LMFDB - Embedded newform 350.6.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("350.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.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:	$56.1343369345$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{79}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.88819$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} -27.7764 q^{3} +16.0000 q^{4} -111.106 q^{6} +49.0000 q^{7} +64.0000 q^{8} +528.528 q^{9} -481.671 q^{11} -444.422 q^{12} -883.515 q^{13} +196.000 q^{14} +256.000 q^{16} +1840.82 q^{17} +2114.11 q^{18} +2440.51 q^{19} -1361.04 q^{21} -1926.68 q^{22} -169.274 q^{23} -1777.69 q^{24} -3534.06 q^{26} -7930.93 q^{27} +784.000 q^{28} -554.156 q^{29} +7734.52 q^{31} +1024.00 q^{32} +13379.1 q^{33} +7363.28 q^{34} +8456.44 q^{36} -11425.8 q^{37} +9762.03 q^{38} +24540.9 q^{39} +10893.6 q^{41} -5444.17 q^{42} -15335.5 q^{43} -7706.73 q^{44} -677.094 q^{46} -9705.85 q^{47} -7110.76 q^{48} +2401.00 q^{49} -51131.3 q^{51} -14136.2 q^{52} -8829.12 q^{53} -31723.7 q^{54} +3136.00 q^{56} -67788.5 q^{57} -2216.62 q^{58} -40580.0 q^{59} -1675.50 q^{61} +30938.1 q^{62} +25897.9 q^{63} +4096.00 q^{64} +53516.3 q^{66} -9885.31 q^{67} +29453.1 q^{68} +4701.81 q^{69} -51350.0 q^{71} +33825.8 q^{72} -26718.3 q^{73} -45703.2 q^{74} +39048.1 q^{76} -23601.9 q^{77} +98163.5 q^{78} -70237.0 q^{79} +91860.4 q^{81} +43574.3 q^{82} -58622.5 q^{83} -21776.7 q^{84} -61342.0 q^{86} +15392.4 q^{87} -30826.9 q^{88} +28370.7 q^{89} -43292.2 q^{91} -2708.38 q^{92} -214837. q^{93} -38823.4 q^{94} -28443.0 q^{96} +117548. q^{97} +9604.00 q^{98} -254576. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 20 q^{3} + 32 q^{4} - 80 q^{6} + 98 q^{7} + 128 q^{8} + 346 q^{9} - 1070 q^{11} - 320 q^{12} - 736 q^{13} + 392 q^{14} + 512 q^{16} + 1904 q^{17} + 1384 q^{18} + 828 q^{19} - 980 q^{21}+ \cdots - 147190 q^{99}+O(q^{100}) $ 2 * q + 8 * q^2 - 20 * q^3 + 32 * q^4 - 80 * q^6 + 98 * q^7 + 128 * q^8 + 346 * q^9 - 1070 * q^11 - 320 * q^12 - 736 * q^13 + 392 * q^14 + 512 * q^16 + 1904 * q^17 + 1384 * q^18 + 828 * q^19 - 980 * q^21 - 4280 * q^22 - 1654 * q^23 - 1280 * q^24 - 2944 * q^26 - 11240 * q^27 + 1568 * q^28 - 2246 * q^29 + 288 * q^31 + 2048 * q^32 + 8804 * q^33 + 7616 * q^34 + 5536 * q^36 - 8986 * q^37 + 3312 * q^38 + 25688 * q^39 + 11228 * q^41 - 3920 * q^42 - 3402 * q^43 - 17120 * q^44 - 6616 * q^46 - 15572 * q^47 - 5120 * q^48 + 4802 * q^49 - 50640 * q^51 - 11776 * q^52 + 16188 * q^53 - 44960 * q^54 + 6272 * q^56 - 80328 * q^57 - 8984 * q^58 - 92928 * q^59 + 15172 * q^61 + 1152 * q^62 + 16954 * q^63 + 8192 * q^64 + 35216 * q^66 - 78966 * q^67 + 30464 * q^68 - 6844 * q^69 - 20822 * q^71 + 22144 * q^72 - 72884 * q^73 - 35944 * q^74 + 13248 * q^76 - 52430 * q^77 + 102752 * q^78 - 72106 * q^79 + 110482 * q^81 + 44912 * q^82 + 47400 * q^83 - 15680 * q^84 - 13608 * q^86 + 2236 * q^87 - 68480 * q^88 - 9920 * q^89 - 36064 * q^91 - 26464 * q^92 - 272744 * q^93 - 62288 * q^94 - 20480 * q^96 + 26792 * q^97 + 19208 * q^98 - 147190 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−27.7764	−1.78186	−0.890928	−	0.454144i	$-0.849945\pi$
$3$	−27.7764	−1.78186	−0.890928	+	0.454144i	$0.849945\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−111.106	−1.25996
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	528.528	2.17501
$10$	0	0
$11$	−481.671	−1.20024	−0.600121	−	0.799909i	$-0.704881\pi$
$11$	−481.671	−1.20024	−0.600121	+	0.799909i	$0.704881\pi$
$12$	−444.422	−0.890928
$13$	−883.515	−1.44996	−0.724979	−	0.688771i	$-0.758151\pi$
$13$	−883.515	−1.44996	−0.724979	+	0.688771i	$0.758151\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1840.82	1.54486	0.772430	−	0.635100i	$-0.219041\pi$
$17$	1840.82	1.54486	0.772430	+	0.635100i	$0.219041\pi$
$18$	2114.11	1.53797
$19$	2440.51	1.55094	0.775472	−	0.631382i	$-0.217512\pi$
$19$	2440.51	1.55094	0.775472	+	0.631382i	$0.217512\pi$
$20$	0	0
$21$	−1361.04	−0.673478
$22$	−1926.68	−0.848699
$23$	−169.274	−0.0667221	−0.0333610	−	0.999443i	$-0.510621\pi$
$23$	−169.274	−0.0667221	−0.0333610	+	0.999443i	$0.510621\pi$
$24$	−1777.69	−0.629981
$25$	0	0
$26$	−3534.06	−1.02528
$27$	−7930.93	−2.09370
$28$	784.000	0.188982
$29$	−554.156	−0.122359	−0.0611796	−	0.998127i	$-0.519486\pi$
$29$	−554.156	−0.122359	−0.0611796	+	0.998127i	$0.519486\pi$
$30$	0	0
$31$	7734.52	1.44554	0.722768	−	0.691091i	$-0.242869\pi$
$31$	7734.52	1.44554	0.722768	+	0.691091i	$0.242869\pi$
$32$	1024.00	0.176777
$33$	13379.1	2.13866
$34$	7363.28	1.09238
$35$	0	0
$36$	8456.44	1.08751
$37$	−11425.8	−1.37209	−0.686044	−	0.727560i	$-0.740654\pi$
$37$	−11425.8	−1.37209	−0.686044	+	0.727560i	$0.740654\pi$
$38$	9762.03	1.09668
$39$	24540.9	2.58362
$40$	0	0
$41$	10893.6	1.01207	0.506036	−	0.862512i	$-0.331110\pi$
$41$	10893.6	1.01207	0.506036	+	0.862512i	$0.331110\pi$
$42$	−5444.17	−0.476221
$43$	−15335.5	−1.26481	−0.632407	−	0.774636i	$-0.717933\pi$
$43$	−15335.5	−1.26481	−0.632407	+	0.774636i	$0.717933\pi$
$44$	−7706.73	−0.600121
$45$	0	0
$46$	−677.094	−0.0471796
$47$	−9705.85	−0.640898	−0.320449	−	0.947266i	$-0.603834\pi$
$47$	−9705.85	−0.640898	−0.320449	+	0.947266i	$0.603834\pi$
$48$	−7110.76	−0.445464
$49$	2401.00	0.142857
$50$	0	0
$51$	−51131.3	−2.75272
$52$	−14136.2	−0.724979
$53$	−8829.12	−0.431746	−0.215873	−	0.976422i	$-0.569260\pi$
$53$	−8829.12	−0.431746	−0.215873	+	0.976422i	$0.569260\pi$
$54$	−31723.7	−1.48047
$55$	0	0
$56$	3136.00	0.133631
$57$	−67788.5	−2.76356
$58$	−2216.62	−0.0865210
$59$	−40580.0	−1.51769	−0.758843	−	0.651273i	$-0.774235\pi$
