LMFDB - Embedded newform 920.4.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,4,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("920.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	920.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:	$54.2817572053$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104 $ x^9 - 3x^8 - 148x^7 + 278x^6 + 6502x^5 - 4928x^4 - 87343x^3 + 42737x^2 + 286800x + 53104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.70964$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70964 q^{3} -5.00000 q^{5} +20.8693 q^{7} -24.0771 q^{9} +O(q^{10})$	$q+1.70964 q^{3} -5.00000 q^{5} +20.8693 q^{7} -24.0771 q^{9} +23.6682 q^{11} +53.7383 q^{13} -8.54819 q^{15} +52.1336 q^{17} +4.88364 q^{19} +35.6789 q^{21} -23.0000 q^{23} +25.0000 q^{25} -87.3234 q^{27} -144.142 q^{29} -288.160 q^{31} +40.4641 q^{33} -104.346 q^{35} +294.532 q^{37} +91.8730 q^{39} +333.215 q^{41} -0.162395 q^{43} +120.386 q^{45} +529.988 q^{47} +92.5270 q^{49} +89.1295 q^{51} +116.147 q^{53} -118.341 q^{55} +8.34926 q^{57} -284.019 q^{59} -120.262 q^{61} -502.473 q^{63} -268.691 q^{65} +194.900 q^{67} -39.3217 q^{69} +201.508 q^{71} +913.140 q^{73} +42.7409 q^{75} +493.939 q^{77} +901.712 q^{79} +500.791 q^{81} +172.627 q^{83} -260.668 q^{85} -246.430 q^{87} -1578.55 q^{89} +1121.48 q^{91} -492.649 q^{93} -24.4182 q^{95} +1720.14 q^{97} -569.863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} - 45 q^{5} + 25 q^{7} + 62 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 - 45 * q^5 + 25 * q^7 + 62 * q^9	$ 9 q - 3 q^{3} - 45 q^{5} + 25 q^{7} + 62 q^{9} + 22 q^{11} + 23 q^{13} + 15 q^{15} - 135 q^{17} + 102 q^{19} + 54 q^{21} - 207 q^{23} + 225 q^{25} - 363 q^{27} + 280 q^{29} + 168 q^{31} + 28 q^{33} - 125 q^{35} + 153 q^{37} - 5 q^{39} - 502 q^{41} + 110 q^{43} - 310 q^{45} - 153 q^{47} + 764 q^{49} + 924 q^{51} - 273 q^{53} - 110 q^{55} + 748 q^{57} + 827 q^{59} + 1976 q^{61} + 2237 q^{63} - 115 q^{65} + 1613 q^{67} + 69 q^{69} + 1370 q^{71} + 425 q^{73} - 75 q^{75} + 2006 q^{77} + 2624 q^{79} + 1729 q^{81} + 2505 q^{83} + 675 q^{85} + 1591 q^{87} + 1120 q^{89} + 2392 q^{91} + 4401 q^{93} - 510 q^{95} + 2026 q^{97} + 3206 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 - 45 * q^5 + 25 * q^7 + 62 * q^9 + 22 * q^11 + 23 * q^13 + 15 * q^15 - 135 * q^17 + 102 * q^19 + 54 * q^21 - 207 * q^23 + 225 * q^25 - 363 * q^27 + 280 * q^29 + 168 * q^31 + 28 * q^33 - 125 * q^35 + 153 * q^37 - 5 * q^39 - 502 * q^41 + 110 * q^43 - 310 * q^45 - 153 * q^47 + 764 * q^49 + 924 * q^51 - 273 * q^53 - 110 * q^55 + 748 * q^57 + 827 * q^59 + 1976 * q^61 + 2237 * q^63 - 115 * q^65 + 1613 * q^67 + 69 * q^69 + 1370 * q^71 + 425 * q^73 - 75 * q^75 + 2006 * q^77 + 2624 * q^79 + 1729 * q^81 + 2505 * q^83 + 675 * q^85 + 1591 * q^87 + 1120 * q^89 + 2392 * q^91 + 4401 * q^93 - 510 * q^95 + 2026 * q^97 + 3206 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.70964	0.329020	0.164510	−	0.986375i	$-0.447396\pi$
$3$	1.70964	0.329020	0.164510	+	0.986375i	$0.447396\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	20.8693	1.12684	0.563418	−	0.826172i	$-0.309486\pi$
$7$	20.8693	1.12684	0.563418	+	0.826172i	$0.309486\pi$
$8$	0	0
$9$	−24.0771	−0.891746
$10$	0	0
$11$	23.6682	0.648749	0.324374	−	0.945929i	$-0.394846\pi$
$11$	23.6682	0.648749	0.324374	+	0.945929i	$0.394846\pi$
$12$	0	0
$13$	53.7383	1.14649	0.573243	−	0.819385i	$-0.305685\pi$
$13$	53.7383	1.14649	0.573243	+	0.819385i	$0.305685\pi$
$14$	0	0
$15$	−8.54819	−0.147142
$16$	0	0
$17$	52.1336	0.743779	0.371890	−	0.928277i	$-0.378710\pi$
$17$	52.1336	0.743779	0.371890	+	0.928277i	$0.378710\pi$
$18$	0	0
$19$	4.88364	0.0589676	0.0294838	−	0.999565i	$-0.490614\pi$
$19$	4.88364	0.0589676	0.0294838	+	0.999565i	$0.490614\pi$
$20$	0	0
$21$	35.6789	0.370751
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−87.3234	−0.622422
$28$	0	0
$29$	−144.142	−0.922980	−0.461490	−	0.887145i	$-0.652685\pi$
$29$	−144.142	−0.922980	−0.461490	+	0.887145i	$0.652685\pi$
$30$	0	0
$31$	−288.160	−1.66952	−0.834759	−	0.550616i	$-0.814393\pi$
$31$	−288.160	−1.66952	−0.834759	+	0.550616i	$0.814393\pi$
$32$	0	0
$33$	40.4641	0.213451
$34$	0	0
$35$	−104.346	−0.503936
$36$	0	0
$37$	294.532	1.30867	0.654334	−	0.756206i	$-0.272949\pi$
$37$	294.532	1.30867	0.654334	+	0.756206i	$0.272949\pi$
$38$	0	0
$39$	91.8730	0.377217
$40$	0	0
$41$	333.215	1.26925	0.634627	−	0.772819i	$-0.281154\pi$
$41$	333.215	1.26925	0.634627	+	0.772819i	$0.281154\pi$
$42$	0	0
$43$	−0.162395	−0.000575930 0	−0.000287965	−	1.00000i	$-0.500092\pi$
$43$	−0.162395	−0.000575930 0	−0.000287965		1.00000i	$0.500092\pi$
$44$	0	0
$45$	120.386	0.398801
$46$	0	0
$47$	529.988	1.64482	0.822412	−	0.568892i	$-0.192628\pi$
$47$	529.988	1.64482	0.822412	+	0.568892i	$0.192628\pi$
$48$	0	0
$49$	92.5270	0.269758
$50$	0	0
$51$	89.1295	0.244718
