LMFDB - Embedded newform 1840.4.a.v.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
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} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 $ x^8 - 2x^7 - 49x^6 + 31x^5 + 750x^4 + 249x^3 - 2892x^2 - 620*x + 2400
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.05234$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.94300 q^{3} +5.00000 q^{5} -32.7645 q^{7} +52.9772 q^{9} +O(q^{10})$	$q+8.94300 q^{3} +5.00000 q^{5} -32.7645 q^{7} +52.9772 q^{9} -45.7496 q^{11} +45.5638 q^{13} +44.7150 q^{15} -76.5649 q^{17} +32.0225 q^{19} -293.013 q^{21} -23.0000 q^{23} +25.0000 q^{25} +232.314 q^{27} +287.046 q^{29} +161.587 q^{31} -409.139 q^{33} -163.823 q^{35} +7.53926 q^{37} +407.476 q^{39} +216.246 q^{41} +469.729 q^{43} +264.886 q^{45} +210.315 q^{47} +730.514 q^{49} -684.719 q^{51} +58.7860 q^{53} -228.748 q^{55} +286.377 q^{57} +376.609 q^{59} -52.7319 q^{61} -1735.77 q^{63} +227.819 q^{65} +16.7678 q^{67} -205.689 q^{69} +209.441 q^{71} -811.910 q^{73} +223.575 q^{75} +1498.96 q^{77} -63.0408 q^{79} +647.197 q^{81} +1062.27 q^{83} -382.824 q^{85} +2567.05 q^{87} +436.069 q^{89} -1492.87 q^{91} +1445.07 q^{93} +160.112 q^{95} +655.840 q^{97} -2423.68 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10}) $ 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9	$ 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100}) $ 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9 - 41 * q^11 + 28 * q^13 + 71 * q^17 - 177 * q^19 + 292 * q^21 - 184 * q^23 + 200 * q^25 + 495 * q^27 + 225 * q^29 + 36 * q^31 - 53 * q^33 - 55 * q^35 - 348 * q^37 + 1077 * q^39 + 620 * q^41 + 390 * q^43 + 790 * q^45 - 123 * q^47 + 881 * q^49 - 957 * q^51 + 1406 * q^53 - 205 * q^55 - 1142 * q^57 + 676 * q^59 + 1447 * q^61 + 58 * q^63 + 140 * q^65 + 1582 * q^67 - 1396 * q^71 + 17 * q^73 + 488 * q^77 - 708 * q^79 + 4316 * q^81 - 1486 * q^83 + 355 * q^85 + 803 * q^87 + 1360 * q^89 - 2693 * q^91 + 3833 * q^93 - 885 * q^95 - 855 * q^97 - 1319 * 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
$3$	8.94300	1.72108	0.860540	−	0.509383i	$-0.170126\pi$
$3$	8.94300	1.72108	0.860540	+	0.509383i	$0.170126\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−32.7645	−1.76912	−0.884559	−	0.466429i	$-0.845540\pi$
$7$	−32.7645	−1.76912	−0.884559	+	0.466429i	$0.845540\pi$
$8$	0	0
$9$	52.9772	1.96212
$10$	0	0
$11$	−45.7496	−1.25400	−0.627001	−	0.779018i	$-0.715718\pi$
$11$	−45.7496	−1.25400	−0.627001	+	0.779018i	$0.715718\pi$
$12$	0	0
$13$	45.5638	0.972086	0.486043	−	0.873935i	$-0.338440\pi$
$13$	45.5638	0.972086	0.486043	+	0.873935i	$0.338440\pi$
$14$	0	0
$15$	44.7150	0.769690
$16$	0	0
$17$	−76.5649	−1.09234	−0.546168	−	0.837676i	$-0.683914\pi$
$17$	−76.5649	−1.09234	−0.546168	+	0.837676i	$0.683914\pi$
$18$	0	0
$19$	32.0225	0.386656	0.193328	−	0.981134i	$-0.438072\pi$
$19$	32.0225	0.386656	0.193328	+	0.981134i	$0.438072\pi$
$20$	0	0
$21$	−293.013	−3.04479
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	232.314	1.65588
$28$	0	0
$29$	287.046	1.83804	0.919019	−	0.394213i	$-0.128983\pi$
$29$	287.046	1.83804	0.919019	+	0.394213i	$0.128983\pi$
$30$	0	0
$31$	161.587	0.936189	0.468095	−	0.883678i	$-0.344941\pi$
$31$	161.587	0.936189	0.468095	+	0.883678i	$0.344941\pi$
$32$	0	0
$33$	−409.139	−2.15824
$34$	0	0
$35$	−163.823	−0.791173
$36$	0	0
$37$	7.53926	0.0334986	0.0167493	−	0.999860i	$-0.494668\pi$
$37$	7.53926	0.0334986	0.0167493	+	0.999860i	$0.494668\pi$
$38$	0	0
$39$	407.476	1.67304
$40$	0	0
$41$	216.246	0.823704	0.411852	−	0.911251i	$-0.364882\pi$
$41$	216.246	0.823704	0.411852	+	0.911251i	$0.364882\pi$
$42$	0	0
$43$	469.729	1.66589	0.832943	−	0.553359i	$-0.186654\pi$
$43$	469.729	1.66589	0.832943	+	0.553359i	$0.186654\pi$
$44$	0	0
$45$	264.886	0.877485
$46$	0	0
$47$	210.315	0.652715	0.326358	−	0.945246i	$-0.394179\pi$
$47$	210.315	0.652715	0.326358	+	0.945246i	$0.394179\pi$
$48$	0	0
$49$	730.514	2.12978
$50$	0	0
$51$	−684.719	−1.88000
$52$	0	0
$53$	58.7860	0.152356	0.0761781	−	0.997094i	$-0.475728\pi$
$53$	58.7860	0.152356	0.0761781	+	0.997094i	$0.475728\pi$
$54$	0	0
$55$	−228.748	−0.560807
$56$	0	0
$57$	286.377	0.665466
$58$	0	0
$59$	376.609	0.831022	0.415511	−	0.909588i	$-0.363603\pi$
$59$	376.609	0.831022	0.415511	+	0.909588i	$0.363603\pi$
$60$	0	0
$61$	−52.7319	−0.110682	−0.0553412	−	0.998467i	$-0.517625\pi$
$61$	−52.7319	−0.110682	−0.0553412	+	0.998467i	$0.517625\pi$
$62$	0	0
$63$	−1735.77	−3.47122
$64$	0	0
$65$	227.819	0.434730
$66$	0	0
$67$	16.7678	0.0305748	0.0152874	−	0.999883i	$-0.495134\pi$
$67$	16.7678	0.0305748	0.0152874	+	0.999883i	$0.495134\pi$
$68$	0	0
$69$	−205.689	−0.358870
