LMFDB - Embedded newform 630.6.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [630,6,Mod(1,630)] 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("630.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(630, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	630.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,32,-50,0,98,-128,0,200,-415] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$101.041806482$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1129}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 282 $ x^2 - x - 282
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$17.3003$ of defining polynomial
Character	$\chi$	$=$	630.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +16.0000 q^{4} -25.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +100.000 q^{10} -425.904 q^{11} +399.303 q^{13} -196.000 q^{14} +256.000 q^{16} -1751.52 q^{17} +2874.23 q^{19} -400.000 q^{20} +1703.62 q^{22} +2313.43 q^{23} +625.000 q^{25} -1597.21 q^{26} +784.000 q^{28} +2127.93 q^{29} -10262.5 q^{31} -1024.00 q^{32} +7006.08 q^{34} -1225.00 q^{35} -7266.05 q^{37} -11496.9 q^{38} +1600.00 q^{40} +5893.44 q^{41} +20157.7 q^{43} -6814.46 q^{44} -9253.72 q^{46} -20056.5 q^{47} +2401.00 q^{49} -2500.00 q^{50} +6388.85 q^{52} +33954.9 q^{53} +10647.6 q^{55} -3136.00 q^{56} -8511.71 q^{58} +4319.34 q^{59} -12253.6 q^{61} +41050.1 q^{62} +4096.00 q^{64} -9982.58 q^{65} -17533.8 q^{67} -28024.3 q^{68} +4900.00 q^{70} -1658.18 q^{71} -8246.91 q^{73} +29064.2 q^{74} +45987.7 q^{76} -20869.3 q^{77} -9168.61 q^{79} -6400.00 q^{80} -23573.7 q^{82} -95203.2 q^{83} +43788.0 q^{85} -80631.0 q^{86} +27257.8 q^{88} +14441.8 q^{89} +19565.9 q^{91} +37014.9 q^{92} +80226.1 q^{94} -71855.8 q^{95} +62132.4 q^{97} -9604.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 50 q^{5} + 98 q^{7} - 128 q^{8} + 200 q^{10} - 415 q^{11} + 429 q^{13} - 392 q^{14} + 512 q^{16} - 1319 q^{17} + 1918 q^{19} - 800 q^{20} + 1660 q^{22} + 1334 q^{23} + 1250 q^{25}+ \cdots - 19208 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 50 * q^5 + 98 * q^7 - 128 * q^8 + 200 * q^10 - 415 * q^11 + 429 * q^13 - 392 * q^14 + 512 * q^16 - 1319 * q^17 + 1918 * q^19 - 800 * q^20 + 1660 * q^22 + 1334 * q^23 + 1250 * q^25 - 1716 * q^26 + 1568 * q^28 + 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 5276 * q^34 - 2450 * q^35 - 9156 * q^37 - 7672 * q^38 + 3200 * q^40 + 7822 * q^41 + 2078 * q^43 - 6640 * q^44 - 5336 * q^46 + 8507 * q^47 + 4802 * q^49 - 5000 * q^50 + 6864 * q^52 + 34242 * q^53 + 10375 * q^55 - 6272 * q^56 - 4524 * q^58 - 6952 * q^59 - 45138 * q^61 + 21888 * q^62 + 8192 * q^64 - 10725 * q^65 - 54556 * q^67 - 21104 * q^68 + 9800 * q^70 + 62272 * q^71 + 40896 * q^73 + 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 62069 * q^79 - 12800 * q^80 - 31288 * q^82 - 189600 * q^83 + 32975 * q^85 - 8312 * q^86 + 26560 * q^88 - 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 34028 * q^94 - 47950 * q^95 + 155547 * q^97 - 19208 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	0	0
$10$	100.000	0.316228
$11$	−425.904	−1.06128	−0.530640	−	0.847597i	$-0.678048\pi$
$11$	−425.904	−1.06128	−0.530640	+	0.847597i	$0.678048\pi$
$12$	0	0
$13$	399.303	0.655307	0.327653	−	0.944798i	$-0.393742\pi$
$13$	399.303	0.655307	0.327653	+	0.944798i	$0.393742\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1751.52	−1.46992	−0.734958	−	0.678112i	$-0.762798\pi$
$17$	−1751.52	−1.46992	−0.734958	+	0.678112i	$0.762798\pi$
$18$	0	0
$19$	2874.23	1.82658	0.913289	−	0.407313i	$-0.133534\pi$
$19$	2874.23	1.82658	0.913289	+	0.407313i	$0.133534\pi$
$20$	−400.000	−0.223607
$21$	0	0
$22$	1703.62	0.750438
$23$	2313.43	0.911878	0.455939	−	0.890011i	$-0.349304\pi$
$23$	2313.43	0.911878	0.455939	+	0.890011i	$0.349304\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1597.21	−0.463372
$27$	0	0
$28$	784.000	0.188982
$29$	2127.93	0.469853	0.234926	−	0.972013i	$-0.424515\pi$
$29$	2127.93	0.469853	0.234926	+	0.972013i	$0.424515\pi$
$30$	0	0
$31$	−10262.5	−1.91801	−0.959004	−	0.283393i	$-0.908540\pi$
$31$	−10262.5	−1.91801	−0.959004	+	0.283393i	$0.908540\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	7006.08	1.03939
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−7266.05	−0.872557	−0.436279	−	0.899812i	$-0.643704\pi$
$37$	−7266.05	−0.872557	−0.436279	+	0.899812i	$0.643704\pi$
$38$	−11496.9	−1.29159
$39$	0	0
$40$	1600.00	0.158114
$41$	5893.44	0.547531	0.273766	−	0.961796i	$-0.411731\pi$
$41$	5893.44	0.547531	0.273766	+	0.961796i	$0.411731\pi$
$42$	0	0
$43$	20157.7	1.66253	0.831267	−	0.555873i	$-0.187616\pi$
$43$	20157.7	1.66253	0.831267	+	0.555873i	$0.187616\pi$
$44$	−6814.46	−0.530640
$45$	0	0
$46$	−9253.72	−0.644795
$47$	−20056.5	−1.32438	−0.662188	−	0.749338i	$-0.730372\pi$
$47$	−20056.5	−1.32438	−0.662188	+	0.749338i	$0.730372\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−2500.00	−0.141421
$51$	0	0
$52$	6388.85	0.327653
$53$	33954.9	1.66040	0.830200	−	0.557466i	$-0.188226\pi$
$53$	33954.9	1.66040	0.830200	+	0.557466i	$0.188226\pi$
$54$	0	0
$55$	10647.6	0.474619
$56$	−3136.00	−0.133631
$57$	0	0
$58$	−8511.71	−0.332236
$59$	4319.34	0.161543	0.0807713	−	0.996733i	$-0.474262\pi$
$59$	4319.34	0.161543	0.0807713	+	0.996733i	$0.474262\pi$
$60$	0	0
