LMFDB - Embedded newform 75.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("75.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.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:	$4.42514325043$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} -24.0000 q^{11} +3.00000 q^{12} -74.0000 q^{13} +60.0000 q^{14} -71.0000 q^{16} -54.0000 q^{17} -27.0000 q^{18} -124.000 q^{19} -60.0000 q^{21} +72.0000 q^{22} +120.000 q^{23} +63.0000 q^{24} +222.000 q^{26} +27.0000 q^{27} -20.0000 q^{28} -78.0000 q^{29} +200.000 q^{31} +45.0000 q^{32} -72.0000 q^{33} +162.000 q^{34} +9.00000 q^{36} +70.0000 q^{37} +372.000 q^{38} -222.000 q^{39} +330.000 q^{41} +180.000 q^{42} -92.0000 q^{43} -24.0000 q^{44} -360.000 q^{46} +24.0000 q^{47} -213.000 q^{48} +57.0000 q^{49} -162.000 q^{51} -74.0000 q^{52} -450.000 q^{53} -81.0000 q^{54} -420.000 q^{56} -372.000 q^{57} +234.000 q^{58} +24.0000 q^{59} -322.000 q^{61} -600.000 q^{62} -180.000 q^{63} +433.000 q^{64} +216.000 q^{66} +196.000 q^{67} -54.0000 q^{68} +360.000 q^{69} -288.000 q^{71} +189.000 q^{72} +430.000 q^{73} -210.000 q^{74} -124.000 q^{76} +480.000 q^{77} +666.000 q^{78} -520.000 q^{79} +81.0000 q^{81} -990.000 q^{82} -156.000 q^{83} -60.0000 q^{84} +276.000 q^{86} -234.000 q^{87} -504.000 q^{88} +1026.00 q^{89} +1480.00 q^{91} +120.000 q^{92} +600.000 q^{93} -72.0000 q^{94} +135.000 q^{96} +286.000 q^{97} -171.000 q^{98} -216.000 q^{99} +O(q^{100})\)

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	1.00000	0.125000
$5$	0	0
$5$	0	0
$6$	−9.00000	−0.612372
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	−20.0000	−1.07990	−0.539949	+	0.841698i	$0.681557\pi$
$8$	21.0000	0.928078
$9$	9.00000	0.333333
$10$	0	0
$11$	−24.0000	−0.657843	−0.328921	−	0.944357i	$-0.606685\pi$
$11$	−24.0000	−0.657843	−0.328921	+	0.944357i	$0.606685\pi$
$12$	3.00000	0.0721688
$13$	−74.0000	−1.57876	−0.789381	−	0.613904i	$-0.789598\pi$
$13$	−74.0000	−1.57876	−0.789381	+	0.613904i	$0.789598\pi$
$14$	60.0000	1.14541
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	−27.0000	−0.353553
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	0	0
$21$	−60.0000	−0.623480
$22$	72.0000	0.697748
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	63.0000	0.535826
$25$	0	0
$26$	222.000	1.67453
$27$	27.0000	0.192450
$28$	−20.0000	−0.134987
$29$	−78.0000	−0.499456	−0.249728	−	0.968316i	$-0.580341\pi$
$29$	−78.0000	−0.499456	−0.249728	+	0.968316i	$0.580341\pi$
$30$	0	0
$31$	200.000	1.15874	0.579372	−	0.815063i	$-0.303298\pi$
$31$	200.000	1.15874	0.579372	+	0.815063i	$0.303298\pi$
$32$	45.0000	0.248592
$33$	−72.0000	−0.379806
$34$	162.000	0.817140
$35$	0	0
$36$	9.00000	0.0416667
$37$	70.0000	0.311025	0.155513	−	0.987834i	$-0.450297\pi$
$37$	70.0000	0.311025	0.155513	+	0.987834i	$0.450297\pi$
$38$	372.000	1.58806
$39$	−222.000	−0.911499
$40$	0	0
$41$	330.000	1.25701	0.628504	−	0.777806i	$-0.283668\pi$
$41$	330.000	1.25701	0.628504	+	0.777806i	$0.283668\pi$
$42$	180.000	0.661300
$43$	−92.0000	−0.326276	−0.163138	−	0.986603i	$-0.552162\pi$
$43$	−92.0000	−0.326276	−0.163138	+	0.986603i	$0.552162\pi$
$44$	−24.0000	−0.0822304
$45$	0	0
$46$	−360.000	−1.15389
$47$	24.0000	0.0744843	0.0372421	−	0.999306i	$-0.488143\pi$
$47$	24.0000	0.0744843	0.0372421	+	0.999306i	$0.488143\pi$
$48$	−213.000	−0.640498
$49$	57.0000	0.166181
$50$	0	0
$51$	−162.000	−0.444795
$52$	−74.0000	−0.197345
$53$	−450.000	−1.16627	−0.583134	−	0.812376i	$-0.698174\pi$
$53$	−450.000	−1.16627	−0.583134	+	0.812376i	$0.698174\pi$
$54$	−81.0000	−0.204124
$55$	0	0
$56$	−420.000	−1.00223
$57$	−372.000	−0.864432
$58$	234.000	0.529754
$59$	24.0000	0.0529582	0.0264791	−	0.999649i	$-0.491570\pi$
$59$	24.0000	0.0529582	0.0264791	+	0.999649i	$0.491570\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	−600.000	−1.22903
$63$	−180.000	−0.359966
$64$	433.000	0.845703
$65$	0	0
$66$	216.000	0.402845
$67$	196.000	0.357391	0.178696	−	0.983904i	$-0.442812\pi$
$67$	196.000	0.357391	0.178696	+	0.983904i	$0.442812\pi$
$68$	−54.0000	−0.0963009
$69$	360.000	0.628100
$70$	0	0
$71$	−288.000	−0.481399	−0.240699	−	0.970600i	$-0.577377\pi$
$71$	−288.000	−0.481399	−0.240699	+	0.970600i	$0.577377\pi$
$72$	189.000	0.309359
$73$	430.000	0.689420	0.344710	−	0.938709i	$-0.387977\pi$
$73$	430.000	0.689420	0.344710	+	0.938709i	$0.387977\pi$
$74$	−210.000	−0.329892
$75$	0	0
$76$	−124.000	−0.187155
$77$	480.000	0.710404
$78$	666.000	0.966790
$79$	−520.000	−0.740564	−0.370282	−	0.928919i	$-0.620739\pi$
$79$	−520.000	−0.740564	−0.370282	+	0.928919i	$0.620739\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−990.000	−1.33326
$83$	−156.000	−0.206304	−0.103152	−	0.994666i	$-0.532893\pi$
$83$	−156.000	−0.206304	−0.103152	+	0.994666i	$0.532893\pi$
$84$	−60.0000	−0.0779350
$85$	0	0
$86$	276.000	0.346068
$87$	−234.000	−0.288361
$88$	−504.000	−0.610529
$89$	1026.00	1.22198	0.610988	−	0.791640i	$-0.290773\pi$
$89$	1026.00	1.22198	0.610988	+	0.791640i	$0.290773\pi$
$90$	0	0
$91$	1480.00	1.70490
