LMFDB - Embedded newform 350.6.a.y.1.4

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 915x^{3} - 2649x^{2} + 122688x - 432576 $ x^5 - x^4 - 915x^3 - 2649x^2 + 122688*x - 432576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.91835$ 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)

$f(q)$	$=$	\(q-4.00000 q^{2} +3.91835 q^{3} +16.0000 q^{4} -15.6734 q^{6} -49.0000 q^{7} -64.0000 q^{8} -227.647 q^{9} +337.260 q^{11} +62.6937 q^{12} -24.3276 q^{13} +196.000 q^{14} +256.000 q^{16} -1445.07 q^{17} +910.586 q^{18} -1197.44 q^{19} -191.999 q^{21} -1349.04 q^{22} +2247.26 q^{23} -250.775 q^{24} +97.3103 q^{26} -1844.16 q^{27} -784.000 q^{28} +1182.78 q^{29} +10458.6 q^{31} -1024.00 q^{32} +1321.50 q^{33} +5780.27 q^{34} -3642.34 q^{36} -825.831 q^{37} +4789.75 q^{38} -95.3241 q^{39} +2843.58 q^{41} +767.997 q^{42} -9617.84 q^{43} +5396.16 q^{44} -8989.03 q^{46} +10435.0 q^{47} +1003.10 q^{48} +2401.00 q^{49} -5662.28 q^{51} -389.241 q^{52} -33934.4 q^{53} +7376.64 q^{54} +3136.00 q^{56} -4691.99 q^{57} -4731.12 q^{58} +18857.8 q^{59} -26489.1 q^{61} -41834.6 q^{62} +11154.7 q^{63} +4096.00 q^{64} -5286.01 q^{66} -10647.5 q^{67} -23121.1 q^{68} +8805.55 q^{69} +47989.5 q^{71} +14569.4 q^{72} +37699.4 q^{73} +3303.32 q^{74} -19159.0 q^{76} -16525.7 q^{77} +381.296 q^{78} +62335.6 q^{79} +48092.0 q^{81} -11374.3 q^{82} +369.796 q^{83} -3071.99 q^{84} +38471.4 q^{86} +4634.56 q^{87} -21584.6 q^{88} +115397. q^{89} +1192.05 q^{91} +35956.1 q^{92} +40980.7 q^{93} -41739.9 q^{94} -4012.39 q^{96} -136508. q^{97} -9604.00 q^{98} -76776.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 20 q^{2} - 14 q^{3} + 80 q^{4} + 56 q^{6} - 245 q^{7} - 320 q^{8} + 655 q^{9} + 748 q^{11} - 224 q^{12} - 456 q^{13} + 980 q^{14} + 1280 q^{16} - 780 q^{17} - 2620 q^{18} - 256 q^{19} + 686 q^{21}+ \cdots + 575944 q^{99}+O(q^{100}) $ 5 * q - 20 * q^2 - 14 * q^3 + 80 * q^4 + 56 * q^6 - 245 * q^7 - 320 * q^8 + 655 * q^9 + 748 * q^11 - 224 * q^12 - 456 * q^13 + 980 * q^14 + 1280 * q^16 - 780 * q^17 - 2620 * q^18 - 256 * q^19 + 686 * q^21 - 2992 * q^22 - 3452 * q^23 + 896 * q^24 + 1824 * q^26 + 910 * q^27 - 3920 * q^28 + 7712 * q^29 + 412 * q^31 - 5120 * q^32 + 2980 * q^33 + 3120 * q^34 + 10480 * q^36 - 14400 * q^37 + 1024 * q^38 + 17668 * q^39 + 7482 * q^41 - 2744 * q^42 - 20668 * q^43 + 11968 * q^44 + 13808 * q^46 - 59202 * q^47 - 3584 * q^48 + 12005 * q^49 + 6628 * q^51 - 7296 * q^52 - 6388 * q^53 - 3640 * q^54 + 15680 * q^56 - 112616 * q^57 - 30848 * q^58 + 79596 * q^59 + 9754 * q^61 - 1648 * q^62 - 32095 * q^63 + 20480 * q^64 - 11920 * q^66 - 49820 * q^67 - 12480 * q^68 + 106152 * q^69 - 61928 * q^71 - 41920 * q^72 + 57248 * q^73 + 57600 * q^74 - 4096 * q^76 - 36652 * q^77 - 70672 * q^78 + 150912 * q^79 + 100825 * q^81 - 29928 * q^82 + 54776 * q^83 + 10976 * q^84 + 82672 * q^86 + 256362 * q^87 - 47872 * q^88 + 268950 * q^89 + 22344 * q^91 - 55232 * q^92 + 187044 * q^93 + 236808 * q^94 + 14336 * q^96 + 296156 * q^97 - 48020 * q^98 + 575944 * 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
$3$	3.91835	0.251363	0.125681	−	0.992071i	$-0.459888\pi$
$3$	3.91835	0.251363	0.125681	+	0.992071i	$0.459888\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−15.6734	−0.177740
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	−227.647	−0.936817
$10$	0	0
$11$	337.260	0.840394	0.420197	−	0.907433i	$-0.361961\pi$
$11$	337.260	0.840394	0.420197	+	0.907433i	$0.361961\pi$
$12$	62.6937	0.125681
$13$	−24.3276	−0.0399246	−0.0199623	−	0.999801i	$-0.506355\pi$
$13$	−24.3276	−0.0399246	−0.0199623	+	0.999801i	$0.506355\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1445.07	−1.21273	−0.606367	−	0.795185i	$-0.707374\pi$
$17$	−1445.07	−1.21273	−0.606367	+	0.795185i	$0.707374\pi$
$18$	910.586	0.662430
$19$	−1197.44	−0.760973	−0.380486	−	0.924786i	$-0.624243\pi$
$19$	−1197.44	−0.760973	−0.380486	+	0.924786i	$0.624243\pi$
$20$	0	0
$21$	−191.999	−0.0950061
$22$	−1349.04	−0.594248
$23$	2247.26	0.885795	0.442898	−	0.896572i	$-0.353950\pi$
$23$	2247.26	0.885795	0.442898	+	0.896572i	$0.353950\pi$
$24$	−250.775	−0.0888701
$25$	0	0
$26$	97.3103	0.0282310
$27$	−1844.16	−0.486843
$28$	−784.000	−0.188982
$29$	1182.78	0.261162	0.130581	−	0.991438i	$-0.458316\pi$
$29$	1182.78	0.261162	0.130581	+	0.991438i	$0.458316\pi$
$30$	0	0
$31$	10458.6	1.95466	0.977330	−	0.211721i	$-0.0679068\pi$
$31$	10458.6	1.95466	0.977330	+	0.211721i	$0.0679068\pi$
$32$	−1024.00	−0.176777
$33$	1321.50	0.211243
$34$	5780.27	0.857533
$35$	0	0
$36$	−3642.34	−0.468408
$37$	−825.831	−0.0991715	−0.0495857	−	0.998770i	$-0.515790\pi$
$37$	−825.831	−0.0991715	−0.0495857	+	0.998770i	$0.515790\pi$
$38$	4789.75	0.538089
$39$	−95.3241	−0.0100355
$40$	0	0
$41$	2843.58	0.264183	0.132092	−	0.991238i	$-0.457831\pi$
$41$	2843.58	0.264183	0.132092	+	0.991238i	$0.457831\pi$
$42$	767.997	0.0671795
$43$	−9617.84	−0.793244	−0.396622	−	0.917982i	$-0.629818\pi$
$43$	−9617.84	−0.793244	−0.396622	+	0.917982i	$0.629818\pi$
$44$	5396.16	0.420197
$45$	0	0
$46$	−8989.03	−0.626352
$47$	10435.0	0.689043	0.344522	−	0.938778i	$-0.388041\pi$
$47$	10435.0	0.689043	0.344522	+	0.938778i	$0.388041\pi$
$48$	1003.10	0.0628406
$49$	2401.00	0.142857
$50$	0	0
$51$	−5662.28	−0.304836
$52$	−389.241	−0.0199623
$53$	−33934.4	−1.65940	−0.829698	−	0.558212i	$-0.811488\pi$
$53$	−33934.4	−1.65940	−0.829698	+	0.558212i	$0.811488\pi$
$54$	7376.64	0.344250
$55$	0	0
$56$	3136.00	0.133631
$57$	−4691.99	−0.191280
$58$	−4731.12	−0.184669
$59$	18857.8	0.705281	0.352640	−	0.935759i	$-0.385284\pi$
$59$	18857.8	0.705281	0.352640	+	0.935759i	$0.385284\pi$
$60$	0	0
