LMFDB - Embedded newform 315.8.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	315.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:	$98.4012830275$
Analytic rank:	$0$
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} - 2x^{3} - 277x^{2} + 80x + 15348 $ x^4 - 2x^3 - 277x^2 + 80*x + 15348
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-12.8099$ 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)

$f(q)$	$=$	\(q+3.41438 q^{2} -116.342 q^{4} -125.000 q^{5} +343.000 q^{7} -834.277 q^{8} -426.798 q^{10} +3379.63 q^{11} -14159.0 q^{13} +1171.13 q^{14} +12043.2 q^{16} +19538.0 q^{17} -38098.7 q^{19} +14542.8 q^{20} +11539.4 q^{22} -103511. q^{23} +15625.0 q^{25} -48344.1 q^{26} -39905.3 q^{28} -59794.5 q^{29} +68349.9 q^{31} +147908. q^{32} +66710.3 q^{34} -42875.0 q^{35} +266512. q^{37} -130084. q^{38} +104285. q^{40} -655437. q^{41} -775223. q^{43} -393193. q^{44} -353426. q^{46} +119729. q^{47} +117649. q^{49} +53349.7 q^{50} +1.64728e6 q^{52} -224102. q^{53} -422454. q^{55} -286157. q^{56} -204161. q^{58} +409602. q^{59} +1.42274e6 q^{61} +233373. q^{62} -1.03652e6 q^{64} +1.76987e6 q^{65} +4.22037e6 q^{67} -2.27309e6 q^{68} -146392. q^{70} -2.17619e6 q^{71} +4.43912e6 q^{73} +909973. q^{74} +4.43248e6 q^{76} +1.15921e6 q^{77} +1.63306e6 q^{79} -1.50540e6 q^{80} -2.23791e6 q^{82} +3.49903e6 q^{83} -2.44225e6 q^{85} -2.64691e6 q^{86} -2.81955e6 q^{88} -2.03583e6 q^{89} -4.85652e6 q^{91} +1.20427e7 q^{92} +408799. q^{94} +4.76234e6 q^{95} +1.27049e7 q^{97} +401698. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) $ 4 * q - 11 * q^2 + 141 * q^4 - 500 * q^5 + 1372 * q^7 - 2133 * q^8 + 1375 * q^10 + 2708 * q^11 - 2212 * q^13 - 3773 * q^14 - 9599 * q^16 + 17016 * q^17 + 32668 * q^19 - 17625 * q^20 - 7196 * q^22 - 87696 * q^23 + 62500 * q^25 - 67354 * q^26 + 48363 * q^28 - 200488 * q^29 + 258852 * q^31 - 39445 * q^32 + 225518 * q^34 - 171500 * q^35 + 666856 * q^37 - 846556 * q^38 + 266625 * q^40 - 833000 * q^41 + 1051896 * q^43 - 1918236 * q^44 + 1786160 * q^46 - 529824 * q^47 + 470596 * q^49 - 171875 * q^50 + 211718 * q^52 - 791764 * q^53 - 338500 * q^55 - 731619 * q^56 + 3500242 * q^58 + 3825904 * q^59 + 212432 * q^61 - 882400 * q^62 - 1515063 * q^64 + 276500 * q^65 + 5165272 * q^67 - 7423202 * q^68 + 471625 * q^70 + 623260 * q^71 - 2083228 * q^73 - 1059642 * q^74 + 6990964 * q^76 + 928844 * q^77 - 1325016 * q^79 + 1199875 * q^80 + 11798982 * q^82 - 11447616 * q^83 - 2127000 * q^85 - 7525948 * q^86 + 19484572 * q^88 + 5805808 * q^89 - 758716 * q^91 + 2716288 * q^92 - 15552560 * q^94 - 4083500 * q^95 + 5857516 * q^97 - 1294139 * 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 $

$2$	3.41438	0.301791	0.150896	−	0.988550i	$-0.451784\pi$
$2$	3.41438	0.301791	0.150896	+	0.988550i	$0.451784\pi$
$3$	0	0
$3$	0	0
$4$	−116.342	−0.908922
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−834.277	−0.576096
$9$	0	0
$10$	−426.798	−0.134965
$11$	3379.63	0.765588	0.382794	−	0.923834i	$-0.374962\pi$
$11$	3379.63	0.765588	0.382794	+	0.923834i	$0.374962\pi$
$12$	0	0
$13$	−14159.0	−1.78743	−0.893716	−	0.448633i	$-0.851911\pi$
$13$	−14159.0	−1.78743	−0.893716	+	0.448633i	$0.851911\pi$
$14$	1171.13	0.114066
$15$	0	0
$16$	12043.2	0.735061
$17$	19538.0	0.964516	0.482258	−	0.876029i	$-0.339817\pi$
$17$	19538.0	0.964516	0.482258	+	0.876029i	$0.339817\pi$
$18$	0	0
$19$	−38098.7	−1.27430	−0.637152	−	0.770738i	$-0.719888\pi$
$19$	−38098.7	−1.27430	−0.637152	+	0.770738i	$0.719888\pi$
$20$	14542.8	0.406482
$21$	0	0
$22$	11539.4	0.231048
$23$	−103511.	−1.77394	−0.886969	−	0.461828i	$-0.847194\pi$
$23$	−103511.	−1.77394	−0.886969	+	0.461828i	$0.847194\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−48344.1	−0.539432
$27$	0	0
$28$	−39905.3	−0.343540
$29$	−59794.5	−0.455269	−0.227635	−	0.973747i	$-0.573099\pi$
$29$	−59794.5	−0.455269	−0.227635	+	0.973747i	$0.573099\pi$