$52$	0	0
$53$	116.147	0.301019	0.150510	−	0.988609i	$-0.451909\pi$
$53$	116.147	0.301019	0.150510	+	0.988609i	$0.451909\pi$
$54$	0	0
$55$	−118.341	−0.290129
$56$	0	0
$57$	8.34926	0.0194015
$58$	0	0
$59$	−284.019	−0.626714	−0.313357	−	0.949635i	$-0.601454\pi$
$59$	−284.019	−0.626714	−0.313357	+	0.949635i	$0.601454\pi$
$60$	0	0
$61$	−120.262	−0.252425	−0.126213	−	0.992003i	$-0.540282\pi$
$61$	−120.262	−0.252425	−0.126213	+	0.992003i	$0.540282\pi$
$62$	0	0
$63$	−502.473	−1.00485
$64$	0	0
$65$	−268.691	−0.512724
$66$	0	0
$67$	194.900	0.355386	0.177693	−	0.984086i	$-0.443137\pi$
$67$	194.900	0.355386	0.177693	+	0.984086i	$0.443137\pi$
$68$	0	0
$69$	−39.3217	−0.0686054
$70$	0	0
$71$	201.508	0.336826	0.168413	−	0.985717i	$-0.446136\pi$
$71$	201.508	0.336826	0.168413	+	0.985717i	$0.446136\pi$
$72$	0	0
$73$	913.140	1.46404	0.732020	−	0.681283i	$-0.238578\pi$
$73$	913.140	1.46404	0.732020	+	0.681283i	$0.238578\pi$
$74$	0	0
$75$	42.7409	0.0658040
$76$	0	0
$77$	493.939	0.731033
$78$	0	0
$79$	901.712	1.28418	0.642092	−	0.766628i	$-0.278067\pi$
$79$	901.712	1.28418	0.642092	+	0.766628i	$0.278067\pi$
$80$	0	0
$81$	500.791	0.686957
$82$	0	0
$83$	172.627	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$83$	172.627	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$84$	0	0
$85$	−260.668	−0.332628
$86$	0	0
$87$	−246.430	−0.303679
$88$	0	0
$89$	−1578.55	−1.88006	−0.940032	−	0.341085i	$-0.889206\pi$
$89$	−1578.55	−1.88006	−0.940032	+	0.341085i	$0.889206\pi$
$90$	0	0
$91$	1121.48	1.29190
$92$	0	0
$93$	−492.649	−0.549305
$94$	0	0
$95$	−24.4182	−0.0263711
$96$	0	0
$97$	1720.14	1.80056	0.900279	−	0.435314i	$-0.143363\pi$
$97$	1720.14	1.80056	0.900279	+	0.435314i	$0.143363\pi$
$98$	0	0
$99$	−569.863	−0.578519
$100$	0	0
$101$	−548.016	−0.539897	−0.269949	−	0.962875i	$-0.587007\pi$
$101$	−548.016	−0.539897	−0.269949	+	0.962875i	$0.587007\pi$
$102$	0	0
$103$	1654.89	1.58311	0.791557	−	0.611095i	$-0.209271\pi$
$103$	1654.89	1.58311	0.791557	+	0.611095i	$0.209271\pi$
$104$	0	0
$105$	−178.395	−0.165805
$106$	0	0
$107$	1556.17	1.40599	0.702995	−	0.711194i	$-0.251845\pi$
$107$	1556.17	1.40599	0.702995	+	0.711194i	$0.251845\pi$
$108$	0	0
$109$	−231.161	−0.203131	−0.101565	−	0.994829i	$-0.532385\pi$
$109$	−231.161	−0.203131	−0.101565	+	0.994829i	$0.532385\pi$
$110$	0	0
$111$	503.542	0.430578
$112$	0	0
$113$	−403.835	−0.336191	−0.168096	−	0.985771i	$-0.553762\pi$
$113$	−403.835	−0.336191	−0.168096	+	0.985771i	$0.553762\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−1293.86	−1.02237
$118$	0	0
$119$	1087.99	0.838117
$120$	0	0
$121$	−770.816	−0.579125
$122$	0	0
$123$	569.676	0.417610
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1842.46	−1.28734	−0.643670	−	0.765303i	$-0.722589\pi$
$127$	−1842.46	−1.28734	−0.643670	+	0.765303i	$0.722589\pi$
$128$	0	0
$129$	−0.277637	−0.000189493 0
$130$	0	0
$131$	1124.87	0.750231	0.375115	−	0.926978i	$-0.377603\pi$
$131$	1124.87	0.750231	0.375115	+	0.926978i	$0.377603\pi$
$132$	0	0
$133$	101.918	0.0664468
$134$	0	0
$135$	436.617	0.278356
$136$	0	0
$137$	2675.29	1.66836	0.834181	−	0.551490i	$-0.185941\pi$
$137$	2675.29	1.66836	0.834181	+	0.551490i	$0.185941\pi$
$138$	0	0
$139$	1060.81	0.647316	0.323658	−	0.946174i	$-0.395087\pi$
$139$	1060.81	0.647316	0.323658	+	0.946174i	$0.395087\pi$
$140$	0	0
$141$	906.088	0.541180
$142$	0	0
$143$	1271.89	0.743781
$144$	0	0
$145$	720.708	0.412769
$146$	0	0
$147$	158.188	0.0887558
$148$	0	0
$149$	1582.20	0.869923	0.434961	−	0.900449i	$-0.356762\pi$
$149$	1582.20	0.869923	0.434961	+	0.900449i	$0.356762\pi$
$150$	0	0
$151$	−1169.01	−0.630020	−0.315010	−	0.949088i	$-0.602008\pi$
$151$	−1169.01	−0.630020	−0.315010	+	0.949088i	$0.602008\pi$
$152$	0	0
$153$	−1255.23	−0.663262
$154$	0	0
$155$	1440.80	0.746631
$156$	0	0
$157$	361.463	0.183744	0.0918722	−	0.995771i	$-0.470715\pi$
$157$	361.463	0.183744	0.0918722	+	0.995771i	$0.470715\pi$
$158$	0	0
$159$	198.569	0.0990414
$160$	0	0
$161$	−479.994	−0.234961
$162$	0	0
$163$	3143.56	1.51057	0.755283	−	0.655399i	$-0.227499\pi$
$163$	3143.56	1.51057	0.755283	+	0.655399i	$0.227499\pi$
$164$	0	0
$165$	−202.320	−0.0954583
$166$	0	0
$167$	−1624.03	−0.752523	−0.376262	−	0.926513i	$-0.622791\pi$
$167$	−1624.03	−0.752523	−0.376262	+	0.926513i	$0.622791\pi$
$168$	0	0
$169$	690.803	0.314430
$170$	0	0
$171$	−117.584	−0.0525841
$172$	0	0
$173$	−2491.15	−1.09479	−0.547395	−	0.836875i	$-0.684380\pi$
$173$	−2491.15	−1.09479	−0.547395	+	0.836875i	$0.684380\pi$
$174$	0	0
$175$	521.732	0.225367
$176$	0	0
$177$	−485.569	−0.206201
$178$	0	0
$179$	−4074.96	−1.70154	−0.850772	−	0.525535i	$-0.823865\pi$
$179$	−4074.96	−1.70154	−0.850772	+	0.525535i	$0.823865\pi$
$180$	0	0
$181$	3304.36	1.35697	0.678484	−	0.734615i	$-0.262637\pi$
$181$	3304.36	1.35697	0.678484	+	0.734615i	$0.262637\pi$
$182$	0	0
$183$	−205.604	−0.0830529
