LMFDB - Embedded newform 75.4.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,4,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$N$	$=$	$75 = 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 10$ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$3.70156$ of defining polynomial
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)

$f(q)$	$=$	$q+4.70156 q^{2} +3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} -16.2094 q^{7} +28.7016 q^{8} +9.00000 q^{9} -40.2094 q^{11} +42.3141 q^{12} +19.7906 q^{13} -76.2094 q^{14} +22.1047 q^{16} +83.0156 q^{17} +42.3141 q^{18} -48.8375 q^{19} -48.6281 q^{21} -189.047 q^{22} +1.61250 q^{23} +86.1047 q^{24} +93.0469 q^{26} +27.0000 q^{27} -228.628 q^{28} -24.5344 q^{29} -12.4187 q^{31} -125.686 q^{32} -120.628 q^{33} +390.303 q^{34} +126.942 q^{36} +325.884 q^{37} -229.612 q^{38} +59.3719 q^{39} -242.419 q^{41} -228.628 q^{42} +367.350 q^{43} -567.141 q^{44} +7.58125 q^{46} -204.544 q^{47} +66.3141 q^{48} -80.2562 q^{49} +249.047 q^{51} +279.141 q^{52} -61.5281 q^{53} +126.942 q^{54} -465.234 q^{56} -146.512 q^{57} -115.350 q^{58} -112.209 q^{59} +477.350 q^{61} -58.3875 q^{62} -145.884 q^{63} -767.758 q^{64} -567.141 q^{66} +558.094 q^{67} +1170.91 q^{68} +4.83749 q^{69} +558.281 q^{71} +258.314 q^{72} +1011.77 q^{73} +1532.17 q^{74} -688.837 q^{76} +651.769 q^{77} +279.141 q^{78} +1150.47 q^{79} +81.0000 q^{81} -1139.75 q^{82} -1157.92 q^{83} -685.884 q^{84} +1727.12 q^{86} -73.6032 q^{87} -1154.07 q^{88} +96.9751 q^{89} -320.794 q^{91} +22.7438 q^{92} -37.2562 q^{93} -961.675 q^{94} -377.058 q^{96} -1152.37 q^{97} -377.330 q^{98} -361.884 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9} - 42 q^{11} + 27 q^{12} + 78 q^{13} - 114 q^{14} + 25 q^{16} + 102 q^{17} + 27 q^{18} + 56 q^{19} + 18 q^{21} - 186 q^{22} - 48 q^{23}+ \cdots - 378 q^{99}+O(q^{100})$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 + 9 * q^6 + 6 * q^7 + 51 * q^8 + 18 * q^9 - 42 * q^11 + 27 * q^12 + 78 * q^13 - 114 * q^14 + 25 * q^16 + 102 * q^17 + 27 * q^18 + 56 * q^19 + 18 * q^21 - 186 * q^22 - 48 * q^23 + 153 * q^24 - 6 * q^26 + 54 * q^27 - 342 * q^28 - 318 * q^29 + 52 * q^31 - 309 * q^32 - 126 * q^33 + 358 * q^34 + 81 * q^36 + 306 * q^37 - 408 * q^38 + 234 * q^39 - 408 * q^41 - 342 * q^42 + 120 * q^43 - 558 * q^44 + 92 * q^46 + 180 * q^47 + 75 * q^48 + 70 * q^49 + 306 * q^51 - 18 * q^52 + 402 * q^53 + 81 * q^54 + 30 * q^56 + 168 * q^57 + 384 * q^58 - 186 * q^59 + 340 * q^61 - 168 * q^62 + 54 * q^63 - 479 * q^64 - 558 * q^66 + 732 * q^67 + 1074 * q^68 - 144 * q^69 - 36 * q^71 + 459 * q^72 + 1332 * q^73 + 1566 * q^74 - 1224 * q^76 + 612 * q^77 - 18 * q^78 + 380 * q^79 + 162 * q^81 - 858 * q^82 - 984 * q^83 - 1026 * q^84 + 2148 * q^86 - 954 * q^87 - 1194 * q^88 + 1116 * q^89 + 972 * q^91 + 276 * q^92 + 156 * q^93 - 1616 * q^94 - 927 * q^96 - 768 * q^97 - 633 * q^98 - 378 * q^99

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

)

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

$2$	4.70156	1.66225	0.831127	−	0.556083i	$-0.187696\pi$
$2$	4.70156	1.66225	0.831127	+	0.556083i	$0.187696\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	14.1047	1.76309
$5$	0	0
$5$	0	0
$6$	14.1047	0.959702
$7$	−16.2094	−0.875224	−0.437612	−	0.899164i	$-0.644176\pi$
$7$	−16.2094	−0.875224	−0.437612	+	0.899164i	$0.644176\pi$
$8$	28.7016	1.26844
$9$	9.00000	0.333333
$10$	0	0
$11$	−40.2094	−1.10214	−0.551072	−	0.834458i	$-0.685781\pi$
$11$	−40.2094	−1.10214	−0.551072	+	0.834458i	$0.685781\pi$
$12$	42.3141	1.01792
$13$	19.7906	0.422226	0.211113	−	0.977462i	$-0.432291\pi$
$13$	19.7906	0.422226	0.211113	+	0.977462i	$0.432291\pi$
$14$	−76.2094	−1.45484
$15$	0	0
$16$	22.1047	0.345386
$17$	83.0156	1.18437	0.592184	−	0.805803i	$-0.298266\pi$
$17$	83.0156	1.18437	0.592184	+	0.805803i	$0.298266\pi$
$18$	42.3141	0.554084
$19$	−48.8375	−0.589689	−0.294844	−	0.955545i	$-0.595268\pi$
$19$	−48.8375	−0.589689	−0.294844	+	0.955545i	$0.595268\pi$
$20$	0	0
$21$	−48.6281	−0.505311
$22$	−189.047	−1.83204
$23$	1.61250	0.0146186	0.00730932	−	0.999973i	$-0.497673\pi$
$23$	1.61250	0.0146186	0.00730932	+	0.999973i	$0.497673\pi$
$24$	86.1047	0.732335
$25$	0	0
$26$	93.0469	0.701846
$27$	27.0000	0.192450
$28$	−228.628	−1.54309
$29$	−24.5344	−0.157101	−0.0785504	−	0.996910i	$-0.525029\pi$
$29$	−24.5344	−0.157101	−0.0785504	+	0.996910i	$0.525029\pi$
$30$	0	0
$31$	−12.4187	−0.0719507	−0.0359754	−	0.999353i	$-0.511454\pi$
$31$	−12.4187	−0.0719507	−0.0359754	+	0.999353i	$0.511454\pi$
$32$	−125.686	−0.694323
$33$	−120.628	−0.636323
$34$	390.303	1.96872
$35$	0	0
$36$	126.942	0.587695
$37$	325.884	1.44797	0.723987	−	0.689813i	$-0.242307\pi$
$37$	325.884	1.44797	0.723987	+	0.689813i	$0.242307\pi$
$38$	−229.612	−0.980212
$39$	59.3719	0.243772
$40$	0	0
$41$	−242.419	−0.923401	−0.461701	−	0.887036i	$-0.652761\pi$
$41$	−242.419	−0.923401	−0.461701	+	0.887036i	$0.652761\pi$
$42$	−228.628	−0.839954
$43$	367.350	1.30280	0.651399	−	0.758735i	$-0.274182\pi$
$43$	367.350	1.30280	0.651399	+	0.758735i	$0.274182\pi$
$44$	−567.141	−1.94317
$45$	0	0
$46$	7.58125	0.0242999
$47$	−204.544	−0.634804	−0.317402	−	0.948291i	$-0.602810\pi$
$47$	−204.544	−0.634804	−0.317402	+	0.948291i	$0.602810\pi$