$61$	−26489.1	−0.911470	−0.455735	−	0.890115i	$-0.650624\pi$
$61$	−26489.1	−0.911470	−0.455735	+	0.890115i	$0.650624\pi$
$62$	−41834.6	−1.38215
$63$	11154.7	0.354083
$64$	4096.00	0.125000
$65$	0	0
$66$	−5286.01	−0.149372
$67$	−10647.5	−0.289774	−0.144887	−	0.989448i	$-0.546282\pi$
$67$	−10647.5	−0.289774	−0.144887	+	0.989448i	$0.546282\pi$
$68$	−23121.1	−0.606367
$69$	8805.55	0.222656
$70$	0	0
$71$	47989.5	1.12980	0.564899	−	0.825160i	$-0.308915\pi$
$71$	47989.5	1.12980	0.564899	+	0.825160i	$0.308915\pi$
$72$	14569.4	0.331215
$73$	37699.4	0.827993	0.413997	−	0.910278i	$-0.364132\pi$
$73$	37699.4	0.827993	0.413997	+	0.910278i	$0.364132\pi$
$74$	3303.32	0.0701248
$75$	0	0
$76$	−19159.0	−0.380486
$77$	−16525.7	−0.317639
$78$	381.296	0.00709620
$79$	62335.6	1.12375	0.561874	−	0.827223i	$-0.310081\pi$
$79$	62335.6	1.12375	0.561874	+	0.827223i	$0.310081\pi$
$80$	0	0
$81$	48092.0	0.814443
$82$	−11374.3	−0.186806
$83$	369.796	0.00589205	0.00294603	−	0.999996i	$-0.499062\pi$
$83$	369.796	0.00589205	0.00294603	+	0.999996i	$0.499062\pi$
$84$	−3071.99	−0.0475031
$85$	0	0
$86$	38471.4	0.560908
$87$	4634.56	0.0656463
$88$	−21584.6	−0.297124
$89$	115397.	1.54425	0.772126	−	0.635469i	$-0.219193\pi$
$89$	115397.	1.54425	0.772126	+	0.635469i	$0.219193\pi$
$90$	0	0
$91$	1192.05	0.0150901
$92$	35956.1	0.442898
$93$	40980.7	0.491328
$94$	−41739.9	−0.487227
$95$	0	0
$96$	−4012.39	−0.0444350
$97$	−136508.	−1.47309	−0.736544	−	0.676390i	$-0.763543\pi$
$97$	−136508.	−1.47309	−0.736544	+	0.676390i	$0.763543\pi$
$98$	−9604.00	−0.101015
$99$	−76776.0	−0.787295
$100$	0	0
$101$	55976.1	0.546008	0.273004	−	0.962013i	$-0.411983\pi$
$101$	55976.1	0.546008	0.273004	+	0.962013i	$0.411983\pi$
$102$	22649.1	0.215552
$103$	165176.	1.53410	0.767052	−	0.641585i	$-0.221723\pi$
$103$	165176.	1.53410	0.767052	+	0.641585i	$0.221723\pi$
$104$	1556.97	0.0141155
$105$	0	0
$106$	135737.	1.17337
$107$	99801.0	0.842705	0.421352	−	0.906897i	$-0.361556\pi$
$107$	99801.0	0.842705	0.421352	+	0.906897i	$0.361556\pi$
$108$	−29506.6	−0.243422
$109$	225386.	1.81703	0.908514	−	0.417855i	$-0.137218\pi$
$109$	225386.	1.81703	0.908514	+	0.417855i	$0.137218\pi$
$110$	0	0
$111$	−3235.90	−0.0249280
$112$	−12544.0	−0.0944911
$113$	−104503.	−0.769895	−0.384948	−	0.922938i	$-0.625781\pi$
$113$	−104503.	−0.769895	−0.384948	+	0.922938i	$0.625781\pi$
$114$	18768.0	0.135255
$115$	0	0
$116$	18924.5	0.130581
$117$	5538.09	0.0374020
$118$	−75431.4	−0.498709
$119$	70808.3	0.458371
$120$	0	0
$121$	−47306.9	−0.293738
$122$	105956.	0.644507
$123$	11142.1	0.0664058
$124$	167338.	0.977330
$125$	0	0
$126$	−44618.7	−0.250375
$127$	245433.	1.35028	0.675141	−	0.737689i	$-0.264083\pi$
$127$	245433.	1.35028	0.675141	+	0.737689i	$0.264083\pi$
$128$	−16384.0	−0.0883883
$129$	−37686.1	−0.199392
$130$	0	0
$131$	−101703.	−0.517792	−0.258896	−	0.965905i	$-0.583359\pi$
$131$	−101703.	−0.517792	−0.258896	+	0.965905i	$0.583359\pi$
$132$	21144.0	0.105622
$133$	58674.5	0.287621
$134$	42589.9	0.204901
$135$	0	0
$136$	92484.3	0.428766
$137$	63654.7	0.289754	0.144877	−	0.989450i	$-0.453721\pi$
$137$	63654.7	0.289754	0.144877	+	0.989450i	$0.453721\pi$
$138$	−35222.2	−0.157441
$139$	277917.	1.22005	0.610025	−	0.792382i	$-0.291159\pi$
$139$	277917.	1.22005	0.610025	+	0.792382i	$0.291159\pi$
$140$	0	0
$141$	40887.9	0.173200
$142$	−191958.	−0.798888
$143$	−8204.71	−0.0335524
$144$	−58277.5	−0.234204
$145$	0	0
$146$	−150797.	−0.585480
$147$	9407.97	0.0359089
$148$	−13213.3	−0.0495857
$149$	456393.	1.68412	0.842060	−	0.539384i	$-0.181343\pi$
$149$	456393.	1.68412	0.842060	+	0.539384i	$0.181343\pi$
$150$	0	0
$151$	318099.	1.13532	0.567662	−	0.823262i	$-0.307848\pi$
$151$	318099.	1.13532	0.567662	+	0.823262i	$0.307848\pi$
$152$	76636.1	0.269045
$153$	328964.	1.13611
$154$	66102.9	0.224605
$155$	0	0
$156$	−1525.19	−0.00501777
$157$	392920.	1.27220	0.636099	−	0.771607i	$-0.280547\pi$
$157$	392920.	1.27220	0.636099	+	0.771607i	$0.280547\pi$
$158$	−249342.	−0.794609
$159$	−132967.	−0.417110
$160$	0	0
$161$	−110116.	−0.334799
$162$	−192368.	−0.575898
$163$	−56353.1	−0.166130	−0.0830652	−	0.996544i	$-0.526471\pi$
$163$	−56353.1	−0.166130	−0.0830652	+	0.996544i	$0.526471\pi$
$164$	45497.2	0.132092
$165$	0	0
$166$	−1479.18	−0.00416631
$167$	312776.	0.867845	0.433922	−	0.900950i	$-0.357129\pi$
$167$	312776.	0.867845	0.433922	+	0.900950i	$0.357129\pi$
$168$	12288.0	0.0335897
$169$	−370701.	−0.998406
$170$	0	0
$171$	272593.	0.712892
$172$	−153885.	−0.396622
$173$	307297.	0.780626	0.390313	−	0.920682i	$-0.372367\pi$
$173$	307297.	0.780626	0.390313	+	0.920682i	$0.372367\pi$
$174$	−18538.2	−0.0464189
$175$	0	0
$176$	86338.5	0.210098
$177$	73891.7	0.177281
$178$	−461587.	−1.09195
$179$	−804792.	−1.87738	−0.938688	−	0.344768i	$-0.887958\pi$
$179$	−804792.	−1.87738	−0.938688	+	0.344768i	$0.887958\pi$
$180$	0	0
$181$	−518678.	−1.17680	−0.588399	−	0.808571i	$-0.700241\pi$
$181$	−518678.	−1.17680	−0.588399	+	0.808571i	$0.700241\pi$
$182$	−4768.21	−0.0106703
$183$	−103794.	−0.229109
$184$	−143825.	−0.313176
$185$	0	0
$186$	−163923.	−0.347422
$187$	−487363.	−1.01917
$188$	166960.	0.344522
$189$	90363.8	0.184009
$190$	0	0
$191$	−204063.	−0.404744	−0.202372	−	0.979309i	$-0.564865\pi$
$191$	−204063.	−0.404744	−0.202372	+	0.979309i	$0.564865\pi$
$192$	16049.6	0.0314203
$193$	873491.	1.68797	0.843986	−	0.536365i	$-0.180203\pi$
$193$	873491.	1.68797	0.843986	+	0.536365i	$0.180203\pi$
$194$	546032.	1.04163
$195$	0	0
$196$	38416.0	0.0714286
$197$	−114332.	−0.209895	−0.104948	−	0.994478i	$-0.533468\pi$
$197$	−114332.	−0.209895	−0.104948	+	0.994478i	$0.533468\pi$
$198$	307104.	0.556702
