LMFDB - Embedded newform 175.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,6,Mod(1,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-9] 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:	$28.0671684673$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 14$ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	175.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.27492 q^{2} +25.6495 q^{3} +36.4743 q^{4} -212.248 q^{6} -49.0000 q^{7} -37.0241 q^{8} +414.897 q^{9} -270.090 q^{11} +935.547 q^{12} -300.640 q^{13} +405.471 q^{14} -860.805 q^{16} -613.106 q^{17} -3433.24 q^{18} -1700.95 q^{19} -1256.83 q^{21} +2234.97 q^{22} -3188.15 q^{23} -949.650 q^{24} +2487.77 q^{26} +4409.07 q^{27} -1787.24 q^{28} +4299.28 q^{29} +2028.46 q^{31} +8307.86 q^{32} -6927.67 q^{33} +5073.40 q^{34} +15133.1 q^{36} -5154.46 q^{37} +14075.2 q^{38} -7711.26 q^{39} -7146.21 q^{41} +10400.1 q^{42} +19584.3 q^{43} -9851.32 q^{44} +26381.7 q^{46} -19998.4 q^{47} -22079.2 q^{48} +2401.00 q^{49} -15725.9 q^{51} -10965.6 q^{52} -3948.82 q^{53} -36484.7 q^{54} +1814.18 q^{56} -43628.5 q^{57} -35576.2 q^{58} -29707.6 q^{59} -50519.3 q^{61} -16785.3 q^{62} -20330.0 q^{63} -41201.1 q^{64} +57325.9 q^{66} -5053.56 q^{67} -22362.6 q^{68} -81774.5 q^{69} +32853.3 q^{71} -15361.2 q^{72} +11115.0 q^{73} +42652.7 q^{74} -62040.8 q^{76} +13234.4 q^{77} +63810.0 q^{78} +81889.4 q^{79} +12270.6 q^{81} +59134.3 q^{82} -118234. q^{83} -45841.8 q^{84} -162058. q^{86} +110274. q^{87} +9999.83 q^{88} -41695.4 q^{89} +14731.3 q^{91} -116286. q^{92} +52028.9 q^{93} +165485. q^{94} +213092. q^{96} -43682.8 q^{97} -19868.1 q^{98} -112059. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 9 q^{2} + 6 q^{3} + 5 q^{4} - 198 q^{6} - 98 q^{7} + 9 q^{8} + 558 q^{9} + 396 q^{11} + 1554 q^{12} + 350 q^{13} + 441 q^{14} + 113 q^{16} - 1800 q^{17} - 3537 q^{18} - 3266 q^{19} - 294 q^{21} + 1752 q^{22}+ \cdots - 16740 q^{99}+O(q^{100})$ 2 * q - 9 * q^2 + 6 * q^3 + 5 * q^4 - 198 * q^6 - 98 * q^7 + 9 * q^8 + 558 * q^9 + 396 * q^11 + 1554 * q^12 + 350 * q^13 + 441 * q^14 + 113 * q^16 - 1800 * q^17 - 3537 * q^18 - 3266 * q^19 - 294 * q^21 + 1752 * q^22 - 2088 * q^23 - 1854 * q^24 + 2016 * q^26 + 6372 * q^27 - 245 * q^28 + 6696 * q^29 - 20 * q^31 + 6129 * q^32 - 20016 * q^33 + 5934 * q^34 + 10629 * q^36 - 6232 * q^37 + 15210 * q^38 - 20496 * q^39 - 6048 * q^41 + 9702 * q^42 + 3020 * q^43 - 30816 * q^44 + 25584 * q^46 - 11700 * q^47 - 41214 * q^48 + 4802 * q^49 + 7596 * q^51 - 31444 * q^52 - 9468 * q^53 - 37908 * q^54 - 441 * q^56 - 12876 * q^57 - 37314 * q^58 - 43938 * q^59 - 64754 * q^61 - 15300 * q^62 - 27342 * q^63 - 70783 * q^64 + 66816 * q^66 - 24784 * q^67 + 14994 * q^68 - 103392 * q^69 + 97416 * q^71 - 8775 * q^72 - 17452 * q^73 + 43434 * q^74 - 12782 * q^76 - 19404 * q^77 + 73080 * q^78 + 51256 * q^79 - 61074 * q^81 + 58338 * q^82 - 117558 * q^83 - 76146 * q^84 - 150048 * q^86 + 63180 * q^87 + 40656 * q^88 + 84276 * q^89 - 17150 * q^91 - 150912 * q^92 + 92280 * q^93 + 159468 * q^94 + 255906 * q^96 - 20776 * q^97 - 21609 * q^98 - 16740 * 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

)

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

$2$	−8.27492	−1.46281	−0.731406	−	0.681942i	$-0.761136\pi$
$2$	−8.27492	−1.46281	−0.731406	+	0.681942i	$0.761136\pi$
$3$	25.6495	1.64542	0.822708	−	0.568464i	$-0.192462\pi$
$3$	25.6495	1.64542	0.822708	+	0.568464i	$0.192462\pi$
$4$	36.4743	1.13982
$5$	0	0
$5$	0	0
$6$	−212.248	−2.40694
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−37.0241	−0.204531
$9$	414.897	1.70740
$10$	0	0
$11$	−270.090	−0.673018	−0.336509	−	0.941680i	$-0.609246\pi$
$11$	−270.090	−0.673018	−0.336509	+	0.941680i	$0.609246\pi$
$12$	935.547	1.87548
$13$	−300.640	−0.493387	−0.246694	−	0.969094i	$-0.579344\pi$
$13$	−300.640	−0.493387	−0.246694	+	0.969094i	$0.579344\pi$
$14$	405.471	0.552891
$15$	0	0
$16$	−860.805	−0.840630
$17$	−613.106	−0.514533	−0.257267	−	0.966340i	$-0.582822\pi$
$17$	−613.106	−0.514533	−0.257267	+	0.966340i	$0.582822\pi$
$18$	−3433.24	−2.49760
$19$	−1700.95	−1.08095	−0.540477	−	0.841359i	$-0.681756\pi$
$19$	−1700.95	−1.08095	−0.540477	+	0.841359i	$0.681756\pi$
$20$	0	0
$21$	−1256.83	−0.621909
$22$	2234.97	0.984498
$23$	−3188.15	−1.25667	−0.628333	−	0.777945i	$-0.716262\pi$
$23$	−3188.15	−1.25667	−0.628333	+	0.777945i	$0.716262\pi$
$24$	−949.650	−0.336539
$25$	0	0
$26$	2487.77	0.721733
$27$	4409.07	1.16396
$28$	−1787.24	−0.430812
$29$	4299.28	0.949294	0.474647	−	0.880176i	$-0.342576\pi$
$29$	4299.28	0.949294	0.474647	+	0.880176i	$0.342576\pi$
$30$	0	0
$31$	2028.46	0.379106	0.189553	−	0.981870i	$-0.439296\pi$
$31$	2028.46	0.379106	0.189553	+	0.981870i	$0.439296\pi$
$32$	8307.86	1.43421
$33$	−6927.67	−1.10739
$34$	5073.40	0.752666
$35$	0	0
$36$	15133.1	1.94612
$37$	−5154.46	−0.618983	−0.309491	−	0.950902i	$-0.600159\pi$
$37$	−5154.46	−0.618983	−0.309491	+	0.950902i	$0.600159\pi$
$38$	14075.2	1.58123
$39$	−7711.26	−0.811827
$40$	0	0
$41$	−7146.21	−0.663921	−0.331960	−	0.943293i	$-0.607710\pi$
$41$	−7146.21	−0.663921	−0.331960	+	0.943293i	$0.607710\pi$
$42$	10400.1	0.909736
$43$	19584.3	1.61524	0.807620	−	0.589703i	$-0.200755\pi$
$43$	19584.3	1.61524	0.807620	+	0.589703i	$0.200755\pi$
$44$	−9851.32	−0.767119
$45$	0	0
$46$	26381.7	1.83827
$47$	−19998.4	−1.32054	−0.660268	−	0.751030i	$-0.729557\pi$