$48$	66.3141	0.199409
$49$	−80.2562	−0.233983
$50$	0	0
$51$	249.047	0.683795
$52$	279.141	0.744420
$53$	−61.5281	−0.159463	−0.0797314	−	0.996816i	$-0.525406\pi$
$53$	−61.5281	−0.159463	−0.0797314	+	0.996816i	$0.525406\pi$
$54$	126.942	0.319901
$55$	0	0
$56$	−465.234	−1.11017
$57$	−146.512	−0.340457
$58$	−115.350	−0.261141
$59$	−112.209	−0.247600	−0.123800	−	0.992307i	$-0.539508\pi$
$59$	−112.209	−0.247600	−0.123800	+	0.992307i	$0.539508\pi$
$60$	0	0
$61$	477.350	1.00194	0.500970	−	0.865464i	$-0.332977\pi$
$61$	477.350	1.00194	0.500970	+	0.865464i	$0.332977\pi$
$62$	−58.3875	−0.119600
$63$	−145.884	−0.291741
$64$	−767.758	−1.49953
$65$	0	0
$66$	−567.141	−1.05773
$67$	558.094	1.01764	0.508821	−	0.860872i	$-0.330082\pi$
$67$	558.094	1.01764	0.508821	+	0.860872i	$0.330082\pi$
$68$	1170.91	2.08814
$69$	4.83749	0.00844008
$70$	0	0
$71$	558.281	0.933180	0.466590	−	0.884474i	$-0.345482\pi$
$71$	558.281	0.933180	0.466590	+	0.884474i	$0.345482\pi$
$72$	258.314	0.422814
$73$	1011.77	1.62217	0.811086	−	0.584927i	$-0.198877\pi$
$73$	1011.77	1.62217	0.811086	+	0.584927i	$0.198877\pi$
$74$	1532.17	2.40690
$75$	0	0
$76$	−688.837	−1.03967
$77$	651.769	0.964623
$78$	279.141	0.405211
$79$	1150.47	1.63845	0.819227	−	0.573470i	$-0.194403\pi$
$79$	1150.47	1.63845	0.819227	+	0.573470i	$0.194403\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−1139.75	−1.53493
$83$	−1157.92	−1.53131	−0.765655	−	0.643251i	$-0.777585\pi$
$83$	−1157.92	−1.53131	−0.765655	+	0.643251i	$0.777585\pi$
$84$	−685.884	−0.890906
$85$	0	0
$86$	1727.12	2.16558
$87$	−73.6032	−0.0907022
$88$	−1154.07	−1.39801
$89$	96.9751	0.115498	0.0577491	−	0.998331i	$-0.481608\pi$
$89$	96.9751	0.115498	0.0577491	+	0.998331i	$0.481608\pi$
$90$	0	0
$91$	−320.794	−0.369542
$92$	22.7438	0.0257739
$93$	−37.2562	−0.0415408
$94$	−961.675	−1.05520
$95$	0	0
$96$	−377.058	−0.400868
$97$	−1152.37	−1.20625	−0.603123	−	0.797648i	$-0.706077\pi$
$97$	−1152.37	−1.20625	−0.603123	+	0.797648i	$0.706077\pi$
$98$	−377.330	−0.388939
$99$	−361.884	−0.367381
$100$	0	0
$101$	−1156.49	−1.13936	−0.569679	−	0.821867i	$-0.692932\pi$
$101$	−1156.49	−1.13936	−0.569679	+	0.821867i	$0.692932\pi$
$102$	1170.91	1.13664
$103$	−1333.70	−1.27585	−0.637927	−	0.770096i	$-0.720208\pi$
$103$	−1333.70	−1.27585	−0.637927	+	0.770096i	$0.720208\pi$
$104$	568.022	0.535569
$105$	0	0
$106$	−289.278	−0.265068
$107$	−798.263	−0.721224	−0.360612	−	0.932716i	$-0.617432\pi$
$107$	−798.263	−0.721224	−0.360612	+	0.932716i	$0.617432\pi$
$108$	380.827	0.339306
$109$	−985.119	−0.865663	−0.432831	−	0.901475i	$-0.642485\pi$
$109$	−985.119	−0.865663	−0.432831	+	0.901475i	$0.642485\pi$
$110$	0	0
$111$	977.653	0.835988
$112$	−358.303	−0.302290
$113$	1888.25	1.57196	0.785979	−	0.618253i	$-0.212159\pi$
$113$	1888.25	1.57196	0.785979	+	0.618253i	$0.212159\pi$
$114$	−688.837	−0.565926
$115$	0	0
$116$	−346.050	−0.276982
$117$	178.116	0.140742
$118$	−527.559	−0.411574
$119$	−1345.63	−1.03659
$120$	0	0
$121$	285.794	0.214721
$122$	2244.29	1.66548
$123$	−727.256	−0.533126
$124$	−175.163	−0.126855
$125$	0	0
$126$	−685.884	−0.484948
$127$	−620.859	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$127$	−620.859	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$128$	−2604.17	−1.79827
$129$	1102.05	0.752171
$130$	0	0
$131$	−2588.35	−1.72630	−0.863151	−	0.504947i	$-0.831512\pi$
$131$	−2588.35	−1.72630	−0.863151	+	0.504947i	$0.831512\pi$
$132$	−1701.42	−1.12189
$133$	791.625	0.516110
$134$	2623.91	1.69158
$135$	0	0
$136$	2382.68	1.50230
$137$	−1656.29	−1.03289	−0.516447	−	0.856319i	$-0.672746\pi$
$137$	−1656.29	−1.03289	−0.516447	+	0.856319i	$0.672746\pi$
$138$	22.7438	0.0140295
$139$	−153.256	−0.0935182	−0.0467591	−	0.998906i	$-0.514889\pi$
$139$	−153.256	−0.0935182	−0.0467591	+	0.998906i	$0.514889\pi$
$140$	0	0
$141$	−613.631	−0.366504
$142$	2624.79	1.55118
$143$	−795.769	−0.465353
$144$	198.942	0.115129
$145$	0	0
$146$	4756.89	2.69646
$147$	−240.769	−0.135090
$148$	4596.50	2.55290
$149$	1483.38	0.815591	0.407795	−	0.913073i	$-0.366298\pi$
$149$	1483.38	0.815591	0.407795	+	0.913073i	$0.366298\pi$
$150$	0	0
$151$	−394.281	−0.212491	−0.106246	−	0.994340i	$-0.533883\pi$
$151$	−394.281	−0.212491	−0.106246	+	0.994340i	$0.533883\pi$
$152$	−1401.71	−0.747986
$153$	747.141	0.394789
$154$	3064.33	1.60345
$155$	0	0
$156$	837.422	0.429791
$157$	1727.05	0.877922	0.438961	−	0.898506i	$-0.355347\pi$
$157$	1727.05	0.877922	0.438961	+	0.898506i	$0.355347\pi$
$158$	5409.00	2.72352
$159$	−184.584	−0.0920659
$160$	0	0
$161$	−26.1376	−0.0127946
$162$	380.827	0.184695
$163$	−2034.28	−0.977529	−0.488764	−	0.872416i	$-0.662552\pi$
$163$	−2034.28	−0.977529	−0.488764	+	0.872416i	$0.662552\pi$
$164$	−3419.24	−1.62804
$165$	0	0
$166$	−5444.06	−2.54543
$167$	192.900	0.0893835	0.0446918	−	0.999001i	$-0.485769\pi$
$167$	192.900	0.0893835	0.0446918	+	0.999001i	$0.485769\pi$
$168$	−1395.70	−0.640957
$169$	−1805.33	−0.821726
$170$	0	0
$171$	−439.537	−0.196563
$172$	5181.36	2.29695
$173$	−1239.91	−0.544905	−0.272452	−	0.962169i	$-0.587835\pi$
$173$	−1239.91	−0.544905	−0.272452	+	0.962169i	$0.587835\pi$
$174$	−346.050	−0.150770
$175$	0	0
$176$	−888.816	−0.380665
$177$	−336.628	−0.142952
$178$	455.934	0.191987
