LMFDB - Embedded newform 315.10.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,10,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$162.236288392$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 307x^{2} - 270x + 8836 $ x^4 - 307x^2 - 270x + 8836
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$17.1051$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+32.2102 q^{2} +525.499 q^{4} +625.000 q^{5} -2401.00 q^{7} +434.806 q^{8} +20131.4 q^{10} +57240.1 q^{11} -72300.8 q^{13} -77336.8 q^{14} -255050. q^{16} -314910. q^{17} +357041. q^{19} +328437. q^{20} +1.84372e6 q^{22} +584389. q^{23} +390625. q^{25} -2.32882e6 q^{26} -1.26172e6 q^{28} -1.31538e6 q^{29} -5.50612e6 q^{31} -8.43785e6 q^{32} -1.01433e7 q^{34} -1.50062e6 q^{35} -1.18399e7 q^{37} +1.15004e7 q^{38} +271753. q^{40} +2.17065e7 q^{41} -1.66343e7 q^{43} +3.00796e7 q^{44} +1.88233e7 q^{46} +1.31502e7 q^{47} +5.76480e6 q^{49} +1.25821e7 q^{50} -3.79940e7 q^{52} -5.91955e7 q^{53} +3.57750e7 q^{55} -1.04397e6 q^{56} -4.23687e7 q^{58} -9.80152e7 q^{59} +7.52813e6 q^{61} -1.77353e8 q^{62} -1.41199e8 q^{64} -4.51880e7 q^{65} +3.44489e7 q^{67} -1.65485e8 q^{68} -4.83355e7 q^{70} -3.30396e8 q^{71} -6.24882e7 q^{73} -3.81365e8 q^{74} +1.87625e8 q^{76} -1.37433e8 q^{77} +1.05279e7 q^{79} -1.59406e8 q^{80} +6.99170e8 q^{82} -2.19370e7 q^{83} -1.96819e8 q^{85} -5.35793e8 q^{86} +2.48883e7 q^{88} -1.91010e8 q^{89} +1.73594e8 q^{91} +3.07096e8 q^{92} +4.23572e8 q^{94} +2.23151e8 q^{95} -6.28904e7 q^{97} +1.85686e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 424 q^{4} + 2500 q^{5} - 9604 q^{7} - 96 q^{8} - 5000 q^{10} - 9832 q^{11} - 68264 q^{13} + 19208 q^{14} - 290784 q^{16} - 86272 q^{17} + 807672 q^{19} + 265000 q^{20} + 4293352 q^{22} - 683032 q^{23}+ \cdots - 46118408 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 424 * q^4 + 2500 * q^5 - 9604 * q^7 - 96 * q^8 - 5000 * q^10 - 9832 * q^11 - 68264 * q^13 + 19208 * q^14 - 290784 * q^16 - 86272 * q^17 + 807672 * q^19 + 265000 * q^20 + 4293352 * q^22 - 683032 * q^23 + 1562500 * q^25 - 4583544 * q^26 - 1018024 * q^28 - 5029312 * q^29 + 9802992 * q^31 - 497280 * q^32 + 429792 * q^34 - 6002500 * q^35 - 4536432 * q^37 - 21534872 * q^38 - 60000 * q^40 - 11081000 * q^41 - 34442248 * q^43 - 35679024 * q^44 + 67652080 * q^46 + 39230168 * q^47 + 23059204 * q^49 - 3125000 * q^50 - 14778976 * q^52 - 76012528 * q^53 - 6145000 * q^55 + 230496 * q^56 + 57813344 * q^58 - 74595424 * q^59 + 101246592 * q^61 - 392291880 * q^62 - 276909952 * q^64 - 42665000 * q^65 - 45694264 * q^67 - 345896144 * q^68 + 12005000 * q^70 - 447187600 * q^71 + 141527624 * q^73 - 480388368 * q^74 + 1081835504 * q^76 + 23606632 * q^77 + 789659136 * q^79 - 181740000 * q^80 + 937199856 * q^82 - 586282352 * q^83 - 53920000 * q^85 - 132711648 * q^86 + 1042925024 * q^88 - 1050230088 * q^89 + 163901864 * q^91 - 44316096 * q^92 - 1211140960 * q^94 + 504795000 * q^95 + 788061288 * q^97 - 46118408 * q^98

Coefficient data

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

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

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

$2$	32.2102	1.42350	0.711752	−	0.702431i	$-0.247902\pi$
$2$	32.2102	1.42350	0.711752	+	0.702431i	$0.247902\pi$
$3$	0	0
$3$	0	0
$4$	525.499	1.02637
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	434.806	0.0375310
$9$	0	0
$10$	20131.4	0.636611
$11$	57240.1	1.17878	0.589390	−	0.807848i	$-0.299368\pi$
$11$	57240.1	1.17878	0.589390	+	0.807848i	$0.299368\pi$
$12$	0	0
$13$	−72300.8	−0.702098	−0.351049	−	0.936357i	$-0.614175\pi$
$13$	−72300.8	−0.702098	−0.351049	+	0.936357i	$0.614175\pi$
$14$	−77336.8	−0.538034
$15$	0	0
$16$	−255050.	−0.972940
$17$	−314910.	−0.914463	−0.457232	−	0.889348i	$-0.651159\pi$
$17$	−314910.	−0.914463	−0.457232	+	0.889348i	$0.651159\pi$
$18$	0	0
$19$	357041.	0.628532	0.314266	−	0.949335i	$-0.398242\pi$
$19$	357041.	0.628532	0.314266	+	0.949335i	$0.398242\pi$
$20$	328437.	0.459004
$21$	0	0
$22$	1.84372e6	1.67800
$23$	584389.	0.435439	0.217719	−	0.976011i	$-0.430138\pi$
$23$	584389.	0.435439	0.217719	+	0.976011i	$0.430138\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−2.32882e6	−0.999440
$27$	0	0
$28$	−1.26172e6	−0.387930
$29$	−1.31538e6	−0.345350	−0.172675	−	0.984979i	$-0.555241\pi$
$29$	−1.31538e6	−0.345350	−0.172675	+	0.984979i	$0.555241\pi$
$30$	0	0
$31$	−5.50612e6	−1.07082	−0.535412	−	0.844591i	$-0.679843\pi$
$31$	−5.50612e6	−1.07082	−0.535412	+	0.844591i	$0.679843\pi$
$32$	−8.43785e6	−1.42252
$33$	0	0
$34$	−1.01433e7	−1.30174
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	−1.18399e7	−1.03858	−0.519290	−	0.854598i	$-0.673803\pi$
$37$	−1.18399e7	−1.03858	−0.519290	+	0.854598i	$0.673803\pi$
$38$	1.15004e7	0.894718
$39$	0	0
$40$	271753.	0.0167844
$41$	2.17065e7	1.19967	0.599835	−	0.800124i	$-0.295233\pi$
$41$	2.17065e7	1.19967	0.599835	+	0.800124i	$0.295233\pi$
$42$	0	0
$43$	−1.66343e7	−0.741986	−0.370993	−	0.928636i	$-0.620983\pi$
$43$	−1.66343e7	−0.741986	−0.370993	+	0.928636i	$0.620983\pi$
$44$	3.00796e7	1.20986
$45$	0	0
$46$	1.88233e7	0.619849
$47$	1.31502e7	0.393091	0.196545	−	0.980495i	$-0.437028\pi$
$47$	1.31502e7	0.393091	0.196545	+	0.980495i	$0.437028\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	1.25821e7	0.284701
$51$	0	0
$52$	−3.79940e7	−0.720609
$53$	−5.91955e7	−1.03050	−0.515249	−	0.857040i	$-0.672301\pi$
$53$	−5.91955e7	−1.03050	−0.515249	+	0.857040i	$0.672301\pi$
$54$	0	0
$55$	3.57750e7	0.527167