$184$	0	0
$185$	−1472.66	−0.585254
$186$	0	0
$187$	1233.91	0.482526
$188$	0	0
$189$	−1822.38	−0.701367
$190$	0	0
$191$	−457.335	−0.173254	−0.0866272	−	0.996241i	$-0.527609\pi$
$191$	−457.335	−0.173254	−0.0866272	+	0.996241i	$0.527609\pi$
$192$	0	0
$193$	230.793	0.0860770	0.0430385	−	0.999073i	$-0.486296\pi$
$193$	230.793	0.0860770	0.0430385	+	0.999073i	$0.486296\pi$
$194$	0	0
$195$	−459.365	−0.168696
$196$	0	0
$197$	−1078.86	−0.390180	−0.195090	−	0.980785i	$-0.562500\pi$
$197$	−1078.86	−0.390180	−0.195090	+	0.980785i	$0.562500\pi$
$198$	0	0
$199$	4525.53	1.61209	0.806046	−	0.591852i	$-0.201603\pi$
$199$	4525.53	1.61209	0.806046	+	0.591852i	$0.201603\pi$
$200$	0	0
$201$	333.209	0.116929
$202$	0	0
$203$	−3008.13	−1.04005
$204$	0	0
$205$	−1666.07	−0.567627
$206$	0	0
$207$	553.774	0.185942
$208$	0	0
$209$	115.587	0.0382551
$210$	0	0
$211$	1359.68	0.443623	0.221812	−	0.975090i	$-0.428803\pi$
$211$	1359.68	0.443623	0.221812	+	0.975090i	$0.428803\pi$
$212$	0	0
$213$	344.506	0.110822
$214$	0	0
$215$	0.811975	0.000257564 0
$216$	0	0
$217$	−6013.69	−1.88127
$218$	0	0
$219$	1561.14	0.481698
$220$	0	0
$221$	2801.57	0.852732
$222$	0	0
$223$	−3484.12	−1.04625	−0.523126	−	0.852256i	$-0.675234\pi$
$223$	−3484.12	−1.04625	−0.523126	+	0.852256i	$0.675234\pi$
$224$	0	0
$225$	−601.928	−0.178349
$226$	0	0
$227$	5022.22	1.46844	0.734222	−	0.678910i	$-0.237547\pi$
$227$	5022.22	1.46844	0.734222	+	0.678910i	$0.237547\pi$
$228$	0	0
$229$	3063.48	0.884020	0.442010	−	0.897010i	$-0.354266\pi$
$229$	3063.48	0.884020	0.442010	+	0.897010i	$0.354266\pi$
$230$	0	0
$231$	844.456	0.240524
$232$	0	0
$233$	591.557	0.166327	0.0831635	−	0.996536i	$-0.473498\pi$
$233$	591.557	0.166327	0.0831635	+	0.996536i	$0.473498\pi$
$234$	0	0
$235$	−2649.94	−0.735588
$236$	0	0
$237$	1541.60	0.422522
$238$	0	0
$239$	864.562	0.233991	0.116996	−	0.993132i	$-0.462674\pi$
$239$	864.562	0.233991	0.116996	+	0.993132i	$0.462674\pi$
$240$	0	0
$241$	−690.020	−0.184432	−0.0922159	−	0.995739i	$-0.529395\pi$
$241$	−690.020	−0.184432	−0.0922159	+	0.995739i	$0.529395\pi$
$242$	0	0
$243$	3213.90	0.848445
$244$	0	0
$245$	−462.635	−0.120639
$246$	0	0
$247$	262.439	0.0676055
$248$	0	0
$249$	295.130	0.0751127
$250$	0	0
$251$	−5057.07	−1.27171	−0.635855	−	0.771808i	$-0.719353\pi$
$251$	−5057.07	−1.27171	−0.635855	+	0.771808i	$0.719353\pi$
$252$	0	0
$253$	−544.369	−0.135273
$254$	0	0
$255$	−445.648	−0.109441
$256$	0	0
$257$	−3562.51	−0.864682	−0.432341	−	0.901710i	$-0.642312\pi$
$257$	−3562.51	−0.864682	−0.432341	+	0.901710i	$0.642312\pi$
$258$	0	0
$259$	6146.66	1.47465
$260$	0	0
$261$	3470.52	0.823064
$262$	0	0
$263$	1139.45	0.267154	0.133577	−	0.991038i	$-0.457354\pi$
$263$	1139.45	0.267154	0.133577	+	0.991038i	$0.457354\pi$
$264$	0	0
$265$	−580.736	−0.134620
$266$	0	0
$267$	−2698.75	−0.618579
$268$	0	0
$269$	−6217.53	−1.40925	−0.704627	−	0.709578i	$-0.748886\pi$
$269$	−6217.53	−1.40925	−0.704627	+	0.709578i	$0.748886\pi$
$270$	0	0
$271$	2169.68	0.486343	0.243172	−	0.969983i	$-0.421812\pi$
$271$	2169.68	0.486343	0.243172	+	0.969983i	$0.421812\pi$
$272$	0	0
$273$	1917.32	0.425061
$274$	0	0
$275$	591.705	0.129750
$276$	0	0
$277$	6335.06	1.37414	0.687070	−	0.726591i	$-0.258896\pi$
$277$	6335.06	1.37414	0.687070	+	0.726591i	$0.258896\pi$
$278$	0	0
$279$	6938.07	1.48879
$280$	0	0
$281$	−4984.66	−1.05822	−0.529110	−	0.848553i	$-0.677474\pi$
$281$	−4984.66	−1.05822	−0.529110	+	0.848553i	$0.677474\pi$
$282$	0	0
$283$	3113.65	0.654018	0.327009	−	0.945021i	$-0.393959\pi$
$283$	3113.65	0.654018	0.327009	+	0.945021i	$0.393959\pi$
$284$	0	0
$285$	−41.7463	−0.00867662
$286$	0	0
$287$	6953.95	1.43024
$288$	0	0
$289$	−2195.09	−0.446792
$290$	0	0
$291$	2940.82	0.592419
$292$	0	0
$293$	−1892.75	−0.377390	−0.188695	−	0.982036i	$-0.560426\pi$
$293$	−1892.75	−0.377390	−0.188695	+	0.982036i	$0.560426\pi$
$294$	0	0
$295$	1420.09	0.280275
$296$	0	0
$297$	−2066.79	−0.403795
$298$	0	0
$299$	−1235.98	−0.239059
$300$	0	0
$301$	−3.38907	−0.000648979 0
$302$	0	0
$303$	−936.909	−0.177637
$304$	0	0
$305$	601.309	0.112888
$306$	0	0
$307$	9988.97	1.85701	0.928503	−	0.371326i	$-0.121097\pi$
$307$	9988.97	1.85701	0.928503	+	0.371326i	$0.121097\pi$
$308$	0	0
$309$	2829.26	0.520876
$310$	0	0
$311$	−1724.91	−0.314503	−0.157252	−	0.987559i	$-0.550263\pi$
$311$	−1724.91	−0.314503	−0.157252	+	0.987559i	$0.550263\pi$
$312$	0	0
$313$	−9143.12	−1.65112	−0.825559	−	0.564316i	$-0.809140\pi$
$313$	−9143.12	−1.65112	−0.825559	+	0.564316i	$0.809140\pi$
$314$	0	0
$315$	2512.36	0.449383
$316$	0	0
$317$	1203.47	0.213229	0.106615	−	0.994300i	$-0.465999\pi$
$317$	1203.47	0.213229	0.106615	+	0.994300i	$0.465999\pi$
$318$	0	0
$319$	−3411.57	−0.598782
$320$	0	0
$321$	2660.49	0.462599
$322$	0	0
$323$	254.602	0.0438589
$324$	0	0
$325$	1343.46	0.229297
$326$	0	0
$327$	−395.202	−0.0668340
$328$	0	0
$329$	11060.5	1.85345
