LMFDB - Embedded newform 630.6.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,6,Mod(1,630)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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{3369}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 842 $ x^2 - x - 842
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$29.5215$ 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)

$f(q)$	$=$	\(q-4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -49.0000 q^{7} -64.0000 q^{8} -100.000 q^{10} -566.565 q^{11} +238.823 q^{13} +196.000 q^{14} +256.000 q^{16} +854.306 q^{17} +1845.13 q^{19} +400.000 q^{20} +2266.26 q^{22} -4128.23 q^{23} +625.000 q^{25} -955.293 q^{26} -784.000 q^{28} +7220.18 q^{29} +4902.39 q^{31} -1024.00 q^{32} -3417.22 q^{34} -1225.00 q^{35} -3401.81 q^{37} -7380.52 q^{38} -1600.00 q^{40} -17843.1 q^{41} -3416.29 q^{43} -9065.03 q^{44} +16512.9 q^{46} +13268.0 q^{47} +2401.00 q^{49} -2500.00 q^{50} +3821.17 q^{52} -23114.4 q^{53} -14164.1 q^{55} +3136.00 q^{56} -28880.7 q^{58} -18129.6 q^{59} -32874.7 q^{61} -19609.6 q^{62} +4096.00 q^{64} +5970.58 q^{65} +66036.8 q^{67} +13668.9 q^{68} +4900.00 q^{70} +10950.5 q^{71} +39440.9 q^{73} +13607.2 q^{74} +29522.1 q^{76} +27761.7 q^{77} +61458.3 q^{79} +6400.00 q^{80} +71372.3 q^{82} -5609.17 q^{83} +21357.7 q^{85} +13665.1 q^{86} +36260.1 q^{88} -126028. q^{89} -11702.3 q^{91} -66051.7 q^{92} -53072.1 q^{94} +46128.2 q^{95} -138769. q^{97} -9604.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 50 q^{5} - 98 q^{7} - 128 q^{8} - 200 q^{10} - 959 q^{11} - 393 q^{13} + 392 q^{14} + 512 q^{16} + 2231 q^{17} + 3342 q^{19} + 800 q^{20} + 3836 q^{22} + 450 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 - 959 * q^11 - 393 * q^13 + 392 * q^14 + 512 * q^16 + 2231 * q^17 + 3342 * q^19 + 800 * q^20 + 3836 * q^22 + 450 * q^23 + 1250 * q^25 + 1572 * q^26 - 1568 * q^28 + 4515 * q^29 - 2036 * q^31 - 2048 * q^32 - 8924 * q^34 - 2450 * q^35 - 1928 * q^37 - 13368 * q^38 - 3200 * q^40 - 19318 * q^41 - 14146 * q^43 - 15344 * q^44 - 1800 * q^46 + 19745 * q^47 + 4802 * q^49 - 5000 * q^50 - 6288 * q^52 - 26378 * q^53 - 23975 * q^55 + 6272 * q^56 - 18060 * q^58 + 11104 * q^59 + 3554 * q^61 + 8144 * q^62 + 8192 * q^64 - 9825 * q^65 + 65208 * q^67 + 35696 * q^68 + 9800 * q^70 - 17104 * q^71 - 36740 * q^73 + 7712 * q^74 + 53472 * q^76 + 46991 * q^77 + 72245 * q^79 + 12800 * q^80 + 77272 * q^82 - 99676 * q^83 + 55775 * q^85 + 56584 * q^86 + 61376 * q^88 - 82454 * q^89 + 19257 * q^91 + 7200 * q^92 - 78980 * q^94 + 83550 * q^95 - 104627 * 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
$11$	−566.565	−1.41178	−0.705891	−	0.708320i	$-0.749453\pi$
$11$	−566.565	−1.41178	−0.705891	+	0.708320i	$0.749453\pi$
$12$	0	0
$13$	238.823	0.391939	0.195969	−	0.980610i	$-0.437215\pi$
$13$	238.823	0.391939	0.195969	+	0.980610i	$0.437215\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	854.306	0.716954	0.358477	−	0.933539i	$-0.383296\pi$
$17$	854.306	0.716954	0.358477	+	0.933539i	$0.383296\pi$
$18$	0	0
$19$	1845.13	1.17258	0.586290	−	0.810101i	$-0.300588\pi$
$19$	1845.13	1.17258	0.586290	+	0.810101i	$0.300588\pi$
$20$	400.000	0.223607
$21$	0	0
$22$	2266.26	0.998281
$23$	−4128.23	−1.62721	−0.813607	−	0.581416i	$-0.802499\pi$
$23$	−4128.23	−1.62721	−0.813607	+	0.581416i	$0.802499\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−955.293	−0.277142
$27$	0	0
$28$	−784.000	−0.188982
$29$	7220.18	1.59424	0.797119	−	0.603822i	$-0.206356\pi$
$29$	7220.18	1.59424	0.797119	+	0.603822i	$0.206356\pi$
$30$	0	0
$31$	4902.39	0.916229	0.458115	−	0.888893i	$-0.348525\pi$
$31$	4902.39	0.916229	0.458115	+	0.888893i	$0.348525\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−3417.22	−0.506963
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−3401.81	−0.408513	−0.204256	−	0.978917i	$-0.565478\pi$
$37$	−3401.81	−0.408513	−0.204256	+	0.978917i	$0.565478\pi$
$38$	−7380.52	−0.829140
$39$	0	0
$40$	−1600.00	−0.158114
$41$	−17843.1	−1.65772	−0.828858	−	0.559459i	$-0.811009\pi$
$41$	−17843.1	−1.65772	−0.828858	+	0.559459i	$0.811009\pi$
$42$	0	0
$43$	−3416.29	−0.281762	−0.140881	−	0.990027i	$-0.544994\pi$
$43$	−3416.29	−0.281762	−0.140881	+	0.990027i	$0.544994\pi$
$44$	−9065.03	−0.705891
$45$	0	0
$46$	16512.9	1.15061
$47$	13268.0	0.876116	0.438058	−	0.898947i	$-0.355666\pi$
$47$	13268.0	0.876116	0.438058	+	0.898947i	$0.355666\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−2500.00	−0.141421
$51$	0	0
$52$	3821.17	0.195969
$53$	−23114.4	−1.13030	−0.565148	−	0.824989i	$-0.691181\pi$
$53$	−23114.4	−1.13030	−0.565148	+	0.824989i	$0.691181\pi$
$54$	0	0
$55$	−14164.1	−0.631368
$56$	3136.00	0.133631
$57$	0	0
$58$	−28880.7	−1.12730
$59$	−18129.6	−0.678044	−0.339022	−	0.940778i	$-0.610096\pi$
$59$	−18129.6	−0.678044	−0.339022	+	0.940778i	$0.610096\pi$
$60$	0	0
$61$	−32874.7	−1.13120	−0.565598	−	0.824681i	$-0.691354\pi$
$61$	−32874.7	−1.13120	−0.565598	+	0.824681i	$0.691354\pi$
$62$	−19609.6	−0.647872
$63$	0	0
$64$	4096.00	0.125000
$65$	5970.58	0.175280
$66$	0	0
$67$	66036.8	1.79721	0.898606	−	0.438757i	$-0.144581\pi$
$67$	66036.8	1.79721	0.898606	+	0.438757i	$0.144581\pi$
$68$	13668.9	0.358477
$69$	0	0
