LMFDB - Embedded newform 45.6.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,6,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,5] 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:	$7.21727189158$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{145}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 36 $ x^2 - x - 36
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.52080$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.52080 q^{2} +40.6040 q^{4} +25.0000 q^{5} +160.416 q^{7} +73.3128 q^{8} +213.020 q^{10} +279.584 q^{11} -541.664 q^{13} +1366.87 q^{14} -674.644 q^{16} +777.334 q^{17} -2682.66 q^{19} +1015.10 q^{20} +2382.28 q^{22} -3694.48 q^{23} +625.000 q^{25} -4615.41 q^{26} +6513.53 q^{28} +8356.62 q^{29} -262.849 q^{31} -8094.51 q^{32} +6623.51 q^{34} +4010.40 q^{35} -14949.9 q^{37} -22858.4 q^{38} +1832.82 q^{40} -7988.28 q^{41} +5133.38 q^{43} +11352.2 q^{44} -31479.9 q^{46} +10567.8 q^{47} +8926.28 q^{49} +5325.50 q^{50} -21993.7 q^{52} +21069.4 q^{53} +6989.60 q^{55} +11760.5 q^{56} +71205.1 q^{58} +25699.6 q^{59} +8195.14 q^{61} -2239.68 q^{62} -47383.1 q^{64} -13541.6 q^{65} +51928.9 q^{67} +31562.9 q^{68} +34171.8 q^{70} +23689.2 q^{71} +20933.5 q^{73} -127385. q^{74} -108926. q^{76} +44849.7 q^{77} -39279.7 q^{79} -16866.1 q^{80} -68066.6 q^{82} +35911.1 q^{83} +19433.4 q^{85} +43740.5 q^{86} +20497.1 q^{88} -94074.7 q^{89} -86891.5 q^{91} -150011. q^{92} +90046.3 q^{94} -67066.4 q^{95} -12249.9 q^{97} +76059.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 21 q^{4} + 50 q^{5} + 80 q^{7} + 255 q^{8} + 125 q^{10} + 800 q^{11} - 120 q^{13} + 1650 q^{14} - 687 q^{16} + 1940 q^{17} - 1512 q^{19} + 525 q^{20} + 550 q^{22} - 1320 q^{23} + 1250 q^{25}+ \cdots + 112465 q^{98}+O(q^{100}) $ 2 * q + 5 * q^2 + 21 * q^4 + 50 * q^5 + 80 * q^7 + 255 * q^8 + 125 * q^10 + 800 * q^11 - 120 * q^13 + 1650 * q^14 - 687 * q^16 + 1940 * q^17 - 1512 * q^19 + 525 * q^20 + 550 * q^22 - 1320 * q^23 + 1250 * q^25 - 6100 * q^26 + 8090 * q^28 + 1300 * q^29 - 5824 * q^31 - 13865 * q^32 + 2530 * q^34 + 2000 * q^35 - 12560 * q^37 - 26980 * q^38 + 6375 * q^40 + 400 * q^41 + 25680 * q^43 + 1150 * q^44 - 39840 * q^46 + 18920 * q^47 - 1414 * q^49 + 3125 * q^50 - 30260 * q^52 + 49460 * q^53 + 20000 * q^55 - 2850 * q^56 + 96050 * q^58 + 63200 * q^59 - 49116 * q^61 + 17340 * q^62 - 26671 * q^64 - 3000 * q^65 + 6080 * q^67 + 8770 * q^68 + 41250 * q^70 + 65200 * q^71 + 97740 * q^73 - 135800 * q^74 - 131876 * q^76 + 3000 * q^77 - 46288 * q^79 - 17175 * q^80 - 97600 * q^82 - 57360 * q^83 + 48500 * q^85 - 28600 * q^86 + 115050 * q^88 - 87000 * q^89 - 120800 * q^91 - 196560 * q^92 + 60640 * q^94 - 37800 * q^95 + 10180 * q^97 + 112465 * q^98

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 $

$2$	8.52080	1.50628	0.753139	−	0.657861i	$-0.228539\pi$
$2$	8.52080	1.50628	0.753139	+	0.657861i	$0.228539\pi$
$3$	0	0
$3$	0	0
$4$	40.6040	1.26887
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	160.416	1.23738	0.618689	−	0.785636i	$-0.287664\pi$
$7$	160.416	1.23738	0.618689	+	0.785636i	$0.287664\pi$
$8$	73.3128	0.405000
$9$	0	0
$10$	213.020	0.673628
$11$	279.584	0.696676	0.348338	−	0.937369i	$-0.386746\pi$
$11$	279.584	0.696676	0.348338	+	0.937369i	$0.386746\pi$
$12$	0	0
$13$	−541.664	−0.888938	−0.444469	−	0.895794i	$-0.646608\pi$
$13$	−541.664	−0.888938	−0.444469	+	0.895794i	$0.646608\pi$
$14$	1366.87	1.86384
$15$	0	0
$16$	−674.644	−0.658832
$17$	777.334	0.652357	0.326179	−	0.945308i	$-0.394239\pi$
$17$	777.334	0.652357	0.326179	+	0.945308i	$0.394239\pi$
$18$	0	0
$19$	−2682.66	−1.70483	−0.852415	−	0.522867i	$-0.824863\pi$
$19$	−2682.66	−1.70483	−0.852415	+	0.522867i	$0.824863\pi$
$20$	1015.10	0.567458
$21$	0	0
$22$	2382.28	1.04939
$23$	−3694.48	−1.45624	−0.728122	−	0.685448i	$-0.759606\pi$
$23$	−3694.48	−1.45624	−0.728122	+	0.685448i	$0.759606\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−4615.41	−1.33899
$27$	0	0
$28$	6513.53	1.57008
$29$	8356.62	1.84517	0.922584	−	0.385797i	$-0.126074\pi$
$29$	8356.62	1.84517	0.922584	+	0.385797i	$0.126074\pi$
$30$	0	0
$31$	−262.849	−0.0491250	−0.0245625	−	0.999698i	$-0.507819\pi$
$31$	−262.849	−0.0491250	−0.0245625	+	0.999698i	$0.507819\pi$
$32$	−8094.51	−1.39738
$33$	0	0
$34$	6623.51	0.982632
$35$	4010.40	0.553372
$36$	0	0
$37$	−14949.9	−1.79529	−0.897647	−	0.440716i	$-0.854725\pi$
$37$	−14949.9	−1.79529	−0.897647	+	0.440716i	$0.854725\pi$
$38$	−22858.4	−2.56795
$39$	0	0
$40$	1832.82	0.181121
$41$	−7988.28	−0.742154	−0.371077	−	0.928602i	$-0.621011\pi$
$41$	−7988.28	−0.742154	−0.371077	+	0.928602i	$0.621011\pi$
$42$	0	0
$43$	5133.38	0.423382	0.211691	−	0.977337i	$-0.432103\pi$
$43$	5133.38	0.423382	0.211691	+	0.977337i	$0.432103\pi$
$44$	11352.2	0.883994
$45$	0	0
$46$	−31479.9	−2.19351
$47$	10567.8	0.697816	0.348908	−	0.937157i	$-0.386553\pi$
$47$	10567.8	0.697816	0.348908	+	0.937157i	$0.386553\pi$
$48$	0	0
$49$	8926.28	0.531105
$50$	5325.50	0.301256
$51$	0	0
$52$	−21993.7	−1.12795
$53$	21069.4	1.03029	0.515147	−	0.857102i	$-0.327737\pi$
$53$	21069.4	1.03029	0.515147	+	0.857102i	$0.327737\pi$
$54$	0	0
$55$	6989.60	0.311563
$56$	11760.5	0.501138
$57$	0	0
$58$	71205.1	2.77934
$59$	25699.6	0.961162	0.480581	−	0.876950i	$-0.340426\pi$
$59$	25699.6	0.961162	0.480581	+	0.876950i	$0.340426\pi$