$92$	120.000	0.135988
$93$	600.000	0.669001
$94$	−72.0000	−0.0790025
$95$	0	0
$96$	135.000	0.143525
$97$	286.000	0.299370	0.149685	−	0.988734i	$-0.452174\pi$
$97$	286.000	0.299370	0.149685	+	0.988734i	$0.452174\pi$
$98$	−171.000	−0.176261
$99$	−216.000	−0.219281
$100$	0	0
$101$	−1734.00	−1.70831	−0.854156	−	0.520017i	$-0.825925\pi$
$101$	−1734.00	−1.70831	−0.854156	+	0.520017i	$0.825925\pi$
$102$	486.000	0.471776
$103$	−452.000	−0.432397	−0.216198	−	0.976349i	$-0.569366\pi$
$103$	−452.000	−0.432397	−0.216198	+	0.976349i	$0.569366\pi$
$104$	−1554.00	−1.46521
$105$	0	0
$106$	1350.00	1.23702
$107$	1404.00	1.26850	0.634251	−	0.773127i	$-0.281308\pi$
$107$	1404.00	1.26850	0.634251	+	0.773127i	$0.281308\pi$
$108$	27.0000	0.0240563
$109$	−1474.00	−1.29526	−0.647631	−	0.761954i	$-0.724240\pi$
$109$	−1474.00	−1.29526	−0.647631	+	0.761954i	$0.724240\pi$
$110$	0	0
$111$	210.000	0.179570
$112$	1420.00	1.19801
$113$	−1086.00	−0.904091	−0.452046	−	0.891995i	$-0.649306\pi$
$113$	−1086.00	−0.904091	−0.452046	+	0.891995i	$0.649306\pi$
$114$	1116.00	0.916868
$115$	0	0
$116$	−78.0000	−0.0624321
$117$	−666.000	−0.526254
$118$	−72.0000	−0.0561707
$119$	1080.00	0.831962
$120$	0	0
$121$	−755.000	−0.567243
$122$	966.000	0.716865
$123$	990.000	0.725734
$124$	200.000	0.144843
$125$	0	0
$126$	540.000	0.381802
$127$	−1244.00	−0.869190	−0.434595	−	0.900626i	$-0.643109\pi$
$127$	−1244.00	−0.869190	−0.434595	+	0.900626i	$0.643109\pi$
$128$	−1659.00	−1.14560
$129$	−276.000	−0.188376
$130$	0	0
$131$	2328.00	1.55266	0.776329	−	0.630327i	$-0.217079\pi$
$131$	2328.00	1.55266	0.776329	+	0.630327i	$0.217079\pi$
$132$	−72.0000	−0.0474757
$133$	2480.00	1.61687
$134$	−588.000	−0.379071
$135$	0	0
$136$	−1134.00	−0.714998
$137$	−2118.00	−1.32082	−0.660412	−	0.750903i	$-0.729618\pi$
$137$	−2118.00	−1.32082	−0.660412	+	0.750903i	$0.729618\pi$
$138$	−1080.00	−0.666201
$139$	2324.00	1.41812	0.709062	−	0.705147i	$-0.249119\pi$
$139$	2324.00	1.41812	0.709062	+	0.705147i	$0.249119\pi$
$140$	0	0
$141$	72.0000	0.0430035
$142$	864.000	0.510600
$143$	1776.00	1.03858
$144$	−639.000	−0.369792
$145$	0	0
$146$	−1290.00	−0.731241
$147$	171.000	0.0959445
$148$	70.0000	0.0388781
$149$	258.000	0.141854	0.0709268	−	0.997482i	$-0.477404\pi$
$149$	258.000	0.141854	0.0709268	+	0.997482i	$0.477404\pi$
$150$	0	0
$151$	−808.000	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$151$	−808.000	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$152$	−2604.00	−1.38955
$153$	−486.000	−0.256802
$154$	−1440.00	−0.753497
$155$	0	0
$156$	−222.000	−0.113937
$157$	−2378.00	−1.20882	−0.604411	−	0.796673i	$-0.706592\pi$
$157$	−2378.00	−1.20882	−0.604411	+	0.796673i	$0.706592\pi$
$158$	1560.00	0.785487
$159$	−1350.00	−0.673346
$160$	0	0
$161$	−2400.00	−1.17482
$162$	−243.000	−0.117851
$163$	52.0000	0.0249874	0.0124937	−	0.999922i	$-0.496023\pi$
$163$	52.0000	0.0249874	0.0124937	+	0.999922i	$0.496023\pi$
$164$	330.000	0.157126
$165$	0	0
$166$	468.000	0.218818
$167$	3720.00	1.72373	0.861863	−	0.507141i	$-0.169298\pi$
$167$	3720.00	1.72373	0.861863	+	0.507141i	$0.169298\pi$
$168$	−1260.00	−0.578638
$169$	3279.00	1.49249
$170$	0	0
$171$	−1116.00	−0.499080
$172$	−92.0000	−0.0407845
$173$	−426.000	−0.187215	−0.0936075	−	0.995609i	$-0.529840\pi$
$173$	−426.000	−0.187215	−0.0936075	+	0.995609i	$0.529840\pi$
$174$	702.000	0.305853
$175$	0	0
$176$	1704.00	0.729795
$177$	72.0000	0.0305754
$178$	−3078.00	−1.29610
$179$	−1440.00	−0.601289	−0.300644	−	0.953736i	$-0.597202\pi$
$179$	−1440.00	−0.601289	−0.300644	+	0.953736i	$0.597202\pi$
$180$	0	0
$181$	−3130.00	−1.28537	−0.642683	−	0.766133i	$-0.722179\pi$
$181$	−3130.00	−1.28537	−0.642683	+	0.766133i	$0.722179\pi$
$182$	−4440.00	−1.80832
$183$	−966.000	−0.390212
$184$	2520.00	1.00966
$185$	0	0
$186$	−1800.00	−0.709583
$187$	1296.00	0.506807
$188$	24.0000	0.00931053
$189$	−540.000	−0.207827
$190$	0	0
$191$	3576.00	1.35471	0.677357	−	0.735655i	$-0.263125\pi$
$191$	3576.00	1.35471	0.677357	+	0.735655i	$0.263125\pi$
$192$	1299.00	0.488267
$193$	−2666.00	−0.994315	−0.497158	−	0.867660i	$-0.665623\pi$
$193$	−2666.00	−0.994315	−0.497158	+	0.867660i	$0.665623\pi$
$194$	−858.000	−0.317530
$195$	0	0
$196$	57.0000	0.0207726
$197$	2718.00	0.982992	0.491496	−	0.870880i	$-0.336450\pi$
$197$	2718.00	0.982992	0.491496	+	0.870880i	$0.336450\pi$
$198$	648.000	0.232583
$199$	−3832.00	−1.36504	−0.682521	−	0.730866i	$-0.739116\pi$
$199$	−3832.00	−1.36504	−0.682521	+	0.730866i	$0.739116\pi$
$200$	0	0
$201$	588.000	0.206340
$202$	5202.00	1.81194
$203$	1560.00	0.539362
$204$	−162.000	−0.0555994
$205$	0	0
$206$	1356.00	0.458626
$207$	1080.00	0.362634
$208$	5254.00	1.75144
$209$	2976.00	0.984948
$210$	0	0
$211$	1100.00	0.358896	0.179448	−	0.983767i	$-0.442569\pi$
$211$	1100.00	0.358896	0.179448	+	0.983767i	$0.442569\pi$
$212$	−450.000	−0.145784
$213$	−864.000	−0.277936
$214$	−4212.00	−1.34545
$215$	0	0
$216$	567.000	0.178609
$217$	−4000.00	−1.25133
$218$	4422.00	1.37383
$219$	1290.00	0.398037
$220$	0	0