$330$	0	0
$331$	−6820.31	−1.13256	−0.566281	−	0.824212i	$-0.691618\pi$
$331$	−6820.31	−1.13256	−0.566281	+	0.824212i	$0.691618\pi$
$332$	0	0
$333$	−7091.48	−1.16700
$334$	0	0
$335$	−974.501	−0.158933
$336$	0	0
$337$	−10495.4	−1.69651	−0.848254	−	0.529589i	$-0.822346\pi$
$337$	−10495.4	−1.69651	−0.848254	+	0.529589i	$0.822346\pi$
$338$	0	0
$339$	−690.411	−0.110614
$340$	0	0
$341$	−6820.23	−1.08310
$342$	0	0
$343$	−5227.19	−0.822862
$344$	0	0
$345$	196.608	0.0306813
$346$	0	0
$347$	−1939.55	−0.300059	−0.150029	−	0.988682i	$-0.547937\pi$
$347$	−1939.55	−0.300059	−0.150029	+	0.988682i	$0.547937\pi$
$348$	0	0
$349$	12226.5	1.87527	0.937636	−	0.347620i	$-0.113010\pi$
$349$	12226.5	1.87527	0.937636	+	0.347620i	$0.113010\pi$
$350$	0	0
$351$	−4692.61	−0.713598
$352$	0	0
$353$	−11190.9	−1.68734	−0.843672	−	0.536859i	$-0.819611\pi$
$353$	−11190.9	−1.68734	−0.843672	+	0.536859i	$0.819611\pi$
$354$	0	0
$355$	−1007.54	−0.150633
$356$	0	0
$357$	1860.07	0.275757
$358$	0	0
$359$	−2874.12	−0.422536	−0.211268	−	0.977428i	$-0.567759\pi$
$359$	−2874.12	−0.422536	−0.211268	+	0.977428i	$0.567759\pi$
$360$	0	0
$361$	−6835.15	−0.996523
$362$	0	0
$363$	−1317.82	−0.190544
$364$	0	0
$365$	−4565.70	−0.654738
$366$	0	0
$367$	2930.03	0.416748	0.208374	−	0.978049i	$-0.433183\pi$
$367$	2930.03	0.416748	0.208374	+	0.978049i	$0.433183\pi$
$368$	0	0
$369$	−8022.86	−1.13185
$370$	0	0
$371$	2423.91	0.339199
$372$	0	0
$373$	−3597.70	−0.499415	−0.249707	−	0.968321i	$-0.580334\pi$
$373$	−3597.70	−0.499415	−0.249707	+	0.968321i	$0.580334\pi$
$374$	0	0
$375$	−213.705	−0.0294284
$376$	0	0
$377$	−7745.92	−1.05818
$378$	0	0
$379$	−7095.95	−0.961727	−0.480864	−	0.876795i	$-0.659677\pi$
$379$	−7095.95	−0.961727	−0.480864	+	0.876795i	$0.659677\pi$
$380$	0	0
$381$	−3149.94	−0.423561
$382$	0	0
$383$	−2459.42	−0.328121	−0.164061	−	0.986450i	$-0.552459\pi$
$383$	−2459.42	−0.328121	−0.164061	+	0.986450i	$0.552459\pi$
$384$	0	0
$385$	−2469.69	−0.326928
$386$	0	0
$387$	3.91001	0.000513584 0
$388$	0	0
$389$	−10638.6	−1.38663	−0.693315	−	0.720635i	$-0.743850\pi$
$389$	−10638.6	−1.38663	−0.693315	+	0.720635i	$0.743850\pi$
$390$	0	0
$391$	−1199.07	−0.155089
$392$	0	0
$393$	1923.12	0.246841
$394$	0	0
$395$	−4508.56	−0.574304
$396$	0	0
$397$	1449.48	0.183242	0.0916212	−	0.995794i	$-0.470795\pi$
$397$	1449.48	0.183242	0.0916212	+	0.995794i	$0.470795\pi$
$398$	0	0
$399$	174.243	0.0218623
$400$	0	0
$401$	12093.8	1.50608	0.753040	−	0.657975i	$-0.228587\pi$
$401$	12093.8	1.50608	0.753040	+	0.657975i	$0.228587\pi$
$402$	0	0
$403$	−15485.2	−1.91408
$404$	0	0
$405$	−2503.96	−0.307216
$406$	0	0
$407$	6971.04	0.848996
$408$	0	0
$409$	−4947.12	−0.598091	−0.299046	−	0.954239i	$-0.596668\pi$
$409$	−4947.12	−0.598091	−0.299046	+	0.954239i	$0.596668\pi$
$410$	0	0
$411$	4573.78	0.548925
$412$	0	0
$413$	−5927.27	−0.706203
$414$	0	0
$415$	−863.135	−0.102095
$416$	0	0
$417$	1813.60	0.212980
$418$	0	0
$419$	−1894.55	−0.220895	−0.110447	−	0.993882i	$-0.535228\pi$
$419$	−1894.55	−0.220895	−0.110447	+	0.993882i	$0.535228\pi$
$420$	0	0
$421$	3663.52	0.424107	0.212054	−	0.977258i	$-0.431985\pi$
$421$	3663.52	0.424107	0.212054	+	0.977258i	$0.431985\pi$
$422$	0	0
$423$	−12760.6	−1.46677
$424$	0	0
$425$	1303.34	0.148756
$426$	0	0
$427$	−2509.78	−0.284442
$428$	0	0
$429$	2174.47	0.244719
$430$	0	0
$431$	6399.83	0.715241	0.357620	−	0.933867i	$-0.383588\pi$
$431$	6399.83	0.715241	0.357620	+	0.933867i	$0.383588\pi$
$432$	0	0
$433$	−12389.9	−1.37511	−0.687553	−	0.726134i	$-0.741315\pi$
$433$	−12389.9	−1.37511	−0.687553	+	0.726134i	$0.741315\pi$
$434$	0	0
$435$	1232.15	0.135809
$436$	0	0
$437$	−112.324	−0.0122956
$438$	0	0
$439$	−1186.74	−0.129021	−0.0645105	−	0.997917i	$-0.520549\pi$
$439$	−1186.74	−0.129021	−0.0645105	+	0.997917i	$0.520549\pi$
$440$	0	0
$441$	−2227.79	−0.240556
$442$	0	0
$443$	−14040.8	−1.50587	−0.752934	−	0.658096i	$-0.771362\pi$
$443$	−14040.8	−1.50587	−0.752934	+	0.658096i	$0.771362\pi$
$444$	0	0
$445$	7892.74	0.840791
$446$	0	0
$447$	2704.98	0.286222
$448$	0	0
$449$	−3724.79	−0.391500	−0.195750	−	0.980654i	$-0.562714\pi$
$449$	−3724.79	−0.391500	−0.195750	+	0.980654i	$0.562714\pi$
$450$	0	0
$451$	7886.60	0.823426
$452$	0	0
$453$	−1998.59	−0.207289
$454$	0	0
$455$	−5607.40	−0.577756
$456$	0	0
$457$	−14267.4	−1.46040	−0.730199	−	0.683235i	$-0.760573\pi$
$457$	−14267.4	−1.46040	−0.730199	+	0.683235i	$0.760573\pi$
$458$	0	0
$459$	−4552.48	−0.462945
$460$	0	0
$461$	15365.9	1.55242	0.776208	−	0.630477i	$-0.217141\pi$
$461$	15365.9	1.55242	0.776208	+	0.630477i	$0.217141\pi$
$462$	0	0
$463$	14467.7	1.45220	0.726101	−	0.687589i	$-0.241331\pi$
$463$	14467.7	1.45220	0.726101	+	0.687589i	$0.241331\pi$
$464$	0	0
$465$	2463.25	0.245656
$466$	0	0
$467$	3374.06	0.334332	0.167166	−	0.985929i	$-0.446538\pi$
$467$	3374.06	0.334332	0.167166	+	0.985929i	$0.446538\pi$
$468$	0	0
