LMFDB - Embedded newform 1815.4.a.bk.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 79x^{10} + 2212x^{8} - 26055x^{6} + 118317x^{4} - 109056x^{2} + 12996 $ x^12 - 79x^10 + 2212x^8 - 26055x^6 + 118317x^4 - 109056*x^2 + 12996
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.373761$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.373761 q^{2} +3.00000 q^{3} -7.86030 q^{4} +5.00000 q^{5} -1.12128 q^{6} +18.2837 q^{7} +5.92796 q^{8} +9.00000 q^{9} -1.86880 q^{10} -23.5809 q^{12} +83.5276 q^{13} -6.83371 q^{14} +15.0000 q^{15} +60.6668 q^{16} +98.0730 q^{17} -3.36385 q^{18} +94.2981 q^{19} -39.3015 q^{20} +54.8510 q^{21} +65.7953 q^{23} +17.7839 q^{24} +25.0000 q^{25} -31.2193 q^{26} +27.0000 q^{27} -143.715 q^{28} +120.655 q^{29} -5.60641 q^{30} +130.246 q^{31} -70.0985 q^{32} -36.6558 q^{34} +91.4183 q^{35} -70.7427 q^{36} -386.662 q^{37} -35.2449 q^{38} +250.583 q^{39} +29.6398 q^{40} -354.489 q^{41} -20.5011 q^{42} -273.737 q^{43} +45.0000 q^{45} -24.5917 q^{46} +548.956 q^{47} +182.000 q^{48} -8.70748 q^{49} -9.34401 q^{50} +294.219 q^{51} -656.552 q^{52} -660.563 q^{53} -10.0915 q^{54} +108.385 q^{56} +282.894 q^{57} -45.0960 q^{58} +283.388 q^{59} -117.905 q^{60} -437.501 q^{61} -48.6808 q^{62} +164.553 q^{63} -459.134 q^{64} +417.638 q^{65} +617.750 q^{67} -770.884 q^{68} +197.386 q^{69} -34.1686 q^{70} +578.571 q^{71} +53.3516 q^{72} +960.165 q^{73} +144.519 q^{74} +75.0000 q^{75} -741.212 q^{76} -93.6579 q^{78} -1008.95 q^{79} +303.334 q^{80} +81.0000 q^{81} +132.494 q^{82} +958.535 q^{83} -431.146 q^{84} +490.365 q^{85} +102.312 q^{86} +361.964 q^{87} -1394.59 q^{89} -16.8192 q^{90} +1527.19 q^{91} -517.171 q^{92} +390.738 q^{93} -205.178 q^{94} +471.491 q^{95} -210.295 q^{96} -63.3140 q^{97} +3.25451 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} + 62 q^{4} + 60 q^{5} + 108 q^{9} + 186 q^{12} + 16 q^{14} + 180 q^{15} + 610 q^{16} + 310 q^{20} - 44 q^{23} + 300 q^{25} + 1016 q^{26} + 324 q^{27} + 288 q^{31} + 710 q^{34} + 558 q^{36}+ \cdots + 1436 q^{97}+O(q^{100}) $ 12 * q + 36 * q^3 + 62 * q^4 + 60 * q^5 + 108 * q^9 + 186 * q^12 + 16 * q^14 + 180 * q^15 + 610 * q^16 + 310 * q^20 - 44 * q^23 + 300 * q^25 + 1016 * q^26 + 324 * q^27 + 288 * q^31 + 710 * q^34 + 558 * q^36 - 488 * q^37 + 872 * q^38 + 48 * q^42 + 540 * q^45 + 264 * q^47 + 1830 * q^48 + 140 * q^49 + 1144 * q^53 + 3172 * q^56 - 488 * q^58 + 1400 * q^59 + 930 * q^60 + 3088 * q^64 + 2032 * q^67 - 132 * q^69 + 80 * q^70 + 4852 * q^71 + 900 * q^75 + 3048 * q^78 + 3050 * q^80 + 972 * q^81 + 816 * q^82 - 2232 * q^86 + 4932 * q^89 + 7336 * q^91 - 2526 * q^92 + 864 * q^93 + 1436 * q^97

Coefficient data

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

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

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

$2$	−0.373761	−0.132144	−0.0660722	−	0.997815i	$-0.521047\pi$
$2$	−0.373761	−0.132144	−0.0660722	+	0.997815i	$0.521047\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.86030	−0.982538
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−1.12128	−0.0762936
$7$	18.2837	0.987225	0.493613	−	0.869682i	$-0.335676\pi$
$7$	18.2837	0.987225	0.493613	+	0.869682i	$0.335676\pi$
$8$	5.92796	0.261981
$9$	9.00000	0.333333
$10$	−1.86880	−0.0590967
$11$	0	0
$11$	0	0
$12$	−23.5809	−0.567269
$13$	83.5276	1.78203	0.891015	−	0.453975i	$-0.149994\pi$
$13$	83.5276	1.78203	0.891015	+	0.453975i	$0.149994\pi$
$14$	−6.83371	−0.130456
$15$	15.0000	0.258199
$16$	60.6668	0.947919
$17$	98.0730	1.39919	0.699594	−	0.714540i	$-0.253364\pi$
$17$	98.0730	1.39919	0.699594	+	0.714540i	$0.253364\pi$
$18$	−3.36385	−0.0440481
$19$	94.2981	1.13860	0.569302	−	0.822128i	$-0.307214\pi$
$19$	94.2981	1.13860	0.569302	+	0.822128i	$0.307214\pi$
$20$	−39.3015	−0.439404
$21$	54.8510	0.569975
$22$	0	0
$23$	65.7953	0.596490	0.298245	−	0.954489i	$-0.403599\pi$
$23$	65.7953	0.596490	0.298245	+	0.954489i	$0.403599\pi$
$24$	17.7839	0.151255
$25$	25.0000	0.200000
$26$	−31.2193	−0.235485
$27$	27.0000	0.192450
$28$	−143.715	−0.969986
$29$	120.655	0.772587	0.386293	−	0.922376i	$-0.373755\pi$
$29$	120.655	0.772587	0.386293	+	0.922376i	$0.373755\pi$
$30$	−5.60641	−0.0341195
$31$	130.246	0.754609	0.377304	−	0.926089i	$-0.376851\pi$
$31$	130.246	0.754609	0.377304	+	0.926089i	$0.376851\pi$
$32$	−70.0985	−0.387243
$33$	0	0
$34$	−36.6558	−0.184895
$35$	91.4183	0.441501
$36$	−70.7427	−0.327513
$37$	−386.662	−1.71802	−0.859012	−	0.511955i	$-0.828921\pi$
$37$	−386.662	−1.71802	−0.859012	+	0.511955i	$0.828921\pi$
$38$	−35.2449	−0.150460
$39$	250.583	1.02886
$40$	29.6398	0.117162
$41$	−354.489	−1.35029	−0.675144	−	0.737686i	$-0.735919\pi$
$41$	−354.489	−1.35029	−0.675144	+	0.737686i	$0.735919\pi$
$42$	−20.5011	−0.0753189
$43$	−273.737	−0.970804	−0.485402	−	0.874291i	$-0.661327\pi$
$43$	−273.737	−0.970804	−0.485402	+	0.874291i	$0.661327\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	−24.5917	−0.0788227
$47$	548.956	1.70369	0.851846	−	0.523792i	$-0.175483\pi$
$47$	548.956	1.70369	0.851846	+	0.523792i	$0.175483\pi$
$48$	182.000	0.547281
$49$	−8.70748	−0.0253862
$50$	−9.34401	−0.0264289
$51$	294.219	0.807822
$52$	−656.552	−1.75091
$53$	−660.563	−1.71199	−0.855993	−	0.516987i	$-0.827054\pi$
$53$	−660.563	−1.71199	−0.855993	+	0.516987i	$0.827054\pi$
$54$	−10.0915	−0.0254312
$55$	0	0
$56$	108.385	0.258634
$57$	282.894	0.657373
$58$	−45.0960	−0.102093
$59$	283.388	0.625321	0.312661	−	0.949865i	$-0.398780\pi$
$59$	283.388	0.625321	0.312661	+	0.949865i	$0.398780\pi$
$60$	−117.905	−0.253690
$61$	−437.501	−0.918300	−0.459150	−	0.888359i	$-0.651846\pi$