$179$	−2636.86	−1.10105	−0.550525	−	0.834818i	$-0.685573\pi$
$179$	−2636.86	−1.10105	−0.550525	+	0.834818i	$0.685573\pi$
$180$	0	0
$181$	3317.58	1.36240	0.681199	−	0.732099i	$-0.261459\pi$
$181$	3317.58	1.36240	0.681199	+	0.732099i	$0.261459\pi$
$182$	−1508.23	−0.614272
$183$	1432.05	0.578471
$184$	46.2812	0.0185429
$185$	0	0
$186$	−175.163	−0.0690513
$187$	−3338.01	−1.30534
$188$	−2885.02	−1.11921
$189$	−437.653	−0.168437
$190$	0	0
$191$	−624.506	−0.236585	−0.118292	−	0.992979i	$-0.537742\pi$
$191$	−624.506	−0.236585	−0.118292	+	0.992979i	$0.537742\pi$
$192$	−2303.27	−0.865752
$193$	−436.144	−0.162665	−0.0813324	−	0.996687i	$-0.525918\pi$
$193$	−436.144	−0.162665	−0.0813324	+	0.996687i	$0.525918\pi$
$194$	−5417.96	−2.00509
$195$	0	0
$196$	−1131.99	−0.412532
$197$	3355.81	1.21366	0.606831	−	0.794831i	$-0.292440\pi$
$197$	3355.81	1.21366	0.606831	+	0.794831i	$0.292440\pi$
$198$	−1701.42	−0.610681
$199$	3799.77	1.35356	0.676780	−	0.736185i	$-0.263375\pi$
$199$	3799.77	1.35356	0.676780	+	0.736185i	$0.263375\pi$
$200$	0	0
$201$	1674.28	0.587536
$202$	−5437.31	−1.89390
$203$	397.687	0.137498
$204$	3512.73	1.20559
$205$	0	0
$206$	−6270.46	−2.12079
$207$	14.5125	0.00487288
$208$	437.466	0.145831
$209$	1963.72	0.649922
$210$	0	0
$211$	2365.27	0.771715	0.385857	−	0.922558i	$-0.373906\pi$
$211$	2365.27	0.771715	0.385857	+	0.922558i	$0.373906\pi$
$212$	−867.834	−0.281147
$213$	1674.84	0.538772
$214$	−3753.08	−1.19886
$215$	0	0
$216$	774.942	0.244112
$217$	201.300	0.0629730
$218$	−4631.60	−1.43895
$219$	3035.31	0.936562
$220$	0	0
$221$	1642.93	0.500070
$222$	4596.50	1.38962
$223$	−3328.58	−0.999545	−0.499772	−	0.866157i	$-0.666583\pi$
$223$	−3328.58	−0.999545	−0.499772	+	0.866157i	$0.666583\pi$
$224$	2037.29	0.607688
$225$	0	0
$226$	8877.71	2.61299
$227$	−527.100	−0.154118	−0.0770592	−	0.997027i	$-0.524553\pi$
$227$	−527.100	−0.154118	−0.0770592	+	0.997027i	$0.524553\pi$
$228$	−2066.51	−0.600255
$229$	2566.06	0.740479	0.370240	−	0.928936i	$-0.379276\pi$
$229$	2566.06	0.740479	0.370240	+	0.928936i	$0.379276\pi$
$230$	0	0
$231$	1955.31	0.556925
$232$	−704.175	−0.199273
$233$	5534.99	1.55626	0.778132	−	0.628101i	$-0.216168\pi$
$233$	5534.99	1.55626	0.778132	+	0.628101i	$0.216168\pi$
$234$	837.422	0.233949
$235$	0	0
$236$	−1582.68	−0.436541
$237$	3451.41	0.945962
$238$	−6326.57	−1.72307
$239$	1010.01	0.273355	0.136678	−	0.990616i	$-0.456358\pi$
$239$	1010.01	0.273355	0.136678	+	0.990616i	$0.456358\pi$
$240$	0	0
$241$	−4074.29	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$241$	−4074.29	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$242$	1343.68	0.356921
$243$	243.000	0.0641500
$244$	6732.87	1.76651
$245$	0	0
$246$	−3419.24	−0.886190
$247$	−966.525	−0.248982
$248$	−356.437	−0.0912653
$249$	−3473.77	−0.884103
$250$	0	0
$251$	1773.98	0.446107	0.223054	−	0.974806i	$-0.428398\pi$
$251$	1773.98	0.446107	0.223054	+	0.974806i	$0.428398\pi$
$252$	−2057.65	−0.514365
$253$	−64.8375	−0.0161119
$254$	−2919.01	−0.721082
$255$	0	0
$256$	−6101.62	−1.48965
$257$	662.784	0.160869	0.0804345	−	0.996760i	$-0.474369\pi$
$257$	662.784	0.160869	0.0804345	+	0.996760i	$0.474369\pi$
$258$	5181.36	1.25030
$259$	−5282.38	−1.26730
$260$	0	0
$261$	−220.810	−0.0523669
$262$	−12169.3	−2.86955
$263$	712.312	0.167008	0.0835039	−	0.996507i	$-0.473389\pi$
$263$	712.312	0.167008	0.0835039	+	0.996507i	$0.473389\pi$
$264$	−3462.22	−0.807139
$265$	0	0
$266$	3721.87	0.857905
$267$	290.925	0.0666829
$268$	7871.74	1.79419
$269$	−3136.41	−0.710894	−0.355447	−	0.934696i	$-0.615671\pi$
$269$	−3136.41	−0.710894	−0.355447	+	0.934696i	$0.615671\pi$
$270$	0	0
$271$	−2275.69	−0.510105	−0.255053	−	0.966927i	$-0.582093\pi$
$271$	−2275.69	−0.510105	−0.255053	+	0.966927i	$0.582093\pi$
$272$	1835.03	0.409064
$273$	−962.381	−0.213355
$274$	−7787.15	−1.71693
$275$	0	0
$276$	68.2313	0.0148806
$277$	5171.00	1.12164	0.560821	−	0.827937i	$-0.310485\pi$
$277$	5171.00	1.12164	0.560821	+	0.827937i	$0.310485\pi$
$278$	−720.544	−0.155451
$279$	−111.769	−0.0239836
$280$	0	0
$281$	2240.14	0.475571	0.237785	−	0.971318i	$-0.423578\pi$
$281$	2240.14	0.475571	0.237785	+	0.971318i	$0.423578\pi$
$282$	−2885.02	−0.609222
$283$	225.244	0.0473123	0.0236561	−	0.999720i	$-0.492469\pi$
$283$	225.244	0.0473123	0.0236561	+	0.999720i	$0.492469\pi$
$284$	7874.38	1.64528
$285$	0	0
$286$	−3741.36	−0.773535
$287$	3929.46	0.808183
$288$	−1131.17	−0.231441
$289$	1978.59	0.402726
$290$	0	0
$291$	−3457.12	−0.696427
$292$	14270.7	2.86003
$293$	1139.86	0.227274	0.113637	−	0.993522i	$-0.463750\pi$
$293$	1139.86	0.227274	0.113637	+	0.993522i	$0.463750\pi$
$294$	−1131.99	−0.224554
$295$	0	0
$296$	9353.39	1.83667
$297$	−1085.65	−0.212108
$298$	6974.19	1.35572
$299$	31.9123	0.00617237
$300$	0	0
$301$	−5954.51	−1.14024
$302$	−1853.74	−0.353214
$303$	−3469.47	−0.657808
$304$	−1079.54	−0.203670
$305$	0	0
$306$	3512.73	0.656240
$307$	5244.86	0.975049	0.487525	−	0.873109i	$-0.337900\pi$
$307$	5244.86	0.975049	0.487525	+	0.873109i	$0.337900\pi$
$308$	9192.99	1.70071
$309$	−4001.09	−0.736615
$310$	0	0
$311$	−5188.26	−0.945977	−0.472989	−	0.881068i	$-0.656825\pi$
$311$	−5188.26	−0.945977	−0.472989	+	0.881068i	$0.656825\pi$
$312$	1704.07	0.309211
$313$	486.656	0.0878832	0.0439416	−	0.999034i	$-0.486008\pi$