$469$	4067.43	0.400462
$470$	0	0
$471$	617.970	0.0604556
$472$	0	0
$473$	−3.84360	−0.000373634 0
$474$	0	0
$475$	122.091	0.0117935
$476$	0	0
$477$	−2796.49	−0.268433
$478$	0	0
$479$	9289.09	0.886074	0.443037	−	0.896503i	$-0.353901\pi$
$479$	9289.09	0.886074	0.443037	+	0.896503i	$0.353901\pi$
$480$	0	0
$481$	15827.6	1.50037
$482$	0	0
$483$	−820.615	−0.0773070
$484$	0	0
$485$	−8600.72	−0.805234
$486$	0	0
$487$	5685.30	0.529005	0.264503	−	0.964385i	$-0.414792\pi$
$487$	5685.30	0.529005	0.264503	+	0.964385i	$0.414792\pi$
$488$	0	0
$489$	5374.34	0.497006
$490$	0	0
$491$	−5514.82	−0.506885	−0.253443	−	0.967350i	$-0.581563\pi$
$491$	−5514.82	−0.506885	−0.253443	+	0.967350i	$0.581563\pi$
$492$	0	0
$493$	−7514.62	−0.686493
$494$	0	0
$495$	2849.31	0.258721
$496$	0	0
$497$	4205.33	0.379547
$498$	0	0
$499$	−17001.4	−1.52523	−0.762613	−	0.646855i	$-0.776084\pi$
$499$	−17001.4	−1.52523	−0.762613	+	0.646855i	$0.776084\pi$
$500$	0	0
$501$	−2776.51	−0.247595
$502$	0	0
$503$	13321.6	1.18088	0.590438	−	0.807083i	$-0.298955\pi$
$503$	13321.6	1.18088	0.590438	+	0.807083i	$0.298955\pi$
$504$	0	0
$505$	2740.08	0.241449
$506$	0	0
$507$	1181.02	0.103454
$508$	0	0
$509$	−17297.4	−1.50628	−0.753138	−	0.657863i	$-0.771461\pi$
$509$	−17297.4	−1.50628	−0.753138	+	0.657863i	$0.771461\pi$
$510$	0	0
$511$	19056.6	1.64973
$512$	0	0
$513$	−426.456	−0.0367027
$514$	0	0
$515$	−8274.43	−0.707990
$516$	0	0
$517$	12543.9	1.06708
$518$	0	0
$519$	−4258.96	−0.360207
$520$	0	0
$521$	−7265.77	−0.610978	−0.305489	−	0.952196i	$-0.598820\pi$
$521$	−7265.77	−0.610978	−0.305489	+	0.952196i	$0.598820\pi$
$522$	0	0
$523$	−12383.0	−1.03532	−0.517658	−	0.855588i	$-0.673196\pi$
$523$	−12383.0	−1.03532	−0.517658	+	0.855588i	$0.673196\pi$
$524$	0	0
$525$	891.973	0.0741503
$526$	0	0
$527$	−15022.8	−1.24175
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	6838.36	0.558869
$532$	0	0
$533$	17906.4	1.45518
$534$	0	0
$535$	−7780.87	−0.628778
$536$	0	0
$537$	−6966.70	−0.559842
$538$	0	0
$539$	2189.95	0.175005
$540$	0	0
$541$	−3946.64	−0.313641	−0.156820	−	0.987627i	$-0.550124\pi$
$541$	−3946.64	−0.313641	−0.156820	+	0.987627i	$0.550124\pi$
$542$	0	0
$543$	5649.26	0.446470
$544$	0	0
$545$	1155.81	0.0908428
$546$	0	0
$547$	20084.1	1.56990	0.784948	−	0.619562i	$-0.212690\pi$
$547$	20084.1	1.56990	0.784948	+	0.619562i	$0.212690\pi$
$548$	0	0
$549$	2895.56	0.225099
$550$	0	0
$551$	−703.936	−0.0544259
$552$	0	0
$553$	18818.1	1.44706
$554$	0	0
$555$	−2517.71	−0.192560
$556$	0	0
$557$	−5182.63	−0.394246	−0.197123	−	0.980379i	$-0.563160\pi$
$557$	−5182.63	−0.394246	−0.197123	+	0.980379i	$0.563160\pi$
$558$	0	0
$559$	−8.72683	−0.000660296 0
$560$	0	0
$561$	2109.54	0.158761
$562$	0	0
$563$	−2114.65	−0.158298	−0.0791492	−	0.996863i	$-0.525220\pi$
$563$	−2114.65	−0.158298	−0.0791492	+	0.996863i	$0.525220\pi$
$564$	0	0
$565$	2019.17	0.150349
$566$	0	0
$567$	10451.2	0.774087
$568$	0	0
$569$	−15202.0	−1.12003	−0.560017	−	0.828481i	$-0.689205\pi$
$569$	−15202.0	−1.12003	−0.560017	+	0.828481i	$0.689205\pi$
$570$	0	0
$571$	−18481.2	−1.35449	−0.677244	−	0.735759i	$-0.736826\pi$
$571$	−18481.2	−1.35449	−0.677244	+	0.735759i	$0.736826\pi$
$572$	0	0
$573$	−781.877	−0.0570042
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−14659.4	−1.05768	−0.528838	−	0.848723i	$-0.677372\pi$
$577$	−14659.4	−1.05768	−0.528838	+	0.848723i	$0.677372\pi$
$578$	0	0
$579$	394.573	0.0283210
$580$	0	0
$581$	3602.60	0.257248
$582$	0	0
$583$	2748.99	0.195286
$584$	0	0
$585$	6469.32	0.457220
$586$	0	0
$587$	3361.40	0.236354	0.118177	−	0.992993i	$-0.462295\pi$
$587$	3361.40	0.236354	0.118177	+	0.992993i	$0.462295\pi$
$588$	0	0
$589$	−1407.27	−0.0984474
$590$	0	0
$591$	−1844.46	−0.128377
$592$	0	0
$593$	9144.05	0.633223	0.316611	−	0.948555i	$-0.397455\pi$
$593$	9144.05	0.633223	0.316611	+	0.948555i	$0.397455\pi$
$594$	0	0
$595$	−5439.95	−0.374817
$596$	0	0
$597$	7737.02	0.530411
$598$	0	0
$599$	−17440.0	−1.18962	−0.594808	−	0.803868i	$-0.702772\pi$
$599$	−17440.0	−1.18962	−0.594808	+	0.803868i	$0.702772\pi$
$600$	0	0
$601$	2840.97	0.192821	0.0964105	−	0.995342i	$-0.469264\pi$
$601$	2840.97	0.192821	0.0964105	+	0.995342i	$0.469264\pi$
$602$	0	0
$603$	−4692.64	−0.316914
$604$	0	0
$605$	3854.08	0.258993
$606$	0	0
$607$	−19233.4	−1.28609	−0.643047	−	0.765827i	$-0.722330\pi$
$607$	−19233.4	−1.28609	−0.643047	+	0.765827i	$0.722330\pi$
$608$	0	0
$609$	−5142.82	−0.342196
$610$	0	0
$611$	28480.7	1.88577
$612$	0	0
$613$	−7979.89	−0.525782	−0.262891	−	0.964825i	$-0.584676\pi$
$613$	−7979.89	−0.525782	−0.262891	+	0.964825i	$0.584676\pi$
$614$	0	0
$615$	−2848.38	−0.186761
$616$	0	0
$617$	12478.3	0.814194	0.407097	−	0.913385i	$-0.366541\pi$
$617$	12478.3	0.814194	0.407097	+	0.913385i	$0.366541\pi$
$618$	0	0
$619$	−2461.50	−0.159832	−0.0799161	−	0.996802i	$-0.525465\pi$
$619$	−2461.50	−0.159832	−0.0799161	+	0.996802i	$0.525465\pi$