$61$	−437.501	−0.918300	−0.459150	+	0.888359i	$0.651846\pi$
$62$	−48.6808	−0.0997172
$63$	164.553	0.329075
$64$	−459.134	−0.896747
$65$	417.638	0.796948
$66$	0	0
$67$	617.750	1.12642	0.563210	−	0.826314i	$-0.309566\pi$
$67$	617.750	1.12642	0.563210	+	0.826314i	$0.309566\pi$
$68$	−770.884	−1.37476
$69$	197.386	0.344384
$70$	−34.1686	−0.0583418
$71$	578.571	0.967095	0.483547	−	0.875318i	$-0.339348\pi$
$71$	578.571	0.967095	0.483547	+	0.875318i	$0.339348\pi$
$72$	53.3516	0.0873270
$73$	960.165	1.53944	0.769718	−	0.638384i	$-0.220397\pi$
$73$	960.165	1.53944	0.769718	+	0.638384i	$0.220397\pi$
$74$	144.519	0.227027
$75$	75.0000	0.115470
$76$	−741.212	−1.11872
$77$	0	0
$78$	−93.6579	−0.135957
$79$	−1008.95	−1.43691	−0.718456	−	0.695572i	$-0.755151\pi$
$79$	−1008.95	−1.43691	−0.718456	+	0.695572i	$0.755151\pi$
$80$	303.334	0.423922
$81$	81.0000	0.111111
$82$	132.494	0.178433
$83$	958.535	1.26762	0.633812	−	0.773487i	$-0.281489\pi$
$83$	958.535	1.26762	0.633812	+	0.773487i	$0.281489\pi$
$84$	−431.146	−0.560022
$85$	490.365	0.625736
$86$	102.312	0.128286
$87$	361.964	0.446053
$88$	0	0
$89$	−1394.59	−1.66097	−0.830483	−	0.557044i	$-0.811935\pi$
$89$	−1394.59	−1.66097	−0.830483	+	0.557044i	$0.811935\pi$
$90$	−16.8192	−0.0196989
$91$	1527.19	1.75926
$92$	−517.171	−0.586074
$93$	390.738	0.435673
$94$	−205.178	−0.225133
$95$	471.491	0.509199
$96$	−210.295	−0.223575
$97$	−63.3140	−0.0662738	−0.0331369	−	0.999451i	$-0.510550\pi$
$97$	−63.3140	−0.0662738	−0.0331369	+	0.999451i	$0.510550\pi$
$98$	3.25451	0.00335465
$99$	0	0
$100$	−196.508	−0.196508
$101$	−179.382	−0.176724	−0.0883622	−	0.996088i	$-0.528163\pi$
$101$	−179.382	−0.176724	−0.0883622	+	0.996088i	$0.528163\pi$
$102$	−109.968	−0.106749
$103$	1099.15	1.05148	0.525740	−	0.850645i	$-0.323789\pi$
$103$	1099.15	1.05148	0.525740	+	0.850645i	$0.323789\pi$
$104$	495.148	0.466858
$105$	274.255	0.254900
$106$	246.892	0.226229
$107$	796.298	0.719449	0.359724	−	0.933059i	$-0.382871\pi$
$107$	796.298	0.719449	0.359724	+	0.933059i	$0.382871\pi$
$108$	−212.228	−0.189090
$109$	−1218.66	−1.07088	−0.535442	−	0.844572i	$-0.679855\pi$
$109$	−1218.66	−1.07088	−0.535442	+	0.844572i	$0.679855\pi$
$110$	0	0
$111$	−1159.99	−0.991902
$112$	1109.21	0.935809
$113$	−1227.93	−1.02225	−0.511123	−	0.859508i	$-0.670770\pi$
$113$	−1227.93	−1.02225	−0.511123	+	0.859508i	$0.670770\pi$
$114$	−105.735	−0.0868681
$115$	328.976	0.266758
$116$	−948.383	−0.759096
$117$	751.748	0.594010
$118$	−105.919	−0.0826326
$119$	1793.13	1.38131
$120$	88.9193	0.0676432
$121$	0	0
$122$	163.521	0.121348
$123$	−1063.47	−0.779589
$124$	−1023.77	−0.741431
$125$	125.000	0.0894427
$126$	−61.5034	−0.0434854
$127$	−455.661	−0.318373	−0.159186	−	0.987249i	$-0.550887\pi$
$127$	−455.661	−0.318373	−0.159186	+	0.987249i	$0.550887\pi$
$128$	732.394	0.505743
$129$	−821.212	−0.560494
$130$	−156.097	−0.105312
$131$	505.566	0.337187	0.168594	−	0.985686i	$-0.446077\pi$
$131$	505.566	0.337187	0.168594	+	0.985686i	$0.446077\pi$
$132$	0	0
$133$	1724.12	1.12406
$134$	−230.890	−0.148850
$135$	135.000	0.0860663
$136$	581.373	0.366561
$137$	−755.472	−0.471127	−0.235563	−	0.971859i	$-0.575694\pi$
$137$	−755.472	−0.471127	−0.235563	+	0.971859i	$0.575694\pi$
$138$	−73.7751	−0.0455083
$139$	−857.604	−0.523317	−0.261658	−	0.965161i	$-0.584269\pi$
$139$	−857.604	−0.523317	−0.261658	+	0.965161i	$0.584269\pi$
$140$	−718.576	−0.433791
$141$	1646.87	0.983627
$142$	−216.247	−0.127796
$143$	0	0
$144$	546.001	0.315973
$145$	603.274	0.345511
$146$	−358.872	−0.203428
$147$	−26.1224	−0.0146568
$148$	3039.28	1.68802
$149$	−2835.66	−1.55910	−0.779551	−	0.626339i	$-0.784553\pi$
$149$	−2835.66	−1.55910	−0.779551	+	0.626339i	$0.784553\pi$
$150$	−28.0320	−0.0152587
$151$	−1810.81	−0.975905	−0.487952	−	0.872870i	$-0.662256\pi$
$151$	−1810.81	−0.975905	−0.487952	+	0.872870i	$0.662256\pi$
$152$	558.995	0.298293
$153$	882.657	0.466396
$154$	0	0
$155$	651.230	0.337471
$156$	−1969.66	−1.01089
$157$	1995.16	1.01421	0.507106	−	0.861884i	$-0.330715\pi$
$157$	1995.16	1.01421	0.507106	+	0.861884i	$0.330715\pi$
$158$	377.107	0.189880
$159$	−1981.69	−0.988416
$160$	−350.492	−0.173180
$161$	1202.98	0.588870
$162$	−30.2746	−0.0146827
$163$	−1317.99	−0.633332	−0.316666	−	0.948537i	$-0.602563\pi$
$163$	−1317.99	−0.633332	−0.316666	+	0.948537i	$0.602563\pi$
$164$	2786.39	1.32671
$165$	0	0
$166$	−358.262	−0.167509
$167$	−2982.63	−1.38205	−0.691026	−	0.722830i	$-0.742841\pi$
$167$	−2982.63	−1.38205	−0.691026	+	0.722830i	$0.742841\pi$
$168$	325.154	0.149323
$169$	4779.85	2.17563
$170$	−183.279	−0.0826875
$171$	848.683	0.379535
$172$	2151.66	0.953851
$173$	−212.362	−0.0933269	−0.0466635	−	0.998911i	$-0.514859\pi$
$173$	−212.362	−0.0933269	−0.0466635	+	0.998911i	$0.514859\pi$
$174$	−135.288	−0.0589434
$175$	457.092	0.197445
$176$	0	0
$177$	850.163	0.361029
$178$	521.242	0.219487
$179$	318.994	0.133200	0.0665998	−	0.997780i	$-0.478785\pi$
$179$	318.994	0.133200	0.0665998	+	0.997780i	$0.478785\pi$
$180$	−353.714	−0.146468
$181$	−1141.87	−0.468922	−0.234461	−	0.972126i	$-0.575332\pi$
$181$	−1141.87	−0.468922	−0.234461	+	0.972126i	$0.575332\pi$
$182$	−570.804	−0.232477
$183$	−1312.50	−0.530181
$184$	390.032	0.156269
$185$	−1933.31	−0.768324
$186$	−146.042	−0.0575718
$187$	0	0
$188$	−4314.96	−1.67394
$189$	493.659	0.189992
$190$	−176.225	−0.0672878
$191$	1303.04	0.493637	0.246818	−	0.969062i	$-0.420615\pi$
$191$	1303.04	0.493637	0.246818	+	0.969062i	$0.420615\pi$
$192$	−1377.40	−0.517737
$193$	468.211	0.174625	0.0873124	−	0.996181i	$-0.472172\pi$