$61$	−12253.6	−0.421638	−0.210819	−	0.977525i	$-0.567613\pi$
$61$	−12253.6	−0.421638	−0.210819	+	0.977525i	$0.567613\pi$
$62$	41050.1	1.35624
$63$	0	0
$64$	4096.00	0.125000
$65$	−9982.58	−0.293062
$66$	0	0
$67$	−17533.8	−0.477188	−0.238594	−	0.971119i	$-0.576687\pi$
$67$	−17533.8	−0.477188	−0.238594	+	0.971119i	$0.576687\pi$
$68$	−28024.3	−0.734958
$69$	0	0
$70$	4900.00	0.119523
$71$	−1658.18	−0.0390379	−0.0195189	−	0.999809i	$-0.506213\pi$
$71$	−1658.18	−0.0390379	−0.0195189	+	0.999809i	$0.506213\pi$
$72$	0	0
$73$	−8246.91	−0.181127	−0.0905637	−	0.995891i	$-0.528867\pi$
$73$	−8246.91	−0.181127	−0.0905637	+	0.995891i	$0.528867\pi$
$74$	29064.2	0.616991
$75$	0	0
$76$	45987.7	0.913289
$77$	−20869.3	−0.401126
$78$	0	0
$79$	−9168.61	−0.165286	−0.0826430	−	0.996579i	$-0.526336\pi$
$79$	−9168.61	−0.165286	−0.0826430	+	0.996579i	$0.526336\pi$
$80$	−6400.00	−0.111803
$81$	0	0
$82$	−23573.7	−0.387163
$83$	−95203.2	−1.51690	−0.758449	−	0.651732i	$-0.774043\pi$
$83$	−95203.2	−1.51690	−0.758449	+	0.651732i	$0.774043\pi$
$84$	0	0
$85$	43788.0	0.657367
$86$	−80631.0	−1.17559
$87$	0	0
$88$	27257.8	0.375219
$89$	14441.8	0.193262	0.0966311	−	0.995320i	$-0.469193\pi$
$89$	14441.8	0.193262	0.0966311	+	0.995320i	$0.469193\pi$
$90$	0	0
$91$	19565.9	0.247683
$92$	37014.9	0.455939
$93$	0	0
$94$	80226.1	0.936475
$95$	−71855.8	−0.816870
$96$	0	0
$97$	62132.4	0.670485	0.335242	−	0.942132i	$-0.391182\pi$
$97$	62132.4	0.670485	0.335242	+	0.942132i	$0.391182\pi$
$98$	−9604.00	−0.101015
$99$	0	0
$100$	10000.0	0.100000
$101$	108138.	1.05481	0.527404	−	0.849615i	$-0.323165\pi$
$101$	108138.	1.05481	0.527404	+	0.849615i	$0.323165\pi$
$102$	0	0
$103$	138034.	1.28201	0.641005	−	0.767536i	$-0.278518\pi$
$103$	138034.	1.28201	0.641005	+	0.767536i	$0.278518\pi$
$104$	−25555.4	−0.231686
$105$	0	0
$106$	−135820.	−1.17408
$107$	−6189.61	−0.0522641	−0.0261321	−	0.999658i	$-0.508319\pi$
$107$	−6189.61	−0.0522641	−0.0261321	+	0.999658i	$0.508319\pi$
$108$	0	0
$109$	−68652.9	−0.553468	−0.276734	−	0.960947i	$-0.589252\pi$
$109$	−68652.9	−0.553468	−0.276734	+	0.960947i	$0.589252\pi$
$110$	−42590.4	−0.335606
$111$	0	0
$112$	12544.0	0.0944911
$113$	62835.9	0.462926	0.231463	−	0.972844i	$-0.425649\pi$
$113$	62835.9	0.462926	0.231463	+	0.972844i	$0.425649\pi$
$114$	0	0
$115$	−57835.7	−0.407804
$116$	34046.8	0.234926
$117$	0	0
$118$	−17277.4	−0.114228
$119$	−85824.4	−0.555576
$120$	0	0
$121$	20343.1	0.126315
$122$	49014.5	0.298143
$123$	0	0
$124$	−164201.	−0.959004
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−198049.	−1.08959	−0.544794	−	0.838570i	$-0.683392\pi$
$127$	−198049.	−1.08959	−0.544794	+	0.838570i	$0.683392\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	39930.3	0.207226
$131$	285132.	1.45167	0.725835	−	0.687868i	$-0.241453\pi$
$131$	285132.	1.45167	0.725835	+	0.687868i	$0.241453\pi$
$132$	0	0
$133$	140837.	0.690381
$134$	70135.3	0.337423
$135$	0	0
$136$	112097.	0.519694
$137$	−406979.	−1.85255	−0.926277	−	0.376844i	$-0.877009\pi$
$137$	−406979.	−1.85255	−0.926277	+	0.376844i	$0.877009\pi$
$138$	0	0
$139$	13530.4	0.0593985	0.0296992	−	0.999559i	$-0.490545\pi$
$139$	13530.4	0.0593985	0.0296992	+	0.999559i	$0.490545\pi$
$140$	−19600.0	−0.0845154
$141$	0	0
$142$	6632.72	0.0276039
$143$	−170065.	−0.695464
$144$	0	0
$145$	−53198.2	−0.210125
$146$	32987.6	0.128076
$147$	0	0
$148$	−116257.	−0.436279
$149$	−77129.8	−0.284614	−0.142307	−	0.989823i	$-0.545452\pi$
$149$	−77129.8	−0.284614	−0.142307	+	0.989823i	$0.545452\pi$
$150$	0	0
$151$	−420464.	−1.50067	−0.750336	−	0.661056i	$-0.770108\pi$
$151$	−420464.	−1.50067	−0.750336	+	0.661056i	$0.770108\pi$
$152$	−183951.	−0.645793
$153$	0	0
$154$	83477.2	0.283639
$155$	256563.	0.857759
$156$	0	0
$157$	−481908.	−1.56032	−0.780162	−	0.625577i	$-0.784864\pi$
$157$	−481908.	−1.56032	−0.780162	+	0.625577i	$0.784864\pi$
$158$	36674.4	0.116875
$159$	0	0
$160$	25600.0	0.0790569
$161$	113358.	0.344657
$162$	0	0
$163$	282039.	0.831458	0.415729	−	0.909488i	$-0.363526\pi$
$163$	282039.	0.831458	0.415729	+	0.909488i	$0.363526\pi$
$164$	94295.0	0.273766
$165$	0	0
$166$	380813.	1.07261
$167$	−131971.	−0.366173	−0.183086	−	0.983097i	$-0.558609\pi$
$167$	−131971.	−0.366173	−0.183086	+	0.983097i	$0.558609\pi$
$168$	0	0
$169$	−211850.	−0.570573
$170$	−175152.	−0.464828
$171$	0	0
$172$	322524.	0.831267
$173$	−55501.5	−0.140990	−0.0704952	−	0.997512i	$-0.522458\pi$
$173$	−55501.5	−0.140990	−0.0704952	+	0.997512i	$0.522458\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	−109031.	−0.265320
$177$	0	0
$178$	−57767.3	−0.136657
$179$	−20768.3	−0.0484471	−0.0242236	−	0.999707i	$-0.507711\pi$
$179$	−20768.3	−0.0484471	−0.0242236	+	0.999707i	$0.507711\pi$
$180$	0	0
$181$	−255866.	−0.580520	−0.290260	−	0.956948i	$-0.593742\pi$
$181$	−255866.	−0.580520	−0.290260	+	0.956948i	$0.593742\pi$
$182$	−78263.4	−0.175138
$183$	0	0
$184$	−148059.	−0.322397
$185$	181651.	0.390219
$186$	0	0
$187$	745979.	1.55999
$188$	−320904.	−0.662188
$189$	0	0
$190$	287423.	0.577615
$191$	−649260.	−1.28776	−0.643881	−	0.765126i	$-0.722677\pi$
$191$	−649260.	−1.28776	−0.643881	+	0.765126i	$0.722677\pi$
$192$	0	0
$193$	−768642.	−1.48536	−0.742678	−	0.669649i	$-0.766445\pi$
$193$	−768642.	−1.48536	−0.742678	+	0.669649i	$0.766445\pi$
$194$	−248530.	−0.474104
$195$	0	0
$196$	38416.0	0.0714286
