$19$	−28.9929	−0.350075	−0.175037	+	0.984562i	$0.556005\pi$
$20$	0	0
$21$	145.901	1.51611
$22$	0	0
$23$	38.3004	0.347226	0.173613	−	0.984814i	$-0.444456\pi$
$23$	38.3004	0.347226	0.173613	+	0.984814i	$0.444456\pi$
$24$	0	0
$25$	−86.8483	−0.694786
$26$	0	0
$27$	151.572	1.08038
$28$	0	0
$29$	39.4345	0.252511	0.126255	−	0.991998i	$-0.459704\pi$
$29$	39.4345	0.252511	0.126255	+	0.991998i	$0.459704\pi$
$30$	0	0
$31$	266.057	1.54146	0.770731	−	0.637161i	$-0.219891\pi$
$31$	266.057	1.54146	0.770731	+	0.637161i	$0.219891\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−198.440	−0.958355
$36$	0	0
$37$	112.232	0.498672	0.249336	−	0.968417i	$-0.419788\pi$
$37$	112.232	0.498672	0.249336	+	0.968417i	$0.419788\pi$
$38$	0	0
$39$	20.4232	0.0838548
$40$	0	0
$41$	−134.223	−0.511271	−0.255635	−	0.966773i	$-0.582285\pi$
$41$	−134.223	−0.511271	−0.255635	+	0.966773i	$0.582285\pi$
$42$	0	0
$43$	252.470	0.895380	0.447690	−	0.894189i	$-0.352247\pi$
$43$	252.470	0.895380	0.447690	+	0.894189i	$0.352247\pi$
$44$	0	0
$45$	−39.3818	−0.130460
$46$	0	0
$47$	182.276	0.565694	0.282847	−	0.959165i	$-0.408721\pi$
$47$	182.276	0.565694	0.282847	+	0.959165i	$0.408721\pi$
$48$	0	0
$49$	689.150	2.00918
$50$	0	0
$51$	−270.104	−0.741611
$52$	0	0
$53$	42.7271	0.110736	0.0553681	−	0.998466i	$-0.482367\pi$
$53$	42.7271	0.110736	0.0553681	+	0.998466i	$0.482367\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	131.668	0.305962
$58$	0	0
$59$	180.626	0.398568	0.199284	−	0.979942i	$-0.436138\pi$
$59$	180.626	0.398568	0.199284	+	0.979942i	$0.436138\pi$
$60$	0	0
$61$	−559.433	−1.17423	−0.587116	−	0.809503i	$-0.699737\pi$
$61$	−559.433	−1.17423	−0.587116	+	0.809503i	$0.699737\pi$
$62$	0	0
$63$	204.838	0.409637
$64$	0	0
$65$	−27.7775	−0.0530058
$66$	0	0
$67$	−770.935	−1.40574	−0.702871	−	0.711318i	$-0.748099\pi$
$67$	−770.935	−1.40574	−0.702871	+	0.711318i	$0.748099\pi$
$68$	0	0
$69$	−173.937	−0.303472
$70$	0	0
$71$	26.5440	0.0443689	0.0221844	−	0.999754i	$-0.492938\pi$
$71$	26.5440	0.0443689	0.0221844	+	0.999754i	$0.492938\pi$
$72$	0	0
$73$	372.155	0.596677	0.298339	−	0.954460i	$-0.403568\pi$
$73$	372.155	0.596677	0.298339	+	0.954460i	$0.403568\pi$
$74$	0	0
$75$	394.411	0.607236
$76$	0	0
$77$	0	0
$78$	0	0
$79$	252.776	0.359994	0.179997	−	0.983667i	$-0.442391\pi$
$79$	252.776	0.359994	0.179997	+	0.983667i	$0.442391\pi$
$80$	0	0
$81$	−516.200	−0.708094
$82$	0	0
$83$	−1055.18	−1.39543	−0.697717	−	0.716373i	$-0.745801\pi$
$83$	−1055.18	−1.39543	−0.697717	+	0.716373i	$0.745801\pi$
$84$	0	0
$85$	367.368	0.468784
$86$	0	0
$87$	−179.087	−0.220691
$88$	0	0
$89$	58.5392	0.0697206	0.0348603	−	0.999392i	$-0.488901\pi$
$89$	58.5392	0.0697206	0.0348603	+	0.999392i	$0.488901\pi$
$90$	0	0
$91$	144.480	0.166436
$92$	0	0
$93$	−1208.27	−1.34722
$94$	0	0
$95$	−179.081	−0.193403
$96$	0	0
$97$	−597.668	−0.625608	−0.312804	−	0.949818i	$-0.601268\pi$
$97$	−597.668	−0.625608	−0.312804	+	0.949818i	$0.601268\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1280.94	1.26196	0.630981	−	0.775798i	$-0.282653\pi$
$101$	1280.94	1.26196	0.630981	+	0.775798i	$0.282653\pi$
$102$	0	0
$103$	−1119.57	−1.07102	−0.535510	−	0.844529i	$-0.679880\pi$
$103$	−1119.57	−1.07102	−0.535510	+	0.844529i	$0.679880\pi$
$104$	0	0
$105$	901.190	0.837592
$106$	0	0
$107$	2.69158	0.00243182	0.00121591	−	0.999999i	$-0.499613\pi$
$107$	2.69158	0.00243182	0.00121591	+	0.999999i	$0.499613\pi$
$108$	0	0
$109$	−10.5392	−0.00926117	−0.00463058	−	0.999989i	$-0.501474\pi$
$109$	−10.5392	−0.00926117	−0.00463058	+	0.999989i	$0.501474\pi$
$110$	0	0
$111$	−509.690	−0.435834
$112$	0	0
$113$	2250.77	1.87375	0.936877	−	0.349658i	$-0.113702\pi$
$113$	2250.77	1.87375	0.936877	+	0.349658i	$0.113702\pi$
$114$	0	0
$115$	236.571	0.191829
$116$	0	0
$117$	28.6731	0.0226567
$118$	0	0
$119$	−1910.80	−1.47196
$120$	0	0
$121$	0	0
$122$	0	0
$123$	609.558	0.446845
$124$	0	0
$125$	−1308.53	−0.936304
$126$	0	0
$127$	−1087.92	−0.760133	−0.380067	−	0.924959i	$-0.624099\pi$
$127$	−1087.92	−0.760133	−0.380067	+	0.924959i	$0.624099\pi$
$128$	0	0
$129$	−1146.56	−0.782553
$130$	0	0
$131$	297.144	0.198180	0.0990901	−	0.995078i	$-0.468407\pi$
$131$	297.144	0.198180	0.0990901	+	0.995078i	$0.468407\pi$
$132$	0	0
$133$	931.457	0.607275
$134$	0	0
$135$	936.219	0.596866
$136$	0	0
$137$	2591.05	1.61583	0.807915	−	0.589300i	$-0.200596\pi$
$137$	2591.05	1.61583	0.807915	+	0.589300i	$0.200596\pi$
$138$	0	0
$139$	−243.567	−0.148626	−0.0743132	−	0.997235i	$-0.523676\pi$