$313$	486.656	0.0878832	0.0439416	+	0.999034i	$0.486008\pi$
$314$	8119.85	1.45933
$315$	0	0
$316$	16227.0	2.88873
$317$	−4218.87	−0.747493	−0.373747	−	0.927531i	$-0.621927\pi$
$317$	−4218.87	−0.747493	−0.373747	+	0.927531i	$0.621927\pi$
$318$	−867.834	−0.153037
$319$	986.512	0.173148
$320$	0	0
$321$	−2394.79	−0.416399
$322$	−122.887	−0.0212678
$323$	−4054.27	−0.698408
$324$	1142.48	0.195898
$325$	0	0
$326$	−9564.30	−1.62490
$327$	−2955.36	−0.499791
$328$	−6957.80	−1.17128
$329$	3315.53	0.555595
$330$	0	0
$331$	7439.94	1.23546	0.617728	−	0.786392i	$-0.288053\pi$
$331$	7439.94	1.23546	0.617728	+	0.786392i	$0.288053\pi$
$332$	−16332.2	−2.69983
$333$	2932.96	0.482658
$334$	906.931	0.148578
$335$	0	0
$336$	−1074.91	−0.174527
$337$	−6555.39	−1.05963	−0.529815	−	0.848113i	$-0.677739\pi$
$337$	−6555.39	−1.05963	−0.529815	+	0.848113i	$0.677739\pi$
$338$	−8487.88	−1.36592
$339$	5664.74	0.907571
$340$	0	0
$341$	499.350	0.0793000
$342$	−2066.51	−0.326737
$343$	6860.72	1.08001
$344$	10543.5	1.65252
$345$	0	0
$346$	−5829.51	−0.905770
$347$	1950.56	0.301763	0.150881	−	0.988552i	$-0.451789\pi$
$347$	1950.56	0.301763	0.150881	+	0.988552i	$0.451789\pi$
$348$	−1038.15	−0.159916
$349$	−1426.74	−0.218830	−0.109415	−	0.993996i	$-0.534898\pi$
$349$	−1426.74	−0.218830	−0.109415	+	0.993996i	$0.534898\pi$
$350$	0	0
$351$	534.347	0.0812573
$352$	5053.75	0.765244
$353$	−7078.96	−1.06735	−0.533676	−	0.845689i	$-0.679190\pi$
$353$	−7078.96	−1.06735	−0.533676	+	0.845689i	$0.679190\pi$
$354$	−1582.68	−0.237623
$355$	0	0
$356$	1367.80	0.203633
$357$	−4036.89	−0.598474
$358$	−12397.4	−1.83023
$359$	−5409.79	−0.795314	−0.397657	−	0.917534i	$-0.630177\pi$
$359$	−5409.79	−0.795314	−0.397657	+	0.917534i	$0.630177\pi$
$360$	0	0
$361$	−4473.90	−0.652267
$362$	15597.8	2.26465
$363$	857.381	0.123969
$364$	−4524.69	−0.651534
$365$	0	0
$366$	6732.87	0.961565
$367$	4940.09	0.702645	0.351322	−	0.936255i	$-0.385732\pi$
$367$	4940.09	0.702645	0.351322	+	0.936255i	$0.385732\pi$
$368$	35.6437	0.00504907
$369$	−2181.77	−0.307800
$370$	0	0
$371$	997.332	0.139566
$372$	−525.488	−0.0732399
$373$	12891.9	1.78959	0.894797	−	0.446473i	$-0.147320\pi$
$373$	12891.9	1.78959	0.894797	+	0.446473i	$0.147320\pi$
$374$	−15693.8	−2.16981
$375$	0	0
$376$	−5870.72	−0.805211
$377$	−485.551	−0.0663320
$378$	−2057.65	−0.279985
$379$	−9475.15	−1.28418	−0.642092	−	0.766627i	$-0.721933\pi$
$379$	−9475.15	−1.28418	−0.642092	+	0.766627i	$0.721933\pi$
$380$	0	0
$381$	−1862.58	−0.250453
$382$	−2936.16	−0.393264
$383$	−5800.97	−0.773931	−0.386966	−	0.922094i	$-0.626477\pi$
$383$	−5800.97	−0.773931	−0.386966	+	0.922094i	$0.626477\pi$
$384$	−7812.52	−1.03823
$385$	0	0
$386$	−2050.56	−0.270390
$387$	3306.15	0.434266
$388$	−16253.9	−2.12672
$389$	13779.7	1.79603	0.898016	−	0.439962i	$-0.145008\pi$
$389$	13779.7	1.79603	0.898016	+	0.439962i	$0.145008\pi$
$390$	0	0
$391$	133.862	0.0173138
$392$	−2303.48	−0.296794
$393$	−7765.06	−0.996680
$394$	15777.5	2.01741
$395$	0	0
$396$	−5104.27	−0.647725
$397$	−2816.46	−0.356056	−0.178028	−	0.984025i	$-0.556972\pi$
$397$	−2816.46	−0.356056	−0.178028	+	0.984025i	$0.556972\pi$
$398$	17864.8	2.24996
$399$	2374.88	0.297976
$400$	0	0
$401$	11986.4	1.49270	0.746352	−	0.665551i	$-0.231804\pi$
$401$	11986.4	1.49270	0.746352	+	0.665551i	$0.231804\pi$
$402$	7871.74	0.976633
$403$	−245.775	−0.0303794
$404$	−16311.9	−2.00879
$405$	0	0
$406$	1869.75	0.228557
$407$	−13103.6	−1.59588
$408$	7148.03	0.867354
$409$	−3339.07	−0.403683	−0.201841	−	0.979418i	$-0.564693\pi$
$409$	−3339.07	−0.403683	−0.201841	+	0.979418i	$0.564693\pi$
$410$	0	0
$411$	−4968.87	−0.596342
$412$	−18811.4	−2.24944
$413$	1818.84	0.216706
$414$	68.2313	0.00809996
$415$	0	0
$416$	−2487.40	−0.293161
$417$	−459.769	−0.0539927
$418$	9232.57	1.08033
$419$	1688.52	0.196873	0.0984363	−	0.995143i	$-0.468616\pi$
$419$	1688.52	0.196873	0.0984363	+	0.995143i	$0.468616\pi$
$420$	0	0
$421$	−2664.27	−0.308429	−0.154214	−	0.988037i	$-0.549285\pi$
$421$	−2664.27	−0.308429	−0.154214	+	0.988037i	$0.549285\pi$
$422$	11120.5	1.28279
$423$	−1840.89	−0.211601
$424$	−1765.95	−0.202269
$425$	0	0
$426$	7874.38	0.895575
$427$	−7737.54	−0.876923
$428$	−11259.2	−1.27158
$429$	−2387.31	−0.268672
$430$	0	0
$431$	12266.0	1.37084	0.685420	−	0.728148i	$-0.259619\pi$
$431$	12266.0	1.37084	0.685420	+	0.728148i	$0.259619\pi$
$432$	596.827	0.0664695
$433$	−15647.3	−1.73664	−0.868318	−	0.496008i	$-0.834799\pi$
$433$	−15647.3	−1.73664	−0.868318	+	0.496008i	$0.834799\pi$
$434$	946.425	0.104677
$435$	0	0
$436$	−13894.8	−1.52624
$437$	−78.7503	−0.00862045
$438$	14270.7	1.55680
$439$	16131.0	1.75373	0.876867	−	0.480733i	$-0.159629\pi$
$439$	16131.0	1.75373	0.876867	+	0.480733i	$0.159629\pi$
$440$	0	0
$441$	−722.306	−0.0779944
$442$	7724.34	0.831243
$443$	10053.7	1.07825	0.539127	−	0.842225i	$-0.318754\pi$
$443$	10053.7	1.07825	0.539127	+	0.842225i	$0.318754\pi$
$444$	13789.5	1.47392
$445$	0	0
$446$	−15649.5	−1.66150
$447$	4450.13	0.470882
$448$	12444.9	1.31242
$449$	7477.71	0.785957	0.392979	−	0.919548i	$-0.371445\pi$
$449$	7477.71	0.785957	0.392979	+	0.919548i	$0.371445\pi$
$450$	0	0
$451$	9747.51	1.01772
$452$	26633.1	2.77150
$453$	−1182.84	−0.122682
$454$	−2478.19	−0.256184