$56$	−1.04397e6	−0.0141854
$57$	0	0
$58$	−4.23687e7	−0.491608
$59$	−9.80152e7	−1.05307	−0.526537	−	0.850152i	$-0.676510\pi$
$59$	−9.80152e7	−1.05307	−0.526537	+	0.850152i	$0.676510\pi$
$60$	0	0
$61$	7.52813e6	0.0696150	0.0348075	−	0.999394i	$-0.488918\pi$
$61$	7.52813e6	0.0696150	0.0348075	+	0.999394i	$0.488918\pi$
$62$	−1.77353e8	−1.52432
$63$	0	0
$64$	−1.41199e8	−1.05202
$65$	−4.51880e7	−0.313988
$66$	0	0
$67$	3.44489e7	0.208852	0.104426	−	0.994533i	$-0.466699\pi$
$67$	3.44489e7	0.208852	0.104426	+	0.994533i	$0.466699\pi$
$68$	−1.65485e8	−0.938573
$69$	0	0
$70$	−4.83355e7	−0.240616
$71$	−3.30396e8	−1.54302	−0.771510	−	0.636217i	$-0.780498\pi$
$71$	−3.30396e8	−1.54302	−0.771510	+	0.636217i	$0.780498\pi$
$72$	0	0
$73$	−6.24882e7	−0.257540	−0.128770	−	0.991674i	$-0.541103\pi$
$73$	−6.24882e7	−0.257540	−0.128770	+	0.991674i	$0.541103\pi$
$74$	−3.81365e8	−1.47842
$75$	0	0
$76$	1.87625e8	0.645103
$77$	−1.37433e8	−0.445537
$78$	0	0
$79$	1.05279e7	0.0304101	0.0152051	−	0.999884i	$-0.495160\pi$
$79$	1.05279e7	0.0304101	0.0152051	+	0.999884i	$0.495160\pi$
$80$	−1.59406e8	−0.435112
$81$	0	0
$82$	6.99170e8	1.70773
$83$	−2.19370e7	−0.0507371	−0.0253686	−	0.999678i	$-0.508076\pi$
$83$	−2.19370e7	−0.0507371	−0.0253686	+	0.999678i	$0.508076\pi$
$84$	0	0
$85$	−1.96819e8	−0.408960
$86$	−5.35793e8	−1.05622
$87$	0	0
$88$	2.48883e7	0.0442408
$89$	−1.91010e8	−0.322701	−0.161351	−	0.986897i	$-0.551585\pi$
$89$	−1.91010e8	−0.322701	−0.161351	+	0.986897i	$0.551585\pi$
$90$	0	0
$91$	1.73594e8	0.265368
$92$	3.07096e8	0.446919
$93$	0	0
$94$	4.23572e8	0.559567
$95$	2.23151e8	0.281088
$96$	0	0
$97$	−6.28904e7	−0.0721293	−0.0360646	−	0.999349i	$-0.511482\pi$
$97$	−6.28904e7	−0.0721293	−0.0360646	+	0.999349i	$0.511482\pi$
$98$	1.85686e8	0.203358
$99$	0	0
$100$	2.05273e8	0.205273
$101$	−1.96333e9	−1.87736	−0.938679	−	0.344793i	$-0.887949\pi$
$101$	−1.96333e9	−1.87736	−0.938679	+	0.344793i	$0.887949\pi$
$102$	0	0
$103$	1.35737e9	1.18832	0.594158	−	0.804348i	$-0.297485\pi$
$103$	1.35737e9	1.18832	0.594158	+	0.804348i	$0.297485\pi$
$104$	−3.14368e7	−0.0263504
$105$	0	0
$106$	−1.90670e9	−1.46692
$107$	−2.42084e9	−1.78541	−0.892707	−	0.450637i	$-0.851197\pi$
$107$	−2.42084e9	−1.78541	−0.892707	+	0.450637i	$0.851197\pi$
$108$	0	0
$109$	8.22030e8	0.557787	0.278894	−	0.960322i	$-0.410032\pi$
$109$	8.22030e8	0.557787	0.278894	+	0.960322i	$0.410032\pi$
$110$	1.15232e9	0.750424
$111$	0	0
$112$	6.12376e8	0.367737
$113$	3.87454e8	0.223546	0.111773	−	0.993734i	$-0.464347\pi$
$113$	3.87454e8	0.223546	0.111773	+	0.993734i	$0.464347\pi$
$114$	0	0
$115$	3.65243e8	0.194734
$116$	−6.91230e8	−0.354456
$117$	0	0
$118$	−3.15709e9	−1.49906
$119$	7.56099e8	0.345635
$120$	0	0
$121$	9.18476e8	0.389523
$122$	2.42483e8	0.0990973
$123$	0	0
$124$	−2.89346e9	−1.09906
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−2.10216e8	−0.0717049	−0.0358525	−	0.999357i	$-0.511415\pi$
$127$	−2.10216e8	−0.0717049	−0.0358525	+	0.999357i	$0.511415\pi$
$128$	−2.27884e8	−0.0750359
$129$	0	0
$130$	−1.45552e9	−0.446963
$131$	−6.16903e9	−1.83019	−0.915094	−	0.403240i	$-0.867884\pi$
$131$	−6.16903e9	−1.83019	−0.915094	+	0.403240i	$0.867884\pi$
$132$	0	0
$133$	−8.57256e8	−0.237563
$134$	1.10961e9	0.297301
$135$	0	0
$136$	−1.36925e8	−0.0343207
$137$	7.32830e9	1.77730	0.888650	−	0.458586i	$-0.151644\pi$
$137$	7.32830e9	1.77730	0.888650	+	0.458586i	$0.151644\pi$
$138$	0	0
$139$	−4.28674e9	−0.974003	−0.487002	−	0.873401i	$-0.661909\pi$
$139$	−4.28674e9	−0.974003	−0.487002	+	0.873401i	$0.661909\pi$
$140$	−7.88577e8	−0.173487
$141$	0	0
$142$	−1.06421e10	−2.19650
$143$	−4.13850e9	−0.827619
$144$	0	0
$145$	−8.22112e8	−0.154445
$146$	−2.01276e9	−0.366610
$147$	0	0
$148$	−6.22185e9	−1.06596
$149$	−8.70461e9	−1.44681	−0.723404	−	0.690425i	$-0.757424\pi$
$149$	−8.70461e9	−1.44681	−0.723404	+	0.690425i	$0.757424\pi$
$150$	0	0
$151$	6.53802e9	1.02341	0.511705	−	0.859161i	$-0.329014\pi$
$151$	6.53802e9	1.02341	0.511705	+	0.859161i	$0.329014\pi$
$152$	1.55244e8	0.0235894
$153$	0	0
$154$	−4.42676e9	−0.634224
$155$	−3.44132e9	−0.478887
$156$	0	0
$157$	1.37436e9	0.180532	0.0902658	−	0.995918i	$-0.471228\pi$
$157$	1.37436e9	0.180532	0.0902658	+	0.995918i	$0.471228\pi$
$158$	3.39105e8	0.0432890
$159$	0	0
$160$	−5.27366e9	−0.636168
$161$	−1.40312e9	−0.164580
$162$	0	0
$163$	7.50693e9	0.832949	0.416474	−	0.909147i	$-0.363266\pi$
$163$	7.50693e9	0.832949	0.416474	+	0.909147i	$0.363266\pi$
$164$	1.14067e10	1.23130
$165$	0	0
$166$	−7.06596e8	−0.0722246
$167$	7.38938e9	0.735163	0.367582	−	0.929991i	$-0.380186\pi$
$167$	7.38938e9	0.735163	0.367582	+	0.929991i	$0.380186\pi$
$168$	0	0
$169$	−5.37710e9	−0.507058
$170$	−6.33958e9	−0.582157
$171$	0	0
$172$	−8.74129e9	−0.761548
$173$	−7.26915e9	−0.616987	−0.308493	−	0.951226i	$-0.599825\pi$
$173$	−7.26915e9	−0.616987	−0.308493	+	0.951226i	$0.599825\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−1.45991e10	−1.14688
$177$	0	0
$178$	−6.15247e9	−0.459367
$179$	−8.72690e9	−0.635362	−0.317681	−	0.948198i	$-0.602904\pi$
$179$	−8.72690e9	−0.635362	−0.317681	+	0.948198i	$0.602904\pi$
$180$	0	0
$181$	5.33555e9	0.369510	0.184755	−	0.982785i	$-0.440851\pi$
$181$	5.33555e9	0.369510	0.184755	+	0.982785i	$0.440851\pi$
$182$	5.59151e9	0.377753
$183$	0	0
$184$	2.54096e8	0.0163424
$185$	−7.39993e9	−0.464467
$186$	0	0
$187$	−1.80255e10	−1.07795
$188$	6.91043e9	0.403455
$189$	0	0
$190$	7.18774e9	0.400130
$191$	−2.57170e10	−1.39820	−0.699100	−	0.715023i	$-0.746416\pi$