$30$	0	0
$31$	68349.9	0.412071	0.206035	−	0.978545i	$-0.433944\pi$
$31$	68349.9	0.412071	0.206035	+	0.978545i	$0.433944\pi$
$32$	147908.	0.797931
$33$	0	0
$34$	66710.3	0.291083
$35$	−42875.0	−0.169031
$36$	0	0
$37$	266512.	0.864989	0.432494	−	0.901637i	$-0.357634\pi$
$37$	266512.	0.864989	0.432494	+	0.901637i	$0.357634\pi$
$38$	−130084.	−0.384574
$39$	0	0
$40$	104285.	0.257638
$41$	−655437.	−1.48521	−0.742604	−	0.669730i	$-0.766410\pi$
$41$	−655437.	−1.48521	−0.742604	+	0.669730i	$0.766410\pi$
$42$	0	0
$43$	−775223.	−1.48692	−0.743459	−	0.668782i	$-0.766816\pi$
$43$	−775223.	−1.48692	−0.743459	+	0.668782i	$0.766816\pi$
$44$	−393193.	−0.695860
$45$	0	0
$46$	−353426.	−0.535360
$47$	119729.	0.168211	0.0841057	−	0.996457i	$-0.473197\pi$
$47$	119729.	0.168211	0.0841057	+	0.996457i	$0.473197\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	53349.7	0.0603583
$51$	0	0
$52$	1.64728e6	1.62464
$53$	−224102.	−0.206767	−0.103383	−	0.994642i	$-0.532967\pi$
$53$	−224102.	−0.206767	−0.103383	+	0.994642i	$0.532967\pi$
$54$	0	0
$55$	−422454.	−0.342382
$56$	−286157.	−0.217744
$57$	0	0
$58$	−204161.	−0.137396
$59$	409602.	0.259645	0.129823	−	0.991537i	$-0.458559\pi$
$59$	409602.	0.259645	0.129823	+	0.991537i	$0.458559\pi$
$60$	0	0
$61$	1.42274e6	0.802548	0.401274	−	0.915958i	$-0.368568\pi$
$61$	1.42274e6	0.802548	0.401274	+	0.915958i	$0.368568\pi$
$62$	233373.	0.124359
$63$	0	0
$64$	−1.03652e6	−0.494252
$65$	1.76987e6	0.799364
$66$	0	0
$67$	4.22037e6	1.71431	0.857153	−	0.515061i	$-0.172231\pi$
$67$	4.22037e6	1.71431	0.857153	+	0.515061i	$0.172231\pi$
$68$	−2.27309e6	−0.876670
$69$	0	0
$70$	−146392.	−0.0510121
$71$	−2.17619e6	−0.721595	−0.360798	−	0.932644i	$-0.617495\pi$
$71$	−2.17619e6	−0.721595	−0.360798	+	0.932644i	$0.617495\pi$
$72$	0	0
$73$	4.43912e6	1.33557	0.667785	−	0.744354i	$-0.267242\pi$
$73$	4.43912e6	1.33557	0.667785	+	0.744354i	$0.267242\pi$
$74$	909973.	0.261046
$75$	0	0
$76$	4.43248e6	1.15824
$77$	1.15921e6	0.289365
$78$	0	0
$79$	1.63306e6	0.372656	0.186328	−	0.982488i	$-0.440341\pi$
$79$	1.63306e6	0.372656	0.186328	+	0.982488i	$0.440341\pi$
$80$	−1.50540e6	−0.328729
$81$	0	0
$82$	−2.23791e6	−0.448223
$83$	3.49903e6	0.671698	0.335849	−	0.941916i	$-0.390977\pi$
$83$	3.49903e6	0.671698	0.335849	+	0.941916i	$0.390977\pi$
$84$	0	0
$85$	−2.44225e6	−0.431345
$86$	−2.64691e6	−0.448739
$87$	0	0
$88$	−2.81955e6	−0.441053
$89$	−2.03583e6	−0.306110	−0.153055	−	0.988218i	$-0.548911\pi$
$89$	−2.03583e6	−0.306110	−0.153055	+	0.988218i	$0.548911\pi$
$90$	0	0
$91$	−4.85652e6	−0.675586
$92$	1.20427e7	1.61237
$93$	0	0
$94$	408799.	0.0507648
$95$	4.76234e6	0.569886
$96$	0	0
$97$	1.27049e7	1.41342	0.706709	−	0.707505i	$-0.250179\pi$
$97$	1.27049e7	1.41342	0.706709	+	0.707505i	$0.250179\pi$
$98$	401698.	0.0431131
$99$	0	0
$100$	−1.81784e6	−0.181784
$101$	1.86609e7	1.80222	0.901110	−	0.433590i	$-0.142753\pi$
$101$	1.86609e7	1.80222	0.901110	+	0.433590i	$0.142753\pi$
$102$	0	0
$103$	5.30772e6	0.478606	0.239303	−	0.970945i	$-0.423081\pi$
$103$	5.30772e6	0.478606	0.239303	+	0.970945i	$0.423081\pi$
$104$	1.18125e7	1.02973
$105$	0	0
$106$	−765171.	−0.0624005
$107$	−1.74818e7	−1.37957	−0.689784	−	0.724015i	$-0.742295\pi$
$107$	−1.74818e7	−1.37957	−0.689784	+	0.724015i	$0.742295\pi$
$108$	0	0
$109$	467681.	0.0345905	0.0172952	−	0.999850i	$-0.494494\pi$
$109$	467681.	0.0345905	0.0172952	+	0.999850i	$0.494494\pi$
$110$	−1.44242e6	−0.103328
$111$	0	0
$112$	4.13083e6	0.277827
$113$	1.45523e7	0.948763	0.474382	−	0.880319i	$-0.342672\pi$
$113$	1.45523e7	0.948763	0.474382	+	0.880319i	$0.342672\pi$
$114$	0	0
$115$	1.29389e7	0.793329
$116$	6.95661e6	0.413804
$117$	0	0
$118$	1.39854e6	0.0783587
$119$	6.70154e6	0.364553
$120$	0	0
$121$	−8.06524e6	−0.413874
$122$	4.85777e6	0.242202
$123$	0	0
$124$	−7.95196e6	−0.374540
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	2.04101e6	0.0884161	0.0442081	−	0.999022i	$-0.485924\pi$