$197$	166160.	0.305044	0.152522	−	0.988300i	$-0.451261\pi$
$197$	166160.	0.305044	0.152522	+	0.988300i	$0.451261\pi$
$198$	0	0
$199$	−74756.1	−0.133818	−0.0669089	−	0.997759i	$-0.521314\pi$
$199$	−74756.1	−0.133818	−0.0669089	+	0.997759i	$0.521314\pi$
$200$	−40000.0	−0.0707107
$201$	0	0
$202$	−432551.	−0.745862
$203$	104268.	0.177588
$204$	0	0
$205$	−147336.	−0.244863
$206$	−552134.	−0.906518
$207$	0	0
$208$	102222.	0.163827
$209$	−1.22415e6	−1.93851
$210$	0	0
$211$	−1.08646e6	−1.67999	−0.839997	−	0.542591i	$-0.817443\pi$
$211$	−1.08646e6	−1.67999	−0.839997	+	0.542591i	$0.817443\pi$
$212$	543278.	0.830200
$213$	0	0
$214$	24758.4	0.0369563
$215$	−503943.	−0.743508
$216$	0	0
$217$	−502864.	−0.724939
$218$	274612.	0.391361
$219$	0	0
$220$	170362.	0.237309
$221$	−699387.	−0.963246
$222$	0	0
$223$	−430809.	−0.580126	−0.290063	−	0.957008i	$-0.593676\pi$
$223$	−430809.	−0.580126	−0.290063	+	0.957008i	$0.593676\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	−251343.	−0.327338
$227$	228724.	0.294610	0.147305	−	0.989091i	$-0.452940\pi$
$227$	228724.	0.294610	0.147305	+	0.989091i	$0.452940\pi$
$228$	0	0
$229$	−1.01797e6	−1.28276	−0.641378	−	0.767225i	$-0.721637\pi$
$229$	−1.01797e6	−1.28276	−0.641378	+	0.767225i	$0.721637\pi$
$230$	231343.	0.288361
$231$	0	0
$232$	−136187.	−0.166118
$233$	236710.	0.285645	0.142822	−	0.989748i	$-0.454382\pi$
$233$	236710.	0.285645	0.142822	+	0.989748i	$0.454382\pi$
$234$	0	0
$235$	501413.	0.592279
$236$	69109.4	0.0807713
$237$	0	0
$238$	343298.	0.392852
$239$	−1.73994e6	−1.97034	−0.985169	−	0.171587i	$-0.945111\pi$
$239$	−1.73994e6	−1.97034	−0.985169	+	0.171587i	$0.945111\pi$
$240$	0	0
$241$	−332840.	−0.369142	−0.184571	−	0.982819i	$-0.559090\pi$
$241$	−332840.	−0.369142	−0.184571	+	0.982819i	$0.559090\pi$
$242$	−81372.4	−0.0893180
$243$	0	0
$244$	−196058.	−0.210819
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	1.14769e6	1.19697
$248$	656802.	0.678118
$249$	0	0
$250$	62500.0	0.0632456
$251$	−1.88585e6	−1.88939	−0.944696	−	0.327948i	$-0.893643\pi$
$251$	−1.88585e6	−1.88939	−0.944696	+	0.327948i	$0.893643\pi$
$252$	0	0
$253$	−985298.	−0.967757
$254$	792194.	0.770455
$255$	0	0
$256$	65536.0	0.0625000
$257$	−432608.	−0.408566	−0.204283	−	0.978912i	$-0.565486\pi$
$257$	−432608.	−0.408566	−0.204283	+	0.978912i	$0.565486\pi$
$258$	0	0
$259$	−356036.	−0.329796
$260$	−159721.	−0.146531
$261$	0	0
$262$	−1.14053e6	−1.02649
$263$	1.20388e6	1.07323	0.536614	−	0.843828i	$-0.319703\pi$
$263$	1.20388e6	1.07323	0.536614	+	0.843828i	$0.319703\pi$
$264$	0	0
$265$	−848872.	−0.742554
$266$	−563350.	−0.488173
$267$	0	0
$268$	−280541.	−0.238594
$269$	−548133.	−0.461855	−0.230927	−	0.972971i	$-0.574176\pi$
$269$	−548133.	−0.461855	−0.230927	+	0.972971i	$0.574176\pi$
$270$	0	0
$271$	−906590.	−0.749873	−0.374936	−	0.927051i	$-0.622335\pi$
$271$	−906590.	−0.749873	−0.374936	+	0.927051i	$0.622335\pi$
$272$	−448389.	−0.367479
$273$	0	0
$274$	1.62792e6	1.30995
$275$	−266190.	−0.212256
$276$	0	0
$277$	1.35338e6	1.05979	0.529895	−	0.848064i	$-0.322231\pi$
$277$	1.35338e6	1.05979	0.529895	+	0.848064i	$0.322231\pi$
$278$	−54121.8	−0.0420011
$279$	0	0
$280$	78400.0	0.0597614
$281$	−752309.	−0.568369	−0.284185	−	0.958770i	$-0.591723\pi$
$281$	−752309.	−0.568369	−0.284185	+	0.958770i	$0.591723\pi$
$282$	0	0
$283$	−820710.	−0.609149	−0.304575	−	0.952489i	$-0.598514\pi$
$283$	−820710.	−0.609149	−0.304575	+	0.952489i	$0.598514\pi$
$284$	−26530.9	−0.0195189
$285$	0	0
$286$	680259.	0.491767
$287$	288778.	0.206947
$288$	0	0
$289$	1.64796e6	1.16065
$290$	212793.	0.148581
$291$	0	0
$292$	−131951.	−0.0905637
$293$	1.92239e6	1.30819	0.654097	−	0.756411i	$-0.273049\pi$
$293$	1.92239e6	1.30819	0.654097	+	0.756411i	$0.273049\pi$
$294$	0	0
$295$	−107983.	−0.0722441
$296$	465027.	0.308496
$297$	0	0
$298$	308519.	0.201253
$299$	923760.	0.597559
$300$	0	0
$301$	987729.	0.628379
$302$	1.68185e6	1.06114
$303$	0	0
$304$	735804.	0.456644
$305$	306340.	0.188562
$306$	0	0
$307$	223229.	0.135177	0.0675887	−	0.997713i	$-0.478469\pi$
$307$	223229.	0.135177	0.0675887	+	0.997713i	$0.478469\pi$
$308$	−333909.	−0.200563
$309$	0	0
$310$	−1.02625e6	−0.606527
$311$	−2.29818e6	−1.34736	−0.673678	−	0.739025i	$-0.735286\pi$
$311$	−2.29818e6	−1.34736	−0.673678	+	0.739025i	$0.735286\pi$
$312$	0	0
$313$	−1.62881e6	−0.939744	−0.469872	−	0.882735i	$-0.655700\pi$
$313$	−1.62881e6	−0.939744	−0.469872	+	0.882735i	$0.655700\pi$
$314$	1.92763e6	1.10332
$315$	0	0
$316$	−146698.	−0.0826430
$317$	−1.13620e6	−0.635049	−0.317524	−	0.948250i	$-0.602851\pi$
$317$	−1.13620e6	−0.635049	−0.317524	+	0.948250i	$0.602851\pi$
$318$	0	0
$319$	−906293.	−0.498645
$320$	−102400.	−0.0559017
$321$	0	0
$322$	−453432.	−0.243710
$323$	−5.03428e6	−2.68492
$324$	0	0
$325$	249565.	0.131061
$326$	−1.12816e6	−0.587930
$327$	0	0
$328$	−377180.	−0.193582
$329$	−982770.	−0.500567
$330$	0	0
$331$	−3.77549e6	−1.89410	−0.947051	−	0.321084i	$-0.895953\pi$
$331$	−3.77549e6	−1.89410	−0.947051	+	0.321084i	$0.895953\pi$
$332$	−1.52325e6	−0.758449
$333$	0	0
$334$	527882.	0.258923
$335$	438346.	0.213405
$336$	0	0
$337$	3.74913e6	1.79827	0.899136	−	0.437669i	$-0.144196\pi$
$337$	3.74913e6	1.79827	0.899136	+	0.437669i	$0.144196\pi$
$338$	847400.	0.403456
$339$	0	0
$340$	700608.	0.328683
$341$	4.37085e6	2.03554