$199$	−262148.	−0.469261	−0.234630	−	0.972085i	$-0.575388\pi$
$199$	−262148.	−0.469261	−0.234630	+	0.972085i	$0.575388\pi$
$200$	0	0
$201$	−41720.6	−0.0728384
$202$	−223904.	−0.386086
$203$	−57956.3	−0.0987098
$204$	−90596.5	−0.152418
$205$	0	0
$206$	−660705.	−1.08478
$207$	−511581.	−0.829828
$208$	−6227.86	−0.00998115
$209$	−403848.	−0.639517
$210$	0	0
$211$	878765.	1.35883	0.679417	−	0.733752i	$-0.262233\pi$
$211$	878765.	1.35883	0.679417	+	0.733752i	$0.262233\pi$
$212$	−542950.	−0.829698
$213$	188040.	0.283989
$214$	−399204.	−0.595882
$215$	0	0
$216$	118026.	0.172125
$217$	−512474.	−0.738792
$218$	−901546.	−1.28483
$219$	147719.	0.208127
$220$	0	0
$221$	35155.0	0.0484179
$222$	12943.6	0.0176268
$223$	−540539.	−0.727888	−0.363944	−	0.931421i	$-0.618570\pi$
$223$	−540539.	−0.727888	−0.363944	+	0.931421i	$0.618570\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	418011.	0.544398
$227$	135234.	0.174189	0.0870947	−	0.996200i	$-0.472242\pi$
$227$	135234.	0.174189	0.0870947	+	0.996200i	$0.472242\pi$
$228$	−75071.8	−0.0956400
$229$	−118878.	−0.149800	−0.0749000	−	0.997191i	$-0.523864\pi$
$229$	−118878.	−0.149800	−0.0749000	+	0.997191i	$0.523864\pi$
$230$	0	0
$231$	−64753.6	−0.0798425
$232$	−75698.0	−0.0923346
$233$	−659512.	−0.795854	−0.397927	−	0.917417i	$-0.630270\pi$
$233$	−659512.	−0.795854	−0.397927	+	0.917417i	$0.630270\pi$
$234$	−22152.4	−0.0264472
$235$	0	0
$236$	301726.	0.352640
$237$	244253.	0.282468
$238$	−283233.	−0.324117
$239$	467946.	0.529908	0.264954	−	0.964261i	$-0.414643\pi$
$239$	467946.	0.529908	0.264954	+	0.964261i	$0.414643\pi$
$240$	0	0
$241$	−215104.	−0.238564	−0.119282	−	0.992860i	$-0.538059\pi$
$241$	−215104.	−0.238564	−0.119282	+	0.992860i	$0.538059\pi$
$242$	189228.	0.207704
$243$	636572.	0.691564
$244$	−423826.	−0.455735
$245$	0	0
$246$	−44568.6	−0.0469560
$247$	29130.8	0.0303815
$248$	−669353.	−0.691077
$249$	1448.99	0.00148104
$250$	0	0
$251$	−874948.	−0.876593	−0.438297	−	0.898830i	$-0.644418\pi$
$251$	−874948.	−0.876593	−0.438297	+	0.898830i	$0.644418\pi$
$252$	178475.	0.177042
$253$	757910.	0.744417
$254$	−981734.	−0.954793
$255$	0	0
$256$	65536.0	0.0625000
$257$	227843.	0.215180	0.107590	−	0.994195i	$-0.465687\pi$
$257$	227843.	0.215180	0.107590	+	0.994195i	$0.465687\pi$
$258$	150744.	0.140991
$259$	40465.7	0.0374833
$260$	0	0
$261$	−269256.	−0.244661
$262$	406812.	0.366134
$263$	−1.56138e6	−1.39194	−0.695968	−	0.718073i	$-0.745024\pi$
$263$	−1.56138e6	−1.39194	−0.695968	+	0.718073i	$0.745024\pi$
$264$	−84576.2	−0.0746858
$265$	0	0
$266$	−234698.	−0.203379
$267$	452165.	0.388167
$268$	−170360.	−0.144887
$269$	−1.58500e6	−1.33551	−0.667756	−	0.744380i	$-0.732745\pi$
$269$	−1.58500e6	−1.33551	−0.667756	+	0.744380i	$0.732745\pi$
$270$	0	0
$271$	477067.	0.394599	0.197300	−	0.980343i	$-0.436783\pi$
$271$	477067.	0.394599	0.197300	+	0.980343i	$0.436783\pi$
$272$	−369937.	−0.303184
$273$	4670.88	0.00379308
$274$	−254619.	−0.204887
$275$	0	0
$276$	140889.	0.111328
$277$	138751.	0.108652	0.0543258	−	0.998523i	$-0.482699\pi$
$277$	138751.	0.108652	0.0543258	+	0.998523i	$0.482699\pi$
$278$	−1.11167e6	−0.862706
$279$	−2.38087e6	−1.83116
$280$	0	0
$281$	−2.51115e6	−1.89717	−0.948585	−	0.316522i	$-0.897485\pi$
$281$	−2.51115e6	−1.89717	−0.948585	+	0.316522i	$0.897485\pi$
$282$	−163552.	−0.122471
$283$	2.29079e6	1.70027	0.850136	−	0.526563i	$-0.176520\pi$
$283$	2.29079e6	1.70027	0.850136	+	0.526563i	$0.176520\pi$
$284$	767833.	0.564899
$285$	0	0
$286$	32818.9	0.0237251
$287$	−139335.	−0.0998519
$288$	233110.	0.165607
$289$	668362.	0.470725
$290$	0	0
$291$	−534886.	−0.370279
$292$	603190.	0.413997
$293$	446400.	0.303777	0.151888	−	0.988398i	$-0.451465\pi$
$293$	446400.	0.303777	0.151888	+	0.988398i	$0.451465\pi$
$294$	−37631.9	−0.0253914
$295$	0	0
$296$	52853.2	0.0350624
$297$	−621961.	−0.409140
$298$	−1.82557e6	−1.19085
$299$	−54670.4	−0.0353650
$300$	0	0
$301$	471274.	0.299818
$302$	−1.27240e6	−0.802795
$303$	219334.	0.137246
$304$	−306544.	−0.190243
$305$	0	0
$306$	−1.31586e6	−0.803351
$307$	132928.	0.0804956	0.0402478	−	0.999190i	$-0.487185\pi$
$307$	132928.	0.0804956	0.0402478	+	0.999190i	$0.487185\pi$
$308$	−264412.	−0.158819
$309$	647219.	0.385616
$310$	0	0
$311$	−1.02347e6	−0.600029	−0.300015	−	0.953935i	$-0.596992\pi$
$311$	−1.02347e6	−0.600029	−0.300015	+	0.953935i	$0.596992\pi$
$312$	6100.74	0.00354810
$313$	2.03277e6	1.17281	0.586404	−	0.810018i	$-0.300543\pi$
$313$	2.03277e6	1.17281	0.586404	+	0.810018i	$0.300543\pi$
$314$	−1.57168e6	−0.899580
$315$	0	0
$316$	997370.	0.561874
$317$	2.42685e6	1.35642	0.678212	−	0.734866i	$-0.262755\pi$
$317$	2.42685e6	1.35642	0.678212	+	0.734866i	$0.262755\pi$
$318$	531868.	0.294941
$319$	398904.	0.219479
$320$	0	0
$321$	391056.	0.211824
$322$	440463.	0.236739
$323$	1.73038e6	0.922858
$324$	769472.	0.407221
$325$	0	0
$326$	225412.	0.117472
$327$	883144.	0.456733
$328$	−181989.	−0.0934029
$329$	−511314.	−0.260434
$330$	0	0
$331$	552981.	0.277422	0.138711	−	0.990333i	$-0.455704\pi$
$331$	552981.	0.277422	0.138711	+	0.990333i	$0.455704\pi$
$332$	5916.73	0.00294603
$333$	187998.	0.0929055
$334$	−1.25110e6	−0.613659
$335$	0	0
$336$	−49151.8	−0.0237515
$337$	−202099.	−0.0969371	−0.0484685	−	0.998825i	$-0.515434\pi$
$337$	−202099.	−0.0969371	−0.0484685	+	0.998825i	$0.515434\pi$
$338$	1.48280e6	0.705980
$339$	−409479.	−0.193523
$340$	0	0
$341$	3.52728e6	1.64268
$342$	−1.09037e6	−0.504091
$343$	−117649.	−0.0539949
$344$	615542.	0.280454
$345$	0	0
$346$	−1.22919e6	−0.551986
$347$	−1.04617e6	−0.466419	−0.233210	−	0.972426i	$-0.574923\pi$
$347$	−1.04617e6	−0.466419	−0.233210	+	0.972426i	$0.574923\pi$