$139$	−243.567	−0.148626	−0.0743132	+	0.997235i	$0.523676\pi$
$140$	0	0
$141$	−827.783	−0.494411
$142$	0	0
$143$	0	0
$144$	0	0
$145$	243.575	0.139502
$146$	0	0
$147$	−3129.69	−1.75600
$148$	0	0
$149$	−1882.28	−1.03492	−0.517458	−	0.855708i	$-0.673122\pi$
$149$	−1882.28	−1.03492	−0.517458	+	0.855708i	$0.673122\pi$
$150$	0	0
$151$	−1618.58	−0.872305	−0.436153	−	0.899873i	$-0.643659\pi$
$151$	−1618.58	−0.872305	−0.436153	+	0.899873i	$0.643659\pi$
$152$	0	0
$153$	−379.212	−0.200376
$154$	0	0
$155$	1643.36	0.851598
$156$	0	0
$157$	−1615.16	−0.821045	−0.410523	−	0.911850i	$-0.634654\pi$
$157$	−1615.16	−0.821045	−0.410523	+	0.911850i	$0.634654\pi$
$158$	0	0
$159$	−194.040	−0.0967823
$160$	0	0
$161$	−1230.48	−0.602332
$162$	0	0
$163$	−414.211	−0.199040	−0.0995199	−	0.995036i	$-0.531731\pi$
$163$	−414.211	−0.199040	−0.0995199	+	0.995036i	$0.531731\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−468.013	−0.216862	−0.108431	−	0.994104i	$-0.534583\pi$
$167$	−468.013	−0.216862	−0.108431	+	0.994104i	$0.534583\pi$
$168$	0	0
$169$	−2176.78	−0.990795
$170$	0	0
$171$	184.854	0.0826677
$172$	0	0
$173$	2302.50	1.01188	0.505941	−	0.862568i	$-0.331145\pi$
$173$	2302.50	1.01188	0.505941	+	0.862568i	$0.331145\pi$
$174$	0	0
$175$	2790.18	1.20525
$176$	0	0
$177$	−820.292	−0.348344
$178$	0	0
$179$	3381.94	1.41217	0.706083	−	0.708129i	$-0.250460\pi$
$179$	3381.94	1.41217	0.706083	+	0.708129i	$0.250460\pi$
$180$	0	0
$181$	−3675.98	−1.50958	−0.754789	−	0.655968i	$-0.772261\pi$
$181$	−3675.98	−1.50958	−0.754789	+	0.655968i	$0.772261\pi$
$182$	0	0
$183$	2540.60	1.02627
$184$	0	0
$185$	693.226	0.275497
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4869.58	−1.87413
$190$	0	0
$191$	1778.66	0.673819	0.336910	−	0.941537i	$-0.390618\pi$
$191$	1778.66	0.673819	0.336910	+	0.941537i	$0.390618\pi$
$192$	0	0
$193$	−3026.33	−1.12870	−0.564352	−	0.825534i	$-0.690874\pi$
$193$	−3026.33	−1.12870	−0.564352	+	0.825534i	$0.690874\pi$
$194$	0	0
$195$	126.148	0.0463265
$196$	0	0
$197$	4415.89	1.59705	0.798527	−	0.601960i	$-0.205613\pi$
$197$	4415.89	1.59705	0.798527	+	0.601960i	$0.205613\pi$
$198$	0	0
$199$	4105.13	1.46234	0.731169	−	0.682196i	$-0.238975\pi$
$199$	4105.13	1.46234	0.731169	+	0.682196i	$0.238975\pi$
$200$	0	0
$201$	3501.11	1.22860
$202$	0	0
$203$	−1266.92	−0.438030
$204$	0	0
$205$	−829.056	−0.282458
$206$	0	0
$207$	−244.198	−0.0819949
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2851.03	−0.930204	−0.465102	−	0.885257i	$-0.653982\pi$
$211$	−2851.03	−0.930204	−0.465102	+	0.885257i	$0.653982\pi$
$212$	0	0
$213$	−120.546	−0.0387779
$214$	0	0
$215$	1559.44	0.494663
$216$	0	0
$217$	−8547.65	−2.67397
$218$	0	0
$219$	−1690.10	−0.521489
$220$	0	0
$221$	−267.473	−0.0814127
$222$	0	0
$223$	−2110.98	−0.633910	−0.316955	−	0.948441i	$-0.602660\pi$
$223$	−2110.98	−0.633910	−0.316955	+	0.948441i	$0.602660\pi$
$224$	0	0
$225$	553.732	0.164069
$226$	0	0
$227$	−1118.76	−0.327113	−0.163556	−	0.986534i	$-0.552297\pi$
$227$	−1118.76	−0.327113	−0.163556	+	0.986534i	$0.552297\pi$
$228$	0	0
$229$	852.481	0.245998	0.122999	−	0.992407i	$-0.460749\pi$
$229$	852.481	0.245998	0.122999	+	0.992407i	$0.460749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3158.86	−0.888171	−0.444086	−	0.895984i	$-0.646471\pi$
$233$	−3158.86	−0.888171	−0.444086	+	0.895984i	$0.646471\pi$
$234$	0	0
$235$	1125.86	0.312524
$236$	0	0
$237$	−1147.95	−0.314631
$238$	0	0
$239$	−4442.88	−1.20245	−0.601226	−	0.799079i	$-0.705321\pi$
$239$	−4442.88	−1.20245	−0.601226	+	0.799079i	$0.705321\pi$
$240$	0	0
$241$	−1903.11	−0.508673	−0.254336	−	0.967116i	$-0.581857\pi$
$241$	−1903.11	−0.508673	−0.254336	+	0.967116i	$0.581857\pi$
$242$	0	0
$243$	−1748.19	−0.461509
$244$	0	0
$245$	4256.68	1.11000
$246$	0	0
$247$	130.385	0.0335879
$248$	0	0
$249$	4791.98	1.21959
$250$	0	0
$251$	1790.11	0.450162	0.225081	−	0.974340i	$-0.427735\pi$
$251$	1790.11	0.450162	0.225081	+	0.974340i	$0.427735\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1668.36	−0.409712
$256$	0	0
$257$	5759.65	1.39797	0.698983	−	0.715138i	$-0.253636\pi$
$257$	5759.65	1.39797	0.698983	+	0.715138i	$0.253636\pi$
$258$	0	0
$259$	−3605.70	−0.865047
$260$	0	0
$261$	−251.429	−0.0596285
$262$	0	0
$263$	−5675.79	−1.33074	−0.665370	−	0.746514i	$-0.731726\pi$
$263$	−5675.79	−1.33074	−0.665370	+	0.746514i	$0.731726\pi$
$264$	0	0
$265$	263.913	0.0611775
$266$	0	0
$267$	−265.849	−0.0609351
$268$	0	0
$269$	−1628.03	−0.369006	−0.184503	−	0.982832i	$-0.559068\pi$