$342$	0	0
$343$	117649.	0.0539949
$344$	−1.29010e6	−0.587795
$345$	0	0
$346$	222006.	0.0996953
$347$	1.14843e6	0.512012	0.256006	−	0.966675i	$-0.417593\pi$
$347$	1.14843e6	0.512012	0.256006	+	0.966675i	$0.417593\pi$
$348$	0	0
$349$	4.03275e6	1.77230	0.886151	−	0.463396i	$-0.153369\pi$
$349$	4.03275e6	1.77230	0.886151	+	0.463396i	$0.153369\pi$
$350$	−122500.	−0.0534522
$351$	0	0
$352$	436126.	0.187610
$353$	−652660.	−0.278773	−0.139386	−	0.990238i	$-0.544513\pi$
$353$	−652660.	−0.278773	−0.139386	+	0.990238i	$0.544513\pi$
$354$	0	0
$355$	41454.5	0.0174583
$356$	231069.	0.0966311
$357$	0	0
$358$	83073.2	0.0342573
$359$	2.84604e6	1.16548	0.582741	−	0.812658i	$-0.301980\pi$
$359$	2.84604e6	1.16548	0.582741	+	0.812658i	$0.301980\pi$
$360$	0	0
$361$	5.78512e6	2.33639
$362$	1.02347e6	0.410489
$363$	0	0
$364$	313054.	0.123841
$365$	206173.	0.0810026
$366$	0	0
$367$	−2.65232e6	−1.02792	−0.513962	−	0.857813i	$-0.671823\pi$
$367$	−2.65232e6	−1.02792	−0.513962	+	0.857813i	$0.671823\pi$
$368$	592238.	0.227969
$369$	0	0
$370$	−726605.	−0.275927
$371$	1.66379e6	0.627572
$372$	0	0
$373$	2.83283e6	1.05426	0.527132	−	0.849784i	$-0.323267\pi$
$373$	2.83283e6	1.05426	0.527132	+	0.849784i	$0.323267\pi$
$374$	−2.98392e6	−1.10308
$375$	0	0
$376$	1.28362e6	0.468237
$377$	849688.	0.307898
$378$	0	0
$379$	−1.20678e6	−0.431551	−0.215775	−	0.976443i	$-0.569228\pi$
$379$	−1.20678e6	−0.431551	−0.215775	+	0.976443i	$0.569228\pi$
$380$	−1.14969e6	−0.408435
$381$	0	0
$382$	2.59704e6	0.910585
$383$	2.22793e6	0.776076	0.388038	−	0.921643i	$-0.373153\pi$
$383$	2.22793e6	0.776076	0.388038	+	0.921643i	$0.373153\pi$
$384$	0	0
$385$	521732.	0.179389
$386$	3.07457e6	1.05031
$387$	0	0
$388$	994119.	0.335242
$389$	3.74004e6	1.25315	0.626574	−	0.779362i	$-0.284457\pi$
$389$	3.74004e6	1.25315	0.626574	+	0.779362i	$0.284457\pi$
$390$	0	0
$391$	−4.05202e6	−1.34038
$392$	−153664.	−0.0505076
$393$	0	0
$394$	−664642.	−0.215699
$395$	229215.	0.0739181
$396$	0	0
$397$	−2.90335e6	−0.924533	−0.462267	−	0.886741i	$-0.652964\pi$
$397$	−2.90335e6	−0.924533	−0.462267	+	0.886741i	$0.652964\pi$
$398$	299024.	0.0946235
$399$	0	0
$400$	160000.	0.0500000
$401$	−1.43777e6	−0.446506	−0.223253	−	0.974761i	$-0.571668\pi$
$401$	−1.43777e6	−0.446506	−0.223253	+	0.974761i	$0.571668\pi$
$402$	0	0
$403$	−4.09786e6	−1.25688
$404$	1.73020e6	0.527404
$405$	0	0
$406$	−417074.	−0.125573
$407$	3.09464e6	0.926027
$408$	0	0
$409$	−4.78825e6	−1.41536	−0.707682	−	0.706531i	$-0.750259\pi$
$409$	−4.78825e6	−1.41536	−0.707682	+	0.706531i	$0.750259\pi$
$410$	589344.	0.173145
$411$	0	0
$412$	2.20854e6	0.641005
$413$	211648.	0.0610574
$414$	0	0
$415$	2.38008e6	0.678377
$416$	−408887.	−0.115843
$417$	0	0
$418$	4.89659e6	1.37073
$419$	2.08411e6	0.579943	0.289972	−	0.957035i	$-0.406354\pi$
$419$	2.08411e6	0.579943	0.289972	+	0.957035i	$0.406354\pi$
$420$	0	0
$421$	−1.00260e6	−0.275690	−0.137845	−	0.990454i	$-0.544018\pi$
$421$	−1.00260e6	−0.275690	−0.137845	+	0.990454i	$0.544018\pi$
$422$	4.34584e6	1.18793
$423$	0	0
$424$	−2.17311e6	−0.587040
$425$	−1.09470e6	−0.293983
$426$	0	0
$427$	−600427.	−0.159364
$428$	−99033.8	−0.0261321
$429$	0	0
$430$	2.01577e6	0.525740
$431$	−1.45620e6	−0.377595	−0.188798	−	0.982016i	$-0.560459\pi$
$431$	−1.45620e6	−0.377595	−0.188798	+	0.982016i	$0.560459\pi$
$432$	0	0
$433$	−1.10726e6	−0.283812	−0.141906	−	0.989880i	$-0.545323\pi$
$433$	−1.10726e6	−0.283812	−0.141906	+	0.989880i	$0.545323\pi$
$434$	2.01146e6	0.512609
$435$	0	0
$436$	−1.09845e6	−0.276734
$437$	6.64934e6	1.66562
$438$	0	0
$439$	4.15410e6	1.02876	0.514382	−	0.857561i	$-0.328021\pi$
$439$	4.15410e6	1.02876	0.514382	+	0.857561i	$0.328021\pi$
$440$	−681446.	−0.167803
$441$	0	0
$442$	2.79755e6	0.681118
$443$	7.20799e6	1.74504	0.872519	−	0.488581i	$-0.162485\pi$
$443$	7.20799e6	1.74504	0.872519	+	0.488581i	$0.162485\pi$
$444$	0	0
$445$	−361046.	−0.0864295
$446$	1.72324e6	0.410211
$447$	0	0
$448$	200704.	0.0472456
$449$	−5.58367e6	−1.30709	−0.653543	−	0.756890i	$-0.726718\pi$
$449$	−5.58367e6	−1.30709	−0.653543	+	0.756890i	$0.726718\pi$
$450$	0	0
$451$	−2.51004e6	−0.581084
$452$	1.00537e6	0.231463
$453$	0	0
$454$	−914895.	−0.208320
$455$	−489147.	−0.110767
$456$	0	0
$457$	−3.54344e6	−0.793661	−0.396831	−	0.917892i	$-0.629890\pi$
$457$	−3.54344e6	−0.793661	−0.396831	+	0.917892i	$0.629890\pi$
$458$	4.07186e6	0.907046
$459$	0	0
$460$	−925372.	−0.203902
$461$	−5.88069e6	−1.28877	−0.644386	−	0.764700i	$-0.722887\pi$
$461$	−5.88069e6	−1.28877	−0.644386	+	0.764700i	$0.722887\pi$
$462$	0	0
$463$	6.65045e6	1.44178	0.720890	−	0.693050i	$-0.243733\pi$
$463$	6.65045e6	1.44178	0.720890	+	0.693050i	$0.243733\pi$
$464$	544749.	0.117463
$465$	0	0
$466$	−946839.	−0.201981
$467$	−4.34743e6	−0.922445	−0.461222	−	0.887285i	$-0.652589\pi$
$467$	−4.34743e6	−0.922445	−0.461222	+	0.887285i	$0.652589\pi$
$468$	0	0
$469$	−859158.	−0.180360
$470$	−2.00565e6	−0.418804
$471$	0	0
$472$	−276438.	−0.0571139
$473$	−8.58526e6	−1.76441
$474$	0	0
$475$	1.79640e6	0.365316
$476$	−1.37319e6	−0.277788
$477$	0	0
$478$	6.95978e6	1.39324
$479$	929680.	0.185138	0.0925688	−	0.995706i	$-0.470492\pi$
$479$	929680.	0.185138	0.0925688	+	0.995706i	$0.470492\pi$
$480$	0	0
$481$	−2.90136e6	−0.571792
$482$	1.33136e6	0.261023
$483$	0	0
$484$	325490.	0.0631573
$485$	−1.55331e6	−0.299850