$191$	−2.57170e10	−1.39820	−0.699100	+	0.715023i	$0.746416\pi$
$192$	0	0
$193$	8.61483e9	0.446929	0.223465	−	0.974712i	$-0.428263\pi$
$193$	8.61483e9	0.446929	0.223465	+	0.974712i	$0.428263\pi$
$194$	−2.02572e9	−0.102676
$195$	0	0
$196$	3.02940e9	0.146624
$197$	4.16507e9	0.197026	0.0985131	−	0.995136i	$-0.468591\pi$
$197$	4.16507e9	0.197026	0.0985131	+	0.995136i	$0.468591\pi$
$198$	0	0
$199$	−9.39656e9	−0.424747	−0.212373	−	0.977189i	$-0.568119\pi$
$199$	−9.39656e9	−0.424747	−0.212373	+	0.977189i	$0.568119\pi$
$200$	1.69846e8	0.00750620
$201$	0	0
$202$	−6.32393e10	−2.67243
$203$	3.15822e9	0.130530
$204$	0	0
$205$	1.35665e10	0.536508
$206$	4.37213e10	1.69157
$207$	0	0
$208$	1.84403e10	0.683099
$209$	2.04371e10	0.740901
$210$	0	0
$211$	−2.15714e10	−0.749216	−0.374608	−	0.927183i	$-0.622223\pi$
$211$	−2.15714e10	−0.749216	−0.374608	+	0.927183i	$0.622223\pi$
$212$	−3.11072e10	−1.05767
$213$	0	0
$214$	−7.79758e10	−2.54155
$215$	−1.03964e10	−0.331826
$216$	0	0
$217$	1.32202e10	0.404733
$218$	2.64778e10	0.794013
$219$	0	0
$220$	1.87997e10	0.541066
$221$	2.27682e10	0.642043
$222$	0	0
$223$	−2.52938e10	−0.684923	−0.342462	−	0.939532i	$-0.611261\pi$
$223$	−2.52938e10	−0.684923	−0.342462	+	0.939532i	$0.611261\pi$
$224$	2.02593e10	0.537660
$225$	0	0
$226$	1.24800e10	0.318219
$227$	2.97607e10	0.743920	0.371960	−	0.928249i	$-0.378686\pi$
$227$	2.97607e10	0.743920	0.371960	+	0.928249i	$0.378686\pi$
$228$	0	0
$229$	−3.64655e9	−0.0876238	−0.0438119	−	0.999040i	$-0.513950\pi$
$229$	−3.64655e9	−0.0876238	−0.0438119	+	0.999040i	$0.513950\pi$
$230$	1.17646e10	0.277205
$231$	0	0
$232$	−5.71934e8	−0.0129613
$233$	3.85362e10	0.856578	0.428289	−	0.903642i	$-0.359117\pi$
$233$	3.85362e10	0.856578	0.428289	+	0.903642i	$0.359117\pi$
$234$	0	0
$235$	8.21889e9	0.175796
$236$	−5.15069e10	−1.08084
$237$	0	0
$238$	2.43541e10	0.492012
$239$	−8.92308e10	−1.76899	−0.884493	−	0.466554i	$-0.845495\pi$
$239$	−8.92308e10	−1.76899	−0.884493	+	0.466554i	$0.845495\pi$
$240$	0	0
$241$	7.60482e10	1.45215	0.726076	−	0.687614i	$-0.241342\pi$
$241$	7.60482e10	1.45215	0.726076	+	0.687614i	$0.241342\pi$
$242$	2.95843e10	0.554488
$243$	0	0
$244$	3.95602e9	0.0714504
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−2.58144e10	−0.441291
$248$	−2.39409e9	−0.0401891
$249$	0	0
$250$	7.86383e9	0.127322
$251$	1.97622e10	0.314270	0.157135	−	0.987577i	$-0.449774\pi$
$251$	1.97622e10	0.314270	0.157135	+	0.987577i	$0.449774\pi$
$252$	0	0
$253$	3.34505e10	0.513286
$254$	−6.77110e9	−0.102072
$255$	0	0
$256$	6.49539e10	0.945203
$257$	9.56309e10	1.36741	0.683706	−	0.729758i	$-0.260367\pi$
$257$	9.56309e10	1.36741	0.683706	+	0.729758i	$0.260367\pi$
$258$	0	0
$259$	2.84276e10	0.392546
$260$	−2.37462e10	−0.322266
$261$	0	0
$262$	−1.98706e11	−2.60528
$263$	3.34459e10	0.431065	0.215532	−	0.976497i	$-0.430851\pi$
$263$	3.34459e10	0.431065	0.215532	+	0.976497i	$0.430851\pi$
$264$	0	0
$265$	−3.69972e10	−0.460853
$266$	−2.76124e10	−0.338172
$267$	0	0
$268$	1.81028e10	0.214358
$269$	−1.43004e11	−1.66519	−0.832595	−	0.553883i	$-0.813146\pi$
$269$	−1.43004e11	−1.66519	−0.832595	+	0.553883i	$0.813146\pi$
$270$	0	0
$271$	1.40660e11	1.58419	0.792097	−	0.610395i	$-0.208989\pi$
$271$	1.40660e11	1.58419	0.792097	+	0.610395i	$0.208989\pi$
$272$	8.03179e10	0.889718
$273$	0	0
$274$	2.36046e11	2.52999
$275$	2.23594e10	0.235756
$276$	0	0
$277$	1.26458e11	1.29058	0.645292	−	0.763936i	$-0.276736\pi$
$277$	1.26458e11	1.29058	0.645292	+	0.763936i	$0.276736\pi$
$278$	−1.38077e11	−1.38650
$279$	0	0
$280$	−6.52480e8	−0.00634390
$281$	−1.43579e11	−1.37377	−0.686885	−	0.726767i	$-0.741022\pi$
$281$	−1.43579e11	−1.37377	−0.686885	+	0.726767i	$0.741022\pi$
$282$	0	0
$283$	−1.56389e11	−1.44933	−0.724667	−	0.689099i	$-0.758006\pi$
$283$	−1.56389e11	−1.44933	−0.724667	+	0.689099i	$0.758006\pi$
$284$	−1.73623e11	−1.58370
$285$	0	0
$286$	−1.33302e11	−1.17812
$287$	−5.21172e10	−0.453432
$288$	0	0
$289$	−1.94196e10	−0.163757
$290$	−2.64804e10	−0.219854
$291$	0	0
$292$	−3.28375e10	−0.264330
$293$	1.68933e11	1.33909	0.669547	−	0.742769i	$-0.266488\pi$
$293$	1.68933e11	1.33909	0.669547	+	0.742769i	$0.266488\pi$
$294$	0	0
$295$	−6.12595e10	−0.470949
$296$	−5.14805e9	−0.0389789
$297$	0	0
$298$	−2.80377e11	−2.05954
$299$	−4.22518e10	−0.305721
$300$	0	0
$301$	3.99389e10	0.280444
$302$	2.10591e11	1.45683
$303$	0	0
$304$	−9.10635e10	−0.611524
$305$	4.70508e9	0.0311328
$306$	0	0
$307$	−8.07485e10	−0.518814	−0.259407	−	0.965768i	$-0.583527\pi$
$307$	−8.07485e10	−0.518814	−0.259407	+	0.965768i	$0.583527\pi$
$308$	−7.22211e10	−0.457284
$309$	0	0
$310$	−1.10846e11	−0.681697
$311$	1.87901e11	1.13896	0.569480	−	0.822005i	$-0.307145\pi$
$311$	1.87901e11	1.13896	0.569480	+	0.822005i	$0.307145\pi$
$312$	0	0
$313$	1.41302e11	0.832143	0.416072	−	0.909332i	$-0.363407\pi$
$313$	1.41302e11	0.832143	0.416072	+	0.909332i	$0.363407\pi$
$314$	4.42686e10	0.256988
$315$	0	0
$316$	5.53238e9	0.0312119
$317$	−7.46593e10	−0.415257	−0.207629	−	0.978208i	$-0.566575\pi$
$317$	−7.46593e10	−0.415257	−0.207629	+	0.978208i	$0.566575\pi$
$318$	0	0
$319$	−7.52924e10	−0.407092
$320$	−8.82496e10	−0.470476
$321$	0	0
$322$	−4.51948e10	−0.234281
$323$	−1.12436e11	−0.574769
$324$	0	0
$325$	−2.82425e10	−0.140420
$326$	2.41800e11	1.18571
$327$	0	0
$328$	9.43809e9	0.0450248
$329$	−3.15737e10	−0.148574
$330$	0	0
$331$	−1.08719e11	−0.497828	−0.248914	−	0.968526i	$-0.580074\pi$
$331$	−1.08719e11	−0.497828	−0.248914	+	0.968526i	$0.580074\pi$
$332$	−1.15279e10	−0.0520748
$333$	0	0
