LMFDB - Embedded newform 350.6.a.u.1.2

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:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 643x - 1542 $ x^3 - 643*x - 1542
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.42018$ 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)

$f(q)$	$=$	\(q-4.00000 q^{2} -5.42018 q^{3} +16.0000 q^{4} +21.6807 q^{6} -49.0000 q^{7} -64.0000 q^{8} -213.622 q^{9} +389.563 q^{11} -86.7229 q^{12} +130.042 q^{13} +196.000 q^{14} +256.000 q^{16} +203.613 q^{17} +854.487 q^{18} +1386.39 q^{19} +265.589 q^{21} -1558.25 q^{22} -3731.86 q^{23} +346.892 q^{24} -520.167 q^{26} +2474.97 q^{27} -784.000 q^{28} +2067.46 q^{29} -937.266 q^{31} -1024.00 q^{32} -2111.50 q^{33} -814.451 q^{34} -3417.95 q^{36} -7349.89 q^{37} -5545.54 q^{38} -704.850 q^{39} +18970.4 q^{41} -1062.36 q^{42} +7641.43 q^{43} +6233.01 q^{44} +14927.5 q^{46} +5361.46 q^{47} -1387.57 q^{48} +2401.00 q^{49} -1103.62 q^{51} +2080.67 q^{52} -5710.73 q^{53} -9899.89 q^{54} +3136.00 q^{56} -7514.46 q^{57} -8269.82 q^{58} -24357.0 q^{59} +28713.8 q^{61} +3749.06 q^{62} +10467.5 q^{63} +4096.00 q^{64} +8446.00 q^{66} -14468.7 q^{67} +3257.80 q^{68} +20227.4 q^{69} +39149.2 q^{71} +13671.8 q^{72} -39651.4 q^{73} +29399.5 q^{74} +22182.2 q^{76} -19088.6 q^{77} +2819.40 q^{78} -103380. q^{79} +38495.3 q^{81} -75881.8 q^{82} -102834. q^{83} +4249.42 q^{84} -30565.7 q^{86} -11206.0 q^{87} -24932.0 q^{88} -94360.3 q^{89} -6372.05 q^{91} -59709.8 q^{92} +5080.15 q^{93} -21445.8 q^{94} +5550.26 q^{96} -43195.2 q^{97} -9604.00 q^{98} -83219.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 12 q^{2} - 9 q^{3} + 48 q^{4} + 36 q^{6} - 147 q^{7} - 192 q^{8} + 584 q^{9} - 107 q^{11} - 144 q^{12} - 842 q^{13} + 588 q^{14} + 768 q^{16} - 251 q^{17} - 2336 q^{18} + 441 q^{19} + 441 q^{21} + 428 q^{22}+ \cdots - 265616 q^{99}+O(q^{100}) $ 3 * q - 12 * q^2 - 9 * q^3 + 48 * q^4 + 36 * q^6 - 147 * q^7 - 192 * q^8 + 584 * q^9 - 107 * q^11 - 144 * q^12 - 842 * q^13 + 588 * q^14 + 768 * q^16 - 251 * q^17 - 2336 * q^18 + 441 * q^19 + 441 * q^21 + 428 * q^22 + 494 * q^23 + 576 * q^24 + 3368 * q^26 - 2655 * q^27 - 2352 * q^28 + 1308 * q^29 + 4724 * q^31 - 3072 * q^32 - 5591 * q^33 + 1004 * q^34 + 9344 * q^36 + 5614 * q^37 - 1764 * q^38 + 4330 * q^39 + 1979 * q^41 - 1764 * q^42 - 19270 * q^43 - 1712 * q^44 - 1976 * q^46 - 7774 * q^47 - 2304 * q^48 + 7203 * q^49 + 68143 * q^51 - 13472 * q^52 - 31800 * q^53 + 10620 * q^54 + 9408 * q^56 + 231 * q^57 - 5232 * q^58 - 16076 * q^59 + 98282 * q^61 - 18896 * q^62 - 28616 * q^63 + 12288 * q^64 + 22364 * q^66 - 32311 * q^67 - 4016 * q^68 - 131404 * q^69 + 154844 * q^71 - 37376 * q^72 - 30761 * q^73 - 22456 * q^74 + 7056 * q^76 + 5243 * q^77 - 17320 * q^78 - 84032 * q^79 + 61043 * q^81 - 7916 * q^82 + 47301 * q^83 + 7056 * q^84 + 77080 * q^86 - 210702 * q^87 + 6848 * q^88 - 210655 * q^89 + 41258 * q^91 + 7904 * q^92 - 219700 * q^93 + 31096 * q^94 + 9216 * q^96 - 249302 * q^97 - 28812 * q^98 - 265616 * 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
$3$	−5.42018	−0.347705	−0.173852	−	0.984772i	$-0.555622\pi$
$3$	−5.42018	−0.347705	−0.173852	+	0.984772i	$0.555622\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	21.6807	0.245864
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	−213.622	−0.879101
$10$	0	0
$11$	389.563	0.970724	0.485362	−	0.874313i	$-0.338688\pi$
$11$	389.563	0.970724	0.485362	+	0.874313i	$0.338688\pi$
$12$	−86.7229	−0.173852
$13$	130.042	0.213415	0.106707	−	0.994290i	$-0.465969\pi$
$13$	130.042	0.213415	0.106707	+	0.994290i	$0.465969\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	203.613	0.170877	0.0854383	−	0.996343i	$-0.472771\pi$
$17$	203.613	0.170877	0.0854383	+	0.996343i	$0.472771\pi$
$18$	854.487	0.621619
$19$	1386.39	0.881049	0.440525	−	0.897741i	$-0.354792\pi$
$19$	1386.39	0.881049	0.440525	+	0.897741i	$0.354792\pi$
$20$	0	0
$21$	265.589	0.131420
$22$	−1558.25	−0.686406
$23$	−3731.86	−1.47098	−0.735489	−	0.677537i	$-0.763047\pi$
$23$	−3731.86	−1.47098	−0.735489	+	0.677537i	$0.763047\pi$
$24$	346.892	0.122932
$25$	0	0
$26$	−520.167	−0.150907
$27$	2474.97	0.653372
$28$	−784.000	−0.188982
$29$	2067.46	0.456500	0.228250	−	0.973603i	$-0.426700\pi$
$29$	2067.46	0.456500	0.228250	+	0.973603i	$0.426700\pi$
$30$	0	0
$31$	−937.266	−0.175170	−0.0875848	−	0.996157i	$-0.527915\pi$
$31$	−937.266	−0.175170	−0.0875848	+	0.996157i	$0.527915\pi$
$32$	−1024.00	−0.176777
$33$	−2111.50	−0.337525
$34$	−814.451	−0.120828
$35$	0	0
$36$	−3417.95	−0.439551
$37$	−7349.89	−0.882625	−0.441313	−	0.897353i	$-0.645487\pi$
$37$	−7349.89	−0.882625	−0.441313	+	0.897353i	$0.645487\pi$
$38$	−5545.54	−0.622996
$39$	−704.850	−0.0742054
$40$	0	0
$41$	18970.4	1.76245	0.881227	−	0.472694i	$-0.156718\pi$
$41$	18970.4	1.76245	0.881227	+	0.472694i	$0.156718\pi$
$42$	−1062.36	−0.0929280
$43$	7641.43	0.630236	0.315118	−	0.949052i	$-0.397956\pi$
$43$	7641.43	0.630236	0.315118	+	0.949052i	$0.397956\pi$
$44$	6233.01	0.485362
$45$	0	0
$46$	14927.5	1.04014
$47$	5361.46	0.354028	0.177014	−	0.984208i	$-0.443356\pi$
$47$	5361.46	0.354028	0.177014	+	0.984208i	$0.443356\pi$
$48$	−1387.57	−0.0869262
$49$	2401.00	0.142857
$50$	0	0
$51$	−1103.62	−0.0594146
$52$	2080.67	0.106707
$53$	−5710.73	−0.279256	−0.139628	−	0.990204i	$-0.544591\pi$
$53$	−5710.73	−0.279256	−0.139628	+	0.990204i	$0.544591\pi$
$54$	−9899.89	−0.462004
$55$	0	0
$56$	3136.00	0.133631
$57$	−7514.46	−0.306345
$58$	−8269.82	−0.322794
$59$	−24357.0	−0.910949	−0.455475	−	0.890249i	$-0.650530\pi$
$59$	−24357.0	−0.910949	−0.455475	+	0.890249i	$0.650530\pi$
$60$	0	0