$455$	0	0
$456$	−4205.14	−0.431850
$457$	1363.46	0.139562	0.0697812	−	0.997562i	$-0.477770\pi$
$457$	1363.46	0.139562	0.0697812	+	0.997562i	$0.477770\pi$
$458$	12064.5	1.23086
$459$	2241.42	0.227932
$460$	0	0
$461$	5276.77	0.533109	0.266555	−	0.963820i	$-0.414115\pi$
$461$	5276.77	0.533109	0.266555	+	0.963820i	$0.414115\pi$
$462$	9192.99	0.925751
$463$	5740.02	0.576159	0.288079	−	0.957607i	$-0.406983\pi$
$463$	5740.02	0.576159	0.288079	+	0.957607i	$0.406983\pi$
$464$	−542.325	−0.0542604
$465$	0	0
$466$	26023.1	2.58690
$467$	6233.36	0.617657	0.308828	−	0.951118i	$-0.400063\pi$
$467$	6233.36	0.617657	0.308828	+	0.951118i	$0.400063\pi$
$468$	2512.27	0.248140
$469$	−9046.35	−0.890664
$470$	0	0
$471$	5181.16	0.506869
$472$	−3220.58	−0.314067
$473$	−14770.9	−1.43587
$474$	16227.0	1.57243
$475$	0	0
$476$	−18979.7	−1.82759
$477$	−553.753	−0.0531543
$478$	4748.61	0.454385
$479$	−19688.2	−1.87803	−0.939013	−	0.343881i	$-0.888258\pi$
$479$	−19688.2	−1.87803	−0.939013	+	0.343881i	$0.888258\pi$
$480$	0	0
$481$	6449.46	0.611372
$482$	−19155.5	−1.81019
$483$	−78.4127	−0.00738696
$484$	4031.03	0.378572
$485$	0	0
$486$	1142.48	0.106634
$487$	−3955.08	−0.368012	−0.184006	−	0.982925i	$-0.558907\pi$
$487$	−3955.08	−0.368012	−0.184006	+	0.982925i	$0.558907\pi$
$488$	13700.7	1.27090
$489$	−6102.84	−0.564377
$490$	0	0
$491$	−13893.5	−1.27699	−0.638497	−	0.769624i	$-0.720443\pi$
$491$	−13893.5	−1.27699	−0.638497	+	0.769624i	$0.720443\pi$
$492$	−10257.7	−0.939947
$493$	−2036.74	−0.186065
$494$	−4544.18	−0.413871
$495$	0	0
$496$	−274.512	−0.0248508
$497$	−9049.39	−0.816741
$498$	−16332.2	−1.46960
$499$	13523.7	1.21324	0.606618	−	0.794993i	$-0.292526\pi$
$499$	13523.7	1.21324	0.606618	+	0.794993i	$0.292526\pi$
$500$	0	0
$501$	578.700	0.0516056
$502$	8340.50	0.741543
$503$	13135.4	1.16437	0.582184	−	0.813057i	$-0.302198\pi$
$503$	13135.4	1.16437	0.582184	+	0.813057i	$0.302198\pi$
$504$	−4187.11	−0.370057
$505$	0	0
$506$	−304.837	−0.0267820
$507$	−5415.99	−0.474423
$508$	−8757.03	−0.764823
$509$	−2222.71	−0.193556	−0.0967778	−	0.995306i	$-0.530854\pi$
$509$	−2222.71	−0.193556	−0.0967778	+	0.995306i	$0.530854\pi$
$510$	0	0
$511$	−16400.1	−1.41976
$512$	−7853.76	−0.677911
$513$	−1318.61	−0.113486
$514$	3116.12	0.267405
$515$	0	0
$516$	15544.1	1.32614
$517$	8224.57	0.699645
$518$	−24835.4	−2.10658
$519$	−3719.73	−0.314601
$520$	0	0
$521$	−4916.42	−0.413421	−0.206710	−	0.978402i	$-0.566276\pi$
$521$	−4916.42	−0.413421	−0.206710	+	0.978402i	$0.566276\pi$
$522$	−1038.15	−0.0870471
$523$	17743.4	1.48349	0.741746	−	0.670681i	$-0.233998\pi$
$523$	17743.4	1.48349	0.741746	+	0.670681i	$0.233998\pi$
$524$	−36507.9	−3.04362
$525$	0	0
$526$	3348.98	0.277609
$527$	−1030.95	−0.0852161
$528$	−2666.45	−0.219777
$529$	−12164.4	−0.999786
$530$	0	0
$531$	−1009.88	−0.0825334
$532$	11165.6	0.909946
$533$	−4797.62	−0.389884
$534$	1367.80	0.110844
$535$	0	0
$536$	16018.2	1.29082
$537$	−7910.58	−0.635692
$538$	−14746.0	−1.18169
$539$	3227.05	0.257883
$540$	0	0
$541$	−12671.3	−1.00699	−0.503495	−	0.863998i	$-0.667953\pi$
$541$	−12671.3	−1.00699	−0.503495	+	0.863998i	$0.667953\pi$
$542$	−10699.3	−0.847924
$543$	9952.74	0.786580
$544$	−10433.9	−0.822334
$545$	0	0
$546$	−4524.69	−0.354650
$547$	5250.90	0.410443	0.205221	−	0.978716i	$-0.434209\pi$
$547$	5250.90	0.410443	0.205221	+	0.978716i	$0.434209\pi$
$548$	−23361.5	−1.82108
$549$	4296.15	0.333980
$550$	0	0
$551$	1198.20	0.0926406
$552$	138.844	0.0107057
$553$	−18648.4	−1.43401
$554$	24311.8	1.86445
$555$	0	0
$556$	−2161.63	−0.164881
$557$	25830.2	1.96492	0.982462	−	0.186465i	$-0.0597032\pi$
$557$	25830.2	1.96492	0.982462	+	0.186465i	$0.0597032\pi$
$558$	−525.488	−0.0398668
$559$	7270.09	0.550075
$560$	0	0
$561$	−10014.0	−0.753640
$562$	10532.1	0.790519
$563$	2021.14	0.151298	0.0756490	−	0.997135i	$-0.475897\pi$
$563$	2021.14	0.151298	0.0756490	+	0.997135i	$0.475897\pi$
$564$	−8655.07	−0.646178
$565$	0	0
$566$	1059.00	0.0786450
$567$	−1312.96	−0.0972471
$568$	16023.5	1.18368
$569$	8706.51	0.641469	0.320734	−	0.947169i	$-0.396070\pi$
$569$	8706.51	0.641469	0.320734	+	0.947169i	$0.396070\pi$
$570$	0	0
$571$	−12194.5	−0.893740	−0.446870	−	0.894599i	$-0.647461\pi$
$571$	−12194.5	−0.893740	−0.446870	+	0.894599i	$0.647461\pi$
$572$	−11224.1	−0.820458
$573$	−1873.52	−0.136592
$574$	18474.6	1.34340
$575$	0	0
$576$	−6909.82	−0.499842
$577$	−15264.0	−1.10130	−0.550649	−	0.834737i	$-0.685620\pi$
$577$	−15264.0	−1.10130	−0.550649	+	0.834737i	$0.685620\pi$
$578$	9302.48	0.669433
$579$	−1308.43	−0.0939146
$580$	0	0
$581$	18769.2	1.34024
$582$	−16253.9	−1.15764
$583$	2474.01	0.175751
$584$	29039.3	2.05763
$585$	0	0
$586$	5359.12	0.377787
$587$	−8456.89	−0.594639	−0.297319	−	0.954778i	$-0.596093\pi$
$587$	−8456.89	−0.594639	−0.297319	+	0.954778i	$0.596093\pi$
$588$	−3395.97	−0.238176
$589$	606.500	0.0424285
$590$	0	0
$591$	10067.4	0.700708
$592$	7203.57	0.500110
$593$	−1225.23	−0.0848467	−0.0424234	−	0.999100i	$-0.513508\pi$
$593$	−1225.23	−0.0848467	−0.0424234	+	0.999100i	$0.513508\pi$
$594$	−5104.27	−0.352577
$595$	0	0
$596$	20922.6	1.43796
$597$	11399.3	0.781478
$598$	150.038	0.0102600
$599$	−16060.0	−1.09548	−0.547741	−	0.836648i	$-0.684512\pi$
$599$	−16060.0	−1.09548	−0.547741	+	0.836648i	$0.684512\pi$