$334$	2.38014e11	1.04651
$335$	2.15305e10	0.0934014
$336$	0	0
$337$	4.04135e11	1.70684	0.853419	−	0.521226i	$-0.174525\pi$
$337$	4.04135e11	1.70684	0.853419	+	0.521226i	$0.174525\pi$
$338$	−1.73198e11	−0.721800
$339$	0	0
$340$	−1.03428e11	−0.419743
$341$	−3.15170e11	−1.26227
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−7.23267e9	−0.0278475
$345$	0	0
$346$	−2.34141e11	−0.878284
$347$	3.00837e11	1.11391	0.556953	−	0.830544i	$-0.311970\pi$
$347$	3.00837e11	1.11391	0.556953	+	0.830544i	$0.311970\pi$
$348$	0	0
$349$	−4.92615e11	−1.77743	−0.888716	−	0.458458i	$-0.848402\pi$
$349$	−4.92615e11	−1.77743	−0.888716	+	0.458458i	$0.848402\pi$
$350$	−3.02097e10	−0.107607
$351$	0	0
$352$	−4.82983e11	−1.67683
$353$	2.81815e11	0.966002	0.483001	−	0.875620i	$-0.339547\pi$
$353$	2.81815e11	0.966002	0.483001	+	0.875620i	$0.339547\pi$
$354$	0	0
$355$	−2.06497e11	−0.690060
$356$	−1.00375e11	−0.331209
$357$	0	0
$358$	−2.81096e11	−0.904441
$359$	−4.52989e11	−1.43934	−0.719669	−	0.694317i	$-0.755706\pi$
$359$	−4.52989e11	−1.43934	−0.719669	+	0.694317i	$0.755706\pi$
$360$	0	0
$361$	−1.95209e11	−0.604948
$362$	1.71859e11	0.525999
$363$	0	0
$364$	9.12235e10	0.272365
$365$	−3.90551e10	−0.115175
$366$	0	0
$367$	3.63463e11	1.04584	0.522918	−	0.852383i	$-0.324844\pi$
$367$	3.63463e11	1.04584	0.522918	+	0.852383i	$0.324844\pi$
$368$	−1.49049e11	−0.423655
$369$	0	0
$370$	−2.38353e11	−0.661171
$371$	1.42128e11	0.389492
$372$	0	0
$373$	−1.72737e11	−0.462058	−0.231029	−	0.972947i	$-0.574209\pi$
$373$	−1.72737e11	−0.462058	−0.231029	+	0.972947i	$0.574209\pi$
$374$	−5.80604e11	−1.53447
$375$	0	0
$376$	5.71779e9	0.0147531
$377$	9.51029e10	0.242470
$378$	0	0
$379$	−3.48679e11	−0.868059	−0.434029	−	0.900899i	$-0.642909\pi$
$379$	−3.48679e11	−0.868059	−0.434029	+	0.900899i	$0.642909\pi$
$380$	1.17266e11	0.288499
$381$	0	0
$382$	−8.28349e11	−1.99035
$383$	3.48271e11	0.827033	0.413516	−	0.910497i	$-0.364300\pi$
$383$	3.48271e11	0.827033	0.413516	+	0.910497i	$0.364300\pi$
$384$	0	0
$385$	−8.58959e10	−0.199250
$386$	2.77486e11	0.636206
$387$	0	0
$388$	−3.30489e10	−0.0740310
$389$	1.50559e11	0.333374	0.166687	−	0.986010i	$-0.446693\pi$
$389$	1.50559e11	0.333374	0.166687	+	0.986010i	$0.446693\pi$
$390$	0	0
$391$	−1.84030e11	−0.398193
$392$	2.50657e9	0.00536157
$393$	0	0
$394$	1.34158e11	0.280468
$395$	6.57992e9	0.0135998
$396$	0	0
$397$	1.16515e11	0.235409	0.117705	−	0.993049i	$-0.462446\pi$
$397$	1.16515e11	0.235409	0.117705	+	0.993049i	$0.462446\pi$
$398$	−3.02665e11	−0.604629
$399$	0	0
$400$	−9.96290e10	−0.194588
$401$	4.72351e11	0.912252	0.456126	−	0.889915i	$-0.349237\pi$
$401$	4.72351e11	0.912252	0.456126	+	0.889915i	$0.349237\pi$
$402$	0	0
$403$	3.98096e11	0.751823
$404$	−1.03173e12	−1.92685
$405$	0	0
$406$	1.01727e11	0.185810
$407$	−6.77716e11	−1.22426
$408$	0	0
$409$	1.05236e12	1.85956	0.929781	−	0.368114i	$-0.119996\pi$
$409$	1.05236e12	1.85956	0.929781	+	0.368114i	$0.119996\pi$
$410$	4.36981e11	0.763722
$411$	0	0
$412$	7.13298e11	1.21965
$413$	2.35334e11	0.398025
$414$	0	0
$415$	−1.37106e10	−0.0226903
$416$	6.10063e11	0.998745
$417$	0	0
$418$	6.58283e11	1.05468
$419$	3.81238e10	0.0604273	0.0302137	−	0.999543i	$-0.490381\pi$
$419$	3.81238e10	0.0604273	0.0302137	+	0.999543i	$0.490381\pi$
$420$	0	0
$421$	−1.89375e11	−0.293802	−0.146901	−	0.989151i	$-0.546930\pi$
$421$	−1.89375e11	−0.293802	−0.146901	+	0.989151i	$0.546930\pi$
$422$	−6.94819e11	−1.06651
$423$	0	0
$424$	−2.57385e10	−0.0386756
$425$	−1.23012e11	−0.182893
$426$	0	0
$427$	−1.80750e10	−0.0263120
$428$	−1.27215e12	−1.83249
$429$	0	0
$430$	−3.34871e11	−0.472356
$431$	1.02244e12	1.42722	0.713612	−	0.700542i	$-0.247058\pi$
$431$	1.02244e12	1.42722	0.713612	+	0.700542i	$0.247058\pi$
$432$	0	0
$433$	1.07146e12	1.46481	0.732404	−	0.680870i	$-0.238398\pi$
$433$	1.07146e12	1.46481	0.732404	+	0.680870i	$0.238398\pi$
$434$	4.25825e11	0.576139
$435$	0	0
$436$	4.31976e11	0.572493
$437$	2.08651e11	0.273687
$438$	0	0
$439$	1.43779e12	1.84759	0.923795	−	0.382886i	$-0.125070\pi$
$439$	1.43779e12	1.84759	0.923795	+	0.382886i	$0.125070\pi$
$440$	1.55552e10	0.0197851
$441$	0	0
$442$	7.33370e11	0.913951
$443$	−1.36924e12	−1.68913	−0.844566	−	0.535451i	$-0.820142\pi$
$443$	−1.36924e12	−1.68913	−0.844566	+	0.535451i	$0.820142\pi$
$444$	0	0
$445$	−1.19381e11	−0.144316
$446$	−8.14719e11	−0.974992
$447$	0	0
$448$	3.39020e11	0.397625
$449$	−2.96585e11	−0.344382	−0.172191	−	0.985064i	$-0.555085\pi$
$449$	−2.96585e11	−0.344382	−0.172191	+	0.985064i	$0.555085\pi$
$450$	0	0
$451$	1.24248e12	1.41415
$452$	2.03607e11	0.229440
$453$	0	0
$454$	9.58598e11	1.05897
$455$	1.08496e11	0.118676
$456$	0	0
$457$	−4.11629e11	−0.441451	−0.220726	−	0.975336i	$-0.570843\pi$
$457$	−4.11629e11	−0.441451	−0.220726	+	0.975336i	$0.570843\pi$
$458$	−1.17456e11	−0.124733
$459$	0	0
$460$	1.91935e11	0.199868
$461$	−4.28275e11	−0.441640	−0.220820	−	0.975315i	$-0.570873\pi$
$461$	−4.28275e11	−0.441640	−0.220820	+	0.975315i	$0.570873\pi$
$462$	0	0
$463$	1.18479e12	1.19820	0.599099	−	0.800675i	$-0.295526\pi$
$463$	1.18479e12	1.19820	0.599099	+	0.800675i	$0.295526\pi$
$464$	3.35488e11	0.336005
$465$	0	0
$466$	1.24126e12	1.21934
$467$	−2.02689e11	−0.197198	−0.0985992	−	0.995127i	$-0.531436\pi$
$467$	−2.02689e11	−0.197198	−0.0985992	+	0.995127i	$0.531436\pi$
$468$	0	0
$469$	−8.27117e10	−0.0789386
$470$	2.64732e11	0.250246
$471$	0	0
$472$	−4.26175e10	−0.0395230
$473$	−9.52146e11	−0.874638
$474$	0	0
$475$	1.39469e11	0.125706
$476$	3.97329e11	0.354747