$193$	468.211	0.174625	0.0873124	+	0.996181i	$0.472172\pi$
$194$	23.6643	0.00875771
$195$	1252.91	0.460118
$196$	68.4434	0.0249429
$197$	3444.95	1.24590	0.622951	−	0.782261i	$-0.285934\pi$
$197$	3444.95	1.24590	0.622951	+	0.782261i	$0.285934\pi$
$198$	0	0
$199$	−1180.48	−0.420514	−0.210257	−	0.977646i	$-0.567430\pi$
$199$	−1180.48	−0.420514	−0.210257	+	0.977646i	$0.567430\pi$
$200$	148.199	0.0523962
$201$	1853.25	0.650339
$202$	67.0459	0.0233531
$203$	2206.01	0.762717
$204$	−2312.65	−0.793716
$205$	−1772.44	−0.603867
$206$	−410.819	−0.138947
$207$	592.158	0.198830
$208$	5067.35	1.68922
$209$	0	0
$210$	−102.506	−0.0336836
$211$	−223.450	−0.0729049	−0.0364525	−	0.999335i	$-0.511606\pi$
$211$	−223.450	−0.0729049	−0.0364525	+	0.999335i	$0.511606\pi$
$212$	5192.22	1.68209
$213$	1735.71	0.558352
$214$	−297.625	−0.0950711
$215$	−1368.69	−0.434157
$216$	160.055	0.0504183
$217$	2381.37	0.744969
$218$	455.486	0.141511
$219$	2880.50	0.888794
$220$	0	0
$221$	8191.80	2.49339
$222$	433.557	0.131074
$223$	5337.99	1.60295	0.801476	−	0.598027i	$-0.204049\pi$
$223$	5337.99	1.60295	0.801476	+	0.598027i	$0.204049\pi$
$224$	−1281.66	−0.382296
$225$	225.000	0.0666667
$226$	458.951	0.135084
$227$	113.876	0.0332961	0.0166480	−	0.999861i	$-0.494701\pi$
$227$	113.876	0.0332961	0.0166480	+	0.999861i	$0.494701\pi$
$228$	−2223.64	−0.645894
$229$	−2121.55	−0.612209	−0.306104	−	0.951998i	$-0.599026\pi$
$229$	−2121.55	−0.612209	−0.306104	+	0.951998i	$0.599026\pi$
$230$	−122.958	−0.0352506
$231$	0	0
$232$	715.236	0.202403
$233$	−1264.80	−0.355622	−0.177811	−	0.984065i	$-0.556902\pi$
$233$	−1264.80	−0.355622	−0.177811	+	0.984065i	$0.556902\pi$
$234$	−280.974	−0.0784950
$235$	2744.78	0.761914
$236$	−2227.51	−0.614402
$237$	−3026.86	−0.829602
$238$	−670.203	−0.182533
$239$	−477.832	−0.129324	−0.0646619	−	0.997907i	$-0.520597\pi$
$239$	−477.832	−0.129324	−0.0646619	+	0.997907i	$0.520597\pi$
$240$	910.002	0.244752
$241$	628.155	0.167896	0.0839481	−	0.996470i	$-0.473247\pi$
$241$	628.155	0.167896	0.0839481	+	0.996470i	$0.473247\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	3438.89	0.902264
$245$	−43.5374	−0.0113531
$246$	397.482	0.103018
$247$	7876.49	2.02903
$248$	772.092	0.197693
$249$	2875.60	0.731863
$250$	−46.7201	−0.0118193
$251$	3824.06	0.961645	0.480822	−	0.876818i	$-0.340338\pi$
$251$	3824.06	0.961645	0.480822	+	0.876818i	$0.340338\pi$
$252$	−1293.44	−0.323329
$253$	0	0
$254$	170.308	0.0420712
$255$	1471.10	0.361269
$256$	3399.33	0.829916
$257$	6243.62	1.51543	0.757717	−	0.652583i	$-0.226315\pi$
$257$	6243.62	1.51543	0.757717	+	0.652583i	$0.226315\pi$
$258$	306.937	0.0740661
$259$	−7069.61	−1.69608
$260$	−3282.76	−0.783031
$261$	1085.89	0.257529
$262$	−188.961	−0.0445573
$263$	3152.33	0.739091	0.369546	−	0.929213i	$-0.379513\pi$
$263$	3152.33	0.739091	0.369546	+	0.929213i	$0.379513\pi$
$264$	0	0
$265$	−3302.81	−0.765624
$266$	−644.406	−0.148538
$267$	−4183.76	−0.958959
$268$	−4855.70	−1.10675
$269$	−4369.52	−0.990387	−0.495194	−	0.868783i	$-0.664903\pi$
$269$	−4369.52	−0.990387	−0.495194	+	0.868783i	$0.664903\pi$
$270$	−50.4577	−0.0113732
$271$	3626.88	0.812979	0.406489	−	0.913655i	$-0.366753\pi$
$271$	3626.88	0.812979	0.406489	+	0.913655i	$0.366753\pi$
$272$	5949.78	1.32632
$273$	4581.57	1.01571
$274$	282.366	0.0622567
$275$	0	0
$276$	−1551.51	−0.338370
$277$	626.932	0.135988	0.0679940	−	0.997686i	$-0.478340\pi$
$277$	626.932	0.135988	0.0679940	+	0.997686i	$0.478340\pi$
$278$	320.539	0.0691533
$279$	1172.21	0.251536
$280$	541.924	0.115665
$281$	−7011.47	−1.48850	−0.744252	−	0.667899i	$-0.767194\pi$
$281$	−7011.47	−1.48850	−0.744252	+	0.667899i	$0.767194\pi$
$282$	−615.535	−0.129981
$283$	−3460.19	−0.726809	−0.363405	−	0.931631i	$-0.618386\pi$
$283$	−3460.19	−0.726809	−0.363405	+	0.931631i	$0.618386\pi$
$284$	−4547.74	−0.950207
$285$	1414.47	0.293986
$286$	0	0
$287$	−6481.35	−1.33304
$288$	−630.886	−0.129081
$289$	4705.32	0.957729
$290$	−225.480	−0.0456574
$291$	−189.942	−0.0382632
$292$	−7547.19	−1.51255
$293$	9546.72	1.90350	0.951749	−	0.306876i	$-0.0992838\pi$
$293$	9546.72	1.90350	0.951749	+	0.306876i	$0.0992838\pi$
$294$	9.76354	0.00193681
$295$	1416.94	0.279652
$296$	−2292.12	−0.450090
$297$	0	0
$298$	1059.86	0.206026
$299$	5495.72	1.06296
$300$	−589.523	−0.113454
$301$	−5004.92	−0.958402
$302$	676.809	0.128960
$303$	−538.146	−0.102032
$304$	5720.76	1.07930
$305$	−2187.51	−0.410676
$306$	−329.903	−0.0616316
$307$	−10005.7	−1.86012	−0.930060	−	0.367407i	$-0.880246\pi$
$307$	−10005.7	−1.86012	−0.930060	+	0.367407i	$0.880246\pi$
$308$	0	0
$309$	3297.45	0.607072
$310$	−243.404	−0.0445949
$311$	−3514.34	−0.640771	−0.320386	−	0.947287i	$-0.603813\pi$
$311$	−3514.34	−0.640771	−0.320386	+	0.947287i	$0.603813\pi$
$312$	1485.44	0.269541
$313$	1396.29	0.252150	0.126075	−	0.992021i	$-0.459762\pi$
$313$	1396.29	0.252150	0.126075	+	0.992021i	$0.459762\pi$
$314$	−745.713	−0.134022
$315$	822.765	0.147167
$316$	7930.67	1.41182
$317$	−649.841	−0.115138	−0.0575690	−	0.998342i	$-0.518335\pi$
$317$	−649.841	−0.115138	−0.0575690	+	0.998342i	$0.518335\pi$
$318$	740.677	0.130614
$319$	0	0
$320$	−2295.67	−0.401037
$321$	2388.89	0.415374
$322$	−449.626	−0.0778158
$323$	9248.10	1.59312
$324$	−636.685	−0.109171
$325$	2088.19	0.356406
$326$	492.613	0.0836912
$327$	−3655.97	−0.618275
$328$	−2101.39	−0.353750
$329$	10036.9	1.68193
$330$	0	0
$331$	−7968.14	−1.32317	−0.661584	−	0.749871i	$-0.730116\pi$
$331$	−7968.14	−1.32317	−0.661584	+	0.749871i	$0.730116\pi$
$332$	−7534.37	−1.24549
$333$	−3479.96	−0.572675
$334$	1114.79	0.182630
$335$	3088.75	0.503750