$47$	−19998.4	−1.32054	−0.660268	+	0.751030i	$0.729557\pi$
$48$	−22079.2	−1.38319
$49$	2401.00	0.142857
$50$	0	0
$51$	−15725.9	−0.846621
$52$	−10965.6	−0.562373
$53$	−3948.82	−0.193098	−0.0965489	−	0.995328i	$-0.530780\pi$
$53$	−3948.82	−0.193098	−0.0965489	+	0.995328i	$0.530780\pi$
$54$	−36484.7	−1.70265
$55$	0	0
$56$	1814.18	0.0773055
$57$	−43628.5	−1.77862
$58$	−35576.2	−1.38864
$59$	−29707.6	−1.11106	−0.555530	−	0.831497i	$-0.687484\pi$
$59$	−29707.6	−1.11106	−0.555530	+	0.831497i	$0.687484\pi$
$60$	0	0
$61$	−50519.3	−1.73833	−0.869165	−	0.494522i	$-0.835343\pi$
$61$	−50519.3	−1.73833	−0.869165	+	0.494522i	$0.835343\pi$
$62$	−16785.3	−0.554562
$63$	−20330.0	−0.645335
$64$	−41201.1	−1.25736
$65$	0	0
$66$	57325.9	1.61991
$67$	−5053.56	−0.137534	−0.0687671	−	0.997633i	$-0.521907\pi$
$67$	−5053.56	−0.137534	−0.0687671	+	0.997633i	$0.521907\pi$
$68$	−22362.6	−0.586476
$69$	−81774.5	−2.06774
$70$	0	0
$71$	32853.3	0.773453	0.386726	−	0.922195i	$-0.373606\pi$
$71$	32853.3	0.773453	0.386726	+	0.922195i	$0.373606\pi$
$72$	−15361.2	−0.349215
$73$	11115.0	0.244119	0.122059	−	0.992523i	$-0.461050\pi$
$73$	11115.0	0.244119	0.122059	+	0.992523i	$0.461050\pi$
$74$	42652.7	0.905456
$75$	0	0
$76$	−62040.8	−1.23209
$77$	13234.4	0.254377
$78$	63810.0	1.18755
$79$	81889.4	1.47625	0.738125	−	0.674664i	$-0.235712\pi$
$79$	81889.4	1.47625	0.738125	+	0.674664i	$0.235712\pi$
$80$	0	0
$81$	12270.6	0.207803
$82$	59134.3	0.971191
$83$	−118234.	−1.88385	−0.941926	−	0.335819i	$-0.890987\pi$
$83$	−118234.	−1.88385	−0.941926	+	0.335819i	$0.890987\pi$
$84$	−45841.8	−0.708865
$85$	0	0
$86$	−162058.	−2.36279
$87$	110274.	1.56198
$88$	9999.83	0.137653
$89$	−41695.4	−0.557972	−0.278986	−	0.960295i	$-0.589998\pi$
$89$	−41695.4	−0.557972	−0.278986	+	0.960295i	$0.589998\pi$
$90$	0	0
$91$	14731.3	0.186483
$92$	−116286.	−1.43237
$93$	52028.9	0.623788
$94$	165485.	1.93170
$95$	0	0
$96$	213092.	2.35988
$97$	−43682.8	−0.471391	−0.235695	−	0.971827i	$-0.575737\pi$
$97$	−43682.8	−0.471391	−0.235695	+	0.971827i	$0.575737\pi$
$98$	−19868.1	−0.208973
$99$	−112059.	−1.14911
$100$	0	0
$101$	25648.1	0.250179	0.125090	−	0.992145i	$-0.460078\pi$
$101$	25648.1	0.250179	0.125090	+	0.992145i	$0.460078\pi$
$102$	130130.	1.23845
$103$	14320.0	0.133000	0.0664999	−	0.997786i	$-0.478817\pi$
$103$	14320.0	0.133000	0.0664999	+	0.997786i	$0.478817\pi$
$104$	11130.9	0.100913
$105$	0	0
$106$	32676.1	0.282466
$107$	−17201.8	−0.145249	−0.0726247	−	0.997359i	$-0.523138\pi$
$107$	−17201.8	−0.145249	−0.0726247	+	0.997359i	$0.523138\pi$
$108$	160818.	1.32670
$109$	−86017.6	−0.693459	−0.346730	−	0.937965i	$-0.612708\pi$
$109$	−86017.6	−0.693459	−0.346730	+	0.937965i	$0.612708\pi$
$110$	0	0
$111$	−132209.	−1.01848
$112$	42179.4	0.317728
$113$	−137568.	−1.01349	−0.506745	−	0.862096i	$-0.669152\pi$
$113$	−137568.	−1.01349	−0.506745	+	0.862096i	$0.669152\pi$
$114$	361022.	2.60179
$115$	0	0
$116$	156813.	1.08202
$117$	−124734.	−0.842407
$118$	245828.	1.62527
$119$	30042.2	0.194475
$120$	0	0
$121$	−88102.5	−0.547047
$122$	418043.	2.54285
$123$	−183297.	−1.09243
$124$	73986.4	0.432113
$125$	0	0
$126$	168229.	0.944004
$127$	70567.1	0.388233	0.194117	−	0.980978i	$-0.437816\pi$
$127$	70567.1	0.388233	0.194117	+	0.980978i	$0.437816\pi$
$128$	75084.2	0.405064
$129$	502328.	2.65774
$130$	0	0
$131$	−173712.	−0.884408	−0.442204	−	0.896914i	$-0.645803\pi$
$131$	−173712.	−0.884408	−0.442204	+	0.896914i	$0.645803\pi$
$132$	−252682.	−1.26223
$133$	83346.5	0.408562
$134$	41817.8	0.201187
$135$	0	0
$136$	22699.7	0.105238
$137$	1989.94	0.00905813	0.00452907	−	0.999990i	$-0.498558\pi$
$137$	1989.94	0.00905813	0.00452907	+	0.999990i	$0.498558\pi$
$138$	676678.	3.02471
$139$	366409.	1.60853	0.804264	−	0.594272i	$-0.202560\pi$
$139$	366409.	1.60853	0.804264	+	0.594272i	$0.202560\pi$
$140$	0	0
$141$	−512949.	−2.17283
$142$	−271859.	−1.13142
$143$	81199.7	0.332058
$144$	−357145.	−1.43529
$145$	0	0
$146$	−91975.4	−0.357100
$147$	61584.5	0.235059
$148$	−188005.	−0.705529
$149$	140719.	0.519261	0.259631	−	0.965708i	$-0.416399\pi$
$149$	140719.	0.519261	0.259631	+	0.965708i	$0.416399\pi$
$150$	0	0
$151$	50064.6	0.178685	0.0893425	−	0.996001i	$-0.471523\pi$
$151$	50064.6	0.178685	0.0893425	+	0.996001i	$0.471523\pi$
$152$	62976.1	0.221089
$153$	−254376.	−0.878512
$154$	−109514.	−0.372105
$155$	0	0
$156$	−281262.	−0.925337
$157$	89794.6	0.290738	0.145369	−	0.989378i	$-0.453563\pi$
$157$	89794.6	0.290738	0.145369	+	0.989378i	$0.453563\pi$
$158$	−677628.	−2.15948
$159$	−101285.	−0.317726
$160$	0	0
$161$	156219.	0.474975
$162$	−101538.	−0.303977
$163$	481230.	1.41868	0.709339	−	0.704867i	$-0.248994\pi$
$163$	481230.	1.41868	0.709339	+	0.704867i	$0.248994\pi$
$164$	−260653.	−0.756750
$165$	0	0
$166$	978376.	2.75572
$167$	86572.7	0.240209	0.120105	−	0.992761i	$-0.461677\pi$
$167$	86572.7	0.240209	0.120105	+	0.992761i	$0.461677\pi$
$168$	46532.8	0.127200
$169$	−280909.	−0.756569
$170$	0	0
$171$	−705718.	−1.84562
$172$	714323.	1.84108
$173$	58137.4	0.147686	0.0738432	−	0.997270i	$-0.476474\pi$
$173$	58137.4	0.147686	0.0738432	+	0.997270i	$0.476474\pi$
$174$	−912511.	−2.28489
$175$	0	0
$176$	232495.	0.565759
$177$	−761985.	−1.82816
$178$	345026.	0.816209
$179$	−209380.	−0.488431	−0.244215	−	0.969721i	$-0.578530\pi$
$179$	−209380.	−0.488431	−0.244215	+	0.969721i	$0.578530\pi$
$180$	0	0
$181$	278996.	0.632996	0.316498	−	0.948593i	$-0.397493\pi$