$59$	−40580.0	−1.51769	−0.758843	+	0.651273i	$0.774235\pi$
$60$	0	0
$61$	−1675.50	−0.0576527	−0.0288263	−	0.999584i	$-0.509177\pi$
$61$	−1675.50	−0.0576527	−0.0288263	+	0.999584i	$0.509177\pi$
$62$	30938.1	1.02215
$63$	25897.9	0.822077
$64$	4096.00	0.125000
$65$	0	0
$66$	53516.3	1.51226
$67$	−9885.31	−0.269032	−0.134516	−	0.990911i	$-0.542948\pi$
$67$	−9885.31	−0.269032	−0.134516	+	0.990911i	$0.542948\pi$
$68$	29453.1	0.772430
$69$	4701.81	0.118889
$70$	0	0
$71$	−51350.0	−1.20891	−0.604456	−	0.796638i	$-0.706610\pi$
$71$	−51350.0	−1.20891	−0.604456	+	0.796638i	$0.706610\pi$
$72$	33825.8	0.768983
$73$	−26718.3	−0.586816	−0.293408	−	0.955987i	$-0.594789\pi$
$73$	−26718.3	−0.586816	−0.293408	+	0.955987i	$0.594789\pi$
$74$	−45703.2	−0.970213
$75$	0	0
$76$	39048.1	0.775472
$77$	−23601.9	−0.453649
$78$	98163.5	1.82689
$79$	−70237.0	−1.26619	−0.633094	−	0.774075i	$-0.718215\pi$
$79$	−70237.0	−1.26619	−0.633094	+	0.774075i	$0.718215\pi$
$80$	0	0
$81$	91860.4	1.55566
$82$	43574.3	0.715643
$83$	−58622.5	−0.934047	−0.467024	−	0.884245i	$-0.654674\pi$
$83$	−58622.5	−0.934047	−0.467024	+	0.884245i	$0.654674\pi$
$84$	−21776.7	−0.336739
$85$	0	0
$86$	−61342.0	−0.894358
$87$	15392.4	0.218027
$88$	−30826.9	−0.424349
$89$	28370.7	0.379661	0.189830	−	0.981817i	$-0.439206\pi$
$89$	28370.7	0.379661	0.189830	+	0.981817i	$0.439206\pi$
$90$	0	0
$91$	−43292.2	−0.548033
$92$	−2708.38	−0.0333610
$93$	−214837.	−2.57574
$94$	−38823.4	−0.453183
$95$	0	0
$96$	−28443.0	−0.314991
$97$	117548.	1.26849	0.634243	−	0.773134i	$-0.281312\pi$
$97$	117548.	1.26849	0.634243	+	0.773134i	$0.281312\pi$
$98$	9604.00	0.101015
$99$	−254576.	−2.61054
$100$	0	0
$101$	−25228.5	−0.246087	−0.123043	−	0.992401i	$-0.539265\pi$
$101$	−25228.5	−0.246087	−0.123043	+	0.992401i	$0.539265\pi$
$102$	−204525.	−1.94646
$103$	−99166.7	−0.921028	−0.460514	−	0.887653i	$-0.652335\pi$
$103$	−99166.7	−0.921028	−0.460514	+	0.887653i	$0.652335\pi$
$104$	−56545.0	−0.512638
$105$	0	0
$106$	−35316.5	−0.305290
$107$	−127493.	−1.07653	−0.538264	−	0.842776i	$-0.680920\pi$
$107$	−127493.	−1.07653	−0.538264	+	0.842776i	$0.680920\pi$
$108$	−126895.	−1.04685
$109$	147420.	1.18848	0.594238	−	0.804289i	$-0.297454\pi$
$109$	147420.	1.18848	0.594238	+	0.804289i	$0.297454\pi$
$110$	0	0
$111$	317367.	2.44486
$112$	12544.0	0.0944911
$113$	88159.3	0.649489	0.324745	−	0.945802i	$-0.394722\pi$
$113$	88159.3	0.649489	0.324745	+	0.945802i	$0.394722\pi$
$114$	−271154.	−1.95413
$115$	0	0
$116$	−8866.49	−0.0611796
$117$	−466962.	−3.15368
$118$	−162320.	−1.07317
$119$	90200.2	0.583902
$120$	0	0
$121$	70955.8	0.440580
$122$	−6701.99	−0.0407666
$123$	−302585.	−1.80337
$124$	123752.	0.722768
$125$	0	0
$126$	103591.	0.581296
$127$	−89172.9	−0.490595	−0.245298	−	0.969448i	$-0.578886\pi$
$127$	−89172.9	−0.490595	−0.245298	+	0.969448i	$0.578886\pi$
$128$	16384.0	0.0883883
$129$	425965.	2.25372
$130$	0	0
$131$	298138.	1.51789	0.758944	−	0.651156i	$-0.225716\pi$
$131$	298138.	1.51789	0.758944	+	0.651156i	$0.225716\pi$
$132$	214065.	1.06933
$133$	119585.	0.586202
$134$	−39541.3	−0.190234
$135$	0	0
$136$	117812.	0.546190
$137$	23992.0	0.109211	0.0546053	−	0.998508i	$-0.482610\pi$
$137$	23992.0	0.109211	0.0546053	+	0.998508i	$0.482610\pi$
$138$	18807.2	0.0840673
$139$	−376423.	−1.65249	−0.826245	−	0.563311i	$-0.809527\pi$
$139$	−376423.	−1.65249	−0.826245	+	0.563311i	$0.809527\pi$
$140$	0	0
$141$	269593.	1.14199
$142$	−205400.	−0.854830
$143$	425564.	1.74030
$144$	135303.	0.543753
$145$	0	0
$146$	−106873.	−0.414942
$147$	−66691.1	−0.254551
$148$	−182813.	−0.686044
$149$	−8717.80	−0.0321693	−0.0160846	−	0.999871i	$-0.505120\pi$
$149$	−8717.80	−0.0321693	−0.0160846	+	0.999871i	$0.505120\pi$
$150$	0	0
$151$	−447551.	−1.59735	−0.798676	−	0.601762i	$-0.794466\pi$
$151$	−447551.	−1.59735	−0.798676	+	0.601762i	$0.794466\pi$
$152$	156193.	0.548342
$153$	972924.	3.36009
$154$	−94407.5	−0.320778
$155$	0	0
$156$	392654.	1.29181
$157$	−63881.6	−0.206836	−0.103418	−	0.994638i	$-0.532978\pi$
$157$	−63881.6	−0.206836	−0.103418	+	0.994638i	$0.532978\pi$
$158$	−280948.	−0.895330
$159$	245241.	0.769308
$160$	0	0
$161$	−8294.41	−0.0252186
$162$	367441.	1.10002
$163$	270208.	0.796581	0.398290	−	0.917259i	$-0.369604\pi$
$163$	270208.	0.796581	0.398290	+	0.917259i	$0.369604\pi$
$164$	174297.	0.506036
$165$	0	0
$166$	−234490.	−0.660471
$167$	−575770.	−1.59756	−0.798781	−	0.601622i	$-0.794521\pi$
$167$	−575770.	−1.59756	−0.798781	+	0.601622i	$0.794521\pi$
$168$	−87106.8	−0.238111
$169$	409306.	1.10238
$170$	0	0
$171$	1.28988e6	3.37332
$172$	−245368.	−0.632407
$173$	−408907.	−1.03875	−0.519373	−	0.854548i	$-0.673835\pi$
$173$	−408907.	−1.03875	−0.519373	+	0.854548i	$0.673835\pi$
$174$	61569.8	0.154168
$175$	0	0
$176$	−123308.	−0.300060
$177$	1.12717e6	2.70430
$178$	113483.	0.268461
$179$	701240.	1.63581	0.817907	−	0.575350i	$-0.195134\pi$
$179$	701240.	1.63581	0.817907	+	0.575350i	$0.195134\pi$
$180$	0	0
$181$	−653879.	−1.48355	−0.741773	−	0.670652i	$-0.766015\pi$
$181$	−653879.	−1.48355	−0.741773	+	0.670652i	$0.766015\pi$
$182$	−173169.	−0.387518
$183$	46539.3	0.102729
$184$	−10833.5	−0.0235898
$185$	0	0
$186$	−859348.	−1.82132
$187$	−886669.	−1.85420
$188$	−155294.	−0.320449
$189$	−388616.	−0.791345
$190$	0	0
$191$	−205677.	−0.407946	−0.203973	−	0.978977i	$-0.565385\pi$
$191$	−205677.	−0.407946	−0.203973	+	0.978977i	$0.565385\pi$
$192$	−113772.	−0.222732
$193$	−163999.	−0.316918	−0.158459	−	0.987366i	$-0.550653\pi$
$193$	−163999.	−0.316918	−0.158459	+	0.987366i	$0.550653\pi$
$194$	470191.	0.896954
$195$	0	0
$196$	38416.0	0.0714286
$197$	77300.2	0.141911	0.0709553	−	0.997479i	$-0.477395\pi$