$336$	3327.63	0.540290
$337$	4214.52	0.681245	0.340623	−	0.940200i	$-0.389362\pi$
$337$	4214.52	0.681245	0.340623	+	0.940200i	$0.389362\pi$
$338$	−1786.52	−0.287497
$339$	−3683.78	−0.590194
$340$	−3854.42	−0.614809
$341$	0	0
$342$	−317.204	−0.0501533
$343$	−6430.50	−1.01229
$344$	−1622.70	−0.254332
$345$	986.929	0.154013
$346$	79.3724	0.0123326
$347$	−978.377	−0.151360	−0.0756801	−	0.997132i	$-0.524113\pi$
$347$	−978.377	−0.151360	−0.0756801	+	0.997132i	$0.524113\pi$
$348$	−2845.15	−0.438264
$349$	16.2172	0.00248736	0.00124368	−	0.999999i	$-0.499604\pi$
$349$	16.2172	0.00248736	0.00124368	+	0.999999i	$0.499604\pi$
$350$	−170.843	−0.0260912
$351$	2255.24	0.342952
$352$	0	0
$353$	9876.02	1.48909	0.744543	−	0.667574i	$-0.232667\pi$
$353$	9876.02	1.48909	0.744543	+	0.667574i	$0.232667\pi$
$354$	−317.757	−0.0477080
$355$	2892.85	0.432498
$356$	10961.9	1.63196
$357$	5379.40	0.797502
$358$	−119.227	−0.0176016
$359$	−7161.66	−1.05286	−0.526432	−	0.850218i	$-0.676470\pi$
$359$	−7161.66	−1.05286	−0.526432	+	0.850218i	$0.676470\pi$
$360$	266.758	0.0390538
$361$	2033.14	0.296419
$362$	426.788	0.0619654
$363$	0	0
$364$	−12004.2	−1.72854
$365$	4800.83	0.688457
$366$	490.562	0.0700604
$367$	1713.76	0.243754	0.121877	−	0.992545i	$-0.461109\pi$
$367$	1713.76	0.243754	0.121877	+	0.992545i	$0.461109\pi$
$368$	3991.59	0.565424
$369$	−3190.40	−0.450096
$370$	722.596	0.101530
$371$	−12077.5	−1.69012
$372$	−3071.32	−0.428066
$373$	−5206.61	−0.722756	−0.361378	−	0.932419i	$-0.617694\pi$
$373$	−5206.61	−0.722756	−0.361378	+	0.932419i	$0.617694\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	3254.19	0.446335
$377$	10078.0	1.37677
$378$	−184.510	−0.0251063
$379$	−7219.74	−0.978504	−0.489252	−	0.872142i	$-0.662730\pi$
$379$	−7219.74	−0.978504	−0.489252	+	0.872142i	$0.662730\pi$
$380$	−3706.06	−0.500307
$381$	−1366.98	−0.183813
$382$	−487.025	−0.0652313
$383$	−856.690	−0.114295	−0.0571473	−	0.998366i	$-0.518200\pi$
$383$	−856.690	−0.114295	−0.0571473	+	0.998366i	$0.518200\pi$
$384$	2197.18	0.291991
$385$	0	0
$386$	−174.999	−0.0230757
$387$	−2463.64	−0.323601
$388$	497.667	0.0651165
$389$	−11928.9	−1.55480	−0.777400	−	0.629007i	$-0.783462\pi$
$389$	−11928.9	−1.55480	−0.777400	+	0.629007i	$0.783462\pi$
$390$	−468.290	−0.0608020
$391$	6452.74	0.834602
$392$	−51.6176	−0.00665071
$393$	1516.70	0.194675
$394$	−1287.59	−0.164639
$395$	−5044.76	−0.642607
$396$	0	0
$397$	−4892.99	−0.618570	−0.309285	−	0.950969i	$-0.600090\pi$
$397$	−4892.99	−0.618570	−0.309285	+	0.950969i	$0.600090\pi$
$398$	441.218	0.0555685
$399$	5172.35	0.648976
$400$	1516.67	0.189584
$401$	7444.84	0.927127	0.463563	−	0.886064i	$-0.346571\pi$
$401$	7444.84	0.927127	0.463563	+	0.886064i	$0.346571\pi$
$402$	−692.671	−0.0859386
$403$	10879.1	1.34473
$404$	1410.00	0.173639
$405$	405.000	0.0496904
$406$	−824.520	−0.100789
$407$	0	0
$408$	1744.12	0.211634
$409$	−8679.87	−1.04937	−0.524685	−	0.851297i	$-0.675817\pi$
$409$	−8679.87	−1.04937	−0.524685	+	0.851297i	$0.675817\pi$
$410$	662.469	0.0797976
$411$	−2266.42	−0.272005
$412$	−8639.65	−1.03312
$413$	5181.37	0.617333
$414$	−221.325	−0.0262742
$415$	4792.67	0.566899
$416$	−5855.16	−0.690079
$417$	−2572.81	−0.302137
$418$	0	0
$419$	−14959.4	−1.74419	−0.872093	−	0.489341i	$-0.837238\pi$
$419$	−14959.4	−1.74419	−0.872093	+	0.489341i	$0.837238\pi$
$420$	−2155.73	−0.250449
$421$	15880.3	1.83838	0.919188	−	0.393818i	$-0.128846\pi$
$421$	15880.3	1.83838	0.919188	+	0.393818i	$0.128846\pi$
$422$	83.5168	0.00963397
$423$	4940.61	0.567897
$424$	−3915.79	−0.448508
$425$	2451.83	0.279838
$426$	−648.741	−0.0737831
$427$	−7999.13	−0.906569
$428$	−6259.14	−0.706886
$429$	0	0
$430$	511.561	0.0573713
$431$	−3614.05	−0.403904	−0.201952	−	0.979395i	$-0.564728\pi$
$431$	−3614.05	−0.403904	−0.201952	+	0.979395i	$0.564728\pi$
$432$	1638.00	0.182427
$433$	1736.20	0.192693	0.0963467	−	0.995348i	$-0.469284\pi$
$433$	1736.20	0.192693	0.0963467	+	0.995348i	$0.469284\pi$
$434$	−890.064	−0.0984434
$435$	1809.82	0.199481
$436$	9579.02	1.05218
$437$	6204.37	0.679166
$438$	−1076.62	−0.117449
$439$	473.123	0.0514372	0.0257186	−	0.999669i	$-0.491813\pi$
$439$	473.123	0.0514372	0.0257186	+	0.999669i	$0.491813\pi$
$440$	0	0
$441$	−78.3673	−0.00846208
$442$	−3061.77	−0.329488
$443$	3485.85	0.373855	0.186927	−	0.982374i	$-0.440147\pi$
$443$	3485.85	0.373855	0.186927	+	0.982374i	$0.440147\pi$
$444$	9117.85	0.974581
$445$	−6972.94	−0.742807
$446$	−1995.13	−0.211821
$447$	−8506.98	−0.900148
$448$	−8394.66	−0.885291
$449$	11006.2	1.15683	0.578413	−	0.815744i	$-0.303672\pi$
$449$	11006.2	1.15683	0.578413	+	0.815744i	$0.303672\pi$
$450$	−84.0961	−0.00880962
$451$	0	0
$452$	9651.89	1.00440
$453$	−5432.43	−0.563439
$454$	−42.5623	−0.00439988
$455$	7635.95	0.786767
$456$	1676.99	0.172219
$457$	9432.42	0.965492	0.482746	−	0.875760i	$-0.339639\pi$
$457$	9432.42	0.965492	0.482746	+	0.875760i	$0.339639\pi$
$458$	792.951	0.0808999
$459$	2647.97	0.269274
$460$	−2585.85	−0.262100
$461$	5591.53	0.564910	0.282455	−	0.959281i	$-0.408851\pi$
$461$	5591.53	0.564910	0.282455	+	0.959281i	$0.408851\pi$
$462$	0	0
$463$	−4857.67	−0.487592	−0.243796	−	0.969827i	$-0.578393\pi$
$463$	−4857.67	−0.487592	−0.243796	+	0.969827i	$0.578393\pi$
$464$	7319.73	0.732349
$465$	1953.69	0.194839
$466$	472.733	0.0469935
$467$	19190.1	1.90153	0.950763	−	0.309920i	$-0.100302\pi$
$467$	19190.1	1.90153	0.950763	+	0.309920i	$0.100302\pi$
$468$	−5908.97	−0.583637
$469$	11294.7	1.11203
$470$	−1025.89	−0.100683
$471$	5985.49	0.585556
$472$	1679.91	0.163822
$473$	0	0
$474$	1131.32	0.109627
$475$	2357.45	0.227721