$477$	0	0
$478$	−2.87414e12	−2.51816
$479$	4.42687e11	0.384226	0.192113	−	0.981373i	$-0.438466\pi$
$479$	4.42687e11	0.384226	0.192113	+	0.981373i	$0.438466\pi$
$480$	0	0
$481$	8.56033e11	0.729185
$482$	2.44953e12	2.06715
$483$	0	0
$484$	4.82658e11	0.399793
$485$	−3.93065e10	−0.0322572
$486$	0	0
$487$	−4.27638e11	−0.344505	−0.172253	−	0.985053i	$-0.555105\pi$
$487$	−4.27638e11	−0.344505	−0.172253	+	0.985053i	$0.555105\pi$
$488$	3.27327e9	0.00261272
$489$	0	0
$490$	1.16053e11	0.0909444
$491$	2.47448e12	1.92140	0.960699	−	0.277591i	$-0.0895359\pi$
$491$	2.47448e12	1.92140	0.960699	+	0.277591i	$0.0895359\pi$
$492$	0	0
$493$	4.14226e11	0.315810
$494$	−8.31487e11	−0.628180
$495$	0	0
$496$	1.40434e12	1.04185
$497$	7.93280e11	0.583207
$498$	0	0
$499$	1.91536e10	0.0138292	0.00691462	−	0.999976i	$-0.497799\pi$
$499$	1.91536e10	0.0138292	0.00691462	+	0.999976i	$0.497799\pi$
$500$	1.28296e11	0.0918009
$501$	0	0
$502$	6.36545e11	0.447365
$503$	−6.62178e11	−0.461231	−0.230616	−	0.973045i	$-0.574074\pi$
$503$	−6.62178e11	−0.461231	−0.230616	+	0.973045i	$0.574074\pi$
$504$	0	0
$505$	−1.22708e12	−0.839580
$506$	1.07745e12	0.730666
$507$	0	0
$508$	−1.10468e11	−0.0735954
$509$	2.00796e12	1.32594	0.662972	−	0.748644i	$-0.269295\pi$
$509$	2.00796e12	1.32594	0.662972	+	0.748644i	$0.269295\pi$
$510$	0	0
$511$	1.50034e11	0.0973410
$512$	2.20886e12	1.42054
$513$	0	0
$514$	3.08029e12	1.94652
$515$	8.48358e11	0.531431
$516$	0	0
$517$	7.52720e11	0.463368
$518$	9.15658e11	0.558791
$519$	0	0
$520$	−1.96480e10	−0.0117843
$521$	−8.43834e10	−0.0501750	−0.0250875	−	0.999685i	$-0.507986\pi$
$521$	−8.43834e10	−0.0501750	−0.0250875	+	0.999685i	$0.507986\pi$
$522$	0	0
$523$	−1.25024e12	−0.730693	−0.365346	−	0.930872i	$-0.619049\pi$
$523$	−1.25024e12	−0.730693	−0.365346	+	0.930872i	$0.619049\pi$
$524$	−3.24182e12	−1.87844
$525$	0	0
$526$	1.07730e12	0.613623
$527$	1.73393e12	0.979228
$528$	0	0
$529$	−1.45964e12	−0.810393
$530$	−1.19169e12	−0.656026
$531$	0	0
$532$	−4.50487e11	−0.243826
$533$	−1.56939e12	−0.842285
$534$	0	0
$535$	−1.51302e12	−0.798462
$536$	1.49786e10	0.00783842
$537$	0	0
$538$	−4.60620e12	−2.37040
$539$	3.29978e11	0.168397
$540$	0	0
$541$	−1.02366e12	−0.513771	−0.256886	−	0.966442i	$-0.582696\pi$
$541$	−1.02366e12	−0.513771	−0.256886	+	0.966442i	$0.582696\pi$
$542$	4.53069e12	2.25511
$543$	0	0
$544$	2.65716e12	1.30084
$545$	5.13769e11	0.249450
$546$	0	0
$547$	9.57028e11	0.457069	0.228534	−	0.973536i	$-0.426607\pi$
$547$	9.57028e11	0.457069	0.228534	+	0.973536i	$0.426607\pi$
$548$	3.85102e12	1.82416
$549$	0	0
$550$	7.20201e11	0.335600
$551$	−4.69645e11	−0.217064
$552$	0	0
$553$	−2.52774e10	−0.0114940
$554$	4.07323e12	1.83715
$555$	0	0
$556$	−2.25268e12	−0.999683
$557$	−1.18726e12	−0.522632	−0.261316	−	0.965253i	$-0.584156\pi$
$557$	−1.18726e12	−0.522632	−0.261316	+	0.965253i	$0.584156\pi$
$558$	0	0
$559$	1.20267e12	0.520947
$560$	3.82735e11	0.164457
$561$	0	0
$562$	−4.62473e12	−1.95557
$563$	2.32001e12	0.973201	0.486601	−	0.873625i	$-0.338237\pi$
$563$	2.32001e12	0.973201	0.486601	+	0.873625i	$0.338237\pi$
$564$	0	0
$565$	2.42159e11	0.0999728
$566$	−5.03734e12	−2.06313
$567$	0	0
$568$	−1.43658e11	−0.0579111
$569$	3.21164e12	1.28446	0.642231	−	0.766511i	$-0.278009\pi$
$569$	3.21164e12	1.28446	0.642231	+	0.766511i	$0.278009\pi$
$570$	0	0
$571$	4.05654e12	1.59696	0.798479	−	0.602023i	$-0.205638\pi$
$571$	4.05654e12	1.59696	0.798479	+	0.602023i	$0.205638\pi$
$572$	−2.17478e12	−0.849440
$573$	0	0
$574$	−1.67871e12	−0.645463
$575$	2.28277e11	0.0870877
$576$	0	0
$577$	2.71448e12	1.01952	0.509759	−	0.860317i	$-0.329734\pi$
$577$	2.71448e12	1.01952	0.509759	+	0.860317i	$0.329734\pi$
$578$	−6.25509e11	−0.233109
$579$	0	0
$580$	−4.32019e11	−0.158517
$581$	5.26708e10	0.0191768
$582$	0	0
$583$	−3.38835e12	−1.21473
$584$	−2.71702e10	−0.00966574
$585$	0	0
$586$	5.44139e12	1.90621
$587$	4.44937e12	1.54678	0.773388	−	0.633933i	$-0.218560\pi$
$587$	4.44937e12	1.54678	0.773388	+	0.633933i	$0.218560\pi$
$588$	0	0
$589$	−1.96591e12	−0.673047
$590$	−1.97318e12	−0.670399
$591$	0	0
$592$	3.01977e12	1.01048
$593$	−5.35101e11	−0.177701	−0.0888505	−	0.996045i	$-0.528319\pi$
$593$	−5.35101e11	−0.177701	−0.0888505	+	0.996045i	$0.528319\pi$
$594$	0	0
$595$	4.72562e11	0.154573
$596$	−4.57426e12	−1.48495
$597$	0	0
$598$	−1.36094e12	−0.435195
$599$	−4.56433e11	−0.144863	−0.0724313	−	0.997373i	$-0.523076\pi$
$599$	−4.56433e11	−0.144863	−0.0724313	+	0.997373i	$0.523076\pi$
$600$	0	0
$601$	5.40101e12	1.68865	0.844326	−	0.535830i	$-0.180001\pi$
$601$	5.40101e12	1.68865	0.844326	+	0.535830i	$0.180001\pi$
$602$	1.28644e12	0.399214
$603$	0	0
$604$	3.43572e12	1.05039
$605$	5.74047e11	0.174200
$606$	0	0
$607$	−2.30047e12	−0.687807	−0.343904	−	0.939005i	$-0.611749\pi$
$607$	−2.30047e12	−0.687807	−0.343904	+	0.939005i	$0.611749\pi$
$608$	−3.01266e12	−0.894096
$609$	0	0
$610$	1.51552e11	0.0443176
$611$	−9.50771e11	−0.275988
$612$	0	0
$613$	−4.45284e12	−1.27369	−0.636846	−	0.770991i	$-0.719761\pi$
$613$	−4.45284e12	−1.27369	−0.636846	+	0.770991i	$0.719761\pi$
$614$	−2.60093e12	−0.738534
$615$	0	0
$616$	−5.97568e10	−0.0167215
$617$	−3.10703e12	−0.863103	−0.431551	−	0.902088i	$-0.642034\pi$
$617$	−3.10703e12	−0.863103	−0.431551	+	0.902088i	$0.642034\pi$
$618$	0	0
$619$	5.32893e11	0.145892	0.0729461	−	0.997336i	$-0.476760\pi$
$619$	5.32893e11	0.145892	0.0729461	+	0.997336i	$0.476760\pi$
$620$	−1.80841e12	−0.491513
$621$	0	0
$622$	6.05235e12	1.62131
$623$	4.58615e11	0.121970