$60$	0	0
$61$	8195.14	0.281989	0.140994	−	0.990010i	$-0.454970\pi$
$61$	8195.14	0.281989	0.140994	+	0.990010i	$0.454970\pi$
$62$	−2239.68	−0.0739959
$63$	0	0
$64$	−47383.1	−1.44602
$65$	−13541.6	−0.397545
$66$	0	0
$67$	51928.9	1.41326	0.706630	−	0.707584i	$-0.250215\pi$
$67$	51928.9	1.41326	0.706630	+	0.707584i	$0.250215\pi$
$68$	31562.9	0.827760
$69$	0	0
$70$	34171.8	0.833533
$71$	23689.2	0.557705	0.278853	−	0.960334i	$-0.410046\pi$
$71$	23689.2	0.557705	0.278853	+	0.960334i	$0.410046\pi$
$72$	0	0
$73$	20933.5	0.459764	0.229882	−	0.973219i	$-0.426166\pi$
$73$	20933.5	0.459764	0.229882	+	0.973219i	$0.426166\pi$
$74$	−127385.	−2.70421
$75$	0	0
$76$	−108926.	−2.16321
$77$	44849.7	0.862051
$78$	0	0
$79$	−39279.7	−0.708110	−0.354055	−	0.935225i	$-0.615197\pi$
$79$	−39279.7	−0.708110	−0.354055	+	0.935225i	$0.615197\pi$
$80$	−16866.1	−0.294639
$81$	0	0
$82$	−68066.6	−1.11789
$83$	35911.1	0.572181	0.286091	−	0.958203i	$-0.407644\pi$
$83$	35911.1	0.572181	0.286091	+	0.958203i	$0.407644\pi$
$84$	0	0
$85$	19433.4	0.291743
$86$	43740.5	0.637731
$87$	0	0
$88$	20497.1	0.282154
$89$	−94074.7	−1.25892	−0.629460	−	0.777033i	$-0.716724\pi$
$89$	−94074.7	−1.25892	−0.629460	+	0.777033i	$0.716724\pi$
$90$	0	0
$91$	−86891.5	−1.09995
$92$	−150011.	−1.84779
$93$	0	0
$94$	90046.3	1.05111
$95$	−67066.4	−0.762423
$96$	0	0
$97$	−12249.9	−0.132191	−0.0660957	−	0.997813i	$-0.521054\pi$
$97$	−12249.9	−0.132191	−0.0660957	+	0.997813i	$0.521054\pi$
$98$	76059.0	0.799991
$99$	0	0
$100$	25377.5	0.253775
$101$	−159360.	−1.55445	−0.777223	−	0.629226i	$-0.783372\pi$
$101$	−159360.	−1.55445	−0.777223	+	0.629226i	$0.783372\pi$
$102$	0	0
$103$	74393.5	0.690942	0.345471	−	0.938429i	$-0.387719\pi$
$103$	74393.5	0.690942	0.345471	+	0.938429i	$0.387719\pi$
$104$	−39710.9	−0.360020
$105$	0	0
$106$	179528.	1.55191
$107$	170203.	1.43717	0.718585	−	0.695439i	$-0.244790\pi$
$107$	170203.	1.43717	0.718585	+	0.695439i	$0.244790\pi$
$108$	0	0
$109$	54355.6	0.438206	0.219103	−	0.975702i	$-0.429687\pi$
$109$	54355.6	0.438206	0.219103	+	0.975702i	$0.429687\pi$
$110$	59557.0	0.469300
$111$	0	0
$112$	−108224.	−0.815224
$113$	−9340.66	−0.0688148	−0.0344074	−	0.999408i	$-0.510954\pi$
$113$	−9340.66	−0.0688148	−0.0344074	+	0.999408i	$0.510954\pi$
$114$	0	0
$115$	−92362.0	−0.651252
$116$	339312.	2.34129
$117$	0	0
$118$	218981.	1.44778
$119$	124697.	0.807213
$120$	0	0
$121$	−82883.8	−0.514643
$122$	69829.1	0.424753
$123$	0	0
$124$	−10672.7	−0.0623334
$125$	15625.0	0.0894427
$126$	0	0
$127$	−22872.4	−0.125835	−0.0629176	−	0.998019i	$-0.520041\pi$
$127$	−22872.4	−0.125835	−0.0629176	+	0.998019i	$0.520041\pi$
$128$	−144717.	−0.780721
$129$	0	0
$130$	−115385.	−0.598814
$131$	−345444.	−1.75873	−0.879366	−	0.476146i	$-0.842033\pi$
$131$	−345444.	−1.75873	−0.879366	+	0.476146i	$0.842033\pi$
$132$	0	0
$133$	−430341.	−2.10952
$134$	442475.	2.12876
$135$	0	0
$136$	56988.6	0.264205
$137$	−170270.	−0.775064	−0.387532	−	0.921856i	$-0.626672\pi$
$137$	−170270.	−0.775064	−0.387532	+	0.921856i	$0.626672\pi$
$138$	0	0
$139$	−268230.	−1.17753	−0.588763	−	0.808305i	$-0.700385\pi$
$139$	−268230.	−1.17753	−0.588763	+	0.808305i	$0.700385\pi$
$140$	162838.	0.702160
$141$	0	0
$142$	201851.	0.840060
$143$	−151441.	−0.619301
$144$	0	0
$145$	208916.	0.825184
$146$	178370.	0.692532
$147$	0	0
$148$	−607027.	−2.27800
$149$	293424.	1.08276	0.541378	−	0.840779i	$-0.317903\pi$
$149$	293424.	1.08276	0.541378	+	0.840779i	$0.317903\pi$
$150$	0	0
$151$	365971.	1.30618	0.653091	−	0.757279i	$-0.273472\pi$
$151$	365971.	1.30618	0.653091	+	0.757279i	$0.273472\pi$
$152$	−196673.	−0.690456
$153$	0	0
$154$	382156.	1.29849
$155$	−6571.23	−0.0219694
$156$	0	0
$157$	503049.	1.62878	0.814388	−	0.580320i	$-0.197073\pi$
$157$	503049.	1.62878	0.814388	+	0.580320i	$0.197073\pi$
$158$	−334695.	−1.06661
$159$	0	0
$160$	−202363.	−0.624929
$161$	−592654.	−1.80192
$162$	0	0
$163$	181660.	0.535538	0.267769	−	0.963483i	$-0.413714\pi$
$163$	181660.	0.535538	0.267769	+	0.963483i	$0.413714\pi$
$164$	−324356.	−0.941700
$165$	0	0
$166$	305991.	0.861864
$167$	440148.	1.22126	0.610629	−	0.791917i	$-0.290917\pi$
$167$	440148.	1.22126	0.610629	+	0.791917i	$0.290917\pi$
$168$	0	0
$169$	−77893.3	−0.209789
$170$	165588.	0.439446
$171$	0	0
$172$	208436.	0.537218
$173$	−552730.	−1.40410	−0.702049	−	0.712129i	$-0.747731\pi$
$173$	−552730.	−1.40410	−0.702049	+	0.712129i	$0.747731\pi$
$174$	0	0
$175$	100260.	0.247476
$176$	−188620.	−0.458992
$177$	0	0
$178$	−801591.	−1.89628
$179$	190220.	0.443735	0.221867	−	0.975077i	$-0.428785\pi$
$179$	190220.	0.443735	0.221867	+	0.975077i	$0.428785\pi$
$180$	0	0
$181$	113139.	0.256694	0.128347	−	0.991729i	$-0.459033\pi$
$181$	113139.	0.256694	0.128347	+	0.991729i	$0.459033\pi$
$182$	−740385.	−1.65683
$183$	0	0
$184$	−270853.	−0.589778
$185$	−373749.	−0.802880
$186$	0	0
$187$	217330.	0.454482
$188$	429096.	0.885441
$189$	0	0
$190$	−571459.	−1.14842
$191$	694078.	1.37665	0.688327	−	0.725401i	$-0.258346\pi$
$191$	694078.	1.37665	0.688327	+	0.725401i	$0.258346\pi$
$192$	0	0
$193$	−51802.2	−0.100105	−0.0500524	−	0.998747i	$-0.515939\pi$
$193$	−51802.2	−0.100105	−0.0500524	+	0.998747i	$0.515939\pi$
$194$	−104379.	−0.199117
$195$	0	0
$196$	362442.	0.673905
$197$	217963.	0.400145	0.200073	−	0.979781i	$-0.435882\pi$