$181$	278996.	0.632996	0.316498	+	0.948593i	$0.397493\pi$
$182$	−121901.	−0.272789
$183$	−1.29579e6	−2.86028
$184$	118038.	0.257027
$185$	0	0
$186$	−430535.	−0.912485
$187$	165594.	0.346290
$188$	−729426.	−1.50517
$189$	−216045.	−0.439935
$190$	0	0
$191$	−445132.	−0.882888	−0.441444	−	0.897289i	$-0.645534\pi$
$191$	−445132.	−0.882888	−0.441444	+	0.897289i	$0.645534\pi$
$192$	−1.05679e6	−2.06888
$193$	726811.	1.40452	0.702260	−	0.711920i	$-0.252174\pi$
$193$	726811.	1.40452	0.702260	+	0.711920i	$0.252174\pi$
$194$	361471.	0.689556
$195$	0	0
$196$	87574.7	0.162831
$197$	364897.	0.669892	0.334946	−	0.942237i	$-0.391282\pi$
$197$	364897.	0.669892	0.334946	+	0.942237i	$0.391282\pi$
$198$	927282.	1.68093
$199$	289307.	0.517877	0.258938	−	0.965894i	$-0.416627\pi$
$199$	289307.	0.517877	0.258938	+	0.965894i	$0.416627\pi$
$200$	0	0
$201$	−129621.	−0.226301
$202$	−212236.	−0.365965
$203$	−210665.	−0.358799
$204$	−573589.	−0.964997
$205$	0	0
$206$	−118497.	−0.194554
$207$	−1.32276e6	−2.14562
$208$	258792.	0.414756
$209$	459409.	0.727501
$210$	0	0
$211$	750147.	1.15995	0.579976	−	0.814633i	$-0.303062\pi$
$211$	750147.	1.15995	0.579976	+	0.814633i	$0.303062\pi$
$212$	−144030.	−0.220097
$213$	842672.	1.27265
$214$	142343.	0.212473
$215$	0	0
$216$	−163242.	−0.238066
$217$	−99394.3	−0.143289
$218$	711788.	1.01440
$219$	285093.	0.401677
$220$	0	0
$221$	184324.	0.253864
$222$	1.09402e6	1.48985
$223$	−534398.	−0.719619	−0.359810	−	0.933026i	$-0.617158\pi$
$223$	−534398.	−0.719619	−0.359810	+	0.933026i	$0.617158\pi$
$224$	−407085.	−0.542082
$225$	0	0
$226$	1.13836e6	1.48255
$227$	410624.	0.528907	0.264453	−	0.964398i	$-0.414808\pi$
$227$	410624.	0.528907	0.264453	+	0.964398i	$0.414808\pi$
$228$	−1.59132e6	−2.02731
$229$	1.03036e6	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$229$	1.03036e6	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$230$	0	0
$231$	339456.	0.418556
$232$	−159177.	−0.194160
$233$	119211.	0.143856	0.0719278	−	0.997410i	$-0.477085\pi$
$233$	119211.	0.143856	0.0719278	+	0.997410i	$0.477085\pi$
$234$	1.03217e6	1.23228
$235$	0	0
$236$	−1.08356e6	−1.26641
$237$	2.10042e6	2.42905
$238$	−248597.	−0.284481
$239$	−254090.	−0.287735	−0.143868	−	0.989597i	$-0.545954\pi$
$239$	−254090.	−0.287735	−0.143868	+	0.989597i	$0.545954\pi$
$240$	0	0
$241$	1.41251e6	1.56656	0.783282	−	0.621667i	$-0.213544\pi$
$241$	1.41251e6	1.56656	0.783282	+	0.621667i	$0.213544\pi$
$242$	729041.	0.800228
$243$	−756671.	−0.822037
$244$	−1.84265e6	−1.98138
$245$	0	0
$246$	1.51677e6	1.59801
$247$	511372.	0.533329
$248$	−75101.7	−0.0775391
$249$	−3.03264e6	−3.09972
$250$	0	0
$251$	−1.67542e6	−1.67857	−0.839286	−	0.543690i	$-0.817027\pi$
$251$	−1.67542e6	−1.67857	−0.839286	+	0.543690i	$0.817027\pi$
$252$	−741520.	−0.735566
$253$	861087.	0.845758
$254$	−583937.	−0.567913
$255$	0	0
$256$	697120.	0.664825
$257$	−726996.	−0.686593	−0.343296	−	0.939227i	$-0.611544\pi$
$257$	−726996.	−0.686593	−0.343296	+	0.939227i	$0.611544\pi$
$258$	−4.15672e6	−3.88778
$259$	252568.	0.233953
$260$	0	0
$261$	1.78376e6	1.62082
$262$	1.43746e6	1.29372
$263$	225880.	0.201367	0.100684	−	0.994918i	$-0.467897\pi$
$263$	225880.	0.201367	0.100684	+	0.994918i	$0.467897\pi$
$264$	256491.	0.226497
$265$	0	0
$266$	−689685.	−0.597650
$267$	−1.06947e6	−0.918097
$268$	−184325.	−0.156764
$269$	1.80527e6	1.52111	0.760557	−	0.649272i	$-0.224926\pi$
$269$	1.80527e6	1.52111	0.760557	+	0.649272i	$0.224926\pi$
$270$	0	0
$271$	−1.71380e6	−1.41754	−0.708771	−	0.705439i	$-0.750750\pi$
$271$	−1.71380e6	−1.41754	−0.708771	+	0.705439i	$0.750750\pi$
$272$	527765.	0.432532
$273$	377852.	0.306842
$274$	−16466.6	−0.0132504
$275$	0	0
$276$	−2.98267e6	−2.35685
$277$	−2.23055e6	−1.74668	−0.873338	−	0.487115i	$-0.838049\pi$
$277$	−2.23055e6	−1.74668	−0.873338	+	0.487115i	$0.838049\pi$
$278$	−3.03200e6	−2.35298
$279$	841600.	0.647285
$280$	0	0
$281$	1.67140e6	1.26274	0.631371	−	0.775481i	$-0.282493\pi$
$281$	1.67140e6	1.26274	0.631371	+	0.775481i	$0.282493\pi$
$282$	4.24461e6	3.17845
$283$	396152.	0.294033	0.147016	−	0.989134i	$-0.453033\pi$
$283$	396152.	0.294033	0.147016	+	0.989134i	$0.453033\pi$
$284$	1.19830e6	0.881597
$285$	0	0
$286$	−671920.	−0.485739
$287$	350164.	0.250938
$288$	3.44691e6	2.44877
$289$	−1.04396e6	−0.735256
$290$	0	0
$291$	−1.12044e6	−0.775634
$292$	405410.	0.278251
$293$	929465.	0.632505	0.316252	−	0.948675i	$-0.397575\pi$
$293$	929465.	0.632505	0.316252	+	0.948675i	$0.397575\pi$
$294$	−509606.	−0.343848
$295$	0	0
$296$	190839.	0.126601
$297$	−1.19085e6	−0.783365
$298$	−1.16443e6	−0.759582
$299$	958485.	0.620022
$300$	0	0
$301$	−959631.	−0.610503
$302$	−414280.	−0.261383
$303$	657860.	0.411649
$304$	1.46418e6	0.908682
$305$	0	0
$306$	2.10494e6	1.28510
$307$	−1.83295e6	−1.10995	−0.554976	−	0.831866i	$-0.687273\pi$
$307$	−1.83295e6	−1.10995	−0.554976	+	0.831866i	$0.687273\pi$
$308$	482715.	0.289944
$309$	367302.	0.218840
$310$	0	0
$311$	−2.29685e6	−1.34658	−0.673289	−	0.739379i	$-0.735119\pi$
$311$	−2.29685e6	−1.34658	−0.673289	+	0.739379i	$0.735119\pi$
$312$	285502.	0.166044
$313$	3.42470e6	1.97589	0.987943	−	0.154817i	$-0.0494787\pi$
$313$	3.42470e6	1.97589	0.987943	+	0.154817i	$0.0494787\pi$
$314$	−743043.	−0.425295
$315$	0	0
$316$	2.98685e6	1.68266
$317$	−2.94305e6	−1.64494	−0.822470	−	0.568808i	$-0.807405\pi$
$317$	−2.94305e6	−1.64494	−0.822470	+	0.568808i	$0.807405\pi$