$348$	74152.9	0.0328231
$349$	−2.72359e6	−1.19696	−0.598479	−	0.801139i	$-0.704228\pi$
$349$	−2.72359e6	−1.19696	−0.598479	+	0.801139i	$0.704228\pi$
$350$	0	0
$351$	44863.9	0.0194370
$352$	−345354.	−0.148562
$353$	−1.73711e6	−0.741979	−0.370989	−	0.928637i	$-0.620981\pi$
$353$	−1.73711e6	−0.741979	−0.370989	+	0.928637i	$0.620981\pi$
$354$	−295567.	−0.125357
$355$	0	0
$356$	1.84635e6	0.772126
$357$	277452.	0.115217
$358$	3.21917e6	1.32751
$359$	1.68699e6	0.690837	0.345419	−	0.938449i	$-0.387737\pi$
$359$	1.68699e6	0.690837	0.345419	+	0.938449i	$0.387737\pi$
$360$	0	0
$361$	−1.04224e6	−0.420920
$362$	2.07471e6	0.832121
$363$	−185365.	−0.0738348
$364$	19072.8	0.00754504
$365$	0	0
$366$	415175.	0.162005
$367$	−3.25743e6	−1.26244	−0.631219	−	0.775605i	$-0.717445\pi$
$367$	−3.25743e6	−1.26244	−0.631219	+	0.775605i	$0.717445\pi$
$368$	575298.	0.221449
$369$	−647330.	−0.247491
$370$	0	0
$371$	1.66278e6	0.627193
$372$	655691.	0.245664
$373$	−3.36622e6	−1.25277	−0.626384	−	0.779514i	$-0.715466\pi$
$373$	−3.36622e6	−1.25277	−0.626384	+	0.779514i	$0.715466\pi$
$374$	1.94945e6	0.720665
$375$	0	0
$376$	−667838.	−0.243614
$377$	−28774.2	−0.0104268
$378$	−361455.	−0.130114
$379$	4.98342e6	1.78209	0.891044	−	0.453917i	$-0.149974\pi$
$379$	4.98342e6	1.78209	0.891044	+	0.453917i	$0.149974\pi$
$380$	0	0
$381$	961695.	0.339410
$382$	816252.	0.286197
$383$	1.14076e6	0.397374	0.198687	−	0.980063i	$-0.436332\pi$
$383$	1.14076e6	0.397374	0.198687	+	0.980063i	$0.436332\pi$
$384$	−64198.3	−0.0222175
$385$	0	0
$386$	−3.49396e6	−1.19358
$387$	2.18947e6	0.743124
$388$	−2.18413e6	−0.736544
$389$	−4.58469e6	−1.53616	−0.768080	−	0.640354i	$-0.778788\pi$
$389$	−4.58469e6	−1.53616	−0.768080	+	0.640354i	$0.778788\pi$
$390$	0	0
$391$	−3.24744e6	−1.07423
$392$	−153664.	−0.0505076
$393$	−398508.	−0.130154
$394$	457329.	0.148418
$395$	0	0
$396$	−1.22842e6	−0.393647
$397$	4.12709e6	1.31422	0.657110	−	0.753795i	$-0.271779\pi$
$397$	4.12709e6	1.31422	0.657110	+	0.753795i	$0.271779\pi$
$398$	1.04859e6	0.331817
$399$	229907.	0.0722971
$400$	0	0
$401$	2.33931e6	0.726487	0.363243	−	0.931694i	$-0.381669\pi$
$401$	2.33931e6	0.726487	0.363243	+	0.931694i	$0.381669\pi$
$402$	166882.	0.0515045
$403$	−254434.	−0.0780390
$404$	895617.	0.273004
$405$	0	0
$406$	231825.	0.0697984
$407$	−278520.	−0.0833431
$408$	362386.	0.107776
$409$	5.04134e6	1.49018	0.745088	−	0.666966i	$-0.232408\pi$
$409$	5.04134e6	1.49018	0.745088	+	0.666966i	$0.232408\pi$
$410$	0	0
$411$	249422.	0.0728333
$412$	2.64282e6	0.767052
$413$	−924034.	−0.266571
$414$	2.04632e6	0.586777
$415$	0	0
$416$	24911.4	0.00705774
$417$	1.08898e6	0.306675
$418$	1.61539e6	0.452207
$419$	66941.5	0.0186277	0.00931387	−	0.999957i	$-0.497035\pi$
$419$	66941.5	0.0186277	0.00931387	+	0.999957i	$0.497035\pi$
$420$	0	0
$421$	670296.	0.184315	0.0921576	−	0.995744i	$-0.470624\pi$
$421$	670296.	0.184315	0.0921576	+	0.995744i	$0.470624\pi$
$422$	−3.51506e6	−0.960841
$423$	−2.37548e6	−0.645508
$424$	2.17180e6	0.586685
$425$	0	0
$426$	−752160.	−0.200810
$427$	1.29797e6	0.344503
$428$	1.59682e6	0.421352
$429$	−32149.0	−0.00843381
$430$	0	0
$431$	−365337.	−0.0947329	−0.0473664	−	0.998878i	$-0.515083\pi$
$431$	−365337.	−0.0947329	−0.0473664	+	0.998878i	$0.515083\pi$
$432$	−472105.	−0.121711
$433$	−1.30294e6	−0.333967	−0.166984	−	0.985960i	$-0.553403\pi$
$433$	−1.30294e6	−0.333967	−0.166984	+	0.985960i	$0.553403\pi$
$434$	2.04989e6	0.522405
$435$	0	0
$436$	3.60618e6	0.908514
$437$	−2.69095e6	−0.674066
$438$	−590878.	−0.147168
$439$	5.16315e6	1.27865	0.639327	−	0.768935i	$-0.279213\pi$
$439$	5.16315e6	1.27865	0.639327	+	0.768935i	$0.279213\pi$
$440$	0	0
$441$	−546579.	−0.133831
$442$	−140620.	−0.0342367
$443$	−7.64624e6	−1.85114	−0.925569	−	0.378579i	$-0.876413\pi$
$443$	−7.64624e6	−1.85114	−0.925569	+	0.378579i	$0.876413\pi$
$444$	−51774.4	−0.0124640
$445$	0	0
$446$	2.16216e6	0.514695
$447$	1.78831e6	0.423325
$448$	−200704.	−0.0472456
$449$	4.83346e6	1.13147	0.565734	−	0.824588i	$-0.308593\pi$
$449$	4.83346e6	1.13147	0.565734	+	0.824588i	$0.308593\pi$
$450$	0	0
$451$	959024.	0.222018
$452$	−1.67204e6	−0.384948
$453$	1.24642e6	0.285378
$454$	−540936.	−0.123170
$455$	0	0
$456$	300287.	0.0676277
$457$	−3.51870e6	−0.788119	−0.394060	−	0.919085i	$-0.628930\pi$
$457$	−3.51870e6	−0.788119	−0.394060	+	0.919085i	$0.628930\pi$
$458$	475511.	0.105925
$459$	2.66493e6	0.590411
$460$	0	0
$461$	3.55938e6	0.780049	0.390024	−	0.920805i	$-0.372467\pi$
$461$	3.55938e6	0.780049	0.390024	+	0.920805i	$0.372467\pi$
$462$	259015.	0.0564572
$463$	4.39317e6	0.952413	0.476207	−	0.879333i	$-0.342011\pi$
$463$	4.39317e6	0.952413	0.476207	+	0.879333i	$0.342011\pi$
$464$	302792.	0.0652904
$465$	0	0
$466$	2.63805e6	0.562754
$467$	−7.46723e6	−1.58441	−0.792205	−	0.610256i	$-0.791067\pi$
$467$	−7.46723e6	−1.58441	−0.792205	+	0.610256i	$0.791067\pi$
$468$	88609.4	0.0187010
$469$	521727.	0.109524
$470$	0	0
$471$	1.53960e6	0.319783
$472$	−1.20690e6	−0.249354
$473$	−3.24371e6	−0.666637
$474$	−977012.	−0.199735
$475$	0	0
$476$	1.13293e6	0.229185
$477$	7.72504e6	1.55455
$478$	−1.87178e6	−0.374702
$479$	−5.07249e6	−1.01014	−0.505070	−	0.863078i	$-0.668534\pi$
$479$	−5.07249e6	−1.01014	−0.505070	+	0.863078i	$0.668534\pi$
$480$	0	0
$481$	20090.5	0.00395938
$482$	860414.	0.168690
$483$	−431472.	−0.0841560
$484$	−756910.	−0.146869
$485$	0	0
$486$	−2.54629e6	−0.489009
$487$	−8.40987e6	−1.60682	−0.803409	−	0.595428i	$-0.796983\pi$
$487$	−8.40987e6	−1.60682	−0.803409	+	0.595428i	$0.796983\pi$
$488$	1.69530e6	0.322253
$489$	−220811.	−0.0417589
$490$	0	0
$491$	−1.07319e6	−0.200896	−0.100448	−	0.994942i	$-0.532028\pi$