$70$	4900.00	0.119523
$71$	10950.5	0.257803	0.128901	−	0.991657i	$-0.458855\pi$
$71$	10950.5	0.257803	0.128901	+	0.991657i	$0.458855\pi$
$72$	0	0
$73$	39440.9	0.866243	0.433122	−	0.901336i	$-0.357412\pi$
$73$	39440.9	0.866243	0.433122	+	0.901336i	$0.357412\pi$
$74$	13607.2	0.288862
$75$	0	0
$76$	29522.1	0.586290
$77$	27761.7	0.533604
$78$	0	0
$79$	61458.3	1.10793	0.553966	−	0.832539i	$-0.313114\pi$
$79$	61458.3	1.10793	0.553966	+	0.832539i	$0.313114\pi$
$80$	6400.00	0.111803
$81$	0	0
$82$	71372.3	1.17218
$83$	−5609.17	−0.0893723	−0.0446862	−	0.999001i	$-0.514229\pi$
$83$	−5609.17	−0.0893723	−0.0446862	+	0.999001i	$0.514229\pi$
$84$	0	0
$85$	21357.7	0.320632
$86$	13665.1	0.199236
$87$	0	0
$88$	36260.1	0.499140
$89$	−126028.	−1.68652	−0.843261	−	0.537505i	$-0.819367\pi$
$89$	−126028.	−1.68652	−0.843261	+	0.537505i	$0.819367\pi$
$90$	0	0
$91$	−11702.3	−0.148139
$92$	−66051.7	−0.813607
$93$	0	0
$94$	−53072.1	−0.619507
$95$	46128.2	0.524394
$96$	0	0
$97$	−138769.	−1.49748	−0.748742	−	0.662862i	$-0.769342\pi$
$97$	−138769.	−1.49748	−0.748742	+	0.662862i	$0.769342\pi$
$98$	−9604.00	−0.101015
$99$	0	0
$100$	10000.0	0.100000
$101$	−63472.7	−0.619133	−0.309566	−	0.950878i	$-0.600184\pi$
$101$	−63472.7	−0.619133	−0.309566	+	0.950878i	$0.600184\pi$
$102$	0	0
$103$	11842.2	0.109986	0.0549931	−	0.998487i	$-0.482486\pi$
$103$	11842.2	0.109986	0.0549931	+	0.998487i	$0.482486\pi$
$104$	−15284.7	−0.138571
$105$	0	0
$106$	92457.5	0.799240
$107$	140875.	1.18953	0.594765	−	0.803900i	$-0.297245\pi$
$107$	140875.	1.18953	0.594765	+	0.803900i	$0.297245\pi$
$108$	0	0
$109$	−92044.5	−0.742047	−0.371024	−	0.928623i	$-0.620993\pi$
$109$	−92044.5	−0.742047	−0.371024	+	0.928623i	$0.620993\pi$
$110$	56656.5	0.446445
$111$	0	0
$112$	−12544.0	−0.0944911
$113$	−10705.3	−0.0788685	−0.0394343	−	0.999222i	$-0.512556\pi$
$113$	−10705.3	−0.0788685	−0.0394343	+	0.999222i	$0.512556\pi$
$114$	0	0
$115$	−103206.	−0.727712
$116$	115523.	0.797119
$117$	0	0
$118$	72518.3	0.479449
$119$	−41861.0	−0.270983
$120$	0	0
$121$	159944.	0.993129
$122$	131499.	0.799876
$123$	0	0
$124$	78438.3	0.458115
$125$	15625.0	0.0894427
$126$	0	0
$127$	−37867.2	−0.208331	−0.104165	−	0.994560i	$-0.533217\pi$
$127$	−37867.2	−0.208331	−0.104165	+	0.994560i	$0.533217\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−23882.3	−0.123942
$131$	−21844.3	−0.111214	−0.0556070	−	0.998453i	$-0.517709\pi$
$131$	−21844.3	−0.111214	−0.0556070	+	0.998453i	$0.517709\pi$
$132$	0	0
$133$	−90411.3	−0.443194
$134$	−264147.	−1.27082
$135$	0	0
$136$	−54675.6	−0.253481
$137$	−279896.	−1.27408	−0.637038	−	0.770832i	$-0.719841\pi$
$137$	−279896.	−1.27408	−0.637038	+	0.770832i	$0.719841\pi$
$138$	0	0
$139$	22107.7	0.0970527	0.0485263	−	0.998822i	$-0.484548\pi$
$139$	22107.7	0.0970527	0.0485263	+	0.998822i	$0.484548\pi$
$140$	−19600.0	−0.0845154
$141$	0	0
$142$	−43801.9	−0.182294
$143$	−135309.	−0.553332
$144$	0	0
$145$	180505.	0.712965
$146$	−157764.	−0.612526
$147$	0	0
$148$	−54429.0	−0.204256
$149$	515664.	1.90284	0.951418	−	0.307903i	$-0.0996272\pi$
$149$	515664.	1.90284	0.951418	+	0.307903i	$0.0996272\pi$
$150$	0	0
$151$	92725.5	0.330946	0.165473	−	0.986214i	$-0.447085\pi$
$151$	92725.5	0.330946	0.165473	+	0.986214i	$0.447085\pi$
$152$	−118088.	−0.414570
$153$	0	0
$154$	−111047.	−0.377315
$155$	122560.	0.409750
$156$	0	0
$157$	−420486.	−1.36145	−0.680726	−	0.732538i	$-0.738335\pi$
$157$	−420486.	−1.36145	−0.680726	+	0.732538i	$0.738335\pi$
$158$	−245833.	−0.783426
$159$	0	0
$160$	−25600.0	−0.0790569
$161$	202283.	0.615029
$162$	0	0
$163$	−473380.	−1.39554	−0.697768	−	0.716324i	$-0.745823\pi$
$163$	−473380.	−1.39554	−0.697768	+	0.716324i	$0.745823\pi$
$164$	−285489.	−0.828858
$165$	0	0
$166$	22436.7	0.0631958
$167$	99103.8	0.274979	0.137489	−	0.990503i	$-0.456097\pi$
$167$	99103.8	0.274979	0.137489	+	0.990503i	$0.456097\pi$
$168$	0	0
$169$	−314257.	−0.846384
$170$	−85430.6	−0.226721
$171$	0	0
$172$	−54660.6	−0.140881
$173$	−491611.	−1.24884	−0.624420	−	0.781089i	$-0.714665\pi$
$173$	−491611.	−1.24884	−0.624420	+	0.781089i	$0.714665\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	−145041.	−0.352946
$177$	0	0
$178$	504112.	1.19255
$179$	495024.	1.15476	0.577382	−	0.816474i	$-0.304074\pi$
$179$	495024.	1.15476	0.577382	+	0.816474i	$0.304074\pi$
$180$	0	0
$181$	−220301.	−0.499828	−0.249914	−	0.968268i	$-0.580402\pi$
$181$	−220301.	−0.499828	−0.249914	+	0.968268i	$0.580402\pi$
$182$	46809.3	0.104750
$183$	0	0
$184$	264207.	0.575307
$185$	−85045.2	−0.182692
$186$	0	0
$187$	−484020.	−1.01218
$188$	212288.	0.438058
$189$	0	0
$190$	−184513.	−0.370803
$191$	−926661.	−1.83797	−0.918983	−	0.394296i	$-0.870988\pi$
$191$	−926661.	−1.83797	−0.918983	+	0.394296i	$0.870988\pi$
$192$	0	0
$193$	−727645.	−1.40613	−0.703066	−	0.711125i	$-0.748186\pi$
$193$	−727645.	−1.40613	−0.703066	+	0.711125i	$0.748186\pi$
$194$	555075.	1.05888
$195$	0	0
$196$	38416.0	0.0714286
$197$	−294843.	−0.541284	−0.270642	−	0.962680i	$-0.587236\pi$
$197$	−294843.	−0.541284	−0.270642	+	0.962680i	$0.587236\pi$
$198$	0	0
$199$	449223.	0.804136	0.402068	−	0.915610i	$-0.368292\pi$
$199$	449223.	0.804136	0.402068	+	0.915610i	$0.368292\pi$
$200$	−40000.0	−0.0707107
$201$	0	0