$197$	217963.	0.400145	0.200073	+	0.979781i	$0.435882\pi$
$198$	0	0
$199$	−385135.	−0.689414	−0.344707	−	0.938710i	$-0.612022\pi$
$199$	−385135.	−0.689414	−0.344707	+	0.938710i	$0.612022\pi$
$200$	45820.5	0.0810000
$201$	0	0
$202$	−1.35787e6	−2.34143
$203$	1.34054e6	2.28317
$204$	0	0
$205$	−199707.	−0.331901
$206$	633892.	1.04075
$207$	0	0
$208$	365430.	0.585661
$209$	−750028.	−1.18771
$210$	0	0
$211$	300290.	0.464339	0.232170	−	0.972675i	$-0.425418\pi$
$211$	300290.	0.464339	0.232170	+	0.972675i	$0.425418\pi$
$212$	855500.	1.30732
$213$	0	0
$214$	1.45027e6	2.16478
$215$	128334.	0.189342
$216$	0	0
$217$	−42165.2	−0.0607862
$218$	463153.	0.660060
$219$	0	0
$220$	283806.	0.395334
$221$	−421054.	−0.579905
$222$	0	0
$223$	−409580.	−0.551539	−0.275769	−	0.961224i	$-0.588933\pi$
$223$	−409580.	−0.551539	−0.275769	+	0.961224i	$0.588933\pi$
$224$	−1.29849e6	−1.72909
$225$	0	0
$226$	−79589.9	−0.103654
$227$	386331.	0.497617	0.248808	−	0.968553i	$-0.419961\pi$
$227$	386331.	0.497617	0.248808	+	0.968553i	$0.419961\pi$
$228$	0	0
$229$	360206.	0.453902	0.226951	−	0.973906i	$-0.427124\pi$
$229$	360206.	0.453902	0.226951	+	0.973906i	$0.427124\pi$
$230$	−786998.	−0.980967
$231$	0	0
$232$	612647.	0.747293
$233$	376766.	0.454655	0.227327	−	0.973818i	$-0.427001\pi$
$233$	376766.	0.454655	0.227327	+	0.973818i	$0.427001\pi$
$234$	0	0
$235$	264196.	0.312073
$236$	1.04351e6	1.21959
$237$	0	0
$238$	1.06252e6	1.21589
$239$	416917.	0.472123	0.236061	−	0.971738i	$-0.424143\pi$
$239$	416917.	0.472123	0.236061	+	0.971738i	$0.424143\pi$
$240$	0	0
$241$	−1.15082e6	−1.27634	−0.638168	−	0.769897i	$-0.720308\pi$
$241$	−1.15082e6	−1.27634	−0.638168	+	0.769897i	$0.720308\pi$
$242$	−706236.	−0.775196
$243$	0	0
$244$	332755.	0.357808
$245$	223157.	0.237517
$246$	0	0
$247$	1.45310e6	1.51549
$248$	−19270.2	−0.0198956
$249$	0	0
$250$	133137.	0.134726
$251$	642245.	0.643452	0.321726	−	0.946833i	$-0.395737\pi$
$251$	642245.	0.643452	0.321726	+	0.946833i	$0.395737\pi$
$252$	0	0
$253$	−1.03292e6	−1.01453
$254$	−194891.	−0.189543
$255$	0	0
$256$	283152.	0.270034
$257$	−126474.	−0.119446	−0.0597228	−	0.998215i	$-0.519022\pi$
$257$	−126474.	−0.119446	−0.0597228	+	0.998215i	$0.519022\pi$
$258$	0	0
$259$	−2.39821e6	−2.22146
$260$	−549843.	−0.504435
$261$	0	0
$262$	−2.94346e6	−2.64914
$263$	366879.	0.327064	0.163532	−	0.986538i	$-0.447711\pi$
$263$	366879.	0.327064	0.163532	+	0.986538i	$0.447711\pi$
$264$	0	0
$265$	526734.	0.460762
$266$	−3.66685e6	−3.17752
$267$	0	0
$268$	2.10852e6	1.79325
$269$	−1.21165e6	−1.02093	−0.510465	−	0.859899i	$-0.670527\pi$
$269$	−1.21165e6	−1.02093	−0.510465	+	0.859899i	$0.670527\pi$
$270$	0	0
$271$	−1.79322e6	−1.48324	−0.741619	−	0.670821i	$-0.765942\pi$
$271$	−1.79322e6	−1.48324	−0.741619	+	0.670821i	$0.765942\pi$
$272$	−524424.	−0.429794
$273$	0	0
$274$	−1.45084e6	−1.16746
$275$	174740.	0.139335
$276$	0	0
$277$	1.46441e6	1.14674	0.573369	−	0.819297i	$-0.305636\pi$
$277$	1.46441e6	1.14674	0.573369	+	0.819297i	$0.305636\pi$
$278$	−2.28554e6	−1.77368
$279$	0	0
$280$	294014.	0.224116
$281$	2.12044e6	1.60199	0.800997	−	0.598669i	$-0.204304\pi$
$281$	2.12044e6	1.60199	0.800997	+	0.598669i	$0.204304\pi$
$282$	0	0
$283$	454281.	0.337177	0.168589	−	0.985687i	$-0.446079\pi$
$283$	454281.	0.337177	0.168589	+	0.985687i	$0.446079\pi$
$284$	961877.	0.707658
$285$	0	0
$286$	−1.29039e6	−0.932840
$287$	−1.28145e6	−0.918325
$288$	0	0
$289$	−815608.	−0.574430
$290$	1.78013e6	1.24296
$291$	0	0
$292$	849984.	0.583383
$293$	−1.93287e6	−1.31533	−0.657664	−	0.753312i	$-0.728455\pi$
$293$	−1.93287e6	−1.31533	−0.657664	+	0.753312i	$0.728455\pi$
$294$	0	0
$295$	642490.	0.429845
$296$	−1.09602e6	−0.727094
$297$	0	0
$298$	2.50021e6	1.63093
$299$	2.00117e6	1.29451
$300$	0	0
$301$	823476.	0.523883
$302$	3.11836e6	1.96747
$303$	0	0
$304$	1.80984e6	1.12320
$305$	204878.	0.126109
$306$	0	0
$307$	224205.	0.135768	0.0678842	−	0.997693i	$-0.478375\pi$
$307$	224205.	0.135768	0.0678842	+	0.997693i	$0.478375\pi$
$308$	1.82108e6	1.09384
$309$	0	0
$310$	−55992.1	−0.0330920
$311$	−592091.	−0.347126	−0.173563	−	0.984823i	$-0.555528\pi$
$311$	−592091.	−0.347126	−0.173563	+	0.984823i	$0.555528\pi$
$312$	0	0
$313$	1.99079e6	1.14859	0.574295	−	0.818648i	$-0.305276\pi$
$313$	1.99079e6	1.14859	0.574295	+	0.818648i	$0.305276\pi$
$314$	4.28638e6	2.45339
$315$	0	0
$316$	−1.59491e6	−0.898503
$317$	−2.89884e6	−1.62023	−0.810113	−	0.586274i	$-0.800594\pi$
$317$	−2.89884e6	−1.62023	−0.810113	+	0.586274i	$0.800594\pi$
$318$	0	0
$319$	2.33638e6	1.28548
$320$	−1.18458e6	−0.646679
$321$	0	0
$322$	−5.04988e6	−2.71420
$323$	−2.08532e6	−1.11216
$324$	0	0
$325$	−338540.	−0.177788
$326$	1.54789e6	0.806669
$327$	0	0
$328$	−585644.	−0.300572
$329$	1.69525e6	0.863462
$330$	0	0
$331$	416722.	0.209063	0.104531	−	0.994522i	$-0.466666\pi$
$331$	416722.	0.209063	0.104531	+	0.994522i	$0.466666\pi$
$332$	1.45813e6	0.726026
$333$	0	0
$334$	3.75041e6	1.83955
$335$	1.29822e6	0.632029
$336$	0	0
$337$	212393.	0.101874	0.0509371	−	0.998702i	$-0.483779\pi$
$337$	212393.	0.101874	0.0509371	+	0.998702i	$0.483779\pi$
$338$	−663713.	−0.316001
$339$	0	0
$340$	789072.	0.370185
$341$	−73488.4	−0.0342242
$342$	0	0
$343$	−1.26419e6	−0.580201
$344$	376343.	0.171470
$345$	0	0
$346$	−4.70970e6	−2.11496
$347$	1.58972e6	0.708755	0.354377	−	0.935103i	$-0.384693\pi$