$127$	2.04101e6	0.0884161	0.0442081	+	0.999022i	$0.485924\pi$
$128$	−2.24713e7	−0.947093
$129$	0	0
$130$	6.04301e6	0.241241
$131$	9.14330e6	0.355348	0.177674	−	0.984089i	$-0.443143\pi$
$131$	9.14330e6	0.355348	0.177674	+	0.984089i	$0.443143\pi$
$132$	0	0
$133$	−1.30679e7	−0.481642
$134$	1.44099e7	0.517363
$135$	0	0
$136$	−1.63001e7	−0.555654
$137$	−3.62725e7	−1.20519	−0.602596	−	0.798047i	$-0.705867\pi$
$137$	−3.62725e7	−1.20519	−0.602596	+	0.798047i	$0.705867\pi$
$138$	0	0
$139$	−2.28823e7	−0.722682	−0.361341	−	0.932434i	$-0.617681\pi$
$139$	−2.28823e7	−0.722682	−0.361341	+	0.932434i	$0.617681\pi$
$140$	4.98816e6	0.153636
$141$	0	0
$142$	−7.43036e6	−0.217771
$143$	−4.78521e7	−1.36844
$144$	0	0
$145$	7.47431e6	0.203603
$146$	1.51568e7	0.403064
$147$	0	0
$148$	−3.10065e7	−0.786207
$149$	7.18870e7	1.78032	0.890161	−	0.455646i	$-0.150592\pi$
$149$	7.18870e7	1.78032	0.890161	+	0.455646i	$0.150592\pi$
$150$	0	0
$151$	6.39974e7	1.51267	0.756333	−	0.654187i	$-0.226989\pi$
$151$	6.39974e7	1.51267	0.756333	+	0.654187i	$0.226989\pi$
$152$	3.17849e7	0.734122
$153$	0	0
$154$	3.95800e6	0.0873280
$155$	−8.54374e6	−0.184284
$156$	0	0
$157$	6.00497e6	0.123840	0.0619202	−	0.998081i	$-0.480278\pi$
$157$	6.00497e6	0.123840	0.0619202	+	0.998081i	$0.480278\pi$
$158$	5.57589e6	0.112464
$159$	0	0
$160$	−1.84885e7	−0.356846
$161$	−3.55042e7	−0.670486
$162$	0	0
$163$	9.83028e7	1.77791	0.888954	−	0.457997i	$-0.151433\pi$
$163$	9.83028e7	1.77791	0.888954	+	0.457997i	$0.151433\pi$
$164$	7.62549e7	1.34994
$165$	0	0
$166$	1.19470e7	0.202713
$167$	−5.80791e7	−0.964966	−0.482483	−	0.875905i	$-0.660265\pi$
$167$	−5.80791e7	−0.964966	−0.482483	+	0.875905i	$0.660265\pi$
$168$	0	0
$169$	1.37728e8	2.19491
$170$	−8.33878e6	−0.130176
$171$	0	0
$172$	9.01909e7	1.35149
$173$	2.31746e7	0.340291	0.170146	−	0.985419i	$-0.445576\pi$
$173$	2.31746e7	0.340291	0.170146	+	0.985419i	$0.445576\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	4.07017e7	0.562754
$177$	0	0
$178$	−6.95111e6	−0.0923814
$179$	−3.77902e7	−0.492486	−0.246243	−	0.969208i	$-0.579196\pi$
$179$	−3.77902e7	−0.492486	−0.246243	+	0.969208i	$0.579196\pi$
$180$	0	0
$181$	−9.09325e6	−0.113984	−0.0569920	−	0.998375i	$-0.518151\pi$
$181$	−9.09325e6	−0.113984	−0.0569920	+	0.998375i	$0.518151\pi$
$182$	−1.65820e7	−0.203886
$183$	0	0
$184$	8.63567e7	1.02196
$185$	−3.33140e7	−0.386835
$186$	0	0
$187$	6.60314e7	0.738422
$188$	−1.39295e7	−0.152891
$189$	0	0
$190$	1.62605e7	0.171987
$191$	5.34900e7	0.555464	0.277732	−	0.960659i	$-0.410417\pi$
$191$	5.34900e7	0.555464	0.277732	+	0.960659i	$0.410417\pi$
$192$	0	0
$193$	−8.24940e7	−0.825984	−0.412992	−	0.910735i	$-0.635516\pi$
$193$	−8.24940e7	−0.825984	−0.412992	+	0.910735i	$0.635516\pi$
$194$	4.33794e7	0.426557
$195$	0	0
$196$	−1.36875e7	−0.129846
$197$	−7.55328e7	−0.703888	−0.351944	−	0.936021i	$-0.614479\pi$
$197$	−7.55328e7	−0.703888	−0.351944	+	0.936021i	$0.614479\pi$
$198$	0	0
$199$	−1.05546e6	−0.00949412	−0.00474706	−	0.999989i	$-0.501511\pi$
$199$	−1.05546e6	−0.00949412	−0.00474706	+	0.999989i	$0.501511\pi$
$200$	−1.30356e7	−0.115219
$201$	0	0
$202$	6.37154e7	0.543895
$203$	−2.05095e7	−0.172076
$204$	0	0
$205$	8.19296e7	0.664205
$206$	1.81226e7	0.144439
$207$	0	0
$208$	−1.70520e8	−1.31387
$209$	−1.28760e8	−0.975593
$210$	0	0
$211$	1.14474e7	0.0838916	0.0419458	−	0.999120i	$-0.486644\pi$
$211$	1.14474e7	0.0838916	0.0419458	+	0.999120i	$0.486644\pi$
$212$	2.60725e7	0.187935
$213$	0	0
$214$	−5.96896e7	−0.416342
$215$	9.69028e7	0.664970
$216$	0	0
$217$	2.34440e7	0.155748
$218$	1.59684e6	0.0104391
$219$	0	0
$220$	4.91492e7	0.311198
$221$	−2.76638e8	−1.72401
$222$	0	0
$223$	2.81598e8	1.70044	0.850222	−	0.526425i	$-0.176468\pi$
$223$	2.81598e8	1.70044	0.850222	+	0.526425i	$0.176468\pi$
$224$	5.07323e7	0.301590
$225$	0	0
$226$	4.96872e7	0.286329