$197$	77300.2	0.141911	0.0709553	+	0.997479i	$0.477395\pi$
$198$	−1.01831e6	−1.84593
$199$	−466915.	−0.835806	−0.417903	−	0.908492i	$-0.637235\pi$
$199$	−466915.	−0.835806	−0.417903	+	0.908492i	$0.637235\pi$
$200$	0	0
$201$	274578.	0.479376
$202$	−100914.	−0.174010
$203$	−27153.6	−0.0462474
$204$	−818101.	−1.37636
$205$	0	0
$206$	−396667.	−0.651265
$207$	−89465.8	−0.145121
$208$	−226180.	−0.362490
$209$	−1.17552e6	−1.86151
$210$	0	0
$211$	1.09847e6	1.69857	0.849284	−	0.527935i	$-0.177034\pi$
$211$	1.09847e6	1.69857	0.849284	+	0.527935i	$0.177034\pi$
$212$	−141266.	−0.215873
$213$	1.42632e6	2.15411
$214$	−509970.	−0.761221
$215$	0	0
$216$	−507580.	−0.740235
$217$	378991.	0.546361
$218$	589680.	0.840379
$219$	742138.	1.04562
$220$	0	0
$221$	−1.62639e6	−2.23998
$222$	1.26947e6	1.72878
$223$	1.33836e6	1.80224	0.901119	−	0.433573i	$-0.142747\pi$
$223$	1.33836e6	1.80224	0.901119	+	0.433573i	$0.142747\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	352637.	0.459258
$227$	1.06702e6	1.37438	0.687189	−	0.726479i	$-0.258844\pi$
$227$	1.06702e6	1.37438	0.687189	+	0.726479i	$0.258844\pi$
$228$	−1.08462e6	−1.38178
$229$	395172.	0.497964	0.248982	−	0.968508i	$-0.419904\pi$
$229$	395172.	0.497964	0.248982	+	0.968508i	$0.419904\pi$
$230$	0	0
$231$	655575.	0.808337
$232$	−35466.0	−0.0432605
$233$	−1.40396e6	−1.69420	−0.847099	−	0.531436i	$-0.821653\pi$
$233$	−1.40396e6	−1.69420	−0.847099	+	0.531436i	$0.821653\pi$
$234$	−1.86785e6	−2.22999
$235$	0	0
$236$	−649280.	−0.758843
$237$	1.95093e6	2.25616
$238$	360801.	0.412881
$239$	−617173.	−0.698896	−0.349448	−	0.936956i	$-0.613631\pi$
$239$	−617173.	−0.698896	−0.349448	+	0.936956i	$0.613631\pi$
$240$	0	0
$241$	−1.16204e6	−1.28878	−0.644390	−	0.764697i	$-0.722889\pi$
$241$	−1.16204e6	−1.28878	−0.644390	+	0.764697i	$0.722889\pi$
$242$	283823.	0.311537
$243$	−624333.	−0.678267
$244$	−26808.0	−0.0288263
$245$	0	0
$246$	−1.21034e6	−1.27517
$247$	−2.15623e6	−2.24881
$248$	495009.	0.511074
$249$	1.62832e6	1.66434
$250$	0	0
$251$	−1.39683e6	−1.39946	−0.699730	−	0.714407i	$-0.746697\pi$
$251$	−1.39683e6	−1.39946	−0.699730	+	0.714407i	$0.746697\pi$
$252$	414366.	0.411039
$253$	81534.2	0.0800826
$254$	−356692.	−0.346903
$255$	0	0
$256$	65536.0	0.0625000
$257$	566476.	0.534994	0.267497	−	0.963559i	$-0.413803\pi$
$257$	566476.	0.534994	0.267497	+	0.963559i	$0.413803\pi$
$258$	1.70386e6	1.59362
$259$	−559864.	−0.518601
$260$	0	0
$261$	−292887.	−0.266133
$262$	1.19255e6	1.07331
$263$	−670643.	−0.597864	−0.298932	−	0.954274i	$-0.596630\pi$
$263$	−670643.	−0.597864	−0.298932	+	0.954274i	$0.596630\pi$
$264$	856261.	0.756130
$265$	0	0
$266$	478340.	0.414507
$267$	−788036.	−0.676501
$268$	−158165.	−0.134516
$269$	19288.8	0.0162526	0.00812632	−	0.999967i	$-0.497413\pi$
$269$	19288.8	0.0162526	0.00812632	+	0.999967i	$0.497413\pi$
$270$	0	0
$271$	−1.92342e6	−1.59093	−0.795466	−	0.605999i	$-0.792774\pi$
$271$	−1.92342e6	−1.59093	−0.795466	+	0.605999i	$0.792774\pi$
$272$	471250.	0.386215
$273$	1.20250e6	0.976516
$274$	95968.0	0.0772236
$275$	0	0
$276$	75229.0	0.0594446
$277$	−1.07324e6	−0.840423	−0.420212	−	0.907426i	$-0.638044\pi$
$277$	−1.07324e6	−0.840423	−0.420212	+	0.907426i	$0.638044\pi$
$278$	−1.50569e6	−1.16849
$279$	4.08791e6	3.14406
$280$	0	0
$281$	1.05444e6	0.796627	0.398313	−	0.917249i	$-0.369596\pi$
$281$	1.05444e6	0.796627	0.398313	+	0.917249i	$0.369596\pi$
$282$	1.07837e6	0.807508
$283$	−2.29016e6	−1.69981	−0.849904	−	0.526937i	$-0.823340\pi$
$283$	−2.29016e6	−1.69981	−0.849904	+	0.526937i	$0.823340\pi$
$284$	−821600.	−0.604456
$285$	0	0
$286$	1.70225e6	1.23058
$287$	533786.	0.382527
$288$	541212.	0.384491
$289$	1.96876e6	1.38659
$290$	0	0
$291$	−3.26506e6	−2.26026
$292$	−427493.	−0.293408
$293$	2.41541e6	1.64370	0.821848	−	0.569707i	$-0.192944\pi$
$293$	2.41541e6	1.64370	0.821848	+	0.569707i	$0.192944\pi$
$294$	−266764.	−0.179995
$295$	0	0
$296$	−731251.	−0.485106
$297$	3.82010e6	2.51295
$298$	−34871.2	−0.0227471
$299$	149556.	0.0967443
$300$	0	0
$301$	−751439.	−0.478055
$302$	−1.79021e6	−1.12950
$303$	700757.	0.438491
$304$	624770.	0.387736
$305$	0	0
$306$	3.89170e6	2.37594
$307$	810075.	0.490545	0.245273	−	0.969454i	$-0.421123\pi$
$307$	810075.	0.490545	0.245273	+	0.969454i	$0.421123\pi$
$308$	−377630.	−0.226824
$309$	2.75449e6	1.64114
$310$	0	0
$311$	2.00540e6	1.17571	0.587856	−	0.808966i	$-0.299972\pi$
$311$	2.00540e6	1.17571	0.587856	+	0.808966i	$0.299972\pi$
$312$	1.57062e6	0.913447
$313$	657277.	0.379217	0.189608	−	0.981860i	$-0.439278\pi$
$313$	657277.	0.379217	0.189608	+	0.981860i	$0.439278\pi$
$314$	−255526.	−0.146255
$315$	0	0
$316$	−1.12379e6	−0.633094
$317$	−312965.	−0.174923	−0.0874616	−	0.996168i	$-0.527876\pi$
$317$	−312965.	−0.174923	−0.0874616	+	0.996168i	$0.527876\pi$
$318$	980965.	0.543983
$319$	266921.	0.146861
$320$	0	0
$321$	3.54128e6	1.91822
$322$	−33177.6	−0.0178322
$323$	4.49254e6	2.39599
$324$	1.46977e6	0.777832
$325$	0	0
$326$	1.08083e6	0.563267
$327$	−4.09480e6	−2.11769
$328$	697190.	0.357821
$329$	−475587.	−0.242237
$330$	0	0
$331$	−252813.	−0.126832	−0.0634162	−	0.997987i	$-0.520200\pi$
$331$	−252813.	−0.126832	−0.0634162	+	0.997987i	$0.520200\pi$
$332$	−937959.	−0.467024
$333$	−6.03885e6	−2.98431
$334$	−2.30308e6	−1.12965
$335$	0	0
$336$	−348427.	−0.168370
$337$	748958.	0.359238	0.179619	−	0.983736i	$-0.442513\pi$
$337$	748958.	0.359238	0.179619	+	0.983736i	$0.442513\pi$
$338$	1.63722e6	0.779501
$339$	−2.44875e6	−1.15730
$340$	0	0
$341$	−3.72549e6	−1.73499
$342$	5.15951e6	2.38530
$343$	117649.	0.0539949
$344$	−981471.	−0.447179
$345$	0	0