$202$	253891.	0.437793
$203$	−353789.	−0.602565
$204$	0	0
$205$	−446077.	−0.741353
$206$	−47368.6	−0.0777719
$207$	0	0
$208$	61138.7	0.0979847
$209$	−1.04538e6	−1.65543
$210$	0	0
$211$	−45152.5	−0.0698193	−0.0349096	−	0.999390i	$-0.511114\pi$
$211$	−45152.5	−0.0698193	−0.0349096	+	0.999390i	$0.511114\pi$
$212$	−369830.	−0.565148
$213$	0	0
$214$	−563501.	−0.841125
$215$	−85407.1	−0.126008
$216$	0	0
$217$	−240217.	−0.346302
$218$	368178.	0.524707
$219$	0	0
$220$	−226626.	−0.315684
$221$	204028.	0.281002
$222$	0	0
$223$	−1.10352e6	−1.48600	−0.743001	−	0.669291i	$-0.766598\pi$
$223$	−1.10352e6	−1.48600	−0.743001	+	0.669291i	$0.766598\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	42821.3	0.0557685
$227$	−108706.	−0.140020	−0.0700101	−	0.997546i	$-0.522303\pi$
$227$	−108706.	−0.140020	−0.0700101	+	0.997546i	$0.522303\pi$
$228$	0	0
$229$	−936787.	−1.18046	−0.590231	−	0.807234i	$-0.700963\pi$
$229$	−936787.	−1.18046	−0.590231	+	0.807234i	$0.700963\pi$
$230$	412823.	0.514570
$231$	0	0
$232$	−462092.	−0.563648
$233$	−1.03380e6	−1.24752	−0.623760	−	0.781616i	$-0.714396\pi$
$233$	−1.03380e6	−1.24752	−0.623760	+	0.781616i	$0.714396\pi$
$234$	0	0
$235$	331701.	0.391811
$236$	−290073.	−0.339022
$237$	0	0
$238$	167444.	0.191614
$239$	−277682.	−0.314451	−0.157226	−	0.987563i	$-0.550255\pi$
$239$	−277682.	−0.314451	−0.157226	+	0.987563i	$0.550255\pi$
$240$	0	0
$241$	48925.2	0.0542613	0.0271307	−	0.999632i	$-0.491363\pi$
$241$	48925.2	0.0542613	0.0271307	+	0.999632i	$0.491363\pi$
$242$	−639778.	−0.702249
$243$	0	0
$244$	−525996.	−0.565598
$245$	60025.0	0.0638877
$246$	0	0
$247$	440660.	0.459580
$248$	−313753.	−0.323936
$249$	0	0
$250$	−62500.0	−0.0632456
$251$	404400.	0.405161	0.202580	−	0.979266i	$-0.435067\pi$
$251$	404400.	0.405161	0.202580	+	0.979266i	$0.435067\pi$
$252$	0	0
$253$	2.33891e6	2.29727
$254$	151469.	0.147312
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.98063e6	−1.87055	−0.935277	−	0.353918i	$-0.884849\pi$
$257$	−1.98063e6	−1.87055	−0.935277	+	0.353918i	$0.884849\pi$
$258$	0	0
$259$	166689.	0.154403
$260$	95529.3	0.0876401
$261$	0	0
$262$	87377.1	0.0786401
$263$	−1.22576e6	−1.09274	−0.546371	−	0.837544i	$-0.683991\pi$
$263$	−1.22576e6	−1.09274	−0.546371	+	0.837544i	$0.683991\pi$
$264$	0	0
$265$	−577859.	−0.505484
$266$	361645.	0.313385
$267$	0	0
$268$	1.05659e6	0.898606
$269$	975012.	0.821541	0.410770	−	0.911739i	$-0.365260\pi$
$269$	975012.	0.821541	0.410770	+	0.911739i	$0.365260\pi$
$270$	0	0
$271$	1.14855e6	0.950007	0.475003	−	0.879984i	$-0.342447\pi$
$271$	1.14855e6	0.950007	0.475003	+	0.879984i	$0.342447\pi$
$272$	218702.	0.179238
$273$	0	0
$274$	1.11958e6	0.900908
$275$	−354103.	−0.282356
$276$	0	0
$277$	−1.13962e6	−0.892405	−0.446203	−	0.894932i	$-0.647224\pi$
$277$	−1.13962e6	−0.892405	−0.446203	+	0.894932i	$0.647224\pi$
$278$	−88431.0	−0.0686266
$279$	0	0
$280$	78400.0	0.0597614
$281$	1.09886e6	0.830185	0.415092	−	0.909779i	$-0.363749\pi$
$281$	1.09886e6	0.830185	0.415092	+	0.909779i	$0.363749\pi$
$282$	0	0
$283$	1.24865e6	0.926779	0.463389	−	0.886155i	$-0.346633\pi$
$283$	1.24865e6	0.926779	0.463389	+	0.886155i	$0.346633\pi$
$284$	175208.	0.128901
$285$	0	0
$286$	541235.	0.391265
$287$	874311.	0.626558
$288$	0	0
$289$	−690018.	−0.485977
$290$	−722018.	−0.504142
$291$	0	0
$292$	631055.	0.433122
$293$	−2.34742e6	−1.59743	−0.798713	−	0.601712i	$-0.794486\pi$
$293$	−2.34742e6	−1.59743	−0.798713	+	0.601712i	$0.794486\pi$
$294$	0	0
$295$	−453239.	−0.303230
$296$	217716.	0.144431
$297$	0	0
$298$	−2.06266e6	−1.34551
$299$	−985917.	−0.637768
$300$	0	0
$301$	167398.	0.106496
$302$	−370902.	−0.234014
$303$	0	0
$304$	472353.	0.293145
$305$	−821868.	−0.505886
$306$	0	0
$307$	−910969.	−0.551642	−0.275821	−	0.961209i	$-0.588950\pi$
$307$	−910969.	−0.551642	−0.275821	+	0.961209i	$0.588950\pi$
$308$	444187.	0.266802
$309$	0	0
$310$	−490239.	−0.289737
$311$	−343622.	−0.201456	−0.100728	−	0.994914i	$-0.532117\pi$
$311$	−343622.	−0.201456	−0.100728	+	0.994914i	$0.532117\pi$
$312$	0	0
$313$	1.17662e6	0.678852	0.339426	−	0.940633i	$-0.389767\pi$
$313$	1.17662e6	0.678852	0.339426	+	0.940633i	$0.389767\pi$
$314$	1.68194e6	0.962691
$315$	0	0
$316$	983333.	0.553966
$317$	−37824.3	−0.0211408	−0.0105704	−	0.999944i	$-0.503365\pi$
$317$	−37824.3	−0.0211408	−0.0105704	+	0.999944i	$0.503365\pi$
$318$	0	0
$319$	−4.09070e6	−2.25072
$320$	102400.	0.0559017
$321$	0	0
$322$	−809133.	−0.434891
$323$	1.57631e6	0.840686
$324$	0	0
$325$	149264.	0.0783877
$326$	1.89352e6	0.986793
$327$	0	0
$328$	1.14196e6	0.586091
$329$	−650133.	−0.331141
$330$	0	0
$331$	−887735.	−0.445362	−0.222681	−	0.974891i	$-0.571481\pi$
$331$	−887735.	−0.445362	−0.222681	+	0.974891i	$0.571481\pi$
$332$	−89746.7	−0.0446862
$333$	0	0
$334$	−396415.	−0.194439
$335$	1.65092e6	0.803737
$336$	0	0
$337$	1.88882e6	0.905972	0.452986	−	0.891518i	$-0.350359\pi$
$337$	1.88882e6	0.905972	0.452986	+	0.891518i	$0.350359\pi$
$338$	1.25703e6	0.598484
$339$	0	0
$340$	341722.	0.160316
$341$	−2.77752e6	−1.29352
$342$	0	0
$343$	−117649.	−0.0539949
$344$	218642.	0.0996180
$345$	0	0
$346$	1.96645e6	0.883063
$347$	−1.45856e6	−0.650280	−0.325140	−	0.945666i	$-0.605412\pi$
$347$	−1.45856e6	−0.650280	−0.325140	+	0.945666i	$0.605412\pi$