$70$	0	0
$71$	209.441	0.350086	0.175043	−	0.984561i	$-0.443994\pi$
$71$	209.441	0.350086	0.175043	+	0.984561i	$0.443994\pi$
$72$	0	0
$73$	−811.910	−1.30174	−0.650869	−	0.759190i	$-0.725595\pi$
$73$	−811.910	−1.30174	−0.650869	+	0.759190i	$0.725595\pi$
$74$	0	0
$75$	223.575	0.344216
$76$	0	0
$77$	1498.96	2.21848
$78$	0	0
$79$	−63.0408	−0.0897803	−0.0448902	−	0.998992i	$-0.514294\pi$
$79$	−63.0408	−0.0897803	−0.0448902	+	0.998992i	$0.514294\pi$
$80$	0	0
$81$	647.197	0.887787
$82$	0	0
$83$	1062.27	1.40481	0.702404	−	0.711779i	$-0.252110\pi$
$83$	1062.27	1.40481	0.702404	+	0.711779i	$0.252110\pi$
$84$	0	0
$85$	−382.824	−0.488508
$86$	0	0
$87$	2567.05	3.16341
$88$	0	0
$89$	436.069	0.519362	0.259681	−	0.965694i	$-0.416383\pi$
$89$	436.069	0.519362	0.259681	+	0.965694i	$0.416383\pi$
$90$	0	0
$91$	−1492.87	−1.71973
$92$	0	0
$93$	1445.07	1.61126
$94$	0	0
$95$	160.112	0.172918
$96$	0	0
$97$	655.840	0.686499	0.343250	−	0.939244i	$-0.388472\pi$
$97$	655.840	0.686499	0.343250	+	0.939244i	$0.388472\pi$
$98$	0	0
$99$	−2423.68	−2.46050
$100$	0	0
$101$	831.308	0.818993	0.409496	−	0.912312i	$-0.365704\pi$
$101$	831.308	0.818993	0.409496	+	0.912312i	$0.365704\pi$
$102$	0	0
$103$	801.275	0.766524	0.383262	−	0.923640i	$-0.374801\pi$
$103$	801.275	0.766524	0.383262	+	0.923640i	$0.374801\pi$
$104$	0	0
$105$	−1465.06	−1.36167
$106$	0	0
$107$	−901.921	−0.814879	−0.407439	−	0.913232i	$-0.633578\pi$
$107$	−901.921	−0.814879	−0.407439	+	0.913232i	$0.633578\pi$
$108$	0	0
$109$	328.880	0.289000	0.144500	−	0.989505i	$-0.453843\pi$
$109$	328.880	0.289000	0.144500	+	0.989505i	$0.453843\pi$
$110$	0	0
$111$	67.4236	0.0576537
$112$	0	0
$113$	−2209.02	−1.83900	−0.919500	−	0.393090i	$-0.871406\pi$
$113$	−2209.02	−1.83900	−0.919500	+	0.393090i	$0.871406\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	2413.84	1.90735
$118$	0	0
$119$	2508.61	1.93247
$120$	0	0
$121$	762.027	0.572522
$122$	0	0
$123$	1933.88	1.41766
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−107.788	−0.0753124	−0.0376562	−	0.999291i	$-0.511989\pi$
$127$	−107.788	−0.0753124	−0.0376562	+	0.999291i	$0.511989\pi$
$128$	0	0
$129$	4200.79	2.86712
$130$	0	0
$131$	985.156	0.657050	0.328525	−	0.944495i	$-0.393449\pi$
$131$	985.156	0.657050	0.328525	+	0.944495i	$0.393449\pi$
$132$	0	0
$133$	−1049.20	−0.684040
$134$	0	0
$135$	1161.57	0.740532
$136$	0	0
$137$	65.3753	0.0407692	0.0203846	−	0.999792i	$-0.493511\pi$
$137$	65.3753	0.0407692	0.0203846	+	0.999792i	$0.493511\pi$
$138$	0	0
$139$	547.498	0.334088	0.167044	−	0.985949i	$-0.446578\pi$
$139$	547.498	0.334088	0.167044	+	0.985949i	$0.446578\pi$
$140$	0	0
$141$	1880.85	1.12338
$142$	0	0
$143$	−2084.52	−1.21900
$144$	0	0
$145$	1435.23	0.821996
$146$	0	0
$147$	6532.98	3.66552
$148$	0	0
$149$	3316.98	1.82374	0.911871	−	0.410476i	$-0.134637\pi$
$149$	3316.98	1.82374	0.911871	+	0.410476i	$0.134637\pi$
$150$	0	0
$151$	1645.53	0.886831	0.443415	−	0.896316i	$-0.353767\pi$
$151$	1645.53	0.886831	0.443415	+	0.896316i	$0.353767\pi$
$152$	0	0
$153$	−4056.19	−2.14329
$154$	0	0
$155$	807.934	0.418676
$156$	0	0
$157$	−2088.75	−1.06179	−0.530893	−	0.847439i	$-0.678143\pi$
$157$	−2088.75	−1.06179	−0.530893	+	0.847439i	$0.678143\pi$
$158$	0	0
$159$	525.723	0.262217
$160$	0	0
$161$	753.584	0.368887
$162$	0	0
$163$	−3160.08	−1.51851	−0.759253	−	0.650796i	$-0.774435\pi$
$163$	−3160.08	−1.51851	−0.759253	+	0.650796i	$0.774435\pi$
$164$	0	0
$165$	−2045.69	−0.965194
$166$	0	0
$167$	1886.30	0.874047	0.437024	−	0.899450i	$-0.356033\pi$
$167$	1886.30	0.874047	0.437024	+	0.899450i	$0.356033\pi$
$168$	0	0
$169$	−120.944	−0.0550495
$170$	0	0
$171$	1696.46	0.758664
$172$	0	0
$173$	−425.628	−0.187051	−0.0935256	−	0.995617i	$-0.529814\pi$
$173$	−425.628	−0.187051	−0.0935256	+	0.995617i	$0.529814\pi$
$174$	0	0
$175$	−819.113	−0.353824
$176$	0	0
$177$	3368.01	1.43026
$178$	0	0
$179$	−4680.83	−1.95453	−0.977266	−	0.212016i	$-0.931997\pi$
$179$	−4680.83	−1.95453	−0.977266	+	0.212016i	$0.931997\pi$
$180$	0	0
$181$	−380.264	−0.156159	−0.0780796	−	0.996947i	$-0.524879\pi$
$181$	−380.264	−0.156159	−0.0780796	+	0.996947i	$0.524879\pi$
$182$	0	0
$183$	−471.581	−0.190493
$184$	0	0
$185$	37.6963	0.0149810
$186$	0	0
$187$	3502.81	1.36979
$188$	0	0
$189$	−7611.64	−2.92945
$190$	0	0
$191$	−1466.03	−0.555383	−0.277691	−	0.960670i	$-0.589569\pi$
$191$	−1466.03	−0.555383	−0.277691	+	0.960670i	$0.589569\pi$
$192$	0	0
$193$	−2129.58	−0.794253	−0.397126	−	0.917764i	$-0.629993\pi$
$193$	−2129.58	−0.794253	−0.397126	+	0.917764i	$0.629993\pi$
$194$	0	0
$195$	2037.38	0.748205
$196$	0	0
$197$	−1146.95	−0.414807	−0.207403	−	0.978256i	$-0.566501\pi$