$347$	1.58972e6	0.708755	0.354377	+	0.935103i	$0.384693\pi$
$348$	0	0
$349$	−3.98771e6	−1.75251	−0.876254	−	0.481850i	$-0.839965\pi$
$349$	−3.98771e6	−1.75251	−0.876254	+	0.481850i	$0.839965\pi$
$350$	854295.	0.372767
$351$	0	0
$352$	−2.26310e6	−0.973524
$353$	−585594.	−0.250127	−0.125063	−	0.992149i	$-0.539913\pi$
$353$	−585594.	−0.250127	−0.125063	+	0.992149i	$0.539913\pi$
$354$	0	0
$355$	592231.	0.249413
$356$	−3.81981e6	−1.59741
$357$	0	0
$358$	1.62082e6	0.668388
$359$	−4.28470e6	−1.75463	−0.877313	−	0.479918i	$-0.840666\pi$
$359$	−4.28470e6	−1.75463	−0.877313	+	0.479918i	$0.840666\pi$
$360$	0	0
$361$	4.72054e6	1.90644
$362$	964033.	0.386652
$363$	0	0
$364$	−3.52814e6	−1.39570
$365$	523338.	0.205613
$366$	0	0
$367$	−524307.	−0.203198	−0.101599	−	0.994825i	$-0.532396\pi$
$367$	−524307.	−0.203198	−0.101599	+	0.994825i	$0.532396\pi$
$368$	2.49246e6	0.959420
$369$	0	0
$370$	−3.18464e6	−1.20936
$371$	3.37986e6	1.27486
$372$	0	0
$373$	−1.16607e6	−0.433964	−0.216982	−	0.976176i	$-0.569621\pi$
$373$	−1.16607e6	−0.433964	−0.216982	+	0.976176i	$0.569621\pi$
$374$	1.85183e6	0.684576
$375$	0	0
$376$	774757.	0.282616
$377$	−4.52648e6	−1.64024
$378$	0	0
$379$	2.48592e6	0.888973	0.444487	−	0.895786i	$-0.353386\pi$
$379$	2.48592e6	0.888973	0.444487	+	0.895786i	$0.353386\pi$
$380$	−2.72316e6	−0.967419
$381$	0	0
$382$	5.91409e6	2.07362
$383$	1.52361e6	0.530732	0.265366	−	0.964148i	$-0.414507\pi$
$383$	1.52361e6	0.530732	0.265366	+	0.964148i	$0.414507\pi$
$384$	0	0
$385$	1.12124e6	0.385521
$386$	−441396.	−0.150786
$387$	0	0
$388$	−497395.	−0.167734
$389$	−4.59972e6	−1.54120	−0.770598	−	0.637322i	$-0.780042\pi$
$389$	−4.59972e6	−1.54120	−0.770598	+	0.637322i	$0.780042\pi$
$390$	0	0
$391$	−2.87185e6	−0.949991
$392$	654410.	0.215097
$393$	0	0
$394$	1.85722e6	0.602730
$395$	−981993.	−0.316677
$396$	0	0
$397$	3.54374e6	1.12846	0.564229	−	0.825618i	$-0.309173\pi$
$397$	3.54374e6	1.12846	0.564229	+	0.825618i	$0.309173\pi$
$398$	−3.28165e6	−1.03845
$399$	0	0
$400$	−421652.	−0.131766
$401$	−1.01833e6	−0.316248	−0.158124	−	0.987419i	$-0.550545\pi$
$401$	−1.01833e6	−0.316248	−0.158124	+	0.987419i	$0.550545\pi$
$402$	0	0
$403$	142376.	0.0436691
$404$	−6.47064e6	−1.97240
$405$	0	0
$406$	1.14224e7	3.43909
$407$	−4.17977e6	−1.25074
$408$	0	0
$409$	−6.32925e6	−1.87087	−0.935436	−	0.353497i	$-0.884992\pi$
$409$	−6.32925e6	−1.87087	−0.935436	+	0.353497i	$0.884992\pi$
$410$	−1.70166e6	−0.499936
$411$	0	0
$412$	3.02067e6	0.876719
$413$	4.12263e6	1.18932
$414$	0	0
$415$	897778.	0.255887
$416$	4.38451e6	1.24219
$417$	0	0
$418$	−6.39083e6	−1.78903
$419$	−2.41445e6	−0.671865	−0.335933	−	0.941886i	$-0.609051\pi$
$419$	−2.41445e6	−0.671865	−0.335933	+	0.941886i	$0.609051\pi$
$420$	0	0
$421$	−3.96045e6	−1.08903	−0.544514	−	0.838752i	$-0.683286\pi$
$421$	−3.96045e6	−1.08903	−0.544514	+	0.838752i	$0.683286\pi$
$422$	2.55871e6	0.699424
$423$	0	0
$424$	1.54465e6	0.417269
$425$	485834.	0.130471
$426$	0	0
$427$	1.31463e6	0.348927
$428$	6.91093e6	1.82359
$429$	0	0
$430$	1.09351e6	0.285202
$431$	1.48955e6	0.386245	0.193123	−	0.981175i	$-0.438139\pi$
$431$	1.48955e6	0.386245	0.193123	+	0.981175i	$0.438139\pi$
$432$	0	0
$433$	−5.66221e6	−1.45133	−0.725665	−	0.688049i	$-0.758468\pi$
$433$	−5.66221e6	−1.45133	−0.725665	+	0.688049i	$0.758468\pi$
$434$	−359281.	−0.0915609
$435$	0	0
$436$	2.20706e6	0.556028
$437$	9.91102e6	2.48265
$438$	0	0
$439$	8764.09	0.00217043	0.00108521	−	0.999999i	$-0.499655\pi$
$439$	8764.09	0.00217043	0.00108521	+	0.999999i	$0.499655\pi$
$440$	512427.	0.126183
$441$	0	0
$442$	−3.58772e6	−0.873499
$443$	−962960.	−0.233130	−0.116565	−	0.993183i	$-0.537188\pi$
$443$	−962960.	−0.233130	−0.116565	+	0.993183i	$0.537188\pi$
$444$	0	0
$445$	−2.35187e6	−0.563006
$446$	−3.48994e6	−0.830771
$447$	0	0
$448$	−7.60101e6	−1.78927
$449$	3.81934e6	0.894071	0.447035	−	0.894516i	$-0.352480\pi$
$449$	3.81934e6	0.894071	0.447035	+	0.894516i	$0.352480\pi$
$450$	0	0
$451$	−2.23340e6	−0.517040
$452$	−379268.	−0.0873173
$453$	0	0
$454$	3.29185e6	0.749549
$455$	−2.17229e6	−0.491914
$456$	0	0
$457$	−2.28095e6	−0.510887	−0.255443	−	0.966824i	$-0.582221\pi$
$457$	−2.28095e6	−0.510887	−0.255443	+	0.966824i	$0.582221\pi$
$458$	3.06924e6	0.683703
$459$	0	0
$460$	−3.75027e6	−0.826357
$461$	2.56688e6	0.562539	0.281270	−	0.959629i	$-0.409245\pi$
$461$	2.56688e6	0.562539	0.281270	+	0.959629i	$0.409245\pi$
$462$	0	0
$463$	−2.16948e6	−0.470330	−0.235165	−	0.971955i	$-0.575563\pi$
$463$	−2.16948e6	−0.470330	−0.235165	+	0.971955i	$0.575563\pi$
$464$	−5.63774e6	−1.21565
$465$	0	0
$466$	3.21035e6	0.684837
$467$	−1.84499e6	−0.391472	−0.195736	−	0.980657i	$-0.562710\pi$
$467$	−1.84499e6	−0.391472	−0.195736	+	0.980657i	$0.562710\pi$
$468$	0	0
$469$	8.33022e6	1.74874
$470$	2.25116e6	0.470069
$471$	0	0
$472$	1.88411e6	0.389271
$473$	1.43521e6	0.294960
$474$	0	0
$475$	−1.67666e6	−0.340966
$476$	5.06319e6	1.02425
$477$	0	0
$478$	3.55247e6	0.711148
$479$	2.88467e6	0.574457	0.287229	−	0.957862i	$-0.407266\pi$
$479$	2.88467e6	0.574457	0.287229	+	0.957862i	$0.407266\pi$
$480$	0	0
$481$	8.09785e6	1.59590
$482$	−9.80591e6	−1.92252
$483$	0	0
$484$	−3.36541e6	−0.653017
$485$	−306247.	−0.0591178
$486$	0	0
$487$	−3.37928e6	−0.645657	−0.322829	−	0.946457i	$-0.604634\pi$
$487$	−3.37928e6	−0.645657	−0.322829	+	0.946457i	$0.604634\pi$