$348$	0	0
$349$	2.10369e6	0.924522	0.462261	−	0.886744i	$-0.347038\pi$
$349$	2.10369e6	0.924522	0.462261	+	0.886744i	$0.347038\pi$
$350$	122500.	0.0534522
$351$	0	0
$352$	580162.	0.249570
$353$	−2.06037e6	−0.880053	−0.440026	−	0.897985i	$-0.645031\pi$
$353$	−2.06037e6	−0.880053	−0.440026	+	0.897985i	$0.645031\pi$
$354$	0	0
$355$	273762.	0.115293
$356$	−2.01645e6	−0.843261
$357$	0	0
$358$	−1.98009e6	−0.816542
$359$	−4.34879e6	−1.78087	−0.890434	−	0.455112i	$-0.849599\pi$
$359$	−4.34879e6	−1.78087	−0.890434	+	0.455112i	$0.849599\pi$
$360$	0	0
$361$	928403.	0.374946
$362$	881204.	0.353431
$363$	0	0
$364$	−187237.	−0.0740694
$365$	986023.	0.387396
$366$	0	0
$367$	−1.04777e6	−0.406072	−0.203036	−	0.979171i	$-0.565081\pi$
$367$	−1.04777e6	−0.406072	−0.203036	+	0.979171i	$0.565081\pi$
$368$	−1.05683e6	−0.406803
$369$	0	0
$370$	340181.	0.129183
$371$	1.13260e6	0.427212
$372$	0	0
$373$	608034.	0.226285	0.113143	−	0.993579i	$-0.463908\pi$
$373$	608034.	0.226285	0.113143	+	0.993579i	$0.463908\pi$
$374$	1.93608e6	0.715721
$375$	0	0
$376$	−849153.	−0.309754
$377$	1.72435e6	0.624844
$378$	0	0
$379$	−3.95319e6	−1.41367	−0.706837	−	0.707376i	$-0.749879\pi$
$379$	−3.95319e6	−1.41367	−0.706837	+	0.707376i	$0.749879\pi$
$380$	738052.	0.262197
$381$	0	0
$382$	3.70665e6	1.29964
$383$	3.63893e6	1.26758	0.633792	−	0.773503i	$-0.281497\pi$
$383$	3.63893e6	1.26758	0.633792	+	0.773503i	$0.281497\pi$
$384$	0	0
$385$	694042.	0.238635
$386$	2.91058e6	0.994285
$387$	0	0
$388$	−2.22030e6	−0.748742
$389$	4.41302e6	1.47864	0.739319	−	0.673355i	$-0.235147\pi$
$389$	4.41302e6	1.47864	0.739319	+	0.673355i	$0.235147\pi$
$390$	0	0
$391$	−3.52677e6	−1.16664
$392$	−153664.	−0.0505076
$393$	0	0
$394$	1.17937e6	0.382746
$395$	1.53646e6	0.495482
$396$	0	0
$397$	700578.	0.223090	0.111545	−	0.993759i	$-0.464420\pi$
$397$	700578.	0.223090	0.111545	+	0.993759i	$0.464420\pi$
$398$	−1.79689e6	−0.568610
$399$	0	0
$400$	160000.	0.0500000
$401$	4.39067e6	1.36355	0.681774	−	0.731563i	$-0.261209\pi$
$401$	4.39067e6	1.36355	0.681774	+	0.731563i	$0.261209\pi$
$402$	0	0
$403$	1.17081e6	0.359106
$404$	−1.01556e6	−0.309566
$405$	0	0
$406$	1.41516e6	0.426078
$407$	1.92735e6	0.576731
$408$	0	0
$409$	2.81587e6	0.832347	0.416173	−	0.909285i	$-0.363371\pi$
$409$	2.81587e6	0.832347	0.416173	+	0.909285i	$0.363371\pi$
$410$	1.78431e6	0.524216
$411$	0	0
$412$	189475.	0.0549931
$413$	888349.	0.256276
$414$	0	0
$415$	−140229.	−0.0399685
$416$	−244555.	−0.0692856
$417$	0	0
$418$	4.18154e6	1.17057
$419$	6.90421e6	1.92123	0.960614	−	0.277887i	$-0.0896342\pi$
$419$	6.90421e6	1.92123	0.960614	+	0.277887i	$0.0896342\pi$
$420$	0	0
$421$	−3.17289e6	−0.872468	−0.436234	−	0.899833i	$-0.643688\pi$
$421$	−3.17289e6	−0.872468	−0.436234	+	0.899833i	$0.643688\pi$
$422$	180610.	0.0493697
$423$	0	0
$424$	1.47932e6	0.399620
$425$	533941.	0.143391
$426$	0	0
$427$	1.61086e6	0.427552
$428$	2.25400e6	0.594765
$429$	0	0
$430$	341629.	0.0891011
$431$	3.06122e6	0.793783	0.396891	−	0.917866i	$-0.370089\pi$
$431$	3.06122e6	0.793783	0.396891	+	0.917866i	$0.370089\pi$
$432$	0	0
$433$	−2.16725e6	−0.555508	−0.277754	−	0.960652i	$-0.589590\pi$
$433$	−2.16725e6	−0.555508	−0.277754	+	0.960652i	$0.589590\pi$
$434$	960869.	0.244873
$435$	0	0
$436$	−1.47271e6	−0.371024
$437$	−7.61712e6	−1.90804
$438$	0	0
$439$	−4.15793e6	−1.02971	−0.514856	−	0.857277i	$-0.672155\pi$
$439$	−4.15793e6	−1.02971	−0.514856	+	0.857277i	$0.672155\pi$
$440$	906503.	0.223222
$441$	0	0
$442$	−816112.	−0.198698
$443$	−4.51661e6	−1.09346	−0.546730	−	0.837309i	$-0.684128\pi$
$443$	−4.51661e6	−1.09346	−0.546730	+	0.837309i	$0.684128\pi$
$444$	0	0
$445$	−3.15070e6	−0.754235
$446$	4.41409e6	1.05076
$447$	0	0
$448$	−200704.	−0.0472456
$449$	5.48241e6	1.28338	0.641691	−	0.766963i	$-0.278233\pi$
$449$	5.48241e6	1.28338	0.641691	+	0.766963i	$0.278233\pi$
$450$	0	0
$451$	1.01093e7	2.34033
$452$	−171285.	−0.0394343
$453$	0	0
$454$	434826.	0.0990092
$455$	−292558.	−0.0662497
$456$	0	0
$457$	−940064.	−0.210556	−0.105278	−	0.994443i	$-0.533573\pi$
$457$	−940064.	−0.210556	−0.105278	+	0.994443i	$0.533573\pi$
$458$	3.74715e6	0.834713
$459$	0	0
$460$	−1.65129e6	−0.363856
$461$	−1.16158e6	−0.254563	−0.127282	−	0.991867i	$-0.540625\pi$
$461$	−1.16158e6	−0.254563	−0.127282	+	0.991867i	$0.540625\pi$
$462$	0	0
$463$	3.77402e6	0.818185	0.409092	−	0.912493i	$-0.365845\pi$
$463$	3.77402e6	0.818185	0.409092	+	0.912493i	$0.365845\pi$
$464$	1.84837e6	0.398560
$465$	0	0
$466$	4.13520e6	0.882129
$467$	−4.28176e6	−0.908510	−0.454255	−	0.890872i	$-0.650094\pi$
$467$	−4.28176e6	−0.908510	−0.454255	+	0.890872i	$0.650094\pi$
$468$	0	0
$469$	−3.23580e6	−0.679282
$470$	−1.32680e6	−0.277052
$471$	0	0
$472$	1.16029e6	0.239725
$473$	1.93555e6	0.397787
$474$	0	0
$475$	1.15321e6	0.234516
$476$	−669776.	−0.135492
$477$	0	0
$478$	1.11073e6	0.222351
$479$	2.56685e6	0.511166	0.255583	−	0.966787i	$-0.417733\pi$
$479$	2.56685e6	0.511166	0.255583	+	0.966787i	$0.417733\pi$
$480$	0	0
$481$	−812431.	−0.160112
$482$	−195701.	−0.0383685
$483$	0	0
$484$	2.55911e6	0.496565
$485$	−3.46922e6	−0.669695
$486$	0	0
$487$	−4.25380e6	−0.812746	−0.406373	−	0.913707i	$-0.633207\pi$
$487$	−4.25380e6	−0.812746	−0.406373	+	0.913707i	$0.633207\pi$