$197$	−1146.95	−0.414807	−0.207403	+	0.978256i	$0.566501\pi$
$198$	0	0
$199$	3857.88	1.37426	0.687130	−	0.726535i	$-0.258870\pi$
$199$	3857.88	1.37426	0.687130	+	0.726535i	$0.258870\pi$
$200$	0	0
$201$	149.954	0.0526217
$202$	0	0
$203$	−9404.93	−3.25171
$204$	0	0
$205$	1081.23	0.368372
$206$	0	0
$207$	−1218.47	−0.409130
$208$	0	0
$209$	−1465.02	−0.484868
$210$	0	0
$211$	−1880.22	−0.613458	−0.306729	−	0.951797i	$-0.599235\pi$
$211$	−1880.22	−0.613458	−0.306729	+	0.951797i	$0.599235\pi$
$212$	0	0
$213$	1873.03	0.602526
$214$	0	0
$215$	2348.65	0.745007
$216$	0	0
$217$	−5294.32	−1.65623
$218$	0	0
$219$	−7260.91	−2.24040
$220$	0	0
$221$	−3488.58	−1.06184
$222$	0	0
$223$	3181.06	0.955246	0.477623	−	0.878565i	$-0.341499\pi$
$223$	3181.06	0.955246	0.477623	+	0.878565i	$0.341499\pi$
$224$	0	0
$225$	1324.43	0.392423
$226$	0	0
$227$	−4817.24	−1.40851	−0.704255	−	0.709947i	$-0.748719\pi$
$227$	−4817.24	−1.40851	−0.704255	+	0.709947i	$0.748719\pi$
$228$	0	0
$229$	−1440.61	−0.415714	−0.207857	−	0.978159i	$-0.566649\pi$
$229$	−1440.61	−0.415714	−0.207857	+	0.978159i	$0.566649\pi$
$230$	0	0
$231$	13405.2	3.81818
$232$	0	0
$233$	2313.68	0.650533	0.325266	−	0.945622i	$-0.394546\pi$
$233$	2313.68	0.650533	0.325266	+	0.945622i	$0.394546\pi$
$234$	0	0
$235$	1051.58	0.291903
$236$	0	0
$237$	−563.774	−0.154519
$238$	0	0
$239$	2798.09	0.757293	0.378647	−	0.925541i	$-0.376390\pi$
$239$	2798.09	0.757293	0.378647	+	0.925541i	$0.376390\pi$
$240$	0	0
$241$	−1809.31	−0.483600	−0.241800	−	0.970326i	$-0.577738\pi$
$241$	−1809.31	−0.483600	−0.241800	+	0.970326i	$0.577738\pi$
$242$	0	0
$243$	−484.593	−0.127929
$244$	0	0
$245$	3652.57	0.952465
$246$	0	0
$247$	1459.07	0.375863
$248$	0	0
$249$	9499.86	2.41779
$250$	0	0
$251$	−5814.80	−1.46226	−0.731129	−	0.682239i	$-0.761006\pi$
$251$	−5814.80	−1.46226	−0.731129	+	0.682239i	$0.761006\pi$
$252$	0	0
$253$	1052.24	0.261478
$254$	0	0
$255$	−3423.60	−0.840761
$256$	0	0
$257$	1886.38	0.457856	0.228928	−	0.973443i	$-0.426478\pi$
$257$	1886.38	0.457856	0.228928	+	0.973443i	$0.426478\pi$
$258$	0	0
$259$	−247.020	−0.0592629
$260$	0	0
$261$	15206.9	3.60645
$262$	0	0
$263$	6017.91	1.41095	0.705476	−	0.708734i	$-0.250733\pi$
$263$	6017.91	1.41095	0.705476	+	0.708734i	$0.250733\pi$
$264$	0	0
$265$	293.930	0.0681358
$266$	0	0
$267$	3899.77	0.893864
$268$	0	0
$269$	−3189.16	−0.722848	−0.361424	−	0.932401i	$-0.617709\pi$
$269$	−3189.16	−0.722848	−0.361424	+	0.932401i	$0.617709\pi$
$270$	0	0
$271$	−1167.43	−0.261683	−0.130841	−	0.991403i	$-0.541768\pi$
$271$	−1167.43	−0.261683	−0.130841	+	0.991403i	$0.541768\pi$
$272$	0	0
$273$	−13350.8	−2.95980
$274$	0	0
$275$	−1143.74	−0.250800
$276$	0	0
$277$	−4164.36	−0.903293	−0.451646	−	0.892197i	$-0.649163\pi$
$277$	−4164.36	−0.903293	−0.451646	+	0.892197i	$0.649163\pi$
$278$	0	0
$279$	8560.42	1.83691
$280$	0	0
$281$	4958.59	1.05269	0.526343	−	0.850273i	$-0.323563\pi$
$281$	4958.59	1.05269	0.526343	+	0.850273i	$0.323563\pi$
$282$	0	0
$283$	−2171.54	−0.456129	−0.228065	−	0.973646i	$-0.573240\pi$
$283$	−2171.54	−0.456129	−0.228065	+	0.973646i	$0.573240\pi$
$284$	0	0
$285$	1431.89	0.297605
$286$	0	0
$287$	−7085.18	−1.45723
$288$	0	0
$289$	949.183	0.193198
$290$	0	0
$291$	5865.17	1.18152
$292$	0	0
$293$	5170.00	1.03083	0.515417	−	0.856939i	$-0.327637\pi$
$293$	5170.00	1.03083	0.515417	+	0.856939i	$0.327637\pi$
$294$	0	0
$295$	1883.04	0.371644
$296$	0	0
$297$	−10628.3	−2.07648
$298$	0	0
$299$	−1047.97	−0.202694
$300$	0	0
$301$	−15390.5	−2.94715
$302$	0	0
$303$	7434.39	1.40955
$304$	0	0
$305$	−263.660	−0.0494987
$306$	0	0
$307$	−9189.38	−1.70836	−0.854178	−	0.519980i	$-0.825939\pi$
$307$	−9189.38	−1.70836	−0.854178	+	0.519980i	$0.825939\pi$
$308$	0	0
$309$	7165.80	1.31925
$310$	0	0
$311$	−2264.59	−0.412903	−0.206452	−	0.978457i	$-0.566192\pi$
$311$	−2264.59	−0.412903	−0.206452	+	0.978457i	$0.566192\pi$
$312$	0	0
$313$	2467.42	0.445581	0.222791	−	0.974866i	$-0.428483\pi$
$313$	2467.42	0.445581	0.222791	+	0.974866i	$0.428483\pi$
$314$	0	0
$315$	−8678.86	−1.55238
$316$	0	0
$317$	7178.08	1.27180	0.635901	−	0.771771i	$-0.280629\pi$
$317$	7178.08	1.27180	0.635901	+	0.771771i	$0.280629\pi$
$318$	0	0
$319$	−13132.2	−2.30490
$320$	0	0
$321$	−8065.88	−1.40247
$322$	0	0
$323$	−2451.80	−0.422358
$324$	0	0
$325$	1139.09	0.194417
$326$	0	0
$327$	2941.17	0.497392
$328$	0	0
$329$	−6890.88	−1.15473
$330$	0	0
$331$	10304.2	1.71109	0.855545	−	0.517728i	$-0.173222\pi$
$331$	10304.2	1.71109	0.855545	+	0.517728i	$0.173222\pi$
$332$	0	0
$333$	399.409	0.0657281
$334$	0	0
$335$	83.8390	0.0136735
$336$	0	0
$337$	−1582.19	−0.255749	−0.127874	−	0.991790i	$-0.540815\pi$
$337$	−1582.19	−0.255749	−0.127874	+	0.991790i	$0.540815\pi$