$61$	28713.8	0.988022	0.494011	−	0.869456i	$-0.335530\pi$
$61$	28713.8	0.988022	0.494011	+	0.869456i	$0.335530\pi$
$62$	3749.06	0.123864
$63$	10467.5	0.332269
$64$	4096.00	0.125000
$65$	0	0
$66$	8446.00	0.238666
$67$	−14468.7	−0.393769	−0.196884	−	0.980427i	$-0.563082\pi$
$67$	−14468.7	−0.393769	−0.196884	+	0.980427i	$0.563082\pi$
$68$	3257.80	0.0854383
$69$	20227.4	0.511466
$70$	0	0
$71$	39149.2	0.921673	0.460836	−	0.887485i	$-0.347549\pi$
$71$	39149.2	0.921673	0.460836	+	0.887485i	$0.347549\pi$
$72$	13671.8	0.310809
$73$	−39651.4	−0.870866	−0.435433	−	0.900221i	$-0.643405\pi$
$73$	−39651.4	−0.870866	−0.435433	+	0.900221i	$0.643405\pi$
$74$	29399.5	0.624110
$75$	0	0
$76$	22182.2	0.440525
$77$	−19088.6	−0.366899
$78$	2819.40	0.0524711
$79$	−103380.	−1.86366	−0.931831	−	0.362893i	$-0.881789\pi$
$79$	−103380.	−1.86366	−0.931831	+	0.362893i	$0.881789\pi$
$80$	0	0
$81$	38495.3	0.651921
$82$	−75881.8	−1.24624
$83$	−102834.	−1.63847	−0.819237	−	0.573455i	$-0.805603\pi$
$83$	−102834.	−1.63847	−0.819237	+	0.573455i	$0.805603\pi$
$84$	4249.42	0.0657100
$85$	0	0
$86$	−30565.7	−0.445644
$87$	−11206.0	−0.158727
$88$	−24932.0	−0.343203
$89$	−94360.3	−1.26274	−0.631370	−	0.775481i	$-0.717507\pi$
$89$	−94360.3	−1.26274	−0.631370	+	0.775481i	$0.717507\pi$
$90$	0	0
$91$	−6372.05	−0.0806632
$92$	−59709.8	−0.735489
$93$	5080.15	0.0609073
$94$	−21445.8	−0.250336
$95$	0	0
$96$	5550.26	0.0614661
$97$	−43195.2	−0.466129	−0.233064	−	0.972461i	$-0.574875\pi$
$97$	−43195.2	−0.466129	−0.233064	+	0.972461i	$0.574875\pi$
$98$	−9604.00	−0.101015
$99$	−83219.1	−0.853365
$100$	0	0
$101$	−4760.78	−0.0464381	−0.0232191	−	0.999730i	$-0.507392\pi$
$101$	−4760.78	−0.0464381	−0.0232191	+	0.999730i	$0.507392\pi$
$102$	4414.47	0.0420125
$103$	83459.4	0.775144	0.387572	−	0.921839i	$-0.373314\pi$
$103$	83459.4	0.775144	0.387572	+	0.921839i	$0.373314\pi$
$104$	−8322.68	−0.0754536
$105$	0	0
$106$	22842.9	0.197464
$107$	−161779.	−1.36604	−0.683018	−	0.730402i	$-0.739333\pi$
$107$	−161779.	−1.36604	−0.683018	+	0.730402i	$0.739333\pi$
$108$	39599.5	0.326686
$109$	−776.049	−0.00625638	−0.00312819	−	0.999995i	$-0.500996\pi$
$109$	−776.049	−0.00625638	−0.00312819	+	0.999995i	$0.500996\pi$
$110$	0	0
$111$	39837.7	0.306893
$112$	−12544.0	−0.0944911
$113$	−76499.9	−0.563592	−0.281796	−	0.959474i	$-0.590930\pi$
$113$	−76499.9	−0.563592	−0.281796	+	0.959474i	$0.590930\pi$
$114$	30057.8	0.216619
$115$	0	0
$116$	33079.3	0.228250
$117$	−27779.8	−0.187613
$118$	97428.1	0.644138
$119$	−9977.02	−0.0645853
$120$	0	0
$121$	−9291.74	−0.0576944
$122$	−114855.	−0.698637
$123$	−102823.	−0.612813
$124$	−14996.3	−0.0875848
$125$	0	0
$126$	−41869.8	−0.234950
$127$	−47837.5	−0.263184	−0.131592	−	0.991304i	$-0.542009\pi$
$127$	−47837.5	−0.263184	−0.131592	+	0.991304i	$0.542009\pi$
$128$	−16384.0	−0.0883883
$129$	−41417.9	−0.219136
$130$	0	0
$131$	−161873.	−0.824132	−0.412066	−	0.911154i	$-0.635193\pi$
$131$	−161873.	−0.824132	−0.412066	+	0.911154i	$0.635193\pi$
$132$	−33784.0	−0.168763
$133$	−67932.9	−0.333005
$134$	57874.6	0.278437
$135$	0	0
$136$	−13031.2	−0.0604140
$137$	−144978.	−0.659936	−0.329968	−	0.943992i	$-0.607038\pi$
$137$	−144978.	−0.659936	−0.329968	+	0.943992i	$0.607038\pi$
$138$	−80909.5	−0.361661
$139$	−132636.	−0.582269	−0.291134	−	0.956682i	$-0.594033\pi$
$139$	−132636.	−0.582269	−0.291134	+	0.956682i	$0.594033\pi$
$140$	0	0
$141$	−29060.1	−0.123097
$142$	−156597.	−0.651721
$143$	50659.5	0.207167
$144$	−54687.1	−0.219775
$145$	0	0
$146$	158606.	0.615795
$147$	−13013.9	−0.0496721
$148$	−117598.	−0.441313
$149$	−228890.	−0.844618	−0.422309	−	0.906452i	$-0.638780\pi$
$149$	−228890.	−0.844618	−0.422309	+	0.906452i	$0.638780\pi$
$150$	0	0
$151$	117085.	0.417886	0.208943	−	0.977928i	$-0.432998\pi$
$151$	117085.	0.417886	0.208943	+	0.977928i	$0.432998\pi$
$152$	−88728.7	−0.311498
$153$	−43496.1	−0.150218
$154$	76354.3	0.259437
$155$	0	0
$156$	−11277.6	−0.0371027
$157$	−510024.	−1.65136	−0.825680	−	0.564139i	$-0.809208\pi$
$157$	−510024.	−1.65136	−0.825680	+	0.564139i	$0.809208\pi$
$158$	413518.	1.31781
$159$	30953.2	0.0970985
$160$	0	0
$161$	182861.	0.555977
$162$	−153981.	−0.460978
$163$	−53477.3	−0.157652	−0.0788261	−	0.996888i	$-0.525117\pi$
$163$	−53477.3	−0.157652	−0.0788261	+	0.996888i	$0.525117\pi$
$164$	303527.	0.881227
$165$	0	0
$166$	411334.	1.15858
$167$	395447.	1.09723	0.548614	−	0.836076i	$-0.315156\pi$
$167$	395447.	1.09723	0.548614	+	0.836076i	$0.315156\pi$
$168$	−16997.7	−0.0464640
$169$	−354382.	−0.954454
$170$	0	0
$171$	−296162.	−0.774532
$172$	122263.	0.315118
$173$	30727.8	0.0780578	0.0390289	−	0.999238i	$-0.487574\pi$
$173$	30727.8	0.0780578	0.0390289	+	0.999238i	$0.487574\pi$
$174$	44823.9	0.112237
$175$	0	0
$176$	99728.1	0.242681
$177$	132019.	0.316741
$178$	377441.	0.892893
$179$	−430398.	−1.00401	−0.502005	−	0.864865i	$-0.667404\pi$
$179$	−430398.	−1.00401	−0.502005	+	0.864865i	$0.667404\pi$
$180$	0	0
$181$	−745718.	−1.69191	−0.845956	−	0.533252i	$-0.820970\pi$
$181$	−745718.	−1.69191	−0.845956	+	0.533252i	$0.820970\pi$
$182$	25488.2	0.0570375
$183$	−155634.	−0.343540
$184$	238839.	0.520069
$185$	0	0
$186$	−20320.6	−0.0430679
$187$	79319.9	0.165874
$188$	85783.3	0.177014
$189$	−121274.	−0.246952
$190$	0	0
$191$	397849.	0.789105	0.394552	−	0.918873i	$-0.370900\pi$
$191$	397849.	0.789105	0.394552	+	0.918873i	$0.370900\pi$
$192$	−22201.1	−0.0434631
$193$	−45711.1	−0.0883340	−0.0441670	−	0.999024i	$-0.514063\pi$
$193$	−45711.1	−0.0883340	−0.0441670	+	0.999024i	$0.514063\pi$
$194$	172781.	0.329603
$195$	0	0
$196$	38416.0	0.0714286
$197$	619236.	1.13682	0.568409	−	0.822746i	$-0.307559\pi$