$346$	−1.63563e6	−0.734505
$347$	772587.	0.344448	0.172224	−	0.985058i	$-0.444905\pi$
$347$	772587.	0.344448	0.172224	+	0.985058i	$0.444905\pi$
$348$	246279.	0.109013
$349$	2.00308e6	0.880309	0.440155	−	0.897922i	$-0.354924\pi$
$349$	2.00308e6	0.880309	0.440155	+	0.897922i	$0.354924\pi$
$350$	0	0
$351$	7.00710e6	3.03578
$352$	−493231.	−0.212175
$353$	1.10664e6	0.472684	0.236342	−	0.971670i	$-0.424052\pi$
$353$	1.10664e6	0.472684	0.236342	+	0.971670i	$0.424052\pi$
$354$	4.50867e6	1.91223
$355$	0	0
$356$	453932.	0.189830
$357$	−2.50543e6	−1.04043
$358$	2.80496e6	1.15670
$359$	−4.22306e6	−1.72938	−0.864692	−	0.502302i	$-0.832486\pi$
$359$	−4.22306e6	−1.72938	−0.864692	+	0.502302i	$0.832486\pi$
$360$	0	0
$361$	3.47998e6	1.40543
$362$	−2.61551e6	−1.04902
$363$	−1.97090e6	−0.785050
$364$	−692676.	−0.274016
$365$	0	0
$366$	186157.	0.0726402
$367$	−721326.	−0.279555	−0.139777	−	0.990183i	$-0.544639\pi$
$367$	−721326.	−0.279555	−0.139777	+	0.990183i	$0.544639\pi$
$368$	−43334.0	−0.0166805
$369$	5.75756e6	2.20127
$370$	0	0
$371$	−432627.	−0.163184
$372$	−3.43739e6	−1.28787
$373$	2.91702e6	1.08559	0.542797	−	0.839864i	$-0.317365\pi$
$373$	2.91702e6	1.08559	0.542797	+	0.839864i	$0.317365\pi$
$374$	−3.54668e6	−1.31112
$375$	0	0
$376$	−621174.	−0.226592
$377$	489605.	0.177416
$378$	−1.55446e6	−0.559565
$379$	−75909.9	−0.0271457	−0.0135728	−	0.999908i	$-0.504321\pi$
$379$	−75909.9	−0.0271457	−0.0135728	+	0.999908i	$0.504321\pi$
$380$	0	0
$381$	2.47690e6	0.874170
$382$	−822708.	−0.288461
$383$	1.48698e6	0.517974	0.258987	−	0.965881i	$-0.416611\pi$
$383$	1.48698e6	0.517974	0.258987	+	0.965881i	$0.416611\pi$
$384$	−455088.	−0.157495
$385$	0	0
$386$	−655995.	−0.224095
$387$	−8.10523e6	−2.75098
$388$	1.88077e6	0.634243
$389$	2.38812e6	0.800169	0.400084	−	0.916478i	$-0.368981\pi$
$389$	2.38812e6	0.800169	0.400084	+	0.916478i	$0.368981\pi$
$390$	0	0
$391$	−311602.	−0.103076
$392$	153664.	0.0505076
$393$	−8.28121e6	−2.70466
$394$	309201.	0.100346
$395$	0	0
$396$	−4.07322e6	−1.30527
$397$	931473.	0.296616	0.148308	−	0.988941i	$-0.452617\pi$
$397$	931473.	0.296616	0.148308	+	0.988941i	$0.452617\pi$
$398$	−1.86766e6	−0.591004
$399$	−3.32164e6	−1.04453
$400$	0	0
$401$	−502048.	−0.155914	−0.0779568	−	0.996957i	$-0.524840\pi$
$401$	−502048.	−0.155914	−0.0779568	+	0.996957i	$0.524840\pi$
$402$	1.09831e6	0.338970
$403$	−6.83356e6	−2.09597
$404$	−403656.	−0.123043
$405$	0	0
$406$	−108614.	−0.0327019
$407$	5.50347e6	1.64684
$408$	−3.27240e6	−0.973232
$409$	2.46221e6	0.727807	0.363903	−	0.931437i	$-0.381444\pi$
$409$	2.46221e6	0.727807	0.363903	+	0.931437i	$0.381444\pi$
$410$	0	0
$411$	−666411.	−0.194598
$412$	−1.58667e6	−0.460514
$413$	−1.98842e6	−0.573632
$414$	−357863.	−0.102616
$415$	0	0
$416$	−904720.	−0.256319
$417$	1.04557e7	2.94450
$418$	−4.70209e6	−1.31628
$419$	−1.72560e6	−0.480182	−0.240091	−	0.970750i	$-0.577177\pi$
$419$	−1.72560e6	−0.480182	−0.240091	+	0.970750i	$0.577177\pi$
$420$	0	0
$421$	−4.78043e6	−1.31450	−0.657251	−	0.753671i	$-0.728281\pi$
$421$	−4.78043e6	−1.31450	−0.657251	+	0.753671i	$0.728281\pi$
$422$	4.39389e6	1.20107
$423$	−5.12981e6	−1.39396
$424$	−565064.	−0.152645
$425$	0	0
$426$	5.70527e6	1.52318
$427$	−82099.4	−0.0217907
$428$	−2.03988e6	−0.538264
$429$	−1.18206e7	−3.10097
$430$	0	0
$431$	−3.18383e6	−0.825574	−0.412787	−	0.910828i	$-0.635445\pi$
$431$	−3.18383e6	−0.825574	−0.412787	+	0.910828i	$0.635445\pi$
$432$	−2.03032e6	−0.523425
$433$	6.03313e6	1.54640	0.773202	−	0.634159i	$-0.218654\pi$
$433$	6.03313e6	1.54640	0.773202	+	0.634159i	$0.218654\pi$
$434$	1.51597e6	0.386336
$435$	0	0
$436$	2.35872e6	0.594238
$437$	−413114.	−0.103482
$438$	2.96855e6	0.739366
$439$	−5.13976e6	−1.27286	−0.636431	−	0.771334i	$-0.719590\pi$
$439$	−5.13976e6	−1.27286	−0.636431	+	0.771334i	$0.719590\pi$
$440$	0	0
$441$	1.26900e6	0.310716
$442$	−6.50557e6	−1.58391
$443$	−1.03796e6	−0.251287	−0.125644	−	0.992075i	$-0.540100\pi$
$443$	−1.03796e6	−0.251287	−0.125644	+	0.992075i	$0.540100\pi$
$444$	5.07788e6	1.22243
$445$	0	0
$446$	5.35345e6	1.27437
$447$	242149.	0.0573210
$448$	200704.	0.0472456
$449$	−7.38979e6	−1.72988	−0.864940	−	0.501875i	$-0.832644\pi$
$449$	−7.38979e6	−1.72988	−0.864940	+	0.501875i	$0.832644\pi$
$450$	0	0
$451$	−5.24712e6	−1.21473
$452$	1.41055e6	0.324745
$453$	1.24314e7	2.84625
$454$	4.26806e6	0.971832
$455$	0	0
$456$	−4.33846e6	−0.977066
$457$	−2.17988e6	−0.488250	−0.244125	−	0.969744i	$-0.578501\pi$
$457$	−2.17988e6	−0.488250	−0.244125	+	0.969744i	$0.578501\pi$
$458$	1.58069e6	0.352113
$459$	−1.45994e7	−3.23447
$460$	0	0
$461$	−7.41009e6	−1.62394	−0.811972	−	0.583696i	$-0.801606\pi$
$461$	−7.41009e6	−1.62394	−0.811972	+	0.583696i	$0.801606\pi$
$462$	2.62230e6	0.571580
$463$	1.73426e6	0.375978	0.187989	−	0.982171i	$-0.439803\pi$
$463$	1.73426e6	0.375978	0.187989	+	0.982171i	$0.439803\pi$
$464$	−141864.	−0.0305898
$465$	0	0
$466$	−5.61583e6	−1.19798
$467$	−1.41704e6	−0.300670	−0.150335	−	0.988635i	$-0.548035\pi$
$467$	−1.41704e6	−0.300670	−0.150335	+	0.988635i	$0.548035\pi$
$468$	−7.47140e6	−1.57684
$469$	−484380.	−0.101684
$470$	0	0
$471$	1.77440e6	0.368553
$472$	−2.59712e6	−0.536583
$473$	7.38666e6	1.51808
$474$	7.80372e6	1.59535
$475$	0	0
$476$	1.44320e6	0.291951
$477$	−4.66644e6	−0.939051
$478$	−2.46869e6	−0.494194
$479$	−3.71353e6	−0.739517	−0.369758	−	0.929128i	$-0.620560\pi$
$479$	−3.71353e6	−0.739517	−0.369758	+	0.929128i	$0.620560\pi$
$480$	0	0
$481$	1.00949e7	1.98947
$482$	−4.64817e6	−0.911306
$483$	230389.	0.0449359
$484$	1.13529e6	0.220290
$485$	0	0
$486$	−2.49733e6	−0.479607
$487$	−6.53152e6	−1.24794	−0.623968	−	0.781450i	$-0.714480\pi$