$269$	−1628.03	−0.369006	−0.184503	+	0.982832i	$0.559068\pi$
$270$	0	0
$271$	5533.83	1.24043	0.620215	−	0.784432i	$-0.287045\pi$
$271$	5533.83	1.24043	0.620215	+	0.784432i	$0.287045\pi$
$272$	0	0
$273$	−656.139	−0.145463
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5357.23	−1.16204	−0.581020	−	0.813889i	$-0.697346\pi$
$277$	−5357.23	−1.16204	−0.581020	+	0.813889i	$0.697346\pi$
$278$	0	0
$279$	−1696.34	−0.364005
$280$	0	0
$281$	−5723.51	−1.21508	−0.607538	−	0.794291i	$-0.707843\pi$
$281$	−5723.51	−1.21508	−0.607538	+	0.794291i	$0.707843\pi$
$282$	0	0
$283$	−5837.12	−1.22608	−0.613041	−	0.790051i	$-0.710054\pi$
$283$	−5837.12	−1.22608	−0.613041	+	0.790051i	$0.710054\pi$
$284$	0	0
$285$	813.273	0.169032
$286$	0	0
$287$	4312.19	0.886902
$288$	0	0
$289$	−1375.57	−0.279987
$290$	0	0
$291$	2714.24	0.546775
$292$	0	0
$293$	−1414.09	−0.281952	−0.140976	−	0.990013i	$-0.545024\pi$
$293$	−1414.09	−0.281952	−0.140976	+	0.990013i	$0.545024\pi$
$294$	0	0
$295$	1115.67	0.220194
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−172.243	−0.0333145
$300$	0	0
$301$	−8111.14	−1.55322
$302$	0	0
$303$	−5817.23	−1.10294
$304$	0	0
$305$	−3455.46	−0.648718
$306$	0	0
$307$	6080.13	1.13033	0.565165	−	0.824978i	$-0.308812\pi$
$307$	6080.13	1.13033	0.565165	+	0.824978i	$0.308812\pi$
$308$	0	0
$309$	5084.42	0.936059
$310$	0	0
$311$	6572.40	1.19835	0.599174	−	0.800619i	$-0.295496\pi$
$311$	6572.40	1.19835	0.599174	+	0.800619i	$0.295496\pi$
$312$	0	0
$313$	−2977.42	−0.537680	−0.268840	−	0.963185i	$-0.586640\pi$
$313$	−2977.42	−0.537680	−0.268840	+	0.963185i	$0.586640\pi$
$314$	0	0
$315$	1265.22	0.226309
$316$	0	0
$317$	2239.07	0.396715	0.198357	−	0.980130i	$-0.436439\pi$
$317$	2239.07	0.396715	0.198357	+	0.980130i	$0.436439\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.2235	−0.00212538
$322$	0	0
$323$	−1724.39	−0.297051
$324$	0	0
$325$	390.569	0.0666612
$326$	0	0
$327$	47.8623	0.00809416
$328$	0	0
$329$	−5855.99	−0.981310
$330$	0	0
$331$	1551.70	0.257671	0.128835	−	0.991666i	$-0.458876\pi$
$331$	1551.70	0.257671	0.128835	+	0.991666i	$0.458876\pi$
$332$	0	0
$333$	−715.577	−0.117758
$334$	0	0
$335$	−4761.84	−0.776618
$336$	0	0
$337$	−6801.54	−1.09942	−0.549708	−	0.835357i	$-0.685261\pi$
$337$	−6801.54	−1.09942	−0.549708	+	0.835357i	$0.685261\pi$
$338$	0	0
$339$	−10221.6	−1.63764
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−11120.8	−1.75063
$344$	0	0
$345$	−1074.36	−0.167656
$346$	0	0
$347$	−8299.68	−1.28401	−0.642003	−	0.766702i	$-0.721896\pi$
$347$	−8299.68	−1.28401	−0.642003	+	0.766702i	$0.721896\pi$
$348$	0	0
$349$	64.3320	0.00986708	0.00493354	−	0.999988i	$-0.498430\pi$
$349$	64.3320	0.00986708	0.00493354	+	0.999988i	$0.498430\pi$
$350$	0	0
$351$	−681.643	−0.103656
$352$	0	0
$353$	5757.22	0.868062	0.434031	−	0.900898i	$-0.357091\pi$
$353$	5757.22	0.868062	0.434031	+	0.900898i	$0.357091\pi$
$354$	0	0
$355$	163.954	0.0245121
$356$	0	0
$357$	8677.67	1.28647
$358$	0	0
$359$	−12590.2	−1.85093	−0.925465	−	0.378833i	$-0.876326\pi$
$359$	−12590.2	−1.85093	−0.925465	+	0.378833i	$0.876326\pi$
$360$	0	0
$361$	−6018.41	−0.877448
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2298.69	0.329641
$366$	0	0
$367$	−7662.22	−1.08982	−0.544911	−	0.838494i	$-0.683436\pi$
$367$	−7662.22	−1.08982	−0.544911	+	0.838494i	$0.683436\pi$
$368$	0	0
$369$	855.787	0.120733
$370$	0	0
$371$	−1372.70	−0.192094
$372$	0	0
$373$	−7421.71	−1.03024	−0.515122	−	0.857117i	$-0.672254\pi$
$373$	−7421.71	−1.03024	−0.515122	+	0.857117i	$0.672254\pi$
$374$	0	0
$375$	5942.51	0.818320
$376$	0	0
$377$	−177.343	−0.0242271
$378$	0	0
$379$	−5526.10	−0.748962	−0.374481	−	0.927235i	$-0.622179\pi$
$379$	−5526.10	−0.748962	−0.374481	+	0.927235i	$0.622179\pi$
$380$	0	0
$381$	4940.64	0.664348
$382$	0	0
$383$	−9043.90	−1.20658	−0.603292	−	0.797520i	$-0.706145\pi$
$383$	−9043.90	−1.20658	−0.603292	+	0.797520i	$0.706145\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1609.71	−0.211438
$388$	0	0
$389$	−11777.7	−1.53510	−0.767552	−	0.640987i	$-0.778525\pi$
$389$	−11777.7	−1.53510	−0.767552	+	0.640987i	$0.778525\pi$
$390$	0	0
$391$	2277.97	0.294634
$392$	0	0
$393$	−1349.45	−0.173207
$394$	0	0
$395$	1561.32	0.198883
$396$	0	0
$397$	8173.72	1.03332	0.516659	−	0.856191i	$-0.327176\pi$
$397$	8173.72	1.03332	0.516659	+	0.856191i	$0.327176\pi$
$398$	0	0
$399$	−4230.10	−0.530752
$400$	0	0
$401$	−8666.14	−1.07922	−0.539609	−	0.841916i	$-0.681428\pi$
$401$	−8666.14	−1.07922	−0.539609	+	0.841916i	$0.681428\pi$