$620$	0	0
$621$	2008.44	0.129784
$622$	0	0
$623$	−32943.2	−2.11852
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	197.612	0.0125867
$628$	0	0
$629$	15355.0	0.973360
$630$	0	0
$631$	22359.4	1.41064	0.705320	−	0.708889i	$-0.250803\pi$
$631$	22359.4	1.41064	0.705320	+	0.708889i	$0.250803\pi$
$632$	0	0
$633$	2324.57	0.145961
$634$	0	0
$635$	9212.32	0.575716
$636$	0	0
$637$	4972.24	0.309274
$638$	0	0
$639$	−4851.74	−0.300363
$640$	0	0
$641$	−1666.84	−0.102709	−0.0513543	−	0.998680i	$-0.516354\pi$
$641$	−1666.84	−0.102709	−0.0513543	+	0.998680i	$0.516354\pi$
$642$	0	0
$643$	−21155.7	−1.29751	−0.648756	−	0.760997i	$-0.724710\pi$
$643$	−21155.7	−1.29751	−0.648756	+	0.760997i	$0.724710\pi$
$644$	0	0
$645$	1.38818	8.47437e−5 0
$646$	0	0
$647$	−1277.13	−0.0776031	−0.0388015	−	0.999247i	$-0.512354\pi$
$647$	−1277.13	−0.0776031	−0.0388015	+	0.999247i	$0.512354\pi$
$648$	0	0
$649$	−6722.22	−0.406580
$650$	0	0
$651$	−10281.2	−0.618976
$652$	0	0
$653$	−3834.25	−0.229779	−0.114890	−	0.993378i	$-0.536651\pi$
$653$	−3834.25	−0.229779	−0.114890	+	0.993378i	$0.536651\pi$
$654$	0	0
$655$	−5624.34	−0.335513
$656$	0	0
$657$	−21985.8	−1.30555
$658$	0	0
$659$	4841.95	0.286215	0.143107	−	0.989707i	$-0.454291\pi$
$659$	4841.95	0.286215	0.143107	+	0.989707i	$0.454291\pi$
$660$	0	0
$661$	22236.9	1.30850	0.654248	−	0.756280i	$-0.272985\pi$
$661$	22236.9	1.30850	0.654248	+	0.756280i	$0.272985\pi$
$662$	0	0
$663$	4789.67	0.280566
$664$	0	0
$665$	−509.591	−0.0297159
$666$	0	0
$667$	3315.26	0.192455
$668$	0	0
$669$	−5956.59	−0.344238
$670$	0	0
$671$	−2846.38	−0.163760
$672$	0	0
$673$	2519.67	0.144318	0.0721591	−	0.997393i	$-0.477011\pi$
$673$	2519.67	0.144318	0.0721591	+	0.997393i	$0.477011\pi$
$674$	0	0
$675$	−2183.09	−0.124484
$676$	0	0
$677$	13978.4	0.793548	0.396774	−	0.917916i	$-0.370130\pi$
$677$	13978.4	0.793548	0.396774	+	0.917916i	$0.370130\pi$
$678$	0	0
$679$	35898.2	2.02893
$680$	0	0
$681$	8586.18	0.483147
$682$	0	0
$683$	13974.4	0.782893	0.391447	−	0.920201i	$-0.371975\pi$
$683$	13974.4	0.782893	0.391447	+	0.920201i	$0.371975\pi$
$684$	0	0
$685$	−13376.5	−0.746115
$686$	0	0
$687$	5237.44	0.290860
$688$	0	0
$689$	6241.55	0.345115
$690$	0	0
$691$	−6018.34	−0.331329	−0.165664	−	0.986182i	$-0.552977\pi$
$691$	−6018.34	−0.331329	−0.165664	+	0.986182i	$0.552977\pi$
$692$	0	0
$693$	−11892.6	−0.651896
$694$	0	0
$695$	−5304.06	−0.289488
$696$	0	0
$697$	17371.7	0.944044
$698$	0	0
$699$	1011.35	0.0547249
$700$	0	0
$701$	17566.5	0.946475	0.473238	−	0.880935i	$-0.343085\pi$
$701$	17566.5	0.946475	0.473238	+	0.880935i	$0.343085\pi$
$702$	0	0
$703$	1438.39	0.0771690
$704$	0	0
$705$	−4530.44	−0.242023
$706$	0	0
$707$	−11436.7	−0.608375
$708$	0	0
$709$	6581.64	0.348630	0.174315	−	0.984690i	$-0.444229\pi$
$709$	6581.64	0.348630	0.174315	+	0.984690i	$0.444229\pi$
$710$	0	0
$711$	−21710.6	−1.14517
$712$	0	0
$713$	6627.68	0.348118
$714$	0	0
$715$	−6359.44	−0.332629
$716$	0	0
$717$	1478.09	0.0769878
$718$	0	0
$719$	17215.6	0.892955	0.446477	−	0.894795i	$-0.352678\pi$
$719$	17215.6	0.892955	0.446477	+	0.894795i	$0.352678\pi$
$720$	0	0
$721$	34536.3	1.78391
$722$	0	0
$723$	−1179.68	−0.0606817
$724$	0	0
$725$	−3603.54	−0.184596
$726$	0	0
$727$	15420.8	0.786693	0.393347	−	0.919390i	$-0.371317\pi$
$727$	15420.8	0.786693	0.393347	+	0.919390i	$0.371317\pi$
$728$	0	0
$729$	−8026.76	−0.407801
$730$	0	0
$731$	−8.46623	−0.000428365 0
$732$	0	0
$733$	−6057.78	−0.305251	−0.152626	−	0.988284i	$-0.548773\pi$
$733$	−6057.78	−0.305251	−0.152626	+	0.988284i	$0.548773\pi$
$734$	0	0
$735$	−790.938	−0.0396928
$736$	0	0
$737$	4612.94	0.230556
$738$	0	0
$739$	4088.83	0.203532	0.101766	−	0.994808i	$-0.467551\pi$
$739$	4088.83	0.203532	0.101766	+	0.994808i	$0.467551\pi$
$740$	0	0
$741$	448.675	0.0222436
$742$	0	0
$743$	−21467.5	−1.05998	−0.529991	−	0.848003i	$-0.677805\pi$
$743$	−21467.5	−1.05998	−0.529991	+	0.848003i	$0.677805\pi$
$744$	0	0
$745$	−7910.98	−0.389041
$746$	0	0
$747$	−4156.36	−0.203579
$748$	0	0
$749$	32476.2	1.58432
$750$	0	0
$751$	5285.55	0.256821	0.128410	−	0.991721i	$-0.459013\pi$
$751$	5285.55	0.256821	0.128410	+	0.991721i	$0.459013\pi$
$752$	0	0
$753$	−8645.76	−0.418418
$754$	0	0
$755$	5845.07	0.281753
$756$	0	0
$757$	1748.30	0.0839405	0.0419702	−	0.999119i	$-0.486637\pi$
$757$	1748.30	0.0839405	0.0419702	+	0.999119i	$0.486637\pi$
$758$	0	0
$759$	−930.674	−0.0445077
$760$	0	0
$761$	6405.11	0.305105	0.152553	−	0.988295i	$-0.451251\pi$
$761$	6405.11	0.305105	0.152553	+	0.988295i	$0.451251\pi$
$762$	0	0
$763$	−4824.17	−0.228895
$764$	0	0
$765$	6276.14	0.296620
$766$	0	0
$767$	−15262.7	−0.718518
$768$	0	0
$769$	32100.5	1.50530	0.752648	−	0.658423i	$-0.228776\pi$
$769$	32100.5	1.50530	0.752648	+	0.658423i	$0.228776\pi$
$770$	0	0
$771$	−6090.60	−0.284498
$772$	0	0
$773$	−17226.1	−0.801527	−0.400764	−	0.916181i	$-0.631255\pi$