$488$	600809.	0.114205
$489$	0	0
$490$	1.90147e6	0.357767
$491$	128325.	0.0240218	0.0120109	−	0.999928i	$-0.496177\pi$
$491$	128325.	0.0240218	0.0120109	+	0.999928i	$0.496177\pi$
$492$	0	0
$493$	6.49589e6	1.20371
$494$	1.23815e7	2.28275
$495$	0	0
$496$	177330.	0.0323651
$497$	3.80013e6	0.690093
$498$	0	0
$499$	−2.15334e6	−0.387135	−0.193567	−	0.981087i	$-0.562006\pi$
$499$	−2.15334e6	−0.387135	−0.193567	+	0.981087i	$0.562006\pi$
$500$	634437.	0.113492
$501$	0	0
$502$	5.47244e6	0.969218
$503$	−7.42765e6	−1.30898	−0.654488	−	0.756072i	$-0.727116\pi$
$503$	−7.42765e6	−1.30898	−0.654488	+	0.756072i	$0.727116\pi$
$504$	0	0
$505$	−3.98400e6	−0.695169
$506$	−8.80129e6	−1.52816
$507$	0	0
$508$	−928710.	−0.159669
$509$	8.01756e6	1.37166	0.685832	−	0.727760i	$-0.259439\pi$
$509$	8.01756e6	1.37166	0.685832	+	0.727760i	$0.259439\pi$
$510$	0	0
$511$	3.35807e6	0.568902
$512$	7.04364e6	1.18747
$513$	0	0
$514$	−1.07766e6	−0.179918
$515$	1.85984e6	0.308999
$516$	0	0
$517$	2.95460e6	0.486152
$518$	−2.04347e7	−3.34613
$519$	0	0
$520$	−992773.	−0.161006
$521$	−2.24266e6	−0.361967	−0.180984	−	0.983486i	$-0.557928\pi$
$521$	−2.24266e6	−0.361967	−0.180984	+	0.983486i	$0.557928\pi$
$522$	0	0
$523$	7.70554e6	1.23182	0.615912	−	0.787815i	$-0.288787\pi$
$523$	7.70554e6	1.23182	0.615912	+	0.787815i	$0.288787\pi$
$524$	−1.40264e7	−2.23161
$525$	0	0
$526$	3.12610e6	0.492650
$527$	−204322.	−0.0320470
$528$	0	0
$529$	7.21285e6	1.12064
$530$	4.48819e6	0.694036
$531$	0	0
$532$	−1.74735e7	−2.67671
$533$	4.32696e6	0.659729
$534$	0	0
$535$	4.25508e6	0.642722
$536$	3.80705e6	0.572370
$537$	0	0
$538$	−1.03242e7	−1.53780
$539$	2.49564e6	0.370008
$540$	0	0
$541$	−7.64969e6	−1.12370	−0.561850	−	0.827239i	$-0.689910\pi$
$541$	−7.64969e6	−1.12370	−0.561850	+	0.827239i	$0.689910\pi$
$542$	−1.52797e7	−2.23417
$543$	0	0
$544$	−6.29214e6	−0.911594
$545$	1.35889e6	0.195972
$546$	0	0
$547$	2.76870e6	0.395647	0.197823	−	0.980238i	$-0.436613\pi$
$547$	2.76870e6	0.395647	0.197823	+	0.980238i	$0.436613\pi$
$548$	−6.91366e6	−0.983459
$549$	0	0
$550$	1.48892e6	0.209878
$551$	−2.24179e7	−3.14569
$552$	0	0
$553$	−6.30110e6	−0.876200
$554$	1.24780e7	1.72731
$555$	0	0
$556$	−1.08912e7	−1.49413
$557$	1.21255e6	0.165600	0.0828002	−	0.996566i	$-0.473614\pi$
$557$	1.21255e6	0.165600	0.0828002	+	0.996566i	$0.473614\pi$
$558$	0	0
$559$	−2.78057e6	−0.376360
$560$	−2.70559e6	−0.364579
$561$	0	0
$562$	1.80679e7	2.41305
$563$	6.55586e6	0.871683	0.435841	−	0.900024i	$-0.356451\pi$
$563$	6.55586e6	0.871683	0.435841	+	0.900024i	$0.356451\pi$
$564$	0	0
$565$	−233517.	−0.0307749
$566$	3.87083e6	0.507883
$567$	0	0
$568$	1.73672e6	0.225871
$569$	8.97209e6	1.16175	0.580875	−	0.813993i	$-0.302710\pi$
$569$	8.97209e6	1.16175	0.580875	+	0.813993i	$0.302710\pi$
$570$	0	0
$571$	−5.46422e6	−0.701354	−0.350677	−	0.936496i	$-0.614049\pi$
$571$	−5.46422e6	−0.701354	−0.350677	+	0.936496i	$0.614049\pi$
$572$	−6.14909e6	−0.785816
$573$	0	0
$574$	−1.09190e7	−1.38325
$575$	−2.30905e6	−0.291249
$576$	0	0
$577$	−6.35363e6	−0.794478	−0.397239	−	0.917715i	$-0.630032\pi$
$577$	−6.35363e6	−0.794478	−0.397239	+	0.917715i	$0.630032\pi$
$578$	−6.94963e6	−0.865251
$579$	0	0
$580$	8.48280e6	1.04705
$581$	5.76072e6	0.708005
$582$	0	0
$583$	5.89066e6	0.717781
$584$	1.53469e6	0.186204
$585$	0	0
$586$	−1.64696e7	−1.98125
$587$	1.33705e7	1.60159	0.800795	−	0.598938i	$-0.204410\pi$
$587$	1.33705e7	1.60159	0.800795	+	0.598938i	$0.204410\pi$
$588$	0	0
$589$	705134.	0.0837497
$590$	5.47453e6	0.647466
$591$	0	0
$592$	1.00859e7	1.18280
$593$	−1.42274e6	−0.166146	−0.0830728	−	0.996543i	$-0.526473\pi$
$593$	−1.42274e6	−0.166146	−0.0830728	+	0.996543i	$0.526473\pi$
$594$	0	0
$595$	3.11742e6	0.360997
$596$	1.19142e7	1.37388
$597$	0	0
$598$	1.70515e7	1.94989
$599$	−5.91805e6	−0.673925	−0.336962	−	0.941518i	$-0.609399\pi$
$599$	−5.91805e6	−0.673925	−0.336962	+	0.941518i	$0.609399\pi$
$600$	0	0
$601$	1.48610e7	1.67827	0.839135	−	0.543924i	$-0.183062\pi$
$601$	1.48610e7	1.67827	0.839135	+	0.543924i	$0.183062\pi$
$602$	7.01667e6	0.789114
$603$	0	0
$604$	1.48599e7	1.65738
$605$	−2.07209e6	−0.230155
$606$	0	0
$607$	−1.16289e7	−1.28105	−0.640526	−	0.767937i	$-0.721283\pi$
$607$	−1.16289e7	−1.28105	−0.640526	+	0.767937i	$0.721283\pi$
$608$	2.17148e7	2.38230
$609$	0	0
$610$	1.74573e6	0.189956
$611$	−5.72421e6	−0.620315
$612$	0	0
$613$	−1.06616e6	−0.114596	−0.0572980	−	0.998357i	$-0.518249\pi$
$613$	−1.06616e6	−0.114596	−0.0572980	+	0.998357i	$0.518249\pi$
$614$	1.91040e6	0.204505
$615$	0	0
$616$	3.28806e6	0.349131
$617$	663411.	0.0701568	0.0350784	−	0.999385i	$-0.488832\pi$
$617$	663411.	0.0701568	0.0350784	+	0.999385i	$0.488832\pi$
$618$	0	0
$619$	2.53444e6	0.265861	0.132930	−	0.991125i	$-0.457561\pi$
$619$	2.53444e6	0.265861	0.132930	+	0.991125i	$0.457561\pi$
$620$	−266818.	−0.0278764
$621$	0	0
$622$	−5.04508e6	−0.522868
$623$	−1.50911e7	−1.55776
$624$	0	0
$625$	390625.	0.0400000
$626$	1.69631e7	1.73010
$627$	0	0
$628$	2.04258e7	2.06671
$629$	−1.16211e7	−1.17117
$630$	0	0
$631$	102057.	0.0102040	0.00510199	−	0.999987i	$-0.498376\pi$
$631$	102057.	0.0102040	0.00510199	+	0.999987i	$0.498376\pi$
$632$	−2.87971e6	−0.286785
$633$	0	0
$634$	−2.47004e7	−2.44051
$635$	−571809.	−0.0562752
$636$	0	0
$637$	−4.83504e6	−0.472119
$638$	1.99078e7	1.93630