$402$	0	0
$403$	−1196.50	−0.147895
$404$	0	0
$405$	−3188.42	−0.391195
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2320.05	0.280486	0.140243	−	0.990117i	$-0.455212\pi$
$409$	2320.05	0.280486	0.140243	+	0.990117i	$0.455212\pi$
$410$	0	0
$411$	−11767.0	−1.41222
$412$	0	0
$413$	−5802.99	−0.691396
$414$	0	0
$415$	−6517.54	−0.770924
$416$	0	0
$417$	1106.13	0.129898
$418$	0	0
$419$	−12620.5	−1.47148	−0.735740	−	0.677264i	$-0.763165\pi$
$419$	−12620.5	−1.47148	−0.735740	+	0.677264i	$0.763165\pi$
$420$	0	0
$421$	8305.13	0.961442	0.480721	−	0.876873i	$-0.340375\pi$
$421$	8305.13	0.961442	0.480721	+	0.876873i	$0.340375\pi$
$422$	0	0
$423$	−1162.16	−0.133585
$424$	0	0
$425$	−5165.41	−0.589551
$426$	0	0
$427$	17973.0	2.03694
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10156.2	1.13505	0.567526	−	0.823356i	$-0.307901\pi$
$431$	10156.2	1.13505	0.567526	+	0.823356i	$0.307901\pi$
$432$	0	0
$433$	−1944.71	−0.215836	−0.107918	−	0.994160i	$-0.534418\pi$
$433$	−1944.71	−0.215836	−0.107918	+	0.994160i	$0.534418\pi$
$434$	0	0
$435$	−1106.17	−0.121924
$436$	0	0
$437$	−1110.44	−0.121555
$438$	0	0
$439$	−15398.2	−1.67407	−0.837035	−	0.547150i	$-0.815713\pi$
$439$	−15398.2	−1.67407	−0.837035	+	0.547150i	$0.815713\pi$
$440$	0	0
$441$	−4393.92	−0.474454
$442$	0	0
$443$	13035.8	1.39808	0.699039	−	0.715083i	$-0.253611\pi$
$443$	13035.8	1.39808	0.699039	+	0.715083i	$0.253611\pi$
$444$	0	0
$445$	361.579	0.0385180
$446$	0	0
$447$	8548.16	0.904506
$448$	0	0
$449$	−7783.29	−0.818076	−0.409038	−	0.912517i	$-0.634136\pi$
$449$	−7783.29	−0.818076	−0.409038	+	0.912517i	$0.634136\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	7350.59	0.762385
$454$	0	0
$455$	892.412	0.0919492
$456$	0	0
$457$	−505.825	−0.0517757	−0.0258878	−	0.999665i	$-0.508241\pi$
$457$	−505.825	−0.0517757	−0.0258878	+	0.999665i	$0.508241\pi$
$458$	0	0
$459$	9014.97	0.916738
$460$	0	0
$461$	9973.13	1.00758	0.503791	−	0.863826i	$-0.331938\pi$
$461$	9973.13	1.00758	0.503791	+	0.863826i	$0.331938\pi$
$462$	0	0
$463$	−5539.97	−0.556078	−0.278039	−	0.960570i	$-0.589685\pi$
$463$	−5539.97	−0.556078	−0.278039	+	0.960570i	$0.589685\pi$
$464$	0	0
$465$	−7463.12	−0.744288
$466$	0	0
$467$	15646.4	1.55039	0.775194	−	0.631723i	$-0.217652\pi$
$467$	15646.4	1.55039	0.775194	+	0.631723i	$0.217652\pi$
$468$	0	0
$469$	24767.9	2.43854
$470$	0	0
$471$	7335.07	0.717585
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2517.98	0.243227
$476$	0	0
$477$	−272.422	−0.0261496
$478$	0	0
$479$	−15820.0	−1.50905	−0.754523	−	0.656273i	$-0.772132\pi$
$479$	−15820.0	−1.50905	−0.754523	+	0.656273i	$0.772132\pi$
$480$	0	0
$481$	−504.725	−0.0478450
$482$	0	0
$483$	5588.08	0.526432
$484$	0	0
$485$	−3691.62	−0.345625
$486$	0	0
$487$	1035.46	0.0963475	0.0481738	−	0.998839i	$-0.484660\pi$
$487$	1035.46	0.0963475	0.0481738	+	0.998839i	$0.484660\pi$
$488$	0	0
$489$	1881.09	0.173959
$490$	0	0
$491$	−9385.84	−0.862682	−0.431341	−	0.902189i	$-0.641959\pi$
$491$	−9385.84	−0.862682	−0.431341	+	0.902189i	$0.641959\pi$
$492$	0	0
$493$	2345.42	0.214264
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−852.781	−0.0769667
$498$	0	0
$499$	4640.55	0.416312	0.208156	−	0.978096i	$-0.433254\pi$
$499$	4640.55	0.416312	0.208156	+	0.978096i	$0.433254\pi$
$500$	0	0
$501$	2125.42	0.189535
$502$	0	0
$503$	4183.00	0.370797	0.185398	−	0.982663i	$-0.440642\pi$
$503$	4183.00	0.370797	0.185398	+	0.982663i	$0.440642\pi$
$504$	0	0
$505$	7911.98	0.697186
$506$	0	0
$507$	9885.57	0.865944
$508$	0	0
$509$	17980.0	1.56572	0.782860	−	0.622198i	$-0.213760\pi$
$509$	17980.0	1.56572	0.782860	+	0.622198i	$0.213760\pi$
$510$	0	0
$511$	−11956.3	−1.03506
$512$	0	0
$513$	−4394.52	−0.378212
$514$	0	0
$515$	−6915.29	−0.591697
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−10456.5	−0.884374
$520$	0	0
$521$	8211.54	0.690507	0.345253	−	0.938510i	$-0.387793\pi$
$521$	8211.54	0.690507	0.345253	+	0.938510i	$0.387793\pi$
$522$	0	0
$523$	13820.0	1.15546	0.577729	−	0.816228i	$-0.303939\pi$
$523$	13820.0	1.15546	0.577729	+	0.816228i	$0.303939\pi$
$524$	0	0
$525$	−12671.3	−1.05337
$526$	0	0
$527$	15824.1	1.30799
$528$	0	0
$529$	−10700.1	−0.879434
$530$	0	0
$531$	−1151.65	−0.0941190
$532$	0	0
$533$	603.620	0.0490538
$534$	0	0
$535$	16.6251	0.00134349
$536$	0	0
$537$	−15358.7	−1.23422
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3747.36	−0.297803	−0.148902	−	0.988852i	$-0.547574\pi$
$541$	−3747.36	−0.297803	−0.148902	+	0.988852i	$0.547574\pi$