$773$	−17226.1	−0.801527	−0.400764	+	0.916181i	$0.631255\pi$
$774$	0	0
$775$	−7204.00	−0.333904
$776$	0	0
$777$	10508.6	0.485190
$778$	0	0
$779$	1627.30	0.0748448
$780$	0	0
$781$	4769.34	0.218515
$782$	0	0
$783$	12586.9	0.574483
$784$	0	0
$785$	−1807.31	−0.0821730
$786$	0	0
$787$	35601.2	1.61251	0.806256	−	0.591566i	$-0.201490\pi$
$787$	35601.2	1.61251	0.806256	+	0.591566i	$0.201490\pi$
$788$	0	0
$789$	1948.05	0.0878990
$790$	0	0
$791$	−8427.74	−0.378832
$792$	0	0
$793$	−6462.66	−0.289402
$794$	0	0
$795$	−992.847	−0.0442927
$796$	0	0
$797$	−96.9760	−0.00431000	−0.00215500	−	0.999998i	$-0.500686\pi$
$797$	−96.9760	−0.00431000	−0.00215500	+	0.999998i	$0.500686\pi$
$798$	0	0
$799$	27630.2	1.22339
$800$	0	0
$801$	38006.9	1.67654
$802$	0	0
$803$	21612.4	0.949794
$804$	0	0
$805$	2399.97	0.105078
$806$	0	0
$807$	−10629.7	−0.463673
$808$	0	0
$809$	−28065.0	−1.21967	−0.609836	−	0.792528i	$-0.708765\pi$
$809$	−28065.0	−1.21967	−0.609836	+	0.792528i	$0.708765\pi$
$810$	0	0
$811$	38011.9	1.64584	0.822921	−	0.568156i	$-0.192343\pi$
$811$	38011.9	1.64584	0.822921	+	0.568156i	$0.192343\pi$
$812$	0	0
$813$	3709.38	0.160017
$814$	0	0
$815$	−15717.8	−0.675546
$816$	0	0
$817$	−0.793079	−3.39612e−5 0
$818$	0	0
$819$	−27002.0	−1.15205
$820$	0	0
$821$	9423.34	0.400581	0.200290	−	0.979737i	$-0.435811\pi$
$821$	9423.34	0.400581	0.200290	+	0.979737i	$0.435811\pi$
$822$	0	0
$823$	−23294.3	−0.986621	−0.493311	−	0.869853i	$-0.664213\pi$
$823$	−23294.3	−0.986621	−0.493311	+	0.869853i	$0.664213\pi$
$824$	0	0
$825$	1011.60	0.0426902
$826$	0	0
$827$	21050.1	0.885106	0.442553	−	0.896742i	$-0.354073\pi$
$827$	21050.1	0.885106	0.442553	+	0.896742i	$0.354073\pi$
$828$	0	0
$829$	30468.6	1.27650	0.638250	−	0.769830i	$-0.279659\pi$
$829$	30468.6	1.27650	0.638250	+	0.769830i	$0.279659\pi$
$830$	0	0
$831$	10830.7	0.452120
$832$	0	0
$833$	4823.76	0.200640
$834$	0	0
$835$	8120.16	0.336539
$836$	0	0
$837$	25163.1	1.03914
$838$	0	0
$839$	20156.4	0.829413	0.414706	−	0.909955i	$-0.363884\pi$
$839$	20156.4	0.829413	0.414706	+	0.909955i	$0.363884\pi$
$840$	0	0
$841$	−3612.20	−0.148108
$842$	0	0
$843$	−8521.97	−0.348176
$844$	0	0
$845$	−3454.01	−0.140617
$846$	0	0
$847$	−16086.4	−0.652579
$848$	0	0
$849$	5323.21	0.215185
$850$	0	0
$851$	−6774.23	−0.272876
$852$	0	0
$853$	48022.4	1.92761	0.963807	−	0.266602i	$-0.0859008\pi$
$853$	48022.4	1.92761	0.963807	+	0.266602i	$0.0859008\pi$
$854$	0	0
$855$	587.921	0.0235163
$856$	0	0
$857$	3811.01	0.151904	0.0759519	−	0.997111i	$-0.475800\pi$
$857$	3811.01	0.151904	0.0759519	+	0.997111i	$0.475800\pi$
$858$	0	0
$859$	2576.98	0.102358	0.0511789	−	0.998690i	$-0.483702\pi$
$859$	2576.98	0.102358	0.0511789	+	0.998690i	$0.483702\pi$
$860$	0	0
$861$	11888.7	0.470577
$862$	0	0
$863$	−10214.8	−0.402915	−0.201457	−	0.979497i	$-0.564568\pi$
$863$	−10214.8	−0.402915	−0.201457	+	0.979497i	$0.564568\pi$
$864$	0	0
$865$	12455.7	0.489605
$866$	0	0
$867$	−3752.81	−0.147004
$868$	0	0
$869$	21341.9	0.833112
$870$	0	0
$871$	10473.6	0.407445
$872$	0	0
$873$	−41416.1	−1.60564
$874$	0	0
$875$	−2608.66	−0.100787
$876$	0	0
$877$	−41225.1	−1.58731	−0.793656	−	0.608367i	$-0.791825\pi$
$877$	−41225.1	−1.58731	−0.793656	+	0.608367i	$0.791825\pi$
$878$	0	0
$879$	−3235.91	−0.124169
$880$	0	0
$881$	−23955.4	−0.916094	−0.458047	−	0.888928i	$-0.651451\pi$
$881$	−23955.4	−0.916094	−0.458047	+	0.888928i	$0.651451\pi$
$882$	0	0
$883$	−28679.2	−1.09302	−0.546508	−	0.837454i	$-0.684043\pi$
$883$	−28679.2	−1.09302	−0.546508	+	0.837454i	$0.684043\pi$
$884$	0	0
$885$	2427.85	0.0922160
$886$	0	0
$887$	29456.1	1.11504	0.557520	−	0.830164i	$-0.311753\pi$
$887$	29456.1	1.11504	0.557520	+	0.830164i	$0.311753\pi$
$888$	0	0
$889$	−38450.9	−1.45062
$890$	0	0
$891$	11852.8	0.445662
$892$	0	0
$893$	2588.27	0.0969913
$894$	0	0
$895$	20374.8	0.760954
$896$	0	0
$897$	−2113.08	−0.0786551
$898$	0	0
$899$	41535.8	1.54093
$900$	0	0
$901$	6055.16	0.223892
$902$	0	0
$903$	−5.79408	−0.000213527 0
$904$	0	0
$905$	−16521.8	−0.606855
$906$	0	0
$907$	38637.7	1.41449	0.707246	−	0.706968i	$-0.249937\pi$
$907$	38637.7	1.41449	0.707246	+	0.706968i	$0.249937\pi$
$908$	0	0
$909$	13194.7	0.481451
$910$	0	0
$911$	−43014.4	−1.56436	−0.782179	−	0.623053i	$-0.785892\pi$
$911$	−43014.4	−1.56436	−0.782179	+	0.623053i	$0.785892\pi$
$912$	0	0
$913$	4085.77	0.148104
$914$	0	0
$915$	1028.02	0.0371424
$916$	0	0
$917$	23475.2	0.845386
$918$	0	0
$919$	−25861.1	−0.928267	−0.464134	−	0.885765i	$-0.653634\pi$
$919$	−25861.1	−0.928267	−0.464134	+	0.885765i	$0.653634\pi$
$920$	0	0
$921$	17077.5	0.610992
$922$	0	0
$923$	10828.7	0.386166
$924$	0	0
$925$	7363.29	0.261734
$926$	0	0
$927$	−39844.9	−1.41174
$928$	0	0
$929$	−50340.0	−1.77783	−0.888914	−	0.458073i	$-0.848540\pi$
$929$	−50340.0	−1.77783	−0.888914	+	0.458073i	$0.848540\pi$
$930$	0	0
$931$	451.869	0.0159070
$932$	0	0
$933$	−2948.97	−0.103478