$491$	−1.07319e6	−0.200896	−0.100448	+	0.994942i	$0.532028\pi$
$492$	178274.	0.0332029
$493$	−1.70920e6	−0.316720
$494$	−116523.	−0.0214830
$495$	0	0
$496$	2.67741e6	0.488665
$497$	−2.35149e6	−0.427023
$498$	−5795.96	−0.00104725
$499$	4.34278e6	0.780758	0.390379	−	0.920654i	$-0.372344\pi$
$499$	4.34278e6	0.780758	0.390379	+	0.920654i	$0.372344\pi$
$500$	0	0
$501$	1.22557e6	0.218144
$502$	3.49979e6	0.619845
$503$	1.03452e6	0.182313	0.0911564	−	0.995837i	$-0.470944\pi$
$503$	1.03452e6	0.182313	0.0911564	+	0.995837i	$0.470944\pi$
$504$	−713899.	−0.125187
$505$	0	0
$506$	−3.03164e6	−0.526382
$507$	−1.45254e6	−0.250962
$508$	3.92693e6	0.675141
$509$	−1.12485e6	−0.192442	−0.0962208	−	0.995360i	$-0.530675\pi$
$509$	−1.12485e6	−0.192442	−0.0962208	+	0.995360i	$0.530675\pi$
$510$	0	0
$511$	−1.84727e6	−0.312952
$512$	−262144.	−0.0441942
$513$	2.20827e6	0.370474
$514$	−911371.	−0.152155
$515$	0	0
$516$	−602978.	−0.0996959
$517$	3.51930e6	0.579068
$518$	−161863.	−0.0265047
$519$	1.20410e6	0.196220
$520$	0	0
$521$	−193877.	−0.0312919	−0.0156460	−	0.999878i	$-0.504980\pi$
$521$	−193877.	−0.0312919	−0.0156460	+	0.999878i	$0.504980\pi$
$522$	1.07702e6	0.173001
$523$	6.31332e6	1.00926	0.504630	−	0.863336i	$-0.331629\pi$
$523$	6.31332e6	1.00926	0.504630	+	0.863336i	$0.331629\pi$
$524$	−1.62725e6	−0.258896
$525$	0	0
$526$	6.24552e6	0.984247
$527$	−1.51134e7	−2.37048
$528$	338305.	0.0528109
$529$	−1.38617e6	−0.215367
$530$	0	0
$531$	−4.29292e6	−0.660719
$532$	938792.	0.143810
$533$	−69177.3	−0.0105474
$534$	−1.80866e6	−0.274476
$535$	0	0
$536$	681439.	0.102451
$537$	−3.15346e6	−0.471902
$538$	6.33999e6	0.944349
$539$	809761.	0.120056
$540$	0	0
$541$	−4.19872e6	−0.616770	−0.308385	−	0.951262i	$-0.599789\pi$
$541$	−4.19872e6	−0.616770	−0.308385	+	0.951262i	$0.599789\pi$
$542$	−1.90827e6	−0.279024
$543$	−2.03237e6	−0.295803
$544$	1.47975e6	0.214383
$545$	0	0
$546$	−18683.5	−0.00268211
$547$	−6.84530e6	−0.978192	−0.489096	−	0.872230i	$-0.662673\pi$
$547$	−6.84530e6	−0.978192	−0.489096	+	0.872230i	$0.662673\pi$
$548$	1.01848e6	0.144877
$549$	6.03015e6	0.853881
$550$	0	0
$551$	−1.41631e6	−0.198737
$552$	−563555.	−0.0787207
$553$	−3.05445e6	−0.424736
$554$	−555003.	−0.0768283
$555$	0	0
$556$	4.44667e6	0.610025
$557$	9.43023e6	1.28791	0.643953	−	0.765065i	$-0.277293\pi$
$557$	9.43023e6	1.28791	0.643953	+	0.765065i	$0.277293\pi$
$558$	9.52350e6	1.29482
$559$	233979.	0.0316699
$560$	0	0
$561$	−1.90966e6	−0.256182
$562$	1.00446e7	1.34150
$563$	−1.27111e7	−1.69010	−0.845052	−	0.534685i	$-0.820430\pi$
$563$	−1.27111e7	−1.69010	−0.845052	+	0.534685i	$0.820430\pi$
$564$	654207.	0.0865998
$565$	0	0
$566$	−9.16315e6	−1.20227
$567$	−2.35651e6	−0.307830
$568$	−3.07133e6	−0.399444
$569$	7.92638e6	1.02635	0.513173	−	0.858285i	$-0.328470\pi$
$569$	7.92638e6	1.02635	0.513173	+	0.858285i	$0.328470\pi$
$570$	0	0
$571$	1.10620e6	0.141985	0.0709927	−	0.997477i	$-0.477383\pi$
$571$	1.10620e6	0.141985	0.0709927	+	0.997477i	$0.477383\pi$
$572$	−131275.	−0.0167762
$573$	−799591.	−0.101738
$574$	557341.	0.0706059
$575$	0	0
$576$	−932440.	−0.117102
$577$	4.53166e6	0.566654	0.283327	−	0.959023i	$-0.408562\pi$
$577$	4.53166e6	0.566654	0.283327	+	0.959023i	$0.408562\pi$
$578$	−2.67345e6	−0.332853
$579$	3.42265e6	0.424293
$580$	0	0
$581$	−18120.0	−0.00222699
$582$	2.13955e6	0.261827
$583$	−1.14447e7	−1.39455
$584$	−2.41276e6	−0.292740
$585$	0	0
$586$	−1.78560e6	−0.214803
$587$	201145.	0.0240943	0.0120472	−	0.999927i	$-0.496165\pi$
$587$	201145.	0.0240943	0.0120472	+	0.999927i	$0.496165\pi$
$588$	150527.	0.0179545
$589$	−1.25236e7	−1.48744
$590$	0	0
$591$	−447994.	−0.0527598
$592$	−211413.	−0.0247929
$593$	1.56210e7	1.82420	0.912098	−	0.409973i	$-0.134462\pi$
$593$	1.56210e7	1.82420	0.912098	+	0.409973i	$0.134462\pi$
$594$	2.48784e6	0.289306
$595$	0	0
$596$	7.30228e6	0.842060
$597$	−1.02719e6	−0.117955
$598$	218681.	0.0250069
$599$	−2.80468e6	−0.319386	−0.159693	−	0.987167i	$-0.551050\pi$
$599$	−2.80468e6	−0.319386	−0.159693	+	0.987167i	$0.551050\pi$
$600$	0	0
$601$	7.57493e6	0.855446	0.427723	−	0.903910i	$-0.359316\pi$
$601$	7.57493e6	0.855446	0.427723	+	0.903910i	$0.359316\pi$
$602$	−1.88510e6	−0.212003
$603$	2.42386e6	0.271466
$604$	5.08958e6	0.567662
$605$	0	0
$606$	−877336.	−0.0970475
$607$	1.71170e7	1.88562	0.942812	−	0.333324i	$-0.108170\pi$
$607$	1.71170e7	1.88562	0.942812	+	0.333324i	$0.108170\pi$
$608$	1.22618e6	0.134522
$609$	−227093.	−0.0248120
$610$	0	0
$611$	−253858.	−0.0275098
$612$	5.26343e6	0.568055
$613$	−1.57603e7	−1.69400	−0.847002	−	0.531590i	$-0.821595\pi$
$613$	−1.57603e7	−1.69400	−0.847002	+	0.531590i	$0.821595\pi$
$614$	−531714.	−0.0569190
$615$	0	0
$616$	1.05765e6	0.112302
$617$	1.16536e7	1.23239	0.616194	−	0.787594i	$-0.288674\pi$
$617$	1.16536e7	1.23239	0.616194	+	0.787594i	$0.288674\pi$
$618$	−2.58888e6	−0.272672
$619$	−9.20111e6	−0.965192	−0.482596	−	0.875843i	$-0.660306\pi$
$619$	−9.20111e6	−0.965192	−0.482596	+	0.875843i	$0.660306\pi$
$620$	0	0
$621$	−4.14430e6	−0.431243
$622$	4.09386e6	0.424285
$623$	−5.65444e6	−0.583673
$624$	−24403.0	−0.00250889
$625$	0	0
$626$	−8.13108e6	−0.829301
$627$	−1.58242e6	−0.160751
$628$	6.28672e6	0.636099
$629$	1.19338e6	0.120269
$630$	0	0
$631$	1.50421e7	1.50395	0.751976	−	0.659191i	$-0.229101\pi$
$631$	1.50421e7	1.50395	0.751976	+	0.659191i	$0.229101\pi$
$632$	−3.98948e6	−0.397305
$633$	3.44331e6	0.341560
$634$	−9.70742e6	−0.959137
$635$	0	0
$636$	−2.12747e6	−0.208555
$637$	−58410.5	−0.00570351
$638$	−1.59562e6	−0.155195
$639$	−1.09247e7	−1.05841
$640$	0	0
$641$	−1.15445e7	−1.10976	−0.554879	−	0.831931i	$-0.687235\pi$
$641$	−1.15445e7	−1.10976	−0.554879	+	0.831931i	$0.687235\pi$