$542$	0	0
$543$	16694.0	1.31935
$544$	0	0
$545$	−65.0973	−0.00511644
$546$	0	0
$547$	12440.2	0.972400	0.486200	−	0.873847i	$-0.338383\pi$
$547$	12440.2	0.972400	0.486200	+	0.873847i	$0.338383\pi$
$548$	0	0
$549$	3566.87	0.277286
$550$	0	0
$551$	−1143.32	−0.0883976
$552$	0	0
$553$	−8120.96	−0.624482
$554$	0	0
$555$	−3148.20	−0.240782
$556$	0	0
$557$	6386.64	0.485836	0.242918	−	0.970047i	$-0.421895\pi$
$557$	6386.64	0.485836	0.242918	+	0.970047i	$0.421895\pi$
$558$	0	0
$559$	−1135.40	−0.0859071
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13966.9	1.04553	0.522765	−	0.852477i	$-0.324901\pi$
$563$	13966.9	1.04553	0.522765	+	0.852477i	$0.324901\pi$
$564$	0	0
$565$	13902.3	1.03518
$566$	0	0
$567$	16584.0	1.22833
$568$	0	0
$569$	19058.3	1.40416	0.702078	−	0.712100i	$-0.252256\pi$
$569$	19058.3	1.40416	0.702078	+	0.712100i	$0.252256\pi$
$570$	0	0
$571$	6805.90	0.498806	0.249403	−	0.968400i	$-0.419766\pi$
$571$	6805.90	0.498806	0.249403	+	0.968400i	$0.419766\pi$
$572$	0	0
$573$	−8077.58	−0.588911
$574$	0	0
$575$	−3326.33	−0.241248
$576$	0	0
$577$	11014.4	0.794689	0.397344	−	0.917670i	$-0.369932\pi$
$577$	11014.4	0.794689	0.397344	+	0.917670i	$0.369932\pi$
$578$	0	0
$579$	13743.7	0.986475
$580$	0	0
$581$	33899.9	2.42066
$582$	0	0
$583$	0	0
$584$	0	0
$585$	177.106	0.0125170
$586$	0	0
$587$	4754.72	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$587$	4754.72	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$588$	0	0
$589$	−7713.77	−0.539627
$590$	0	0
$591$	−20054.3	−1.39581
$592$	0	0
$593$	8479.59	0.587209	0.293604	−	0.955927i	$-0.405145\pi$
$593$	8479.59	0.587209	0.293604	+	0.955927i	$0.405145\pi$
$594$	0	0
$595$	−11802.5	−0.813199
$596$	0	0
$597$	−18643.0	−1.27807
$598$	0	0
$599$	7838.58	0.534684	0.267342	−	0.963602i	$-0.413855\pi$
$599$	7838.58	0.534684	0.267342	+	0.963602i	$0.413855\pi$
$600$	0	0
$601$	−27421.1	−1.86112	−0.930558	−	0.366146i	$-0.880677\pi$
$601$	−27421.1	−1.86112	−0.930558	+	0.366146i	$0.880677\pi$
$602$	0	0
$603$	4915.37	0.331956
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24778.0	−1.65685	−0.828423	−	0.560102i	$-0.810762\pi$
$607$	−24778.0	−1.65685	−0.828423	+	0.560102i	$0.810762\pi$
$608$	0	0
$609$	5753.55	0.382833
$610$	0	0
$611$	−819.720	−0.0542755
$612$	0	0
$613$	15133.1	0.997098	0.498549	−	0.866862i	$-0.333866\pi$
$613$	15133.1	0.997098	0.498549	+	0.866862i	$0.333866\pi$
$614$	0	0
$615$	3765.06	0.246865
$616$	0	0
$617$	15497.8	1.01121	0.505605	−	0.862765i	$-0.331269\pi$
$617$	15497.8	1.01121	0.505605	+	0.862765i	$0.331269\pi$
$618$	0	0
$619$	9208.55	0.597937	0.298969	−	0.954263i	$-0.403357\pi$
$619$	9208.55	0.597937	0.298969	+	0.954263i	$0.403357\pi$
$620$	0	0
$621$	5805.29	0.375134
$622$	0	0
$623$	−1880.69	−0.120944
$624$	0	0
$625$	2773.66	0.177514
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6675.16	0.423141
$630$	0	0
$631$	−3182.55	−0.200785	−0.100393	−	0.994948i	$-0.532010\pi$
$631$	−3182.55	−0.200785	−0.100393	+	0.994948i	$0.532010\pi$
$632$	0	0
$633$	12947.6	0.812988
$634$	0	0
$635$	−6719.74	−0.419944
$636$	0	0
$637$	−3099.21	−0.192771
$638$	0	0
$639$	−169.241	−0.0104774
$640$	0	0
$641$	−2153.33	−0.132686	−0.0663428	−	0.997797i	$-0.521133\pi$
$641$	−2153.33	−0.132686	−0.0663428	+	0.997797i	$0.521133\pi$
$642$	0	0
$643$	23264.7	1.42686	0.713429	−	0.700727i	$-0.247141\pi$
$643$	23264.7	1.42686	0.713429	+	0.700727i	$0.247141\pi$
$644$	0	0
$645$	−7081.99	−0.432330
$646$	0	0
$647$	14758.9	0.896806	0.448403	−	0.893832i	$-0.351993\pi$
$647$	14758.9	0.896806	0.448403	+	0.893832i	$0.351993\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	38818.1	2.33702
$652$	0	0
$653$	−3774.08	−0.226173	−0.113087	−	0.993585i	$-0.536074\pi$
$653$	−3774.08	−0.226173	−0.113087	+	0.993585i	$0.536074\pi$
$654$	0	0
$655$	1835.37	0.109487
$656$	0	0
$657$	−2372.81	−0.140901
$658$	0	0
$659$	9973.47	0.589547	0.294773	−	0.955567i	$-0.404756\pi$
$659$	9973.47	0.589547	0.294773	+	0.955567i	$0.404756\pi$
$660$	0	0
$661$	−24856.8	−1.46266	−0.731329	−	0.682025i	$-0.761100\pi$
$661$	−24856.8	−1.46266	−0.731329	+	0.682025i	$0.761100\pi$
$662$	0	0
$663$	1214.70	0.0711538
$664$	0	0
$665$	5753.34	0.335496
$666$	0	0
$667$	1510.36	0.0876781
$668$	0	0
$669$	9586.78	0.554030
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14191.4	−0.812834	−0.406417	−	0.913688i	$-0.633222\pi$
$673$	−14191.4	−0.812834	−0.406417	+	0.913688i	$0.633222\pi$
$674$	0	0
$675$	−13163.8	−0.750630
$676$	0	0