$338$	0	0
$339$	−19755.2	−3.16507
$340$	0	0
$341$	−7392.54	−1.17398
$342$	0	0
$343$	−12696.7	−1.99871
$344$	0	0
$345$	−1028.44	−0.160492
$346$	0	0
$347$	−2566.92	−0.397117	−0.198559	−	0.980089i	$-0.563626\pi$
$347$	−2566.92	−0.397117	−0.198559	+	0.980089i	$0.563626\pi$
$348$	0	0
$349$	−215.226	−0.0330108	−0.0165054	−	0.999864i	$-0.505254\pi$
$349$	−215.226	−0.0330108	−0.0165054	+	0.999864i	$0.505254\pi$
$350$	0	0
$351$	10585.1	1.60966
$352$	0	0
$353$	−5111.32	−0.770675	−0.385338	−	0.922776i	$-0.625915\pi$
$353$	−5111.32	−0.770675	−0.385338	+	0.922776i	$0.625915\pi$
$354$	0	0
$355$	1047.21	0.156563
$356$	0	0
$357$	22434.5	3.32594
$358$	0	0
$359$	4317.14	0.634680	0.317340	−	0.948312i	$-0.397210\pi$
$359$	4317.14	0.634680	0.317340	+	0.948312i	$0.397210\pi$
$360$	0	0
$361$	−5833.56	−0.850497
$362$	0	0
$363$	6814.80	0.985356
$364$	0	0
$365$	−4059.55	−0.582155
$366$	0	0
$367$	7628.24	1.08499	0.542495	−	0.840059i	$-0.317480\pi$
$367$	7628.24	1.08499	0.542495	+	0.840059i	$0.317480\pi$
$368$	0	0
$369$	11456.1	1.61620
$370$	0	0
$371$	−1926.10	−0.269536
$372$	0	0
$373$	8487.60	1.17821	0.589104	−	0.808057i	$-0.299481\pi$
$373$	8487.60	1.17821	0.589104	+	0.808057i	$0.299481\pi$
$374$	0	0
$375$	1117.87	0.153938
$376$	0	0
$377$	13078.9	1.78673
$378$	0	0
$379$	1227.48	0.166363	0.0831815	−	0.996534i	$-0.473492\pi$
$379$	1227.48	0.166363	0.0831815	+	0.996534i	$0.473492\pi$
$380$	0	0
$381$	−963.951	−0.129619
$382$	0	0
$383$	4106.30	0.547838	0.273919	−	0.961753i	$-0.411680\pi$
$383$	4106.30	0.547838	0.273919	+	0.961753i	$0.411680\pi$
$384$	0	0
$385$	7494.82	0.992133
$386$	0	0
$387$	24884.9	3.26866
$388$	0	0
$389$	−11206.6	−1.46066	−0.730328	−	0.683097i	$-0.760633\pi$
$389$	−11206.6	−1.46066	−0.730328	+	0.683097i	$0.760633\pi$
$390$	0	0
$391$	1760.99	0.227768
$392$	0	0
$393$	8810.25	1.13084
$394$	0	0
$395$	−315.204	−0.0401510
$396$	0	0
$397$	7629.81	0.964557	0.482278	−	0.876018i	$-0.339809\pi$
$397$	7629.81	0.964557	0.482278	+	0.876018i	$0.339809\pi$
$398$	0	0
$399$	−9383.01	−1.17729
$400$	0	0
$401$	1221.46	0.152112	0.0760560	−	0.997104i	$-0.475767\pi$
$401$	1221.46	0.152112	0.0760560	+	0.997104i	$0.475767\pi$
$402$	0	0
$403$	7362.51	0.910056
$404$	0	0
$405$	3235.98	0.397030
$406$	0	0
$407$	−344.918	−0.0420073
$408$	0	0
$409$	16112.2	1.94791	0.973957	−	0.226731i	$-0.0728038\pi$
$409$	16112.2	1.94791	0.973957	+	0.226731i	$0.0728038\pi$
$410$	0	0
$411$	584.651	0.0701672
$412$	0	0
$413$	−12339.4	−1.47018
$414$	0	0
$415$	5311.34	0.628249
$416$	0	0
$417$	4896.27	0.574992
$418$	0	0
$419$	−1997.98	−0.232954	−0.116477	−	0.993193i	$-0.537160\pi$
$419$	−1997.98	−0.232954	−0.116477	+	0.993193i	$0.537160\pi$
$420$	0	0
$421$	−4849.75	−0.561431	−0.280715	−	0.959791i	$-0.590572\pi$
$421$	−4849.75	−0.561431	−0.280715	+	0.959791i	$0.590572\pi$
$422$	0	0
$423$	11141.9	1.28070
$424$	0	0
$425$	−1914.12	−0.218467
$426$	0	0
$427$	1727.74	0.195810
$428$	0	0
$429$	−18641.9	−2.09799
$430$	0	0
$431$	−13262.3	−1.48219	−0.741094	−	0.671402i	$-0.765693\pi$
$431$	−13262.3	−1.48219	−0.741094	+	0.671402i	$0.765693\pi$
$432$	0	0
$433$	13656.0	1.51562	0.757812	−	0.652473i	$-0.226269\pi$
$433$	13656.0	1.51562	0.757812	+	0.652473i	$0.226269\pi$
$434$	0	0
$435$	12835.3	1.41472
$436$	0	0
$437$	−736.517	−0.0806234
$438$	0	0
$439$	1132.68	0.123143	0.0615717	−	0.998103i	$-0.480389\pi$
$439$	1132.68	0.123143	0.0615717	+	0.998103i	$0.480389\pi$
$440$	0	0
$441$	38700.5	4.17887
$442$	0	0
$443$	−6583.81	−0.706109	−0.353054	−	0.935603i	$-0.614857\pi$
$443$	−6583.81	−0.706109	−0.353054	+	0.935603i	$0.614857\pi$
$444$	0	0
$445$	2180.35	0.232266
$446$	0	0
$447$	29663.7	3.13881
$448$	0	0
$449$	−12407.9	−1.30416	−0.652078	−	0.758152i	$-0.726103\pi$
$449$	−12407.9	−1.30416	−0.652078	+	0.758152i	$0.726103\pi$
$450$	0	0
$451$	−9893.15	−1.03293
$452$	0	0
$453$	14716.0	1.52631
$454$	0	0
$455$	−7464.37	−0.769088
$456$	0	0
$457$	17336.1	1.77450	0.887251	−	0.461286i	$-0.152612\pi$
$457$	17336.1	1.77450	0.887251	+	0.461286i	$0.152612\pi$
$458$	0	0
$459$	−17787.1	−1.80878
$460$	0	0
$461$	9940.64	1.00430	0.502150	−	0.864781i	$-0.332543\pi$
$461$	9940.64	1.00430	0.502150	+	0.864781i	$0.332543\pi$
$462$	0	0
$463$	−6459.56	−0.648383	−0.324191	−	0.945991i	$-0.605092\pi$
$463$	−6459.56	−0.648383	−0.324191	+	0.945991i	$0.605092\pi$
$464$	0	0
$465$	7225.35	0.720576
$466$	0	0
$467$	−8151.66	−0.807739	−0.403869	−	0.914817i	$-0.632335\pi$
$467$	−8151.66	−0.807739	−0.403869	+	0.914817i	$0.632335\pi$
$468$	0	0
$469$	−549.389	−0.0540905
$470$	0	0
$471$	−18679.7	−1.82742
$472$	0	0
$473$	−21489.9	−2.08902
$474$	0	0
$475$	800.562	0.0773312
$476$	0	0
$477$	3114.32	0.298941
$478$	0	0