$642$	−1.56422e6	−0.149782
$643$	4.82038e6	0.459784	0.229892	−	0.973216i	$-0.426163\pi$
$643$	4.82038e6	0.459784	0.229892	+	0.973216i	$0.426163\pi$
$644$	−1.76185e6	−0.167400
$645$	0	0
$646$	−6.92152e6	−0.652559
$647$	7.53202e6	0.707377	0.353688	−	0.935363i	$-0.384927\pi$
$647$	7.53202e6	0.707377	0.353688	+	0.935363i	$0.384927\pi$
$648$	−3.07789e6	−0.287949
$649$	6.35999e6	0.592713
$650$	0	0
$651$	−2.00805e6	−0.185705
$652$	−901650.	−0.0830652
$653$	4.07749e6	0.374205	0.187103	−	0.982340i	$-0.440090\pi$
$653$	4.07749e6	0.374205	0.187103	+	0.982340i	$0.440090\pi$
$654$	−3.53257e6	−0.322959
$655$	0	0
$656$	727956.	0.0660458
$657$	−8.58213e6	−0.775678
$658$	2.04525e6	0.184155
$659$	−1.52918e7	−1.37165	−0.685826	−	0.727765i	$-0.740559\pi$
$659$	−1.52918e7	−1.37165	−0.685826	+	0.727765i	$0.740559\pi$
$660$	0	0
$661$	2.03724e7	1.81359	0.906796	−	0.421570i	$-0.138521\pi$
$661$	2.03724e7	1.81359	0.906796	+	0.421570i	$0.138521\pi$
$662$	−2.21193e6	−0.196167
$663$	137750.	0.0121705
$664$	−23666.9	−0.00208315
$665$	0	0
$666$	−751990.	−0.0656941
$667$	2.65802e6	0.231336
$668$	5.00442e6	0.433922
$669$	−2.11802e6	−0.182964
$670$	0	0
$671$	−8.93371e6	−0.765994
$672$	196607.	0.0167949
$673$	4.09627e6	0.348619	0.174309	−	0.984691i	$-0.444231\pi$
$673$	4.09627e6	0.348619	0.174309	+	0.984691i	$0.444231\pi$
$674$	808397.	0.0685449
$675$	0	0
$676$	−5.93122e6	−0.499203
$677$	7.73865e6	0.648924	0.324462	−	0.945899i	$-0.394817\pi$
$677$	7.73865e6	0.648924	0.324462	+	0.945899i	$0.394817\pi$
$678$	1.63792e6	0.136841
$679$	6.68889e6	0.556775
$680$	0	0
$681$	529895.	0.0437847
$682$	−1.41091e7	−1.16155
$683$	−2.24027e7	−1.83759	−0.918795	−	0.394734i	$-0.870837\pi$
$683$	−2.24027e7	−1.83759	−0.918795	+	0.394734i	$0.870837\pi$
$684$	4.36148e6	0.356446
$685$	0	0
$686$	470596.	0.0381802
$687$	−465805.	−0.0376541
$688$	−2.46217e6	−0.198311
$689$	825541.	0.0662507
$690$	0	0
$691$	−2.60639e6	−0.207656	−0.103828	−	0.994595i	$-0.533109\pi$
$691$	−2.60639e6	−0.207656	−0.103828	+	0.994595i	$0.533109\pi$
$692$	4.91675e6	0.390313
$693$	3.76202e6	0.297570
$694$	4.18466e6	0.329808
$695$	0	0
$696$	−296612.	−0.0232095
$697$	−4.10916e6	−0.320384
$698$	1.08944e7	0.846377
$699$	−2.58420e6	−0.200048
$700$	0	0
$701$	−5.34202e6	−0.410592	−0.205296	−	0.978700i	$-0.565816\pi$
$701$	−5.34202e6	−0.410592	−0.205296	+	0.978700i	$0.565816\pi$
$702$	−179456.	−0.0137440
$703$	988882.	0.0754668
$704$	1.38142e6	0.105049
$705$	0	0
$706$	6.94846e6	0.524658
$707$	−2.74283e6	−0.206372
$708$	1.18227e6	0.0886406
$709$	−1.78376e7	−1.33266	−0.666332	−	0.745656i	$-0.732136\pi$
$709$	−1.78376e7	−1.33266	−0.666332	+	0.745656i	$0.732136\pi$
$710$	0	0
$711$	−1.41905e7	−1.05275
$712$	−7.38539e6	−0.545976
$713$	2.35033e7	1.73143
$714$	−1.10981e6	−0.0814708
$715$	0	0
$716$	−1.28767e7	−0.938688
$717$	1.83358e6	0.133199
$718$	−6.74795e6	−0.488496
$719$	−1.93072e7	−1.39283	−0.696414	−	0.717640i	$-0.745222\pi$
$719$	−1.93072e7	−1.39283	−0.696414	+	0.717640i	$0.745222\pi$
$720$	0	0
$721$	−8.09364e6	−0.579837
$722$	4.16896e6	0.297636
$723$	−842852.	−0.0599660
$724$	−8.29885e6	−0.588399
$725$	0	0
$726$	741460.	0.0522091
$727$	−2.11119e7	−1.48147	−0.740733	−	0.671800i	$-0.765522\pi$
$727$	−2.11119e7	−1.48147	−0.740733	+	0.671800i	$0.765522\pi$
$728$	−76291.3	−0.00533515
$729$	−9.19205e6	−0.640610
$730$	0	0
$731$	1.38984e7	0.961994
$732$	−1.66070e6	−0.114555
$733$	2.35579e7	1.61948	0.809742	−	0.586786i	$-0.199607\pi$
$733$	2.35579e7	1.61948	0.809742	+	0.586786i	$0.199607\pi$
$734$	1.30297e7	0.892678
$735$	0	0
$736$	−2.30119e6	−0.156588
$737$	−3.59097e6	−0.243525
$738$	2.58932e6	0.175003
$739$	−8.49978e6	−0.572528	−0.286264	−	0.958151i	$-0.592413\pi$
$739$	−8.49978e6	−0.572528	−0.286264	+	0.958151i	$0.592413\pi$
$740$	0	0
$741$	114145.	0.00763678
$742$	−6.65114e6	−0.443492
$743$	5.28179e6	0.351001	0.175501	−	0.984479i	$-0.443846\pi$
$743$	5.28179e6	0.351001	0.175501	+	0.984479i	$0.443846\pi$
$744$	−2.62276e6	−0.173711
$745$	0	0
$746$	1.34649e7	0.885841
$747$	−84182.7	−0.00551977
$748$	−7.79781e6	−0.509587
$749$	−4.89025e6	−0.318512
$750$	0	0
$751$	1.72132e7	1.11368	0.556842	−	0.830619i	$-0.312013\pi$
$751$	1.72132e7	1.11368	0.556842	+	0.830619i	$0.312013\pi$
$752$	2.67135e6	0.172261
$753$	−3.42836e6	−0.220343
$754$	115097.	0.00737284
$755$	0	0
$756$	1.44582e6	0.0920047
$757$	1.95610e7	1.24066	0.620328	−	0.784343i	$-0.287000\pi$
$757$	1.95610e7	1.24066	0.620328	+	0.784343i	$0.287000\pi$
$758$	−1.99337e7	−1.26013
$759$	2.96976e6	0.187118
$760$	0	0
$761$	−2.65917e7	−1.66450	−0.832250	−	0.554401i	$-0.812948\pi$
$761$	−2.65917e7	−1.66450	−0.832250	+	0.554401i	$0.812948\pi$
$762$	−3.84678e6	−0.239999
$763$	−1.10439e7	−0.686772
$764$	−3.26501e6	−0.202372
$765$	0	0
$766$	−4.56306e6	−0.280986
$767$	−458766.	−0.0281581
$768$	256793.	0.0157102
$769$	3.32124e6	0.202528	0.101264	−	0.994860i	$-0.467711\pi$
$769$	3.32124e6	0.202528	0.101264	+	0.994860i	$0.467711\pi$
$770$	0	0
$771$	892769.	0.0540883
$772$	1.39759e7	0.843986
$773$	−3.03574e7	−1.82732	−0.913662	−	0.406476i	$-0.866758\pi$
$773$	−3.03574e7	−1.82732	−0.913662	+	0.406476i	$0.866758\pi$
$774$	−8.75787e6	−0.525468
$775$	0	0
$776$	8.73651e6	0.520815
$777$	158559.	0.00942190
$778$	1.83388e7	1.08623
$779$	−3.40501e6	−0.201036
$780$	0	0
$781$	1.61849e7	0.949475
$782$	1.29898e7	0.759599
$783$	−2.18124e6	−0.127145
$784$	614656.	0.0357143
$785$	0	0
$786$	1.59403e6	0.0920325
$787$	−1.37541e7	−0.791583	−0.395792	−	0.918340i	$-0.629530\pi$
$787$	−1.37541e7	−0.791583	−0.395792	+	0.918340i	$0.629530\pi$
$788$	−1.82932e6	−0.104948
$789$	−6.11804e6	−0.349880
$790$	0	0