$677$	12041.6	0.683599	0.341799	−	0.939773i	$-0.388964\pi$
$677$	12041.6	0.683599	0.341799	+	0.939773i	$0.388964\pi$
$678$	0	0
$679$	19201.3	1.08524
$680$	0	0
$681$	5080.71	0.285893
$682$	0	0
$683$	−3428.81	−0.192093	−0.0960467	−	0.995377i	$-0.530620\pi$
$683$	−3428.81	−0.192093	−0.0960467	+	0.995377i	$0.530620\pi$
$684$	0	0
$685$	16004.2	0.892684
$686$	0	0
$687$	−3871.44	−0.215000
$688$	0	0
$689$	−192.150	−0.0106246
$690$	0	0
$691$	−26382.2	−1.45243	−0.726213	−	0.687469i	$-0.758722\pi$
$691$	−26382.2	−1.45243	−0.726213	+	0.687469i	$0.758722\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1504.44	−0.0821104
$696$	0	0
$697$	−7983.08	−0.433832
$698$	0	0
$699$	14345.6	0.776252
$700$	0	0
$701$	−3052.43	−0.164463	−0.0822317	−	0.996613i	$-0.526205\pi$
$701$	−3052.43	−0.164463	−0.0822317	+	0.996613i	$0.526205\pi$
$702$	0	0
$703$	−3253.94	−0.174573
$704$	0	0
$705$	−5112.98	−0.273143
$706$	0	0
$707$	−41152.8	−2.18913
$708$	0	0
$709$	−6702.32	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$709$	−6702.32	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$710$	0	0
$711$	−1611.66	−0.0850100
$712$	0	0
$713$	10190.1	0.535235
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20176.8	1.05093
$718$	0	0
$719$	16564.6	0.859188	0.429594	−	0.903022i	$-0.358657\pi$
$719$	16564.6	0.859188	0.429594	+	0.903022i	$0.358657\pi$
$720$	0	0
$721$	35968.7	1.85790
$722$	0	0
$723$	8642.75	0.444574
$724$	0	0
$725$	−3424.82	−0.175441
$726$	0	0
$727$	26489.8	1.35138	0.675689	−	0.737187i	$-0.263846\pi$
$727$	26489.8	1.35138	0.675689	+	0.737187i	$0.263846\pi$
$728$	0	0
$729$	21876.6	1.11145
$730$	0	0
$731$	15016.0	0.759763
$732$	0	0
$733$	−12401.8	−0.624924	−0.312462	−	0.949930i	$-0.601154\pi$
$733$	−12401.8	−0.624924	−0.312462	+	0.949930i	$0.601154\pi$
$734$	0	0
$735$	−19331.2	−0.970125
$736$	0	0
$737$	0	0
$738$	0	0
$739$	15374.2	0.765291	0.382645	−	0.923895i	$-0.375013\pi$
$739$	15374.2	0.765291	0.382645	+	0.923895i	$0.375013\pi$
$740$	0	0
$741$	−592.129	−0.0293554
$742$	0	0
$743$	13918.4	0.687238	0.343619	−	0.939109i	$-0.388347\pi$
$743$	13918.4	0.687238	0.343619	+	0.939109i	$0.388347\pi$
$744$	0	0
$745$	−11626.3	−0.571752
$746$	0	0
$747$	6727.68	0.329522
$748$	0	0
$749$	−86.4725	−0.00421847
$750$	0	0
$751$	−16460.6	−0.799806	−0.399903	−	0.916557i	$-0.630956\pi$
$751$	−16460.6	−0.799806	−0.399903	+	0.916557i	$0.630956\pi$
$752$	0	0
$753$	−8129.57	−0.393437
$754$	0	0
$755$	−9997.49	−0.481915
$756$	0	0
$757$	−19863.9	−0.953718	−0.476859	−	0.878980i	$-0.658225\pi$
$757$	−19863.9	−0.953718	−0.476859	+	0.878980i	$0.658225\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	860.397	0.0409847	0.0204924	−	0.999790i	$-0.493477\pi$
$761$	860.397	0.0409847	0.0204924	+	0.999790i	$0.493477\pi$
$762$	0	0
$763$	338.592	0.0160653
$764$	0	0
$765$	−2342.28	−0.110700
$766$	0	0
$767$	−812.301	−0.0382406
$768$	0	0
$769$	−18337.1	−0.859885	−0.429942	−	0.902856i	$-0.641466\pi$
$769$	−18337.1	−0.859885	−0.429942	+	0.902856i	$0.641466\pi$
$770$	0	0
$771$	−26156.8	−1.22181
$772$	0	0
$773$	−40910.3	−1.90354	−0.951772	−	0.306806i	$-0.900740\pi$
$773$	−40910.3	−1.90354	−0.951772	+	0.306806i	$0.900740\pi$
$774$	0	0
$775$	−23106.6	−1.07099
$776$	0	0
$777$	16374.8	0.756042
$778$	0	0
$779$	3891.51	0.178983
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5977.19	0.272806
$784$	0	0
$785$	−9976.39	−0.453596
$786$	0	0
$787$	−33328.3	−1.50956	−0.754781	−	0.655977i	$-0.772257\pi$
$787$	−33328.3	−1.50956	−0.754781	+	0.655977i	$0.772257\pi$
$788$	0	0
$789$	25775.9	1.16305
$790$	0	0
$791$	−72310.6	−3.25040
$792$	0	0
$793$	2515.85	0.112661
$794$	0	0
$795$	−1198.53	−0.0534685
$796$	0	0
$797$	−11904.7	−0.529090	−0.264545	−	0.964373i	$-0.585222\pi$
$797$	−11904.7	−0.529090	−0.264545	+	0.964373i	$0.585222\pi$
$798$	0	0
$799$	10841.1	0.480012
$800$	0	0
$801$	−373.237	−0.0164640
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−7600.32	−0.332766
$806$	0	0
$807$	7393.50	0.322507
$808$	0	0
$809$	−43695.8	−1.89897	−0.949483	−	0.313818i	$-0.898392\pi$
$809$	−43695.8	−1.89897	−0.949483	+	0.313818i	$0.898392\pi$
$810$	0	0
$811$	32650.0	1.41368	0.706842	−	0.707371i	$-0.250119\pi$
$811$	32650.0	1.41368	0.706842	+	0.707371i	$0.250119\pi$
$812$	0	0
$813$	−25131.3	−1.08412
$814$	0	0
$815$	−2558.46	−0.109962
$816$	0	0
$817$	−7319.84	−0.313450
$818$	0	0
$819$	−921.184	−0.0393026
$820$	0	0
$821$	−16030.9	−0.681464	−0.340732	−	0.940160i	$-0.610675\pi$
$821$	−16030.9	−0.681464	−0.340732	+	0.940160i	$0.610675\pi$
$822$	0	0