$197$	619236.	1.13682	0.568409	+	0.822746i	$0.307559\pi$
$198$	332876.	0.603420
$199$	150491.	0.269387	0.134694	−	0.990887i	$-0.456995\pi$
$199$	150491.	0.269387	0.134694	+	0.990887i	$0.456995\pi$
$200$	0	0
$201$	78422.7	0.136915
$202$	19043.1	0.0328367
$203$	−101305.	−0.172541
$204$	−17657.9	−0.0297073
$205$	0	0
$206$	−333838.	−0.548110
$207$	797207.	1.29314
$208$	33290.7	0.0533537
$209$	540085.	0.855256
$210$	0	0
$211$	484288.	0.748855	0.374428	−	0.927256i	$-0.377839\pi$
$211$	484288.	0.748855	0.374428	+	0.927256i	$0.377839\pi$
$212$	−91371.7	−0.139628
$213$	−212196.	−0.320470
$214$	647115.	0.965933
$215$	0	0
$216$	−158398.	−0.231002
$217$	45926.0	0.0662079
$218$	3104.20	0.00442393
$219$	214918.	0.302804
$220$	0	0
$221$	26478.2	0.0364676
$222$	−159351.	−0.217006
$223$	−648978.	−0.873912	−0.436956	−	0.899483i	$-0.643943\pi$
$223$	−648978.	−0.873912	−0.436956	+	0.899483i	$0.643943\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	306000.	0.398520
$227$	739300.	0.952261	0.476130	−	0.879375i	$-0.342039\pi$
$227$	739300.	0.952261	0.476130	+	0.879375i	$0.342039\pi$
$228$	−120231.	−0.153172
$229$	−959630.	−1.20925	−0.604624	−	0.796511i	$-0.706677\pi$
$229$	−959630.	−1.20925	−0.604624	+	0.796511i	$0.706677\pi$
$230$	0	0
$231$	103464.	0.127573
$232$	−132317.	−0.161397
$233$	103718.	0.125160	0.0625799	−	0.998040i	$-0.480067\pi$
$233$	103718.	0.125160	0.0625799	+	0.998040i	$0.480067\pi$
$234$	111119.	0.132663
$235$	0	0
$236$	−389712.	−0.455475
$237$	560336.	0.648004
$238$	39908.1	0.0456687
$239$	761491.	0.862324	0.431162	−	0.902275i	$-0.358104\pi$
$239$	761491.	0.862324	0.431162	+	0.902275i	$0.358104\pi$
$240$	0	0
$241$	−535459.	−0.593860	−0.296930	−	0.954899i	$-0.595963\pi$
$241$	−535459.	−0.593860	−0.296930	+	0.954899i	$0.595963\pi$
$242$	37167.0	0.0407961
$243$	−810069.	−0.880048
$244$	459421.	0.494011
$245$	0	0
$246$	411293.	0.433325
$247$	180288.	0.188029
$248$	59985.0	0.0619318
$249$	557377.	0.569705
$250$	0	0
$251$	−1.09418e6	−1.09624	−0.548118	−	0.836401i	$-0.684656\pi$
$251$	−1.09418e6	−1.09624	−0.548118	+	0.836401i	$0.684656\pi$
$252$	167479.	0.166135
$253$	−1.45380e6	−1.42791
$254$	191350.	0.186099
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.54020e6	1.45461	0.727303	−	0.686317i	$-0.240774\pi$
$257$	1.54020e6	1.45461	0.727303	+	0.686317i	$0.240774\pi$
$258$	165672.	0.154953
$259$	360144.	0.333601
$260$	0	0
$261$	−441653.	−0.401310
$262$	647493.	0.582749
$263$	1.49556e6	1.33326	0.666629	−	0.745389i	$-0.267736\pi$
$263$	1.49556e6	1.33326	0.666629	+	0.745389i	$0.267736\pi$
$264$	135136.	0.119333
$265$	0	0
$266$	271732.	0.235470
$267$	511450.	0.439061
$268$	−231499.	−0.196884
$269$	−395161.	−0.332961	−0.166481	−	0.986045i	$-0.553240\pi$
$269$	−395161.	−0.332961	−0.166481	+	0.986045i	$0.553240\pi$
$270$	0	0
$271$	−780625.	−0.645683	−0.322841	−	0.946453i	$-0.604638\pi$
$271$	−780625.	−0.645683	−0.322841	+	0.946453i	$0.604638\pi$
$272$	52124.8	0.0427191
$273$	34537.7	0.0280470
$274$	579914.	0.466645
$275$	0	0
$276$	323638.	0.255733
$277$	1.25065e6	0.979344	0.489672	−	0.871907i	$-0.337117\pi$
$277$	1.25065e6	0.979344	0.489672	+	0.871907i	$0.337117\pi$
$278$	530543.	0.411726
$279$	200220.	0.153992
$280$	0	0
$281$	−370579.	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$281$	−370579.	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$282$	116240.	0.0870430
$283$	674799.	0.500851	0.250425	−	0.968136i	$-0.419430\pi$
$283$	674799.	0.500851	0.250425	+	0.968136i	$0.419430\pi$
$284$	626387.	0.460836
$285$	0	0
$286$	−202638.	−0.146489
$287$	−929551.	−0.666145
$288$	218749.	0.155405
$289$	−1.37840e6	−0.970801
$290$	0	0
$291$	234126.	0.162075
$292$	−634422.	−0.435433
$293$	2.32160e6	1.57986	0.789931	−	0.613195i	$-0.210116\pi$
$293$	2.32160e6	1.57986	0.789931	+	0.613195i	$0.210116\pi$
$294$	52055.4	0.0351235
$295$	0	0
$296$	470393.	0.312055
$297$	964157.	0.634244
$298$	915558.	0.597235
$299$	−485298.	−0.313929
$300$	0	0
$301$	−374430.	−0.238207
$302$	−468339.	−0.295490
$303$	25804.3	0.0161468
$304$	354915.	0.220262
$305$	0	0
$306$	173984.	0.106220
$307$	3.05261e6	1.84852	0.924261	−	0.381761i	$-0.124682\pi$
$307$	3.05261e6	1.84852	0.924261	+	0.381761i	$0.124682\pi$
$308$	−305417.	−0.183450
$309$	−452365.	−0.269521
$310$	0	0
$311$	−1.15377e6	−0.676422	−0.338211	−	0.941070i	$-0.609822\pi$
$311$	−1.15377e6	−0.676422	−0.338211	+	0.941070i	$0.609822\pi$
$312$	45110.4	0.0262356
$313$	−386260.	−0.222853	−0.111427	−	0.993773i	$-0.535542\pi$
$313$	−386260.	−0.222853	−0.111427	+	0.993773i	$0.535542\pi$
$314$	2.04010e6	1.16769
$315$	0	0
$316$	−1.65407e6	−0.931831
$317$	−2.62495e6	−1.46714	−0.733572	−	0.679612i	$-0.762148\pi$
$317$	−2.62495e6	−1.46714	−0.733572	+	0.679612i	$0.762148\pi$
$318$	−123813.	−0.0686590
$319$	805404.	0.443136
$320$	0	0
$321$	876870.	0.474977
$322$	−731445.	−0.393135
$323$	282286.	0.150551
$324$	615924.	0.325960
$325$	0	0
$326$	213909.	0.111477
$327$	4206.32	0.00217537
$328$	−1.21411e6	−0.623121
$329$	−262711.	−0.133810
$330$	0	0
$331$	−956737.	−0.479979	−0.239990	−	0.970775i	$-0.577144\pi$
$331$	−956737.	−0.479979	−0.239990	+	0.970775i	$0.577144\pi$
$332$	−1.64534e6	−0.819237
$333$	1.57009e6	0.775917
$334$	−1.58179e6	−0.775857
$335$	0	0
$336$	67990.7	0.0328550
$337$	−2.07803e6	−0.996730	−0.498365	−	0.866967i	$-0.666066\pi$
$337$	−2.07803e6	−0.996730	−0.498365	+	0.866967i	$0.666066\pi$
$338$	1.41753e6	0.674901
$339$	414643.	0.195964
$340$	0	0
$341$	−365124.	−0.170041
$342$	1.18465e6	0.547677
$343$	−117649.	−0.0539949
$344$	−489051.	−0.222822
$345$	0	0
$346$	−122911.	−0.0551952