$486$	0	0
$487$	−5.44887e6	−1.04108	−0.520540	−	0.853837i	$-0.674269\pi$
$487$	−5.44887e6	−1.04108	−0.520540	+	0.853837i	$0.674269\pi$
$488$	784232.	0.149072
$489$	0	0
$490$	240100.	0.0451754
$491$	670070.	0.125434	0.0627172	−	0.998031i	$-0.480023\pi$
$491$	670070.	0.125434	0.0627172	+	0.998031i	$0.480023\pi$
$492$	0	0
$493$	−3.72711e6	−0.690644
$494$	−4.59076e6	−0.846384
$495$	0	0
$496$	−2.62721e6	−0.479502
$497$	−81250.9	−0.0147549
$498$	0	0
$499$	6.58017e6	1.18300	0.591501	−	0.806304i	$-0.298535\pi$
$499$	6.58017e6	1.18300	0.591501	+	0.806304i	$0.298535\pi$
$500$	−250000.	−0.0447214
$501$	0	0
$502$	7.54339e6	1.33600
$503$	1.47714e6	0.260317	0.130159	−	0.991493i	$-0.458451\pi$
$503$	1.47714e6	0.260317	0.130159	+	0.991493i	$0.458451\pi$
$504$	0	0
$505$	−2.70344e6	−0.471724
$506$	3.94119e6	0.684308
$507$	0	0
$508$	−3.16878e6	−0.544794
$509$	−5.16474e6	−0.883597	−0.441799	−	0.897114i	$-0.645659\pi$
$509$	−5.16474e6	−0.883597	−0.441799	+	0.897114i	$0.645659\pi$
$510$	0	0
$511$	−404099.	−0.0684597
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.73043e6	0.288899
$515$	−3.45084e6	−0.573333
$516$	0	0
$517$	8.54215e6	1.40553
$518$	1.42415e6	0.233201
$519$	0	0
$520$	638885.	0.103613
$521$	−294103.	−0.0474684	−0.0237342	−	0.999718i	$-0.507556\pi$
$521$	−294103.	−0.0474684	−0.0237342	+	0.999718i	$0.507556\pi$
$522$	0	0
$523$	1.39098e6	0.222365	0.111182	−	0.993800i	$-0.464536\pi$
$523$	1.39098e6	0.222365	0.111182	+	0.993800i	$0.464536\pi$
$524$	4.56212e6	0.725835
$525$	0	0
$526$	−4.81550e6	−0.758887
$527$	1.79750e7	2.81931
$528$	0	0
$529$	−1.08439e6	−0.168479
$530$	3.39549e6	0.525065
$531$	0	0
$532$	2.25340e6	0.345191
$533$	2.35327e6	0.358801
$534$	0	0
$535$	154740.	0.0233732
$536$	1.12216e6	0.168712
$537$	0	0
$538$	2.19253e6	0.326581
$539$	−1.02260e6	−0.151611
$540$	0	0
$541$	2.36315e6	0.347134	0.173567	−	0.984822i	$-0.444471\pi$
$541$	2.36315e6	0.347134	0.173567	+	0.984822i	$0.444471\pi$
$542$	3.62636e6	0.530240
$543$	0	0
$544$	1.79356e6	0.259847
$545$	1.71632e6	0.247519
$546$	0	0
$547$	4.52298e6	0.646333	0.323167	−	0.946342i	$-0.395253\pi$
$547$	4.52298e6	0.646333	0.323167	+	0.946342i	$0.395253\pi$
$548$	−6.51167e6	−0.926277
$549$	0	0
$550$	1.06476e6	0.150088
$551$	6.11616e6	0.858223
$552$	0	0
$553$	−449262.	−0.0624722
$554$	−5.41351e6	−0.749384
$555$	0	0
$556$	216487.	0.0296992
$557$	8.40670e6	1.14812	0.574060	−	0.818813i	$-0.305367\pi$
$557$	8.40670e6	1.14812	0.574060	+	0.818813i	$0.305367\pi$
$558$	0	0
$559$	8.04905e6	1.08947
$560$	−313600.	−0.0422577
$561$	0	0
$562$	3.00924e6	0.401898
$563$	2.22527e6	0.295877	0.147938	−	0.988997i	$-0.452736\pi$
$563$	2.22527e6	0.295877	0.147938	+	0.988997i	$0.452736\pi$
$564$	0	0
$565$	−1.57090e6	−0.207027
$566$	3.28284e6	0.430733
$567$	0	0
$568$	106124.	0.0138020
$569$	−1.58379e6	−0.205078	−0.102539	−	0.994729i	$-0.532697\pi$
$569$	−1.58379e6	−0.205078	−0.102539	+	0.994729i	$0.532697\pi$
$570$	0	0
$571$	1.25695e7	1.61335	0.806676	−	0.590994i	$-0.201264\pi$
$571$	1.25695e7	1.61335	0.806676	+	0.590994i	$0.201264\pi$
$572$	−2.72104e6	−0.347732
$573$	0	0
$574$	−1.15511e6	−0.146334
$575$	1.44589e6	0.182376
$576$	0	0
$577$	1.37636e7	1.72104	0.860521	−	0.509414i	$-0.170138\pi$
$577$	1.37636e7	1.72104	0.860521	+	0.509414i	$0.170138\pi$
$578$	−6.59185e6	−0.820706
$579$	0	0
$580$	−851171.	−0.105062
$581$	−4.66496e6	−0.573334
$582$	0	0
$583$	−1.44615e7	−1.76215
$584$	527802.	0.0640382
$585$	0	0
$586$	−7.68956e6	−0.925033
$587$	−1.43475e6	−0.171862	−0.0859311	−	0.996301i	$-0.527386\pi$
$587$	−1.43475e6	−0.171862	−0.0859311	+	0.996301i	$0.527386\pi$
$588$	0	0
$589$	−2.94969e7	−3.50339
$590$	431934.	0.0510843
$591$	0	0
$592$	−1.86011e6	−0.218139
$593$	−2.58486e6	−0.301856	−0.150928	−	0.988545i	$-0.548226\pi$
$593$	−2.58486e6	−0.301856	−0.150928	+	0.988545i	$0.548226\pi$
$594$	0	0
$595$	2.14561e6	0.248461
$596$	−1.23408e6	−0.142307
$597$	0	0
$598$	−3.69504e6	−0.422538
$599$	5.51209e6	0.627696	0.313848	−	0.949473i	$-0.398382\pi$
$599$	5.51209e6	0.627696	0.313848	+	0.949473i	$0.398382\pi$
$600$	0	0
$601$	6.92526e6	0.782077	0.391039	−	0.920374i	$-0.372116\pi$
$601$	6.92526e6	0.782077	0.391039	+	0.920374i	$0.372116\pi$
$602$	−3.95092e6	−0.444331
$603$	0	0
$604$	−6.72742e6	−0.750336
$605$	−508578.	−0.0564896
$606$	0	0
$607$	1.19865e7	1.32045	0.660225	−	0.751067i	$-0.270461\pi$
$607$	1.19865e7	1.32045	0.660225	+	0.751067i	$0.270461\pi$
$608$	−2.94322e6	−0.322896
$609$	0	0
$610$	−1.22536e6	−0.133334
$611$	−8.00864e6	−0.867872
$612$	0	0
$613$	−4.84046e6	−0.520278	−0.260139	−	0.965571i	$-0.583768\pi$
$613$	−4.84046e6	−0.520278	−0.260139	+	0.965571i	$0.583768\pi$
$614$	−892915.	−0.0955849
$615$	0	0
$616$	1.33563e6	0.141819
$617$	9.94864e6	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$617$	9.94864e6	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$618$	0	0
$619$	−1.38913e7	−1.45719	−0.728597	−	0.684943i	$-0.759827\pi$
$619$	−1.38913e7	−1.45719	−0.728597	+	0.684943i	$0.759827\pi$
$620$	4.10501e6	0.428880
$621$	0	0
$622$	9.19270e6	0.952724
$623$	707649.	0.0730463
$624$	0	0
$625$	390625.	0.0400000
$626$	6.51524e6	0.664499
$627$	0	0
$628$	−7.71053e6	−0.780162
$629$	1.27266e7	1.28259
$630$	0	0
$631$	−7.29060e6	−0.728937	−0.364469	−	0.931216i	$-0.618749\pi$
$631$	−7.29060e6	−0.728937	−0.364469	+	0.931216i	$0.618749\pi$