$227$	4.00301e7	0.227141	0.113571	−	0.993530i	$-0.463771\pi$
$227$	4.00301e7	0.227141	0.113571	+	0.993530i	$0.463771\pi$
$228$	0	0
$229$	−1.24592e8	−0.685591	−0.342796	−	0.939410i	$-0.611374\pi$
$229$	−1.24592e8	−0.685591	−0.342796	+	0.939410i	$0.611374\pi$
$230$	4.41782e7	0.239420
$231$	0	0
$232$	4.98852e7	0.262279
$233$	3.21485e8	1.66500	0.832500	−	0.554025i	$-0.186909\pi$
$233$	3.21485e8	1.66500	0.832500	+	0.554025i	$0.186909\pi$
$234$	0	0
$235$	−1.49661e7	−0.0752264
$236$	−4.76540e7	−0.235997
$237$	0	0
$238$	2.28816e7	0.110019
$239$	2.95076e8	1.39811	0.699054	−	0.715068i	$-0.253605\pi$
$239$	2.95076e8	1.39811	0.699054	+	0.715068i	$0.253605\pi$
$240$	0	0
$241$	1.00887e8	0.464273	0.232137	−	0.972683i	$-0.425428\pi$
$241$	1.00887e8	0.464273	0.232137	+	0.972683i	$0.425428\pi$
$242$	−2.75378e7	−0.124904
$243$	0	0
$244$	−1.65524e8	−0.729453
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	5.39439e8	2.27773
$248$	−5.70227e7	−0.237393
$249$	0	0
$250$	−6.66871e6	−0.0269931
$251$	9.30364e7	0.371360	0.185680	−	0.982610i	$-0.440551\pi$
$251$	9.30364e7	0.371360	0.185680	+	0.982610i	$0.440551\pi$
$252$	0	0
$253$	−3.49829e8	−1.35811
$254$	6.96877e6	0.0266832
$255$	0	0
$256$	5.59493e7	0.208428
$257$	−3.64584e8	−1.33978	−0.669888	−	0.742462i	$-0.733658\pi$
$257$	−3.64584e8	−1.33978	−0.669888	+	0.742462i	$0.733658\pi$
$258$	0	0
$259$	9.14136e7	0.326935
$260$	−2.05910e8	−0.726559
$261$	0	0
$262$	3.12187e7	0.107241
$263$	−1.11633e8	−0.378396	−0.189198	−	0.981939i	$-0.560589\pi$
$263$	−1.11633e8	−0.378396	−0.189198	+	0.981939i	$0.560589\pi$
$264$	0	0
$265$	2.80128e7	0.0924690
$266$	−4.46187e7	−0.145355
$267$	0	0
$268$	−4.91006e8	−1.55817
$269$	1.53396e8	0.480487	0.240244	−	0.970713i	$-0.422773\pi$
$269$	1.53396e8	0.480487	0.240244	+	0.970713i	$0.422773\pi$
$270$	0	0
$271$	−4.17128e8	−1.27314	−0.636571	−	0.771218i	$-0.719648\pi$
$271$	−4.17128e8	−1.27314	−0.636571	+	0.771218i	$0.719648\pi$
$272$	2.35301e8	0.708978
$273$	0	0
$274$	−1.23848e8	−0.363717
$275$	5.28068e7	0.153118
$276$	0	0
$277$	−4.05683e8	−1.14685	−0.573426	−	0.819257i	$-0.694386\pi$
$277$	−4.05683e8	−1.14685	−0.573426	+	0.819257i	$0.694386\pi$
$278$	−7.81288e7	−0.218099
$279$	0	0
$280$	3.57696e7	0.0973781
$281$	1.50774e8	0.405371	0.202686	−	0.979244i	$-0.435033\pi$
$281$	1.50774e8	0.405371	0.202686	+	0.979244i	$0.435033\pi$
$282$	0	0
$283$	−3.69597e7	−0.0969338	−0.0484669	−	0.998825i	$-0.515434\pi$
$283$	−3.69597e7	−0.0969338	−0.0484669	+	0.998825i	$0.515434\pi$
$284$	2.53183e8	0.655874
$285$	0	0
$286$	−1.63385e8	−0.412983
$287$	−2.24815e8	−0.561356
$288$	0	0
$289$	−2.86042e7	−0.0697088
$290$	2.55202e7	0.0614455
$291$	0	0
$292$	−5.16456e8	−1.21393
$293$	6.00426e8	1.39451	0.697257	−	0.716822i	$-0.254404\pi$
$293$	6.00426e8	1.39451	0.697257	+	0.716822i	$0.254404\pi$
$294$	0	0
$295$	−5.12003e7	−0.116117
$296$	−2.22345e8	−0.498317
$297$	0	0
$298$	2.45450e8	0.537286
$299$	1.46561e9	3.17079
$300$	0	0
$301$	−2.65901e8	−0.562002
$302$	2.18511e8	0.456510
$303$	0	0
$304$	−4.58832e8	−0.936691
$305$	−1.77842e8	−0.358910
$306$	0	0
$307$	9.08890e8	1.79278	0.896390	−	0.443267i	$-0.146181\pi$
$307$	9.08890e8	1.79278	0.896390	+	0.443267i	$0.146181\pi$
$308$	−1.34865e8	−0.263010
$309$	0	0
$310$	−2.91716e7	−0.0556153
$311$	−6.45862e8	−1.21753	−0.608763	−	0.793352i	$-0.708334\pi$
$311$	−6.45862e8	−1.21753	−0.608763	+	0.793352i	$0.708334\pi$
$312$	0	0
$313$	−8.96603e8	−1.65270	−0.826352	−	0.563154i	$-0.809588\pi$
$313$	−8.96603e8	−1.65270	−0.826352	+	0.563154i	$0.809588\pi$
$314$	2.05033e7	0.0373740
$315$	0	0
$316$	−1.89994e8	−0.338715
$317$	−8.47295e8	−1.49392	−0.746959	−	0.664870i	$-0.768487\pi$
$317$	−8.47295e8	−1.49392	−0.746959	+	0.664870i	$0.768487\pi$
$318$	0	0
$319$	−2.02084e8	−0.348549
$320$	1.29565e8	0.221036
$321$	0	0
$322$	−1.21225e8	−0.202347
$323$	−7.44374e8	−1.22909
$324$	0	0
$325$	−2.21234e8	−0.357486
$326$	3.35643e8	0.536557