$487$	−6.53152e6	−1.24794	−0.623968	+	0.781450i	$0.714480\pi$
$488$	−107232.	−0.0203833
$489$	−7.50541e6	−1.41939
$490$	0	0
$491$	1.03633e7	1.93997	0.969986	−	0.243159i	$-0.0781836\pi$
$491$	1.03633e7	1.93997	0.969986	+	0.243159i	$0.0781836\pi$
$492$	−4.84135e6	−0.901683
$493$	−1.02010e6	−0.189028
$494$	−8.62491e6	−1.59015
$495$	0	0
$496$	1.98004e6	0.361384
$497$	−2.51615e6	−0.456926
$498$	6.51328e6	1.17686
$499$	−1.34074e6	−0.241043	−0.120521	−	0.992711i	$-0.538457\pi$
$499$	−1.34074e6	−0.241043	−0.120521	+	0.992711i	$0.538457\pi$
$500$	0	0
$501$	1.59928e7	2.84663
$502$	−5.58734e6	−0.989568
$503$	−1.07226e6	−0.188964	−0.0944820	−	0.995527i	$-0.530119\pi$
$503$	−1.07226e6	−0.188964	−0.0944820	+	0.995527i	$0.530119\pi$
$504$	1.65746e6	0.290648
$505$	0	0
$506$	326137.	0.0566270
$507$	−1.13690e7	−1.96428
$508$	−1.42677e6	−0.245298
$509$	7.26315e6	1.24260	0.621299	−	0.783574i	$-0.286605\pi$
$509$	7.26315e6	1.24260	0.621299	+	0.783574i	$0.286605\pi$
$510$	0	0
$511$	−1.30920e6	−0.221796
$512$	262144.	0.0441942
$513$	−1.93555e7	−3.24721
$514$	2.26591e6	0.378298
$515$	0	0
$516$	6.81543e6	1.12686
$517$	4.67502e6	0.769232
$518$	−2.23946e6	−0.366706
$519$	1.13580e7	1.85090
$520$	0	0
$521$	−3.31930e6	−0.535738	−0.267869	−	0.963455i	$-0.586320\pi$
$521$	−3.31930e6	−0.535738	−0.267869	+	0.963455i	$0.586320\pi$
$522$	−1.17155e6	−0.188184
$523$	4.01229e6	0.641414	0.320707	−	0.947178i	$-0.396080\pi$
$523$	4.01229e6	0.641414	0.320707	+	0.947178i	$0.396080\pi$
$524$	4.77021e6	0.758944
$525$	0	0
$526$	−2.68257e6	−0.422754
$527$	1.42379e7	2.23315
$528$	3.42504e6	0.534664
$529$	−6.40769e6	−0.995548
$530$	0	0
$531$	−2.14477e7	−3.30099
$532$	1.91336e6	0.293101
$533$	−9.62465e6	−1.46746
$534$	−3.15215e6	−0.478358
$535$	0	0
$536$	−632660.	−0.0951171
$537$	−1.94779e7	−2.91479
$538$	77155.1	0.0114923
$539$	−1.15649e6	−0.171463
$540$	0	0
$541$	−1.20893e7	−1.77586	−0.887929	−	0.459981i	$-0.847856\pi$
$541$	−1.20893e7	−1.77586	−0.887929	+	0.459981i	$0.847856\pi$
$542$	−7.69369e6	−1.12496
$543$	1.81624e7	2.64346
$544$	1.88500e6	0.273095
$545$	0	0
$546$	4.81001e6	0.690501
$547$	4.43431e6	0.633663	0.316831	−	0.948482i	$-0.397381\pi$
$547$	4.43431e6	0.633663	0.316831	+	0.948482i	$0.397381\pi$
$548$	383872.	0.0546053
$549$	−885548.	−0.125395
$550$	0	0
$551$	−1.35242e6	−0.189772
$552$	300916.	0.0420337
$553$	−3.44161e6	−0.478574
$554$	−4.29297e6	−0.594269
$555$	0	0
$556$	−6.02276e6	−0.826245
$557$	390925.	0.0533895	0.0266947	−	0.999644i	$-0.491502\pi$
$557$	390925.	0.0533895	0.0266947	+	0.999644i	$0.491502\pi$
$558$	1.63516e7	2.22318
$559$	1.35491e7	1.83393
$560$	0	0
$561$	2.46285e7	3.30392
$562$	4.21775e6	0.563300
$563$	2.64029e6	0.351059	0.175530	−	0.984474i	$-0.443836\pi$
$563$	2.64029e6	0.351059	0.175530	+	0.984474i	$0.443836\pi$
$564$	4.31350e6	0.570994
$565$	0	0
$566$	−9.16065e6	−1.20195
$567$	4.50116e6	0.587985
$568$	−3.28640e6	−0.427415
$569$	−228685.	−0.0296112	−0.0148056	−	0.999890i	$-0.504713\pi$
$569$	−228685.	−0.0296112	−0.0148056	+	0.999890i	$0.504713\pi$
$570$	0	0
$571$	5.96584e6	0.765740	0.382870	−	0.923802i	$-0.374936\pi$
$571$	5.96584e6	0.765740	0.382870	+	0.923802i	$0.374936\pi$
$572$	6.80902e6	0.870150
$573$	5.71296e6	0.726900
$574$	2.13514e6	0.270488
$575$	0	0
$576$	2.16485e6	0.271876
$577$	7.86646e6	0.983648	0.491824	−	0.870695i	$-0.336330\pi$
$577$	7.86646e6	0.983648	0.491824	+	0.870695i	$0.336330\pi$
$578$	7.87504e6	0.980467
$579$	4.55530e6	0.564703
$580$	0	0
$581$	−2.87250e6	−0.353037
$582$	−1.30602e7	−1.59824
$583$	4.25273e6	0.518199
$584$	−1.70997e6	−0.207471
$585$	0	0
$586$	9.66163e6	1.16227
$587$	−1.19474e7	−1.43113	−0.715565	−	0.698546i	$-0.753831\pi$
$587$	−1.19474e7	−1.43113	−0.715565	+	0.698546i	$0.753831\pi$
$588$	−1.06706e6	−0.127275
$589$	1.88762e7	2.24195
$590$	0	0
$591$	−2.14712e6	−0.252864
$592$	−2.92500e6	−0.343022
$593$	−7.64277e6	−0.892512	−0.446256	−	0.894905i	$-0.647243\pi$
$593$	−7.64277e6	−0.892512	−0.446256	+	0.894905i	$0.647243\pi$
$594$	1.52804e7	1.77692
$595$	0	0
$596$	−139485.	−0.0160846
$597$	1.29692e7	1.48929
$598$	598223.	0.0684085
$599$	615573.	0.0700991	0.0350496	−	0.999386i	$-0.488841\pi$
$599$	615573.	0.0700991	0.0350496	+	0.999386i	$0.488841\pi$
$600$	0	0
$601$	−6.68100e6	−0.754493	−0.377247	−	0.926113i	$-0.623129\pi$
$601$	−6.68100e6	−0.754493	−0.377247	+	0.926113i	$0.623129\pi$
$602$	−3.00576e6	−0.338036
$603$	−5.22466e6	−0.585147
$604$	−7.16082e6	−0.798676
$605$	0	0
$606$	2.80303e6	0.310060
$607$	810442.	0.0892792	0.0446396	−	0.999003i	$-0.485786\pi$
$607$	810442.	0.0892792	0.0446396	+	0.999003i	$0.485786\pi$
$608$	2.49908e6	0.274171
$609$	754230.	0.0824063
$610$	0	0
$611$	8.57527e6	0.929276
$612$	1.55668e7	1.68004
$613$	4.96233e6	0.533377	0.266688	−	0.963783i	$-0.414071\pi$
$613$	4.96233e6	0.533377	0.266688	+	0.963783i	$0.414071\pi$
$614$	3.24030e6	0.346868
$615$	0	0
$616$	−1.51052e6	−0.160389
$617$	−2.35809e6	−0.249371	−0.124686	−	0.992196i	$-0.539792\pi$
$617$	−2.35809e6	−0.249371	−0.124686	+	0.992196i	$0.539792\pi$
$618$	1.10180e7	1.16046
$619$	7.68885e6	0.806556	0.403278	−	0.915077i	$-0.367871\pi$
$619$	7.68885e6	0.806556	0.403278	+	0.915077i	$0.367871\pi$
$620$	0	0
$621$	1.34250e6	0.139696
$622$	8.02162e6	0.831354
$623$	1.39017e6	0.143498
$624$	6.28246e6	0.645905
$625$	0	0
$626$	2.62911e6	0.268147
$627$	3.26517e7	3.31694
$628$	−1.02211e6	−0.103418
$629$	−2.10328e7	−2.11968
$630$	0	0
$631$	−1.65595e7	−1.65567	−0.827835	−	0.560971i	$-0.810428\pi$
$631$	−1.65595e7	−1.65567	−0.827835	+	0.560971i	$0.810428\pi$
$632$	−4.49517e6	−0.447665
$633$	−3.05116e7	−3.02661