$600$	0	0
$601$	9699.93	0.658350	0.329175	−	0.944269i	$-0.393229\pi$
$601$	9699.93	0.658350	0.329175	+	0.944269i	$0.393229\pi$
$602$	−27995.5	−1.89537
$603$	5022.84	0.339214
$604$	−5561.21	−0.374640
$605$	0	0
$606$	−16311.9	−1.09344
$607$	23661.2	1.58217	0.791087	−	0.611703i	$-0.209515\pi$
$607$	23661.2	1.58217	0.791087	+	0.611703i	$0.209515\pi$
$608$	6138.19	0.409435
$609$	1193.06	0.0793847
$610$	0	0
$611$	−4048.05	−0.268030
$612$	10538.2	0.696047
$613$	−8085.63	−0.532749	−0.266375	−	0.963870i	$-0.585826\pi$
$613$	−8085.63	−0.532749	−0.266375	+	0.963870i	$0.585826\pi$
$614$	24659.0	1.62078
$615$	0	0
$616$	18706.8	1.22357
$617$	−11035.1	−0.720029	−0.360014	−	0.932947i	$-0.617228\pi$
$617$	−11035.1	−0.720029	−0.360014	+	0.932947i	$0.617228\pi$
$618$	−18811.4	−1.22444
$619$	−16826.3	−1.09258	−0.546290	−	0.837596i	$-0.683960\pi$
$619$	−16826.3	−1.09258	−0.546290	+	0.837596i	$0.683960\pi$
$620$	0	0
$621$	43.5374	0.00281336
$622$	−24392.9	−1.57245
$623$	−1571.90	−0.101087
$624$	1312.40	0.0841954
$625$	0	0
$626$	2288.04	0.146084
$627$	5891.17	0.375233
$628$	24359.5	1.54785
$629$	27053.5	1.71493
$630$	0	0
$631$	−3705.91	−0.233803	−0.116902	−	0.993144i	$-0.537296\pi$
$631$	−3705.91	−0.233803	−0.116902	+	0.993144i	$0.537296\pi$
$632$	33020.2	2.07828
$633$	7095.81	0.445550
$634$	−19835.3	−1.24252
$635$	0	0
$636$	−2603.50	−0.162320
$637$	−1588.32	−0.0987937
$638$	4638.15	0.287815
$639$	5024.53	0.311060
$640$	0	0
$641$	−24597.4	−1.51566	−0.757829	−	0.652453i	$-0.773740\pi$
$641$	−24597.4	−1.51566	−0.757829	+	0.652453i	$0.773740\pi$
$642$	−11259.2	−0.692160
$643$	−21479.5	−1.31737	−0.658685	−	0.752419i	$-0.728887\pi$
$643$	−21479.5	−1.31737	−0.658685	+	0.752419i	$0.728887\pi$
$644$	−368.662	−0.0225580
$645$	0	0
$646$	−19061.4	−1.16093
$647$	27119.7	1.64789	0.823946	−	0.566668i	$-0.191768\pi$
$647$	27119.7	1.64789	0.823946	+	0.566668i	$0.191768\pi$
$648$	2324.83	0.140938
$649$	4511.87	0.272891
$650$	0	0
$651$	603.900	0.0363575
$652$	−28692.9	−1.72347
$653$	18476.4	1.10725	0.553627	−	0.832765i	$-0.313243\pi$
$653$	18476.4	1.10725	0.553627	+	0.832765i	$0.313243\pi$
$654$	−13894.8	−0.830779
$655$	0	0
$656$	−5358.59	−0.318930
$657$	9105.92	0.540724
$658$	15588.1	0.923540
$659$	−19273.5	−1.13928	−0.569641	−	0.821894i	$-0.692918\pi$
$659$	−19273.5	−1.13928	−0.569641	+	0.821894i	$0.692918\pi$
$660$	0	0
$661$	25605.3	1.50670	0.753352	−	0.657618i	$-0.228436\pi$
$661$	25605.3	1.50670	0.753352	+	0.657618i	$0.228436\pi$
$662$	34979.3	2.05364
$663$	4928.79	0.288716
$664$	−33234.3	−1.94238
$665$	0	0
$666$	13789.5	0.802300
$667$	−39.5616	−0.00229660
$668$	2720.79	0.157591
$669$	−9985.75	−0.577087
$670$	0	0
$671$	−19193.9	−1.10428
$672$	6111.87	0.350849
$673$	7855.52	0.449938	0.224969	−	0.974366i	$-0.427772\pi$
$673$	7855.52	0.449938	0.224969	+	0.974366i	$0.427772\pi$
$674$	−30820.6	−1.76137
$675$	0	0
$676$	−25463.6	−1.44877
$677$	−6763.09	−0.383939	−0.191970	−	0.981401i	$-0.561488\pi$
$677$	−6763.09	−0.383939	−0.191970	+	0.981401i	$0.561488\pi$
$678$	26633.1	1.50861
$679$	18679.3	1.05574
$680$	0	0
$681$	−1581.30	−0.0889803
$682$	2347.72	0.131817
$683$	15608.6	0.874447	0.437224	−	0.899353i	$-0.355962\pi$
$683$	15608.6	0.874447	0.437224	+	0.899353i	$0.355962\pi$
$684$	−6199.54	−0.346557
$685$	0	0
$686$	32256.1	1.79525
$687$	7698.17	0.427516
$688$	8120.16	0.449968
$689$	−1217.68	−0.0673293
$690$	0	0
$691$	6203.15	0.341504	0.170752	−	0.985314i	$-0.445380\pi$
$691$	6203.15	0.341504	0.170752	+	0.985314i	$0.445380\pi$
$692$	−17488.5	−0.960714
$693$	5865.92	0.321541
$694$	9170.69	0.501606
$695$	0	0
$696$	−2112.53	−0.115050
$697$	−20124.5	−1.09365
$698$	−6707.90	−0.363750
$699$	16605.0	0.898509
$700$	0	0
$701$	−16507.9	−0.889435	−0.444718	−	0.895671i	$-0.646696\pi$
$701$	−16507.9	−0.889435	−0.444718	+	0.895671i	$0.646696\pi$
$702$	2512.27	0.135070
$703$	−15915.4	−0.853854
$704$	30871.1	1.65269
$705$	0	0
$706$	−33282.2	−1.77421
$707$	18746.0	0.997193
$708$	−4748.03	−0.252037
$709$	−25539.6	−1.35283	−0.676416	−	0.736520i	$-0.736468\pi$
$709$	−25539.6	−1.35283	−0.676416	+	0.736520i	$0.736468\pi$
$710$	0	0
$711$	10354.2	0.546151
$712$	2783.34	0.146503
$713$	−20.0252	−0.00105182
$714$	−18979.7	−0.994815
$715$	0	0
$716$	−37192.1	−1.94125
$717$	3030.02	0.157822
$718$	−25434.4	−1.32201
$719$	−7353.45	−0.381415	−0.190708	−	0.981647i	$-0.561078\pi$
$719$	−7353.45	−0.381415	−0.190708	+	0.981647i	$0.561078\pi$
$720$	0	0
$721$	21618.4	1.11666
$722$	−21034.3	−1.08423
$723$	−12222.9	−0.628732
$724$	46793.4	2.40202
$725$	0	0
$726$	4031.03	0.206068
$727$	−21696.5	−1.10685	−0.553424	−	0.832900i	$-0.686679\pi$
$727$	−21696.5	−1.10685	−0.553424	+	0.832900i	$0.686679\pi$
$728$	−9207.28	−0.468742
$729$	729.000	0.0370370
$730$	0	0
$731$	30495.8	1.54299
$732$	20198.6	1.01989
$733$	90.2714	0.00454877	0.00227439	−	0.999997i	$-0.499276\pi$
$733$	90.2714	0.00454877	0.00227439	+	0.999997i	$0.499276\pi$
$734$	23226.1	1.16797
$735$	0	0
$736$	−202.668	−0.0101501
$737$	−22440.6	−1.12159
$738$	−10257.7	−0.511642
$739$	14273.1	0.710479	0.355239	−	0.934775i	$-0.384399\pi$
$739$	14273.1	0.710479	0.355239	+	0.934775i	$0.384399\pi$
$740$	0	0
$741$	−2899.57	−0.143750
$742$	4689.02	0.231994