$639$	0	0
$640$	−3.61794e6	−0.349149
$641$	−3.67995e6	−0.353750	−0.176875	−	0.984233i	$-0.556599\pi$
$641$	−3.67995e6	−0.353750	−0.176875	+	0.984233i	$0.556599\pi$
$642$	0	0
$643$	8.95581e6	0.854235	0.427118	−	0.904196i	$-0.359529\pi$
$643$	8.95581e6	0.854235	0.427118	+	0.904196i	$0.359529\pi$
$644$	−2.40641e7	−2.28642
$645$	0	0
$646$	−1.77686e7	−1.67522
$647$	1.36816e6	0.128492	0.0642458	−	0.997934i	$-0.479536\pi$
$647$	1.36816e6	0.128492	0.0642458	+	0.997934i	$0.479536\pi$
$648$	0	0
$649$	7.18520e6	0.669618
$650$	−2.88463e6	−0.267798
$651$	0	0
$652$	7.37612e6	0.679530
$653$	3.23673e6	0.297046	0.148523	−	0.988909i	$-0.452548\pi$
$653$	3.23673e6	0.297046	0.148523	+	0.988909i	$0.452548\pi$
$654$	0	0
$655$	−8.63611e6	−0.786529
$656$	5.38925e6	0.488955
$657$	0	0
$658$	1.44449e7	1.30061
$659$	5.17492e6	0.464184	0.232092	−	0.972694i	$-0.425443\pi$
$659$	5.17492e6	0.464184	0.232092	+	0.972694i	$0.425443\pi$
$660$	0	0
$661$	−1.07060e7	−0.953064	−0.476532	−	0.879157i	$-0.658106\pi$
$661$	−1.07060e7	−0.953064	−0.476532	+	0.879157i	$0.658106\pi$
$662$	3.55080e6	0.314906
$663$	0	0
$664$	2.63275e6	0.231733
$665$	−1.07585e7	−0.943405
$666$	0	0
$667$	−3.08734e7	−2.68701
$668$	1.78717e7	1.54962
$669$	0	0
$670$	1.10619e7	0.952011
$671$	2.29123e6	0.196455
$672$	0	0
$673$	−2.15653e7	−1.83535	−0.917673	−	0.397338i	$-0.869934\pi$
$673$	−2.15653e7	−1.83535	−0.917673	+	0.397338i	$0.869934\pi$
$674$	1.80975e6	0.153451
$675$	0	0
$676$	−3.16278e6	−0.266196
$677$	6.09762e6	0.511315	0.255658	−	0.966767i	$-0.417708\pi$
$677$	6.09762e6	0.511315	0.255658	+	0.966767i	$0.417708\pi$
$678$	0	0
$679$	−1.96508e6	−0.163571
$680$	1.42471e6	0.118156
$681$	0	0
$682$	−626180.	−0.0515511
$683$	2.04240e7	1.67528	0.837642	−	0.546220i	$-0.183934\pi$
$683$	2.04240e7	1.67528	0.837642	+	0.546220i	$0.183934\pi$
$684$	0	0
$685$	−4.25676e6	−0.346619
$686$	−1.07719e7	−0.873944
$687$	0	0
$688$	−3.46320e6	−0.278937
$689$	−1.14125e7	−0.915868
$690$	0	0
$691$	−2.91819e6	−0.232497	−0.116249	−	0.993220i	$-0.537087\pi$
$691$	−2.91819e6	−0.232497	−0.116249	+	0.993220i	$0.537087\pi$
$692$	−2.24430e7	−1.78162
$693$	0	0
$694$	1.35456e7	1.06758
$695$	−6.70576e6	−0.526606
$696$	0	0
$697$	−6.20957e6	−0.484150
$698$	−3.39785e7	−2.63976
$699$	0	0
$700$	4.07095e6	0.314016
$701$	−8.78394e6	−0.675141	−0.337570	−	0.941300i	$-0.609605\pi$
$701$	−8.78394e6	−0.675141	−0.337570	+	0.941300i	$0.609605\pi$
$702$	0	0
$703$	4.01056e7	3.06067
$704$	−1.32476e7	−1.00741
$705$	0	0
$706$	−4.98973e6	−0.376760
$707$	−2.55639e7	−1.92344
$708$	0	0
$709$	1.66475e7	1.24375	0.621875	−	0.783116i	$-0.286371\pi$
$709$	1.66475e7	1.24375	0.621875	+	0.783116i	$0.286371\pi$
$710$	5.04628e6	0.375686
$711$	0	0
$712$	−6.89688e6	−0.509862
$713$	971092.	0.0715379
$714$	0	0
$715$	−3.78601e6	−0.276960
$716$	7.72368e6	0.563043
$717$	0	0
$718$	−3.65091e7	−2.64296
$719$	−2.55062e7	−1.84003	−0.920013	−	0.391888i	$-0.871822\pi$
$719$	−2.55062e7	−1.84003	−0.920013	+	0.391888i	$0.871822\pi$
$720$	0	0
$721$	1.19339e7	0.854957
$722$	4.02228e7	2.87163
$723$	0	0
$724$	4.59389e6	0.325712
$725$	5.22289e6	0.369033
$726$	0	0
$727$	1.36506e7	0.957890	0.478945	−	0.877845i	$-0.341019\pi$
$727$	1.36506e7	0.957890	0.478945	+	0.877845i	$0.341019\pi$
$728$	−6.37026e6	−0.445481
$729$	0	0
$730$	4.45925e6	0.309710
$731$	3.99035e6	0.276196
$732$	0	0
$733$	−1.38922e7	−0.955018	−0.477509	−	0.878627i	$-0.658460\pi$
$733$	−1.38922e7	−0.955018	−0.477509	+	0.878627i	$0.658460\pi$
$734$	−4.46751e6	−0.306073
$735$	0	0
$736$	2.99050e7	2.03493
$737$	1.45185e7	0.984584
$738$	0	0
$739$	837888.	0.0564384	0.0282192	−	0.999602i	$-0.491016\pi$
$739$	837888.	0.0564384	0.0282192	+	0.999602i	$0.491016\pi$
$740$	−1.51757e7	−1.01875
$741$	0	0
$742$	2.87991e7	1.92030
$743$	−8.07258e6	−0.536464	−0.268232	−	0.963354i	$-0.586439\pi$
$743$	−8.07258e6	−0.536464	−0.268232	+	0.963354i	$0.586439\pi$
$744$	0	0
$745$	7.33561e6	0.484223
$746$	−9.93588e6	−0.653671
$747$	0	0
$748$	8.82448e6	0.576680
$749$	2.73033e7	1.77832
$750$	0	0
$751$	−1.74892e7	−1.13154	−0.565770	−	0.824563i	$-0.691421\pi$
$751$	−1.74892e7	−1.13154	−0.565770	+	0.824563i	$0.691421\pi$
$752$	−7.12952e6	−0.459744
$753$	0	0
$754$	−3.85692e7	−2.47066
$755$	9.14927e6	0.584143
$756$	0	0
$757$	2.19231e7	1.39047	0.695236	−	0.718781i	$-0.255300\pi$
$757$	2.19231e7	1.39047	0.695236	+	0.718781i	$0.255300\pi$
$758$	2.11820e7	1.33904
$759$	0	0
$760$	−4.91683e6	−0.308781
$761$	1.18585e7	0.742278	0.371139	−	0.928577i	$-0.378967\pi$
$761$	1.18585e7	0.742278	0.371139	+	0.928577i	$0.378967\pi$
$762$	0	0
$763$	8.71951e6	0.542227
$764$	2.81823e7	1.74680
$765$	0	0
$766$	1.29823e7	0.799431
$767$	−1.39206e7	−0.854413
$768$	0	0
$769$	−2.94181e6	−0.179390	−0.0896950	−	0.995969i	$-0.528589\pi$
$769$	−2.94181e6	−0.179390	−0.0896950	+	0.995969i	$0.528589\pi$
$770$	9.55389e6	0.580702
$771$	0	0
$772$	−2.10337e6	−0.127020
$773$	−2.01745e7	−1.21438	−0.607188	−	0.794558i	$-0.707703\pi$
$773$	−2.01745e7	−1.21438	−0.607188	+	0.794558i	$0.707703\pi$
$774$	0	0
$775$	−164281.	−0.00982500
$776$	−898074.	−0.0535375
$777$	0	0
$778$	−3.91933e7	−2.32147
$779$	2.14298e7	1.26525
$780$	0	0
$781$	6.62313e6	0.388540
$782$	−2.44704e7	−1.43095
$783$	0	0
$784$	−6.02206e6	−0.349909
$785$	1.25762e7	0.728411
$786$	0	0
$787$	8.17269e6	0.470358	0.235179	−	0.971952i	$-0.424432\pi$