$634$	−1.25186e6	−0.123689
$635$	0	0
$636$	3.92386e6	0.384654
$637$	−2.12132e6	−0.207137
$638$	1.06768e6	0.103846
$639$	−2.71399e7	−2.62940
$640$	0	0
$641$	3.81994e6	0.367207	0.183604	−	0.983000i	$-0.441224\pi$
$641$	3.81994e6	0.367207	0.183604	+	0.983000i	$0.441224\pi$
$642$	1.41651e7	1.35639
$643$	−1.54152e7	−1.47035	−0.735176	−	0.677877i	$-0.762900\pi$
$643$	−1.54152e7	−1.47035	−0.735176	+	0.677877i	$0.762900\pi$
$644$	−132711.	−0.0126093
$645$	0	0
$646$	1.79701e7	1.69422
$647$	1.24057e7	1.16509	0.582546	−	0.812798i	$-0.302057\pi$
$647$	1.24057e7	1.16509	0.582546	+	0.812798i	$0.302057\pi$
$648$	5.87906e6	0.550010
$649$	1.95462e7	1.82159
$650$	0	0
$651$	−1.05270e7	−0.973537
$652$	4.32333e6	0.398290
$653$	−9.76650e6	−0.896306	−0.448153	−	0.893957i	$-0.647918\pi$
$653$	−9.76650e6	−0.896306	−0.448153	+	0.893957i	$0.647918\pi$
$654$	−1.63792e7	−1.49744
$655$	0	0
$656$	2.78876e6	0.253018
$657$	−1.41214e7	−1.27633
$658$	−1.90235e6	−0.171287
$659$	1.51635e7	1.36014	0.680072	−	0.733146i	$-0.261949\pi$
$659$	1.51635e7	1.36014	0.680072	+	0.733146i	$0.261949\pi$
$660$	0	0
$661$	1.84762e6	0.164479	0.0822394	−	0.996613i	$-0.473793\pi$
$661$	1.84762e6	0.164479	0.0822394	+	0.996613i	$0.473793\pi$
$662$	−1.01125e6	−0.0896840
$663$	4.51753e7	3.99133
$664$	−3.75184e6	−0.330236
$665$	0	0
$666$	−2.41554e7	−2.11022
$667$	93803.9	0.00816406
$668$	−9.21232e6	−0.798781
$669$	−3.71749e7	−3.21133
$670$	0	0
$671$	807039.	0.0691971
$672$	−1.39371e6	−0.119055
$673$	−1.70205e7	−1.44855	−0.724275	−	0.689511i	$-0.757825\pi$
$673$	−1.70205e7	−1.44855	−0.724275	+	0.689511i	$0.757825\pi$
$674$	2.99583e6	0.254020
$675$	0	0
$676$	6.54890e6	0.551190
$677$	5.61056e6	0.470472	0.235236	−	0.971938i	$-0.424414\pi$
$677$	5.61056e6	0.470472	0.235236	+	0.971938i	$0.424414\pi$
$678$	−9.79499e6	−0.818332
$679$	5.75985e6	0.479442
$680$	0	0
$681$	−2.96378e7	−2.44894
$682$	−1.49020e7	−1.22683
$683$	−1.73009e7	−1.41911	−0.709556	−	0.704649i	$-0.751105\pi$
$683$	−1.73009e7	−1.41911	−0.709556	+	0.704649i	$0.751105\pi$
$684$	2.06380e7	1.68666
$685$	0	0
$686$	470596.	0.0381802
$687$	−1.09765e7	−0.887299
$688$	−3.92589e6	−0.316203
$689$	7.80066e6	0.626013
$690$	0	0
$691$	−9.30689e6	−0.741497	−0.370748	−	0.928733i	$-0.620899\pi$
$691$	−9.30689e6	−0.741497	−0.370748	+	0.928733i	$0.620899\pi$
$692$	−6.54251e6	−0.519373
$693$	−1.24742e7	−0.986691
$694$	3.09035e6	0.243562
$695$	0	0
$696$	985116.	0.0770840
$697$	2.00531e7	1.56351
$698$	8.01233e6	0.622473
$699$	3.89969e7	3.01882
$700$	0	0
$701$	1.44180e7	1.10818	0.554090	−	0.832457i	$-0.313066\pi$
$701$	1.44180e7	1.10818	0.554090	+	0.832457i	$0.313066\pi$
$702$	2.80284e7	2.14662
$703$	−2.78847e7	−2.12803
$704$	−1.97292e6	−0.150030
$705$	0	0
$706$	4.42657e6	0.334238
$707$	−1.23620e6	−0.0930120
$708$	1.80347e7	1.35215
$709$	−1.00341e7	−0.749656	−0.374828	−	0.927094i	$-0.622298\pi$
$709$	−1.00341e7	−0.749656	−0.374828	+	0.927094i	$0.622298\pi$
$710$	0	0
$711$	−3.71222e7	−2.75397
$712$	1.81573e6	0.134230
$713$	−1.30925e6	−0.0964492
$714$	−1.00217e7	−0.735695
$715$	0	0
$716$	1.12198e7	0.817907
$717$	1.71428e7	1.24533
$718$	−1.68923e7	−1.22286
$719$	−1.43364e7	−1.03423	−0.517114	−	0.855916i	$-0.672994\pi$
$719$	−1.43364e7	−1.03423	−0.517114	+	0.855916i	$0.672994\pi$
$720$	0	0
$721$	−4.85917e6	−0.348116
$722$	1.39199e7	0.993789
$723$	3.22773e7	2.29642
$724$	−1.04621e7	−0.741773
$725$	0	0
$726$	−7.88358e6	−0.555114
$727$	−577386.	−0.0405164	−0.0202582	−	0.999795i	$-0.506449\pi$
$727$	−577386.	−0.0405164	−0.0202582	+	0.999795i	$0.506449\pi$
$728$	−2.77070e6	−0.193759
$729$	−4.98035e6	−0.347089
$730$	0	0
$731$	−2.82299e7	−1.95396
$732$	744629.	0.0513644
$733$	−2.12449e7	−1.46048	−0.730240	−	0.683191i	$-0.760592\pi$
$733$	−2.12449e7	−1.46048	−0.730240	+	0.683191i	$0.760592\pi$
$734$	−2.88530e6	−0.197675
$735$	0	0
$736$	−173336.	−0.0117949
$737$	4.76147e6	0.322903
$738$	2.30303e7	1.55653
$739$	1.09365e7	0.736662	0.368331	−	0.929695i	$-0.379929\pi$
$739$	1.09365e7	0.736662	0.368331	+	0.929695i	$0.379929\pi$
$740$	0	0
$741$	5.98922e7	4.00705
$742$	−1.73051e6	−0.115389
$743$	2.42027e7	1.60839	0.804195	−	0.594366i	$-0.202597\pi$
$743$	2.42027e7	1.60839	0.804195	+	0.594366i	$0.202597\pi$
$744$	−1.37496e7	−0.910661
$745$	0	0
$746$	1.16681e7	0.767631
$747$	−3.09836e7	−2.03156
$748$	−1.41867e7	−0.927102
$749$	−6.24714e6	−0.406890
$750$	0	0
$751$	−1.22083e7	−0.789871	−0.394936	−	0.918709i	$-0.629233\pi$
$751$	−1.22083e7	−0.789871	−0.394936	+	0.918709i	$0.629233\pi$
$752$	−2.48470e6	−0.160225
$753$	3.87990e7	2.49364
$754$	1.95842e6	0.125452
$755$	0	0
$756$	−6.21785e6	−0.395672
$757$	6.73453e6	0.427138	0.213569	−	0.976928i	$-0.431491\pi$
$757$	6.73453e6	0.427138	0.213569	+	0.976928i	$0.431491\pi$
$758$	−303640.	−0.0191949
$759$	−2.26472e6	−0.142696
$760$	0	0
$761$	1.28864e7	0.806625	0.403312	−	0.915062i	$-0.367859\pi$
$761$	1.28864e7	0.806625	0.403312	+	0.915062i	$0.367859\pi$
$762$	9.90760e6	0.618132
$763$	7.22358e6	0.449202
$764$	−3.29083e6	−0.203973
$765$	0	0
$766$	5.94791e6	0.366263
$767$	3.58531e7	2.20058
$768$	−1.82035e6	−0.111366
$769$	1.11806e7	0.681787	0.340893	−	0.940102i	$-0.389270\pi$
$769$	1.11806e7	0.681787	0.340893	+	0.940102i	$0.389270\pi$
$770$	0	0
$771$	−1.57347e7	−0.953283
$772$	−2.62398e6	−0.158459
$773$	−2.70973e6	−0.163109	−0.0815543	−	0.996669i	$-0.525988\pi$
$773$	−2.70973e6	−0.163109	−0.0815543	+	0.996669i	$0.525988\pi$
$774$	−3.24209e7	−1.94524
$775$	0	0
$776$	7.52306e6	0.448477
$777$	1.55510e7	0.924072
$778$	9.55247e6	0.565805
$779$	2.65859e7	1.56967
$780$	0	0
$781$	2.47338e7	1.45099
$782$	−1.24641e6	−0.0728859
$783$	4.39497e6	0.256184