$632$	586791.	0.0584374
$633$	0	0
$634$	4.54480e6	0.449047
$635$	4.95121e6	0.487279
$636$	0	0
$637$	958727.	0.0936152
$638$	3.62517e6	0.352595
$639$	0	0
$640$	409600.	0.0395285
$641$	−1.58175e7	−1.52052	−0.760259	−	0.649620i	$-0.774928\pi$
$641$	−1.58175e7	−1.52052	−0.760259	+	0.649620i	$0.774928\pi$
$642$	0	0
$643$	−7.35926e6	−0.701951	−0.350975	−	0.936385i	$-0.614150\pi$
$643$	−7.35926e6	−0.701951	−0.350975	+	0.936385i	$0.614150\pi$
$644$	1.81373e6	0.172329
$645$	0	0
$646$	2.01371e7	1.89852
$647$	−1.08644e7	−1.02034	−0.510172	−	0.860072i	$-0.670418\pi$
$647$	−1.08644e7	−1.02034	−0.510172	+	0.860072i	$0.670418\pi$
$648$	0	0
$649$	−1.83962e6	−0.171442
$650$	−998258.	−0.0926743
$651$	0	0
$652$	4.51263e6	0.415729
$653$	2.85006e6	0.261560	0.130780	−	0.991411i	$-0.458252\pi$
$653$	2.85006e6	0.261560	0.130780	+	0.991411i	$0.458252\pi$
$654$	0	0
$655$	−7.12831e6	−0.649207
$656$	1.50872e6	0.136883
$657$	0	0
$658$	3.93108e6	0.353954
$659$	214583.	0.0192479	0.00962393	−	0.999954i	$-0.496937\pi$
$659$	214583.	0.0192479	0.00962393	+	0.999954i	$0.496937\pi$
$660$	0	0
$661$	2.26579e6	0.201704	0.100852	−	0.994901i	$-0.467843\pi$
$661$	2.26579e6	0.201704	0.100852	+	0.994901i	$0.467843\pi$
$662$	1.51020e7	1.33933
$663$	0	0
$664$	6.09301e6	0.536304
$665$	−3.52094e6	−0.308748
$666$	0	0
$667$	4.92281e6	0.428448
$668$	−2.11153e6	−0.183086
$669$	0	0
$670$	−1.75338e6	−0.150900
$671$	5.21886e6	0.447476
$672$	0	0
$673$	−9.52991e6	−0.811057	−0.405528	−	0.914082i	$-0.632912\pi$
$673$	−9.52991e6	−0.811057	−0.405528	+	0.914082i	$0.632912\pi$
$674$	−1.49965e7	−1.27157
$675$	0	0
$676$	−3.38960e6	−0.285287
$677$	−2.14827e7	−1.80143	−0.900714	−	0.434412i	$-0.856956\pi$
$677$	−2.14827e7	−1.80143	−0.900714	+	0.434412i	$0.856956\pi$
$678$	0	0
$679$	3.04449e6	0.253419
$680$	−2.80243e6	−0.232414
$681$	0	0
$682$	−1.74834e7	−1.43935
$683$	2.89672e6	0.237605	0.118802	−	0.992918i	$-0.462095\pi$
$683$	2.89672e6	0.237605	0.118802	+	0.992918i	$0.462095\pi$
$684$	0	0
$685$	1.01745e7	0.828487
$686$	−470596.	−0.0381802
$687$	0	0
$688$	5.16038e6	0.415634
$689$	1.35583e7	1.08807
$690$	0	0
$691$	−2.09710e6	−0.167080	−0.0835399	−	0.996504i	$-0.526623\pi$
$691$	−2.09710e6	−0.167080	−0.0835399	+	0.996504i	$0.526623\pi$
$692$	−888024.	−0.0704952
$693$	0	0
$694$	−4.59371e6	−0.362047
$695$	−338261.	−0.0265638
$696$	0	0
$697$	−1.03225e7	−0.804825
$698$	−1.61310e7	−1.25321
$699$	0	0
$700$	490000.	0.0377964
$701$	−6.59970e6	−0.507258	−0.253629	−	0.967302i	$-0.581624\pi$
$701$	−6.59970e6	−0.507258	−0.253629	+	0.967302i	$0.581624\pi$
$702$	0	0
$703$	−2.08843e7	−1.59379
$704$	−1.74450e6	−0.132660
$705$	0	0
$706$	2.61064e6	0.197122
$707$	5.29874e6	0.398680
$708$	0	0
$709$	−2.22011e7	−1.65866	−0.829332	−	0.558756i	$-0.811279\pi$
$709$	−2.22011e7	−1.65866	−0.829332	+	0.558756i	$0.811279\pi$
$710$	−165818.	−0.0123449
$711$	0	0
$712$	−924277.	−0.0683285
$713$	−2.37416e7	−1.74899
$714$	0	0
$715$	4.25162e6	0.311021
$716$	−332293.	−0.0242236
$717$	0	0
$718$	−1.13842e7	−0.824120
$719$	−1.87540e7	−1.35292	−0.676460	−	0.736479i	$-0.736487\pi$
$719$	−1.87540e7	−1.35292	−0.676460	+	0.736479i	$0.736487\pi$
$720$	0	0
$721$	6.76365e6	0.484555
$722$	−2.31405e7	−1.65207
$723$	0	0
$724$	−4.09386e6	−0.290260
$725$	1.32995e6	0.0939706
$726$	0	0
$727$	1.38593e7	0.972537	0.486269	−	0.873809i	$-0.338358\pi$
$727$	1.38593e7	0.972537	0.486269	+	0.873809i	$0.338358\pi$
$728$	−1.25222e6	−0.0875690
$729$	0	0
$730$	−824691.	−0.0572775
$731$	−3.53067e7	−2.44379
$732$	0	0
$733$	1.08280e7	0.744372	0.372186	−	0.928158i	$-0.378608\pi$
$733$	1.08280e7	0.744372	0.372186	+	0.928158i	$0.378608\pi$
$734$	1.06093e7	0.726851
$735$	0	0
$736$	−2.36895e6	−0.161199
$737$	7.46772e6	0.506430
$738$	0	0
$739$	−906866.	−0.0610846	−0.0305423	−	0.999533i	$-0.509723\pi$
$739$	−906866.	−0.0610846	−0.0305423	+	0.999533i	$0.509723\pi$
$740$	2.90642e6	0.195110
$741$	0	0
$742$	−6.65516e6	−0.443761
$743$	1.76804e7	1.17495	0.587475	−	0.809242i	$-0.300122\pi$
$743$	1.76804e7	1.17495	0.587475	+	0.809242i	$0.300122\pi$
$744$	0	0
$745$	1.92825e6	0.127283
$746$	−1.13313e7	−0.745477
$747$	0	0
$748$	1.19357e7	0.779996
$749$	−303291.	−0.0197540
$750$	0	0
$751$	1.96975e7	1.27441	0.637207	−	0.770692i	$-0.280089\pi$
$751$	1.96975e7	1.27441	0.637207	+	0.770692i	$0.280089\pi$
$752$	−5.13447e6	−0.331094
$753$	0	0
$754$	−3.39875e6	−0.217717
$755$	1.05116e7	0.671121
$756$	0	0
$757$	−1.88790e7	−1.19740	−0.598699	−	0.800974i	$-0.704315\pi$
$757$	−1.88790e7	−1.19740	−0.598699	+	0.800974i	$0.704315\pi$
$758$	4.82714e6	0.305152
$759$	0	0
$760$	4.59877e6	0.288807
$761$	−9.74849e6	−0.610205	−0.305102	−	0.952320i	$-0.598691\pi$
$761$	−9.74849e6	−0.610205	−0.305102	+	0.952320i	$0.598691\pi$
$762$	0	0
$763$	−3.36399e6	−0.209191
$764$	−1.03882e7	−0.643881
$765$	0	0
$766$	−8.91171e6	−0.548768
$767$	1.72473e6	0.105860
$768$	0	0
$769$	6.11920e6	0.373146	0.186573	−	0.982441i	$-0.440262\pi$
$769$	6.11920e6	0.373146	0.186573	+	0.982441i	$0.440262\pi$
$770$	−2.08693e6	−0.126847
$771$	0	0
$772$	−1.22983e7	−0.742678
$773$	−1.47893e6	−0.0890223	−0.0445111	−	0.999009i	$-0.514173\pi$
$773$	−1.47893e6	−0.0890223	−0.0445111	+	0.999009i	$0.514173\pi$
$774$	0	0
$775$	−6.41408e6	−0.383602
$776$	−3.97648e6	−0.237052
$777$	0	0
$778$	−1.49602e7	−0.886109
$779$	1.69391e7	1.00011