$743$	15866.6	0.783429	0.391715	−	0.920087i	$-0.371882\pi$
$743$	15866.6	0.783429	0.391715	+	0.920087i	$0.371882\pi$
$744$	−1069.31	−0.0526921
$745$	0	0
$746$	60612.2	2.97476
$747$	−10421.3	−0.510437
$748$	−47081.5	−2.30143
$749$	12939.3	0.631232
$750$	0	0
$751$	26776.9	1.30107	0.650534	−	0.759477i	$-0.274545\pi$
$751$	26776.9	1.30107	0.650534	+	0.759477i	$0.274545\pi$
$752$	−4521.37	−0.219252
$753$	5321.95	0.257560
$754$	−2282.85	−0.110261
$755$	0	0
$756$	−6172.96	−0.296969
$757$	30478.0	1.46333	0.731666	−	0.681663i	$-0.238743\pi$
$757$	30478.0	1.46333	0.731666	+	0.681663i	$0.238743\pi$
$758$	−44548.0	−2.13464
$759$	−194.512	−0.00930218
$760$	0	0
$761$	29104.7	1.38639	0.693195	−	0.720750i	$-0.256202\pi$
$761$	29104.7	1.38639	0.693195	+	0.720750i	$0.256202\pi$
$762$	−8757.03	−0.416317
$763$	15968.2	0.757649
$764$	−8808.47	−0.417119
$765$	0	0
$766$	−27273.6	−1.28647
$767$	−2220.69	−0.104543
$768$	−18304.9	−0.860052
$769$	4170.65	0.195575	0.0977876	−	0.995207i	$-0.468823\pi$
$769$	4170.65	0.195575	0.0977876	+	0.995207i	$0.468823\pi$
$770$	0	0
$771$	1988.35	0.0928778
$772$	−6151.67	−0.286792
$773$	−17738.5	−0.825367	−0.412684	−	0.910874i	$-0.635409\pi$
$773$	−17738.5	−0.825367	−0.412684	+	0.910874i	$0.635409\pi$
$774$	15544.1	0.721860
$775$	0	0
$776$	−33075.0	−1.53005
$777$	−15847.1	−0.731677
$778$	64786.0	2.98546
$779$	11839.1	0.544519
$780$	0	0
$781$	−22448.1	−1.02850
$782$	629.363	0.0287800
$783$	−662.429	−0.0302341
$784$	−1774.04	−0.0808145
$785$	0	0
$786$	−36507.9	−1.65674
$787$	−3807.92	−0.172475	−0.0862374	−	0.996275i	$-0.527484\pi$
$787$	−3807.92	−0.172475	−0.0862374	+	0.996275i	$0.527484\pi$
$788$	47332.6	2.13979
$789$	2136.94	0.0964220
$790$	0	0
$791$	−30607.3	−1.37582
$792$	−10386.6	−0.466002
$793$	9447.06	0.423045
$794$	−13241.8	−0.591856
$795$	0	0
$796$	53594.5	2.38644
$797$	23840.3	1.05956	0.529779	−	0.848136i	$-0.322275\pi$
$797$	23840.3	1.05956	0.529779	+	0.848136i	$0.322275\pi$
$798$	11165.6	0.495312
$799$	−16980.3	−0.751841
$800$	0	0
$801$	872.775	0.0384994
$802$	56355.0	2.48125
$803$	−40682.6	−1.78787
$804$	23615.2	1.03588
$805$	0	0
$806$	−1155.53	−0.0504983
$807$	−9409.24	−0.410435
$808$	−33193.1	−1.44521
$809$	−1984.22	−0.0862316	−0.0431158	−	0.999070i	$-0.513728\pi$
$809$	−1984.22	−0.0862316	−0.0431158	+	0.999070i	$0.513728\pi$
$810$	0	0
$811$	−9713.78	−0.420588	−0.210294	−	0.977638i	$-0.567442\pi$
$811$	−9713.78	−0.420588	−0.210294	+	0.977638i	$0.567442\pi$
$812$	5609.25	0.242421
$813$	−6827.08	−0.294509
$814$	−61607.4	−2.65275
$815$	0	0
$816$	5505.10	0.236173
$817$	−17940.5	−0.768246
$818$	−15698.8	−0.671023
$819$	−2887.14	−0.123181
$820$	0	0
$821$	−19235.4	−0.817686	−0.408843	−	0.912605i	$-0.634068\pi$
$821$	−19235.4	−0.817686	−0.408843	+	0.912605i	$0.634068\pi$
$822$	−23361.5	−0.991271
$823$	−12717.6	−0.538650	−0.269325	−	0.963049i	$-0.586801\pi$
$823$	−12717.6	−0.538650	−0.269325	+	0.963049i	$0.586801\pi$
$824$	−38279.2	−1.61835
$825$	0	0
$826$	8551.41	0.360220
$827$	−6744.75	−0.283601	−0.141800	−	0.989895i	$-0.545289\pi$
$827$	−6744.75	−0.283601	−0.141800	+	0.989895i	$0.545289\pi$
$828$	204.694	0.00859131
$829$	3404.22	0.142622	0.0713108	−	0.997454i	$-0.477282\pi$
$829$	3404.22	0.142622	0.0713108	+	0.997454i	$0.477282\pi$
$830$	0	0
$831$	15513.0	0.647581
$832$	−15194.4	−0.633139
$833$	−6662.52	−0.277122
$834$	−2161.63	−0.0897496
$835$	0	0
$836$	27697.7	1.14587
$837$	−335.306	−0.0138469
$838$	7938.69	0.327252
$839$	−21361.9	−0.879015	−0.439508	−	0.898239i	$-0.644847\pi$
$839$	−21361.9	−0.879015	−0.439508	+	0.898239i	$0.644847\pi$
$840$	0	0
$841$	−23787.1	−0.975319
$842$	−12526.2	−0.512687
$843$	6720.41	0.274571
$844$	33361.4	1.36060
$845$	0	0
$846$	−8655.07	−0.351735
$847$	−4632.54	−0.187929
$848$	−1360.06	−0.0550762
$849$	675.732	0.0273158
$850$	0	0
$851$	525.488	0.0211674
$852$	23623.1	0.949901
$853$	−10728.9	−0.430657	−0.215328	−	0.976542i	$-0.569082\pi$
$853$	−10728.9	−0.430657	−0.215328	+	0.976542i	$0.569082\pi$
$854$	−36378.5	−1.45767
$855$	0	0
$856$	−22911.4	−0.914830
$857$	−42895.2	−1.70977	−0.854885	−	0.518817i	$-0.826373\pi$
$857$	−42895.2	−1.70977	−0.854885	+	0.518817i	$0.826373\pi$
$858$	−11224.1	−0.446601
$859$	35530.5	1.41127	0.705637	−	0.708574i	$-0.250661\pi$
$859$	35530.5	1.41127	0.705637	+	0.708574i	$0.250661\pi$
$860$	0	0
$861$	11788.4	0.466605
$862$	57669.3	2.27868
$863$	5704.35	0.225004	0.112502	−	0.993652i	$-0.464114\pi$
$863$	5704.35	0.225004	0.112502	+	0.993652i	$0.464114\pi$
$864$	−3393.52	−0.133623
$865$	0	0
$866$	−73567.0	−2.88673
$867$	5935.78	0.232514
$868$	2839.27	0.111027
$869$	−46259.6	−1.80581
$870$	0	0
$871$	11045.0	0.429674
$872$	−28274.4	−1.09804
$873$	−10371.4	−0.402082
$874$	−370.249	−0.0143294
$875$	0	0
$876$	42812.0	1.65124
$877$	50249.0	1.93476	0.967382	−	0.253324i	$-0.0815237\pi$
$877$	50249.0	1.93476	0.967382	+	0.253324i	$0.0815237\pi$
$878$	75840.7	2.91515
$879$	3419.58	0.131217
$880$	0	0
$881$	−26864.5	−1.02734	−0.513672	−	0.857987i	$-0.671715\pi$
$881$	−26864.5	−1.02734	−0.513672	+	0.857987i	$0.671715\pi$
$882$	−3395.97	−0.129646
$883$	18942.1	0.721918	0.360959	−	0.932582i	$-0.382449\pi$
$883$	18942.1	0.721918	0.360959	+	0.932582i	$0.382449\pi$
$884$	23173.0	0.881667
$885$	0	0
$886$	47268.2	1.79233