$787$	8.17269e6	0.470358	0.235179	+	0.971952i	$0.424432\pi$
$788$	8.85018e6	0.507734
$789$	0	0
$790$	−8.36737e6	−0.477003
$791$	−1.49839e6	−0.0851499
$792$	0	0
$793$	−4.43901e6	−0.250670
$794$	3.01955e7	1.69977
$795$	0	0
$796$	−1.56380e7	−0.874779
$797$	−1.66457e7	−0.928234	−0.464117	−	0.885774i	$-0.653628\pi$
$797$	−1.66457e7	−0.928234	−0.464117	+	0.885774i	$0.653628\pi$
$798$	0	0
$799$	8.21474e6	0.455226
$800$	−5.05907e6	−0.279477
$801$	0	0
$802$	−8.67699e6	−0.476358
$803$	5.85267e6	0.320306
$804$	0	0
$805$	−1.48163e7	−0.805845
$806$	1.21316e6	0.0657778
$807$	0	0
$808$	−1.16831e7	−0.629550
$809$	−7.91242e6	−0.425048	−0.212524	−	0.977156i	$-0.568168\pi$
$809$	−7.91242e6	−0.425048	−0.212524	+	0.977156i	$0.568168\pi$
$810$	0	0
$811$	−4.07133e6	−0.217362	−0.108681	−	0.994077i	$-0.534663\pi$
$811$	−4.07133e6	−0.217362	−0.108681	+	0.994077i	$0.534663\pi$
$812$	5.44311e7	2.89706
$813$	0	0
$814$	−3.56149e7	−1.88396
$815$	4.54150e6	0.239500
$816$	0	0
$817$	−1.37711e7	−0.721794
$818$	−5.39302e7	−2.81805
$819$	0	0
$820$	−8.10890e6	−0.421141
$821$	−1.86139e7	−0.963782	−0.481891	−	0.876231i	$-0.660050\pi$
$821$	−1.86139e7	−0.963782	−0.481891	+	0.876231i	$0.660050\pi$
$822$	0	0
$823$	−3.54074e6	−0.182220	−0.0911098	−	0.995841i	$-0.529041\pi$
$823$	−3.54074e6	−0.182220	−0.0911098	+	0.995841i	$0.529041\pi$
$824$	5.45400e6	0.279832
$825$	0	0
$826$	3.51281e7	1.79145
$827$	3.76580e7	1.91467	0.957335	−	0.288981i	$-0.0933164\pi$
$827$	3.76580e7	1.91467	0.957335	+	0.288981i	$0.0933164\pi$
$828$	0	0
$829$	1.16650e7	0.589518	0.294759	−	0.955572i	$-0.404761\pi$
$829$	1.16650e7	0.589518	0.294759	+	0.955572i	$0.404761\pi$
$830$	7.64978e6	0.385437
$831$	0	0
$832$	2.56657e7	1.28542
$833$	6.93870e6	0.346470
$834$	0	0
$835$	1.10037e7	0.546163
$836$	−3.04541e7	−1.50706
$837$	0	0
$838$	−2.05730e7	−1.01202
$839$	2.07091e7	1.01568	0.507839	−	0.861452i	$-0.330444\pi$
$839$	2.07091e7	1.01568	0.507839	+	0.861452i	$0.330444\pi$
$840$	0	0
$841$	4.93220e7	2.40464
$842$	−3.37462e7	−1.64038
$843$	0	0
$844$	1.21930e7	0.589188
$845$	−1.94733e6	−0.0938207
$846$	0	0
$847$	−1.32959e7	−0.636808
$848$	−1.42143e7	−0.678791
$849$	0	0
$850$	4.13969e6	0.196526
$851$	5.52323e7	2.61438
$852$	0	0
$853$	−1.94473e7	−0.915137	−0.457569	−	0.889174i	$-0.651280\pi$
$853$	−1.94473e7	−0.915137	−0.457569	+	0.889174i	$0.651280\pi$
$854$	1.12017e7	0.525581
$855$	0	0
$856$	1.24781e7	0.582054
$857$	−2.10458e7	−0.978843	−0.489421	−	0.872047i	$-0.662792\pi$
$857$	−2.10458e7	−0.978843	−0.489421	+	0.872047i	$0.662792\pi$
$858$	0	0
$859$	2.40179e7	1.11058	0.555292	−	0.831656i	$-0.312607\pi$
$859$	2.40179e7	1.11058	0.555292	+	0.831656i	$0.312607\pi$
$860$	5.21089e6	0.240251
$861$	0	0
$862$	1.26922e7	0.581793
$863$	5.25620e6	0.240240	0.120120	−	0.992759i	$-0.461672\pi$
$863$	5.25620e6	0.240240	0.120120	+	0.992759i	$0.461672\pi$
$864$	0	0
$865$	−1.38182e7	−0.627932
$866$	−4.82465e7	−2.18611
$867$	0	0
$868$	−1.71208e6	−0.0771300
$869$	−1.09820e7	−0.493323
$870$	0	0
$871$	−2.81280e7	−1.25630
$872$	3.98497e6	0.177473
$873$	0	0
$874$	8.44498e7	3.73956
$875$	2.50650e6	0.110674
$876$	0	0
$877$	−2.99490e6	−0.131487	−0.0657436	−	0.997837i	$-0.520942\pi$
$877$	−2.99490e6	−0.131487	−0.0657436	+	0.997837i	$0.520942\pi$
$878$	74677.0	0.00326927
$879$	0	0
$880$	−4.71549e6	−0.205268
$881$	2.19714e7	0.953714	0.476857	−	0.878981i	$-0.341776\pi$
$881$	2.19714e7	0.953714	0.476857	+	0.878981i	$0.341776\pi$
$882$	0	0
$883$	−1.05962e7	−0.457348	−0.228674	−	0.973503i	$-0.573439\pi$
$883$	−1.05962e7	−0.457348	−0.228674	+	0.973503i	$0.573439\pi$
$884$	−1.70965e7	−0.735827
$885$	0	0
$886$	−8.20519e6	−0.351159
$887$	−1.57778e6	−0.0673343	−0.0336672	−	0.999433i	$-0.510719\pi$
$887$	−1.57778e6	−0.0673343	−0.0336672	+	0.999433i	$0.510719\pi$
$888$	0	0
$889$	−3.66909e6	−0.155706
$890$	−2.00398e7	−0.848043
$891$	0	0
$892$	−1.66306e7	−0.699834
$893$	−2.83498e7	−1.18966
$894$	0	0
$895$	4.75550e6	0.198444
$896$	−2.32150e7	−0.966047
$897$	0	0
$898$	3.25438e7	1.34672
$899$	−2.19653e6	−0.0906438
$900$	0	0
$901$	1.63779e7	0.672121
$902$	−1.90303e7	−0.778807
$903$	0	0
$904$	−684790.	−0.0278700
$905$	2.82847e6	0.114797
$906$	0	0
$907$	−3.03048e7	−1.22319	−0.611594	−	0.791171i	$-0.709472\pi$
$907$	−3.03048e7	−1.22319	−0.611594	+	0.791171i	$0.709472\pi$
$908$	1.56866e7	0.631413
$909$	0	0
$910$	−1.85096e7	−0.740959
$911$	1.14226e7	0.456005	0.228003	−	0.973661i	$-0.426781\pi$
$911$	1.14226e7	0.456005	0.228003	+	0.973661i	$0.426781\pi$
$912$	0	0
$913$	1.00402e7	0.398625
$914$	−1.94355e7	−0.769538
$915$	0	0
$916$	1.46258e7	0.575945
$917$	−5.54148e7	−2.17622
$918$	0	0
$919$	−4.43959e7	−1.73402	−0.867010	−	0.498291i	$-0.833961\pi$
$919$	−4.43959e7	−1.73402	−0.867010	+	0.498291i	$0.833961\pi$
$920$	−6.77132e6	−0.263757
$921$	0	0
$922$	2.18718e7	0.847340
$923$	−1.28316e7	−0.495766
$924$	0	0
$925$	−9.34372e6	−0.359059
$926$	−1.84857e7	−0.708448
$927$	0	0
$928$	−6.76428e7	−2.57841
$929$	−2.73604e7	−1.04012	−0.520059	−	0.854130i	$-0.674090\pi$
$929$	−2.73604e7	−1.04012	−0.520059	+	0.854130i	$0.674090\pi$
$930$	0	0
$931$	−2.39461e7	−0.905443
$932$	1.52982e7	0.576900
$933$	0	0
$934$	−1.57208e7	−0.589666
$935$	5.43326e6	0.203250
$936$	0	0
$937$	−1.23707e7	−0.460306	−0.230153	−	0.973154i	$-0.573923\pi$
$937$	−1.23707e7	−0.460306	−0.230153	+	0.973154i	$0.573923\pi$