$327$	0	0
$328$	5.46816e8	0.855623
$329$	4.10669e7	0.0635779
$330$	0	0
$331$	7.42401e8	1.12523	0.562614	−	0.826720i	$-0.309796\pi$
$331$	7.42401e8	1.12523	0.562614	+	0.826720i	$0.309796\pi$
$332$	−4.07084e8	−0.610521
$333$	0	0
$334$	−1.98304e8	−0.291219
$335$	−5.27546e8	−0.766661
$336$	0	0
$337$	−7.24512e8	−1.03120	−0.515598	−	0.856831i	$-0.672430\pi$
$337$	−7.24512e8	−1.03120	−0.515598	+	0.856831i	$0.672430\pi$
$338$	4.70254e8	0.662406
$339$	0	0
$340$	2.84137e8	0.392059
$341$	2.30998e8	0.315477
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	6.46750e8	0.856608
$345$	0	0
$346$	7.91269e7	0.102697
$347$	−5.09846e8	−0.655068	−0.327534	−	0.944839i	$-0.606218\pi$
$347$	−5.09846e8	−0.655068	−0.327534	+	0.944839i	$0.606218\pi$
$348$	0	0
$349$	−8.76052e8	−1.10317	−0.551583	−	0.834120i	$-0.685976\pi$
$349$	−8.76052e8	−1.10317	−0.551583	+	0.834120i	$0.685976\pi$
$350$	1.82989e7	0.0228133
$351$	0	0
$352$	4.99874e8	0.610887
$353$	−1.26670e9	−1.53272	−0.766358	−	0.642413i	$-0.777933\pi$
$353$	−1.26670e9	−1.53272	−0.766358	+	0.642413i	$0.777933\pi$
$354$	0	0
$355$	2.72024e8	0.322707
$356$	2.36853e8	0.278230
$357$	0	0
$358$	−1.29030e8	−0.148628
$359$	−1.66558e9	−1.89992	−0.949961	−	0.312369i	$-0.898878\pi$
$359$	−1.66558e9	−1.89992	−0.949961	+	0.312369i	$0.898878\pi$
$360$	0	0
$361$	5.57643e8	0.623851
$362$	−3.10478e7	−0.0343994
$363$	0	0
$364$	5.65018e8	0.614055
$365$	−5.54890e8	−0.597285
$366$	0	0
$367$	1.67575e9	1.76961	0.884807	−	0.465957i	$-0.154290\pi$
$367$	1.67575e9	1.76961	0.884807	+	0.465957i	$0.154290\pi$
$368$	−1.24661e9	−1.30395
$369$	0	0
$370$	−1.13747e8	−0.116743
$371$	−7.68672e7	−0.0781506
$372$	0	0
$373$	1.47694e9	1.47360	0.736802	−	0.676109i	$-0.236335\pi$
$373$	1.47694e9	1.47360	0.736802	+	0.676109i	$0.236335\pi$
$374$	2.25456e8	0.222850
$375$	0	0
$376$	−9.98868e7	−0.0969060
$377$	8.46628e8	0.813763
$378$	0	0
$379$	−1.64279e9	−1.55005	−0.775024	−	0.631932i	$-0.782262\pi$
$379$	−1.64279e9	−1.55005	−0.775024	+	0.631932i	$0.782262\pi$
$380$	−5.54061e8	−0.517982
$381$	0	0
$382$	1.82635e8	0.167634
$383$	1.89784e9	1.72609	0.863044	−	0.505129i	$-0.168555\pi$
$383$	1.89784e9	1.72609	0.863044	+	0.505129i	$0.168555\pi$
$384$	0	0
$385$	−1.44902e8	−0.129408
$386$	−2.81666e8	−0.249275
$387$	0	0
$388$	−1.47811e9	−1.28469
$389$	−1.05432e9	−0.908131	−0.454065	−	0.890968i	$-0.650027\pi$
$389$	−1.05432e9	−0.908131	−0.454065	+	0.890968i	$0.650027\pi$
$390$	0	0
$391$	−2.02240e9	−1.71099
$392$	−9.81518e7	−0.0822995
$393$	0	0
$394$	−2.57898e8	−0.212428
$395$	−2.04133e8	−0.166657
$396$	0	0
$397$	2.96848e8	0.238104	0.119052	−	0.992888i	$-0.462014\pi$
$397$	2.96848e8	0.238104	0.119052	+	0.992888i	$0.462014\pi$
$398$	−3.60373e6	−0.00286525
$399$	0	0
$400$	1.88176e8	0.147012
$401$	7.79941e8	0.604028	0.302014	−	0.953304i	$-0.402341\pi$
$401$	7.79941e8	0.604028	0.302014	+	0.953304i	$0.402341\pi$
$402$	0	0
$403$	−9.67763e8	−0.736549
$404$	−2.17105e9	−1.63808
$405$	0	0
$406$	−7.00273e7	−0.0519310
$407$	9.00713e8	0.662225
$408$	0	0
$409$	8.75152e8	0.632488	0.316244	−	0.948678i	$-0.397578\pi$
$409$	8.75152e8	0.632488	0.316244	+	0.948678i	$0.397578\pi$
$410$	2.79739e8	0.200452
$411$	0	0
$412$	−6.17511e8	−0.435015
$413$	1.40494e8	0.0981367
$414$	0	0
$415$	−4.37378e8	−0.300392
$416$	−2.09422e9	−1.42625
$417$	0	0
$418$	−4.39635e8	−0.294426
$419$	−1.07058e9	−0.711000	−0.355500	−	0.934676i	$-0.615689\pi$
$419$	−1.07058e9	−0.711000	−0.355500	+	0.934676i	$0.615689\pi$
$420$	0	0
$421$	2.54184e9	1.66020	0.830100	−	0.557614i	$-0.188283\pi$
$421$	2.54184e9	1.66020	0.830100	+	0.557614i	$0.188283\pi$
$422$	3.90858e7	0.0253178
$423$	0	0
$424$	1.86963e8	0.119118
$425$	3.05282e8	0.192903
$426$	0	0
$427$	4.87999e8	0.303335
$428$	2.03387e9	1.25392
$429$	0	0
$430$	3.30863e8	0.200682
$431$	−8.91747e8	−0.536502	−0.268251	−	0.963349i	$-0.586446\pi$
$431$	−8.91747e8	−0.536502	−0.268251	+	0.963349i	$0.586446\pi$