$318$	838127.	0.464774
$319$	−1.16119e6	−0.638891
$320$	0	0
$321$	−441217.	−0.238996
$322$	−1.29270e6	−0.694799
$323$	1.04286e6	0.556187
$324$	447560.	0.236858
$325$	0	0
$326$	−3.98214e6	−2.07526
$327$	−2.20631e6	−1.14103
$328$	264582.	0.135792
$329$	979921.	0.499116
$330$	0	0
$331$	966164.	0.484709	0.242354	−	0.970188i	$-0.422080\pi$
$331$	966164.	0.484709	0.242354	+	0.970188i	$0.422080\pi$
$332$	−4.31250e6	−2.14725
$333$	−2.13857e6	−1.05685
$334$	−716382.	−0.351381
$335$	0	0
$336$	1.08188e6	0.522795
$337$	−136417.	−0.0654327	−0.0327163	−	0.999465i	$-0.510416\pi$
$337$	−136417.	−0.0654327	−0.0327163	+	0.999465i	$0.510416\pi$
$338$	2.32450e6	1.10672
$339$	−3.52854e6	−1.66761
$340$	0	0
$341$	−547865.	−0.255145
$342$	5.83976e6	2.69979
$343$	−117649.	−0.0539949
$344$	−725091.	−0.330367
$345$	0	0
$346$	−481082.	−0.216038
$347$	−355408.	−0.158454	−0.0792270	−	0.996857i	$-0.525245\pi$
$347$	−355408.	−0.158454	−0.0792270	+	0.996857i	$0.525245\pi$
$348$	4.02218e6	1.78038
$349$	−140128.	−0.0615830	−0.0307915	−	0.999526i	$-0.509803\pi$
$349$	−140128.	−0.0615830	−0.0307915	+	0.999526i	$0.509803\pi$
$350$	0	0
$351$	−1.32554e6	−0.574283
$352$	−2.24387e6	−0.965252
$353$	−3.48141e6	−1.48703	−0.743514	−	0.668721i	$-0.766842\pi$
$353$	−3.48141e6	−1.48703	−0.743514	+	0.668721i	$0.766842\pi$
$354$	6.30536e6	2.67425
$355$	0	0
$356$	−1.52081e6	−0.635988
$357$	770568.	0.319993
$358$	1.73260e6	0.714482
$359$	1.75285e6	0.717810	0.358905	−	0.933374i	$-0.383150\pi$
$359$	1.75285e6	0.717810	0.358905	+	0.933374i	$0.383150\pi$
$360$	0	0
$361$	417127.	0.168461
$362$	−2.30867e6	−0.925955
$363$	−2.25979e6	−0.900121
$364$	537315.	0.212557
$365$	0	0
$366$	1.07226e7	4.18405
$367$	1.76939e6	0.685738	0.342869	−	0.939383i	$-0.388601\pi$
$367$	1.76939e6	0.685738	0.342869	+	0.939383i	$0.388601\pi$
$368$	2.74438e6	1.05639
$369$	−2.96494e6	−1.13357
$370$	0	0
$371$	193492.	0.0729841
$372$	1.89771e6	0.711006
$373$	4.16212e6	1.54897	0.774485	−	0.632592i	$-0.218009\pi$
$373$	4.16212e6	1.54897	0.774485	+	0.632592i	$0.218009\pi$
$374$	−1.37027e6	−0.506557
$375$	0	0
$376$	740422.	0.270091
$377$	−1.29253e6	−0.468369
$378$	1.78775e6	0.643543
$379$	618163.	0.221057	0.110529	−	0.993873i	$-0.464746\pi$
$379$	618163.	0.221057	0.110529	+	0.993873i	$0.464746\pi$
$380$	0	0
$381$	1.81001e6	0.638805
$382$	3.68343e6	1.29150
$383$	4.11163e6	1.43225	0.716123	−	0.697974i	$-0.245915\pi$
$383$	4.11163e6	1.43225	0.716123	+	0.697974i	$0.245915\pi$
$384$	1.92587e6	0.666498
$385$	0	0
$386$	−6.01430e6	−2.05455
$387$	8.12547e6	2.75785
$388$	−1.59330e6	−0.537301
$389$	4.62076e6	1.54824	0.774122	−	0.633037i	$-0.218192\pi$
$389$	4.62076e6	1.54824	0.774122	+	0.633037i	$0.218192\pi$
$390$	0	0
$391$	1.95468e6	0.646596
$392$	−88894.8	−0.0292187
$393$	−4.45564e6	−1.45522
$394$	−3.01949e6	−0.979926
$395$	0	0
$396$	−4.08728e6	−1.30978
$397$	−5.07349e6	−1.61559	−0.807794	−	0.589465i	$-0.799339\pi$
$397$	−5.07349e6	−1.61559	−0.807794	+	0.589465i	$0.799339\pi$
$398$	−2.39399e6	−0.757557
$399$	2.13780e6	0.672255
$400$	0	0
$401$	−1.48056e6	−0.459795	−0.229898	−	0.973215i	$-0.573839\pi$
$401$	−1.48056e6	−0.459795	−0.229898	+	0.973215i	$0.573839\pi$
$402$	1.07261e6	0.331036
$403$	−609834.	−0.187046
$404$	935495.	0.285160
$405$	0	0
$406$	1.74323e6	0.524856
$407$	1.39217e6	0.416586
$408$	582236.	0.173160
$409$	−4.53379e6	−1.34015	−0.670075	−	0.742294i	$-0.733738\pi$
$409$	−4.53379e6	−1.34015	−0.670075	+	0.742294i	$0.733738\pi$
$410$	0	0
$411$	51041.0	0.0149044
$412$	522313.	0.151596
$413$	1.45567e6	0.419941
$414$	1.09457e7	3.13865
$415$	0	0
$416$	−2.49767e6	−0.707623
$417$	9.39820e6	2.64670
$418$	−3.80157e6	−1.06420
$419$	111026.	0.0308952	0.0154476	−	0.999881i	$-0.495083\pi$
$419$	111026.	0.0308952	0.0154476	+	0.999881i	$0.495083\pi$
$420$	0	0
$421$	−1.41151e6	−0.388132	−0.194066	−	0.980988i	$-0.562168\pi$
$421$	−1.41151e6	−0.388132	−0.194066	+	0.980988i	$0.562168\pi$
$422$	−6.20740e6	−1.69679
$423$	−8.29727e6	−2.25468
$424$	146201.	0.0394945
$425$	0	0
$426$	−6.97304e6	−1.86165
$427$	2.47544e6	0.657027
$428$	−627422.	−0.165558
$429$	2.08273e6	0.546374
$430$	0	0
$431$	1.07640e6	0.279113	0.139557	−	0.990214i	$-0.455432\pi$
$431$	1.07640e6	0.279113	0.139557	+	0.990214i	$0.455432\pi$
$432$	−3.79535e6	−0.978459
$433$	310172.	0.0795029	0.0397515	−	0.999210i	$-0.487343\pi$
$433$	310172.	0.0795029	0.0397515	+	0.999210i	$0.487343\pi$
$434$	822480.	0.209605
$435$	0	0
$436$	−3.13743e6	−0.790419
$437$	5.42288e6	1.35840
$438$	−2.35912e6	−0.587578
$439$	5.67650e6	1.40579	0.702893	−	0.711296i	$-0.251891\pi$
$439$	5.67650e6	1.40579	0.702893	+	0.711296i	$0.251891\pi$
$440$	0	0
$441$	996168.	0.243914
$442$	−1.52527e6	−0.371356
$443$	−4.05966e6	−0.982834	−0.491417	−	0.870924i	$-0.663521\pi$
$443$	−4.05966e6	−0.982834	−0.491417	+	0.870924i	$0.663521\pi$
$444$	−4.82223e6	−1.16089
$445$	0	0
$446$	4.42210e6	1.05267
$447$	3.60936e6	0.854401
$448$	2.01885e6	0.475237
$449$	−6.96544e6	−1.63054	−0.815272	−	0.579078i	$-0.803413\pi$
$449$	−6.96544e6	−1.63054	−0.815272	+	0.579078i	$0.803413\pi$
$450$	0	0
$451$	1.93012e6	0.446830
$452$	−5.01767e6	−1.15520
$453$	1.28413e6	0.294011
$454$	−3.39788e6	−0.773692
$455$	0	0
$456$	1.61530e6	0.363783
$457$	−1.79523e6	−0.402096	−0.201048	−	0.979581i	$-0.564435\pi$
$457$	−1.79523e6	−0.402096	−0.201048	+	0.979581i	$0.564435\pi$
$458$	−8.52616e6	−1.89928
$459$	−2.70323e6	−0.598896
$460$	0	0