$938$	7.09801e7	2.63408
$939$	0	0
$940$	1.07274e7	0.395981
$941$	1.77596e7	0.653822	0.326911	−	0.945055i	$-0.393992\pi$
$941$	1.77596e7	0.653822	0.326911	+	0.945055i	$0.393992\pi$
$942$	0	0
$943$	2.95126e7	1.08076
$944$	−1.73381e7	−0.633244
$945$	0	0
$946$	1.22291e7	0.444292
$947$	4.50046e7	1.63073	0.815364	−	0.578949i	$-0.196537\pi$
$947$	4.50046e7	1.63073	0.815364	+	0.578949i	$0.196537\pi$
$948$	0	0
$949$	−1.13389e7	−0.408701
$950$	−1.42865e7	−0.513589
$951$	0	0
$952$	9.14188e6	0.326921
$953$	4.10110e7	1.46274	0.731372	−	0.681979i	$-0.238881\pi$
$953$	4.10110e7	1.46274	0.731372	+	0.681979i	$0.238881\pi$
$954$	0	0
$955$	1.73519e7	0.615658
$956$	1.69285e7	0.599065
$957$	0	0
$958$	2.45797e7	0.865293
$959$	−2.73141e7	−0.959048
$960$	0	0
$961$	−2.85601e7	−0.997587
$962$	6.90001e7	2.40388
$963$	0	0
$964$	−4.67279e7	−1.61951
$965$	−1.29505e6	−0.0447682
$966$	0	0
$967$	2.23360e7	0.768137	0.384069	−	0.923305i	$-0.374523\pi$
$967$	2.23360e7	0.768137	0.384069	+	0.923305i	$0.374523\pi$
$968$	−6.07644e6	−0.208430
$969$	0	0
$970$	−2.60947e6	−0.0890478
$971$	3.34728e7	1.13932	0.569658	−	0.821882i	$-0.307076\pi$
$971$	3.34728e7	1.13932	0.569658	+	0.821882i	$0.307076\pi$
$972$	0	0
$973$	−4.30284e7	−1.45705
$974$	−2.87942e7	−0.972539
$975$	0	0
$976$	−5.52880e6	−0.185783
$977$	−8.52076e6	−0.285589	−0.142795	−	0.989752i	$-0.545609\pi$
$977$	−8.52076e6	−0.285589	−0.142795	+	0.989752i	$0.545609\pi$
$978$	0	0
$979$	−2.63018e7	−0.877058
$980$	9.06106e6	0.301380
$981$	0	0
$982$	1.09343e6	0.0361836
$983$	2.01155e7	0.663969	0.331985	−	0.943285i	$-0.392282\pi$
$983$	2.01155e7	0.663969	0.331985	+	0.943285i	$0.392282\pi$
$984$	0	0
$985$	5.44908e6	0.178950
$986$	5.53502e7	1.81312
$987$	0	0
$988$	5.90015e7	1.92296
$989$	−1.89652e7	−0.616547
$990$	0	0
$991$	3.24195e7	1.04863	0.524315	−	0.851524i	$-0.324321\pi$
$991$	3.24195e7	1.04863	0.524315	+	0.851524i	$0.324321\pi$
$992$	2.12764e6	0.0686465
$993$	0	0
$994$	3.23801e7	1.03947
$995$	−9.62836e6	−0.308315
$996$	0	0
$997$	4.47995e7	1.42737	0.713683	−	0.700469i	$-0.247026\pi$
$997$	4.47995e7	1.42737	0.713683	+	0.700469i	$0.247026\pi$
$998$	−1.83482e7	−0.583133
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.6.a.f.1.2	yes	2
3.2	odd	2		45.6.a.d.1.1	✓	2
4.3	odd	2		720.6.a.be.1.1		2
5.2	odd	4		225.6.b.k.199.4		4
5.3	odd	4		225.6.b.k.199.1		4
5.4	even	2		225.6.a.k.1.1		2
12.11	even	2		720.6.a.y.1.1		2
15.2	even	4		225.6.b.j.199.1		4
15.8	even	4		225.6.b.j.199.4		4
15.14	odd	2		225.6.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.a.d.1.1	✓	2	3.2	odd	2
45.6.a.f.1.2	yes	2	1.1	even	1	trivial
225.6.a.k.1.1		2	5.4	even	2
225.6.a.r.1.2		2	15.14	odd	2
225.6.b.j.199.1		4	15.2	even	4
225.6.b.j.199.4		4	15.8	even	4
225.6.b.k.199.1		4	5.3	odd	4
225.6.b.k.199.4		4	5.2	odd	4
720.6.a.y.1.1		2	12.11	even	2
720.6.a.be.1.1		2	4.3	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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q+8.52080 q^{2} +40.6040 q^{4} +25.0000 q^{5} +160.416 q^{7} +73.3128 q^{8} +213.020 q^{10} +279.584 q^{11} -541.664 q^{13} +1366.87 q^{14} -674.644 q^{16} +777.334 q^{17} -2682.66 q^{19} +1015.10 q^{20} +2382.28 q^{22} -3694.48 q^{23} +625.000 q^{25} -4615.41 q^{26} +6513.53 q^{28} +8356.62 q^{29} -262.849 q^{31} -8094.51 q^{32} +6623.51 q^{34} +4010.40 q^{35} -14949.9 q^{37} -22858.4 q^{38} +1832.82 q^{40} -7988.28 q^{41} +5133.38 q^{43} +11352.2 q^{44} -31479.9 q^{46} +10567.8 q^{47} +8926.28 q^{49} +5325.50 q^{50} -21993.7 q^{52} +21069.4 q^{53} +6989.60 q^{55} +11760.5 q^{56} +71205.1 q^{58} +25699.6 q^{59} +8195.14 q^{61} -2239.68 q^{62} -47383.1 q^{64} -13541.6 q^{65} +51928.9 q^{67} +31562.9 q^{68} +34171.8 q^{70} +23689.2 q^{71} +20933.5 q^{73} -127385. q^{74} -108926. q^{76} +44849.7 q^{77} -39279.7 q^{79} -16866.1 q^{80} -68066.6 q^{82} +35911.1 q^{83} +19433.4 q^{85} +43740.5 q^{86} +20497.1 q^{88} -94074.7 q^{89} -86891.5 q^{91} -150011. q^{92} +90046.3 q^{94} -67066.4 q^{95} -12249.9 q^{97} +76059.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 21 q^{4} + 50 q^{5} + 80 q^{7} + 255 q^{8} + 125 q^{10} + 800 q^{11} - 120 q^{13} + 1650 q^{14} - 687 q^{16} + 1940 q^{17} - 1512 q^{19} + 525 q^{20} + 550 q^{22} - 1320 q^{23} + 1250 q^{25}+ \cdots + 112465 q^{98}+O(q^{100}) \) 2 * q + 5 * q^2 + 21 * q^4 + 50 * q^5 + 80 * q^7 + 255 * q^8 + 125 * q^10 + 800 * q^11 - 120 * q^13 + 1650 * q^14 - 687 * q^16 + 1940 * q^17 - 1512 * q^19 + 525 * q^20 + 550 * q^22 - 1320 * q^23 + 1250 * q^25 - 6100 * q^26 + 8090 * q^28 + 1300 * q^29 - 5824 * q^31 - 13865 * q^32 + 2530 * q^34 + 2000 * q^35 - 12560 * q^37 - 26980 * q^38 + 6375 * q^40 + 400 * q^41 + 25680 * q^43 + 1150 * q^44 - 39840 * q^46 + 18920 * q^47 - 1414 * q^49 + 3125 * q^50 - 30260 * q^52 + 49460 * q^53 + 20000 * q^55 - 2850 * q^56 + 96050 * q^58 + 63200 * q^59 - 49116 * q^61 + 17340 * q^62 - 26671 * q^64 - 3000 * q^65 + 6080 * q^67 + 8770 * q^68 + 41250 * q^70 + 65200 * q^71 + 97740 * q^73 - 135800 * q^74 - 131876 * q^76 + 3000 * q^77 - 46288 * q^79 - 17175 * q^80 - 97600 * q^82 - 57360 * q^83 + 48500 * q^85 - 28600 * q^86 + 115050 * q^88 - 87000 * q^89 - 120800 * q^91 - 196560 * q^92 + 60640 * q^94 - 37800 * q^95 + 10180 * q^97 + 112465 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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