$347$	−203577.	−0.0907620	−0.0453810	−	0.998970i	$-0.514450\pi$
$347$	−203577.	−0.0907620	−0.0453810	+	0.998970i	$0.514450\pi$
$348$	−179296.	−0.0793637
$349$	1.29100e6	0.567367	0.283683	−	0.958918i	$-0.408443\pi$
$349$	1.29100e6	0.567367	0.283683	+	0.958918i	$0.408443\pi$
$350$	0	0
$351$	321850.	0.139439
$352$	−398912.	−0.171601
$353$	−1.02964e6	−0.439792	−0.219896	−	0.975523i	$-0.570572\pi$
$353$	−1.02964e6	−0.439792	−0.219896	+	0.975523i	$0.570572\pi$
$354$	−528078.	−0.223970
$355$	0	0
$356$	−1.50976e6	−0.631370
$357$	54077.2	0.0224566
$358$	1.72159e6	0.709942
$359$	2.88442e6	1.18120	0.590599	−	0.806965i	$-0.298891\pi$
$359$	2.88442e6	1.18120	0.590599	+	0.806965i	$0.298891\pi$
$360$	0	0
$361$	−554033.	−0.223752
$362$	2.98287e6	1.19636
$363$	50362.9	0.0200606
$364$	−101953.	−0.0403316
$365$	0	0
$366$	622537.	0.242919
$367$	1.41890e6	0.549903	0.274951	−	0.961458i	$-0.411338\pi$
$367$	1.41890e6	0.549903	0.274951	+	0.961458i	$0.411338\pi$
$368$	−955357.	−0.367744
$369$	−4.05250e6	−1.54938
$370$	0	0
$371$	279826.	0.105549
$372$	81282.4	0.0304536
$373$	706757.	0.263026	0.131513	−	0.991314i	$-0.458017\pi$
$373$	706757.	0.263026	0.131513	+	0.991314i	$0.458017\pi$
$374$	−317280.	−0.117291
$375$	0	0
$376$	−343133.	−0.125168
$377$	268856.	0.0974240
$378$	485094.	0.174621
$379$	−5.04524e6	−1.80419	−0.902097	−	0.431533i	$-0.857973\pi$
$379$	−5.04524e6	−1.80419	−0.902097	+	0.431533i	$0.857973\pi$
$380$	0	0
$381$	259288.	0.0915102
$382$	−1.59140e6	−0.557981
$383$	−1.42707e6	−0.497106	−0.248553	−	0.968618i	$-0.579955\pi$
$383$	−1.42707e6	−0.497106	−0.248553	+	0.968618i	$0.579955\pi$
$384$	88804.2	0.0307330
$385$	0	0
$386$	182844.	0.0624616
$387$	−1.63237e6	−0.554042
$388$	−691123.	−0.233064
$389$	2.82715e6	0.947272	0.473636	−	0.880721i	$-0.342941\pi$
$389$	2.82715e6	0.947272	0.473636	+	0.880721i	$0.342941\pi$
$390$	0	0
$391$	−759854.	−0.251356
$392$	−153664.	−0.0505076
$393$	877382.	0.286555
$394$	−2.47694e6	−0.803851
$395$	0	0
$396$	−1.33151e6	−0.426683
$397$	−4.28294e6	−1.36385	−0.681924	−	0.731423i	$-0.738857\pi$
$397$	−4.28294e6	−1.36385	−0.681924	+	0.731423i	$0.738857\pi$
$398$	−601963.	−0.190486
$399$	368209.	0.115788
$400$	0	0
$401$	4.38739e6	1.36253	0.681263	−	0.732038i	$-0.261431\pi$
$401$	4.38739e6	1.36253	0.681263	+	0.732038i	$0.261431\pi$
$402$	−313691.	−0.0968137
$403$	−121884.	−0.0373838
$404$	−76172.5	−0.0232191
$405$	0	0
$406$	405221.	0.122005
$407$	−2.86324e6	−0.856786
$408$	70631.5	0.0210062
$409$	5.23967e6	1.54880	0.774401	−	0.632696i	$-0.218052\pi$
$409$	5.23967e6	1.54880	0.774401	+	0.632696i	$0.218052\pi$
$410$	0	0
$411$	785809.	0.229463
$412$	1.33535e6	0.387572
$413$	1.19349e6	0.344306
$414$	−3.18883e6	−0.914387
$415$	0	0
$416$	−133163.	−0.0377268
$417$	718909.	0.202458
$418$	−2.16034e6	−0.604757
$419$	−2.41299e6	−0.671460	−0.335730	−	0.941958i	$-0.608983\pi$
$419$	−2.41299e6	−0.671460	−0.335730	+	0.941958i	$0.608983\pi$
$420$	0	0
$421$	−1.02513e6	−0.281886	−0.140943	−	0.990018i	$-0.545013\pi$
$421$	−1.02513e6	−0.281886	−0.140943	+	0.990018i	$0.545013\pi$
$422$	−1.93715e6	−0.529521
$423$	−1.14532e6	−0.311227
$424$	365487.	0.0987318
$425$	0	0
$426$	848782.	0.226606
$427$	−1.40698e6	−0.373437
$428$	−2.58846e6	−0.683018
$429$	−274583.	−0.0720329
$430$	0	0
$431$	1.99902e6	0.518350	0.259175	−	0.965830i	$-0.416549\pi$
$431$	1.99902e6	0.518350	0.259175	+	0.965830i	$0.416549\pi$
$432$	633593.	0.163343
$433$	−3.14718e6	−0.806681	−0.403340	−	0.915050i	$-0.632151\pi$
$433$	−3.14718e6	−0.806681	−0.403340	+	0.915050i	$0.632151\pi$
$434$	−183704.	−0.0468160
$435$	0	0
$436$	−12416.8	−0.00312819
$437$	−5.17380e6	−1.29600
$438$	−859671.	−0.214115
$439$	−4.92278e6	−1.21913	−0.609564	−	0.792737i	$-0.708656\pi$
$439$	−4.92278e6	−1.21913	−0.609564	+	0.792737i	$0.708656\pi$
$440$	0	0
$441$	−512906.	−0.125586
$442$	−105913.	−0.0257865
$443$	−2.09947e6	−0.508277	−0.254138	−	0.967168i	$-0.581792\pi$
$443$	−2.09947e6	−0.508277	−0.254138	+	0.967168i	$0.581792\pi$
$444$	637403.	0.153446
$445$	0	0
$446$	2.59591e6	0.617949
$447$	1.24062e6	0.293678
$448$	−200704.	−0.0472456
$449$	−3.02677e6	−0.708539	−0.354270	−	0.935143i	$-0.615270\pi$
$449$	−3.02677e6	−0.708539	−0.354270	+	0.935143i	$0.615270\pi$
$450$	0	0
$451$	7.39018e6	1.71086
$452$	−1.22400e6	−0.281796
$453$	−634620.	−0.145301
$454$	−2.95720e6	−0.673350
$455$	0	0
$456$	480926.	0.108309
$457$	−5.87932e6	−1.31685	−0.658426	−	0.752646i	$-0.728777\pi$
$457$	−5.87932e6	−1.31685	−0.658426	+	0.752646i	$0.728777\pi$
$458$	3.83852e6	0.855067
$459$	503936.	0.111646
$460$	0	0
$461$	−6.55456e6	−1.43645	−0.718226	−	0.695810i	$-0.755045\pi$
$461$	−6.55456e6	−1.43645	−0.718226	+	0.695810i	$0.755045\pi$
$462$	−413854.	−0.0902075
$463$	−7.87451e6	−1.70715	−0.853573	−	0.520973i	$-0.825569\pi$
$463$	−7.87451e6	−1.70715	−0.853573	+	0.520973i	$0.825569\pi$
$464$	529269.	0.114125
$465$	0	0
$466$	−414873.	−0.0885014
$467$	84786.5	0.0179901	0.00899507	−	0.999960i	$-0.497137\pi$
$467$	84786.5	0.0179901	0.00899507	+	0.999960i	$0.497137\pi$
$468$	−444476.	−0.0938067
$469$	708964.	0.148831
$470$	0	0
$471$	2.76442e6	0.574185
$472$	1.55885e6	0.322069
$473$	2.97682e6	0.611786
$474$	−2.24134e6	−0.458208
$475$	0	0
$476$	−159632.	−0.0322926
$477$	1.21994e6	0.245494
$478$	−3.04597e6	−0.609755
$479$	7.73272e6	1.53990	0.769952	−	0.638102i	$-0.220280\pi$
$479$	7.73272e6	1.53990	0.769952	+	0.638102i	$0.220280\pi$
$480$	0	0
$481$	−955793.	−0.188365
$482$	2.14184e6	0.419922
$483$	−991141.	−0.193316
$484$	−148668.	−0.0288472
$485$	0	0
$486$	3.24028e6	0.622288
$487$	−3.03835e6	−0.580518	−0.290259	−	0.956948i	$-0.593741\pi$