$221$	3996.00	1.21629
$222$	−630.000	−0.190463
$223$	−1964.00	−0.589772	−0.294886	−	0.955532i	$-0.595282\pi$
$223$	−1964.00	−0.589772	−0.294886	+	0.955532i	$0.595282\pi$
$224$	−900.000	−0.268454
$225$	0	0
$226$	3258.00	0.958933
$227$	−660.000	−0.192977	−0.0964884	−	0.995334i	$-0.530761\pi$
$227$	−660.000	−0.192977	−0.0964884	+	0.995334i	$0.530761\pi$
$228$	−372.000	−0.108054
$229$	−1906.00	−0.550009	−0.275004	−	0.961443i	$-0.588679\pi$
$229$	−1906.00	−0.550009	−0.275004	+	0.961443i	$0.588679\pi$
$230$	0	0
$231$	1440.00	0.410152
$232$	−1638.00	−0.463534
$233$	1458.00	0.409943	0.204972	−	0.978768i	$-0.434290\pi$
$233$	1458.00	0.409943	0.204972	+	0.978768i	$0.434290\pi$
$234$	1998.00	0.558177
$235$	0	0
$236$	24.0000	0.00661978
$237$	−1560.00	−0.427565
$238$	−3240.00	−0.882429
$239$	1176.00	0.318281	0.159140	−	0.987256i	$-0.449128\pi$
$239$	1176.00	0.318281	0.159140	+	0.987256i	$0.449128\pi$
$240$	0	0
$241$	866.000	0.231469	0.115734	−	0.993280i	$-0.463078\pi$
$241$	866.000	0.231469	0.115734	+	0.993280i	$0.463078\pi$
$242$	2265.00	0.601652
$243$	243.000	0.0641500
$244$	−322.000	−0.0844834
$245$	0	0
$246$	−2970.00	−0.769757
$247$	9176.00	2.36379
$248$	4200.00	1.07540
$249$	−468.000	−0.119110
$250$	0	0
$251$	432.000	0.108636	0.0543179	−	0.998524i	$-0.482702\pi$
$251$	432.000	0.108636	0.0543179	+	0.998524i	$0.482702\pi$
$252$	−180.000	−0.0449958
$253$	−2880.00	−0.715668
$254$	3732.00	0.921915
$255$	0	0
$256$	1513.00	0.369385
$257$	−2526.00	−0.613103	−0.306552	−	0.951854i	$-0.599175\pi$
$257$	−2526.00	−0.613103	−0.306552	+	0.951854i	$0.599175\pi$
$258$	828.000	0.199802
$259$	−1400.00	−0.335876
$260$	0	0
$261$	−702.000	−0.166485
$262$	−6984.00	−1.64684
$263$	−5448.00	−1.27733	−0.638666	−	0.769484i	$-0.720513\pi$
$263$	−5448.00	−1.27733	−0.638666	+	0.769484i	$0.720513\pi$
$264$	−1512.00	−0.352489
$265$	0	0
$266$	−7440.00	−1.71495
$267$	3078.00	0.705508
$268$	196.000	0.0446739
$269$	−2574.00	−0.583418	−0.291709	−	0.956507i	$-0.594224\pi$
$269$	−2574.00	−0.583418	−0.291709	+	0.956507i	$0.594224\pi$
$270$	0	0
$271$	−3184.00	−0.713706	−0.356853	−	0.934161i	$-0.616150\pi$
$271$	−3184.00	−0.713706	−0.356853	+	0.934161i	$0.616150\pi$
$272$	3834.00	0.854671
$273$	4440.00	0.984326
$274$	6354.00	1.40095
$275$	0	0
$276$	360.000	0.0785125
$277$	−3962.00	−0.859399	−0.429699	−	0.902972i	$-0.641380\pi$
$277$	−3962.00	−0.859399	−0.429699	+	0.902972i	$0.641380\pi$
$278$	−6972.00	−1.50415
$279$	1800.00	0.386248
$280$	0	0
$281$	−8286.00	−1.75908	−0.879540	−	0.475825i	$-0.842149\pi$
$281$	−8286.00	−1.75908	−0.879540	+	0.475825i	$0.842149\pi$
$282$	−216.000	−0.0456121
$283$	2716.00	0.570493	0.285246	−	0.958454i	$-0.407925\pi$
$283$	2716.00	0.570493	0.285246	+	0.958454i	$0.407925\pi$
$284$	−288.000	−0.0601748
$285$	0	0
$286$	−5328.00	−1.10158
$287$	−6600.00	−1.35744
$288$	405.000	0.0828641
$289$	−1997.00	−0.406473
$290$	0	0
$291$	858.000	0.172841
$292$	430.000	0.0861776
$293$	−6018.00	−1.19992	−0.599958	−	0.800032i	$-0.704816\pi$
$293$	−6018.00	−1.19992	−0.599958	+	0.800032i	$0.704816\pi$
$294$	−513.000	−0.101765
$295$	0	0
$296$	1470.00	0.288655
$297$	−648.000	−0.126602
$298$	−774.000	−0.150458
$299$	−8880.00	−1.71754
$300$	0	0
$301$	1840.00	0.352345
$302$	2424.00	0.461873
$303$	−5202.00	−0.986294
$304$	8804.00	1.66100
$305$	0	0
$306$	1458.00	0.272380
$307$	−9236.00	−1.71702	−0.858512	−	0.512793i	$-0.828611\pi$
$307$	−9236.00	−1.71702	−0.858512	+	0.512793i	$0.828611\pi$
$308$	480.000	0.0888004
$309$	−1356.00	−0.249644
$310$	0	0
$311$	1536.00	0.280060	0.140030	−	0.990147i	$-0.455280\pi$
$311$	1536.00	0.280060	0.140030	+	0.990147i	$0.455280\pi$
$312$	−4662.00	−0.845942
$313$	7342.00	1.32586	0.662930	−	0.748681i	$-0.269313\pi$
$313$	7342.00	1.32586	0.662930	+	0.748681i	$0.269313\pi$
$314$	7134.00	1.28215
$315$	0	0
$316$	−520.000	−0.0925705
$317$	3894.00	0.689933	0.344967	−	0.938615i	$-0.387890\pi$
$317$	3894.00	0.689933	0.344967	+	0.938615i	$0.387890\pi$
$318$	4050.00	0.714191
$319$	1872.00	0.328564
$320$	0	0
$321$	4212.00	0.732370
$322$	7200.00	1.24609
$323$	6696.00	1.15348
$324$	81.0000	0.0138889
$325$	0	0
$326$	−156.000	−0.0265032
$327$	−4422.00	−0.747820
$328$	6930.00	1.16660
$329$	−480.000	−0.0804354
$330$	0	0
$331$	3692.00	0.613084	0.306542	−	0.951857i	$-0.400828\pi$
$331$	3692.00	0.613084	0.306542	+	0.951857i	$0.400828\pi$
$332$	−156.000	−0.0257880
$333$	630.000	0.103675
$334$	−11160.0	−1.82829
$335$	0	0
$336$	4260.00	0.691673
$337$	8998.00	1.45446	0.727229	−	0.686395i	$-0.240808\pi$
$337$	8998.00	1.45446	0.727229	+	0.686395i	$0.240808\pi$
$338$	−9837.00	−1.58302
$339$	−3258.00	−0.521977
$340$	0	0
$341$	−4800.00	−0.762271
$342$	3348.00	0.529354
$343$	5720.00	0.900440
$344$	−1932.00	−0.302809
$345$	0	0
$346$	1278.00	0.198571
$347$	−5244.00	−0.811276	−0.405638	−	0.914034i	$-0.632951\pi$
$347$	−5244.00	−0.811276	−0.405638	+	0.914034i	$0.632951\pi$
$348$	−234.000	−0.0360452
$349$	6302.00	0.966585	0.483293	−	0.875459i	$-0.339441\pi$
$349$	6302.00	0.966585	0.483293	+	0.875459i	$0.339441\pi$