$461$	−2.11294e6	−0.463058	−0.231529	−	0.972828i	$-0.574373\pi$
$461$	−2.11294e6	−0.463058	−0.231529	+	0.972828i	$0.574373\pi$
$462$	−2.80897e6	−0.612268
$463$	−1.26223e6	−0.273643	−0.136822	−	0.990596i	$-0.543689\pi$
$463$	−1.26223e6	−0.273643	−0.136822	+	0.990596i	$0.543689\pi$
$464$	−3.70084e6	−0.798005
$465$	0	0
$466$	−986462.	−0.210434
$467$	3.58926e6	0.761576	0.380788	−	0.924662i	$-0.375653\pi$
$467$	3.58926e6	0.761576	0.380788	+	0.924662i	$0.375653\pi$
$468$	−4.54960e6	−0.960192
$469$	247624.	0.0519830
$470$	0	0
$471$	2.30319e6	0.478384
$472$	1.09990e6	0.227246
$473$	−5.28952e6	−1.08708
$474$	−1.73808e7	−3.55324
$475$	0	0
$476$	1.09577e6	0.221667
$477$	−1.63835e6	−0.329694
$478$	2.10257e6	0.420903
$479$	−2.41693e6	−0.481311	−0.240655	−	0.970611i	$-0.577362\pi$
$479$	−2.41693e6	−0.481311	−0.240655	+	0.970611i	$0.577362\pi$
$480$	0	0
$481$	1.54963e6	0.305398
$482$	−1.16884e7	−2.29159
$483$	4.00695e6	0.781531
$484$	−3.21347e6	−0.623536
$485$	0	0
$486$	6.26139e6	1.20249
$487$	5.19403e6	0.992388	0.496194	−	0.868212i	$-0.334730\pi$
$487$	5.19403e6	0.992388	0.496194	+	0.868212i	$0.334730\pi$
$488$	1.87043e6	0.355543
$489$	1.23433e7	2.33432
$490$	0	0
$491$	5.38961e6	1.00891	0.504456	−	0.863437i	$-0.331693\pi$
$491$	5.38961e6	1.00891	0.504456	+	0.863437i	$0.331693\pi$
$492$	−6.68561e6	−1.24517
$493$	−2.63592e6	−0.488443
$494$	−4.23156e6	−0.780160
$495$	0	0
$496$	−1.74610e6	−0.318688
$497$	−1.60981e6	−0.292338
$498$	2.50949e7	4.53431
$499$	−3.29606e6	−0.592576	−0.296288	−	0.955099i	$-0.595749\pi$
$499$	−3.29606e6	−0.592576	−0.296288	+	0.955099i	$0.595749\pi$
$500$	0	0
$501$	2.22055e6	0.395244
$502$	1.38640e7	2.45544
$503$	1.06512e7	1.87706	0.938528	−	0.345204i	$-0.112190\pi$
$503$	1.06512e7	1.87706	0.938528	+	0.345204i	$0.112190\pi$
$504$	752698.	0.131991
$505$	0	0
$506$	−7.12543e6	−1.23718
$507$	−7.20517e6	−1.24487
$508$	2.57388e6	0.442516
$509$	−2.74268e6	−0.469225	−0.234612	−	0.972089i	$-0.575382\pi$
$509$	−2.74268e6	−0.469225	−0.234612	+	0.972089i	$0.575382\pi$
$510$	0	0
$511$	−544633.	−0.0922682
$512$	−8.17130e6	−1.37758
$513$	−7.49961e6	−1.25819
$514$	6.01583e6	1.00436
$515$	0	0
$516$	1.83220e7	3.02935
$517$	5.40136e6	0.888744
$518$	−2.08998e6	−0.342230
$519$	1.49120e6	0.243006
$520$	0	0
$521$	4.97077e6	0.802286	0.401143	−	0.916015i	$-0.368613\pi$
$521$	4.97077e6	0.802286	0.401143	+	0.916015i	$0.368613\pi$
$522$	−1.47605e7	−2.37096
$523$	−2.41579e6	−0.386193	−0.193096	−	0.981180i	$-0.561853\pi$
$523$	−2.41579e6	−0.386193	−0.193096	+	0.981180i	$0.561853\pi$
$524$	−6.33603e6	−1.00807
$525$	0	0
$526$	−1.86914e6	−0.294563
$527$	−1.24366e6	−0.195063
$528$	5.96337e6	0.930908
$529$	3.72798e6	0.579207
$530$	0	0
$531$	−1.23256e7	−1.89702
$532$	3.04000e6	0.465688
$533$	2.14843e6	0.327570
$534$	8.84974e6	1.34300
$535$	0	0
$536$	187103.	0.0281300
$537$	−5.37050e6	−0.803672
$538$	−1.49385e7	−2.22510
$539$	−648485.	−0.0961454
$540$	0	0
$541$	472165.	0.0693587	0.0346794	−	0.999398i	$-0.488959\pi$
$541$	472165.	0.0693587	0.0346794	+	0.999398i	$0.488959\pi$
$542$	1.41815e7	2.07360
$543$	7.15610e6	1.04154
$544$	−5.09360e6	−0.737951
$545$	0	0
$546$	−3.12669e6	−0.448852
$547$	−7.63716e6	−1.09135	−0.545675	−	0.837997i	$-0.683727\pi$
$547$	−7.63716e6	−1.09135	−0.545675	+	0.837997i	$0.683727\pi$
$548$	72581.6	0.0103246
$549$	−2.09603e7	−2.96802
$550$	0	0
$551$	−7.31285e6	−1.02614
$552$	3.02763e6	0.422917
$553$	−4.01258e6	−0.557970
$554$	1.84576e7	2.55506
$555$	0	0
$556$	1.33645e7	1.83343
$557$	4.48807e6	0.612946	0.306473	−	0.951879i	$-0.400851\pi$
$557$	4.48807e6	0.612946	0.306473	+	0.951879i	$0.400851\pi$
$558$	−6.96417e6	−0.946856
$559$	−5.88782e6	−0.796938
$560$	0	0
$561$	4.24740e6	0.569791
$562$	−1.38307e7	−1.84715
$563$	2.16500e6	0.287864	0.143932	−	0.989588i	$-0.454025\pi$
$563$	2.16500e6	0.287864	0.143932	+	0.989588i	$0.454025\pi$
$564$	−1.87094e7	−2.47664
$565$	0	0
$566$	−3.27812e6	−0.430115
$567$	−601258.	−0.0785422
$568$	−1.21637e6	−0.158195
$569$	−1.13325e7	−1.46739	−0.733696	−	0.679478i	$-0.762206\pi$
$569$	−1.13325e7	−1.46739	−0.733696	+	0.679478i	$0.762206\pi$
$570$	0	0
$571$	−843773.	−0.108302	−0.0541509	−	0.998533i	$-0.517245\pi$
$571$	−843773.	−0.108302	−0.0541509	+	0.998533i	$0.517245\pi$
$572$	2.96170e6	0.378487
$573$	−1.14174e7	−1.45272
$574$	−2.89758e6	−0.367076
$575$	0	0
$576$	−1.70942e7	−2.14681
$577$	2.23784e6	0.279827	0.139914	−	0.990164i	$-0.455318\pi$
$577$	2.23784e6	0.279827	0.139914	+	0.990164i	$0.455318\pi$
$578$	8.63866e6	1.07554
$579$	1.86423e7	2.31102
$580$	0	0
$581$	5.79346e6	0.712029
$582$	9.27156e6	1.13461
$583$	1.06653e6	0.129958
$584$	−411521.	−0.0499299
$585$	0	0
$586$	−7.69124e6	−0.925236
$587$	−1.21190e7	−1.45168	−0.725839	−	0.687864i	$-0.758548\pi$
$587$	−1.21190e7	−1.45168	−0.725839	+	0.687864i	$0.758548\pi$
$588$	2.24625e6	0.267926
$589$	−3.45030e6	−0.409797
$590$	0	0
$591$	9.35942e6	1.10225
$592$	4.43698e6	0.520335
$593$	−8.00167e6	−0.934424	−0.467212	−	0.884145i	$-0.654742\pi$
$593$	−8.00167e6	−0.934424	−0.467212	+	0.884145i	$0.654742\pi$
$594$	9.85415e6	1.14592
$595$	0	0
$596$	5.13261e6	0.591865
$597$	7.42058e6	0.852123
$598$	−7.93138e6	−0.906976
$599$	1.45899e7	1.66144	0.830719	−	0.556692i	$-0.187930\pi$
$599$	1.45899e7	1.66144	0.830719	+	0.556692i	$0.187930\pi$
$600$	0	0
$601$	−8.67178e6	−0.979314	−0.489657	−	0.871915i	$-0.662878\pi$
$601$	−8.67178e6	−0.979314	−0.489657	+	0.871915i	$0.662878\pi$