$479$	214.878	0.0204969	0.0102485	−	0.999947i	$-0.496738\pi$
$479$	214.878	0.0204969	0.0102485	+	0.999947i	$0.496738\pi$
$480$	0	0
$481$	343.517	0.0325635
$482$	0	0
$483$	6739.30	0.634883
$484$	0	0
$485$	3279.20	0.307012
$486$	0	0
$487$	−6216.61	−0.578442	−0.289221	−	0.957262i	$-0.593396\pi$
$487$	−6216.61	−0.578442	−0.289221	+	0.957262i	$0.593396\pi$
$488$	0	0
$489$	−28260.6	−2.61347
$490$	0	0
$491$	17480.8	1.60671	0.803356	−	0.595499i	$-0.203046\pi$
$491$	17480.8	1.60671	0.803356	+	0.595499i	$0.203046\pi$
$492$	0	0
$493$	−21977.7	−2.00776
$494$	0	0
$495$	−12118.4	−1.10037
$496$	0	0
$497$	−6862.24	−0.619343
$498$	0	0
$499$	17884.0	1.60440	0.802200	−	0.597055i	$-0.203663\pi$
$499$	17884.0	1.60440	0.802200	+	0.597055i	$0.203663\pi$
$500$	0	0
$501$	16869.1	1.50431
$502$	0	0
$503$	−3215.45	−0.285029	−0.142515	−	0.989793i	$-0.545519\pi$
$503$	−3215.45	−0.285029	−0.142515	+	0.989793i	$0.545519\pi$
$504$	0	0
$505$	4156.54	0.366265
$506$	0	0
$507$	−1081.60	−0.0947446
$508$	0	0
$509$	6737.65	0.586721	0.293360	−	0.956002i	$-0.405226\pi$
$509$	6737.65	0.586721	0.293360	+	0.956002i	$0.405226\pi$
$510$	0	0
$511$	26601.9	2.30293
$512$	0	0
$513$	7439.26	0.640256
$514$	0	0
$515$	4006.38	0.342800
$516$	0	0
$517$	−9621.84	−0.818507
$518$	0	0
$519$	−3806.39	−0.321930
$520$	0	0
$521$	13464.5	1.13223	0.566114	−	0.824327i	$-0.308446\pi$
$521$	13464.5	1.13223	0.566114	+	0.824327i	$0.308446\pi$
$522$	0	0
$523$	21370.5	1.78674	0.893372	−	0.449317i	$-0.148333\pi$
$523$	21370.5	1.78674	0.893372	+	0.449317i	$0.148333\pi$
$524$	0	0
$525$	−7325.32	−0.608959
$526$	0	0
$527$	−12371.9	−1.02263
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	19951.7	1.63056
$532$	0	0
$533$	9852.96	0.800711
$534$	0	0
$535$	−4509.61	−0.364425
$536$	0	0
$537$	−41860.6	−3.36391
$538$	0	0
$539$	−33420.7	−2.67075
$540$	0	0
$541$	−14485.2	−1.15114	−0.575572	−	0.817751i	$-0.695221\pi$
$541$	−14485.2	−1.15114	−0.575572	+	0.817751i	$0.695221\pi$
$542$	0	0
$543$	−3400.70	−0.268763
$544$	0	0
$545$	1644.40	0.129245
$546$	0	0
$547$	18926.7	1.47943	0.739714	−	0.672922i	$-0.234961\pi$
$547$	18926.7	1.47943	0.739714	+	0.672922i	$0.234961\pi$
$548$	0	0
$549$	−2793.59	−0.217172
$550$	0	0
$551$	9191.93	0.710689
$552$	0	0
$553$	2065.50	0.158832
$554$	0	0
$555$	337.118	0.0257835
$556$	0	0
$557$	2146.15	0.163259	0.0816294	−	0.996663i	$-0.473988\pi$
$557$	2146.15	0.163259	0.0816294	+	0.996663i	$0.473988\pi$
$558$	0	0
$559$	21402.6	1.61938
$560$	0	0
$561$	31325.6	2.35752
$562$	0	0
$563$	14066.3	1.05297	0.526487	−	0.850183i	$-0.323509\pi$
$563$	14066.3	1.05297	0.526487	+	0.850183i	$0.323509\pi$
$564$	0	0
$565$	−11045.1	−0.822426
$566$	0	0
$567$	−21205.1	−1.57060
$568$	0	0
$569$	−1253.22	−0.0923337	−0.0461668	−	0.998934i	$-0.514701\pi$
$569$	−1253.22	−0.0923337	−0.0461668	+	0.998934i	$0.514701\pi$
$570$	0	0
$571$	−12499.6	−0.916097	−0.458048	−	0.888927i	$-0.651451\pi$
$571$	−12499.6	−0.916097	−0.458048	+	0.888927i	$0.651451\pi$
$572$	0	0
$573$	−13110.7	−0.955858
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−7260.56	−0.523849	−0.261925	−	0.965088i	$-0.584357\pi$
$577$	−7260.56	−0.523849	−0.261925	+	0.965088i	$0.584357\pi$
$578$	0	0
$579$	−19044.9	−1.36697
$580$	0	0
$581$	−34804.7	−2.48527
$582$	0	0
$583$	−2689.44	−0.191055
$584$	0	0
$585$	12069.2	0.852991
$586$	0	0
$587$	−14411.6	−1.01334	−0.506671	−	0.862139i	$-0.669124\pi$
$587$	−14411.6	−1.01334	−0.506671	+	0.862139i	$0.669124\pi$
$588$	0	0
$589$	5174.42	0.361983
$590$	0	0
$591$	−10257.2	−0.713916
$592$	0	0
$593$	21665.4	1.50032	0.750160	−	0.661256i	$-0.229976\pi$
$593$	21665.4	1.50032	0.750160	+	0.661256i	$0.229976\pi$
$594$	0	0
$595$	12543.1	0.864227
$596$	0	0
$597$	34501.0	2.36521
$598$	0	0
$599$	23174.4	1.58077	0.790384	−	0.612612i	$-0.209881\pi$
$599$	23174.4	1.58077	0.790384	+	0.612612i	$0.209881\pi$
$600$	0	0
$601$	−22969.1	−1.55895	−0.779475	−	0.626433i	$-0.784514\pi$
$601$	−22969.1	−1.55895	−0.779475	+	0.626433i	$0.784514\pi$
$602$	0	0
$603$	888.311	0.0599914
$604$	0	0
$605$	3810.13	0.256040
$606$	0	0
$607$	4606.78	0.308045	0.154023	−	0.988067i	$-0.450777\pi$
$607$	4606.78	0.308045	0.154023	+	0.988067i	$0.450777\pi$
$608$	0	0
$609$	−84108.2	−5.59645
$610$	0	0
$611$	9582.75	0.634495
$612$	0	0
$613$	7780.72	0.512660	0.256330	−	0.966589i	$-0.417487\pi$
$613$	7780.72	0.512660	0.256330	+	0.966589i	$0.417487\pi$
$614$	0	0
$615$	9669.41	0.633997
$616$	0	0
$617$	2438.43	0.159105	0.0795523	−	0.996831i	$-0.474651\pi$
$617$	2438.43	0.159105	0.0795523	+	0.996831i	$0.474651\pi$
$618$	0	0
$619$	−6821.13	−0.442915	−0.221458	−	0.975170i	$-0.571081\pi$
$619$	−6821.13	−0.442915	−0.221458	+	0.975170i	$0.571081\pi$
$620$	0	0
$621$	−5343.21	−0.345275