$476$	−14094.6	−1.35719
$477$	−5945.07	−0.570662
$478$	178.595	0.0170894
$479$	7027.31	0.670326	0.335163	−	0.942160i	$-0.391209\pi$
$479$	7027.31	0.670326	0.335163	+	0.942160i	$0.391209\pi$
$480$	−1051.48	−0.0999858
$481$	−32297.0	−3.06157
$482$	−234.779	−0.0221865
$483$	3608.94	0.339984
$484$	0	0
$485$	−316.570	−0.0296386
$486$	−90.8238	−0.00847706
$487$	−11069.5	−1.02999	−0.514995	−	0.857193i	$-0.672206\pi$
$487$	−11069.5	−1.02999	−0.514995	+	0.857193i	$0.672206\pi$
$488$	−2593.49	−0.240577
$489$	−3953.97	−0.365654
$490$	16.2726	0.00150024
$491$	3498.85	0.321590	0.160795	−	0.986988i	$-0.448594\pi$
$491$	3498.85	0.321590	0.160795	+	0.986988i	$0.448594\pi$
$492$	8359.16	0.765976
$493$	11833.0	1.08099
$494$	−2943.92	−0.268124
$495$	0	0
$496$	7901.60	0.715307
$497$	10578.4	0.954740
$498$	−1074.79	−0.0967116
$499$	−9378.44	−0.841356	−0.420678	−	0.907210i	$-0.638208\pi$
$499$	−9378.44	−0.841356	−0.420678	+	0.907210i	$0.638208\pi$
$500$	−982.538	−0.0878809
$501$	−8947.89	−0.797928
$502$	−1429.28	−0.127076
$503$	12948.7	1.14782	0.573912	−	0.818917i	$-0.305425\pi$
$503$	12948.7	1.14782	0.573912	+	0.818917i	$0.305425\pi$
$504$	975.463	0.0862115
$505$	−896.910	−0.0790336
$506$	0	0
$507$	14339.6	1.25610
$508$	3581.63	0.312813
$509$	−3798.99	−0.330819	−0.165410	−	0.986225i	$-0.552895\pi$
$509$	−3798.99	−0.330819	−0.165410	+	0.986225i	$0.552895\pi$
$510$	−549.838	−0.0477396
$511$	17555.3	1.51977
$512$	−7129.69	−0.615412
$513$	2546.05	0.219124
$514$	−2333.62	−0.200256
$515$	5495.75	0.470236
$516$	6454.98	0.550706
$517$	0	0
$518$	2642.34	0.224127
$519$	−637.085	−0.0538823
$520$	2475.74	0.208785
$521$	10109.8	0.850128	0.425064	−	0.905163i	$-0.360252\pi$
$521$	10109.8	0.850128	0.425064	+	0.905163i	$0.360252\pi$
$522$	−405.864	−0.0340310
$523$	6102.75	0.510238	0.255119	−	0.966910i	$-0.417885\pi$
$523$	6102.75	0.510238	0.255119	+	0.966910i	$0.417885\pi$
$524$	−3973.90	−0.331299
$525$	1371.28	0.113995
$526$	−1178.22	−0.0976667
$527$	12773.6	1.05584
$528$	0	0
$529$	−7837.98	−0.644200
$530$	1234.46	0.101173
$531$	2550.49	0.208440
$532$	−13552.1	−1.10443
$533$	−29609.6	−2.40625
$534$	1563.73	0.126721
$535$	3981.49	0.321747
$536$	3661.99	0.295101
$537$	956.982	0.0769028
$538$	1633.15	0.130874
$539$	0	0
$540$	−1061.14	−0.0845634
$541$	−757.373	−0.0601886	−0.0300943	−	0.999547i	$-0.509581\pi$
$541$	−757.373	−0.0601886	−0.0300943	+	0.999547i	$0.509581\pi$
$542$	−1355.58	−0.107431
$543$	−3425.62	−0.270732
$544$	−6874.77	−0.541826
$545$	−6093.29	−0.478914
$546$	−1712.41	−0.134221
$547$	4637.82	0.362521	0.181260	−	0.983435i	$-0.441982\pi$
$547$	4637.82	0.362521	0.181260	+	0.983435i	$0.441982\pi$
$548$	5938.24	0.462900
$549$	−3937.51	−0.306100
$550$	0	0
$551$	11377.5	0.879670
$552$	1170.09	0.0902220
$553$	−18447.4	−1.41856
$554$	−234.322	−0.0179700
$555$	−5799.94	−0.443592
$556$	6741.03	0.514179
$557$	13464.1	1.02422	0.512111	−	0.858919i	$-0.328864\pi$
$557$	13464.1	1.02422	0.512111	+	0.858919i	$0.328864\pi$
$558$	−438.127	−0.0332391
$559$	−22864.6	−1.73000
$560$	5546.06	0.418507
$561$	0	0
$562$	2620.61	0.196697
$563$	−17077.4	−1.27838	−0.639190	−	0.769049i	$-0.720730\pi$
$563$	−17077.4	−1.27838	−0.639190	+	0.769049i	$0.720730\pi$
$564$	−12944.9	−0.966451
$565$	−6139.64	−0.457162
$566$	1293.28	0.0960437
$567$	1480.98	0.109692
$568$	3429.74	0.253361
$569$	−11163.1	−0.822465	−0.411233	−	0.911530i	$-0.634902\pi$
$569$	−11163.1	−0.822465	−0.411233	+	0.911530i	$0.634902\pi$
$570$	−528.674	−0.0388486
$571$	−733.675	−0.0537712	−0.0268856	−	0.999639i	$-0.508559\pi$
$571$	−733.675	−0.0537712	−0.0268856	+	0.999639i	$0.508559\pi$
$572$	0	0
$573$	3909.12	0.285001
$574$	2422.47	0.176153
$575$	1644.88	0.119298
$576$	−4132.21	−0.298916
$577$	8345.82	0.602151	0.301075	−	0.953600i	$-0.402654\pi$
$577$	8345.82	0.602151	0.301075	+	0.953600i	$0.402654\pi$
$578$	−1758.66	−0.126558
$579$	1404.63	0.100820
$580$	−4741.91	−0.339478
$581$	17525.5	1.25143
$582$	70.9928	0.00505627
$583$	0	0
$584$	5691.82	0.403303
$585$	3758.74	0.265649
$586$	−3568.19	−0.251537
$587$	−24731.2	−1.73895	−0.869476	−	0.493975i	$-0.835543\pi$
$587$	−24731.2	−1.73895	−0.869476	+	0.493975i	$0.835543\pi$
$588$	205.330	0.0144008
$589$	12282.0	0.859200
$590$	−529.596	−0.0369544
$591$	10334.9	0.719321
$592$	−23457.6	−1.62855
$593$	−3868.75	−0.267910	−0.133955	−	0.990987i	$-0.542768\pi$
$593$	−3868.75	−0.267910	−0.133955	+	0.990987i	$0.542768\pi$
$594$	0	0
$595$	8965.67	0.617743
$596$	22289.1	1.53188
$597$	−3541.45	−0.242784
$598$	−2054.08	−0.140464
$599$	15180.8	1.03551	0.517755	−	0.855529i	$-0.326768\pi$
$599$	15180.8	1.03551	0.517755	+	0.855529i	$0.326768\pi$
$600$	444.597	0.0302510
$601$	−14402.5	−0.977522	−0.488761	−	0.872418i	$-0.662551\pi$
$601$	−14402.5	−0.977522	−0.488761	+	0.872418i	$0.662551\pi$
$602$	1870.64	0.126647
$603$	5559.75	0.375473
$604$	14233.5	0.958863
$605$	0	0
$606$	201.138	0.0134829
$607$	11557.5	0.772822	0.386411	−	0.922327i	$-0.373715\pi$
$607$	11557.5	0.772822	0.386411	+	0.922327i	$0.373715\pi$
$608$	−6610.16	−0.440917
$609$	6618.03	0.440355
$610$	817.603	0.0542685
$611$	45853.0	3.03603
$612$	−6937.95	−0.458252
$613$	−7956.02	−0.524210	−0.262105	−	0.965039i	$-0.584417\pi$
$613$	−7956.02	−0.524210	−0.262105	+	0.965039i	$0.584417\pi$
$614$	3739.75	0.245804
$615$	−5317.33	−0.348643
$616$	0	0
$617$	12760.7	0.832619	0.416309	−	0.909223i	$-0.363323\pi$
$617$	12760.7	0.832619	0.416309	+	0.909223i	$0.363323\pi$
$618$	−1232.46	−0.0802211
$619$	−27103.5	−1.75991	−0.879953	−	0.475061i	$-0.842426\pi$
$619$	−27103.5	−1.75991	−0.879953	+	0.475061i	$0.842426\pi$
$620$	−5118.86	−0.331578