$488$	2.10398e6	0.399938
$489$	0	0
$490$	−240100.	−0.0451754
$491$	1.94579e6	0.364244	0.182122	−	0.983276i	$-0.441703\pi$
$491$	1.94579e6	0.364244	0.182122	+	0.983276i	$0.441703\pi$
$492$	0	0
$493$	6.16825e6	1.14300
$494$	−1.76264e6	−0.324972
$495$	0	0
$496$	1.25501e6	0.229057
$497$	−536573.	−0.0974402
$498$	0	0
$499$	−1.88194e6	−0.338341	−0.169170	−	0.985587i	$-0.554109\pi$
$499$	−1.88194e6	−0.338341	−0.169170	+	0.985587i	$0.554109\pi$
$500$	250000.	0.0447214
$501$	0	0
$502$	−1.61760e6	−0.286492
$503$	5.19370e6	0.915287	0.457643	−	0.889136i	$-0.348694\pi$
$503$	5.19370e6	0.915287	0.457643	+	0.889136i	$0.348694\pi$
$504$	0	0
$505$	−1.58682e6	−0.276885
$506$	−9.35564e6	−1.62442
$507$	0	0
$508$	−605875.	−0.104165
$509$	−4.04678e6	−0.692334	−0.346167	−	0.938173i	$-0.612517\pi$
$509$	−4.04678e6	−0.692334	−0.346167	+	0.938173i	$0.612517\pi$
$510$	0	0
$511$	−1.93260e6	−0.327409
$512$	−262144.	−0.0441942
$513$	0	0
$514$	7.92251e6	1.32268
$515$	296054.	0.0491873
$516$	0	0
$517$	−7.51719e6	−1.23688
$518$	−666755.	−0.109180
$519$	0	0
$520$	−382117.	−0.0619709
$521$	6.27733e6	1.01317	0.506583	−	0.862191i	$-0.330908\pi$
$521$	6.27733e6	1.01317	0.506583	+	0.862191i	$0.330908\pi$
$522$	0	0
$523$	−654780.	−0.104675	−0.0523373	−	0.998629i	$-0.516667\pi$
$523$	−654780.	−0.104675	−0.0523373	+	0.998629i	$0.516667\pi$
$524$	−349508.	−0.0556070
$525$	0	0
$526$	4.90305e6	0.772685
$527$	4.18815e6	0.656894
$528$	0	0
$529$	1.06060e7	1.64782
$530$	2.31144e6	0.357431
$531$	0	0
$532$	−1.44658e6	−0.221597
$533$	−4.26134e6	−0.649723
$534$	0	0
$535$	3.52188e6	0.531974
$536$	−4.22636e6	−0.635410
$537$	0	0
$538$	−3.90005e6	−0.580917
$539$	−1.36032e6	−0.201683
$540$	0	0
$541$	6.93592e6	1.01885	0.509426	−	0.860515i	$-0.329858\pi$
$541$	6.93592e6	1.01885	0.509426	+	0.860515i	$0.329858\pi$
$542$	−4.59420e6	−0.671756
$543$	0	0
$544$	−874809.	−0.126741
$545$	−2.30111e6	−0.331854
$546$	0	0
$547$	−5.90397e6	−0.843676	−0.421838	−	0.906671i	$-0.638615\pi$
$547$	−5.90397e6	−0.843676	−0.421838	+	0.906671i	$0.638615\pi$
$548$	−4.47834e6	−0.637038
$549$	0	0
$550$	1.41641e6	0.199656
$551$	1.33222e7	1.86937
$552$	0	0
$553$	−3.01146e6	−0.418759
$554$	4.55850e6	0.631026
$555$	0	0
$556$	353724.	0.0485263
$557$	7.75606e6	1.05926	0.529631	−	0.848228i	$-0.322330\pi$
$557$	7.75606e6	1.05926	0.529631	+	0.848228i	$0.322330\pi$
$558$	0	0
$559$	−815888.	−0.110434
$560$	−313600.	−0.0422577
$561$	0	0
$562$	−4.39542e6	−0.587029
$563$	3.99224e6	0.530818	0.265409	−	0.964136i	$-0.414493\pi$
$563$	3.99224e6	0.530818	0.265409	+	0.964136i	$0.414493\pi$
$564$	0	0
$565$	−267633.	−0.0352711
$566$	−4.99462e6	−0.655331
$567$	0	0
$568$	−700831.	−0.0911470
$569$	−9.00859e6	−1.16648	−0.583238	−	0.812301i	$-0.698215\pi$
$569$	−9.00859e6	−1.16648	−0.583238	+	0.812301i	$0.698215\pi$
$570$	0	0
$571$	5.46033e6	0.700856	0.350428	−	0.936590i	$-0.386036\pi$
$571$	5.46033e6	0.700856	0.350428	+	0.936590i	$0.386036\pi$
$572$	−2.16494e6	−0.276666
$573$	0	0
$574$	−3.49724e6	−0.443043
$575$	−2.58014e6	−0.325443
$576$	0	0
$577$	−3.09221e6	−0.386660	−0.193330	−	0.981134i	$-0.561929\pi$
$577$	−3.09221e6	−0.386660	−0.193330	+	0.981134i	$0.561929\pi$
$578$	2.76007e6	0.343638
$579$	0	0
$580$	2.88807e6	0.356483
$581$	274849.	0.0337796
$582$	0	0
$583$	1.30958e7	1.59573
$584$	−2.52422e6	−0.306263
$585$	0	0
$586$	9.38966e6	1.12955
$587$	−1.20567e7	−1.44423	−0.722113	−	0.691776i	$-0.756829\pi$
$587$	−1.20567e7	−1.44423	−0.722113	+	0.691776i	$0.756829\pi$
$588$	0	0
$589$	9.04555e6	1.07435
$590$	1.81296e6	0.214416
$591$	0	0
$592$	−870863.	−0.102128
$593$	−1.93528e6	−0.225999	−0.113000	−	0.993595i	$-0.536046\pi$
$593$	−1.93528e6	−0.225999	−0.113000	+	0.993595i	$0.536046\pi$
$594$	0	0
$595$	−1.04652e6	−0.121187
$596$	8.25063e6	0.951418
$597$	0	0
$598$	3.94367e6	0.450970
$599$	1.72671e7	1.96631	0.983157	−	0.182762i	$-0.0585037\pi$
$599$	1.72671e7	1.96631	0.983157	+	0.182762i	$0.0585037\pi$
$600$	0	0
$601$	−1.65345e7	−1.86726	−0.933632	−	0.358234i	$-0.883379\pi$
$601$	−1.65345e7	−1.86726	−0.933632	+	0.358234i	$0.883379\pi$
$602$	−669592.	−0.0753042
$603$	0	0
$604$	1.48361e6	0.165473
$605$	3.99861e6	0.444141
$606$	0	0
$607$	−1.09509e7	−1.20636	−0.603179	−	0.797606i	$-0.706100\pi$
$607$	−1.09509e7	−1.20636	−0.603179	+	0.797606i	$0.706100\pi$
$608$	−1.88941e6	−0.207285
$609$	0	0
$610$	3.28747e6	0.357715
$611$	3.16871e6	0.343384
$612$	0	0
$613$	−1.61328e6	−0.173404	−0.0867018	−	0.996234i	$-0.527633\pi$
$613$	−1.61328e6	−0.173404	−0.0867018	+	0.996234i	$0.527633\pi$
$614$	3.64387e6	0.390070
$615$	0	0
$616$	−1.77675e6	−0.188657
$617$	−7.38770e6	−0.781262	−0.390631	−	0.920547i	$-0.627743\pi$
$617$	−7.38770e6	−0.781262	−0.390631	+	0.920547i	$0.627743\pi$
$618$	0	0
$619$	−1.19703e6	−0.125568	−0.0627838	−	0.998027i	$-0.519998\pi$
$619$	−1.19703e6	−0.125568	−0.0627838	+	0.998027i	$0.519998\pi$
$620$	1.96096e6	0.204875
$621$	0	0
$622$	1.37449e6	0.142451
$623$	6.17537e6	0.637445
$624$	0	0
$625$	390625.	0.0400000
$626$	−4.70648e6	−0.480021
$627$	0	0
$628$	−6.72777e6	−0.680726
$629$	−2.90619e6	−0.292885
$630$	0	0
$631$	−2.53030e6	−0.252988	−0.126494	−	0.991967i	$-0.540372\pi$
$631$	−2.53030e6	−0.252988	−0.126494	+	0.991967i	$0.540372\pi$