$622$	0	0
$623$	−14287.6	−0.918813
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−13101.6	−0.834496
$628$	0	0
$629$	−577.242	−0.0365917
$630$	0	0
$631$	15698.7	0.990420	0.495210	−	0.868773i	$-0.335091\pi$
$631$	15698.7	0.990420	0.495210	+	0.868773i	$0.335091\pi$
$632$	0	0
$633$	−16814.8	−1.05581
$634$	0	0
$635$	−538.942	−0.0336807
$636$	0	0
$637$	33284.9	2.07033
$638$	0	0
$639$	11095.6	0.686909
$640$	0	0
$641$	15360.1	0.946472	0.473236	−	0.880936i	$-0.343086\pi$
$641$	15360.1	0.946472	0.473236	+	0.880936i	$0.343086\pi$
$642$	0	0
$643$	14107.3	0.865221	0.432610	−	0.901581i	$-0.357593\pi$
$643$	14107.3	0.865221	0.432610	+	0.901581i	$0.357593\pi$
$644$	0	0
$645$	21003.9	1.28222
$646$	0	0
$647$	−8107.22	−0.492624	−0.246312	−	0.969191i	$-0.579219\pi$
$647$	−8107.22	−0.492624	−0.246312	+	0.969191i	$0.579219\pi$
$648$	0	0
$649$	−17229.7	−1.04210
$650$	0	0
$651$	−47347.0	−2.85050
$652$	0	0
$653$	−25123.5	−1.50560	−0.752802	−	0.658247i	$-0.771298\pi$
$653$	−25123.5	−1.50560	−0.752802	+	0.658247i	$0.771298\pi$
$654$	0	0
$655$	4925.78	0.293842
$656$	0	0
$657$	−43012.7	−2.55416
$658$	0	0
$659$	809.773	0.0478669	0.0239334	−	0.999714i	$-0.492381\pi$
$659$	809.773	0.0478669	0.0239334	+	0.999714i	$0.492381\pi$
$660$	0	0
$661$	4086.36	0.240455	0.120228	−	0.992746i	$-0.461638\pi$
$661$	4086.36	0.240455	0.120228	+	0.992746i	$0.461638\pi$
$662$	0	0
$663$	−31198.4	−1.82752
$664$	0	0
$665$	−5246.01	−0.305912
$666$	0	0
$667$	−6602.06	−0.383258
$668$	0	0
$669$	28448.2	1.64405
$670$	0	0
$671$	2412.47	0.138796
$672$	0	0
$673$	−8985.56	−0.514662	−0.257331	−	0.966323i	$-0.582843\pi$
$673$	−8985.56	−0.514662	−0.257331	+	0.966323i	$0.582843\pi$
$674$	0	0
$675$	5807.84	0.331176
$676$	0	0
$677$	10328.0	0.586316	0.293158	−	0.956064i	$-0.405294\pi$
$677$	10328.0	0.586316	0.293158	+	0.956064i	$0.405294\pi$
$678$	0	0
$679$	−21488.3	−1.21450
$680$	0	0
$681$	−43080.6	−2.42416
$682$	0	0
$683$	−4516.74	−0.253043	−0.126521	−	0.991964i	$-0.540381\pi$
$683$	−4516.74	−0.253043	−0.126521	+	0.991964i	$0.540381\pi$
$684$	0	0
$685$	326.876	0.0182326
$686$	0	0
$687$	−12883.4	−0.715476
$688$	0	0
$689$	2678.51	0.148103
$690$	0	0
$691$	−7957.06	−0.438062	−0.219031	−	0.975718i	$-0.570290\pi$
$691$	−7957.06	−0.438062	−0.219031	+	0.975718i	$0.570290\pi$
$692$	0	0
$693$	79410.9	4.35291
$694$	0	0
$695$	2737.49	0.149409
$696$	0	0
$697$	−16556.8	−0.899762
$698$	0	0
$699$	20691.2	1.11962
$700$	0	0
$701$	7970.58	0.429450	0.214725	−	0.976675i	$-0.431114\pi$
$701$	7970.58	0.429450	0.214725	+	0.976675i	$0.431114\pi$
$702$	0	0
$703$	241.426	0.0129524
$704$	0	0
$705$	9404.24	0.502389
$706$	0	0
$707$	−27237.4	−1.44889
$708$	0	0
$709$	2079.18	0.110134	0.0550672	−	0.998483i	$-0.482463\pi$
$709$	2079.18	0.110134	0.0550672	+	0.998483i	$0.482463\pi$
$710$	0	0
$711$	−3339.72	−0.176159
$712$	0	0
$713$	−3716.50	−0.195209
$714$	0	0
$715$	−10422.6	−0.545152
$716$	0	0
$717$	25023.3	1.30336
$718$	0	0
$719$	−29046.5	−1.50661	−0.753305	−	0.657671i	$-0.771542\pi$
$719$	−29046.5	−1.50661	−0.753305	+	0.657671i	$0.771542\pi$
$720$	0	0
$721$	−26253.4	−1.35607
$722$	0	0
$723$	−16180.6	−0.832315
$724$	0	0
$725$	7176.15	0.367608
$726$	0	0
$727$	−14037.7	−0.716133	−0.358067	−	0.933696i	$-0.616564\pi$
$727$	−14037.7	−0.716133	−0.358067	+	0.933696i	$0.616564\pi$
$728$	0	0
$729$	−21808.0	−1.10796
$730$	0	0
$731$	−35964.8	−1.81971
$732$	0	0
$733$	10862.5	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$733$	10862.5	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$734$	0	0
$735$	32664.9	1.63927
$736$	0	0
$737$	−767.121	−0.0383409
$738$	0	0
$739$	8660.65	0.431106	0.215553	−	0.976492i	$-0.430845\pi$
$739$	8660.65	0.431106	0.215553	+	0.976492i	$0.430845\pi$
$740$	0	0
$741$	13048.4	0.646890
$742$	0	0
$743$	33484.3	1.65332	0.826661	−	0.562700i	$-0.190238\pi$
$743$	33484.3	1.65332	0.826661	+	0.562700i	$0.190238\pi$
$744$	0	0
$745$	16584.9	0.815603
$746$	0	0
$747$	56275.9	2.75640
$748$	0	0
$749$	29551.0	1.44162
$750$	0	0
$751$	−29209.3	−1.41926	−0.709628	−	0.704577i	$-0.751137\pi$
$751$	−29209.3	−1.41926	−0.709628	+	0.704577i	$0.751137\pi$
$752$	0	0
$753$	−52001.7	−2.51666
$754$	0	0
$755$	8227.66	0.396603
$756$	0	0
$757$	3283.09	0.157630	0.0788151	−	0.996889i	$-0.474886\pi$
$757$	3283.09	0.157630	0.0788151	+	0.996889i	$0.474886\pi$
$758$	0	0
$759$	9410.19	0.450024
$760$	0	0
$761$	22957.1	1.09355	0.546776	−	0.837279i	$-0.315855\pi$
$761$	22957.1	1.09355	0.546776	+	0.837279i	$0.315855\pi$
$762$	0	0
$763$	−10775.6	−0.511275
$764$	0	0
$765$	−20281.0	−0.958509
$766$	0	0
$767$	17159.7	0.807825
$768$	0	0
$769$	−2510.87	−0.117743	−0.0588714	−	0.998266i	$-0.518750\pi$