$432$	0	0
$433$	−1.03966e9	−0.615440	−0.307720	−	0.951477i	$-0.599566\pi$
$433$	−1.03966e9	−0.615440	−0.307720	+	0.951477i	$0.599566\pi$
$434$	8.00468e7	0.0470035
$435$	0	0
$436$	−5.44109e7	−0.0314400
$437$	3.94363e9	2.26054
$438$	0	0
$439$	−1.55931e9	−0.879641	−0.439821	−	0.898086i	$-0.644958\pi$
$439$	−1.55931e9	−0.879641	−0.439821	+	0.898086i	$0.644958\pi$
$440$	3.52444e8	0.197245
$441$	0	0
$442$	−9.44548e8	−0.520291
$443$	2.79970e8	0.153002	0.0765012	−	0.997069i	$-0.475625\pi$
$443$	2.79970e8	0.153002	0.0765012	+	0.997069i	$0.475625\pi$
$444$	0	0
$445$	2.54479e8	0.136897
$446$	9.61482e8	0.513179
$447$	0	0
$448$	−3.55527e8	−0.186810
$449$	−6.18452e8	−0.322436	−0.161218	−	0.986919i	$-0.551542\pi$
$449$	−6.18452e8	−0.322436	−0.161218	+	0.986919i	$0.551542\pi$
$450$	0	0
$451$	−2.21514e9	−1.13706
$452$	−1.69305e9	−0.862352
$453$	0	0
$454$	1.36678e8	0.0685494
$455$	6.07065e8	0.302131
$456$	0	0
$457$	9.07329e8	0.444691	0.222345	−	0.974968i	$-0.428629\pi$
$457$	9.07329e8	0.444691	0.222345	+	0.974968i	$0.428629\pi$
$458$	−4.25404e8	−0.206906
$459$	0	0
$460$	−1.50533e9	−0.721075
$461$	2.41729e9	1.14915	0.574573	−	0.818453i	$-0.305168\pi$
$461$	2.41729e9	1.14915	0.574573	+	0.818453i	$0.305168\pi$
$462$	0	0
$463$	−3.14336e8	−0.147184	−0.0735921	−	0.997288i	$-0.523446\pi$
$463$	−3.14336e8	−0.147184	−0.0735921	+	0.997288i	$0.523446\pi$
$464$	−7.20120e8	−0.334651
$465$	0	0
$466$	1.09767e9	0.502483
$467$	−4.74332e7	−0.0215513	−0.0107757	−	0.999942i	$-0.503430\pi$
$467$	−4.74332e7	−0.0215513	−0.0107757	+	0.999942i	$0.503430\pi$
$468$	0	0
$469$	1.44759e9	0.647947
$470$	−5.10999e7	−0.0227027
$471$	0	0
$472$	−3.41722e8	−0.149581
$473$	−2.61997e9	−1.13837
$474$	0	0
$475$	−5.95293e8	−0.254861
$476$	−7.79671e8	−0.331350
$477$	0	0
$478$	1.00750e9	0.421937
$479$	2.08794e9	0.868048	0.434024	−	0.900901i	$-0.357093\pi$
$479$	2.08794e9	0.868048	0.434024	+	0.900901i	$0.357093\pi$
$480$	0	0
$481$	−3.77353e9	−1.54611
$482$	3.44465e8	0.140114
$483$	0	0
$484$	9.38326e8	0.376179
$485$	−1.58811e9	−0.632099
$486$	0	0
$487$	−2.19273e9	−0.860267	−0.430134	−	0.902765i	$-0.641534\pi$
$487$	−2.19273e9	−0.860267	−0.430134	+	0.902765i	$0.641534\pi$
$488$	−1.18696e9	−0.462345
$489$	0	0
$490$	−5.02123e7	−0.0192808
$491$	−2.19027e9	−0.835050	−0.417525	−	0.908666i	$-0.637102\pi$
$491$	−2.19027e9	−0.835050	−0.417525	+	0.908666i	$0.637102\pi$
$492$	0	0
$493$	−1.16827e9	−0.439115
$494$	1.84185e9	0.687400
$495$	0	0
$496$	8.23154e8	0.302897
$497$	−7.46435e8	−0.272737
$498$	0	0
$499$	6.56737e8	0.236614	0.118307	−	0.992977i	$-0.462253\pi$
$499$	6.56737e8	0.236614	0.118307	+	0.992977i	$0.462253\pi$
$500$	2.27230e8	0.0812964
$501$	0	0
$502$	3.17662e8	0.112073
$503$	2.58040e9	0.904065	0.452033	−	0.892001i	$-0.350699\pi$
$503$	2.58040e9	0.904065	0.452033	+	0.892001i	$0.350699\pi$
$504$	0	0
$505$	−2.33261e9	−0.805977
$506$	−1.19445e9	−0.409865
$507$	0	0
$508$	−2.37455e8	−0.0803633
$509$	2.07943e9	0.698928	0.349464	−	0.936950i	$-0.386364\pi$
$509$	2.07943e9	0.698928	0.349464	+	0.936950i	$0.386364\pi$
$510$	0	0
$511$	1.52262e9	0.504798
$512$	3.06735e9	1.00999
$513$	0	0
$514$	−1.24483e9	−0.404333
$515$	−6.63465e8	−0.214039
$516$	0	0
$517$	4.04639e8	0.128781
$518$	3.12121e8	0.0986662
$519$	0	0
$520$	−1.47656e9	−0.460511
$521$	2.82646e9	0.875609	0.437804	−	0.899070i	$-0.355756\pi$
$521$	2.82646e9	0.875609	0.437804	+	0.899070i	$0.355756\pi$
$522$	0	0
$523$	4.81610e9	1.47211	0.736055	−	0.676922i	$-0.236687\pi$
$523$	4.81610e9	1.47211	0.736055	+	0.676922i	$0.236687\pi$
$524$	−1.06375e9	−0.322983
$525$	0	0
$526$	−3.81157e8	−0.114197
$527$	1.33542e9	0.397449
$528$	0	0
$529$	7.30968e9	2.14686
$530$	9.56464e7	0.0279064
$531$	0	0
$532$	1.52034e9	0.437775
$533$	9.28030e9	2.65471
$534$	0	0
$535$	2.18523e9	0.616962
$536$	−3.52096e9	−0.987606
$537$	0	0
$538$	5.23754e8	0.145007
$539$	3.97611e8	0.109370
$540$	0	0