$934$	0	0
$935$	−6169.54	−0.215792
$936$	0	0
$937$	−51360.8	−1.79070	−0.895349	−	0.445365i	$-0.853074\pi$
$937$	−51360.8	−1.79070	−0.895349	+	0.445365i	$0.853074\pi$
$938$	0	0
$939$	−15631.4	−0.543251
$940$	0	0
$941$	−4590.77	−0.159038	−0.0795190	−	0.996833i	$-0.525338\pi$
$941$	−4590.77	−0.159038	−0.0795190	+	0.996833i	$0.525338\pi$
$942$	0	0
$943$	−7663.94	−0.264658
$944$	0	0
$945$	9111.88	0.313661
$946$	0	0
$947$	−8532.06	−0.292772	−0.146386	−	0.989228i	$-0.546764\pi$
$947$	−8532.06	−0.292772	−0.146386	+	0.989228i	$0.546764\pi$
$948$	0	0
$949$	49070.5	1.67850
$950$	0	0
$951$	2057.50	0.0701567
$952$	0	0
$953$	−38218.1	−1.29906	−0.649531	−	0.760335i	$-0.725035\pi$
$953$	−38218.1	−1.29906	−0.649531	+	0.760335i	$0.725035\pi$
$954$	0	0
$955$	2286.67	0.0774817
$956$	0	0
$957$	−5832.56	−0.197011
$958$	0	0
$959$	55831.4	1.87997
$960$	0	0
$961$	53245.1	1.78729
$962$	0	0
$963$	−37468.2	−1.25379
$964$	0	0
$965$	−1153.97	−0.0384948
$966$	0	0
$967$	581.187	0.0193275	0.00966376	−	0.999953i	$-0.496924\pi$
$967$	581.187	0.0193275	0.00966376	+	0.999953i	$0.496924\pi$
$968$	0	0
$969$	435.277	0.0144304
$970$	0	0
$971$	−41718.0	−1.37878	−0.689389	−	0.724391i	$-0.742121\pi$
$971$	−41718.0	−1.37878	−0.689389	+	0.724391i	$0.742121\pi$
$972$	0	0
$973$	22138.4	0.729419
$974$	0	0
$975$	2296.82	0.0754433
$976$	0	0
$977$	34468.0	1.12869	0.564344	−	0.825540i	$-0.309129\pi$
$977$	34468.0	1.12869	0.564344	+	0.825540i	$0.309129\pi$
$978$	0	0
$979$	−37361.4	−1.21969
$980$	0	0
$981$	5565.70	0.181141
$982$	0	0
$983$	−43972.2	−1.42675	−0.713375	−	0.700783i	$-0.752834\pi$
$983$	−43972.2	−1.42675	−0.713375	+	0.700783i	$0.752834\pi$
$984$	0	0
$985$	5394.29	0.174494
$986$	0	0
$987$	18909.4	0.609821
$988$	0	0
$989$	3.73509	0.000120090 0
$990$	0	0
$991$	−9819.21	−0.314750	−0.157375	−	0.987539i	$-0.550303\pi$
$991$	−9819.21	−0.314750	−0.157375	+	0.987539i	$0.550303\pi$
$992$	0	0
$993$	−11660.3	−0.372636
$994$	0	0
$995$	−22627.7	−0.720950
$996$	0	0
$997$	2688.77	0.0854106	0.0427053	−	0.999088i	$-0.486402\pi$
$997$	2688.77	0.0854106	0.0427053	+	0.999088i	$0.486402\pi$
$998$	0	0
$999$	−25719.5	−0.814544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.4.a.e.1.6	✓	9
4.3	odd	2		1840.4.a.z.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.e.1.6	✓	9	1.1	even	1	trivial
1840.4.a.z.1.4		9	4.3	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70964 q^{3} -5.00000 q^{5} +20.8693 q^{7} -24.0771 q^{9} +O(q^{10})\)	\(q+1.70964 q^{3} -5.00000 q^{5} +20.8693 q^{7} -24.0771 q^{9} +23.6682 q^{11} +53.7383 q^{13} -8.54819 q^{15} +52.1336 q^{17} +4.88364 q^{19} +35.6789 q^{21} -23.0000 q^{23} +25.0000 q^{25} -87.3234 q^{27} -144.142 q^{29} -288.160 q^{31} +40.4641 q^{33} -104.346 q^{35} +294.532 q^{37} +91.8730 q^{39} +333.215 q^{41} -0.162395 q^{43} +120.386 q^{45} +529.988 q^{47} +92.5270 q^{49} +89.1295 q^{51} +116.147 q^{53} -118.341 q^{55} +8.34926 q^{57} -284.019 q^{59} -120.262 q^{61} -502.473 q^{63} -268.691 q^{65} +194.900 q^{67} -39.3217 q^{69} +201.508 q^{71} +913.140 q^{73} +42.7409 q^{75} +493.939 q^{77} +901.712 q^{79} +500.791 q^{81} +172.627 q^{83} -260.668 q^{85} -246.430 q^{87} -1578.55 q^{89} +1121.48 q^{91} -492.649 q^{93} -24.4182 q^{95} +1720.14 q^{97} -569.863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} - 45 q^{5} + 25 q^{7} + 62 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 - 45 * q^5 + 25 * q^7 + 62 * q^9	\( 9 q - 3 q^{3} - 45 q^{5} + 25 q^{7} + 62 q^{9} + 22 q^{11} + 23 q^{13} + 15 q^{15} - 135 q^{17} + 102 q^{19} + 54 q^{21} - 207 q^{23} + 225 q^{25} - 363 q^{27} + 280 q^{29} + 168 q^{31} + 28 q^{33} - 125 q^{35} + 153 q^{37} - 5 q^{39} - 502 q^{41} + 110 q^{43} - 310 q^{45} - 153 q^{47} + 764 q^{49} + 924 q^{51} - 273 q^{53} - 110 q^{55} + 748 q^{57} + 827 q^{59} + 1976 q^{61} + 2237 q^{63} - 115 q^{65} + 1613 q^{67} + 69 q^{69} + 1370 q^{71} + 425 q^{73} - 75 q^{75} + 2006 q^{77} + 2624 q^{79} + 1729 q^{81} + 2505 q^{83} + 675 q^{85} + 1591 q^{87} + 1120 q^{89} + 2392 q^{91} + 4401 q^{93} - 510 q^{95} + 2026 q^{97} + 3206 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 - 45 * q^5 + 25 * q^7 + 62 * q^9 + 22 * q^11 + 23 * q^13 + 15 * q^15 - 135 * q^17 + 102 * q^19 + 54 * q^21 - 207 * q^23 + 225 * q^25 - 363 * q^27 + 280 * q^29 + 168 * q^31 + 28 * q^33 - 125 * q^35 + 153 * q^37 - 5 * q^39 - 502 * q^41 + 110 * q^43 - 310 * q^45 - 153 * q^47 + 764 * q^49 + 924 * q^51 - 273 * q^53 - 110 * q^55 + 748 * q^57 + 827 * q^59 + 1976 * q^61 + 2237 * q^63 - 115 * q^65 + 1613 * q^67 + 69 * q^69 + 1370 * q^71 + 425 * q^73 - 75 * q^75 + 2006 * q^77 + 2624 * q^79 + 1729 * q^81 + 2505 * q^83 + 675 * q^85 + 1591 * q^87 + 1120 * q^89 + 2392 * q^91 + 4401 * q^93 - 510 * q^95 + 2026 * q^97 + 3206 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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