$791$	5.12064e6	0.290993
$792$	4.91366e6	0.278351
$793$	644416.	0.0363901
$794$	−1.65084e7	−0.929294
$795$	0	0
$796$	−4.19437e6	−0.234630
$797$	2.68171e6	0.149543	0.0747715	−	0.997201i	$-0.476177\pi$
$797$	2.68171e6	0.149543	0.0747715	+	0.997201i	$0.476177\pi$
$798$	−919630.	−0.0511217
$799$	−1.50792e7	−0.835627
$800$	0	0
$801$	−2.62697e7	−1.44668
$802$	−9.35726e6	−0.513704
$803$	1.27145e7	0.695840
$804$	−667530.	−0.0364192
$805$	0	0
$806$	1.01773e6	0.0551819
$807$	−6.21058e6	−0.335698
$808$	−3.58247e6	−0.193043
$809$	3.35759e7	1.80367	0.901834	−	0.432083i	$-0.142221\pi$
$809$	3.35759e7	1.80367	0.901834	+	0.432083i	$0.142221\pi$
$810$	0	0
$811$	5.24303e6	0.279917	0.139959	−	0.990157i	$-0.455303\pi$
$811$	5.24303e6	0.279917	0.139959	+	0.990157i	$0.455303\pi$
$812$	−927300.	−0.0493549
$813$	1.86932e6	0.0991875
$814$	1.11408e6	0.0589325
$815$	0	0
$816$	−1.44954e6	−0.0762090
$817$	1.15168e7	0.603637
$818$	−2.01653e7	−1.05371
$819$	−271366.	−0.0141366
$820$	0	0
$821$	−1.95688e7	−1.01323	−0.506614	−	0.862173i	$-0.669103\pi$
$821$	−1.95688e7	−1.01323	−0.506614	+	0.862173i	$0.669103\pi$
$822$	−997687.	−0.0515009
$823$	6.20206e6	0.319181	0.159590	−	0.987183i	$-0.448983\pi$
$823$	6.20206e6	0.319181	0.159590	+	0.987183i	$0.448983\pi$
$824$	−1.05713e7	−0.542388
$825$	0	0
$826$	3.69614e6	0.188494
$827$	−2.30838e7	−1.17366	−0.586831	−	0.809710i	$-0.699625\pi$
$827$	−2.30838e7	−1.17366	−0.586831	+	0.809710i	$0.699625\pi$
$828$	−8.18529e6	−0.414914
$829$	2.73598e7	1.38270	0.691348	−	0.722522i	$-0.257017\pi$
$829$	2.73598e7	1.38270	0.691348	+	0.722522i	$0.257017\pi$
$830$	0	0
$831$	543675.	0.0273109
$832$	−99645.8	−0.00499058
$833$	−3.46961e6	−0.173248
$834$	−4.35590e6	−0.216852
$835$	0	0
$836$	−6.46156e6	−0.319758
$837$	−1.92874e7	−0.951613
$838$	−267766.	−0.0131718
$839$	1.15303e7	0.565504	0.282752	−	0.959193i	$-0.408753\pi$
$839$	1.15303e7	0.565504	0.282752	+	0.959193i	$0.408753\pi$
$840$	0	0
$841$	−1.91122e7	−0.931795
$842$	−2.68118e6	−0.130331
$843$	−9.83956e6	−0.476878
$844$	1.40602e7	0.679417
$845$	0	0
$846$	9.50194e6	0.456443
$847$	2.31804e6	0.111023
$848$	−8.68720e6	−0.414849
$849$	8.97611e6	0.427385
$850$	0	0
$851$	−1.85586e6	−0.0878456
$852$	3.00864e6	0.141994
$853$	1.31074e7	0.616800	0.308400	−	0.951257i	$-0.400206\pi$
$853$	1.31074e7	0.616800	0.308400	+	0.951257i	$0.400206\pi$
$854$	−5.19186e6	−0.243601
$855$	0	0
$856$	−6.38726e6	−0.297941
$857$	3.48607e7	1.62138	0.810689	−	0.585477i	$-0.199093\pi$
$857$	3.48607e7	1.62138	0.810689	+	0.585477i	$0.199093\pi$
$858$	128596.	0.00596361
$859$	2.14838e7	0.993410	0.496705	−	0.867919i	$-0.334543\pi$
$859$	2.14838e7	0.993410	0.496705	+	0.867919i	$0.334543\pi$
$860$	0	0
$861$	−545965.	−0.0250990
$862$	1.46135e6	0.0669863
$863$	3.37660e6	0.154331	0.0771654	−	0.997018i	$-0.475413\pi$
$863$	3.37660e6	0.154331	0.0771654	+	0.997018i	$0.475413\pi$
$864$	1.88842e6	0.0860625
$865$	0	0
$866$	5.21175e6	0.236151
$867$	2.61888e6	0.118323
$868$	−8.19958e6	−0.369396
$869$	2.10233e7	0.944390
$870$	0	0
$871$	259028.	0.0115691
$872$	−1.44247e7	−0.642416
$873$	3.10756e7	1.38001
$874$	1.07638e7	0.476637
$875$	0	0
$876$	2.36351e6	0.104063
$877$	−1.58283e7	−0.694921	−0.347461	−	0.937695i	$-0.612956\pi$
$877$	−1.58283e7	−0.694921	−0.347461	+	0.937695i	$0.612956\pi$
$878$	−2.06526e7	−0.904145
$879$	1.74915e6	0.0763581
$880$	0	0
$881$	−1.50941e7	−0.655191	−0.327596	−	0.944818i	$-0.606238\pi$
$881$	−1.50941e7	−0.655191	−0.327596	+	0.944818i	$0.606238\pi$
$882$	2.18632e6	0.0946328
$883$	4.20608e7	1.81542	0.907708	−	0.419603i	$-0.137831\pi$
$883$	4.20608e7	1.81542	0.907708	+	0.419603i	$0.137831\pi$
$884$	562480.	0.0242090
$885$	0	0
$886$	3.05850e7	1.30895
$887$	2.83691e7	1.21070	0.605351	−	0.795959i	$-0.293033\pi$
$887$	2.83691e7	1.21070	0.605351	+	0.795959i	$0.293033\pi$
$888$	207097.	0.00881338
$889$	−1.20262e7	−0.510358
$890$	0	0
$891$	1.62195e7	0.684453
$892$	−8.64862e6	−0.363944
$893$	−1.24952e7	−0.524343
$894$	−7.15323e6	−0.299336
$895$	0	0
$896$	802816.	0.0334077
$897$	−214218.	−0.00888944
$898$	−1.93338e7	−0.800068
$899$	1.23703e7	0.510482
$900$	0	0
$901$	4.90374e7	2.01241
$902$	−3.83610e6	−0.156990
$903$	1.84662e6	0.0753630
$904$	6.68818e6	0.272199
$905$	0	0
$906$	−4.98569e6	−0.201793
$907$	−1.00151e7	−0.404236	−0.202118	−	0.979361i	$-0.564783\pi$
$907$	−1.00151e7	−0.404236	−0.202118	+	0.979361i	$0.564783\pi$
$908$	2.16375e6	0.0870947
$909$	−1.27428e7	−0.511509
$910$	0	0
$911$	−2.06987e7	−0.826317	−0.413158	−	0.910659i	$-0.635574\pi$
$911$	−2.06987e7	−0.826317	−0.413158	+	0.910659i	$0.635574\pi$
$912$	−1.20115e6	−0.0478200
$913$	124717.	0.00495164
$914$	1.40748e7	0.557284
$915$	0	0
$916$	−1.90204e6	−0.0749000
$917$	4.98345e6	0.195707
$918$	−1.06597e7	−0.417484
$919$	6.79248e6	0.265301	0.132651	−	0.991163i	$-0.457651\pi$
$919$	6.79248e6	0.265301	0.132651	+	0.991163i	$0.457651\pi$
$920$	0	0
$921$	520861.	0.0202336
$922$	−1.42375e7	−0.551578
$923$	−1.16747e6	−0.0451067
$924$	−1.03606e6	−0.0399213
$925$	0	0
$926$	−1.75727e7	−0.673458
$927$	−3.76018e7	−1.43717
$928$	−1.21117e6	−0.0461673
$929$	−2.18354e7	−0.830085	−0.415042	−	0.909802i	$-0.636233\pi$
$929$	−2.18354e7	−0.830085	−0.415042	+	0.909802i	$0.636233\pi$
$930$	0	0
$931$	−2.87505e6	−0.108710
$932$	−1.05522e7	−0.397927
$933$	−4.01030e6	−0.150825
$934$	2.98689e7	1.12035
$935$	0	0
$936$	−354438.	−0.0132236
$937$	3.76114e7	1.39949	0.699747	−	0.714391i	$-0.253296\pi$
$937$	3.76114e7	1.39949	0.699747	+	0.714391i	$0.253296\pi$
$938$	−2.08691e6	−0.0774455
$939$	7.96511e6	0.294800
$940$	0	0
$941$	3.13453e7	1.15398	0.576989	−	0.816752i	$-0.304227\pi$
$941$	3.13453e7	1.15398	0.576989	+	0.816752i	$0.304227\pi$