$350$	0	0
$351$	−1998.00	−0.303833
$352$	−1080.00	−0.163535
$353$	−3414.00	−0.514756	−0.257378	−	0.966311i	$-0.582859\pi$
$353$	−3414.00	−0.514756	−0.257378	+	0.966311i	$0.582859\pi$
$354$	−216.000	−0.0324301
$355$	0	0
$356$	1026.00	0.152747
$357$	3240.00	0.480333
$358$	4320.00	0.637763
$359$	4824.00	0.709195	0.354597	−	0.935019i	$-0.384618\pi$
$359$	4824.00	0.709195	0.354597	+	0.935019i	$0.384618\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	9390.00	1.36334
$363$	−2265.00	−0.327498
$364$	1480.00	0.213113
$365$	0	0
$366$	2898.00	0.413882
$367$	3508.00	0.498954	0.249477	−	0.968381i	$-0.419741\pi$
$367$	3508.00	0.498954	0.249477	+	0.968381i	$0.419741\pi$
$368$	−8520.00	−1.20689
$369$	2970.00	0.419003
$370$	0	0
$371$	9000.00	1.25945
$372$	600.000	0.0836251
$373$	−10802.0	−1.49948	−0.749740	−	0.661732i	$-0.769822\pi$
$373$	−10802.0	−1.49948	−0.749740	+	0.661732i	$0.769822\pi$
$374$	−3888.00	−0.537550
$375$	0	0
$376$	504.000	0.0691272
$377$	5772.00	0.788523
$378$	1620.00	0.220433
$379$	1460.00	0.197876	0.0989382	−	0.995094i	$-0.468455\pi$
$379$	1460.00	0.197876	0.0989382	+	0.995094i	$0.468455\pi$
$380$	0	0
$381$	−3732.00	−0.501827
$382$	−10728.0	−1.43689
$383$	4872.00	0.649994	0.324997	−	0.945715i	$-0.394637\pi$
$383$	4872.00	0.649994	0.324997	+	0.945715i	$0.394637\pi$
$384$	−4977.00	−0.661410
$385$	0	0
$386$	7998.00	1.05463
$387$	−828.000	−0.108759
$388$	286.000	0.0374213
$389$	−14046.0	−1.83075	−0.915373	−	0.402606i	$-0.868104\pi$
$389$	−14046.0	−1.83075	−0.915373	+	0.402606i	$0.868104\pi$
$390$	0	0
$391$	−6480.00	−0.838127
$392$	1197.00	0.154229
$393$	6984.00	0.896428
$394$	−8154.00	−1.04262
$395$	0	0
$396$	−216.000	−0.0274101
$397$	2734.00	0.345631	0.172816	−	0.984954i	$-0.444714\pi$
$397$	2734.00	0.345631	0.172816	+	0.984954i	$0.444714\pi$
$398$	11496.0	1.44785
$399$	7440.00	0.933498
$400$	0	0
$401$	−15942.0	−1.98530	−0.992650	−	0.121019i	$-0.961384\pi$
$401$	−15942.0	−1.98530	−0.992650	+	0.121019i	$0.961384\pi$
$402$	−1764.00	−0.218857
$403$	−14800.0	−1.82938
$404$	−1734.00	−0.213539
$405$	0	0
$406$	−4680.00	−0.572080
$407$	−1680.00	−0.204606
$408$	−3402.00	−0.412804
$409$	8714.00	1.05350	0.526748	−	0.850022i	$-0.323411\pi$
$409$	8714.00	1.05350	0.526748	+	0.850022i	$0.323411\pi$
$410$	0	0
$411$	−6354.00	−0.762578
$412$	−452.000	−0.0540496
$413$	−480.000	−0.0571895
$414$	−3240.00	−0.384631
$415$	0	0
$416$	−3330.00	−0.392468
$417$	6972.00	0.818754
$418$	−8928.00	−1.04470
$419$	11976.0	1.39634	0.698169	−	0.715933i	$-0.253998\pi$
$419$	11976.0	1.39634	0.698169	+	0.715933i	$0.253998\pi$
$420$	0	0
$421$	11054.0	1.27967	0.639833	−	0.768514i	$-0.279004\pi$
$421$	11054.0	1.27967	0.639833	+	0.768514i	$0.279004\pi$
$422$	−3300.00	−0.380667
$423$	216.000	0.0248281
$424$	−9450.00	−1.08239
$425$	0	0
$426$	2592.00	0.294795
$427$	6440.00	0.729868
$428$	1404.00	0.158563
$429$	5328.00	0.599623
$430$	0	0
$431$	720.000	0.0804668	0.0402334	−	0.999190i	$-0.487190\pi$
$431$	720.000	0.0804668	0.0402334	+	0.999190i	$0.487190\pi$
$432$	−1917.00	−0.213499
$433$	15622.0	1.73382	0.866912	−	0.498462i	$-0.166102\pi$
$433$	15622.0	1.73382	0.866912	+	0.498462i	$0.166102\pi$
$434$	12000.0	1.32723
$435$	0	0
$436$	−1474.00	−0.161908
$437$	−14880.0	−1.62885
$438$	−3870.00	−0.422182
$439$	−9880.00	−1.07414	−0.537069	−	0.843538i	$-0.680469\pi$
$439$	−9880.00	−1.07414	−0.537069	+	0.843538i	$0.680469\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	−11988.0	−1.29007
$443$	16116.0	1.72843	0.864215	−	0.503123i	$-0.167816\pi$
$443$	16116.0	1.72843	0.864215	+	0.503123i	$0.167816\pi$
$444$	210.000	0.0224463
$445$	0	0
$446$	5892.00	0.625548
$447$	774.000	0.0818992
$448$	−8660.00	−0.913274
$449$	9018.00	0.947852	0.473926	−	0.880565i	$-0.342836\pi$
$449$	9018.00	0.947852	0.473926	+	0.880565i	$0.342836\pi$
$450$	0	0
$451$	−7920.00	−0.826914
$452$	−1086.00	−0.113011
$453$	−2424.00	−0.251412
$454$	1980.00	0.204683
$455$	0	0
$456$	−7812.00	−0.802260
$457$	3670.00	0.375657	0.187829	−	0.982202i	$-0.439855\pi$
$457$	3670.00	0.375657	0.187829	+	0.982202i	$0.439855\pi$
$458$	5718.00	0.583372
$459$	−1458.00	−0.148265
$460$	0	0
$461$	17562.0	1.77428	0.887141	−	0.461499i	$-0.152688\pi$
$461$	17562.0	1.77428	0.887141	+	0.461499i	$0.152688\pi$
$462$	−4320.00	−0.435032
$463$	−1172.00	−0.117640	−0.0588202	−	0.998269i	$-0.518734\pi$
$463$	−1172.00	−0.117640	−0.0588202	+	0.998269i	$0.518734\pi$
$464$	5538.00	0.554084
$465$	0	0
$466$	−4374.00	−0.434810
$467$	−6876.00	−0.681335	−0.340667	−	0.940184i	$-0.610653\pi$
$467$	−6876.00	−0.681335	−0.340667	+	0.940184i	$0.610653\pi$
$468$	−666.000	−0.0657818
$469$	−3920.00	−0.385946
$470$	0	0
$471$	−7134.00	−0.697914
$472$	504.000	0.0491493
$473$	2208.00	0.214638
$474$	4680.00	0.453501
$475$	0	0
$476$	1080.00	0.103995
$477$	−4050.00	−0.388756
$478$	−3528.00	−0.337588
$479$	2280.00	0.217486	0.108743	−	0.994070i	$-0.465317\pi$
$479$	2280.00	0.217486	0.108743	+	0.994070i	$0.465317\pi$
$480$	0	0
$481$	−5180.00	−0.491035
$482$	−2598.00	−0.245510