$784$	614656.	0.0357143
$785$	0	0
$786$	−3.31248e7	−1.91248
$787$	7.17969e6	0.413208	0.206604	−	0.978425i	$-0.433759\pi$
$787$	7.17969e6	0.413208	0.206604	+	0.978425i	$0.433759\pi$
$788$	1.23680e6	0.0709553
$789$	1.86281e7	1.06531
$790$	0	0
$791$	4.31981e6	0.245484
$792$	−1.62929e7	−0.922965
$793$	1.48033e6	0.0835940
$794$	3.72589e6	0.209739
$795$	0	0
$796$	−7.47064e6	−0.417903
$797$	2.09215e7	1.16667	0.583335	−	0.812232i	$-0.301748\pi$
$797$	2.09215e7	1.16667	0.583335	+	0.812232i	$0.301748\pi$
$798$	−1.32865e7	−0.738593
$799$	−1.78667e7	−0.990097
$800$	0	0
$801$	1.49947e7	0.825766
$802$	−2.00819e6	−0.110248
$803$	1.28694e7	0.704321
$804$	4.39325e6	0.239688
$805$	0	0
$806$	−2.73343e7	−1.48207
$807$	−535772.	−0.0289599
$808$	−1.61462e6	−0.0870048
$809$	−8.14429e6	−0.437504	−0.218752	−	0.975780i	$-0.570199\pi$
$809$	−8.14429e6	−0.437504	−0.218752	+	0.975780i	$0.570199\pi$
$810$	0	0
$811$	−1.99391e7	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$811$	−1.99391e7	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$812$	−434458.	−0.0231237
$813$	5.34257e7	2.83481
$814$	2.20139e7	1.16449
$815$	0	0
$816$	−1.30896e7	−0.688179
$817$	−3.74264e7	−1.96166
$818$	9.84882e6	0.514637
$819$	−2.28812e7	−1.19198
$820$	0	0
$821$	3.57285e7	1.84994	0.924968	−	0.380046i	$-0.124092\pi$
$821$	3.57285e7	1.84994	0.924968	+	0.380046i	$0.124092\pi$
$822$	−2.66564e6	−0.137601
$823$	−1.29131e7	−0.664556	−0.332278	−	0.943182i	$-0.607817\pi$
$823$	−1.29131e7	−0.664556	−0.332278	+	0.943182i	$0.607817\pi$
$824$	−6.34667e6	−0.325632
$825$	0	0
$826$	−7.95368e6	−0.405619
$827$	−365916.	−0.0186045	−0.00930225	−	0.999957i	$-0.502961\pi$
$827$	−365916.	−0.0186045	−0.00930225	+	0.999957i	$0.502961\pi$
$828$	−1.43145e6	−0.0725607
$829$	2.46028e7	1.24336	0.621682	−	0.783270i	$-0.286450\pi$
$829$	2.46028e7	1.24336	0.621682	+	0.783270i	$0.286450\pi$
$830$	0	0
$831$	2.98108e7	1.49751
$832$	−3.61888e6	−0.181245
$833$	4.41981e6	0.220694
$834$	4.18227e7	2.08208
$835$	0	0
$836$	−1.88083e7	−0.930754
$837$	−6.13419e7	−3.02652
$838$	−6.90241e6	−0.339540
$839$	−1.60455e7	−0.786950	−0.393475	−	0.919335i	$-0.628727\pi$
$839$	−1.60455e7	−0.786950	−0.393475	+	0.919335i	$0.628727\pi$
$840$	0	0
$841$	−2.02041e7	−0.985028
$842$	−1.91217e7	−0.929494
$843$	−2.92885e7	−1.41947
$844$	1.75756e7	0.849284
$845$	0	0
$846$	−2.05192e7	−0.985679
$847$	3.47683e6	0.166523
$848$	−2.26026e6	−0.107936
$849$	6.36124e7	3.02881
$850$	0	0
$851$	1.93409e6	0.0915486
$852$	2.28211e7	1.07705
$853$	−9.50336e6	−0.447203	−0.223601	−	0.974681i	$-0.571781\pi$
$853$	−9.50336e6	−0.447203	−0.223601	+	0.974681i	$0.571781\pi$
$854$	−328398.	−0.0154083
$855$	0	0
$856$	−8.15953e6	−0.380610
$857$	8.55821e6	0.398044	0.199022	−	0.979995i	$-0.436223\pi$
$857$	8.55821e6	0.398044	0.199022	+	0.979995i	$0.436223\pi$
$858$	−4.72825e7	−2.19271
$859$	3.73000e7	1.72475	0.862376	−	0.506269i	$-0.168976\pi$
$859$	3.73000e7	1.72475	0.862376	+	0.506269i	$0.168976\pi$
$860$	0	0
$861$	−1.48266e7	−0.681608
$862$	−1.27353e7	−0.583769
$863$	1.41708e7	0.647690	0.323845	−	0.946110i	$-0.395024\pi$
$863$	1.41708e7	0.647690	0.323845	+	0.946110i	$0.395024\pi$
$864$	−8.12127e6	−0.370118
$865$	0	0
$866$	2.41325e7	1.09347
$867$	−5.46850e7	−2.47070
$868$	6.06386e6	0.273181
$869$	3.38311e7	1.51973
$870$	0	0
$871$	8.73382e6	0.390085
$872$	9.43488e6	0.420190
$873$	6.21273e7	2.75897
$874$	−1.65245e6	−0.0731730
$875$	0	0
$876$	1.18742e7	0.522811
$877$	−2.66505e7	−1.17006	−0.585028	−	0.811013i	$-0.698917\pi$
$877$	−2.66505e7	−1.17006	−0.585028	+	0.811013i	$0.698917\pi$
$878$	−2.05590e7	−0.900049
$879$	−6.70913e7	−2.92883
$880$	0	0
$881$	2.56753e7	1.11449	0.557245	−	0.830348i	$-0.311859\pi$
$881$	2.56753e7	1.11449	0.557245	+	0.830348i	$0.311859\pi$
$882$	5.07598e6	0.219709
$883$	9.63276e6	0.415766	0.207883	−	0.978154i	$-0.433343\pi$
$883$	9.63276e6	0.415766	0.207883	+	0.978154i	$0.433343\pi$
$884$	−2.60223e7	−1.11999
$885$	0	0
$886$	−4.15183e6	−0.177687
$887$	6.02461e6	0.257110	0.128555	−	0.991702i	$-0.458966\pi$
$887$	6.02461e6	0.257110	0.128555	+	0.991702i	$0.458966\pi$
$888$	2.03115e7	0.864390
$889$	−4.36947e6	−0.185428
$890$	0	0
$891$	−4.42465e7	−1.86717
$892$	2.14138e7	0.901119
$893$	−2.36872e7	−0.993997
$894$	968596.	0.0405321
$895$	0	0
$896$	802816.	0.0334077
$897$	−4.15412e6	−0.172384
$898$	−2.95592e7	−1.22321
$899$	−4.28613e6	−0.176875
$900$	0	0
$901$	−1.62528e7	−0.666986
$902$	−2.09885e7	−0.858944
$903$	2.08723e7	0.851825
$904$	5.64219e6	0.229629
$905$	0	0
$906$	4.97254e7	2.01260
$907$	−1.62666e7	−0.656566	−0.328283	−	0.944579i	$-0.606470\pi$
$907$	−1.62666e7	−0.656566	−0.328283	+	0.944579i	$0.606470\pi$
$908$	1.70723e7	0.687189
$909$	−1.33340e7	−0.535241
$910$	0	0
$911$	2.90438e7	1.15946	0.579732	−	0.814807i	$-0.303157\pi$
$911$	2.90438e7	1.15946	0.579732	+	0.814807i	$0.303157\pi$
$912$	−1.73539e7	−0.690890
$913$	2.82367e7	1.12108
$914$	−8.71952e6	−0.345245
$915$	0	0
$916$	6.32275e6	0.248982
$917$	1.46088e7	0.573708
$918$	−5.83976e7	−2.28712
$919$	−4.01091e7	−1.56659	−0.783293	−	0.621652i	$-0.786462\pi$
$919$	−4.01091e7	−1.56659	−0.783293	+	0.621652i	$0.786462\pi$
$920$	0	0
$921$	−2.25009e7	−0.874081
$922$	−2.96404e7	−1.14830
$923$	4.53685e7	1.75287
$924$	1.04892e7	0.404168
$925$	0	0
$926$	6.93705e6	0.265857
$927$	−5.24123e7	−2.00325
$928$	−567455.	−0.0216303
$929$	1.54551e7	0.587533	0.293766	−	0.955877i	$-0.405091\pi$
$929$	1.54551e7	0.587533	0.293766	+	0.955877i	$0.405091\pi$
$930$	0	0
$931$	5.85966e6	0.221564
$932$	−2.24633e7	−0.847099
$933$	−5.57029e7	−2.09495
$934$	−5.66816e6	−0.212606
$935$	0	0
$936$	−2.98856e7	−1.11499