$621$	1776.47	0.114795
$622$	1313.52	0.0846743
$623$	−25498.2	−1.63975
$624$	15202.0	0.975271
$625$	625.000	0.0400000
$626$	−521.879	−0.0333202
$627$	0	0
$628$	−15682.6	−0.996502
$629$	−37921.2	−2.40384
$630$	−307.517	−0.0194473
$631$	21856.3	1.37890	0.689449	−	0.724334i	$-0.257853\pi$
$631$	21856.3	1.37890	0.689449	+	0.724334i	$0.257853\pi$
$632$	−5981.03	−0.376444
$633$	−670.350	−0.0420917
$634$	242.885	0.0152148
$635$	−2278.30	−0.142381
$636$	15576.7	0.971156
$637$	−727.315	−0.0452390
$638$	0	0
$639$	5207.14	0.322365
$640$	3661.97	0.226175
$641$	30723.5	1.89314	0.946571	−	0.322495i	$-0.104522\pi$
$641$	30723.5	1.89314	0.946571	+	0.322495i	$0.104522\pi$
$642$	−892.874	−0.0548893
$643$	5303.56	0.325275	0.162638	−	0.986686i	$-0.448000\pi$
$643$	5303.56	0.325275	0.162638	+	0.986686i	$0.448000\pi$
$644$	−9455.78	−0.578587
$645$	−4106.06	−0.250660
$646$	−3456.58	−0.210522
$647$	6370.06	0.387068	0.193534	−	0.981094i	$-0.438005\pi$
$647$	6370.06	0.387068	0.193534	+	0.981094i	$0.438005\pi$
$648$	480.164	0.0291090
$649$	0	0
$650$	−780.483	−0.0470970
$651$	7144.12	0.430108
$652$	10359.8	0.622272
$653$	−8358.64	−0.500917	−0.250459	−	0.968127i	$-0.580581\pi$
$653$	−8358.64	−0.500917	−0.250459	+	0.968127i	$0.580581\pi$
$654$	1366.46	0.0817015
$655$	2527.83	0.150795
$656$	−21505.7	−1.27996
$657$	8641.49	0.513146
$658$	−3751.41	−0.222257
$659$	−12765.3	−0.754577	−0.377288	−	0.926096i	$-0.623143\pi$
$659$	−12765.3	−0.754577	−0.377288	+	0.926096i	$0.623143\pi$
$660$	0	0
$661$	−16327.4	−0.960761	−0.480381	−	0.877060i	$-0.659501\pi$
$661$	−16327.4	−0.960761	−0.480381	+	0.877060i	$0.659501\pi$
$662$	2978.18	0.174849
$663$	24575.4	1.43956
$664$	5682.15	0.332094
$665$	8620.58	0.502694
$666$	1300.67	0.0756757
$667$	7938.51	0.460840
$668$	23444.4	1.35792
$669$	16014.0	0.925465
$670$	−1154.45	−0.0665677
$671$	0	0
$672$	−3844.97	−0.220719
$673$	−29466.8	−1.68776	−0.843880	−	0.536532i	$-0.819734\pi$
$673$	−29466.8	−1.68776	−0.843880	+	0.536532i	$0.819734\pi$
$674$	−1575.22	−0.0900227
$675$	675.000	0.0384900
$676$	−37571.1	−2.13764
$677$	−15078.9	−0.856028	−0.428014	−	0.903772i	$-0.640787\pi$
$677$	−15078.9	−0.856028	−0.428014	+	0.903772i	$0.640787\pi$
$678$	1376.85	0.0779908
$679$	−1157.61	−0.0654272
$680$	2906.86	0.163931
$681$	341.627	0.0192235
$682$	0	0
$683$	25073.7	1.40471	0.702356	−	0.711825i	$-0.252131\pi$
$683$	25073.7	1.40471	0.702356	+	0.711825i	$0.252131\pi$
$684$	−6670.91	−0.372907
$685$	−3777.36	−0.210694
$686$	2403.47	0.133768
$687$	−6364.64	−0.353459
$688$	−16606.8	−0.920243
$689$	−55175.2	−3.05081
$690$	−368.875	−0.0203519
$691$	−8342.46	−0.459280	−0.229640	−	0.973276i	$-0.573755\pi$
$691$	−8342.46	−0.459280	−0.229640	+	0.973276i	$0.573755\pi$
$692$	1669.23	0.0916972
$693$	0	0
$694$	365.679	0.0200014
$695$	−4288.02	−0.234034
$696$	2145.71	0.116858
$697$	−34765.8	−1.88931
$698$	−6.06136	−0.000328690 0
$699$	−3794.41	−0.205319
$700$	−3592.88	−0.193997
$701$	8547.73	0.460547	0.230273	−	0.973126i	$-0.426038\pi$
$701$	8547.73	0.460547	0.230273	+	0.973126i	$0.426038\pi$
$702$	−842.921	−0.0453191
$703$	−36461.5	−1.95615
$704$	0	0
$705$	8234.35	0.439891
$706$	−3691.27	−0.196774
$707$	−3279.76	−0.174467
$708$	−6682.54	−0.354725
$709$	−2552.49	−0.135206	−0.0676029	−	0.997712i	$-0.521535\pi$
$709$	−2552.49	−0.135206	−0.0676029	+	0.997712i	$0.521535\pi$
$710$	−1081.23	−0.0571521
$711$	−9080.57	−0.478971
$712$	−8267.05	−0.435142
$713$	8569.57	0.450116
$714$	−2010.61	−0.105385
$715$	0	0
$716$	−2507.39	−0.130874
$717$	−1433.50	−0.0746652
$718$	2676.75	0.139130
$719$	7756.97	0.402345	0.201172	−	0.979556i	$-0.435525\pi$
$719$	7756.97	0.402345	0.201172	+	0.979556i	$0.435525\pi$
$720$	2730.01	0.141307
$721$	20096.5	1.03805
$722$	−759.907	−0.0391701
$723$	1884.46	0.0969349
$724$	8975.48	0.460734
$725$	3016.37	0.154517
$726$	0	0
$727$	4636.62	0.236538	0.118269	−	0.992982i	$-0.462266\pi$
$727$	4636.62	0.236538	0.118269	+	0.992982i	$0.462266\pi$
$728$	9053.12	0.460894
$729$	729.000	0.0370370
$730$	−1794.36	−0.0909757
$731$	−26846.3	−1.35834
$732$	10316.7	0.520923
$733$	7663.01	0.386139	0.193069	−	0.981185i	$-0.438156\pi$
$733$	7663.01	0.386139	0.193069	+	0.981185i	$0.438156\pi$
$734$	−640.537	−0.0322107
$735$	−130.612	−0.00655470
$736$	−4612.15	−0.230987
$737$	0	0
$738$	1192.44	0.0594776
$739$	20031.1	0.997099	0.498549	−	0.866861i	$-0.333866\pi$
$739$	20031.1	0.997099	0.498549	+	0.866861i	$0.333866\pi$
$740$	15196.4	0.754907
$741$	23629.5	1.17146
$742$	4514.10	0.223339
$743$	6555.56	0.323688	0.161844	−	0.986816i	$-0.448256\pi$
$743$	6555.56	0.323688	0.161844	+	0.986816i	$0.448256\pi$
$744$	2316.28	0.114138
$745$	−14178.3	−0.697252
$746$	1946.03	0.0955082
$747$	8626.81	0.422542
$748$	0	0
$749$	14559.2	0.710258
$750$	−140.160	−0.00682390
$751$	−37910.9	−1.84206	−0.921031	−	0.389490i	$-0.872651\pi$
$751$	−37910.9	−1.84206	−0.921031	+	0.389490i	$0.872651\pi$
$752$	33303.4	1.61496
$753$	11472.2	0.555206
$754$	−3766.76	−0.181933
$755$	−9054.05	−0.436438
$756$	−3880.31	−0.186674
$757$	−14200.6	−0.681811	−0.340906	−	0.940098i	$-0.610734\pi$
$757$	−14200.6	−0.681811	−0.340906	+	0.940098i	$0.610734\pi$
$758$	2698.45	0.129304
$759$	0	0
$760$	2794.98	0.133401
$761$	29571.9	1.40865	0.704324	−	0.709879i	$-0.251250\pi$
$761$	29571.9	1.40865	0.704324	+	0.709879i	$0.251250\pi$
$762$	510.924	0.0242898
$763$	−22281.5	−1.05720
$764$	−10242.3	−0.485017
$765$	4413.29	0.208579
$766$	320.197	0.0151034
$767$	23670.7	1.11434
$768$	10198.0	0.479152
$769$	−18130.9	−0.850218	−0.425109	−	0.905142i	$-0.639764\pi$
$769$	−18130.9	−0.850218	−0.425109	+	0.905142i	$0.639764\pi$