$483$	−7200.00	−0.678284
$484$	−755.000	−0.0709053
$485$	0	0
$486$	−729.000	−0.0680414
$487$	3076.00	0.286215	0.143108	−	0.989707i	$-0.454290\pi$
$487$	3076.00	0.286215	0.143108	+	0.989707i	$0.454290\pi$
$488$	−6762.00	−0.627257
$489$	156.000	0.0144265
$490$	0	0
$491$	−18912.0	−1.73826	−0.869131	−	0.494582i	$-0.835321\pi$
$491$	−18912.0	−1.73826	−0.869131	+	0.494582i	$0.835321\pi$
$492$	990.000	0.0907168
$493$	4212.00	0.384785
$494$	−27528.0	−2.50717
$495$	0	0
$496$	−14200.0	−1.28548
$497$	5760.00	0.519862
$498$	1404.00	0.126335
$499$	9956.00	0.893170	0.446585	−	0.894741i	$-0.352640\pi$
$499$	9956.00	0.893170	0.446585	+	0.894741i	$0.352640\pi$
$500$	0	0
$501$	11160.0	0.995194
$502$	−1296.00	−0.115226
$503$	10656.0	0.944588	0.472294	−	0.881441i	$-0.343426\pi$
$503$	10656.0	0.944588	0.472294	+	0.881441i	$0.343426\pi$
$504$	−3780.00	−0.334077
$505$	0	0
$506$	8640.00	0.759081
$507$	9837.00	0.861689
$508$	−1244.00	−0.108649
$509$	−2766.00	−0.240866	−0.120433	−	0.992721i	$-0.538428\pi$
$509$	−2766.00	−0.240866	−0.120433	+	0.992721i	$0.538428\pi$
$510$	0	0
$511$	−8600.00	−0.744504
$512$	8733.00	0.753804
$513$	−3348.00	−0.288144
$514$	7578.00	0.650294
$515$	0	0
$516$	−276.000	−0.0235469
$517$	−576.000	−0.0489989
$518$	4200.00	0.356250
$519$	−1278.00	−0.108089
$520$	0	0
$521$	10530.0	0.885466	0.442733	−	0.896654i	$-0.354009\pi$
$521$	10530.0	0.885466	0.442733	+	0.896654i	$0.354009\pi$
$522$	2106.00	0.176585
$523$	−12692.0	−1.06115	−0.530576	−	0.847637i	$-0.678024\pi$
$523$	−12692.0	−1.06115	−0.530576	+	0.847637i	$0.678024\pi$
$524$	2328.00	0.194082
$525$	0	0
$526$	16344.0	1.35481
$527$	−10800.0	−0.892705
$528$	5112.00	0.421347
$529$	2233.00	0.183529
$530$	0	0
$531$	216.000	0.0176527
$532$	2480.00	0.202108
$533$	−24420.0	−1.98452
$534$	−9234.00	−0.748304
$535$	0	0
$536$	4116.00	0.331687
$537$	−4320.00	−0.347154
$538$	7722.00	0.618809
$539$	−1368.00	−0.109321
$540$	0	0
$541$	18110.0	1.43920	0.719602	−	0.694386i	$-0.244324\pi$
$541$	18110.0	1.43920	0.719602	+	0.694386i	$0.244324\pi$
$542$	9552.00	0.756999
$543$	−9390.00	−0.742106
$544$	−2430.00	−0.191517
$545$	0	0
$546$	−13320.0	−1.04404
$547$	−3620.00	−0.282962	−0.141481	−	0.989941i	$-0.545186\pi$
$547$	−3620.00	−0.282962	−0.141481	+	0.989941i	$0.545186\pi$
$548$	−2118.00	−0.165103
$549$	−2898.00	−0.225289
$550$	0	0
$551$	9672.00	0.747806
$552$	7560.00	0.582926
$553$	10400.0	0.799734
$554$	11886.0	0.911530
$555$	0	0
$556$	2324.00	0.177265
$557$	14166.0	1.07762	0.538809	−	0.842428i	$-0.318875\pi$
$557$	14166.0	1.07762	0.538809	+	0.842428i	$0.318875\pi$
$558$	−5400.00	−0.409678
$559$	6808.00	0.515112
$560$	0	0
$561$	3888.00	0.292605
$562$	24858.0	1.86579
$563$	13404.0	1.00339	0.501697	−	0.865043i	$-0.332709\pi$
$563$	13404.0	1.00339	0.501697	+	0.865043i	$0.332709\pi$
$564$	72.0000	0.00537544
$565$	0	0
$566$	−8148.00	−0.605099
$567$	−1620.00	−0.119989
$568$	−6048.00	−0.446775
$569$	−18654.0	−1.37437	−0.687185	−	0.726483i	$-0.741154\pi$
$569$	−18654.0	−1.37437	−0.687185	+	0.726483i	$0.741154\pi$
$570$	0	0
$571$	−7684.00	−0.563162	−0.281581	−	0.959537i	$-0.590859\pi$
$571$	−7684.00	−0.563162	−0.281581	+	0.959537i	$0.590859\pi$
$572$	1776.00	0.129822
$573$	10728.0	0.782144
$574$	19800.0	1.43978
$575$	0	0
$576$	3897.00	0.281901
$577$	1726.00	0.124531	0.0622654	−	0.998060i	$-0.480167\pi$
$577$	1726.00	0.124531	0.0622654	+	0.998060i	$0.480167\pi$
$578$	5991.00	0.431129
$579$	−7998.00	−0.574068
$580$	0	0
$581$	3120.00	0.222787
$582$	−2574.00	−0.183326
$583$	10800.0	0.767222
$584$	9030.00	0.639836
$585$	0	0
$586$	18054.0	1.27270
$587$	−10596.0	−0.745049	−0.372524	−	0.928022i	$-0.621508\pi$
$587$	−10596.0	−0.745049	−0.372524	+	0.928022i	$0.621508\pi$
$588$	171.000	0.0119931
$589$	−24800.0	−1.73492
$590$	0	0
$591$	8154.00	0.567531
$592$	−4970.00	−0.345043
$593$	−2862.00	−0.198193	−0.0990963	−	0.995078i	$-0.531595\pi$
$593$	−2862.00	−0.198193	−0.0990963	+	0.995078i	$0.531595\pi$
$594$	1944.00	0.134282
$595$	0	0
$596$	258.000	0.0177317
$597$	−11496.0	−0.788107
$598$	26640.0	1.82172
$599$	−23592.0	−1.60925	−0.804627	−	0.593781i	$-0.797635\pi$
$599$	−23592.0	−1.60925	−0.804627	+	0.593781i	$0.797635\pi$
$600$	0	0
$601$	−9574.00	−0.649803	−0.324902	−	0.945748i	$-0.605331\pi$
$601$	−9574.00	−0.649803	−0.324902	+	0.945748i	$0.605331\pi$
$602$	−5520.00	−0.373718
$603$	1764.00	0.119130
$604$	−808.000	−0.0544322
$605$	0	0
$606$	15606.0	1.04612
$607$	−17444.0	−1.16644	−0.583221	−	0.812314i	$-0.698208\pi$
$607$	−17444.0	−1.16644	−0.583221	+	0.812314i	$0.698208\pi$
$608$	−5580.00	−0.372202
$609$	4680.00	0.311401
$610$	0	0
$611$	−1776.00	−0.117593
$612$	−486.000	−0.0321003
$613$	2374.00	0.156419	0.0782096	−	0.996937i	$-0.475080\pi$
$613$	2374.00	0.156419	0.0782096	+	0.996937i	$0.475080\pi$
$614$	27708.0	1.82118
$615$	0	0
$616$	10080.0	0.659310
$617$	12162.0	0.793555	0.396778	−	0.917915i	$-0.370128\pi$
$617$	12162.0	0.793555	0.396778	+	0.917915i	$0.370128\pi$
$618$	4068.00	0.264788
$619$	8804.00	0.571668	0.285834	−	0.958279i	$-0.407729\pi$