$769$	−2510.87	−0.117743	−0.0588714	+	0.998266i	$0.518750\pi$
$770$	0	0
$771$	16869.9	0.788007
$772$	0	0
$773$	−11566.1	−0.538167	−0.269084	−	0.963117i	$-0.586721\pi$
$773$	−11566.1	−0.538167	−0.269084	+	0.963117i	$0.586721\pi$
$774$	0	0
$775$	4039.67	0.187238
$776$	0	0
$777$	−2209.10	−0.101996
$778$	0	0
$779$	6924.72	0.318490
$780$	0	0
$781$	−9581.85	−0.439008
$782$	0	0
$783$	66684.7	3.04357
$784$	0	0
$785$	−10443.7	−0.474845
$786$	0	0
$787$	751.934	0.0340579	0.0170289	−	0.999855i	$-0.494579\pi$
$787$	751.934	0.0340579	0.0170289	+	0.999855i	$0.494579\pi$
$788$	0	0
$789$	53818.1	2.42836
$790$	0	0
$791$	72377.4	3.25341
$792$	0	0
$793$	−2402.66	−0.107593
$794$	0	0
$795$	2628.62	0.117267
$796$	0	0
$797$	−15898.3	−0.706585	−0.353292	−	0.935513i	$-0.614938\pi$
$797$	−15898.3	−0.706585	−0.353292	+	0.935513i	$0.614938\pi$
$798$	0	0
$799$	−16102.8	−0.712985
$800$	0	0
$801$	23101.7	1.01905
$802$	0	0
$803$	37144.6	1.63238
$804$	0	0
$805$	3767.92	0.164971
$806$	0	0
$807$	−28520.6	−1.24408
$808$	0	0
$809$	−30018.9	−1.30458	−0.652292	−	0.757967i	$-0.726193\pi$
$809$	−30018.9	−1.30458	−0.652292	+	0.757967i	$0.726193\pi$
$810$	0	0
$811$	−5012.47	−0.217030	−0.108515	−	0.994095i	$-0.534610\pi$
$811$	−5012.47	−0.217030	−0.108515	+	0.994095i	$0.534610\pi$
$812$	0	0
$813$	−10440.3	−0.450377
$814$	0	0
$815$	−15800.4	−0.679096
$816$	0	0
$817$	15041.9	0.644125
$818$	0	0
$819$	−79088.3	−3.37432
$820$	0	0
$821$	−2707.44	−0.115092	−0.0575459	−	0.998343i	$-0.518328\pi$
$821$	−2707.44	−0.115092	−0.0575459	+	0.998343i	$0.518328\pi$
$822$	0	0
$823$	−15781.8	−0.668431	−0.334215	−	0.942497i	$-0.608471\pi$
$823$	−15781.8	−0.668431	−0.334215	+	0.942497i	$0.608471\pi$
$824$	0	0
$825$	−10228.5	−0.431648
$826$	0	0
$827$	21176.8	0.890435	0.445218	−	0.895422i	$-0.353126\pi$
$827$	21176.8	0.890435	0.445218	+	0.895422i	$0.353126\pi$
$828$	0	0
$829$	20905.6	0.875852	0.437926	−	0.899011i	$-0.355713\pi$
$829$	20905.6	0.875852	0.437926	+	0.899011i	$0.355713\pi$
$830$	0	0
$831$	−37241.8	−1.55464
$832$	0	0
$833$	−55931.7	−2.32643
$834$	0	0
$835$	9431.48	0.390886
$836$	0	0
$837$	37538.8	1.55022
$838$	0	0
$839$	−28724.8	−1.18199	−0.590995	−	0.806675i	$-0.701265\pi$
$839$	−28724.8	−1.18199	−0.590995	+	0.806675i	$0.701265\pi$
$840$	0	0
$841$	58006.4	2.37839
$842$	0	0
$843$	44344.6	1.81176
$844$	0	0
$845$	−604.719	−0.0246189
$846$	0	0
$847$	−24967.4	−1.01286
$848$	0	0
$849$	−19420.1	−0.785035
$850$	0	0
$851$	−173.403	−0.00698493
$852$	0	0
$853$	−37511.5	−1.50571	−0.752854	−	0.658188i	$-0.771323\pi$
$853$	−37511.5	−1.50571	−0.752854	+	0.658188i	$0.771323\pi$
$854$	0	0
$855$	8482.31	0.339285
$856$	0	0
$857$	26541.3	1.05792	0.528958	−	0.848648i	$-0.322583\pi$
$857$	26541.3	1.05792	0.528958	+	0.848648i	$0.322583\pi$
$858$	0	0
$859$	−20535.4	−0.815667	−0.407833	−	0.913056i	$-0.633716\pi$
$859$	−20535.4	−0.815667	−0.407833	+	0.913056i	$0.633716\pi$
$860$	0	0
$861$	−63362.7	−2.50801
$862$	0	0
$863$	−21337.9	−0.841660	−0.420830	−	0.907140i	$-0.638261\pi$
$863$	−21337.9	−0.841660	−0.420830	+	0.907140i	$0.638261\pi$
$864$	0	0
$865$	−2128.14	−0.0836519
$866$	0	0
$867$	8488.54	0.332510
$868$	0	0
$869$	2884.09	0.112585
$870$	0	0
$871$	764.004	0.0297214
$872$	0	0
$873$	34744.5	1.34699
$874$	0	0
$875$	−4095.56	−0.158235
$876$	0	0
$877$	−42216.5	−1.62548	−0.812742	−	0.582624i	$-0.802026\pi$
$877$	−42216.5	−1.62548	−0.812742	+	0.582624i	$0.802026\pi$
$878$	0	0
$879$	46235.3	1.77415
$880$	0	0
$881$	−33457.6	−1.27947	−0.639737	−	0.768594i	$-0.720957\pi$
$881$	−33457.6	−1.27947	−0.639737	+	0.768594i	$0.720957\pi$
$882$	0	0
$883$	−22192.2	−0.845785	−0.422893	−	0.906180i	$-0.638985\pi$
$883$	−22192.2	−0.845785	−0.422893	+	0.906180i	$0.638985\pi$
$884$	0	0
$885$	16840.1	0.639630
$886$	0	0
$887$	−17904.9	−0.677777	−0.338889	−	0.940826i	$-0.610051\pi$
$887$	−17904.9	−0.677777	−0.338889	+	0.940826i	$0.610051\pi$
$888$	0	0
$889$	3531.64	0.133237
$890$	0	0
$891$	−29609.0	−1.11329
$892$	0	0
$893$	6734.82	0.252376
$894$	0	0
$895$	−23404.1	−0.874093
$896$	0	0
$897$	−9371.96	−0.348852
$898$	0	0
$899$	46382.9	1.72075
$900$	0	0
$901$	−4500.95	−0.166424
$902$	0	0
$903$	−137637.	−5.07228
$904$	0	0
$905$	−1901.32	−0.0698365
$906$	0	0
$907$	−9482.20	−0.347135	−0.173567	−	0.984822i	$-0.555529\pi$
$907$	−9482.20	−0.347135	−0.173567	+	0.984822i	$0.555529\pi$
$908$	0	0
$909$	44040.4	1.60696
$910$	0	0
$911$	−24018.2	−0.873499	−0.436750	−	0.899583i	$-0.643870\pi$
$911$	−24018.2	−0.873499	−0.436750	+	0.899583i	$0.643870\pi$
$912$	0	0
$913$	−48598.3	−1.76163
$914$	0	0
$915$	−2357.91	−0.0851913
$916$	0	0
$917$	−32278.2	−1.16240
$918$	0	0
$919$	39263.9	1.40935	0.704677	−	0.709528i	$-0.251092\pi$
$919$	39263.9	1.40935	0.704677	+	0.709528i	$0.251092\pi$