$780$	0	0
$781$	706226.	0.0414301
$782$	1.62081e7	0.947795
$783$	0	0
$784$	614656.	0.0357143
$785$	1.20477e7	0.697799
$786$	0	0
$787$	2.39360e7	1.37758	0.688788	−	0.724963i	$-0.258143\pi$
$787$	2.39360e7	1.37758	0.688788	+	0.724963i	$0.258143\pi$
$788$	2.65857e6	0.152522
$789$	0	0
$790$	−916861.	−0.0522680
$791$	3.07896e6	0.174970
$792$	0	0
$793$	−4.89291e6	−0.276302
$794$	1.16134e7	0.653744
$795$	0	0
$796$	−1.19610e6	−0.0669089
$797$	−6.21796e6	−0.346738	−0.173369	−	0.984857i	$-0.555465\pi$
$797$	−6.21796e6	−0.346738	−0.173369	+	0.984857i	$0.555465\pi$
$798$	0	0
$799$	3.51294e7	1.94672
$800$	−640000.	−0.0353553
$801$	0	0
$802$	5.75106e6	0.315727
$803$	3.51239e6	0.192227
$804$	0	0
$805$	−2.83395e6	−0.154135
$806$	1.63915e7	0.888750
$807$	0	0
$808$	−6.92081e6	−0.372931
$809$	4.98100e6	0.267575	0.133787	−	0.991010i	$-0.457286\pi$
$809$	4.98100e6	0.267575	0.133787	+	0.991010i	$0.457286\pi$
$810$	0	0
$811$	−3.09722e7	−1.65356	−0.826780	−	0.562525i	$-0.809830\pi$
$811$	−3.09722e7	−1.65356	−0.826780	+	0.562525i	$0.809830\pi$
$812$	1.66830e6	0.0887938
$813$	0	0
$814$	−1.23786e7	−0.654800
$815$	−7.05098e6	−0.371839
$816$	0	0
$817$	5.79381e7	3.03675
$818$	1.91530e7	1.00081
$819$	0	0
$820$	−2.35737e6	−0.122432
$821$	2.87667e7	1.48947	0.744734	−	0.667361i	$-0.232576\pi$
$821$	2.87667e7	1.48947	0.744734	+	0.667361i	$0.232576\pi$
$822$	0	0
$823$	3.55713e7	1.83063	0.915313	−	0.402743i	$-0.131943\pi$
$823$	3.55713e7	1.83063	0.915313	+	0.402743i	$0.131943\pi$
$824$	−8.83415e6	−0.453259
$825$	0	0
$826$	−846590.	−0.0431741
$827$	−3.01542e7	−1.53315	−0.766575	−	0.642155i	$-0.778040\pi$
$827$	−3.01542e7	−1.53315	−0.766575	+	0.642155i	$0.778040\pi$
$828$	0	0
$829$	−3.11697e7	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$829$	−3.11697e7	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$830$	−9.52032e6	−0.479685
$831$	0	0
$832$	1.63555e6	0.0819133
$833$	−4.20540e6	−0.209988
$834$	0	0
$835$	3.29926e6	0.163757
$836$	−1.95864e7	−0.969255
$837$	0	0
$838$	−8.33644e6	−0.410082
$839$	−3.63050e7	−1.78058	−0.890291	−	0.455393i	$-0.849499\pi$
$839$	−3.63050e7	−1.78058	−0.890291	+	0.455393i	$0.849499\pi$
$840$	0	0
$841$	−1.59831e7	−0.779238
$842$	4.01039e6	0.194942
$843$	0	0
$844$	−1.73834e7	−0.839997
$845$	5.29625e6	0.255168
$846$	0	0
$847$	996812.	0.0477425
$848$	8.69245e6	0.415100
$849$	0	0
$850$	4.37880e6	0.207878
$851$	−1.68095e7	−0.795666
$852$	0	0
$853$	1.00191e7	0.471472	0.235736	−	0.971817i	$-0.424250\pi$
$853$	1.00191e7	0.471472	0.235736	+	0.971817i	$0.424250\pi$
$854$	2.40171e6	0.112687
$855$	0	0
$856$	396135.	0.0184782
$857$	1.17397e7	0.546014	0.273007	−	0.962012i	$-0.411982\pi$
$857$	1.17397e7	0.546014	0.273007	+	0.962012i	$0.411982\pi$
$858$	0	0
$859$	−1.69498e7	−0.783756	−0.391878	−	0.920017i	$-0.628175\pi$
$859$	−1.69498e7	−0.783756	−0.391878	+	0.920017i	$0.628175\pi$
$860$	−8.06310e6	−0.371754
$861$	0	0
$862$	5.82478e6	0.267000
$863$	2.11590e7	0.967091	0.483546	−	0.875319i	$-0.339349\pi$
$863$	2.11590e7	0.967091	0.483546	+	0.875319i	$0.339349\pi$
$864$	0	0
$865$	1.38754e6	0.0630528
$866$	4.42905e6	0.200686
$867$	0	0
$868$	−8.04583e6	−0.362469
$869$	3.90495e6	0.175415
$870$	0	0
$871$	−7.00131e6	−0.312705
$872$	4.39379e6	0.195681
$873$	0	0
$874$	−2.65973e7	−1.17777
$875$	−765625.	−0.0338062
$876$	0	0
$877$	1.01611e7	0.446109	0.223055	−	0.974806i	$-0.428397\pi$
$877$	1.01611e7	0.446109	0.223055	+	0.974806i	$0.428397\pi$
$878$	−1.66164e7	−0.727446
$879$	0	0
$880$	2.72578e6	0.118655
$881$	3.50223e7	1.52021	0.760106	−	0.649799i	$-0.225147\pi$
$881$	3.50223e7	1.52021	0.760106	+	0.649799i	$0.225147\pi$
$882$	0	0
$883$	−2.15600e7	−0.930566	−0.465283	−	0.885162i	$-0.654047\pi$
$883$	−2.15600e7	−0.930566	−0.465283	+	0.885162i	$0.654047\pi$
$884$	−1.11902e7	−0.481623
$885$	0	0
$886$	−2.88319e7	−1.23393
$887$	1.29121e7	0.551048	0.275524	−	0.961294i	$-0.411149\pi$
$887$	1.29121e7	0.551048	0.275524	+	0.961294i	$0.411149\pi$
$888$	0	0
$889$	−9.70438e6	−0.411826
$890$	1.44418e6	0.0611149
$891$	0	0
$892$	−6.89294e6	−0.290063
$893$	−5.76472e7	−2.41907
$894$	0	0
$895$	519207.	0.0216662
$896$	−802816.	−0.0334077
$897$	0	0
$898$	2.23347e7	0.924249
$899$	−2.18379e7	−0.901181
$900$	0	0
$901$	−5.94727e7	−2.44065
$902$	1.00401e7	0.410888
$903$	0	0
$904$	−4.02149e6	−0.163669
$905$	6.39666e6	0.259616
$906$	0	0
$907$	2.63522e7	1.06365	0.531825	−	0.846854i	$-0.321506\pi$
$907$	2.63522e7	1.06365	0.531825	+	0.846854i	$0.321506\pi$
$908$	3.65958e6	0.147305
$909$	0	0
$910$	1.95659e6	0.0783241
$911$	4.82227e6	0.192511	0.0962555	−	0.995357i	$-0.469313\pi$
$911$	4.82227e6	0.192511	0.0962555	+	0.995357i	$0.469313\pi$
$912$	0	0
$913$	4.05474e7	1.60985
$914$	1.41738e7	0.561203
$915$	0	0
$916$	−1.62874e7	−0.641378
$917$	1.39715e7	0.548680
$918$	0	0
$919$	−121562.	−0.00474798	−0.00237399	−	0.999997i	$-0.500756\pi$
$919$	−121562.	−0.00474798	−0.00237399	+	0.999997i	$0.500756\pi$
$920$	3.70149e6	0.144181
$921$	0	0
$922$	2.35228e7	0.911299
$923$	−662117.	−0.0255818
$924$	0	0
$925$	−4.54128e6	−0.174511
$926$	−2.66018e7	−1.01949
$927$	0	0
$928$	−2.17900e6	−0.0830590
$929$	−1.74856e7	−0.664725	−0.332362	−	0.943152i	$-0.607846\pi$
$929$	−1.74856e7	−0.664725	−0.332362	+	0.943152i	$0.607846\pi$
$930$	0	0
$931$	6.90104e6	0.260940
$932$	3.78736e6	0.142822
$933$	0	0
$934$	1.73897e7	0.652267