$487$	−3.03835e6	−0.580518	−0.290259	+	0.956948i	$0.593741\pi$
$488$	−1.83769e6	−0.349318
$489$	289856.	0.0548164
$490$	0	0
$491$	9.03161e6	1.69068	0.845340	−	0.534228i	$-0.179398\pi$
$491$	9.03161e6	1.69068	0.845340	+	0.534228i	$0.179398\pi$
$492$	−1.64517e6	−0.306407
$493$	420960.	0.0780052
$494$	−721153.	−0.132957
$495$	0	0
$496$	−239940.	−0.0437924
$497$	−1.91831e6	−0.348360
$498$	−2.22951e6	−0.402843
$499$	6.29394e6	1.13154	0.565771	−	0.824562i	$-0.308578\pi$
$499$	6.29394e6	1.13154	0.565771	+	0.824562i	$0.308578\pi$
$500$	0	0
$501$	−2.14339e6	−0.381511
$502$	4.37672e6	0.775157
$503$	9.76827e6	1.72146	0.860732	−	0.509059i	$-0.170006\pi$
$503$	9.76827e6	1.72146	0.860732	+	0.509059i	$0.170006\pi$
$504$	−669917.	−0.117475
$505$	0	0
$506$	5.81518e6	1.00969
$507$	1.92081e6	0.331868
$508$	−765400.	−0.131592
$509$	−7.00867e6	−1.19906	−0.599530	−	0.800352i	$-0.704646\pi$
$509$	−7.00867e6	−1.19906	−0.599530	+	0.800352i	$0.704646\pi$
$510$	0	0
$511$	1.94292e6	0.329156
$512$	−262144.	−0.0441942
$513$	3.43127e6	0.575653
$514$	−6.16081e6	−1.02856
$515$	0	0
$516$	−662687.	−0.109568
$517$	2.08862e6	0.343664
$518$	−1.44058e6	−0.235891
$519$	−166550.	−0.0271411
$520$	0	0
$521$	5.23145e6	0.844359	0.422180	−	0.906512i	$-0.361265\pi$
$521$	5.23145e6	0.844359	0.422180	+	0.906512i	$0.361265\pi$
$522$	1.76661e6	0.283769
$523$	−1.07626e7	−1.72054	−0.860270	−	0.509838i	$-0.829705\pi$
$523$	−1.07626e7	−1.72054	−0.860270	+	0.509838i	$0.829705\pi$
$524$	−2.58997e6	−0.412066
$525$	0	0
$526$	−5.98224e6	−0.942756
$527$	−190839.	−0.0299324
$528$	−540544.	−0.0843813
$529$	7.49046e6	1.16378
$530$	0	0
$531$	5.20319e6	0.800817
$532$	−1.08693e6	−0.166503
$533$	2.46695e6	0.376134
$534$	−2.04580e6	−0.310463
$535$	0	0
$536$	925994.	0.139218
$537$	2.33284e6	0.349099
$538$	1.58064e6	0.235439
$539$	935341.	0.138675
$540$	0	0
$541$	2.36347e6	0.347181	0.173591	−	0.984818i	$-0.444463\pi$
$541$	2.36347e6	0.347181	0.173591	+	0.984818i	$0.444463\pi$
$542$	3.12250e6	0.456567
$543$	4.04192e6	0.588286
$544$	−208499.	−0.0302070
$545$	0	0
$546$	−138151.	−0.0198322
$547$	2.96142e6	0.423186	0.211593	−	0.977358i	$-0.432135\pi$
$547$	2.96142e6	0.423186	0.211593	+	0.977358i	$0.432135\pi$
$548$	−2.31965e6	−0.329968
$549$	−6.13390e6	−0.868571
$550$	0	0
$551$	2.86629e6	0.402199
$552$	−1.29455e6	−0.180830
$553$	5.06560e6	0.704398
$554$	−5.00259e6	−0.692501
$555$	0	0
$556$	−2.12217e6	−0.291134
$557$	−1.22485e7	−1.67280	−0.836401	−	0.548119i	$-0.815344\pi$
$557$	−1.22485e7	−1.67280	−0.836401	+	0.548119i	$0.815344\pi$
$558$	−800881.	−0.108889
$559$	993705.	0.134502
$560$	0	0
$561$	−429928.	−0.0576752
$562$	1.48232e6	0.197970
$563$	7.14138e6	0.949536	0.474768	−	0.880111i	$-0.342532\pi$
$563$	7.14138e6	0.949536	0.474768	+	0.880111i	$0.342532\pi$
$564$	−464961.	−0.0615487
$565$	0	0
$566$	−2.69919e6	−0.354155
$567$	−1.88627e6	−0.246403
$568$	−2.50555e6	−0.325861
$569$	6.72084e6	0.870247	0.435124	−	0.900371i	$-0.356705\pi$
$569$	6.72084e6	0.870247	0.435124	+	0.900371i	$0.356705\pi$
$570$	0	0
$571$	4.22595e6	0.542418	0.271209	−	0.962520i	$-0.412576\pi$
$571$	4.22595e6	0.542418	0.271209	+	0.962520i	$0.412576\pi$
$572$	810552.	0.103584
$573$	−2.15641e6	−0.274375
$574$	3.71821e6	0.471036
$575$	0	0
$576$	−874994.	−0.109888
$577$	−9.55624e6	−1.19494	−0.597472	−	0.801890i	$-0.703828\pi$
$577$	−9.55624e6	−1.19494	−0.597472	+	0.801890i	$0.703828\pi$
$578$	5.51360e6	0.686460
$579$	247762.	0.0307142
$580$	0	0
$581$	5.03885e6	0.619285
$582$	−936503.	−0.114604
$583$	−2.22469e6	−0.271080
$584$	2.53769e6	0.307898
$585$	0	0
$586$	−9.28642e6	−1.11713
$587$	−6.81230e6	−0.816016	−0.408008	−	0.912978i	$-0.633776\pi$
$587$	−6.81230e6	−0.816016	−0.408008	+	0.912978i	$0.633776\pi$
$588$	−208222.	−0.0248360
$589$	−1.29941e6	−0.154333
$590$	0	0
$591$	−3.35637e6	−0.395277
$592$	−1.88157e6	−0.220656
$593$	−6.67613e6	−0.779629	−0.389815	−	0.920893i	$-0.627461\pi$
$593$	−6.67613e6	−0.779629	−0.389815	+	0.920893i	$0.627461\pi$
$594$	−3.85663e6	−0.448479
$595$	0	0
$596$	−3.66223e6	−0.422309
$597$	−815687.	−0.0936672
$598$	1.94119e6	0.221981
$599$	1.50864e7	1.71798	0.858992	−	0.511989i	$-0.171091\pi$
$599$	1.50864e7	1.71798	0.858992	+	0.511989i	$0.171091\pi$
$600$	0	0
$601$	−1.56764e7	−1.77036	−0.885178	−	0.465253i	$-0.845963\pi$
$601$	−1.56764e7	−1.77036	−0.885178	+	0.465253i	$0.845963\pi$
$602$	1.49772e6	0.168438
$603$	3.09082e6	0.346163
$604$	1.87335e6	0.208943
$605$	0	0
$606$	−103217.	−0.0114175
$607$	1.22561e7	1.35015	0.675074	−	0.737750i	$-0.264112\pi$
$607$	1.22561e7	1.35015	0.675074	+	0.737750i	$0.264112\pi$
$608$	−1.41966e6	−0.155749
$609$	549093.	0.0599933
$610$	0	0
$611$	697214.	0.0755549
$612$	−695937.	−0.0751089
$613$	4.01747e6	0.431818	0.215909	−	0.976413i	$-0.430729\pi$
$613$	4.01747e6	0.431818	0.215909	+	0.976413i	$0.430729\pi$
$614$	−1.22104e7	−1.30710
$615$	0	0
$616$	1.22167e6	0.129718
$617$	6.78797e6	0.717839	0.358920	−	0.933369i	$-0.383145\pi$
$617$	6.78797e6	0.717839	0.358920	+	0.933369i	$0.383145\pi$
$618$	1.80946e6	0.190580
$619$	1.53189e7	1.60694	0.803471	−	0.595344i	$-0.202984\pi$
$619$	1.53189e7	1.60694	0.803471	+	0.595344i	$0.202984\pi$
$620$	0	0
$621$	−9.23625e6	−0.961096
$622$	4.61507e6	0.478302
$623$	4.62365e6	0.477271
$624$	−180442.	−0.0185513
$625$	0	0
$626$	1.54504e6	0.157581
$627$	−2.92736e6	−0.297376
$628$	−8.16039e6	−0.825680
$629$	−1.49653e6	−0.150820
$630$	0	0
$631$	−4.77017e6	−0.476936	−0.238468	−	0.971150i	$-0.576645\pi$
$631$	−4.77017e6	−0.476936	−0.238468	+	0.971150i	$0.576645\pi$
$632$	6.61629e6	0.658904
$633$	−2.62493e6	−0.260380
$634$	1.04998e7	1.03743
$635$	0	0
$636$	495251.	0.0485492