$602$	7.94087e6	0.893052
$603$	−2.09671e6	−0.234825
$604$	1.82607e6	0.203669
$605$	0	0
$606$	−5.44374e6	−0.602166
$607$	1.33059e7	1.46580	0.732898	−	0.680339i	$-0.238167\pi$
$607$	1.33059e7	1.46580	0.732898	+	0.680339i	$0.238167\pi$
$608$	−1.41312e7	−1.55032
$609$	−5.40344e6	−0.590374
$610$	0	0
$611$	6.01231e6	0.651536
$612$	−9.27817e6	−1.00135
$613$	−2.35101e6	−0.252699	−0.126350	−	0.991986i	$-0.540326\pi$
$613$	−2.35101e6	−0.252699	−0.126350	+	0.991986i	$0.540326\pi$
$614$	1.51675e7	1.62365
$615$	0	0
$616$	−489991.	−0.0520280
$617$	−9.63523e6	−1.01894	−0.509470	−	0.860488i	$-0.670159\pi$
$617$	−9.63523e6	−1.01894	−0.509470	+	0.860488i	$0.670159\pi$
$618$	−3.03939e6	−0.320122
$619$	−4.86148e6	−0.509967	−0.254983	−	0.966945i	$-0.582070\pi$
$619$	−4.86148e6	−0.509967	−0.254983	+	0.966945i	$0.582070\pi$
$620$	0	0
$621$	−1.40568e7	−1.46271
$622$	1.90062e7	1.96979
$623$	2.04307e6	0.210894
$624$	6.63789e6	0.682446
$625$	0	0
$626$	−2.83391e7	−2.89035
$627$	1.17836e7	1.19704
$628$	3.27519e6	0.331389
$629$	3.16023e6	0.318487
$630$	0	0
$631$	−6.59770e6	−0.659659	−0.329829	−	0.944041i	$-0.606991\pi$
$631$	−6.59770e6	−0.659659	−0.329829	+	0.944041i	$0.606991\pi$
$632$	−3.03188e6	−0.301939
$633$	1.92409e7	1.90860
$634$	2.43535e7	2.40624
$635$	0	0
$636$	−3.69430e6	−0.362151
$637$	−721836.	−0.0704839
$638$	9.60876e6	0.934578
$639$	1.36308e7	1.32059
$640$	0	0
$641$	1.44525e7	1.38930	0.694651	−	0.719347i	$-0.255559\pi$
$641$	1.44525e7	1.38930	0.694651	+	0.719347i	$0.255559\pi$
$642$	3.65104e6	0.349606
$643$	1.54720e7	1.47577	0.737886	−	0.674926i	$-0.235824\pi$
$643$	1.54720e7	1.47577	0.737886	+	0.674926i	$0.235824\pi$
$644$	5.69799e6	0.541386
$645$	0	0
$646$	−8.62960e6	−0.813597
$647$	−1.66647e7	−1.56508	−0.782540	−	0.622601i	$-0.786076\pi$
$647$	−1.66647e7	−1.56508	−0.782540	+	0.622601i	$0.786076\pi$
$648$	−454306.	−0.0425022
$649$	8.02371e6	0.747762
$650$	0	0
$651$	−2.54941e6	−0.235770
$652$	1.75525e7	1.61704
$653$	1.33451e7	1.22472	0.612361	−	0.790578i	$-0.290220\pi$
$653$	1.33451e7	1.22472	0.612361	+	0.790578i	$0.290220\pi$
$654$	1.82570e7	1.66911
$655$	0	0
$656$	6.15149e6	0.558111
$657$	4.61157e6	0.416807
$658$	−8.10877e6	−0.730113
$659$	−4.00667e6	−0.359393	−0.179697	−	0.983722i	$-0.557512\pi$
$659$	−4.00667e6	−0.359393	−0.179697	+	0.983722i	$0.557512\pi$
$660$	0	0
$661$	1.08005e7	0.961478	0.480739	−	0.876864i	$-0.340368\pi$
$661$	1.08005e7	0.961478	0.480739	+	0.876864i	$0.340368\pi$
$662$	−7.99493e6	−0.709038
$663$	4.72782e6	0.417712
$664$	4.37750e6	0.385307
$665$	0	0
$666$	1.76965e7	1.54597
$667$	−1.37068e7	−1.19294
$668$	3.15767e6	0.273795
$669$	−1.37070e7	−1.18407
$670$	0	0
$671$	1.36447e7	1.16993
$672$	−1.04415e7	−0.891951
$673$	−1.09119e7	−0.928676	−0.464338	−	0.885658i	$-0.653708\pi$
$673$	−1.09119e7	−0.928676	−0.464338	+	0.885658i	$0.653708\pi$
$674$	1.12884e6	0.0957158
$675$	0	0
$676$	−1.02459e7	−0.862353
$677$	1.35765e7	1.13846	0.569229	−	0.822179i	$-0.307242\pi$
$677$	1.35765e7	1.13846	0.569229	+	0.822179i	$0.307242\pi$
$678$	2.91984e7	2.43941
$679$	2.14046e6	0.178169
$680$	0	0
$681$	1.05323e7	0.870272
$682$	4.53354e6	0.373230
$683$	1.26726e7	1.03948	0.519738	−	0.854326i	$-0.326030\pi$
$683$	1.26726e7	1.03948	0.519738	+	0.854326i	$0.326030\pi$
$684$	−2.57406e7	−2.10367
$685$	0	0
$686$	973536.	0.0789845
$687$	2.64283e7	2.13637
$688$	−1.68583e7	−1.35782
$689$	1.18717e6	0.0952720
$690$	0	0
$691$	7.11964e6	0.567235	0.283617	−	0.958938i	$-0.408465\pi$
$691$	7.11964e6	0.567235	0.283617	+	0.958938i	$0.408465\pi$
$692$	2.12052e6	0.168336
$693$	5.49091e6	0.434322
$694$	2.94097e6	0.231789
$695$	0	0
$696$	−4.08281e6	−0.319474
$697$	4.38139e6	0.341609
$698$	1.15955e6	0.0900844
$699$	3.05770e6	0.236702
$700$	0	0
$701$	−1.00155e7	−0.769803	−0.384902	−	0.922958i	$-0.625765\pi$
$701$	−1.00155e7	−0.769803	−0.384902	+	0.922958i	$0.625765\pi$
$702$	1.09687e7	0.840068
$703$	8.76746e6	0.669092
$704$	1.11280e7	0.846224
$705$	0	0
$706$	2.88084e7	2.17524
$707$	−1.25676e6	−0.0945589
$708$	−2.77928e7	−2.08377
$709$	−8.84454e6	−0.660784	−0.330392	−	0.943844i	$-0.607181\pi$
$709$	−8.84454e6	−0.660784	−0.330392	+	0.943844i	$0.607181\pi$
$710$	0	0
$711$	3.39757e7	2.52054
$712$	1.54373e6	0.114123
$713$	−6.46703e6	−0.476410
$714$	−6.37638e6	−0.468090
$715$	0	0
$716$	−7.63698e6	−0.556723
$717$	−6.51728e6	−0.473444
$718$	−1.45047e7	−1.05002
$719$	6.58086e6	0.474745	0.237373	−	0.971419i	$-0.423714\pi$
$719$	6.58086e6	0.474745	0.237373	+	0.971419i	$0.423714\pi$
$720$	0	0
$721$	−701682.	−0.0502692
$722$	−3.45169e6	−0.246427
$723$	3.62301e7	2.57765
$724$	1.01762e7	0.721502
$725$	0	0
$726$	1.86995e7	1.31671
$727$	−1.88401e7	−1.32205	−0.661023	−	0.750365i	$-0.729878\pi$
$727$	−1.88401e7	−1.32205	−0.661023	+	0.750365i	$0.729878\pi$
$728$	−545414.	−0.0381415
$729$	−2.23900e7	−1.56040
$730$	0	0
$731$	−1.20073e7	−0.831095
$732$	−4.72631e7	−3.26020
$733$	2.78330e6	0.191337	0.0956687	−	0.995413i	$-0.469501\pi$
$733$	2.78330e6	0.191337	0.0956687	+	0.995413i	$0.469501\pi$
$734$	−1.46416e7	−1.00311
$735$	0	0
$736$	−2.64867e7	−1.80233
$737$	1.36491e6	0.0925629
$738$	2.45346e7	1.65821
$739$	−2.48970e7	−1.67701	−0.838505	−	0.544894i	$-0.816570\pi$
$739$	−2.48970e7	−1.67701	−0.838505	+	0.544894i	$0.816570\pi$
$740$	0	0
$741$	1.31164e7	0.877548
$742$	−1.60113e6	−0.106762
$743$	3.86085e6	0.256573	0.128286	−	0.991737i	$-0.459052\pi$
$743$	3.86085e6	0.256573	0.128286	+	0.991737i	$0.459052\pi$