$935$	−1.86495e7	−0.697650
$936$	0	0
$937$	−1.97882e7	−0.736304	−0.368152	−	0.929766i	$-0.620009\pi$
$937$	−1.97882e7	−0.736304	−0.368152	+	0.929766i	$0.620009\pi$
$938$	3.43663e6	0.127534
$939$	0	0
$940$	8.02261e6	0.296139
$941$	−2.05948e7	−0.758199	−0.379100	−	0.925356i	$-0.623766\pi$
$941$	−2.05948e7	−0.758199	−0.379100	+	0.925356i	$0.623766\pi$
$942$	0	0
$943$	1.36340e7	0.499282
$944$	1.10575e6	0.0403857
$945$	0	0
$946$	3.43410e7	1.24763
$947$	−6.90803e6	−0.250311	−0.125155	−	0.992137i	$-0.539943\pi$
$947$	−6.90803e6	−0.250311	−0.125155	+	0.992137i	$0.539943\pi$
$948$	0	0
$949$	−3.29302e6	−0.118694
$950$	−7.18558e6	−0.258317
$951$	0	0
$952$	5.49276e6	0.196426
$953$	−3.72040e7	−1.32696	−0.663480	−	0.748194i	$-0.730921\pi$
$953$	−3.72040e7	−1.32696	−0.663480	+	0.748194i	$0.730921\pi$
$954$	0	0
$955$	1.62315e7	0.575905
$956$	−2.78391e7	−0.985169
$957$	0	0
$958$	−3.71872e6	−0.130912
$959$	−1.99420e7	−0.700199
$960$	0	0
$961$	7.66904e7	2.67875
$962$	1.16054e7	0.404318
$963$	0	0
$964$	−5.32545e6	−0.184571
$965$	1.92160e7	0.664272
$966$	0	0
$967$	1.88878e7	0.649555	0.324778	−	0.945790i	$-0.394711\pi$
$967$	1.88878e7	0.649555	0.324778	+	0.945790i	$0.394711\pi$
$968$	−1.30196e6	−0.0446590
$969$	0	0
$970$	6.21324e6	0.212026
$971$	4.20566e7	1.43148	0.715741	−	0.698366i	$-0.246089\pi$
$971$	4.20566e7	1.43148	0.715741	+	0.698366i	$0.246089\pi$
$972$	0	0
$973$	662992.	0.0224505
$974$	2.17955e7	0.736155
$975$	0	0
$976$	−3.13693e6	−0.105409
$977$	−4.61174e7	−1.54571	−0.772855	−	0.634583i	$-0.781172\pi$
$977$	−4.61174e7	−1.54571	−0.772855	+	0.634583i	$0.781172\pi$
$978$	0	0
$979$	−6.15083e6	−0.205105
$980$	−960400.	−0.0319438
$981$	0	0
$982$	−2.68028e6	−0.0886954
$983$	−3.25827e7	−1.07548	−0.537741	−	0.843110i	$-0.680722\pi$
$983$	−3.25827e7	−1.07548	−0.537741	+	0.843110i	$0.680722\pi$
$984$	0	0
$985$	−4.15401e6	−0.136420
$986$	1.49084e7	0.488359
$987$	0	0
$988$	1.83631e7	0.598484
$989$	4.66335e7	1.51603
$990$	0	0
$991$	1.68209e7	0.544084	0.272042	−	0.962285i	$-0.412301\pi$
$991$	1.68209e7	0.544084	0.272042	+	0.962285i	$0.412301\pi$
$992$	1.05088e7	0.339059
$993$	0	0
$994$	325003.	0.0104333
$995$	1.86890e6	0.0598451
$996$	0	0
$997$	7.67044e6	0.244389	0.122195	−	0.992506i	$-0.461007\pi$
$997$	7.67044e6	0.244389	0.122195	+	0.992506i	$0.461007\pi$
$998$	−2.63207e7	−0.836509
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.6.a.s.1.1		2
3.2	odd	2		70.6.a.h.1.1	✓	2
12.11	even	2		560.6.a.k.1.2		2
15.2	even	4		350.6.c.k.99.4		4
15.8	even	4		350.6.c.k.99.1		4
15.14	odd	2		350.6.a.p.1.2		2
21.20	even	2		490.6.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.h.1.1	✓	2	3.2	odd	2
350.6.a.p.1.2		2	15.14	odd	2
350.6.c.k.99.1		4	15.8	even	4
350.6.c.k.99.4		4	15.2	even	4
490.6.a.u.1.2		2	21.20	even	2
560.6.a.k.1.2		2	12.11	even	2
630.6.a.s.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	630.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -25.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +100.000 q^{10} -425.904 q^{11} +399.303 q^{13} -196.000 q^{14} +256.000 q^{16} -1751.52 q^{17} +2874.23 q^{19} -400.000 q^{20} +1703.62 q^{22} +2313.43 q^{23} +625.000 q^{25} -1597.21 q^{26} +784.000 q^{28} +2127.93 q^{29} -10262.5 q^{31} -1024.00 q^{32} +7006.08 q^{34} -1225.00 q^{35} -7266.05 q^{37} -11496.9 q^{38} +1600.00 q^{40} +5893.44 q^{41} +20157.7 q^{43} -6814.46 q^{44} -9253.72 q^{46} -20056.5 q^{47} +2401.00 q^{49} -2500.00 q^{50} +6388.85 q^{52} +33954.9 q^{53} +10647.6 q^{55} -3136.00 q^{56} -8511.71 q^{58} +4319.34 q^{59} -12253.6 q^{61} +41050.1 q^{62} +4096.00 q^{64} -9982.58 q^{65} -17533.8 q^{67} -28024.3 q^{68} +4900.00 q^{70} -1658.18 q^{71} -8246.91 q^{73} +29064.2 q^{74} +45987.7 q^{76} -20869.3 q^{77} -9168.61 q^{79} -6400.00 q^{80} -23573.7 q^{82} -95203.2 q^{83} +43788.0 q^{85} -80631.0 q^{86} +27257.8 q^{88} +14441.8 q^{89} +19565.9 q^{91} +37014.9 q^{92} +80226.1 q^{94} -71855.8 q^{95} +62132.4 q^{97} -9604.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 50 q^{5} + 98 q^{7} - 128 q^{8} + 200 q^{10} - 415 q^{11} + 429 q^{13} - 392 q^{14} + 512 q^{16} - 1319 q^{17} + 1918 q^{19} - 800 q^{20} + 1660 q^{22} + 1334 q^{23} + 1250 q^{25}+ \cdots - 19208 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 50 * q^5 + 98 * q^7 - 128 * q^8 + 200 * q^10 - 415 * q^11 + 429 * q^13 - 392 * q^14 + 512 * q^16 - 1319 * q^17 + 1918 * q^19 - 800 * q^20 + 1660 * q^22 + 1334 * q^23 + 1250 * q^25 - 1716 * q^26 + 1568 * q^28 + 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 5276 * q^34 - 2450 * q^35 - 9156 * q^37 - 7672 * q^38 + 3200 * q^40 + 7822 * q^41 + 2078 * q^43 - 6640 * q^44 - 5336 * q^46 + 8507 * q^47 + 4802 * q^49 - 5000 * q^50 + 6864 * q^52 + 34242 * q^53 + 10375 * q^55 - 6272 * q^56 - 4524 * q^58 - 6952 * q^59 - 45138 * q^61 + 21888 * q^62 + 8192 * q^64 - 10725 * q^65 - 54556 * q^67 - 21104 * q^68 + 9800 * q^70 + 62272 * q^71 + 40896 * q^73 + 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 62069 * q^79 - 12800 * q^80 - 31288 * q^82 - 189600 * q^83 + 32975 * q^85 - 8312 * q^86 + 26560 * q^88 - 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 34028 * q^94 - 47950 * q^95 + 155547 * q^97 - 19208 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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