$619$	8804.00	0.571668	0.285834	+	0.958279i	$0.407729\pi$
$620$	0	0
$621$	3240.00	0.209367
$622$	−4608.00	−0.297048
$623$	−20520.0	−1.31961
$624$	15762.0	1.01119
$625$	0	0
$626$	−22026.0	−1.40629
$627$	8928.00	0.568660
$628$	−2378.00	−0.151103
$629$	−3780.00	−0.239616
$630$	0	0
$631$	−12688.0	−0.800478	−0.400239	−	0.916411i	$-0.631073\pi$
$631$	−12688.0	−0.800478	−0.400239	+	0.916411i	$0.631073\pi$
$632$	−10920.0	−0.687301
$633$	3300.00	0.207209
$634$	−11682.0	−0.731785
$635$	0	0
$636$	−1350.00	−0.0841682
$637$	−4218.00	−0.262360
$638$	−5616.00	−0.348495
$639$	−2592.00	−0.160466
$640$	0	0
$641$	−9150.00	−0.563812	−0.281906	−	0.959442i	$-0.590967\pi$
$641$	−9150.00	−0.563812	−0.281906	+	0.959442i	$0.590967\pi$
$642$	−12636.0	−0.776796
$643$	−25292.0	−1.55120	−0.775598	−	0.631227i	$-0.782552\pi$
$643$	−25292.0	−1.55120	−0.775598	+	0.631227i	$0.782552\pi$
$644$	−2400.00	−0.146853
$645$	0	0
$646$	−20088.0	−1.22345
$647$	2736.00	0.166249	0.0831246	−	0.996539i	$-0.473510\pi$
$647$	2736.00	0.166249	0.0831246	+	0.996539i	$0.473510\pi$
$648$	1701.00	0.103120
$649$	−576.000	−0.0348382
$650$	0	0
$651$	−12000.0	−0.722453
$652$	52.0000	0.00312343
$653$	−22218.0	−1.33148	−0.665741	−	0.746183i	$-0.731884\pi$
$653$	−22218.0	−1.33148	−0.665741	+	0.746183i	$0.731884\pi$
$654$	13266.0	0.793183
$655$	0	0
$656$	−23430.0	−1.39449
$657$	3870.00	0.229807
$658$	1440.00	0.0853147
$659$	14520.0	0.858299	0.429149	−	0.903234i	$-0.358813\pi$
$659$	14520.0	0.858299	0.429149	+	0.903234i	$0.358813\pi$
$660$	0	0
$661$	−10618.0	−0.624799	−0.312400	−	0.949951i	$-0.601133\pi$
$661$	−10618.0	−0.624799	−0.312400	+	0.949951i	$0.601133\pi$
$662$	−11076.0	−0.650273
$663$	11988.0	0.702225
$664$	−3276.00	−0.191466
$665$	0	0
$666$	−1890.00	−0.109964
$667$	−9360.00	−0.543359
$668$	3720.00	0.215466
$669$	−5892.00	−0.340505
$670$	0	0
$671$	7728.00	0.444614
$672$	−2700.00	−0.154992
$673$	−1370.00	−0.0784690	−0.0392345	−	0.999230i	$-0.512492\pi$
$673$	−1370.00	−0.0784690	−0.0392345	+	0.999230i	$0.512492\pi$
$674$	−26994.0	−1.54269
$675$	0	0
$676$	3279.00	0.186561
$677$	13758.0	0.781038	0.390519	−	0.920595i	$-0.372296\pi$
$677$	13758.0	0.781038	0.390519	+	0.920595i	$0.372296\pi$
$678$	9774.00	0.553640
$679$	−5720.00	−0.323289
$680$	0	0
$681$	−1980.00	−0.111415
$682$	14400.0	0.808511
$683$	−11988.0	−0.671608	−0.335804	−	0.941932i	$-0.609008\pi$
$683$	−11988.0	−0.671608	−0.335804	+	0.941932i	$0.609008\pi$
$684$	−1116.00	−0.0623850
$685$	0	0
$686$	−17160.0	−0.955061
$687$	−5718.00	−0.317548
$688$	6532.00	0.361962
$689$	33300.0	1.84126
$690$	0	0
$691$	32996.0	1.81654	0.908268	−	0.418388i	$-0.137405\pi$
$691$	32996.0	1.81654	0.908268	+	0.418388i	$0.137405\pi$
$692$	−426.000	−0.0234019
$693$	4320.00	0.236801
$694$	15732.0	0.860488
$695$	0	0
$696$	−4914.00	−0.267622
$697$	−17820.0	−0.968408
$698$	−18906.0	−1.02522
$699$	4374.00	0.236681
$700$	0	0
$701$	−25902.0	−1.39558	−0.697792	−	0.716300i	$-0.745834\pi$
$701$	−25902.0	−1.39558	−0.697792	+	0.716300i	$0.745834\pi$
$702$	5994.00	0.322263
$703$	−8680.00	−0.465679
$704$	−10392.0	−0.556340
$705$	0	0
$706$	10242.0	0.545981
$707$	34680.0	1.84480
$708$	72.0000	0.00382193
$709$	−27394.0	−1.45106	−0.725531	−	0.688189i	$-0.758406\pi$
$709$	−27394.0	−1.45106	−0.725531	+	0.688189i	$0.758406\pi$
$710$	0	0
$711$	−4680.00	−0.246855
$712$	21546.0	1.13409
$713$	24000.0	1.26060
$714$	−9720.00	−0.509470
$715$	0	0
$716$	−1440.00	−0.0751611
$717$	3528.00	0.183760
$718$	−14472.0	−0.752215
$719$	34848.0	1.80753	0.903763	−	0.428033i	$-0.140793\pi$
$719$	34848.0	1.80753	0.903763	+	0.428033i	$0.140793\pi$
$720$	0	0
$721$	9040.00	0.466945
$722$	−25551.0	−1.31705
$723$	2598.00	0.133639
$724$	−3130.00	−0.160671
$725$	0	0
$726$	6795.00	0.347364
$727$	−28028.0	−1.42985	−0.714925	−	0.699201i	$-0.753539\pi$
$727$	−28028.0	−1.42985	−0.714925	+	0.699201i	$0.753539\pi$
$728$	31080.0	1.58228
$729$	729.000	0.0370370
$730$	0	0
$731$	4968.00	0.251365
$732$	−966.000	−0.0487765
$733$	−18002.0	−0.907120	−0.453560	−	0.891226i	$-0.649846\pi$
$733$	−18002.0	−0.907120	−0.453560	+	0.891226i	$0.649846\pi$
$734$	−10524.0	−0.529221
$735$	0	0
$736$	5400.00	0.270444
$737$	−4704.00	−0.235107
$738$	−8910.00	−0.444420
$739$	15284.0	0.760800	0.380400	−	0.924822i	$-0.375786\pi$
$739$	15284.0	0.760800	0.380400	+	0.924822i	$0.375786\pi$
$740$	0	0
$741$	27528.0	1.36473
$742$	−27000.0	−1.33585
$743$	18768.0	0.926691	0.463345	−	0.886178i	$-0.346649\pi$
$743$	18768.0	0.926691	0.463345	+	0.886178i	$0.346649\pi$
$744$	12600.0	0.620885
$745$	0	0
$746$	32406.0	1.59044
$747$	−1404.00	−0.0687680
$748$	1296.00	0.0633509
$749$	−28080.0	−1.36985
$750$	0	0
$751$	8696.00	0.422532	0.211266	−	0.977429i	$-0.432241\pi$
$751$	8696.00	0.422532	0.211266	+	0.977429i	$0.432241\pi$
$752$	−1704.00	−0.0826310
$753$	1296.00	0.0627209
$754$	−17316.0	−0.836355
$755$	0	0
$756$	−540.000	−0.0259783
$757$	38662.0	1.85627	0.928134	−	0.372247i	$-0.121413\pi$
$757$	38662.0	1.85627	0.928134	+	0.372247i	$0.121413\pi$
$758$	−4380.00	−0.209880