$744$	−1.92632e6	−0.127584
$745$	0	0
$746$	−3.44412e7	−2.26585
$747$	−4.90549e7	−3.21648
$748$	6.03991e6	0.394708
$749$	842888.	0.0548991
$750$	0	0
$751$	6.72737e6	0.435257	0.217628	−	0.976032i	$-0.430168\pi$
$751$	6.72737e6	0.435257	0.217628	+	0.976032i	$0.430168\pi$
$752$	1.72147e7	1.11008
$753$	−4.29737e7	−2.76195
$754$	1.06956e7	0.685136
$755$	0	0
$756$	−7.88007e6	−0.501447
$757$	−2.17782e7	−1.38128	−0.690642	−	0.723197i	$-0.742672\pi$
$757$	−2.17782e7	−1.38128	−0.690642	+	0.723197i	$0.742672\pi$
$758$	−5.11525e6	−0.323366
$759$	2.20865e7	1.39162
$760$	0	0
$761$	−2.57074e7	−1.60915	−0.804575	−	0.593851i	$-0.797607\pi$
$761$	−2.57074e7	−1.60915	−0.804575	+	0.593851i	$0.797607\pi$
$762$	−1.49777e7	−0.934453
$763$	4.21486e6	0.262103
$764$	−1.62359e7	−1.00633
$765$	0	0
$766$	−3.40234e7	−2.09511
$767$	8.93127e6	0.548182
$768$	1.78808e7	1.09391
$769$	−1.34375e7	−0.819413	−0.409706	−	0.912217i	$-0.634369\pi$
$769$	−1.34375e7	−0.819413	−0.409706	+	0.912217i	$0.634369\pi$
$770$	0	0
$771$	−1.86471e7	−1.12973
$772$	2.65099e7	1.60090
$773$	−3.05572e7	−1.83935	−0.919674	−	0.392682i	$-0.871547\pi$
$773$	−3.05572e7	−1.83935	−0.919674	+	0.392682i	$0.871547\pi$
$774$	−6.72376e7	−4.03422
$775$	0	0
$776$	1.61731e6	0.0964140
$777$	6.47825e6	0.384951
$778$	−3.82364e7	−2.26479
$779$	1.21553e7	0.717667
$780$	0	0
$781$	−8.87335e6	−0.520547
$782$	−1.61748e7	−0.945849
$783$	1.89558e7	1.10494
$784$	−2.06679e6	−0.120090
$785$	0	0
$786$	3.68700e7	2.12871
$787$	2.07672e6	0.119520	0.0597602	−	0.998213i	$-0.480966\pi$
$787$	2.07672e6	0.119520	0.0597602	+	0.998213i	$0.480966\pi$
$788$	1.33093e7	0.763556
$789$	5.79372e6	0.331333
$790$	0	0
$791$	6.74081e6	0.383064
$792$	4.14890e6	0.235028
$793$	1.51881e7	0.857670
$794$	4.19827e7	2.36330
$795$	0	0
$796$	1.05523e7	0.590286
$797$	5.98563e6	0.333783	0.166892	−	0.985975i	$-0.446627\pi$
$797$	5.98563e6	0.333783	0.166892	+	0.985975i	$0.446627\pi$
$798$	−1.76901e7	−0.983383
$799$	1.22611e7	0.679460
$800$	0	0
$801$	−1.72993e7	−0.952679
$802$	1.22515e7	0.672594
$803$	−3.00204e6	−0.164296
$804$	−4.72784e6	−0.257942
$805$	0	0
$806$	5.04633e6	0.273614
$807$	4.63043e7	2.50286
$808$	−949597.	−0.0511695
$809$	1.96864e7	1.05754	0.528769	−	0.848766i	$-0.322654\pi$
$809$	1.96864e7	1.05754	0.528769	+	0.848766i	$0.322654\pi$
$810$	0	0
$811$	8.50101e6	0.453856	0.226928	−	0.973912i	$-0.427132\pi$
$811$	8.50101e6	0.453856	0.226928	+	0.973912i	$0.427132\pi$
$812$	−7.68384e6	−0.408967
$813$	−4.39580e7	−2.33245
$814$	−1.15201e7	−0.609387
$815$	0	0
$816$	1.35369e7	0.711695
$817$	−3.33119e7	−1.74600
$818$	3.75168e7	1.96039
$819$	6.11199e6	0.318400
$820$	0	0
$821$	−1.36199e6	−0.0705204	−0.0352602	−	0.999378i	$-0.511226\pi$
$821$	−1.36199e6	−0.0705204	−0.0352602	+	0.999378i	$0.511226\pi$
$822$	−422360.	−0.0218023
$823$	1.35934e6	0.0699566	0.0349783	−	0.999388i	$-0.488864\pi$
$823$	1.35934e6	0.0699566	0.0349783	+	0.999388i	$0.488864\pi$
$824$	−530186.	−0.0272026
$825$	0	0
$826$	−1.20456e7	−0.614295
$827$	−1.00727e7	−0.512132	−0.256066	−	0.966659i	$-0.582426\pi$
$827$	−1.00727e7	−0.512132	−0.256066	+	0.966659i	$0.582426\pi$
$828$	−4.82465e7	−2.44563
$829$	−5.63984e6	−0.285023	−0.142512	−	0.989793i	$-0.545518\pi$
$829$	−5.63984e6	−0.285023	−0.142512	+	0.989793i	$0.545518\pi$
$830$	0	0
$831$	−5.72125e7	−2.87401
$832$	1.23867e7	0.620364
$833$	−1.47207e6	−0.0735048
$834$	−7.77693e7	−3.87163
$835$	0	0
$836$	1.67566e7	0.829220
$837$	8.94361e6	0.441265
$838$	−918733.	−0.0451938
$839$	−1.16351e7	−0.570642	−0.285321	−	0.958432i	$-0.592100\pi$
$839$	−1.16351e7	−0.570642	−0.285321	+	0.958432i	$0.592100\pi$
$840$	0	0
$841$	−2.02735e6	−0.0988413
$842$	1.16801e7	0.567764
$843$	4.28706e7	2.07774
$844$	2.73610e7	1.32214
$845$	0	0
$846$	6.86592e7	3.29817
$847$	4.31702e6	0.206764
$848$	3.39916e6	0.162324
$849$	1.01611e7	0.483806
$850$	0	0
$851$	1.64332e7	0.777854
$852$	3.07358e7	1.45059
$853$	−2.85205e7	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$853$	−2.85205e7	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$854$	−2.04841e7	−0.961108
$855$	0	0
$856$	636880.	0.0297080
$857$	9.95725e6	0.463113	0.231557	−	0.972821i	$-0.425618\pi$
$857$	9.95725e6	0.463113	0.231557	+	0.972821i	$0.425618\pi$
$858$	−1.72344e7	−0.799243
$859$	−1.49322e7	−0.690463	−0.345232	−	0.938517i	$-0.612200\pi$
$859$	−1.49322e7	−0.690463	−0.345232	+	0.938517i	$0.612200\pi$
$860$	0	0
$861$	8.98154e6	0.412898
$862$	−8.90711e6	−0.408290
$863$	−3.84933e7	−1.75937	−0.879687	−	0.475553i	$-0.842248\pi$
$863$	−3.84933e7	−1.75937	−0.879687	+	0.475553i	$0.842248\pi$
$864$	3.66300e7	1.66937
$865$	0	0
$866$	−2.56665e6	−0.116298
$867$	−2.67770e7	−1.20980
$868$	−3.62533e6	−0.163323
$869$	−2.21175e7	−0.993542
$870$	0	0
$871$	1.51930e6	0.0678576
$872$	3.18472e6	0.141834
$873$	−1.81239e7	−0.804850
$874$	−4.48739e7	−1.98708
$875$	0	0
$876$	1.03986e7	0.457839
$877$	−9.40311e6	−0.412831	−0.206416	−	0.978464i	$-0.566180\pi$
$877$	−9.40311e6	−0.412831	−0.206416	+	0.978464i	$0.566180\pi$
$878$	−4.69726e7	−2.05640
$879$	2.38403e7	1.04073
$880$	0	0
$881$	1.10395e6	0.0479194	0.0239597	−	0.999713i	$-0.492373\pi$
$881$	1.10395e6	0.0479194	0.0239597	+	0.999713i	$0.492373\pi$
$882$	−8.24321e6	−0.356800
$883$	−8.06579e6	−0.348133	−0.174067	−	0.984734i	$-0.555691\pi$
$883$	−8.06579e6	−0.348133	−0.174067	+	0.984734i	$0.555691\pi$
$884$	6.72308e6	0.289359
$885$	0	0