$823$	15092.4	0.639232	0.319616	−	0.947547i	$-0.396446\pi$
$823$	15092.4	0.639232	0.319616	+	0.947547i	$0.396446\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11534.5	0.485000	0.242500	−	0.970151i	$-0.422033\pi$
$827$	11534.5	0.485000	0.242500	+	0.970151i	$0.422033\pi$
$828$	0	0
$829$	39259.4	1.64479	0.822397	−	0.568915i	$-0.192636\pi$
$829$	39259.4	1.64479	0.822397	+	0.568915i	$0.192636\pi$
$830$	0	0
$831$	24329.2	1.01561
$832$	0	0
$833$	40988.1	1.70486
$834$	0	0
$835$	−2890.78	−0.119808
$836$	0	0
$837$	40327.0	1.66536
$838$	0	0
$839$	−23334.3	−0.960180	−0.480090	−	0.877219i	$-0.659396\pi$
$839$	−23334.3	−0.960180	−0.480090	+	0.877219i	$0.659396\pi$
$840$	0	0
$841$	−22833.9	−0.936238
$842$	0	0
$843$	25992.7	1.06196
$844$	0	0
$845$	−13445.3	−0.547376
$846$	0	0
$847$	0	0
$848$	0	0
$849$	26508.6	1.07158
$850$	0	0
$851$	4298.55	0.173152
$852$	0	0
$853$	−2585.70	−0.103790	−0.0518949	−	0.998653i	$-0.516526\pi$
$853$	−2585.70	−0.103790	−0.0518949	+	0.998653i	$0.516526\pi$
$854$	0	0
$855$	1141.79	0.0456707
$856$	0	0
$857$	−15421.4	−0.614687	−0.307343	−	0.951599i	$-0.599440\pi$
$857$	−15421.4	−0.614687	−0.307343	+	0.951599i	$0.599440\pi$
$858$	0	0
$859$	−10191.5	−0.404807	−0.202403	−	0.979302i	$-0.564875\pi$
$859$	−10191.5	−0.404807	−0.202403	+	0.979302i	$0.564875\pi$
$860$	0	0
$861$	−19583.3	−0.775143
$862$	0	0
$863$	−35480.0	−1.39948	−0.699742	−	0.714396i	$-0.746702\pi$
$863$	−35480.0	−1.39948	−0.699742	+	0.714396i	$0.746702\pi$
$864$	0	0
$865$	14221.8	0.559026
$866$	0	0
$867$	6247.01	0.244705
$868$	0	0
$869$	0	0
$870$	0	0
$871$	3467.00	0.134874
$872$	0	0
$873$	3810.65	0.147733
$874$	0	0
$875$	42039.1	1.62421
$876$	0	0
$877$	18992.9	0.731296	0.365648	−	0.930753i	$-0.380847\pi$
$877$	18992.9	0.731296	0.365648	+	0.930753i	$0.380847\pi$
$878$	0	0
$879$	6421.91	0.246423
$880$	0	0
$881$	−43661.0	−1.66967	−0.834833	−	0.550504i	$-0.814436\pi$
$881$	−43661.0	−1.66967	−0.834833	+	0.550504i	$0.814436\pi$
$882$	0	0
$883$	20242.0	0.771458	0.385729	−	0.922612i	$-0.373950\pi$
$883$	20242.0	0.771458	0.385729	+	0.922612i	$0.373950\pi$
$884$	0	0
$885$	−5066.70	−0.192447
$886$	0	0
$887$	−20063.9	−0.759503	−0.379752	−	0.925088i	$-0.623991\pi$
$887$	−20063.9	−0.759503	−0.379752	+	0.925088i	$0.623991\pi$
$888$	0	0
$889$	34951.6	1.31860
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5284.70	−0.198035
$894$	0	0
$895$	20889.2	0.780168
$896$	0	0
$897$	782.219	0.0291165
$898$	0	0
$899$	10491.8	0.389235
$900$	0	0
$901$	2541.25	0.0939637
$902$	0	0
$903$	36835.8	1.35749
$904$	0	0
$905$	−22705.5	−0.833983
$906$	0	0
$907$	9772.55	0.357764	0.178882	−	0.983870i	$-0.442752\pi$
$907$	9772.55	0.357764	0.178882	+	0.983870i	$0.442752\pi$
$908$	0	0
$909$	−8167.08	−0.298003
$910$	0	0
$911$	9864.72	0.358763	0.179381	−	0.983780i	$-0.442590\pi$
$911$	9864.72	0.358763	0.179381	+	0.983780i	$0.442590\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	15692.5	0.566972
$916$	0	0
$917$	−9546.38	−0.343783
$918$	0	0
$919$	−12906.0	−0.463253	−0.231627	−	0.972805i	$-0.574405\pi$
$919$	−12906.0	−0.463253	−0.231627	+	0.972805i	$0.574405\pi$
$920$	0	0
$921$	−27612.2	−0.987897
$922$	0	0
$923$	−119.372	−0.00425696
$924$	0	0
$925$	−9747.18	−0.346471
$926$	0	0
$927$	7138.25	0.252914
$928$	0	0
$929$	51082.5	1.80405	0.902025	−	0.431684i	$-0.142080\pi$
$929$	51082.5	1.80405	0.902025	+	0.431684i	$0.142080\pi$
$930$	0	0
$931$	−19980.4	−0.703365
$932$	0	0
$933$	−29847.8	−1.04734
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10712.2	0.373482	0.186741	−	0.982409i	$-0.440208\pi$
$937$	10712.2	0.373482	0.186741	+	0.982409i	$0.440208\pi$
$938$	0	0
$939$	13521.6	0.469926
$940$	0	0
$941$	46943.3	1.62626	0.813128	−	0.582085i	$-0.197763\pi$
$941$	46943.3	1.62626	0.813128	+	0.582085i	$0.197763\pi$
$942$	0	0
$943$	−5140.80	−0.177526
$944$	0	0
$945$	−30078.0	−1.03538
$946$	0	0
$947$	−8917.62	−0.306002	−0.153001	−	0.988226i	$-0.548894\pi$
$947$	−8917.62	−0.306002	−0.153001	+	0.988226i	$0.548894\pi$
$948$	0	0
$949$	−1673.63	−0.0572481
$950$	0	0
$951$	−10168.5	−0.346724
$952$	0	0
$953$	44396.9	1.50908	0.754541	−	0.656253i	$-0.227859\pi$
$953$	44396.9	1.50908	0.754541	+	0.656253i	$0.227859\pi$
$954$	0	0
$955$	10986.3	0.372259
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−83243.0	−2.80298
$960$	0	0
$961$	40995.5	1.37610
$962$	0	0
$963$	−17.1611	−0.000574256 0
$964$	0	0
$965$	−18692.8	−0.623566
$966$	0	0
$967$	−21049.8	−0.700016	−0.350008	−	0.936747i	$-0.613821\pi$