$887$	−25344.8	−0.959409	−0.479705	−	0.877430i	$-0.659256\pi$
$887$	−25344.8	−0.959409	−0.479705	+	0.877430i	$0.659256\pi$
$888$	28060.2	1.06040
$889$	10063.7	0.379670
$890$	0	0
$891$	−3256.96	−0.122460
$892$	−46948.6	−1.76228
$893$	9989.40	0.374337
$894$	20922.6	0.782725
$895$	0	0
$896$	42212.0	1.57389
$897$	95.7370	0.00356362
$898$	35156.9	1.30646
$899$	304.686	0.0113035
$900$	0	0
$901$	−5107.79	−0.188863
$902$	45828.5	1.69171
$903$	−17863.5	−0.658318
$904$	54195.6	1.99394
$905$	0	0
$906$	−5561.21	−0.203928
$907$	4800.11	0.175728	0.0878639	−	0.996132i	$-0.471996\pi$
$907$	4800.11	0.175728	0.0878639	+	0.996132i	$0.471996\pi$
$908$	−7434.58	−0.271724
$909$	−10408.4	−0.379786
$910$	0	0
$911$	−25731.7	−0.935819	−0.467909	−	0.883776i	$-0.654993\pi$
$911$	−25731.7	−0.935819	−0.467909	+	0.883776i	$0.654993\pi$
$912$	−3238.61	−0.117589
$913$	46559.4	1.68772
$914$	6410.40	0.231988
$915$	0	0
$916$	36193.4	1.30553
$917$	41955.6	1.51090
$918$	10538.2	0.378880
$919$	−12751.9	−0.457722	−0.228861	−	0.973459i	$-0.573500\pi$
$919$	−12751.9	−0.457722	−0.228861	+	0.973459i	$0.573500\pi$
$920$	0	0
$921$	15734.6	0.562945
$922$	24809.0	0.886163
$923$	11048.7	0.394012
$924$	27579.0	0.981907
$925$	0	0
$926$	26987.1	0.957722
$927$	−12003.3	−0.425285
$928$	3083.63	0.109079
$929$	15557.8	0.549444	0.274722	−	0.961524i	$-0.411414\pi$
$929$	15557.8	0.549444	0.274722	+	0.961524i	$0.411414\pi$
$930$	0	0
$931$	3919.51	0.137977
$932$	78069.3	2.74383
$933$	−15564.8	−0.546160
$934$	29306.5	1.02670
$935$	0	0
$936$	5112.20	0.178523
$937$	−23858.0	−0.831811	−0.415905	−	0.909408i	$-0.636535\pi$
$937$	−23858.0	−0.831811	−0.415905	+	0.909408i	$0.636535\pi$
$938$	−42532.0	−1.48051
$939$	1459.97	0.0507394
$940$	0	0
$941$	9748.00	0.337700	0.168850	−	0.985642i	$-0.445995\pi$
$941$	9748.00	0.337700	0.168850	+	0.985642i	$0.445995\pi$
$942$	24359.5	0.842544
$943$	−390.899	−0.0134989
$944$	−2480.35	−0.0855176
$945$	0	0
$946$	−69446.4	−2.38678
$947$	−51537.0	−1.76845	−0.884227	−	0.467057i	$-0.845314\pi$
$947$	−51537.0	−1.76845	−0.884227	+	0.467057i	$0.845314\pi$
$948$	48681.0	1.66781
$949$	20023.5	0.684923
$950$	0	0
$951$	−12656.6	−0.431566
$952$	−38621.7	−1.31485
$953$	−5631.36	−0.191414	−0.0957071	−	0.995410i	$-0.530511\pi$
$953$	−5631.36	−0.191414	−0.0957071	+	0.995410i	$0.530511\pi$
$954$	−2603.50	−0.0883559
$955$	0	0
$956$	14245.8	0.481948
$957$	2959.54	0.0999668
$958$	−92565.1	−3.12176
$959$	26847.4	0.904013
$960$	0	0
$961$	−29636.8	−0.994823
$962$	30322.5	1.01625
$963$	−7184.36	−0.240408
$964$	−57466.5	−1.91999
$965$	0	0
$966$	−368.662	−0.0122790
$967$	−43360.9	−1.44198	−0.720989	−	0.692946i	$-0.756312\pi$
$967$	−43360.9	−1.44198	−0.720989	+	0.692946i	$0.756312\pi$
$968$	8202.72	0.272361
$969$	−12162.8	−0.403226
$970$	0	0
$971$	12920.0	0.427007	0.213503	−	0.976942i	$-0.431513\pi$
$971$	12920.0	0.427007	0.213503	+	0.976942i	$0.431513\pi$
$972$	3427.44	0.113102
$973$	2484.19	0.0818493
$974$	−18595.0	−0.611728
$975$	0	0
$976$	10551.7	0.346056
$977$	−10650.4	−0.348759	−0.174379	−	0.984679i	$-0.555792\pi$
$977$	−10650.4	−0.348759	−0.174379	+	0.984679i	$0.555792\pi$
$978$	−28692.9	−0.938137
$979$	−3899.31	−0.127296
$980$	0	0
$981$	−8866.07	−0.288554
$982$	−65321.0	−2.12269
$983$	−49450.3	−1.60450	−0.802248	−	0.596991i	$-0.796363\pi$
$983$	−49450.3	−1.60450	−0.802248	+	0.596991i	$0.796363\pi$
$984$	−20873.4	−0.676239
$985$	0	0
$986$	−9575.85	−0.309287
$987$	9946.58	0.320773
$988$	−13632.5	−0.438976
$989$	592.351	0.0190452
$990$	0	0
$991$	9410.47	0.301648	0.150824	−	0.988561i	$-0.451807\pi$
$991$	9410.47	0.301648	0.150824	+	0.988561i	$0.451807\pi$
$992$	1560.86	0.0499571
$993$	22319.8	0.713291
$994$	−42546.3	−1.35763
$995$	0	0
$996$	−48996.5	−1.55875
$997$	532.117	0.0169030	0.00845151	−	0.999964i	$-0.497310\pi$
$997$	532.117	0.0169030	0.00845151	+	0.999964i	$0.497310\pi$
$998$	63582.6	2.01671
$999$	8798.88	0.278663

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.a.f.1.2		2
3.2	odd	2		225.4.a.i.1.1		2
4.3	odd	2		1200.4.a.bn.1.2		2
5.2	odd	4		15.4.b.a.4.4	yes	4
5.3	odd	4		15.4.b.a.4.1	✓	4
5.4	even	2		75.4.a.c.1.1		2
15.2	even	4		45.4.b.b.19.1		4
15.8	even	4		45.4.b.b.19.4		4
15.14	odd	2		225.4.a.o.1.2		2
20.3	even	4		240.4.f.f.49.2		4
20.7	even	4		240.4.f.f.49.4		4
20.19	odd	2		1200.4.a.bt.1.1		2
40.3	even	4		960.4.f.p.769.3		4
40.13	odd	4		960.4.f.q.769.1		4
40.27	even	4		960.4.f.p.769.1		4
40.37	odd	4		960.4.f.q.769.3		4
60.23	odd	4		720.4.f.j.289.2		4
60.47	odd	4		720.4.f.j.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.1	✓	4	5.3	odd	4
15.4.b.a.4.4	yes	4	5.2	odd	4
45.4.b.b.19.1		4	15.2	even	4
45.4.b.b.19.4		4	15.8	even	4
75.4.a.c.1.1		2	5.4	even	2
75.4.a.f.1.2		2	1.1	even	1	trivial
225.4.a.i.1.1		2	3.2	odd	2
225.4.a.o.1.2		2	15.14	odd	2
240.4.f.f.49.2		4	20.3	even	4
240.4.f.f.49.4		4	20.7	even	4
720.4.f.j.289.1		4	60.47	odd	4
720.4.f.j.289.2		4	60.23	odd	4
960.4.f.p.769.1		4	40.27	even	4
960.4.f.p.769.3		4	40.3	even	4
960.4.f.q.769.1		4	40.13	odd	4
960.4.f.q.769.3		4	40.37	odd	4
1200.4.a.bn.1.2		2	4.3	odd	2
1200.4.a.bt.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	75.4.a.f.1.2

Level	$75$
Weight	$4$
Character	75.1
Self dual	yes
Analytic conductor	$4.425$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$