$759$	−8640.00	−0.413191
$760$	0	0
$761$	23874.0	1.13723	0.568615	−	0.822604i	$-0.307479\pi$
$761$	23874.0	1.13723	0.568615	+	0.822604i	$0.307479\pi$
$762$	11196.0	0.532268
$763$	29480.0	1.39875
$764$	3576.00	0.169339
$765$	0	0
$766$	−14616.0	−0.689422
$767$	−1776.00	−0.0836084
$768$	4539.00	0.213264
$769$	23618.0	1.10753	0.553763	−	0.832675i	$-0.313192\pi$
$769$	23618.0	1.10753	0.553763	+	0.832675i	$0.313192\pi$
$770$	0	0
$771$	−7578.00	−0.353975
$772$	−2666.00	−0.124289
$773$	−11538.0	−0.536860	−0.268430	−	0.963299i	$-0.586505\pi$
$773$	−11538.0	−0.536860	−0.268430	+	0.963299i	$0.586505\pi$
$774$	2484.00	0.115356
$775$	0	0
$776$	6006.00	0.277839
$777$	−4200.00	−0.193918
$778$	42138.0	1.94180
$779$	−40920.0	−1.88204
$780$	0	0
$781$	6912.00	0.316685
$782$	19440.0	0.888968
$783$	−2106.00	−0.0961204
$784$	−4047.00	−0.184357
$785$	0	0
$786$	−20952.0	−0.950805
$787$	14884.0	0.674152	0.337076	−	0.941478i	$-0.390562\pi$
$787$	14884.0	0.674152	0.337076	+	0.941478i	$0.390562\pi$
$788$	2718.00	0.122874
$789$	−16344.0	−0.737467
$790$	0	0
$791$	21720.0	0.976327
$792$	−4536.00	−0.203510
$793$	23828.0	1.06703
$794$	−8202.00	−0.366597
$795$	0	0
$796$	−3832.00	−0.170630
$797$	11334.0	0.503728	0.251864	−	0.967763i	$-0.418957\pi$
$797$	11334.0	0.503728	0.251864	+	0.967763i	$0.418957\pi$
$798$	−22320.0	−0.990125
$799$	−1296.00	−0.0573832
$800$	0	0
$801$	9234.00	0.407325
$802$	47826.0	2.10573
$803$	−10320.0	−0.453530
$804$	588.000	0.0257925
$805$	0	0
$806$	44400.0	1.94035
$807$	−7722.00	−0.336837
$808$	−36414.0	−1.58545
$809$	44730.0	1.94391	0.971955	−	0.235167i	$-0.0755638\pi$
$809$	44730.0	1.94391	0.971955	+	0.235167i	$0.0755638\pi$
$810$	0	0
$811$	−42748.0	−1.85091	−0.925453	−	0.378862i	$-0.876316\pi$
$811$	−42748.0	−1.85091	−0.925453	+	0.378862i	$0.876316\pi$
$812$	1560.00	0.0674203
$813$	−9552.00	−0.412058
$814$	5040.00	0.217017
$815$	0	0
$816$	11502.0	0.493444
$817$	11408.0	0.488513
$818$	−26142.0	−1.11740
$819$	13320.0	0.568301
$820$	0	0
$821$	−31686.0	−1.34695	−0.673477	−	0.739208i	$-0.735200\pi$
$821$	−31686.0	−1.34695	−0.673477	+	0.739208i	$0.735200\pi$
$822$	19062.0	0.808836
$823$	−11036.0	−0.467425	−0.233713	−	0.972306i	$-0.575087\pi$
$823$	−11036.0	−0.467425	−0.233713	+	0.972306i	$0.575087\pi$
$824$	−9492.00	−0.401298
$825$	0	0
$826$	1440.00	0.0606586
$827$	−25884.0	−1.08836	−0.544181	−	0.838968i	$-0.683159\pi$
$827$	−25884.0	−1.08836	−0.544181	+	0.838968i	$0.683159\pi$
$828$	1080.00	0.0453292
$829$	15950.0	0.668234	0.334117	−	0.942532i	$-0.391562\pi$
$829$	15950.0	0.668234	0.334117	+	0.942532i	$0.391562\pi$
$830$	0	0
$831$	−11886.0	−0.496174
$832$	−32042.0	−1.33516
$833$	−3078.00	−0.128027
$834$	−20916.0	−0.868419
$835$	0	0
$836$	2976.00	0.123119
$837$	5400.00	0.223000
$838$	−35928.0	−1.48104
$839$	13800.0	0.567853	0.283927	−	0.958846i	$-0.408363\pi$
$839$	13800.0	0.567853	0.283927	+	0.958846i	$0.408363\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	−33162.0	−1.35729
$843$	−24858.0	−1.01560
$844$	1100.00	0.0448620
$845$	0	0
$846$	−648.000	−0.0263342
$847$	15100.0	0.612565
$848$	31950.0	1.29383
$849$	8148.00	0.329374
$850$	0	0
$851$	8400.00	0.338365
$852$	−864.000	−0.0347420
$853$	27862.0	1.11838	0.559189	−	0.829040i	$-0.311113\pi$
$853$	27862.0	1.11838	0.559189	+	0.829040i	$0.311113\pi$
$854$	−19320.0	−0.774141
$855$	0	0
$856$	29484.0	1.17727
$857$	7314.00	0.291530	0.145765	−	0.989319i	$-0.453436\pi$
$857$	7314.00	0.291530	0.145765	+	0.989319i	$0.453436\pi$
$858$	−15984.0	−0.635996
$859$	−28780.0	−1.14314	−0.571572	−	0.820552i	$-0.693666\pi$
$859$	−28780.0	−1.14314	−0.571572	+	0.820552i	$0.693666\pi$
$860$	0	0
$861$	−19800.0	−0.783719
$862$	−2160.00	−0.0853479
$863$	32688.0	1.28935	0.644677	−	0.764455i	$-0.276992\pi$
$863$	32688.0	1.28935	0.644677	+	0.764455i	$0.276992\pi$
$864$	1215.00	0.0478416
$865$	0	0
$866$	−46866.0	−1.83900
$867$	−5991.00	−0.234677
$868$	−4000.00	−0.156416
$869$	12480.0	0.487175
$870$	0	0
$871$	−14504.0	−0.564236
$872$	−30954.0	−1.20210
$873$	2574.00	0.0997900
$874$	44640.0	1.72766
$875$	0	0
$876$	1290.00	0.0497546
$877$	−36650.0	−1.41115	−0.705577	−	0.708633i	$-0.749312\pi$
$877$	−36650.0	−1.41115	−0.705577	+	0.708633i	$0.749312\pi$
$878$	29640.0	1.13930
$879$	−18054.0	−0.692772
$880$	0	0
$881$	−2646.00	−0.101187	−0.0505936	−	0.998719i	$-0.516111\pi$
$881$	−2646.00	−0.101187	−0.0505936	+	0.998719i	$0.516111\pi$
$882$	−1539.00	−0.0587538
$883$	−10892.0	−0.415113	−0.207557	−	0.978223i	$-0.566551\pi$
$883$	−10892.0	−0.415113	−0.207557	+	0.978223i	$0.566551\pi$
$884$	3996.00	0.152036
$885$	0	0
$886$	−48348.0	−1.83328
$887$	43464.0	1.64530	0.822648	−	0.568550i	$-0.192496\pi$
$887$	43464.0	1.64530	0.822648	+	0.568550i	$0.192496\pi$
$888$	4410.00	0.166655
$889$	24880.0	0.938637
$890$	0	0
$891$	−1944.00	−0.0730937
$892$	−1964.00	−0.0737215
$893$	−2976.00	−0.111521
$894$	−2322.00	−0.0868672
$895$	0	0
$896$	33180.0	1.23713
$897$	−26640.0	−0.991621
$898$	−27054.0	−1.00535
$899$	−15600.0	−0.578742
$900$	0	0
$901$	24300.0	0.898502