$967$	−21049.8	−0.700016	−0.350008	+	0.936747i	$0.613821\pi$
$968$	0	0
$969$	7831.10	0.259620
$970$	0	0
$971$	5436.91	0.179690	0.0898450	−	0.995956i	$-0.471363\pi$
$971$	5436.91	0.179690	0.0898450	+	0.995956i	$0.471363\pi$
$972$	0	0
$973$	7825.09	0.257822
$974$	0	0
$975$	−1773.72	−0.0582611
$976$	0	0
$977$	34449.4	1.12808	0.564040	−	0.825747i	$-0.309246\pi$
$977$	34449.4	1.12808	0.564040	+	0.825747i	$0.309246\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	67.1961	0.00218696
$982$	0	0
$983$	16668.6	0.540840	0.270420	−	0.962742i	$-0.412837\pi$
$983$	16668.6	0.540840	0.270420	+	0.962742i	$0.412837\pi$
$984$	0	0
$985$	27275.7	0.882311
$986$	0	0
$987$	26594.3	0.857654
$988$	0	0
$989$	9669.72	0.310899
$990$	0	0
$991$	−26999.0	−0.865441	−0.432721	−	0.901528i	$-0.642446\pi$
$991$	−26999.0	−0.865441	−0.432721	+	0.901528i	$0.642446\pi$
$992$	0	0
$993$	−7046.85	−0.225201
$994$	0	0
$995$	25356.2	0.807886
$996$	0	0
$997$	−19974.4	−0.634499	−0.317249	−	0.948342i	$-0.602759\pi$
$997$	−19974.4	−0.634499	−0.317249	+	0.948342i	$0.602759\pi$
$998$	0	0
$999$	17011.3	0.538753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bk.1.2		4
4.3	odd	2		121.4.a.g.1.3		4
11.3	even	5		176.4.m.c.97.2		8
11.4	even	5		176.4.m.c.49.2		8
11.10	odd	2		1936.4.a.bl.1.2		4
12.11	even	2		1089.4.a.y.1.2		4
44.3	odd	10		11.4.c.a.9.2	yes	8
44.7	even	10		121.4.c.i.27.1		8
44.15	odd	10		11.4.c.a.5.2	✓	8
44.19	even	10		121.4.c.i.9.1		8
44.27	odd	10		121.4.c.h.3.1		8
44.31	odd	10		121.4.c.h.81.1		8
44.35	even	10		121.4.c.b.81.2		8
44.39	even	10		121.4.c.b.3.2		8
44.43	even	2		121.4.a.f.1.2		4
132.47	even	10		99.4.f.c.64.1		8
132.59	even	10		99.4.f.c.82.1		8
132.131	odd	2		1089.4.a.bh.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.c.a.5.2	✓	8	44.15	odd	10
11.4.c.a.9.2	yes	8	44.3	odd	10
99.4.f.c.64.1		8	132.47	even	10
99.4.f.c.82.1		8	132.59	even	10
121.4.a.f.1.2		4	44.43	even	2
121.4.a.g.1.3		4	4.3	odd	2
121.4.c.b.3.2		8	44.39	even	10
121.4.c.b.81.2		8	44.35	even	10
121.4.c.h.3.1		8	44.27	odd	10
121.4.c.h.81.1		8	44.31	odd	10
121.4.c.i.9.1		8	44.19	even	10
121.4.c.i.27.1		8	44.7	even	10
176.4.m.c.49.2		8	11.4	even	5
176.4.m.c.97.2		8	11.3	even	5
1089.4.a.y.1.2		4	12.11	even	2
1089.4.a.bh.1.3		4	132.131	odd	2
1936.4.a.bk.1.2		4	1.1	even	1	trivial
1936.4.a.bl.1.2		4	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.54138 q^{3} +6.17671 q^{5} -32.1271 q^{7} -6.37586 q^{9} +O(q^{10})\)	\(q-4.54138 q^{3} +6.17671 q^{5} -32.1271 q^{7} -6.37586 q^{9} -4.49714 q^{13} -28.0508 q^{15} +59.4763 q^{17} -28.9929 q^{19} +145.901 q^{21} +38.3004 q^{23} -86.8483 q^{25} +151.572 q^{27} +39.4345 q^{29} +266.057 q^{31} -198.440 q^{35} +112.232 q^{37} +20.4232 q^{39} -134.223 q^{41} +252.470 q^{43} -39.3818 q^{45} +182.276 q^{47} +689.150 q^{49} -270.104 q^{51} +42.7271 q^{53} +131.668 q^{57} +180.626 q^{59} -559.433 q^{61} +204.838 q^{63} -27.7775 q^{65} -770.935 q^{67} -173.937 q^{69} +26.5440 q^{71} +372.155 q^{73} +394.411 q^{75} +252.776 q^{79} -516.200 q^{81} -1055.18 q^{83} +367.368 q^{85} -179.087 q^{87} +58.5392 q^{89} +144.480 q^{91} -1208.27 q^{93} -179.081 q^{95} -597.668 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} - 11 q^{5} - 25 q^{7} - 62 q^{9}+O(q^{10}) \) 4 * q - 6 * q^3 - 11 * q^5 - 25 * q^7 - 62 * q^9	\( 4 q - 6 q^{3} - 11 q^{5} - 25 q^{7} - 62 q^{9} + 25 q^{13} - 2 q^{15} + 232 q^{17} - 154 q^{19} + 167 q^{21} + 6 q^{23} - 13 q^{25} + 144 q^{27} + 363 q^{29} - 37 q^{31} - 356 q^{35} + 93 q^{37} + 240 q^{39} + 152 q^{41} + 325 q^{43} + 226 q^{45} + 869 q^{47} - 245 q^{49} - 52 q^{51} + 811 q^{53} + 231 q^{57} - 178 q^{59} - 105 q^{61} - q^{63} + 895 q^{65} - 43 q^{67} - 1156 q^{69} - 629 q^{71} - 270 q^{73} + 815 q^{75} - 977 q^{79} + 52 q^{81} - 1686 q^{83} - 721 q^{85} - 1155 q^{87} + 1891 q^{89} + 80 q^{91} - 666 q^{93} - 1804 q^{95} - 1772 q^{97}+O(q^{100}) \) 4 * q - 6 * q^3 - 11 * q^5 - 25 * q^7 - 62 * q^9 + 25 * q^13 - 2 * q^15 + 232 * q^17 - 154 * q^19 + 167 * q^21 + 6 * q^23 - 13 * q^25 + 144 * q^27 + 363 * q^29 - 37 * q^31 - 356 * q^35 + 93 * q^37 + 240 * q^39 + 152 * q^41 + 325 * q^43 + 226 * q^45 + 869 * q^47 - 245 * q^49 - 52 * q^51 + 811 * q^53 + 231 * q^57 - 178 * q^59 - 105 * q^61 - q^63 + 895 * q^65 - 43 * q^67 - 1156 * q^69 - 629 * q^71 - 270 * q^73 + 815 * q^75 - 977 * q^79 + 52 * q^81 - 1686 * q^83 - 721 * q^85 - 1155 * q^87 + 1891 * q^89 + 80 * q^91 - 666 * q^93 - 1804 * q^95 - 1772 * q^97

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