$942$	−6.15840e6	−0.226121
$943$	6.39025e6	0.234012
$944$	4.82761e6	0.176320
$945$	0	0
$946$	1.29748e7	0.471383
$947$	−2.16274e6	−0.0783664	−0.0391832	−	0.999232i	$-0.512476\pi$
$947$	−2.16274e6	−0.0783664	−0.0391832	+	0.999232i	$0.512476\pi$
$948$	3.90805e6	0.141234
$949$	−917134.	−0.0330573
$950$	0	0
$951$	9.50927e6	0.340954
$952$	−4.53173e6	−0.162058
$953$	2.60248e7	0.928230	0.464115	−	0.885775i	$-0.346372\pi$
$953$	2.60248e7	0.928230	0.464115	+	0.885775i	$0.346372\pi$
$954$	−3.09002e7	−1.09923
$955$	0	0
$956$	7.48713e6	0.264954
$957$	1.56305e6	0.0551687
$958$	2.02899e7	0.714278
$959$	−3.11908e6	−0.109517
$960$	0	0
$961$	8.07542e7	2.82070
$962$	−80361.9	−0.00279971
$963$	−2.27193e7	−0.789460
$964$	−3.44166e6	−0.119282
$965$	0	0
$966$	1.72589e6	0.0595073
$967$	−3.85997e7	−1.32745	−0.663725	−	0.747977i	$-0.731025\pi$
$967$	−3.85997e7	−1.32745	−0.663725	+	0.747977i	$0.731025\pi$
$968$	3.02764e6	0.103852
$969$	6.78024e6	0.231972
$970$	0	0
$971$	1.98202e7	0.674621	0.337310	−	0.941393i	$-0.390483\pi$
$971$	1.98202e7	0.674621	0.337310	+	0.941393i	$0.390483\pi$
$972$	1.01852e7	0.345782
$973$	−1.36179e7	−0.461136
$974$	3.36395e7	1.13619
$975$	0	0
$976$	−6.78121e6	−0.227868
$977$	6.29461e6	0.210976	0.105488	−	0.994421i	$-0.466360\pi$
$977$	6.29461e6	0.210976	0.105488	+	0.994421i	$0.466360\pi$
$978$	883246.	0.0295280
$979$	3.89187e7	1.29778
$980$	0	0
$981$	−5.13084e7	−1.70222
$982$	4.29275e6	0.142055
$983$	3.54716e7	1.17084	0.585419	−	0.810731i	$-0.300930\pi$
$983$	3.54716e7	1.17084	0.585419	+	0.810731i	$0.300930\pi$
$984$	−713097.	−0.0234780
$985$	0	0
$986$	6.83679e6	0.223955
$987$	−2.00351e6	−0.0654633
$988$	466093.	0.0151908
$989$	−2.16138e7	−0.702652
$990$	0	0
$991$	−1.49772e7	−0.484446	−0.242223	−	0.970221i	$-0.577877\pi$
$991$	−1.49772e7	−0.484446	−0.242223	+	0.970221i	$0.577877\pi$
$992$	−1.07097e7	−0.345538
$993$	2.16678e6	0.0697334
$994$	9.40595e6	0.301951
$995$	0	0
$996$	23183.8	0.000740520 0
$997$	3.79210e7	1.20821	0.604104	−	0.796905i	$-0.293531\pi$
$997$	3.79210e7	1.20821	0.604104	+	0.796905i	$0.293531\pi$
$998$	−1.73711e7	−0.552079
$999$	1.52296e6	0.0482810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.y.1.4		5
5.2	odd	4		70.6.c.d.29.2	✓	10
5.3	odd	4		70.6.c.d.29.9	yes	10
5.4	even	2		350.6.a.z.1.2		5
15.2	even	4		630.6.g.h.379.7		10
15.8	even	4		630.6.g.h.379.2		10
20.3	even	4		560.6.g.d.449.4		10
20.7	even	4		560.6.g.d.449.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.c.d.29.2	✓	10	5.2	odd	4
70.6.c.d.29.9	yes	10	5.3	odd	4
350.6.a.y.1.4		5	1.1	even	1	trivial
350.6.a.z.1.2		5	5.4	even	2
560.6.g.d.449.4		10	20.3	even	4
560.6.g.d.449.7		10	20.7	even	4
630.6.g.h.379.2		10	15.8	even	4
630.6.g.h.379.7		10	15.2	even	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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +3.91835 q^{3} +16.0000 q^{4} -15.6734 q^{6} -49.0000 q^{7} -64.0000 q^{8} -227.647 q^{9} +337.260 q^{11} +62.6937 q^{12} -24.3276 q^{13} +196.000 q^{14} +256.000 q^{16} -1445.07 q^{17} +910.586 q^{18} -1197.44 q^{19} -191.999 q^{21} -1349.04 q^{22} +2247.26 q^{23} -250.775 q^{24} +97.3103 q^{26} -1844.16 q^{27} -784.000 q^{28} +1182.78 q^{29} +10458.6 q^{31} -1024.00 q^{32} +1321.50 q^{33} +5780.27 q^{34} -3642.34 q^{36} -825.831 q^{37} +4789.75 q^{38} -95.3241 q^{39} +2843.58 q^{41} +767.997 q^{42} -9617.84 q^{43} +5396.16 q^{44} -8989.03 q^{46} +10435.0 q^{47} +1003.10 q^{48} +2401.00 q^{49} -5662.28 q^{51} -389.241 q^{52} -33934.4 q^{53} +7376.64 q^{54} +3136.00 q^{56} -4691.99 q^{57} -4731.12 q^{58} +18857.8 q^{59} -26489.1 q^{61} -41834.6 q^{62} +11154.7 q^{63} +4096.00 q^{64} -5286.01 q^{66} -10647.5 q^{67} -23121.1 q^{68} +8805.55 q^{69} +47989.5 q^{71} +14569.4 q^{72} +37699.4 q^{73} +3303.32 q^{74} -19159.0 q^{76} -16525.7 q^{77} +381.296 q^{78} +62335.6 q^{79} +48092.0 q^{81} -11374.3 q^{82} +369.796 q^{83} -3071.99 q^{84} +38471.4 q^{86} +4634.56 q^{87} -21584.6 q^{88} +115397. q^{89} +1192.05 q^{91} +35956.1 q^{92} +40980.7 q^{93} -41739.9 q^{94} -4012.39 q^{96} -136508. q^{97} -9604.00 q^{98} -76776.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 20 q^{2} - 14 q^{3} + 80 q^{4} + 56 q^{6} - 245 q^{7} - 320 q^{8} + 655 q^{9} + 748 q^{11} - 224 q^{12} - 456 q^{13} + 980 q^{14} + 1280 q^{16} - 780 q^{17} - 2620 q^{18} - 256 q^{19} + 686 q^{21}+ \cdots + 575944 q^{99}+O(q^{100}) \) 5 * q - 20 * q^2 - 14 * q^3 + 80 * q^4 + 56 * q^6 - 245 * q^7 - 320 * q^8 + 655 * q^9 + 748 * q^11 - 224 * q^12 - 456 * q^13 + 980 * q^14 + 1280 * q^16 - 780 * q^17 - 2620 * q^18 - 256 * q^19 + 686 * q^21 - 2992 * q^22 - 3452 * q^23 + 896 * q^24 + 1824 * q^26 + 910 * q^27 - 3920 * q^28 + 7712 * q^29 + 412 * q^31 - 5120 * q^32 + 2980 * q^33 + 3120 * q^34 + 10480 * q^36 - 14400 * q^37 + 1024 * q^38 + 17668 * q^39 + 7482 * q^41 - 2744 * q^42 - 20668 * q^43 + 11968 * q^44 + 13808 * q^46 - 59202 * q^47 - 3584 * q^48 + 12005 * q^49 + 6628 * q^51 - 7296 * q^52 - 6388 * q^53 - 3640 * q^54 + 15680 * q^56 - 112616 * q^57 - 30848 * q^58 + 79596 * q^59 + 9754 * q^61 - 1648 * q^62 - 32095 * q^63 + 20480 * q^64 - 11920 * q^66 - 49820 * q^67 - 12480 * q^68 + 106152 * q^69 - 61928 * q^71 - 41920 * q^72 + 57248 * q^73 + 57600 * q^74 - 4096 * q^76 - 36652 * q^77 - 70672 * q^78 + 150912 * q^79 + 100825 * q^81 - 29928 * q^82 + 54776 * q^83 + 10976 * q^84 + 82672 * q^86 + 256362 * q^87 - 47872 * q^88 + 268950 * q^89 + 22344 * q^91 - 55232 * q^92 + 187044 * q^93 + 236808 * q^94 + 14336 * q^96 + 296156 * q^97 - 48020 * q^98 + 575944 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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