$632$	−3.93333e6	−0.391713
$633$	0	0
$634$	151297.	0.0149488
$635$	−946679.	−0.0931684
$636$	0	0
$637$	573414.	0.0559912
$638$	1.63628e7	1.59150
$639$	0	0
$640$	−409600.	−0.0395285
$641$	−1.34506e7	−1.29299	−0.646497	−	0.762916i	$-0.723767\pi$
$641$	−1.34506e7	−1.29299	−0.646497	+	0.762916i	$0.723767\pi$
$642$	0	0
$643$	6.77639e6	0.646355	0.323177	−	0.946338i	$-0.395249\pi$
$643$	6.77639e6	0.646355	0.323177	+	0.946338i	$0.395249\pi$
$644$	3.23653e6	0.307514
$645$	0	0
$646$	−6.30522e6	−0.594455
$647$	−9.58548e6	−0.900229	−0.450115	−	0.892971i	$-0.648617\pi$
$647$	−9.58548e6	−0.900229	−0.450115	+	0.892971i	$0.648617\pi$
$648$	0	0
$649$	1.02716e7	0.957250
$650$	−597058.	−0.0554285
$651$	0	0
$652$	−7.57409e6	−0.697768
$653$	3.89619e6	0.357567	0.178783	−	0.983888i	$-0.442784\pi$
$653$	3.89619e6	0.357567	0.178783	+	0.983888i	$0.442784\pi$
$654$	0	0
$655$	−546107.	−0.0497364
$656$	−4.56783e6	−0.414429
$657$	0	0
$658$	2.60053e6	0.234152
$659$	−1.32170e7	−1.18555	−0.592773	−	0.805370i	$-0.701967\pi$
$659$	−1.32170e7	−1.18555	−0.592773	+	0.805370i	$0.701967\pi$
$660$	0	0
$661$	−7.08193e6	−0.630446	−0.315223	−	0.949018i	$-0.602079\pi$
$661$	−7.08193e6	−0.630446	−0.315223	+	0.949018i	$0.602079\pi$
$662$	3.55094e6	0.314919
$663$	0	0
$664$	358987.	0.0315979
$665$	−2.26028e6	−0.198202
$666$	0	0
$667$	−2.98066e7	−2.59417
$668$	1.58566e6	0.137489
$669$	0	0
$670$	−6.60368e6	−0.568328
$671$	1.86257e7	1.59700
$672$	0	0
$673$	2.43353e6	0.207109	0.103554	−	0.994624i	$-0.466978\pi$
$673$	2.43353e6	0.207109	0.103554	+	0.994624i	$0.466978\pi$
$674$	−7.55526e6	−0.640619
$675$	0	0
$676$	−5.02810e6	−0.423192
$677$	6.52078e6	0.546800	0.273400	−	0.961900i	$-0.411852\pi$
$677$	6.52078e6	0.546800	0.273400	+	0.961900i	$0.411852\pi$
$678$	0	0
$679$	6.79967e6	0.565996
$680$	−1.36689e6	−0.113360
$681$	0	0
$682$	1.11101e7	0.914654
$683$	−1.41845e7	−1.16349	−0.581746	−	0.813370i	$-0.697630\pi$
$683$	−1.41845e7	−1.16349	−0.581746	+	0.813370i	$0.697630\pi$
$684$	0	0
$685$	−6.99740e6	−0.569784
$686$	470596.	0.0381802
$687$	0	0
$688$	−874569.	−0.0704406
$689$	−5.52025e6	−0.443007
$690$	0	0
$691$	−1.77069e7	−1.41074	−0.705371	−	0.708839i	$-0.749219\pi$
$691$	−1.77069e7	−1.41074	−0.705371	+	0.708839i	$0.749219\pi$
$692$	−7.86578e6	−0.624420
$693$	0	0
$694$	5.83424e6	0.459818
$695$	552694.	0.0434033
$696$	0	0
$697$	−1.52434e7	−1.18851
$698$	−8.41474e6	−0.653736
$699$	0	0
$700$	−490000.	−0.0377964
$701$	977570.	0.0751369	0.0375684	−	0.999294i	$-0.488039\pi$
$701$	977570.	0.0751369	0.0375684	+	0.999294i	$0.488039\pi$
$702$	0	0
$703$	−6.27678e6	−0.479014
$704$	−2.32065e6	−0.176473
$705$	0	0
$706$	8.24148e6	0.622291
$707$	3.11016e6	0.234010
$708$	0	0
$709$	1.92882e6	0.144104	0.0720520	−	0.997401i	$-0.477045\pi$
$709$	1.92882e6	0.144104	0.0720520	+	0.997401i	$0.477045\pi$
$710$	−1.09505e6	−0.0815243
$711$	0	0
$712$	8.06579e6	0.596275
$713$	−2.02382e7	−1.49090
$714$	0	0
$715$	−3.38272e6	−0.247458
$716$	7.92038e6	0.577382
$717$	0	0
$718$	1.73951e7	1.25926
$719$	−6.44948e6	−0.465267	−0.232634	−	0.972564i	$-0.574734\pi$
$719$	−6.44948e6	−0.465267	−0.232634	+	0.972564i	$0.574734\pi$
$720$	0	0
$721$	−580266.	−0.0415708
$722$	−3.71361e6	−0.265127
$723$	0	0
$724$	−3.52482e6	−0.249914
$725$	4.51261e6	0.318848
$726$	0	0
$727$	−7.11269e6	−0.499112	−0.249556	−	0.968360i	$-0.580285\pi$
$727$	−7.11269e6	−0.499112	−0.249556	+	0.968360i	$0.580285\pi$
$728$	748949.	0.0523750
$729$	0	0
$730$	−3.94409e6	−0.273930
$731$	−2.91855e6	−0.202011
$732$	0	0
$733$	2.02361e6	0.139113	0.0695565	−	0.997578i	$-0.477842\pi$
$733$	2.02361e6	0.139113	0.0695565	+	0.997578i	$0.477842\pi$
$734$	4.19110e6	0.287136
$735$	0	0
$736$	4.22731e6	0.287653
$737$	−3.74141e7	−2.53727
$738$	0	0
$739$	1.64319e7	1.10682	0.553409	−	0.832910i	$-0.313327\pi$
$739$	1.64319e7	1.10682	0.553409	+	0.832910i	$0.313327\pi$
$740$	−1.36072e6	−0.0913462
$741$	0	0
$742$	−4.53042e6	−0.302084
$743$	1.22465e7	0.813841	0.406921	−	0.913464i	$-0.366603\pi$
$743$	1.22465e7	0.813841	0.406921	+	0.913464i	$0.366603\pi$
$744$	0	0
$745$	1.28916e7	0.850974
$746$	−2.43214e6	−0.160008
$747$	0	0
$748$	−7.74431e6	−0.506091
$749$	−6.90289e6	−0.449600
$750$	0	0
$751$	1.30720e7	0.845748	0.422874	−	0.906189i	$-0.361021\pi$
$751$	1.30720e7	0.845748	0.422874	+	0.906189i	$0.361021\pi$
$752$	3.39661e6	0.219029
$753$	0	0
$754$	−6.89739e6	−0.441831
$755$	2.31814e6	0.148003
$756$	0	0
$757$	1.27287e7	0.807317	0.403659	−	0.914910i	$-0.367738\pi$
$757$	1.27287e7	0.807317	0.403659	+	0.914910i	$0.367738\pi$
$758$	1.58127e7	0.999619
$759$	0	0
$760$	−2.95221e6	−0.185401
$761$	2.82767e7	1.76997	0.884987	−	0.465616i	$-0.154167\pi$
$761$	2.82767e7	1.76997	0.884987	+	0.465616i	$0.154167\pi$
$762$	0	0
$763$	4.51018e6	0.280468
$764$	−1.48266e7	−0.918983
$765$	0	0
$766$	−1.45557e7	−0.896318
$767$	−4.32976e6	−0.265752
$768$	0	0
$769$	9.22579e6	0.562584	0.281292	−	0.959622i	$-0.409237\pi$
$769$	9.22579e6	0.562584	0.281292	+	0.959622i	$0.409237\pi$
$770$	−2.77617e6	−0.168740
$771$	0	0
$772$	−1.16423e7	−0.703066
$773$	2.03038e7	1.22216	0.611081	−	0.791568i	$-0.290735\pi$
$773$	2.03038e7	1.22216	0.611081	+	0.791568i	$0.290735\pi$
$774$	0	0
$775$	3.06400e6	0.183246
$776$	8.88120e6	0.529440
$777$	0	0