$902$	23760.0	0.877075
$903$	5520.00	0.203426
$904$	−22806.0	−0.839067
$905$	0	0
$906$	7272.00	0.266662
$907$	14884.0	0.544890	0.272445	−	0.962171i	$-0.412168\pi$
$907$	14884.0	0.544890	0.272445	+	0.962171i	$0.412168\pi$
$908$	−660.000	−0.0241221
$909$	−15606.0	−0.569437
$910$	0	0
$911$	−1248.00	−0.0453876	−0.0226938	−	0.999742i	$-0.507224\pi$
$911$	−1248.00	−0.0453876	−0.0226938	+	0.999742i	$0.507224\pi$
$912$	26412.0	0.958979
$913$	3744.00	0.135716
$914$	−11010.0	−0.398445
$915$	0	0
$916$	−1906.00	−0.0687511
$917$	−46560.0	−1.67671
$918$	4374.00	0.157259
$919$	−6640.00	−0.238339	−0.119169	−	0.992874i	$-0.538023\pi$
$919$	−6640.00	−0.238339	−0.119169	+	0.992874i	$0.538023\pi$
$920$	0	0
$921$	−27708.0	−0.991324
$922$	−52686.0	−1.88191
$923$	21312.0	0.760014
$924$	1440.00	0.0512690
$925$	0	0
$926$	3516.00	0.124776
$927$	−4068.00	−0.144132
$928$	−3510.00	−0.124161
$929$	29946.0	1.05758	0.528792	−	0.848751i	$-0.322645\pi$
$929$	29946.0	1.05758	0.528792	+	0.848751i	$0.322645\pi$
$930$	0	0
$931$	−7068.00	−0.248812
$932$	1458.00	0.0512429
$933$	4608.00	0.161693
$934$	20628.0	0.722665
$935$	0	0
$936$	−13986.0	−0.488405
$937$	−45002.0	−1.56900	−0.784499	−	0.620130i	$-0.787080\pi$
$937$	−45002.0	−1.56900	−0.784499	+	0.620130i	$0.787080\pi$
$938$	11760.0	0.409358
$939$	22026.0	0.765486
$940$	0	0
$941$	6090.00	0.210976	0.105488	−	0.994421i	$-0.466360\pi$
$941$	6090.00	0.210976	0.105488	+	0.994421i	$0.466360\pi$
$942$	21402.0	0.740249
$943$	39600.0	1.36750
$944$	−1704.00	−0.0587505
$945$	0	0
$946$	−6624.00	−0.227658
$947$	−56388.0	−1.93491	−0.967457	−	0.253035i	$-0.918571\pi$
$947$	−56388.0	−1.93491	−0.967457	+	0.253035i	$0.918571\pi$
$948$	−1560.00	−0.0534456
$949$	−31820.0	−1.08843
$950$	0	0
$951$	11682.0	0.398333
$952$	22680.0	0.772125
$953$	−10854.0	−0.368936	−0.184468	−	0.982839i	$-0.559056\pi$
$953$	−10854.0	−0.368936	−0.184468	+	0.982839i	$0.559056\pi$
$954$	12150.0	0.412338
$955$	0	0
$956$	1176.00	0.0397851
$957$	5616.00	0.189696
$958$	−6840.00	−0.230679
$959$	42360.0	1.42636
$960$	0	0
$961$	10209.0	0.342687
$962$	15540.0	0.520821
$963$	12636.0	0.422834
$964$	866.000	0.0289336
$965$	0	0
$966$	21600.0	0.719429
$967$	42316.0	1.40723	0.703615	−	0.710582i	$-0.251568\pi$
$967$	42316.0	1.40723	0.703615	+	0.710582i	$0.251568\pi$
$968$	−15855.0	−0.526445
$969$	20088.0	0.665964
$970$	0	0
$971$	24480.0	0.809063	0.404532	−	0.914524i	$-0.367435\pi$
$971$	24480.0	0.809063	0.404532	+	0.914524i	$0.367435\pi$
$972$	243.000	0.00801875
$973$	−46480.0	−1.53143
$974$	−9228.00	−0.303577
$975$	0	0
$976$	22862.0	0.749790
$977$	6906.00	0.226144	0.113072	−	0.993587i	$-0.463931\pi$
$977$	6906.00	0.226144	0.113072	+	0.993587i	$0.463931\pi$
$978$	−468.000	−0.0153016
$979$	−24624.0	−0.803868
$980$	0	0
$981$	−13266.0	−0.431754
$982$	56736.0	1.84371
$983$	−6960.00	−0.225829	−0.112914	−	0.993605i	$-0.536019\pi$
$983$	−6960.00	−0.225829	−0.112914	+	0.993605i	$0.536019\pi$
$984$	20790.0	0.673538
$985$	0	0
$986$	−12636.0	−0.408126
$987$	−1440.00	−0.0464394
$988$	9176.00	0.295473
$989$	−11040.0	−0.354956
$990$	0	0
$991$	47792.0	1.53195	0.765975	−	0.642870i	$-0.222256\pi$
$991$	47792.0	1.53195	0.765975	+	0.642870i	$0.222256\pi$
$992$	9000.00	0.288055
$993$	11076.0	0.353964
$994$	−17280.0	−0.551397
$995$	0	0
$996$	−468.000	−0.0148887
$997$	−9938.00	−0.315687	−0.157843	−	0.987464i	$-0.550454\pi$
$997$	−9938.00	−0.315687	−0.157843	+	0.987464i	$0.550454\pi$
$998$	−29868.0	−0.947350
$999$	1890.00	0.0598568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.a.a.1.1		1
3.2	odd	2		225.4.a.g.1.1		1
4.3	odd	2		1200.4.a.o.1.1		1
5.2	odd	4		75.4.b.a.49.1		2
5.3	odd	4		75.4.b.a.49.2		2
5.4	even	2		15.4.a.b.1.1	✓	1
15.2	even	4		225.4.b.d.199.2		2
15.8	even	4		225.4.b.d.199.1		2
15.14	odd	2		45.4.a.b.1.1		1
20.3	even	4		1200.4.f.m.49.1		2
20.7	even	4		1200.4.f.m.49.2		2
20.19	odd	2		240.4.a.f.1.1		1
35.34	odd	2		735.4.a.i.1.1		1
40.19	odd	2		960.4.a.l.1.1		1
40.29	even	2		960.4.a.bi.1.1		1
45.4	even	6		405.4.e.d.136.1		2
45.14	odd	6		405.4.e.k.136.1		2
45.29	odd	6		405.4.e.k.271.1		2
45.34	even	6		405.4.e.d.271.1		2
55.54	odd	2		1815.4.a.a.1.1		1
60.59	even	2		720.4.a.r.1.1		1
105.104	even	2		2205.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	5.4	even	2
45.4.a.b.1.1		1	15.14	odd	2
75.4.a.a.1.1		1	1.1	even	1	trivial
75.4.b.a.49.1		2	5.2	odd	4
75.4.b.a.49.2		2	5.3	odd	4
225.4.a.g.1.1		1	3.2	odd	2
225.4.b.d.199.1		2	15.8	even	4
225.4.b.d.199.2		2	15.2	even	4
240.4.a.f.1.1		1	20.19	odd	2
405.4.e.d.136.1		2	45.4	even	6
405.4.e.d.271.1		2	45.34	even	6
405.4.e.k.136.1		2	45.14	odd	6
405.4.e.k.271.1		2	45.29	odd	6
720.4.a.r.1.1		1	60.59	even	2
735.4.a.i.1.1		1	35.34	odd	2
960.4.a.l.1.1		1	40.19	odd	2
960.4.a.bi.1.1		1	40.29	even	2
1200.4.a.o.1.1		1	4.3	odd	2
1200.4.f.m.49.1		2	20.3	even	4
1200.4.f.m.49.2		2	20.7	even	4
1815.4.a.a.1.1		1	55.54	odd	2
2205.4.a.c.1.1		1	105.104	even	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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