$637$	312230.	0.0304878
$638$	−3.22162e6	−0.313344
$639$	−8.36311e6	−0.810244
$640$	0	0
$641$	−1.03680e7	−0.996665	−0.498332	−	0.866986i	$-0.666054\pi$
$641$	−1.03680e7	−0.996665	−0.498332	+	0.866986i	$0.666054\pi$
$642$	−3.50748e6	−0.335859
$643$	−5.44801e6	−0.519649	−0.259825	−	0.965656i	$-0.583665\pi$
$643$	−5.44801e6	−0.519649	−0.259825	+	0.965656i	$0.583665\pi$
$644$	2.92578e6	0.277989
$645$	0	0
$646$	−1.12914e6	−0.106455
$647$	9.89370e6	0.929176	0.464588	−	0.885527i	$-0.346202\pi$
$647$	9.89370e6	0.929176	0.464588	+	0.885527i	$0.346202\pi$
$648$	−2.46370e6	−0.230489
$649$	−9.48859e6	−0.884281
$650$	0	0
$651$	−248927.	−0.0230208
$652$	−855636.	−0.0788261
$653$	1.62774e7	1.49383	0.746915	−	0.664920i	$-0.231534\pi$
$653$	1.62774e7	1.49383	0.746915	+	0.664920i	$0.231534\pi$
$654$	−16825.3	−0.00153822
$655$	0	0
$656$	4.85643e6	0.440613
$657$	8.47040e6	0.765580
$658$	1.05085e6	0.0946181
$659$	−7.80614e6	−0.700201	−0.350100	−	0.936712i	$-0.613853\pi$
$659$	−7.80614e6	−0.700201	−0.350100	+	0.936712i	$0.613853\pi$
$660$	0	0
$661$	−1.77568e7	−1.58074	−0.790371	−	0.612629i	$-0.790112\pi$
$661$	−1.77568e7	−1.58074	−0.790371	+	0.612629i	$0.790112\pi$
$662$	3.82695e6	0.339397
$663$	−143516.	−0.0126800
$664$	6.58135e6	0.579288
$665$	0	0
$666$	−6.28038e6	−0.548656
$667$	−7.71546e6	−0.671502
$668$	6.32715e6	0.548614
$669$	3.51758e6	0.303863
$670$	0	0
$671$	1.11858e7	0.959097
$672$	−271963.	−0.0232320
$673$	1.32796e7	1.13018	0.565089	−	0.825030i	$-0.308842\pi$
$673$	1.32796e7	1.13018	0.565089	+	0.825030i	$0.308842\pi$
$674$	8.31213e6	0.704794
$675$	0	0
$676$	−5.67011e6	−0.477227
$677$	1.12446e7	0.942916	0.471458	−	0.881889i	$-0.343728\pi$
$677$	1.12446e7	0.942916	0.471458	+	0.881889i	$0.343728\pi$
$678$	−1.65857e6	−0.138567
$679$	2.11656e6	0.176180
$680$	0	0
$681$	−4.00714e6	−0.331106
$682$	1.46050e6	0.120237
$683$	1.04160e7	0.854375	0.427187	−	0.904163i	$-0.359504\pi$
$683$	1.04160e7	0.854375	0.427187	+	0.904163i	$0.359504\pi$
$684$	−4.73859e6	−0.387266
$685$	0	0
$686$	470596.	0.0381802
$687$	5.20137e6	0.420461
$688$	1.95621e6	0.157559
$689$	−742634.	−0.0595973
$690$	0	0
$691$	−1.96002e7	−1.56159	−0.780793	−	0.624790i	$-0.785184\pi$
$691$	−1.96002e7	−1.56159	−0.780793	+	0.624790i	$0.785184\pi$
$692$	491645.	0.0390289
$693$	4.07773e6	0.322542
$694$	814306.	0.0641784
$695$	0	0
$696$	717183.	0.0561186
$697$	3.86262e6	0.301162
$698$	−5.16402e6	−0.401189
$699$	−562171.	−0.0435187
$700$	0	0
$701$	2.05673e6	0.158082	0.0790409	−	0.996871i	$-0.474814\pi$
$701$	2.05673e6	0.158082	0.0790409	+	0.996871i	$0.474814\pi$
$702$	−1.28740e6	−0.0985985
$703$	−1.01898e7	−0.777636
$704$	1.59565e6	0.121341
$705$	0	0
$706$	4.11854e6	0.310980
$707$	233278.	0.0175520
$708$	2.11231e6	0.158371
$709$	2.19571e7	1.64044	0.820219	−	0.572049i	$-0.193851\pi$
$709$	2.19571e7	1.64044	0.820219	+	0.572049i	$0.193851\pi$
$710$	0	0
$711$	2.20841e7	1.63835
$712$	6.03906e6	0.446446
$713$	3.49775e6	0.257670
$714$	−216309.	−0.0158792
$715$	0	0
$716$	−6.88637e6	−0.502005
$717$	−4.12742e6	−0.299834
$718$	−1.15377e7	−0.835233
$719$	−1.59909e7	−1.15359	−0.576795	−	0.816889i	$-0.695697\pi$
$719$	−1.59909e7	−1.15359	−0.576795	+	0.816889i	$0.695697\pi$
$720$	0	0
$721$	−4.08951e6	−0.292977
$722$	2.21613e6	0.158217
$723$	2.90229e6	0.206488
$724$	−1.19315e7	−0.845956
$725$	0	0
$726$	−201452.	−0.0141850
$727$	1.38270e7	0.970267	0.485134	−	0.874440i	$-0.338771\pi$
$727$	1.38270e7	0.970267	0.485134	+	0.874440i	$0.338771\pi$
$728$	407811.	0.0285188
$729$	−4.96363e6	−0.345924
$730$	0	0
$731$	1.55589e6	0.107693
$732$	−2.49015e6	−0.171770
$733$	−7.38390e6	−0.507605	−0.253803	−	0.967256i	$-0.581681\pi$
$733$	−7.38390e6	−0.507605	−0.253803	+	0.967256i	$0.581681\pi$
$734$	−5.67559e6	−0.388840
$735$	0	0
$736$	3.82143e6	0.260035
$737$	−5.63645e6	−0.382241
$738$	1.62100e7	1.09557
$739$	7.73969e6	0.521330	0.260665	−	0.965429i	$-0.416058\pi$
$739$	7.73969e6	0.521330	0.260665	+	0.965429i	$0.416058\pi$
$740$	0	0
$741$	−977194.	−0.0653786
$742$	−1.11930e6	−0.0746342
$743$	−2.07000e7	−1.37562	−0.687811	−	0.725890i	$-0.741428\pi$
$743$	−2.07000e7	−1.37562	−0.687811	+	0.725890i	$0.741428\pi$
$744$	−325130.	−0.0215340
$745$	0	0
$746$	−2.82703e6	−0.185987
$747$	2.19675e7	1.44039
$748$	1.26912e6	0.0829370
$749$	7.92716e6	0.516313
$750$	0	0
$751$	−6.25772e6	−0.404870	−0.202435	−	0.979296i	$-0.564886\pi$
$751$	−6.25772e6	−0.404870	−0.202435	+	0.979296i	$0.564886\pi$
$752$	1.37253e6	0.0885071
$753$	5.93065e6	0.381167
$754$	−1.07542e6	−0.0688891
$755$	0	0
$756$	−1.94038e6	−0.123476
$757$	2.65505e7	1.68397	0.841984	−	0.539503i	$-0.181388\pi$
$757$	2.65505e7	1.68397	0.841984	+	0.539503i	$0.181388\pi$
$758$	2.01809e7	1.27576
$759$	7.87983e6	0.496492
$760$	0	0
$761$	−2.87851e7	−1.80180	−0.900900	−	0.434026i	$-0.857093\pi$
$761$	−2.87851e7	−1.80180	−0.900900	+	0.434026i	$0.857093\pi$
$762$	−1.03715e6	−0.0647075
$763$	38026.4	0.00236469
$764$	6.36558e6	0.394552
$765$	0	0
$766$	5.70829e6	0.351507
$767$	−3.16743e6	−0.194410
$768$	−355217.	−0.0217315
$769$	−2.00357e7	−1.22177	−0.610884	−	0.791720i	$-0.709186\pi$
$769$	−2.00357e7	−1.22177	−0.610884	+	0.791720i	$0.709186\pi$
$770$	0	0
$771$	−8.34818e6	−0.505773
$772$	−731377.	−0.0441670
$773$	1.80213e7	1.08477	0.542383	−	0.840131i	$-0.317522\pi$
$773$	1.80213e7	1.08477	0.542383	+	0.840131i	$0.317522\pi$
$774$	6.52950e6	0.391767
$775$	0	0
$776$	2.76449e6	0.164801
$777$	−1.95205e6	−0.115995
$778$	−1.13086e7	−0.669823
$779$	2.63004e7	1.55281
$780$	0	0
$781$	1.52511e7	0.894690
$782$	3.03942e6	0.177735
$783$	5.11689e6	0.298265
$784$	614656.	0.0357143
$785$	0	0
$786$	−3.50953e6	−0.202625