$541$	1.56108e9	0.423871	0.211936	−	0.977284i	$-0.432023\pi$
$541$	1.56108e9	0.423871	0.211936	+	0.977284i	$0.432023\pi$
$542$	−1.42423e9	−0.384223
$543$	0	0
$544$	2.88982e9	0.769618
$545$	−5.84601e7	−0.0154693
$546$	0	0
$547$	−3.82511e8	−0.0999282	−0.0499641	−	0.998751i	$-0.515911\pi$
$547$	−3.82511e8	−0.0999282	−0.0499641	+	0.998751i	$0.515911\pi$
$548$	4.22002e9	1.09543
$549$	0	0
$550$	1.80303e8	0.0462096
$551$	2.27810e9	0.580152
$552$	0	0
$553$	5.60140e8	0.140851
$554$	−1.38516e9	−0.346110
$555$	0	0
$556$	2.66217e9	0.656861
$557$	7.35207e9	1.80267	0.901335	−	0.433122i	$-0.142588\pi$
$557$	7.35207e9	1.80267	0.901335	+	0.433122i	$0.142588\pi$
$558$	0	0
$559$	1.09763e10	2.65776
$560$	−5.16354e8	−0.124248
$561$	0	0
$562$	5.14798e8	0.122338
$563$	−7.91584e9	−1.86947	−0.934733	−	0.355352i	$-0.884361\pi$
$563$	−7.91584e9	−1.86947	−0.934733	+	0.355352i	$0.884361\pi$
$564$	0	0
$565$	−1.81904e9	−0.424300
$566$	−1.26194e8	−0.0292538
$567$	0	0
$568$	1.81555e9	0.415708
$569$	−4.49464e9	−1.02283	−0.511414	−	0.859335i	$-0.670878\pi$
$569$	−4.49464e9	−1.02283	−0.511414	+	0.859335i	$0.670878\pi$
$570$	0	0
$571$	−2.15052e9	−0.483412	−0.241706	−	0.970350i	$-0.577707\pi$
$571$	−2.15052e9	−0.483412	−0.241706	+	0.970350i	$0.577707\pi$
$572$	5.56721e9	1.24380
$573$	0	0
$574$	−7.67604e8	−0.169412
$575$	−1.61736e9	−0.354788
$576$	0	0
$577$	−1.73438e9	−0.375862	−0.187931	−	0.982182i	$-0.560178\pi$
$577$	−1.73438e9	−0.375862	−0.187931	+	0.982182i	$0.560178\pi$
$578$	−9.76656e7	−0.0210375
$579$	0	0
$580$	−8.69577e8	−0.185059
$581$	1.20017e9	0.253878
$582$	0	0
$583$	−7.57385e8	−0.158298
$584$	−3.70345e9	−0.769418
$585$	0	0
$586$	2.05008e9	0.420852
$587$	2.98775e9	0.609692	0.304846	−	0.952402i	$-0.401395\pi$
$587$	2.98775e9	0.609692	0.304846	+	0.952402i	$0.401395\pi$
$588$	0	0
$589$	−2.60405e9	−0.525104
$590$	−1.74817e8	−0.0350431
$591$	0	0
$592$	3.20967e9	0.635819
$593$	−5.56485e9	−1.09588	−0.547938	−	0.836519i	$-0.684587\pi$
$593$	−5.56485e9	−1.09588	−0.547938	+	0.836519i	$0.684587\pi$
$594$	0	0
$595$	−8.37693e8	−0.163033
$596$	−8.36348e9	−1.61817
$597$	0	0
$598$	5.00414e9	0.956919
$599$	5.39399e9	1.02545	0.512727	−	0.858552i	$-0.328635\pi$
$599$	5.39399e9	1.02545	0.512727	+	0.858552i	$0.328635\pi$
$600$	0	0
$601$	−5.88616e8	−0.110604	−0.0553021	−	0.998470i	$-0.517612\pi$
$601$	−5.88616e8	−0.110604	−0.0553021	+	0.998470i	$0.517612\pi$
$602$	−9.07888e8	−0.169607
$603$	0	0
$604$	−7.44558e9	−1.37489
$605$	1.00815e9	0.185090
$606$	0	0
$607$	5.38304e9	0.976938	0.488469	−	0.872581i	$-0.337556\pi$
$607$	5.38304e9	0.976938	0.488469	+	0.872581i	$0.337556\pi$
$608$	−5.63509e9	−1.01681
$609$	0	0
$610$	−6.07222e8	−0.108316
$611$	−1.69523e9	−0.300667
$612$	0	0
$613$	8.87502e9	1.55617	0.778086	−	0.628158i	$-0.216191\pi$
$613$	8.87502e9	1.55617	0.778086	+	0.628158i	$0.216191\pi$
$614$	3.10330e9	0.541046
$615$	0	0
$616$	−9.67106e8	−0.166702
$617$	−5.94996e6	−0.00101980	−0.000509901	−	1.00000i	$-0.500162\pi$
$617$	−5.94996e6	−0.00101980	−0.000509901		1.00000i	$0.500162\pi$
$618$	0	0
$619$	8.21425e9	1.39204	0.696019	−	0.718024i	$-0.254953\pi$
$619$	8.21425e9	1.39204	0.696019	+	0.718024i	$0.254953\pi$
$620$	9.93996e8	0.167500
$621$	0	0
$622$	−2.20522e9	−0.367439
$623$	−6.98291e8	−0.115699
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−3.06134e9	−0.498772
$627$	0	0
$628$	−6.98631e8	−0.112561
$629$	5.20711e9	0.834295
$630$	0	0
$631$	2.53051e9	0.400964	0.200482	−	0.979697i	$-0.435749\pi$
$631$	2.53051e9	0.400964	0.200482	+	0.979697i	$0.435749\pi$
$632$	−1.36243e9	−0.214686
$633$	0	0
$634$	−2.89299e9	−0.450852
$635$	−2.55126e8	−0.0395409
$636$	0	0
$637$	−1.66579e9	−0.255347
$638$	−6.89990e8	−0.105189
$639$	0	0
$640$	2.80891e9	0.423553
$641$	−6.96698e9	−1.04482	−0.522410	−	0.852694i	$-0.674967\pi$
$641$	−6.96698e9	−1.04482	−0.522410	+	0.852694i	$0.674967\pi$
$642$	0	0
$643$	1.21823e10	1.80714	0.903572	−	0.428437i	$-0.140936\pi$