$778$	−1.76521e7	−1.04556
$779$	−3.29228e7	−1.94381
$780$	0	0
$781$	−6.20415e6	−0.363961
$782$	1.41071e7	0.824937
$783$	0	0
$784$	614656.	0.0357143
$785$	−1.05121e7	−0.608860
$786$	0	0
$787$	9.16529e6	0.527484	0.263742	−	0.964593i	$-0.415043\pi$
$787$	9.16529e6	0.527484	0.263742	+	0.964593i	$0.415043\pi$
$788$	−4.71749e6	−0.270642
$789$	0	0
$790$	−6.14583e6	−0.350359
$791$	524561.	0.0298095
$792$	0	0
$793$	−7.85125e6	−0.443359
$794$	−2.80231e6	−0.157749
$795$	0	0
$796$	7.18757e6	0.402068
$797$	3.11124e7	1.73495	0.867477	−	0.497477i	$-0.165740\pi$
$797$	3.11124e7	1.73495	0.867477	+	0.497477i	$0.165740\pi$
$798$	0	0
$799$	1.13350e7	0.628135
$800$	−640000.	−0.0353553
$801$	0	0
$802$	−1.75627e7	−0.964174
$803$	−2.23458e7	−1.22295
$804$	0	0
$805$	5.05708e6	0.275049
$806$	−4.68322e6	−0.253926
$807$	0	0
$808$	4.06225e6	0.218896
$809$	2.06869e7	1.11128	0.555642	−	0.831422i	$-0.312472\pi$
$809$	2.06869e7	1.11128	0.555642	+	0.831422i	$0.312472\pi$
$810$	0	0
$811$	−9.61505e6	−0.513333	−0.256667	−	0.966500i	$-0.582624\pi$
$811$	−9.61505e6	−0.513333	−0.256667	+	0.966500i	$0.582624\pi$
$812$	−5.66062e6	−0.301283
$813$	0	0
$814$	−7.70938e6	−0.407811
$815$	−1.18345e7	−0.624103
$816$	0	0
$817$	−6.30349e6	−0.330389
$818$	−1.12635e7	−0.588558
$819$	0	0
$820$	−7.13723e6	−0.370677
$821$	−2.47979e7	−1.28398	−0.641989	−	0.766714i	$-0.721891\pi$
$821$	−2.47979e7	−1.28398	−0.641989	+	0.766714i	$0.721891\pi$
$822$	0	0
$823$	1.03071e6	0.0530441	0.0265220	−	0.999648i	$-0.491557\pi$
$823$	1.03071e6	0.0530441	0.0265220	+	0.999648i	$0.491557\pi$
$824$	−757898.	−0.0388860
$825$	0	0
$826$	−3.55340e6	−0.181215
$827$	1.95090e7	0.991907	0.495953	−	0.868349i	$-0.334819\pi$
$827$	1.95090e7	0.991907	0.495953	+	0.868349i	$0.334819\pi$
$828$	0	0
$829$	8.24989e6	0.416929	0.208464	−	0.978030i	$-0.433153\pi$
$829$	8.24989e6	0.416929	0.208464	+	0.978030i	$0.433153\pi$
$830$	560917.	0.0282620
$831$	0	0
$832$	978220.	0.0489923
$833$	2.05119e6	0.102422
$834$	0	0
$835$	2.47760e6	0.122974
$836$	−1.67262e7	−0.827714
$837$	0	0
$838$	−2.76168e7	−1.35851
$839$	−3.41432e6	−0.167456	−0.0837278	−	0.996489i	$-0.526683\pi$
$839$	−3.41432e6	−0.167456	−0.0837278	+	0.996489i	$0.526683\pi$
$840$	0	0
$841$	3.16199e7	1.54160
$842$	1.26916e7	0.616928
$843$	0	0
$844$	−722440.	−0.0349096
$845$	−7.85641e6	−0.378514
$846$	0	0
$847$	−7.83728e6	−0.375368
$848$	−5.91728e6	−0.282574
$849$	0	0
$850$	−2.13577e6	−0.101393
$851$	1.40435e7	0.664737
$852$	0	0
$853$	−1.46769e7	−0.690657	−0.345328	−	0.938482i	$-0.612232\pi$
$853$	−1.46769e7	−0.690657	−0.345328	+	0.938482i	$0.612232\pi$
$854$	−6.44345e6	−0.302325
$855$	0	0
$856$	−9.01602e6	−0.420562
$857$	3.15902e7	1.46927	0.734634	−	0.678464i	$-0.237354\pi$
$857$	3.15902e7	1.46927	0.734634	+	0.678464i	$0.237354\pi$
$858$	0	0
$859$	7.92194e6	0.366310	0.183155	−	0.983084i	$-0.441369\pi$
$859$	7.92194e6	0.366310	0.183155	+	0.983084i	$0.441369\pi$
$860$	−1.36651e6	−0.0630040
$861$	0	0
$862$	−1.22449e7	−0.561289
$863$	9.96301e6	0.455369	0.227685	−	0.973735i	$-0.426884\pi$
$863$	9.96301e6	0.455369	0.227685	+	0.973735i	$0.426884\pi$
$864$	0	0
$865$	−1.22903e7	−0.558498
$866$	8.66901e6	0.392803
$867$	0	0
$868$	−3.84348e6	−0.173151
$869$	−3.48201e7	−1.56416
$870$	0	0
$871$	1.57711e7	0.704396
$872$	5.89085e6	0.262353
$873$	0	0
$874$	3.04685e7	1.34919
$875$	−765625.	−0.0338062
$876$	0	0
$877$	−3.11495e6	−0.136758	−0.0683789	−	0.997659i	$-0.521783\pi$
$877$	−3.11495e6	−0.136758	−0.0683789	+	0.997659i	$0.521783\pi$
$878$	1.66317e7	0.728116
$879$	0	0
$880$	−3.62601e6	−0.157842
$881$	−2.47277e7	−1.07335	−0.536677	−	0.843787i	$-0.680321\pi$
$881$	−2.47277e7	−1.07335	−0.536677	+	0.843787i	$0.680321\pi$
$882$	0	0
$883$	1.95110e7	0.842127	0.421063	−	0.907031i	$-0.361657\pi$
$883$	1.95110e7	0.842127	0.421063	+	0.907031i	$0.361657\pi$
$884$	3.26445e6	0.140501
$885$	0	0
$886$	1.80664e7	0.773194
$887$	−3.49032e7	−1.48955	−0.744777	−	0.667314i	$-0.767444\pi$
$887$	−3.49032e7	−1.48955	−0.744777	+	0.667314i	$0.767444\pi$
$888$	0	0
$889$	1.85549e6	0.0787416
$890$	1.26028e7	0.533325
$891$	0	0
$892$	−1.76564e7	−0.743001
$893$	2.44812e7	1.02732
$894$	0	0
$895$	1.23756e7	0.516426
$896$	802816.	0.0334077
$897$	0	0
$898$	−2.19296e7	−0.907488
$899$	3.53962e7	1.46069
$900$	0	0
$901$	−1.97467e7	−0.810370
$902$	−4.04370e7	−1.65487
$903$	0	0
$904$	685141.	0.0278842
$905$	−5.50753e6	−0.223530
$906$	0	0
$907$	802947.	0.0324092	0.0162046	−	0.999869i	$-0.494842\pi$
$907$	802947.	0.0324092	0.0162046	+	0.999869i	$0.494842\pi$
$908$	−1.73930e6	−0.0700101
$909$	0	0
$910$	1.17023e6	0.0468456
$911$	−1.53636e6	−0.0613335	−0.0306667	−	0.999530i	$-0.509763\pi$
$911$	−1.53636e6	−0.0613335	−0.0306667	+	0.999530i	$0.509763\pi$
$912$	0	0
$913$	3.17796e6	0.126174
$914$	3.76026e6	0.148885
$915$	0	0
$916$	−1.49886e7	−0.590231
$917$	1.07037e6	0.0420349
$918$	0	0
$919$	−1.94637e7	−0.760214	−0.380107	−	0.924943i	$-0.624113\pi$
$919$	−1.94637e7	−0.760214	−0.380107	+	0.924943i	$0.624113\pi$
$920$	6.60517e6	0.257285
$921$	0	0
$922$	4.64631e6	0.180003
$923$	2.61523e6	0.101043
$924$	0	0
$925$	−2.12613e6	−0.0817026
$926$	−1.50961e7	−0.578544
$927$	0	0
$928$	−7.39347e6	−0.281824
$929$	1.60720e7	0.610987	0.305494	−	0.952194i	$-0.401179\pi$