$770$	0	0
$771$	18730.9	0.874936
$772$	−3680.28	−0.171576
$773$	−33348.0	−1.55168	−0.775838	−	0.630932i	$-0.782673\pi$
$773$	−33348.0	−1.55168	−0.775838	+	0.630932i	$0.782673\pi$
$774$	920.810	0.0427621
$775$	3256.15	0.150922
$776$	−375.323	−0.0173625
$777$	−21208.8	−0.979231
$778$	4458.54	0.205458
$779$	−33427.6	−1.53744
$780$	−9848.28	−0.452083
$781$	0	0
$782$	−2411.78	−0.110288
$783$	3257.68	0.148684
$784$	−528.255	−0.0240641
$785$	9975.81	0.453569
$786$	−566.882	−0.0257252
$787$	−1659.65	−0.0751716	−0.0375858	−	0.999293i	$-0.511967\pi$
$787$	−1659.65	−0.0751716	−0.0375858	+	0.999293i	$0.511967\pi$
$788$	−27078.4	−1.22415
$789$	9456.99	0.426715
$790$	1885.53	0.0849168
$791$	−22451.0	−1.00919
$792$	0	0
$793$	−36543.4	−1.63644
$794$	1828.81	0.0817405
$795$	−9908.44	−0.442033
$796$	9278.96	0.413171
$797$	20532.3	0.912538	0.456269	−	0.889842i	$-0.349186\pi$
$797$	20532.3	0.912538	0.456269	+	0.889842i	$0.349186\pi$
$798$	−1933.22	−0.0857584
$799$	53837.8	2.38379
$800$	−1752.46	−0.0774486
$801$	−12551.3	−0.553655
$802$	−2782.59	−0.122515
$803$	0	0
$804$	−14567.1	−0.638982
$805$	6014.90	0.263351
$806$	−4066.19	−0.177699
$807$	−13108.6	−0.571800
$808$	−1063.37	−0.0462985
$809$	11092.0	0.482045	0.241023	−	0.970519i	$-0.422517\pi$
$809$	11092.0	0.482045	0.241023	+	0.970519i	$0.422517\pi$
$810$	−151.373	−0.00656630
$811$	−24351.1	−1.05435	−0.527177	−	0.849755i	$-0.676750\pi$
$811$	−24351.1	−1.05435	−0.527177	+	0.849755i	$0.676750\pi$
$812$	−17339.9	−0.749399
$813$	10880.6	0.469374
$814$	0	0
$815$	−6589.96	−0.283235
$816$	17849.3	0.765749
$817$	−25812.9	−1.10536
$818$	3244.19	0.138668
$819$	13744.7	0.586421
$820$	13931.9	0.593323
$821$	13624.0	0.579148	0.289574	−	0.957156i	$-0.406486\pi$
$821$	13624.0	0.579148	0.289574	+	0.957156i	$0.406486\pi$
$822$	847.097	0.0359439
$823$	−26857.0	−1.13752	−0.568758	−	0.822505i	$-0.692576\pi$
$823$	−26857.0	−1.13752	−0.568758	+	0.822505i	$0.692576\pi$
$824$	6515.71	0.275468
$825$	0	0
$826$	−1936.59	−0.0815770
$827$	−29552.3	−1.24261	−0.621303	−	0.783570i	$-0.713396\pi$
$827$	−29552.3	−1.24261	−0.621303	+	0.783570i	$0.713396\pi$
$828$	−4654.54	−0.195358
$829$	−30296.4	−1.26929	−0.634643	−	0.772806i	$-0.718853\pi$
$829$	−30296.4	−1.26929	−0.634643	+	0.772806i	$0.718853\pi$
$830$	−1791.31	−0.0749125
$831$	1880.80	0.0785127
$832$	−38350.4	−1.59803
$833$	−853.969	−0.0355201
$834$	961.616	0.0399257
$835$	−14913.1	−0.618073
$836$	0	0
$837$	3516.64	0.145224
$838$	5591.23	0.230484
$839$	−37607.6	−1.54751	−0.773753	−	0.633488i	$-0.781623\pi$
$839$	−37607.6	−1.54751	−0.773753	+	0.633488i	$0.781623\pi$
$840$	1625.77	0.0667791
$841$	−9831.44	−0.403110
$842$	−5935.42	−0.242931
$843$	−21034.4	−0.859388
$844$	1756.39	0.0716318
$845$	23899.3	0.972970
$846$	−1846.60	−0.0750444
$847$	0	0
$848$	−40074.2	−1.62282
$849$	−10380.6	−0.419624
$850$	−916.396	−0.0369790
$851$	−25440.6	−1.02478
$852$	−13643.2	−0.548602
$853$	1998.25	0.0802095	0.0401048	−	0.999195i	$-0.487231\pi$
$853$	1998.25	0.0802095	0.0401048	+	0.999195i	$0.487231\pi$
$854$	2989.76	0.119798
$855$	4243.42	0.169733
$856$	4720.42	0.188482
$857$	−541.203	−0.0215719	−0.0107860	−	0.999942i	$-0.503433\pi$
$857$	−541.203	−0.0215719	−0.0107860	+	0.999942i	$0.503433\pi$
$858$	0	0
$859$	42679.5	1.69523	0.847617	−	0.530609i	$-0.178037\pi$
$859$	42679.5	1.69523	0.847617	+	0.530609i	$0.178037\pi$
$860$	10758.3	0.426575
$861$	−19444.1	−0.769630
$862$	1350.79	0.0533736
$863$	−14089.0	−0.555729	−0.277864	−	0.960620i	$-0.589627\pi$
$863$	−14089.0	−0.555729	−0.277864	+	0.960620i	$0.589627\pi$
$864$	−1892.66	−0.0745250
$865$	−1061.81	−0.0417371
$866$	−648.922	−0.0254633
$867$	14116.0	0.552945
$868$	−18718.3	−0.731960
$869$	0	0
$870$	−676.440	−0.0263603
$871$	51599.1	2.00731
$872$	−7224.15	−0.280551
$873$	−569.826	−0.0220913
$874$	−2318.95	−0.0897479
$875$	2285.46	0.0883001
$876$	−22641.6	−0.873274
$877$	35724.5	1.37552	0.687760	−	0.725938i	$-0.258594\pi$
$877$	35724.5	1.37552	0.687760	+	0.725938i	$0.258594\pi$
$878$	−176.835	−0.00679713
$879$	28640.1	1.09899
$880$	0	0
$881$	−5000.09	−0.191211	−0.0956057	−	0.995419i	$-0.530479\pi$
$881$	−5000.09	−0.191211	−0.0956057	+	0.995419i	$0.530479\pi$
$882$	29.2906	0.00111822
$883$	6227.85	0.237354	0.118677	−	0.992933i	$-0.462135\pi$
$883$	6227.85	0.237354	0.118677	+	0.992933i	$0.462135\pi$
$884$	−64390.1	−2.44985
$885$	4250.82	0.161457
$886$	−1302.87	−0.0494028
$887$	28201.3	1.06754	0.533770	−	0.845630i	$-0.320775\pi$
$887$	28201.3	1.06754	0.533770	+	0.845630i	$0.320775\pi$
$888$	−6876.35	−0.259860
$889$	−8331.15	−0.314306
$890$	2606.21	0.0981577
$891$	0	0
$892$	−41958.2	−1.57496
$893$	51765.6	1.93983
$894$	3179.57	0.118949
$895$	1594.97	0.0595687
$896$	13390.9	0.499282
$897$	16487.2	0.613702
$898$	−4113.68	−0.152868
$899$	15714.8	0.583001
$900$	−1768.57	−0.0655025
$901$	−64783.4	−2.39539
$902$	0	0
$903$	−15014.8	−0.553334
$904$	−7279.10	−0.267809
$905$	−5709.37	−0.209708
$906$	2030.43	0.0744552
$907$	−48308.3	−1.76852	−0.884261	−	0.466993i	$-0.845337\pi$
$907$	−48308.3	−1.76852	−0.884261	+	0.466993i	$0.845337\pi$
$908$	−895.098	−0.0327146
$909$	−1614.44	−0.0589082
$910$	−2854.02	−0.103967
$911$	31175.8	1.13381	0.566905	−	0.823783i	$-0.308140\pi$
$911$	31175.8	1.13381	0.566905	+	0.823783i	$0.308140\pi$
$912$	17162.3	0.623136
$913$	0	0
$914$	−3525.46	−0.127584
$915$	−6562.52	−0.237104
$916$	16676.0	0.601518
$917$	9243.60	0.332880
$918$	−989.708	−0.0355830
$919$	45907.1	1.64781	0.823904	−	0.566729i	$-0.191791\pi$
$919$	45907.1	1.64781	0.823904	+	0.566729i	$0.191791\pi$
$920$	1950.16	0.0698857