$886$	3.35933e7	1.43770
$887$	1.49902e7	0.639732	0.319866	−	0.947463i	$-0.396362\pi$
$887$	1.49902e7	0.639732	0.319866	+	0.947463i	$0.396362\pi$
$888$	4.89493e6	0.208312
$889$	−3.45779e6	−0.146738
$890$	0	0
$891$	−3.31415e6	−0.139855
$892$	−1.94918e7	−0.820237
$893$	3.40162e7	1.42744
$894$	−2.98672e7	−1.24983
$895$	0	0
$896$	−3.67912e6	−0.153100
$897$	2.45847e7	1.02019
$898$	5.76384e7	2.38518
$899$	8.72090e6	0.359883
$900$	0	0
$901$	2.42104e6	0.0993553
$902$	−1.59716e7	−0.653629
$903$	−2.46141e7	−1.00453
$904$	5.09331e6	0.207290
$905$	0	0
$906$	−1.06261e7	−0.430083
$907$	−4.12622e6	−0.166546	−0.0832730	−	0.996527i	$-0.526537\pi$
$907$	−4.12622e6	−0.166546	−0.0832730	+	0.996527i	$0.526537\pi$
$908$	1.49772e7	0.602859
$909$	1.06413e7	0.427155
$910$	0	0
$911$	4.04272e7	1.61391	0.806953	−	0.590616i	$-0.201115\pi$
$911$	4.04272e7	1.61391	0.806953	+	0.590616i	$0.201115\pi$
$912$	3.75556e7	1.49516
$913$	3.19338e7	1.26787
$914$	1.48554e7	0.588191
$915$	0	0
$916$	3.75817e7	1.47992
$917$	8.51191e6	0.334275
$918$	2.23690e7	0.876072
$919$	2.18546e7	0.853600	0.426800	−	0.904346i	$-0.359641\pi$
$919$	2.18546e7	0.853600	0.426800	+	0.904346i	$0.359641\pi$
$920$	0	0
$921$	−4.70142e7	−1.82633
$922$	1.74844e7	0.677366
$923$	−9.87702e6	−0.381612
$924$	1.23814e7	0.477078
$925$	0	0
$926$	1.04448e7	0.400289
$927$	5.94134e6	0.227083
$928$	3.57178e7	1.36149
$929$	1.06843e7	0.406169	0.203085	−	0.979161i	$-0.434903\pi$
$929$	1.06843e7	0.406169	0.203085	+	0.979161i	$0.434903\pi$
$930$	0	0
$931$	−4.08398e6	−0.154422
$932$	4.34813e6	0.163970
$933$	−5.89130e7	−2.21568
$934$	−2.97009e7	−1.11404
$935$	0	0
$936$	4.61818e6	0.172298
$937$	3.99105e7	1.48504	0.742521	−	0.669823i	$-0.233630\pi$
$937$	3.99105e7	1.48504	0.742521	+	0.669823i	$0.233630\pi$
$938$	−2.04907e6	−0.0760414
$939$	8.78419e7	3.25116
$940$	0	0
$941$	1.32350e6	0.0487248	0.0243624	−	0.999703i	$-0.492244\pi$
$941$	1.32350e6	0.0487248	0.0243624	+	0.999703i	$0.492244\pi$
$942$	−1.90587e7	−0.699787
$943$	2.27832e7	0.834326
$944$	2.55724e7	0.933990
$945$	0	0
$946$	4.37703e7	1.59020
$947$	2.76322e7	1.00124	0.500622	−	0.865666i	$-0.333105\pi$
$947$	2.76322e7	1.00124	0.500622	+	0.865666i	$0.333105\pi$
$948$	7.66113e7	2.76868
$949$	−3.34160e6	−0.120445
$950$	0	0
$951$	−7.54879e7	−2.70661
$952$	−1.11229e6	−0.0397763
$953$	3.07901e7	1.09819	0.549096	−	0.835759i	$-0.314972\pi$
$953$	3.07901e7	1.09819	0.549096	+	0.835759i	$0.314972\pi$
$954$	1.35572e7	0.482281
$955$	0	0
$956$	−9.26775e6	−0.327966
$957$	−2.97840e7	−1.05124
$958$	1.99999e7	0.704068
$959$	−97507.1	−0.00342365
$960$	0	0
$961$	−2.45145e7	−0.856278
$962$	−1.28231e7	−0.446740
$963$	−7.13697e6	−0.247998
$964$	5.15202e7	1.78560
$965$	0	0
$966$	−3.31572e7	−1.14323
$967$	−2.92557e6	−0.100611	−0.0503055	−	0.998734i	$-0.516019\pi$
$967$	−2.92557e6	−0.100611	−0.0503055	+	0.998734i	$0.516019\pi$
$968$	3.26192e6	0.111888
$969$	2.67489e7	0.915159
$970$	0	0
$971$	2.78109e6	0.0946601	0.0473301	−	0.998879i	$-0.484929\pi$
$971$	2.78109e6	0.0946601	0.0473301	+	0.998879i	$0.484929\pi$
$972$	−2.75990e7	−0.936975
$973$	−1.79540e7	−0.607967
$974$	−4.29801e7	−1.45168
$975$	0	0
$976$	4.34872e7	1.46129
$977$	−7.48673e6	−0.250932	−0.125466	−	0.992098i	$-0.540043\pi$
$977$	−7.48673e6	−0.250932	−0.125466	+	0.992098i	$0.540043\pi$
$978$	−1.02140e8	−3.41467
$979$	1.12615e7	0.375525
$980$	0	0
$981$	−3.56884e7	−1.18401
$982$	−4.45985e7	−1.47585
$983$	−1.79815e7	−0.593528	−0.296764	−	0.954951i	$-0.595907\pi$
$983$	−1.79815e7	−0.593528	−0.296764	+	0.954951i	$0.595907\pi$
$984$	6.78639e6	0.223435
$985$	0	0
$986$	2.18120e7	0.714501
$987$	2.51345e7	0.821253
$988$	1.86519e7	0.607899
$989$	−6.24378e7	−2.02982
$990$	0	0
$991$	3.72778e7	1.20578	0.602888	−	0.797826i	$-0.294017\pi$
$991$	3.72778e7	1.20578	0.602888	+	0.797826i	$0.294017\pi$
$992$	1.68521e7	0.543720
$993$	2.47816e7	0.797548
$994$	1.33211e7	0.427635
$995$	0	0
$996$	−1.10613e8	−3.53313
$997$	−4.87422e7	−1.55298	−0.776492	−	0.630128i	$-0.783003\pi$
$997$	−4.87422e7	−1.55298	−0.776492	+	0.630128i	$0.783003\pi$
$998$	2.72747e7	0.866828
$999$	−2.27264e7	−0.720471

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	175.6.a.c.1.1		2
5.2	odd	4		175.6.b.c.99.1		4
5.3	odd	4		175.6.b.c.99.4		4
5.4	even	2		7.6.a.b.1.2	✓	2
15.14	odd	2		63.6.a.f.1.1		2
20.19	odd	2		112.6.a.h.1.2		2
35.4	even	6		49.6.c.e.30.1		4
35.9	even	6		49.6.c.e.18.1		4
35.19	odd	6		49.6.c.d.18.1		4
35.24	odd	6		49.6.c.d.30.1		4
35.34	odd	2		49.6.a.f.1.2		2
40.19	odd	2		448.6.a.u.1.1		2
40.29	even	2		448.6.a.w.1.2		2
55.54	odd	2		847.6.a.c.1.1		2
60.59	even	2		1008.6.a.bq.1.1		2
105.104	even	2		441.6.a.l.1.1		2
140.139	even	2		784.6.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.2	✓	2	5.4	even	2
49.6.a.f.1.2		2	35.34	odd	2
49.6.c.d.18.1		4	35.19	odd	6
49.6.c.d.30.1		4	35.24	odd	6
49.6.c.e.18.1		4	35.9	even	6
49.6.c.e.30.1		4	35.4	even	6
63.6.a.f.1.1		2	15.14	odd	2
112.6.a.h.1.2		2	20.19	odd	2
175.6.a.c.1.1		2	1.1	even	1	trivial
175.6.b.c.99.1		4	5.2	odd	4
175.6.b.c.99.4		4	5.3	odd	4
441.6.a.l.1.1		2	105.104	even	2
448.6.a.u.1.1		2	40.19	odd	2
448.6.a.w.1.2		2	40.29	even	2
784.6.a.v.1.1		2	140.139	even	2
847.6.a.c.1.1		2	55.54	odd	2
1008.6.a.bq.1.1		2	60.59	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	175.6.a.c.1.1

Level	$175$
Weight	$6$
Character	175.1
Self dual	yes
Analytic conductor	$28.067$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$