$937$	−4.63024e7	−1.72288	−0.861439	−	0.507861i	$-0.830436\pi$
$937$	−4.63024e7	−1.72288	−0.861439	+	0.507861i	$0.830436\pi$
$938$	−1.93752e6	−0.0719017
$939$	−1.82568e7	−0.675710
$940$	0	0
$941$	1.24284e7	0.457553	0.228777	−	0.973479i	$-0.426527\pi$
$941$	1.24284e7	0.457553	0.228777	+	0.973479i	$0.426527\pi$
$942$	7.09760e6	0.260606
$943$	−1.84400e6	−0.0675275
$944$	−1.03885e7	−0.379422
$945$	0	0
$946$	2.95466e7	1.07345
$947$	3.59653e7	1.30319	0.651597	−	0.758565i	$-0.274099\pi$
$947$	3.59653e7	1.30319	0.651597	+	0.758565i	$0.274099\pi$
$948$	3.12149e7	1.12808
$949$	2.36060e7	0.850859
$950$	0	0
$951$	8.69303e6	0.311688
$952$	5.77281e6	0.206440
$953$	−1.07512e7	−0.383465	−0.191733	−	0.981447i	$-0.561411\pi$
$953$	−1.07512e7	−0.383465	−0.191733	+	0.981447i	$0.561411\pi$
$954$	−1.86657e7	−0.664010
$955$	0	0
$956$	−9.87477e6	−0.349448
$957$	−7.41409e6	−0.261684
$958$	−1.48541e7	−0.522917
$959$	1.17561e6	0.0412777
$960$	0	0
$961$	3.11936e7	1.08958
$962$	4.03794e7	1.40677
$963$	−6.73834e7	−2.34146
$964$	−1.85927e7	−0.644390
$965$	0	0
$966$	921555.	0.0317745
$967$	1.57963e7	0.543235	0.271618	−	0.962405i	$-0.412441\pi$
$967$	1.57963e7	0.543235	0.271618	+	0.962405i	$0.412441\pi$
$968$	4.54117e6	0.155768
$969$	−1.24786e8	−4.26931
$970$	0	0
$971$	2.98682e7	1.01662	0.508312	−	0.861173i	$-0.330270\pi$
$971$	2.98682e7	1.01662	0.508312	+	0.861173i	$0.330270\pi$
$972$	−9.98933e6	−0.339133
$973$	−1.84447e7	−0.624582
$974$	−2.61261e7	−0.882424
$975$	0	0
$976$	−428928.	−0.0144132
$977$	−8.43017e6	−0.282553	−0.141276	−	0.989970i	$-0.545121\pi$
$977$	−8.43017e6	−0.282553	−0.141276	+	0.989970i	$0.545121\pi$
$978$	−3.00216e7	−1.00366
$979$	−1.36654e7	−0.455684
$980$	0	0
$981$	7.79156e7	2.58495
$982$	4.14533e7	1.37177
$983$	4.42301e7	1.45994	0.729969	−	0.683480i	$-0.239535\pi$
$983$	4.42301e7	1.45994	0.729969	+	0.683480i	$0.239535\pi$
$984$	−1.93654e7	−0.637586
$985$	0	0
$986$	−4.08040e6	−0.133663
$987$	1.32101e7	0.431631
$988$	−3.44996e7	−1.12440
$989$	2.59589e6	0.0843910
$990$	0	0
$991$	−7.69521e6	−0.248907	−0.124453	−	0.992225i	$-0.539718\pi$
$991$	−7.69521e6	−0.248907	−0.124453	+	0.992225i	$0.539718\pi$
$992$	7.92015e6	0.255537
$993$	7.02224e6	0.225997
$994$	−1.00646e7	−0.323095
$995$	0	0
$996$	2.60531e7	0.832169
$997$	−4.71077e6	−0.150091	−0.0750453	−	0.997180i	$-0.523910\pi$
$997$	−4.71077e6	−0.150091	−0.0750453	+	0.997180i	$0.523910\pi$
$998$	−5.36297e6	−0.170443
$999$	9.06172e7	2.87274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.s.1.1	yes	2
5.2	odd	4		350.6.c.i.99.4		4
5.3	odd	4		350.6.c.i.99.1		4
5.4	even	2		350.6.a.r.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.r.1.2	✓	2	5.4	even	2
350.6.a.s.1.1	yes	2	1.1	even	1	trivial
350.6.c.i.99.1		4	5.3	odd	4
350.6.c.i.99.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -27.7764 q^{3} +16.0000 q^{4} -111.106 q^{6} +49.0000 q^{7} +64.0000 q^{8} +528.528 q^{9} -481.671 q^{11} -444.422 q^{12} -883.515 q^{13} +196.000 q^{14} +256.000 q^{16} +1840.82 q^{17} +2114.11 q^{18} +2440.51 q^{19} -1361.04 q^{21} -1926.68 q^{22} -169.274 q^{23} -1777.69 q^{24} -3534.06 q^{26} -7930.93 q^{27} +784.000 q^{28} -554.156 q^{29} +7734.52 q^{31} +1024.00 q^{32} +13379.1 q^{33} +7363.28 q^{34} +8456.44 q^{36} -11425.8 q^{37} +9762.03 q^{38} +24540.9 q^{39} +10893.6 q^{41} -5444.17 q^{42} -15335.5 q^{43} -7706.73 q^{44} -677.094 q^{46} -9705.85 q^{47} -7110.76 q^{48} +2401.00 q^{49} -51131.3 q^{51} -14136.2 q^{52} -8829.12 q^{53} -31723.7 q^{54} +3136.00 q^{56} -67788.5 q^{57} -2216.62 q^{58} -40580.0 q^{59} -1675.50 q^{61} +30938.1 q^{62} +25897.9 q^{63} +4096.00 q^{64} +53516.3 q^{66} -9885.31 q^{67} +29453.1 q^{68} +4701.81 q^{69} -51350.0 q^{71} +33825.8 q^{72} -26718.3 q^{73} -45703.2 q^{74} +39048.1 q^{76} -23601.9 q^{77} +98163.5 q^{78} -70237.0 q^{79} +91860.4 q^{81} +43574.3 q^{82} -58622.5 q^{83} -21776.7 q^{84} -61342.0 q^{86} +15392.4 q^{87} -30826.9 q^{88} +28370.7 q^{89} -43292.2 q^{91} -2708.38 q^{92} -214837. q^{93} -38823.4 q^{94} -28443.0 q^{96} +117548. q^{97} +9604.00 q^{98} -254576. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 20 q^{3} + 32 q^{4} - 80 q^{6} + 98 q^{7} + 128 q^{8} + 346 q^{9} - 1070 q^{11} - 320 q^{12} - 736 q^{13} + 392 q^{14} + 512 q^{16} + 1904 q^{17} + 1384 q^{18} + 828 q^{19} - 980 q^{21}+ \cdots - 147190 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 - 20 * q^3 + 32 * q^4 - 80 * q^6 + 98 * q^7 + 128 * q^8 + 346 * q^9 - 1070 * q^11 - 320 * q^12 - 736 * q^13 + 392 * q^14 + 512 * q^16 + 1904 * q^17 + 1384 * q^18 + 828 * q^19 - 980 * q^21 - 4280 * q^22 - 1654 * q^23 - 1280 * q^24 - 2944 * q^26 - 11240 * q^27 + 1568 * q^28 - 2246 * q^29 + 288 * q^31 + 2048 * q^32 + 8804 * q^33 + 7616 * q^34 + 5536 * q^36 - 8986 * q^37 + 3312 * q^38 + 25688 * q^39 + 11228 * q^41 - 3920 * q^42 - 3402 * q^43 - 17120 * q^44 - 6616 * q^46 - 15572 * q^47 - 5120 * q^48 + 4802 * q^49 - 50640 * q^51 - 11776 * q^52 + 16188 * q^53 - 44960 * q^54 + 6272 * q^56 - 80328 * q^57 - 8984 * q^58 - 92928 * q^59 + 15172 * q^61 + 1152 * q^62 + 16954 * q^63 + 8192 * q^64 + 35216 * q^66 - 78966 * q^67 + 30464 * q^68 - 6844 * q^69 - 20822 * q^71 + 22144 * q^72 - 72884 * q^73 - 35944 * q^74 + 13248 * q^76 - 52430 * q^77 + 102752 * q^78 - 72106 * q^79 + 110482 * q^81 + 44912 * q^82 + 47400 * q^83 - 15680 * q^84 - 13608 * q^86 + 2236 * q^87 - 68480 * q^88 - 9920 * q^89 - 36064 * q^91 - 26464 * q^92 - 272744 * q^93 - 62288 * q^94 - 20480 * q^96 + 26792 * q^97 + 19208 * q^98 - 147190 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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