$787$	−1.62585e7	−0.935716	−0.467858	−	0.883804i	$-0.654974\pi$
$787$	−1.62585e7	−0.935716	−0.467858	+	0.883804i	$0.654974\pi$
$788$	9.90778e6	0.568409
$789$	−8.10620e6	−0.463580
$790$	0	0
$791$	3.74850e6	0.213018
$792$	5.32602e6	0.301710
$793$	3.73400e6	0.210859
$794$	1.71318e7	0.964386
$795$	0	0
$796$	2.40785e6	0.134694
$797$	−5.70146e6	−0.317936	−0.158968	−	0.987284i	$-0.550817\pi$
$797$	−5.70146e6	−0.317936	−0.158968	+	0.987284i	$0.550817\pi$
$798$	−1.47283e6	−0.0818741
$799$	1.09166e6	0.0604952
$800$	0	0
$801$	2.01574e7	1.11008
$802$	−1.75495e7	−0.963452
$803$	−1.54467e7	−0.845371
$804$	1.25476e6	0.0684576
$805$	0	0
$806$	487535.	0.0264343
$807$	2.14184e6	0.115772
$808$	304690.	0.0164184
$809$	7.88934e6	0.423808	0.211904	−	0.977290i	$-0.432034\pi$
$809$	7.88934e6	0.423808	0.211904	+	0.977290i	$0.432034\pi$
$810$	0	0
$811$	3.04845e7	1.62752	0.813761	−	0.581199i	$-0.197416\pi$
$811$	3.04845e7	1.62752	0.813761	+	0.581199i	$0.197416\pi$
$812$	−1.62088e6	−0.0862705
$813$	4.23113e6	0.224507
$814$	1.14530e7	0.605839
$815$	0	0
$816$	−282526.	−0.0148536
$817$	1.05940e7	0.555269
$818$	−2.09587e7	−1.09517
$819$	1.36121e6	0.0709112
$820$	0	0
$821$	−8.10036e6	−0.419417	−0.209709	−	0.977764i	$-0.567252\pi$
$821$	−8.10036e6	−0.419417	−0.209709	+	0.977764i	$0.567252\pi$
$822$	−3.14324e6	−0.162255
$823$	−2.62965e7	−1.35331	−0.676657	−	0.736299i	$-0.736572\pi$
$823$	−2.62965e7	−1.35331	−0.676657	+	0.736299i	$0.736572\pi$
$824$	−5.34140e6	−0.274055
$825$	0	0
$826$	−4.77398e6	−0.243461
$827$	2.20314e7	1.12015	0.560077	−	0.828441i	$-0.310772\pi$
$827$	2.20314e7	1.12015	0.560077	+	0.828441i	$0.310772\pi$
$828$	1.27553e7	0.646569
$829$	1.70715e7	0.862748	0.431374	−	0.902173i	$-0.358029\pi$
$829$	1.70715e7	0.862748	0.431374	+	0.902173i	$0.358029\pi$
$830$	0	0
$831$	−6.77873e6	−0.340522
$832$	532651.	0.0266769
$833$	488874.	0.0244109
$834$	−2.87564e6	−0.143159
$835$	0	0
$836$	8.64135e6	0.427628
$837$	−2.31971e6	−0.114451
$838$	9.65195e6	0.474794
$839$	−2.52085e7	−1.23635	−0.618176	−	0.786040i	$-0.712128\pi$
$839$	−2.52085e7	−1.23635	−0.618176	+	0.786040i	$0.712128\pi$
$840$	0	0
$841$	−1.62368e7	−0.791607
$842$	4.10052e6	0.199324
$843$	2.00861e6	0.0973477
$844$	7.74861e6	0.374428
$845$	0	0
$846$	4.58129e6	0.220071
$847$	455295.	0.0218064
$848$	−1.46195e6	−0.0698139
$849$	−3.65753e6	−0.174148
$850$	0	0
$851$	2.74288e7	1.29832
$852$	−3.39513e6	−0.160235
$853$	−3.02011e7	−1.42118	−0.710591	−	0.703606i	$-0.751572\pi$
$853$	−3.02011e7	−1.42118	−0.710591	+	0.703606i	$0.751572\pi$
$854$	5.62791e6	0.264060
$855$	0	0
$856$	1.03538e7	0.482967
$857$	−2.14185e7	−0.996179	−0.498090	−	0.867126i	$-0.665965\pi$
$857$	−2.14185e7	−0.996179	−0.498090	+	0.867126i	$0.665965\pi$
$858$	1.09833e6	0.0509350
$859$	−2.80803e7	−1.29843	−0.649216	−	0.760604i	$-0.724903\pi$
$859$	−2.80803e7	−1.29843	−0.649216	+	0.760604i	$0.724903\pi$
$860$	0	0
$861$	5.03834e6	0.231622
$862$	−7.99606e6	−0.366529
$863$	−1.30287e7	−0.595490	−0.297745	−	0.954645i	$-0.596235\pi$
$863$	−1.30287e7	−0.595490	−0.297745	+	0.954645i	$0.596235\pi$
$864$	−2.53437e6	−0.115501
$865$	0	0
$866$	1.25887e7	0.570409
$867$	7.47117e6	0.337552
$868$	734816.	0.0331039
$869$	−4.02728e7	−1.80910
$870$	0	0
$871$	−1.88153e6	−0.0840361
$872$	49667.1	0.00221196
$873$	9.22743e6	0.409775
$874$	2.06952e7	0.916413
$875$	0	0
$876$	3.43868e6	0.151402
$877$	−3.24985e7	−1.42680	−0.713401	−	0.700756i	$-0.752846\pi$
$877$	−3.24985e7	−1.42680	−0.713401	+	0.700756i	$0.752846\pi$
$878$	1.96911e7	0.862054
$879$	−1.25835e7	−0.549326
$880$	0	0
$881$	−1.08088e6	−0.0469177	−0.0234589	−	0.999725i	$-0.507468\pi$
$881$	−1.08088e6	−0.0469177	−0.0234589	+	0.999725i	$0.507468\pi$
$882$	2.05162e6	0.0888027
$883$	−2.44497e7	−1.05529	−0.527645	−	0.849465i	$-0.676925\pi$
$883$	−2.44497e7	−1.05529	−0.527645	+	0.849465i	$0.676925\pi$
$884$	423651.	0.0182338
$885$	0	0
$886$	8.39788e6	0.359406
$887$	2.32866e7	0.993796	0.496898	−	0.867809i	$-0.334472\pi$
$887$	2.32866e7	0.993796	0.496898	+	0.867809i	$0.334472\pi$
$888$	−2.54961e6	−0.108503
$889$	2.34404e6	0.0994741
$890$	0	0
$891$	1.49963e7	0.632835
$892$	−1.03836e7	−0.436956
$893$	7.43305e6	0.311917
$894$	−4.96249e6	−0.207661
$895$	0	0
$896$	802816.	0.0334077
$897$	2.63040e6	0.109154
$898$	1.21071e7	0.501013
$899$	−1.93776e6	−0.0799649
$900$	0	0
$901$	−1.16278e6	−0.0477182
$902$	−2.95607e7	−1.20976
$903$	2.02948e6	0.0828257
$904$	4.89599e6	0.199260
$905$	0	0
$906$	2.53848e6	0.102743
$907$	−1.24678e6	−0.0503237	−0.0251618	−	0.999683i	$-0.508010\pi$
$907$	−1.24678e6	−0.0503237	−0.0251618	+	0.999683i	$0.508010\pi$
$908$	1.18288e7	0.476130
$909$	1.01701e6	0.0408238
$910$	0	0
$911$	1.36704e7	0.545738	0.272869	−	0.962051i	$-0.412027\pi$
$911$	1.36704e7	0.545738	0.272869	+	0.962051i	$0.412027\pi$
$912$	−1.92370e6	−0.0765862
$913$	−4.00601e7	−1.59051
$914$	2.35173e7	0.931154
$915$	0	0
$916$	−1.53541e7	−0.604624
$917$	7.93179e6	0.311493
$918$	−2.01574e6	−0.0789457
$919$	4.94453e7	1.93124	0.965620	−	0.259957i	$-0.0837085\pi$
$919$	4.94453e7	1.93124	0.965620	+	0.259957i	$0.0837085\pi$
$920$	0	0
$921$	−1.65457e7	−0.642740
$922$	2.62182e7	1.01572
$923$	5.09103e6	0.196699
$924$	1.65542e6	0.0637863
$925$	0	0
$926$	3.14980e7	1.20713
$927$	−1.78287e7	−0.681430
$928$	−2.11707e6	−0.0806986
$929$	−3.88943e7	−1.47859	−0.739294	−	0.673383i	$-0.764840\pi$
$929$	−3.88943e7	−1.47859	−0.739294	+	0.673383i	$0.764840\pi$
$930$	0	0
$931$	3.32871e6	0.125864
$932$	1.65949e6	0.0625799
$933$	6.25363e6	0.235195
$934$	−339146.	−0.0127210
$935$	0	0
$936$	1.77790e6	0.0663313
$937$	−4.89669e7	−1.82202	−0.911011	−	0.412382i	$-0.864697\pi$