$921$	−30017.2	−1.07394
$922$	−2089.89	−0.0746496
$923$	48326.6	1.72339
$924$	0	0
$925$	−9666.56	−0.343605
$926$	1815.61	0.0644325
$927$	9892.34	0.350493
$928$	−8457.71	−0.299179
$929$	17464.1	0.616767	0.308384	−	0.951262i	$-0.400212\pi$
$929$	17464.1	0.616767	0.308384	+	0.951262i	$0.400212\pi$
$930$	−730.212	−0.0257469
$931$	−821.099	−0.0289049
$932$	9941.73	0.349412
$933$	−10543.0	−0.369950
$934$	−7172.50	−0.251276
$935$	0	0
$936$	4456.33	0.155619
$937$	−36048.3	−1.25683	−0.628413	−	0.777880i	$-0.716295\pi$
$937$	−36048.3	−1.25683	−0.628413	+	0.777880i	$0.716295\pi$
$938$	−4221.52	−0.146948
$939$	4188.88	0.145579
$940$	−21574.8	−0.748610
$941$	56899.2	1.97116	0.985580	−	0.169207i	$-0.0541207\pi$
$941$	56899.2	1.97116	0.985580	+	0.169207i	$0.0541207\pi$
$942$	−2237.14	−0.0773778
$943$	−23323.7	−0.805433
$944$	17192.2	0.592753
$945$	2468.30	0.0849668
$946$	0	0
$947$	3351.74	0.115013	0.0575063	−	0.998345i	$-0.481685\pi$
$947$	3351.74	0.115013	0.0575063	+	0.998345i	$0.481685\pi$
$948$	23792.0	0.815115
$949$	80200.3	2.74332
$950$	−881.123	−0.0300920
$951$	−1949.52	−0.0664749
$952$	10629.6	0.361878
$953$	−1701.88	−0.0578483	−0.0289242	−	0.999582i	$-0.509208\pi$
$953$	−1701.88	−0.0578483	−0.0289242	+	0.999582i	$0.509208\pi$
$954$	2222.03	0.0754098
$955$	6515.20	0.220761
$956$	3755.91	0.127066
$957$	0	0
$958$	−2626.53	−0.0885797
$959$	−13812.8	−0.465108
$960$	−6887.01	−0.231539
$961$	−12827.0	−0.430566
$962$	12071.3	0.404569
$963$	7166.68	0.239816
$964$	−4937.49	−0.164964
$965$	2341.06	0.0780946
$966$	−1348.88	−0.0449270
$967$	−52244.3	−1.73740	−0.868699	−	0.495340i	$-0.835044\pi$
$967$	−52244.3	−1.73740	−0.868699	+	0.495340i	$0.835044\pi$
$968$	0	0
$969$	27744.3	0.919789
$970$	118.321	0.00391657
$971$	24422.6	0.807166	0.403583	−	0.914943i	$-0.367765\pi$
$971$	24422.6	0.807166	0.403583	+	0.914943i	$0.367765\pi$
$972$	−1910.05	−0.0630298
$973$	−15680.1	−0.516632
$974$	4137.33	0.136107
$975$	6264.57	0.205771
$976$	−26541.8	−0.870473
$977$	−57863.9	−1.89481	−0.947406	−	0.320035i	$-0.896305\pi$
$977$	−57863.9	−1.89481	−0.947406	+	0.320035i	$0.896305\pi$
$978$	1477.84	0.0483191
$979$	0	0
$980$	342.217	0.0111548
$981$	−10967.9	−0.356961
$982$	−1307.73	−0.0424963
$983$	−15617.5	−0.506736	−0.253368	−	0.967370i	$-0.581538\pi$
$983$	−15617.5	−0.506736	−0.253368	+	0.967370i	$0.581538\pi$
$984$	−6304.18	−0.204238
$985$	17224.8	0.557184
$986$	−4422.70	−0.142847
$987$	30110.8	0.971061
$988$	−61911.6	−1.99359
$989$	−18010.6	−0.579075
$990$	0	0
$991$	−6332.84	−0.202996	−0.101498	−	0.994836i	$-0.532364\pi$
$991$	−6332.84	−0.202996	−0.101498	+	0.994836i	$0.532364\pi$
$992$	−9130.05	−0.292217
$993$	−23904.4	−0.763932
$994$	−3953.79	−0.126164
$995$	−5902.42	−0.188060
$996$	−22603.1	−0.719084
$997$	44697.7	1.41985	0.709926	−	0.704277i	$-0.248729\pi$
$997$	44697.7	1.41985	0.709926	+	0.704277i	$0.248729\pi$
$998$	3505.29	0.111180
$999$	−10439.9	−0.330634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bk.1.6	✓	12
11.10	odd	2	inner	1815.4.a.bk.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.bk.1.6	✓	12	1.1	even	1	trivial
1815.4.a.bk.1.7	yes	12	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-0.373761 q^{2} +3.00000 q^{3} -7.86030 q^{4} +5.00000 q^{5} -1.12128 q^{6} +18.2837 q^{7} +5.92796 q^{8} +9.00000 q^{9} -1.86880 q^{10} -23.5809 q^{12} +83.5276 q^{13} -6.83371 q^{14} +15.0000 q^{15} +60.6668 q^{16} +98.0730 q^{17} -3.36385 q^{18} +94.2981 q^{19} -39.3015 q^{20} +54.8510 q^{21} +65.7953 q^{23} +17.7839 q^{24} +25.0000 q^{25} -31.2193 q^{26} +27.0000 q^{27} -143.715 q^{28} +120.655 q^{29} -5.60641 q^{30} +130.246 q^{31} -70.0985 q^{32} -36.6558 q^{34} +91.4183 q^{35} -70.7427 q^{36} -386.662 q^{37} -35.2449 q^{38} +250.583 q^{39} +29.6398 q^{40} -354.489 q^{41} -20.5011 q^{42} -273.737 q^{43} +45.0000 q^{45} -24.5917 q^{46} +548.956 q^{47} +182.000 q^{48} -8.70748 q^{49} -9.34401 q^{50} +294.219 q^{51} -656.552 q^{52} -660.563 q^{53} -10.0915 q^{54} +108.385 q^{56} +282.894 q^{57} -45.0960 q^{58} +283.388 q^{59} -117.905 q^{60} -437.501 q^{61} -48.6808 q^{62} +164.553 q^{63} -459.134 q^{64} +417.638 q^{65} +617.750 q^{67} -770.884 q^{68} +197.386 q^{69} -34.1686 q^{70} +578.571 q^{71} +53.3516 q^{72} +960.165 q^{73} +144.519 q^{74} +75.0000 q^{75} -741.212 q^{76} -93.6579 q^{78} -1008.95 q^{79} +303.334 q^{80} +81.0000 q^{81} +132.494 q^{82} +958.535 q^{83} -431.146 q^{84} +490.365 q^{85} +102.312 q^{86} +361.964 q^{87} -1394.59 q^{89} -16.8192 q^{90} +1527.19 q^{91} -517.171 q^{92} +390.738 q^{93} -205.178 q^{94} +471.491 q^{95} -210.295 q^{96} -63.3140 q^{97} +3.25451 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} + 62 q^{4} + 60 q^{5} + 108 q^{9} + 186 q^{12} + 16 q^{14} + 180 q^{15} + 610 q^{16} + 310 q^{20} - 44 q^{23} + 300 q^{25} + 1016 q^{26} + 324 q^{27} + 288 q^{31} + 710 q^{34} + 558 q^{36}+ \cdots + 1436 q^{97}+O(q^{100}) \) 12 * q + 36 * q^3 + 62 * q^4 + 60 * q^5 + 108 * q^9 + 186 * q^12 + 16 * q^14 + 180 * q^15 + 610 * q^16 + 310 * q^20 - 44 * q^23 + 300 * q^25 + 1016 * q^26 + 324 * q^27 + 288 * q^31 + 710 * q^34 + 558 * q^36 - 488 * q^37 + 872 * q^38 + 48 * q^42 + 540 * q^45 + 264 * q^47 + 1830 * q^48 + 140 * q^49 + 1144 * q^53 + 3172 * q^56 - 488 * q^58 + 1400 * q^59 + 930 * q^60 + 3088 * q^64 + 2032 * q^67 - 132 * q^69 + 80 * q^70 + 4852 * q^71 + 900 * q^75 + 3048 * q^78 + 3050 * q^80 + 972 * q^81 + 816 * q^82 - 2232 * q^86 + 4932 * q^89 + 7336 * q^91 - 2526 * q^92 + 864 * q^93 + 1436 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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