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	4.55136e12	1.18456
$627$	0	0
$628$	7.22227e11	0.185291
$629$	3.72850e12	0.949743
$630$	0	0
$631$	−1.67342e12	−0.420216	−0.210108	−	0.977678i	$-0.567382\pi$
$631$	−1.67342e12	−0.420216	−0.210108	+	0.977678i	$0.567382\pi$
$632$	4.57758e9	0.00114132
$633$	0	0
$634$	−2.40479e12	−0.591120
$635$	−1.31385e11	−0.0320674
$636$	0	0
$637$	−4.16799e11	−0.100300
$638$	−2.42518e12	−0.579498
$639$	0	0
$640$	−1.42428e11	−0.0335571
$641$	−5.43083e12	−1.27059	−0.635295	−	0.772270i	$-0.719121\pi$
$641$	−5.43083e12	−1.27059	−0.635295	+	0.772270i	$0.719121\pi$
$642$	0	0
$643$	5.07073e11	0.116983	0.0584913	−	0.998288i	$-0.481371\pi$
$643$	5.07073e11	0.116983	0.0584913	+	0.998288i	$0.481371\pi$
$644$	−7.37337e11	−0.168919
$645$	0	0
$646$	−3.62159e12	−0.818187
$647$	−1.21169e12	−0.271845	−0.135922	−	0.990719i	$-0.543400\pi$
$647$	−1.21169e12	−0.271845	−0.135922	+	0.990719i	$0.543400\pi$
$648$	0	0
$649$	−5.61039e12	−1.24134
$650$	−9.09697e11	−0.199888
$651$	0	0
$652$	3.94488e12	0.854909
$653$	−1.76564e12	−0.380008	−0.190004	−	0.981783i	$-0.560850\pi$
$653$	−1.76564e12	−0.380008	−0.190004	+	0.981783i	$0.560850\pi$
$654$	0	0
$655$	−3.85564e12	−0.818485
$656$	−5.53624e12	−1.16721
$657$	0	0
$658$	−1.01700e12	−0.211496
$659$	1.80608e12	0.373038	0.186519	−	0.982451i	$-0.440279\pi$
$659$	1.80608e12	0.373038	0.186519	+	0.982451i	$0.440279\pi$
$660$	0	0
$661$	−3.74182e12	−0.762389	−0.381194	−	0.924495i	$-0.624487\pi$
$661$	−3.74182e12	−0.762389	−0.381194	+	0.924495i	$0.624487\pi$
$662$	−3.50186e12	−0.708661
$663$	0	0
$664$	−9.53833e9	−0.00190422
$665$	−5.35785e11	−0.106241
$666$	0	0
$667$	−7.68693e11	−0.150379
$668$	3.88311e12	0.754546
$669$	0	0
$670$	6.93504e11	0.132957
$671$	4.30911e11	0.0820608
$672$	0	0
$673$	−9.07915e12	−1.70599	−0.852997	−	0.521916i	$-0.825218\pi$
$673$	−9.07915e12	−1.70599	−0.852997	+	0.521916i	$0.825218\pi$
$674$	1.30173e13	2.42969
$675$	0	0
$676$	−2.82566e12	−0.520427
$677$	5.62344e12	1.02885	0.514426	−	0.857535i	$-0.328005\pi$
$677$	5.62344e12	1.02885	0.514426	+	0.857535i	$0.328005\pi$
$678$	0	0
$679$	1.51000e11	0.0272623
$680$	−8.55779e10	−0.0153487
$681$	0	0
$682$	−1.01517e13	−1.79684
$683$	−8.49074e12	−1.49297	−0.746487	−	0.665399i	$-0.768261\pi$
$683$	−8.49074e12	−1.49297	−0.746487	+	0.665399i	$0.768261\pi$
$684$	0	0
$685$	4.58019e12	0.794833
$686$	−4.45831e11	−0.0768620
$687$	0	0
$688$	4.24257e12	0.721907
$689$	4.27988e12	0.723511
$690$	0	0
$691$	6.94021e12	1.15803	0.579017	−	0.815315i	$-0.303436\pi$
$691$	6.94021e12	1.15803	0.579017	+	0.815315i	$0.303436\pi$
$692$	−3.81993e12	−0.633254
$693$	0	0
$694$	9.69003e12	1.58565
$695$	−2.67921e12	−0.435587
$696$	0	0
$697$	−6.83558e12	−1.09705
$698$	−1.58672e13	−2.53018
$699$	0	0
$700$	−4.92861e11	−0.0775859
$701$	9.12070e12	1.42658	0.713292	−	0.700867i	$-0.247203\pi$
$701$	9.12070e12	1.42658	0.713292	+	0.700867i	$0.247203\pi$
$702$	0	0
$703$	−4.22733e12	−0.652780
$704$	−8.08226e12	−1.24010
$705$	0	0
$706$	9.07733e12	1.37511
$707$	4.71395e12	0.709574
$708$	0	0
$709$	−6.32560e12	−0.940143	−0.470072	−	0.882628i	$-0.655772\pi$
$709$	−6.32560e12	−0.940143	−0.470072	+	0.882628i	$0.655772\pi$
$710$	−6.65132e12	−0.982303
$711$	0	0
$712$	−8.30521e10	−0.0121113
$713$	−3.21771e12	−0.466278
$714$	0	0
$715$	−2.58656e12	−0.370123
$716$	−4.58598e12	−0.652114
$717$	0	0
$718$	−1.45909e13	−2.04890
$719$	−1.10129e13	−1.53682	−0.768408	−	0.639960i	$-0.778951\pi$
$719$	−1.10129e13	−1.53682	−0.768408	+	0.639960i	$0.778951\pi$
$720$	0	0
$721$	−3.25905e12	−0.449141
$722$	−6.28773e12	−0.861146
$723$	0	0
$724$	2.80383e12	0.379252
$725$	−5.13820e11	−0.0690701
$726$	0	0
$727$	5.93415e12	0.787869	0.393934	−	0.919139i	$-0.371114\pi$
$727$	5.93415e12	0.787869	0.393934	+	0.919139i	$0.371114\pi$
$728$	7.54797e10	0.00995953
$729$	0	0
$730$	−1.25797e12	−0.163953
$731$	5.23829e12	0.678519
$732$	0	0
$733$	−1.07262e12	−0.137239	−0.0686193	−	0.997643i	$-0.521859\pi$
$733$	−1.07262e12	−0.137239	−0.0686193	+	0.997643i	$0.521859\pi$
$734$	1.17072e13	1.48875
$735$	0	0
$736$	−4.93099e12	−0.619418
$737$	1.97185e12	0.246190
$738$	0	0
$739$	−9.60184e12	−1.18428	−0.592140	−	0.805835i	$-0.701717\pi$
$739$	−9.60184e12	−1.18428	−0.592140	+	0.805835i	$0.701717\pi$
$740$	−3.88866e12	−0.476713
$741$	0	0
$742$	4.57799e12	0.554443
$743$	−2.37163e12	−0.285494	−0.142747	−	0.989759i	$-0.545593\pi$
$743$	−2.37163e12	−0.285494	−0.142747	+	0.989759i	$0.545593\pi$
$744$	0	0
$745$	−5.44038e12	−0.647032
$746$	−5.56391e12	−0.657742
$747$	0	0
$748$	−9.47236e12	−1.10637
$749$	5.81244e12	0.674823
$750$	0	0
$751$	1.21180e13	1.39012	0.695061	−	0.718951i	$-0.255377\pi$
$751$	1.21180e13	1.39012	0.695061	+	0.718951i	$0.255377\pi$
$752$	−3.35397e12	−0.382454
$753$	0	0
$754$	3.06329e12	0.345157
$755$	4.08626e12	0.457683
$756$	0	0
$757$	5.38650e12	0.596177	0.298088	−	0.954538i	$-0.403651\pi$
$757$	5.38650e12	0.596177	0.298088	+	0.954538i	$0.403651\pi$
$758$	−1.12310e13	−1.23569
$759$	0	0
$760$	9.70272e10	0.0105495
$761$	−7.10742e12	−0.768212	−0.384106	−	0.923289i	$-0.625490\pi$
$761$	−7.10742e12	−0.768212	−0.384106	+	0.923289i	$0.625490\pi$
$762$	0	0
$763$	−1.97369e12	−0.210824
$764$	−1.35142e13	−1.43506
$765$	0	0
$766$	1.12179e13	1.17729
$767$	7.08657e12	0.739362
$768$	0	0
$769$	−1.06791e13	−1.10120	−0.550602	−	0.834768i	$-0.685602\pi$
$769$	−1.06791e13	−1.10120	−0.550602	+	0.834768i	$0.685602\pi$
$770$	−2.76673e12	−0.283634
$771$	0	0
$772$	4.52709e12	0.458713
$773$	−1.57045e13	−1.58204	−0.791020	−	0.611791i	$-0.790449\pi$