$937$	−4.89669e7	−1.82202	−0.911011	+	0.412382i	$0.864697\pi$
$938$	−2.83586e6	−0.105239
$939$	2.09360e6	0.0774871
$940$	0	0
$941$	−5.08317e7	−1.87137	−0.935687	−	0.352832i	$-0.885219\pi$
$941$	−5.08317e7	−1.87137	−0.935687	+	0.352832i	$0.885219\pi$
$942$	−1.10577e7	−0.406010
$943$	−7.07951e7	−2.59253
$944$	−6.23540e6	−0.227737
$945$	0	0
$946$	−1.19073e7	−0.432598
$947$	3.04613e7	1.10376	0.551878	−	0.833925i	$-0.313911\pi$
$947$	3.04613e7	1.10376	0.551878	+	0.833925i	$0.313911\pi$
$948$	8.96537e6	0.324002
$949$	−5.15634e6	−0.185856
$950$	0	0
$951$	1.42277e7	0.510133
$952$	638529.	0.0228343
$953$	2.80569e7	1.00071	0.500354	−	0.865821i	$-0.333203\pi$
$953$	2.80569e7	1.00071	0.500354	+	0.865821i	$0.333203\pi$
$954$	−4.87974e6	−0.173590
$955$	0	0
$956$	1.21839e7	0.431162
$957$	−4.36543e6	−0.154080
$958$	−3.09309e7	−1.08888
$959$	7.10394e6	0.249432
$960$	0	0
$961$	−2.77507e7	−0.969316
$962$	3.82317e6	0.133194
$963$	3.45594e7	1.20088
$964$	−8.56735e6	−0.296930
$965$	0	0
$966$	3.96456e6	0.136695
$967$	−7.83841e6	−0.269564	−0.134782	−	0.990875i	$-0.543033\pi$
$967$	−7.83841e6	−0.269564	−0.134782	+	0.990875i	$0.543033\pi$
$968$	594671.	0.0203980
$969$	−1.53004e6	−0.0523472
$970$	0	0
$971$	2.54639e7	0.866717	0.433359	−	0.901222i	$-0.357328\pi$
$971$	2.54639e7	0.866717	0.433359	+	0.901222i	$0.357328\pi$
$972$	−1.29611e7	−0.440024
$973$	6.49915e6	0.220077
$974$	1.21534e7	0.410488
$975$	0	0
$976$	7.35074e6	0.247005
$977$	−4.78641e7	−1.60426	−0.802128	−	0.597152i	$-0.796299\pi$
$977$	−4.78641e7	−1.60426	−0.802128	+	0.597152i	$0.796299\pi$
$978$	−1.15943e6	−0.0387611
$979$	−3.67593e7	−1.22577
$980$	0	0
$981$	165781.	0.00549999
$982$	−3.61264e7	−1.19549
$983$	−2.23671e7	−0.738289	−0.369144	−	0.929372i	$-0.620349\pi$
$983$	−2.23671e7	−0.738289	−0.369144	+	0.929372i	$0.620349\pi$
$984$	6.58068e6	0.216662
$985$	0	0
$986$	−1.68384e6	−0.0551580
$987$	1.42394e6	0.0465264
$988$	2.88461e6	0.0940145
$989$	−2.85168e7	−0.927063
$990$	0	0
$991$	1.92246e6	0.0621831	0.0310916	−	0.999517i	$-0.490102\pi$
$991$	1.92246e6	0.0621831	0.0310916	+	0.999517i	$0.490102\pi$
$992$	959760.	0.0309659
$993$	5.18568e6	0.166891
$994$	7.67324e6	0.246327
$995$	0	0
$996$	8.91802e6	0.284853
$997$	−1.75771e7	−0.560027	−0.280013	−	0.959996i	$-0.590339\pi$
$997$	−1.75771e7	−0.560027	−0.280013	+	0.959996i	$0.590339\pi$
$998$	−2.51757e7	−0.800122
$999$	−1.81908e7	−0.576683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.u.1.2	✓	3
5.2	odd	4		350.6.c.n.99.2		6
5.3	odd	4		350.6.c.n.99.5		6
5.4	even	2		350.6.a.x.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.u.1.2	✓	3	1.1	even	1	trivial
350.6.a.x.1.2	yes	3	5.4	even	2
350.6.c.n.99.2		6	5.2	odd	4
350.6.c.n.99.5		6	5.3	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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -5.42018 q^{3} +16.0000 q^{4} +21.6807 q^{6} -49.0000 q^{7} -64.0000 q^{8} -213.622 q^{9} +389.563 q^{11} -86.7229 q^{12} +130.042 q^{13} +196.000 q^{14} +256.000 q^{16} +203.613 q^{17} +854.487 q^{18} +1386.39 q^{19} +265.589 q^{21} -1558.25 q^{22} -3731.86 q^{23} +346.892 q^{24} -520.167 q^{26} +2474.97 q^{27} -784.000 q^{28} +2067.46 q^{29} -937.266 q^{31} -1024.00 q^{32} -2111.50 q^{33} -814.451 q^{34} -3417.95 q^{36} -7349.89 q^{37} -5545.54 q^{38} -704.850 q^{39} +18970.4 q^{41} -1062.36 q^{42} +7641.43 q^{43} +6233.01 q^{44} +14927.5 q^{46} +5361.46 q^{47} -1387.57 q^{48} +2401.00 q^{49} -1103.62 q^{51} +2080.67 q^{52} -5710.73 q^{53} -9899.89 q^{54} +3136.00 q^{56} -7514.46 q^{57} -8269.82 q^{58} -24357.0 q^{59} +28713.8 q^{61} +3749.06 q^{62} +10467.5 q^{63} +4096.00 q^{64} +8446.00 q^{66} -14468.7 q^{67} +3257.80 q^{68} +20227.4 q^{69} +39149.2 q^{71} +13671.8 q^{72} -39651.4 q^{73} +29399.5 q^{74} +22182.2 q^{76} -19088.6 q^{77} +2819.40 q^{78} -103380. q^{79} +38495.3 q^{81} -75881.8 q^{82} -102834. q^{83} +4249.42 q^{84} -30565.7 q^{86} -11206.0 q^{87} -24932.0 q^{88} -94360.3 q^{89} -6372.05 q^{91} -59709.8 q^{92} +5080.15 q^{93} -21445.8 q^{94} +5550.26 q^{96} -43195.2 q^{97} -9604.00 q^{98} -83219.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 12 q^{2} - 9 q^{3} + 48 q^{4} + 36 q^{6} - 147 q^{7} - 192 q^{8} + 584 q^{9} - 107 q^{11} - 144 q^{12} - 842 q^{13} + 588 q^{14} + 768 q^{16} - 251 q^{17} - 2336 q^{18} + 441 q^{19} + 441 q^{21} + 428 q^{22}+ \cdots - 265616 q^{99}+O(q^{100}) \) 3 * q - 12 * q^2 - 9 * q^3 + 48 * q^4 + 36 * q^6 - 147 * q^7 - 192 * q^8 + 584 * q^9 - 107 * q^11 - 144 * q^12 - 842 * q^13 + 588 * q^14 + 768 * q^16 - 251 * q^17 - 2336 * q^18 + 441 * q^19 + 441 * q^21 + 428 * q^22 + 494 * q^23 + 576 * q^24 + 3368 * q^26 - 2655 * q^27 - 2352 * q^28 + 1308 * q^29 + 4724 * q^31 - 3072 * q^32 - 5591 * q^33 + 1004 * q^34 + 9344 * q^36 + 5614 * q^37 - 1764 * q^38 + 4330 * q^39 + 1979 * q^41 - 1764 * q^42 - 19270 * q^43 - 1712 * q^44 - 1976 * q^46 - 7774 * q^47 - 2304 * q^48 + 7203 * q^49 + 68143 * q^51 - 13472 * q^52 - 31800 * q^53 + 10620 * q^54 + 9408 * q^56 + 231 * q^57 - 5232 * q^58 - 16076 * q^59 + 98282 * q^61 - 18896 * q^62 - 28616 * q^63 + 12288 * q^64 + 22364 * q^66 - 32311 * q^67 - 4016 * q^68 - 131404 * q^69 + 154844 * q^71 - 37376 * q^72 - 30761 * q^73 - 22456 * q^74 + 7056 * q^76 + 5243 * q^77 - 17320 * q^78 - 84032 * q^79 + 61043 * q^81 - 7916 * q^82 + 47301 * q^83 + 7056 * q^84 + 77080 * q^86 - 210702 * q^87 + 6848 * q^88 - 210655 * q^89 + 41258 * q^91 + 7904 * q^92 - 219700 * q^93 + 31096 * q^94 + 9216 * q^96 - 249302 * q^97 - 28812 * q^98 - 265616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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