$929$	1.60720e7	0.610987	0.305494	+	0.952194i	$0.401179\pi$
$930$	0	0
$931$	4.43016e6	0.167512
$932$	−1.65408e7	−0.623760
$933$	0	0
$934$	1.71270e7	0.642413
$935$	−1.21005e7	−0.452662
$936$	0	0
$937$	−3.32237e7	−1.23623	−0.618114	−	0.786089i	$-0.712103\pi$
$937$	−3.32237e7	−1.23623	−0.618114	+	0.786089i	$0.712103\pi$
$938$	1.29432e7	0.480325
$939$	0	0
$940$	5.30721e6	0.195905
$941$	−1.30088e6	−0.0478921	−0.0239461	−	0.999713i	$-0.507623\pi$
$941$	−1.30088e6	−0.0478921	−0.0239461	+	0.999713i	$0.507623\pi$
$942$	0	0
$943$	7.36603e7	2.69746
$944$	−4.64117e6	−0.169511
$945$	0	0
$946$	−7.74219e6	−0.281278
$947$	−9.04734e6	−0.327828	−0.163914	−	0.986475i	$-0.552412\pi$
$947$	−9.04734e6	−0.327828	−0.163914	+	0.986475i	$0.552412\pi$
$948$	0	0
$949$	9.41940e6	0.339514
$950$	−4.61282e6	−0.165828
$951$	0	0
$952$	2.67910e6	0.0958070
$953$	2.41731e6	0.0862185	0.0431092	−	0.999070i	$-0.486274\pi$
$953$	2.41731e6	0.0862185	0.0431092	+	0.999070i	$0.486274\pi$
$954$	0	0
$955$	−2.31665e7	−0.821964
$956$	−4.44291e6	−0.157226
$957$	0	0
$958$	−1.02674e7	−0.361449
$959$	1.37149e7	0.481555
$960$	0	0
$961$	−4.59568e6	−0.160524
$962$	3.24972e6	0.113216
$963$	0	0
$964$	782804.	0.0271307
$965$	−1.81911e7	−0.628841
$966$	0	0
$967$	1.04638e7	0.359850	0.179925	−	0.983680i	$-0.442414\pi$
$967$	1.04638e7	0.359850	0.179925	+	0.983680i	$0.442414\pi$
$968$	−1.02364e7	−0.351124
$969$	0	0
$970$	1.38769e7	0.473546
$971$	−4.16932e7	−1.41911	−0.709556	−	0.704649i	$-0.751105\pi$
$971$	−4.16932e7	−1.41911	−0.709556	+	0.704649i	$0.751105\pi$
$972$	0	0
$973$	−1.08328e6	−0.0366825
$974$	1.70152e7	0.574698
$975$	0	0
$976$	−8.41593e6	−0.282799
$977$	−2.60092e7	−0.871748	−0.435874	−	0.900008i	$-0.643561\pi$
$977$	−2.60092e7	−0.871748	−0.435874	+	0.900008i	$0.643561\pi$
$978$	0	0
$979$	7.14030e7	2.38100
$980$	960400.	0.0319438
$981$	0	0
$982$	−7.78317e6	−0.257560
$983$	2.12634e7	0.701858	0.350929	−	0.936402i	$-0.385866\pi$
$983$	2.12634e7	0.701858	0.350929	+	0.936402i	$0.385866\pi$
$984$	0	0
$985$	−7.37108e6	−0.242070
$986$	−2.46730e7	−0.808220
$987$	0	0
$988$	7.05055e6	0.229790
$989$	1.41032e7	0.458487
$990$	0	0
$991$	−3.56443e6	−0.115294	−0.0576468	−	0.998337i	$-0.518360\pi$
$991$	−3.56443e6	−0.115294	−0.0576468	+	0.998337i	$0.518360\pi$
$992$	−5.02005e6	−0.161968
$993$	0	0
$994$	2.14629e6	0.0689006
$995$	1.12306e7	0.359620
$996$	0	0
$997$	−721404.	−0.0229848	−0.0114924	−	0.999934i	$-0.503658\pi$
$997$	−721404.	−0.0229848	−0.0114924	+	0.999934i	$0.503658\pi$
$998$	7.52776e6	0.239243
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.g.1.1	✓	2	3.2	odd	2
350.6.a.q.1.2		2	15.14	odd	2
350.6.c.j.99.1		4	15.8	even	4
350.6.c.j.99.4		4	15.2	even	4
490.6.a.v.1.2		2	21.20	even	2
560.6.a.m.1.2		2	12.11	even	2
630.6.a.u.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -49.0000 q^{7} -64.0000 q^{8} -100.000 q^{10} -566.565 q^{11} +238.823 q^{13} +196.000 q^{14} +256.000 q^{16} +854.306 q^{17} +1845.13 q^{19} +400.000 q^{20} +2266.26 q^{22} -4128.23 q^{23} +625.000 q^{25} -955.293 q^{26} -784.000 q^{28} +7220.18 q^{29} +4902.39 q^{31} -1024.00 q^{32} -3417.22 q^{34} -1225.00 q^{35} -3401.81 q^{37} -7380.52 q^{38} -1600.00 q^{40} -17843.1 q^{41} -3416.29 q^{43} -9065.03 q^{44} +16512.9 q^{46} +13268.0 q^{47} +2401.00 q^{49} -2500.00 q^{50} +3821.17 q^{52} -23114.4 q^{53} -14164.1 q^{55} +3136.00 q^{56} -28880.7 q^{58} -18129.6 q^{59} -32874.7 q^{61} -19609.6 q^{62} +4096.00 q^{64} +5970.58 q^{65} +66036.8 q^{67} +13668.9 q^{68} +4900.00 q^{70} +10950.5 q^{71} +39440.9 q^{73} +13607.2 q^{74} +29522.1 q^{76} +27761.7 q^{77} +61458.3 q^{79} +6400.00 q^{80} +71372.3 q^{82} -5609.17 q^{83} +21357.7 q^{85} +13665.1 q^{86} +36260.1 q^{88} -126028. q^{89} -11702.3 q^{91} -66051.7 q^{92} -53072.1 q^{94} +46128.2 q^{95} -138769. q^{97} -9604.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 50 q^{5} - 98 q^{7} - 128 q^{8} - 200 q^{10} - 959 q^{11} - 393 q^{13} + 392 q^{14} + 512 q^{16} + 2231 q^{17} + 3342 q^{19} + 800 q^{20} + 3836 q^{22} + 450 q^{23} + 1250 q^{25}+ \cdots - 19208 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 50 * q^5 - 98 * q^7 - 128 * q^8 - 200 * q^10 - 959 * q^11 - 393 * q^13 + 392 * q^14 + 512 * q^16 + 2231 * q^17 + 3342 * q^19 + 800 * q^20 + 3836 * q^22 + 450 * q^23 + 1250 * q^25 + 1572 * q^26 - 1568 * q^28 + 4515 * q^29 - 2036 * q^31 - 2048 * q^32 - 8924 * q^34 - 2450 * q^35 - 1928 * q^37 - 13368 * q^38 - 3200 * q^40 - 19318 * q^41 - 14146 * q^43 - 15344 * q^44 - 1800 * q^46 + 19745 * q^47 + 4802 * q^49 - 5000 * q^50 - 6288 * q^52 - 26378 * q^53 - 23975 * q^55 + 6272 * q^56 - 18060 * q^58 + 11104 * q^59 + 3554 * q^61 + 8144 * q^62 + 8192 * q^64 - 9825 * q^65 + 65208 * q^67 + 35696 * q^68 + 9800 * q^70 - 17104 * q^71 - 36740 * q^73 + 7712 * q^74 + 53472 * q^76 + 46991 * q^77 + 72245 * q^79 + 12800 * q^80 + 77272 * q^82 - 99676 * q^83 + 55775 * q^85 + 56584 * q^86 + 61376 * q^88 - 82454 * q^89 + 19257 * q^91 + 7200 * q^92 - 78980 * q^94 + 83550 * q^95 - 104627 * q^97 - 19208 * q^98

Introduction

L-functions

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Varieties

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Database

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