$773$	−1.57045e13	−1.58204	−0.791020	+	0.611791i	$0.790449\pi$
$774$	0	0
$775$	−2.15083e12	−0.214165
$776$	−2.73451e10	−0.00270708
$777$	0	0
$778$	4.84953e12	0.474560
$779$	7.75010e12	0.754030
$780$	0	0
$781$	−1.89119e13	−1.81888
$782$	−5.92765e12	−0.566829
$783$	0	0
$784$	−1.47031e12	−0.138991
$785$	8.58977e11	0.0807362
$786$	0	0
$787$	9.58762e12	0.890891	0.445445	−	0.895309i	$-0.353045\pi$
$787$	9.58762e12	0.890891	0.445445	+	0.895309i	$0.353045\pi$
$788$	2.18874e12	0.202221
$789$	0	0
$790$	2.11941e11	0.0193594
$791$	−9.30276e11	−0.0844924
$792$	0	0
$793$	−5.44289e11	−0.0488766
$794$	3.75297e12	0.335106
$795$	0	0
$796$	−4.93788e12	−0.435945
$797$	−4.40502e12	−0.386710	−0.193355	−	0.981129i	$-0.561937\pi$
$797$	−4.40502e12	−0.386710	−0.193355	+	0.981129i	$0.561937\pi$
$798$	0	0
$799$	−4.14114e12	−0.359467
$800$	−3.29603e12	−0.284503
$801$	0	0
$802$	1.52145e13	1.29859
$803$	−3.57683e12	−0.303583
$804$	0	0
$805$	−8.76949e11	−0.0736025
$806$	1.28228e13	1.07022
$807$	0	0
$808$	−8.53666e11	−0.0704591
$809$	−3.91499e12	−0.321338	−0.160669	−	0.987008i	$-0.551365\pi$
$809$	−3.91499e12	−0.321338	−0.160669	+	0.987008i	$0.551365\pi$
$810$	0	0
$811$	−3.87497e12	−0.314539	−0.157269	−	0.987556i	$-0.550269\pi$
$811$	−3.87497e12	−0.314539	−0.157269	+	0.987556i	$0.550269\pi$
$812$	1.65964e12	0.133972
$813$	0	0
$814$	−2.18294e13	−1.74274
$815$	4.69183e12	0.372506
$816$	0	0
$817$	−5.93912e12	−0.466362
$818$	3.38968e13	2.64709
$819$	0	0
$820$	7.12920e12	0.550654
$821$	−1.93633e13	−1.48743	−0.743715	−	0.668497i	$-0.766938\pi$
$821$	−1.93633e13	−1.48743	−0.743715	+	0.668497i	$0.766938\pi$
$822$	0	0
$823$	−1.60460e13	−1.21918	−0.609589	−	0.792717i	$-0.708666\pi$
$823$	−1.60460e13	−1.21918	−0.609589	+	0.792717i	$0.708666\pi$
$824$	5.90193e11	0.0445987
$825$	0	0
$826$	7.58018e12	0.566590
$827$	−1.36528e13	−1.01496	−0.507478	−	0.861665i	$-0.669422\pi$
$827$	−1.36528e13	−1.01496	−0.507478	+	0.861665i	$0.669422\pi$
$828$	0	0
$829$	1.61464e13	1.18736	0.593678	−	0.804703i	$-0.297675\pi$
$829$	1.61464e13	1.18736	0.593678	+	0.804703i	$0.297675\pi$
$830$	−4.41623e11	−0.0322998
$831$	0	0
$832$	1.02088e13	0.738619
$833$	−1.81539e12	−0.130638
$834$	0	0
$835$	4.61836e12	0.328775
$836$	1.07397e13	0.760435
$837$	0	0
$838$	1.22798e12	0.0860186
$839$	1.78914e13	1.24657	0.623284	−	0.781996i	$-0.285798\pi$
$839$	1.78914e13	1.24657	0.623284	+	0.781996i	$0.285798\pi$
$840$	0	0
$841$	−1.27769e13	−0.880733
$842$	−6.09983e12	−0.418228
$843$	0	0
$844$	−1.13357e13	−0.768969
$845$	−3.36069e12	−0.226763
$846$	0	0
$847$	−2.20526e12	−0.147226
$848$	1.50978e13	1.00261
$849$	0	0
$850$	−3.96224e12	−0.260349
$851$	−6.91910e12	−0.452238
$852$	0	0
$853$	−1.21767e13	−0.787518	−0.393759	−	0.919214i	$-0.628825\pi$
$853$	−1.21767e13	−0.787518	−0.393759	+	0.919214i	$0.628825\pi$
$854$	−5.82201e11	−0.0374552
$855$	0	0
$856$	−1.05259e12	−0.0670084
$857$	−3.29355e12	−0.208569	−0.104285	−	0.994547i	$-0.533255\pi$
$857$	−3.29355e12	−0.208569	−0.104285	+	0.994547i	$0.533255\pi$
$858$	0	0
$859$	−1.71452e13	−1.07442	−0.537209	−	0.843449i	$-0.680521\pi$
$859$	−1.71452e13	−1.07442	−0.537209	+	0.843449i	$0.680521\pi$
$860$	−5.46330e12	−0.340575
$861$	0	0
$862$	3.29332e13	2.03166
$863$	−1.62825e12	−0.0999249	−0.0499624	−	0.998751i	$-0.515910\pi$
$863$	−1.62825e12	−0.0999249	−0.0499624	+	0.998751i	$0.515910\pi$
$864$	0	0
$865$	−4.54322e12	−0.275925
$866$	3.45120e13	2.08516
$867$	0	0
$868$	6.94719e12	0.415404
$869$	6.02616e11	0.0358469
$870$	0	0
$871$	−2.49068e12	−0.146634
$872$	3.57423e11	0.0209343
$873$	0	0
$874$	6.72070e12	0.389595
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	2.97042e13	1.69559	0.847794	−	0.530326i	$-0.177930\pi$
$877$	2.97042e13	1.69559	0.847794	+	0.530326i	$0.177930\pi$
$878$	4.63116e13	2.63005
$879$	0	0
$880$	−9.12443e12	−0.512901
$881$	7.16593e12	0.400757	0.200378	−	0.979719i	$-0.435783\pi$
$881$	7.16593e12	0.400757	0.200378	+	0.979719i	$0.435783\pi$
$882$	0	0
$883$	1.10773e13	0.613211	0.306605	−	0.951837i	$-0.400807\pi$
$883$	1.10773e13	0.613211	0.306605	+	0.951837i	$0.400807\pi$
$884$	1.19647e13	0.658970
$885$	0	0
$886$	−4.41036e13	−2.40449
$887$	−2.10318e13	−1.14083	−0.570414	−	0.821358i	$-0.693217\pi$
$887$	−2.10318e13	−1.14083	−0.570414	+	0.821358i	$0.693217\pi$
$888$	0	0
$889$	5.04729e11	0.0271019
$890$	−3.84529e12	−0.205435
$891$	0	0
$892$	−1.32919e13	−0.702982
$893$	4.69518e12	0.247070
$894$	0	0
$895$	−5.45432e12	−0.284143
$896$	5.47150e11	0.0283609
$897$	0	0
$898$	−9.55308e12	−0.490230
$899$	7.24263e12	0.369809
$900$	0	0
$901$	1.86412e13	0.942353
$902$	4.00205e13	2.01304
$903$	0	0
$904$	1.68467e11	0.00838990
$905$	3.33472e12	0.165250
$906$	0	0
$907$	5.40076e12	0.264985	0.132493	−	0.991184i	$-0.457702\pi$
$907$	5.40076e12	0.264985	0.132493	+	0.991184i	$0.457702\pi$
$908$	1.56392e13	0.763534
$909$	0	0
$910$	3.49469e12	0.168936
$911$	7.77746e12	0.374115	0.187058	−	0.982349i	$-0.440105\pi$
$911$	7.77746e12	0.374115	0.187058	+	0.982349i	$0.440105\pi$
$912$	0	0
$913$	−1.25568e12	−0.0598080
$914$	−1.32587e13	−0.628408
$915$	0	0
$916$	−1.91626e12	−0.0899340
$917$	1.48118e13	0.691746
$918$	0	0
$919$	2.65722e13	1.22888	0.614438	−	0.788965i	$-0.289383\pi$
$919$	2.65722e13	1.22888	0.614438	+	0.788965i	$0.289383\pi$
$920$	1.58810e11	0.00730856
$921$	0	0
$922$	−1.37948e13	−0.628677
$923$	2.38879e13	1.08335
$924$	0	0
$925$	−4.62496e12	−0.207716
$926$	3.81625e13	1.70564
$927$	0	0
$928$	1.10990e13	0.491266
$929$	−5.08029e12	−0.223778	−0.111889	−	0.993721i	$-0.535690\pi$