$920$	0	0
$921$	−82180.5	−2.94022
$922$	0	0
$923$	9542.92	0.340313
$924$	0	0
$925$	188.481	0.00669971
$926$	0	0
$927$	42449.3	1.50401
$928$	0	0
$929$	30536.7	1.07845	0.539223	−	0.842163i	$-0.318718\pi$
$929$	30536.7	1.07845	0.539223	+	0.842163i	$0.318718\pi$
$930$	0	0
$931$	23392.9	0.823491
$932$	0	0
$933$	−20252.2	−0.710640
$934$	0	0
$935$	17514.1	0.612590
$936$	0	0
$937$	−37724.9	−1.31528	−0.657640	−	0.753332i	$-0.728445\pi$
$937$	−37724.9	−1.31528	−0.657640	+	0.753332i	$0.728445\pi$
$938$	0	0
$939$	22066.1	0.766881
$940$	0	0
$941$	33414.9	1.15759	0.578797	−	0.815472i	$-0.303522\pi$
$941$	33414.9	1.15759	0.578797	+	0.815472i	$0.303522\pi$
$942$	0	0
$943$	−4973.65	−0.171754
$944$	0	0
$945$	−38058.2	−1.31009
$946$	0	0
$947$	11604.7	0.398205	0.199103	−	0.979979i	$-0.436197\pi$
$947$	11604.7	0.398205	0.199103	+	0.979979i	$0.436197\pi$
$948$	0	0
$949$	−36993.7	−1.26540
$950$	0	0
$951$	64193.5	2.18887
$952$	0	0
$953$	33381.8	1.13467	0.567336	−	0.823487i	$-0.307974\pi$
$953$	33381.8	1.13467	0.567336	+	0.823487i	$0.307974\pi$
$954$	0	0
$955$	−7330.14	−0.248375
$956$	0	0
$957$	−117442.	−3.96693
$958$	0	0
$959$	−2141.99	−0.0721256
$960$	0	0
$961$	−3680.68	−0.123550
$962$	0	0
$963$	−47781.2	−1.59889
$964$	0	0
$965$	−10647.9	−0.355201
$966$	0	0
$967$	−44013.3	−1.46367	−0.731837	−	0.681479i	$-0.761337\pi$
$967$	−44013.3	−1.46367	−0.731837	+	0.681479i	$0.761337\pi$
$968$	0	0
$969$	−21926.4	−0.726913
$970$	0	0
$971$	−54941.3	−1.81581	−0.907905	−	0.419176i	$-0.862319\pi$
$971$	−54941.3	−1.81581	−0.907905	+	0.419176i	$0.862319\pi$
$972$	0	0
$973$	−17938.5	−0.591040
$974$	0	0
$975$	10186.9	0.334607
$976$	0	0
$977$	52377.7	1.71516	0.857580	−	0.514351i	$-0.171967\pi$
$977$	52377.7	1.71516	0.857580	+	0.514351i	$0.171967\pi$
$978$	0	0
$979$	−19950.0	−0.651282
$980$	0	0
$981$	17423.1	0.567052
$982$	0	0
$983$	18746.9	0.608275	0.304137	−	0.952628i	$-0.401632\pi$
$983$	18746.9	0.608275	0.304137	+	0.952628i	$0.401632\pi$
$984$	0	0
$985$	−5734.76	−0.185507
$986$	0	0
$987$	−61625.1	−1.98738
$988$	0	0
$989$	−10803.8	−0.347361
$990$	0	0
$991$	−51116.6	−1.63852	−0.819260	−	0.573423i	$-0.805615\pi$
$991$	−51116.6	−1.63852	−0.819260	+	0.573423i	$0.805615\pi$
$992$	0	0
$993$	92150.5	2.94492
$994$	0	0
$995$	19289.4	0.614587
$996$	0	0
$997$	−36176.5	−1.14917	−0.574584	−	0.818445i	$-0.694836\pi$
$997$	−36176.5	−1.14917	−0.574584	+	0.818445i	$0.694836\pi$
$998$	0	0
$999$	1751.47	0.0554696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.v.1.7		8
4.3	odd	2		115.4.a.f.1.6	✓	8
12.11	even	2		1035.4.a.r.1.3		8
20.3	even	4		575.4.b.k.24.4		16
20.7	even	4		575.4.b.k.24.13		16
20.19	odd	2		575.4.a.n.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.f.1.6	✓	8	4.3	odd	2
575.4.a.n.1.3		8	20.19	odd	2
575.4.b.k.24.4		16	20.3	even	4
575.4.b.k.24.13		16	20.7	even	4
1035.4.a.r.1.3		8	12.11	even	2
1840.4.a.v.1.7		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+8.94300 q^{3} +5.00000 q^{5} -32.7645 q^{7} +52.9772 q^{9} +O(q^{10})\)	\(q+8.94300 q^{3} +5.00000 q^{5} -32.7645 q^{7} +52.9772 q^{9} -45.7496 q^{11} +45.5638 q^{13} +44.7150 q^{15} -76.5649 q^{17} +32.0225 q^{19} -293.013 q^{21} -23.0000 q^{23} +25.0000 q^{25} +232.314 q^{27} +287.046 q^{29} +161.587 q^{31} -409.139 q^{33} -163.823 q^{35} +7.53926 q^{37} +407.476 q^{39} +216.246 q^{41} +469.729 q^{43} +264.886 q^{45} +210.315 q^{47} +730.514 q^{49} -684.719 q^{51} +58.7860 q^{53} -228.748 q^{55} +286.377 q^{57} +376.609 q^{59} -52.7319 q^{61} -1735.77 q^{63} +227.819 q^{65} +16.7678 q^{67} -205.689 q^{69} +209.441 q^{71} -811.910 q^{73} +223.575 q^{75} +1498.96 q^{77} -63.0408 q^{79} +647.197 q^{81} +1062.27 q^{83} -382.824 q^{85} +2567.05 q^{87} +436.069 q^{89} -1492.87 q^{91} +1445.07 q^{93} +160.112 q^{95} +655.840 q^{97} -2423.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10}) \) 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9	\( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100}) \) 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9 - 41 * q^11 + 28 * q^13 + 71 * q^17 - 177 * q^19 + 292 * q^21 - 184 * q^23 + 200 * q^25 + 495 * q^27 + 225 * q^29 + 36 * q^31 - 53 * q^33 - 55 * q^35 - 348 * q^37 + 1077 * q^39 + 620 * q^41 + 390 * q^43 + 790 * q^45 - 123 * q^47 + 881 * q^49 - 957 * q^51 + 1406 * q^53 - 205 * q^55 - 1142 * q^57 + 676 * q^59 + 1447 * q^61 + 58 * q^63 + 140 * q^65 + 1582 * q^67 - 1396 * q^71 + 17 * q^73 + 488 * q^77 - 708 * q^79 + 4316 * q^81 - 1486 * q^83 + 355 * q^85 + 803 * q^87 + 1360 * q^89 - 2693 * q^91 + 3833 * q^93 - 885 * q^95 - 855 * q^97 - 1319 * q^99

Introduction

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