$929$	−5.08029e12	−0.223778	−0.111889	+	0.993721i	$0.535690\pi$
$930$	0	0
$931$	2.05827e12	0.0897903
$932$	2.02507e13	0.879162
$933$	0	0
$934$	−6.52865e12	−0.280713
$935$	−1.12659e13	−0.482075
$936$	0	0
$937$	1.30381e13	0.552567	0.276283	−	0.961076i	$-0.410897\pi$
$937$	1.30381e13	0.552567	0.276283	+	0.961076i	$0.410897\pi$
$938$	−2.66416e12	−0.112369
$939$	0	0
$940$	4.31902e12	0.180430
$941$	−9.26900e12	−0.385372	−0.192686	−	0.981261i	$-0.561720\pi$
$941$	−9.26900e12	−0.385372	−0.192686	+	0.981261i	$0.561720\pi$
$942$	0	0
$943$	1.26850e13	0.522382
$944$	2.49988e13	1.02458
$945$	0	0
$946$	−3.06688e13	−1.24505
$947$	−4.28675e13	−1.73202	−0.866011	−	0.500026i	$-0.833324\pi$
$947$	−4.28675e13	−1.73202	−0.866011	+	0.500026i	$0.833324\pi$
$948$	0	0
$949$	4.51794e12	0.180818
$950$	4.49234e12	0.178944
$951$	0	0
$952$	3.28756e11	0.0129720
$953$	2.73559e13	1.07432	0.537158	−	0.843481i	$-0.319498\pi$
$953$	2.73559e13	1.07432	0.537158	+	0.843481i	$0.319498\pi$
$954$	0	0
$955$	−1.60731e13	−0.625294
$956$	−4.68907e13	−1.81563
$957$	0	0
$958$	1.42590e13	0.546948
$959$	−1.75953e13	−0.671756
$960$	0	0
$961$	3.87769e12	0.146662
$962$	2.75730e13	1.03800
$963$	0	0
$964$	3.99632e13	1.49044
$965$	5.38427e12	0.199873
$966$	0	0
$967$	3.50479e13	1.28897	0.644485	−	0.764617i	$-0.277072\pi$
$967$	3.50479e13	1.28897	0.644485	+	0.764617i	$0.277072\pi$
$968$	3.99358e11	0.0146192
$969$	0	0
$970$	−1.26607e12	−0.0459183
$971$	−4.18449e13	−1.51062	−0.755312	−	0.655366i	$-0.772514\pi$
$971$	−4.18449e13	−1.51062	−0.755312	+	0.655366i	$0.772514\pi$
$972$	0	0
$973$	1.02925e13	0.368139
$974$	−1.37743e13	−0.490405
$975$	0	0
$976$	−1.92005e12	−0.0677312
$977$	−8.72268e12	−0.306284	−0.153142	−	0.988204i	$-0.548939\pi$
$977$	−8.72268e12	−0.306284	−0.153142	+	0.988204i	$0.548939\pi$
$978$	0	0
$979$	−1.09334e13	−0.380394
$980$	1.89337e12	0.0655721
$981$	0	0
$982$	7.97036e13	2.73512
$983$	−3.58281e13	−1.22386	−0.611931	−	0.790911i	$-0.709607\pi$
$983$	−3.58281e13	−1.22386	−0.611931	+	0.790911i	$0.709607\pi$
$984$	0	0
$985$	2.60317e12	0.0881128
$986$	1.33423e13	0.449557
$987$	0	0
$988$	−1.35654e13	−0.452926
$989$	−9.72088e12	−0.323089
$990$	0	0
$991$	2.58270e13	0.850634	0.425317	−	0.905044i	$-0.360163\pi$
$991$	2.58270e13	0.850634	0.425317	+	0.905044i	$0.360163\pi$
$992$	4.64598e13	1.52326
$993$	0	0
$994$	2.55517e13	0.830198
$995$	−5.87285e12	−0.189953
$996$	0	0
$997$	−5.35364e13	−1.71602	−0.858008	−	0.513636i	$-0.828298\pi$
$997$	−5.35364e13	−1.71602	−0.858008	+	0.513636i	$0.828298\pi$
$998$	6.16943e11	0.0196860
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.c.1.4		4
3.2	odd	2		105.10.a.f.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.f.1.1	✓	4	3.2	odd	2
315.10.a.c.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+32.2102 q^{2} +525.499 q^{4} +625.000 q^{5} -2401.00 q^{7} +434.806 q^{8} +20131.4 q^{10} +57240.1 q^{11} -72300.8 q^{13} -77336.8 q^{14} -255050. q^{16} -314910. q^{17} +357041. q^{19} +328437. q^{20} +1.84372e6 q^{22} +584389. q^{23} +390625. q^{25} -2.32882e6 q^{26} -1.26172e6 q^{28} -1.31538e6 q^{29} -5.50612e6 q^{31} -8.43785e6 q^{32} -1.01433e7 q^{34} -1.50062e6 q^{35} -1.18399e7 q^{37} +1.15004e7 q^{38} +271753. q^{40} +2.17065e7 q^{41} -1.66343e7 q^{43} +3.00796e7 q^{44} +1.88233e7 q^{46} +1.31502e7 q^{47} +5.76480e6 q^{49} +1.25821e7 q^{50} -3.79940e7 q^{52} -5.91955e7 q^{53} +3.57750e7 q^{55} -1.04397e6 q^{56} -4.23687e7 q^{58} -9.80152e7 q^{59} +7.52813e6 q^{61} -1.77353e8 q^{62} -1.41199e8 q^{64} -4.51880e7 q^{65} +3.44489e7 q^{67} -1.65485e8 q^{68} -4.83355e7 q^{70} -3.30396e8 q^{71} -6.24882e7 q^{73} -3.81365e8 q^{74} +1.87625e8 q^{76} -1.37433e8 q^{77} +1.05279e7 q^{79} -1.59406e8 q^{80} +6.99170e8 q^{82} -2.19370e7 q^{83} -1.96819e8 q^{85} -5.35793e8 q^{86} +2.48883e7 q^{88} -1.91010e8 q^{89} +1.73594e8 q^{91} +3.07096e8 q^{92} +4.23572e8 q^{94} +2.23151e8 q^{95} -6.28904e7 q^{97} +1.85686e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 424 q^{4} + 2500 q^{5} - 9604 q^{7} - 96 q^{8} - 5000 q^{10} - 9832 q^{11} - 68264 q^{13} + 19208 q^{14} - 290784 q^{16} - 86272 q^{17} + 807672 q^{19} + 265000 q^{20} + 4293352 q^{22} - 683032 q^{23}+ \cdots - 46118408 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 424 * q^4 + 2500 * q^5 - 9604 * q^7 - 96 * q^8 - 5000 * q^10 - 9832 * q^11 - 68264 * q^13 + 19208 * q^14 - 290784 * q^16 - 86272 * q^17 + 807672 * q^19 + 265000 * q^20 + 4293352 * q^22 - 683032 * q^23 + 1562500 * q^25 - 4583544 * q^26 - 1018024 * q^28 - 5029312 * q^29 + 9802992 * q^31 - 497280 * q^32 + 429792 * q^34 - 6002500 * q^35 - 4536432 * q^37 - 21534872 * q^38 - 60000 * q^40 - 11081000 * q^41 - 34442248 * q^43 - 35679024 * q^44 + 67652080 * q^46 + 39230168 * q^47 + 23059204 * q^49 - 3125000 * q^50 - 14778976 * q^52 - 76012528 * q^53 - 6145000 * q^55 + 230496 * q^56 + 57813344 * q^58 - 74595424 * q^59 + 101246592 * q^61 - 392291880 * q^62 - 276909952 * q^64 - 42665000 * q^65 - 45694264 * q^67 - 345896144 * q^68 + 12005000 * q^70 - 447187600 * q^71 + 141527624 * q^73 - 480388368 * q^74 + 1081835504 * q^76 + 23606632 * q^77 + 789659136 * q^79 - 181740000 * q^80 + 937199856 * q^82 - 586282352 * q^83 - 53920000 * q^85 - 132711648 * q^86 + 1042925024 * q^88 - 1050230088 * q^89 + 163901864 * q^91 - 44316096 * q^92 - 1211140960 * q^94 + 504795000 * q^95 + 788061288 * q^97 - 46118408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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