$643$	1.21823e10	1.80714	0.903572	+	0.428437i	$0.140936\pi$
$644$	4.13063e9	0.609419
$645$	0	0
$646$	−2.54158e9	−0.370928
$647$	9.95454e9	1.44496	0.722481	−	0.691391i	$-0.243002\pi$
$647$	9.95454e9	1.44496	0.722481	+	0.691391i	$0.243002\pi$
$648$	0	0
$649$	1.38431e9	0.198781
$650$	−7.55376e8	−0.107886
$651$	0	0
$652$	−1.14367e10	−1.61598
$653$	−2.38576e9	−0.335297	−0.167649	−	0.985847i	$-0.553617\pi$
$653$	−2.38576e9	−0.335297	−0.167649	+	0.985847i	$0.553617\pi$
$654$	0	0
$655$	−1.14291e9	−0.158916
$656$	−7.89358e9	−1.09172
$657$	0	0
$658$	1.40218e8	0.0191873
$659$	−1.98573e8	−0.0270284	−0.0135142	−	0.999909i	$-0.504302\pi$
$659$	−1.98573e8	−0.0270284	−0.0135142	+	0.999909i	$0.504302\pi$
$660$	0	0
$661$	1.68735e9	0.227247	0.113624	−	0.993524i	$-0.463754\pi$
$661$	1.68735e9	0.227247	0.113624	+	0.993524i	$0.463754\pi$
$662$	2.53484e9	0.339584
$663$	0	0
$664$	−2.91916e9	−0.386963
$665$	1.63348e9	0.215397
$666$	0	0
$667$	6.18938e9	0.807620
$668$	6.75704e9	0.877079
$669$	0	0
$670$	−1.80124e9	−0.231372
$671$	4.80834e9	0.614421
$672$	0	0
$673$	−5.02352e9	−0.635266	−0.317633	−	0.948214i	$-0.602888\pi$
$673$	−5.02352e9	−0.635266	−0.317633	+	0.948214i	$0.602888\pi$
$674$	−2.47376e9	−0.311206
$675$	0	0
$676$	−1.60235e10	−1.99501
$677$	1.03907e10	1.28702	0.643510	−	0.765438i	$-0.277477\pi$
$677$	1.03907e10	1.28702	0.643510	+	0.765438i	$0.277477\pi$
$678$	0	0
$679$	4.35778e9	0.534222
$680$	2.03751e9	0.248496
$681$	0	0
$682$	7.88714e8	0.0952082
$683$	−3.61608e9	−0.434276	−0.217138	−	0.976141i	$-0.569672\pi$
$683$	−3.61608e9	−0.434276	−0.217138	+	0.976141i	$0.569672\pi$
$684$	0	0
$685$	4.53407e9	0.538978
$686$	1.37783e8	0.0162952
$687$	0	0
$688$	−9.33619e9	−1.09297
$689$	3.17306e9	0.369582
$690$	0	0
$691$	1.65856e10	1.91231	0.956153	−	0.292869i	$-0.0946100\pi$
$691$	1.65856e10	1.91231	0.956153	+	0.292869i	$0.0946100\pi$
$692$	−2.69618e9	−0.309298
$693$	0	0
$694$	−1.74081e9	−0.197694
$695$	2.86028e9	0.323193
$696$	0	0
$697$	−1.28059e10	−1.43251
$698$	−2.99117e9	−0.332926
$699$	0	0
$700$	−6.23520e8	−0.0687080
$701$	−1.42971e10	−1.56760	−0.783798	−	0.621015i	$-0.786720\pi$
$701$	−1.42971e10	−1.56760	−0.783798	+	0.621015i	$0.786720\pi$
$702$	0	0
$703$	−1.01538e10	−1.10226
$704$	−3.50306e9	−0.378394
$705$	0	0
$706$	−4.32499e9	−0.462561
$707$	6.40069e9	0.681175
$708$	0	0
$709$	4.66391e9	0.491460	0.245730	−	0.969338i	$-0.420972\pi$
$709$	4.66391e9	0.491460	0.245730	+	0.969338i	$0.420972\pi$
$710$	9.28795e8	0.0973903
$711$	0	0
$712$	1.69845e9	0.176349
$713$	−7.07496e9	−0.730989
$714$	0	0
$715$	5.98151e9	0.611984
$716$	4.39659e9	0.447631
$717$	0	0
$718$	−5.68694e9	−0.573380
$719$	−4.82173e9	−0.483784	−0.241892	−	0.970303i	$-0.577768\pi$
$719$	−4.82173e9	−0.483784	−0.241892	+	0.970303i	$0.577768\pi$
$720$	0	0
$721$	1.82055e9	0.180896
$722$	1.90400e9	0.188273
$723$	0	0
$724$	1.05793e9	0.103603
$725$	−9.34289e8	−0.0910539
$726$	0	0
$727$	2.17621e9	0.210054	0.105027	−	0.994469i	$-0.466507\pi$
$727$	2.17621e9	0.210054	0.105027	+	0.994469i	$0.466507\pi$
$728$	4.05168e9	0.389203
$729$	0	0
$730$	−1.89461e9	−0.180256
$731$	−1.51463e10	−1.43416
$732$	0	0
$733$	4.65228e9	0.436317	0.218158	−	0.975913i	$-0.429995\pi$
$733$	4.65228e9	0.436317	0.218158	+	0.975913i	$0.429995\pi$
$734$	5.72166e9	0.534055
$735$	0	0
$736$	−1.53100e10	−1.41548
$737$	1.42633e10	1.31245
$738$	0	0
$739$	4.05338e8	0.0369456	0.0184728	−	0.999829i	$-0.494120\pi$
$739$	4.05338e8	0.0369456	0.0184728	+	0.999829i	$0.494120\pi$
$740$	3.87581e9	0.351603
$741$	0	0
$742$	−2.62454e8	−0.0235852
$743$	−1.08140e10	−0.967223	−0.483611	−	0.875283i	$-0.660675\pi$
$743$	−1.08140e10	−0.967223	−0.483611	+	0.875283i	$0.660675\pi$
$744$	0	0
$745$	−8.98588e9	−0.796184
$746$	5.04282e9	0.444721
$747$	0	0
$748$	−7.68222e9	−0.671168
$749$	−5.99626e9	−0.521428
$750$	0	0
$751$	−1.42070e10	−1.22394	−0.611972	−	0.790880i	$-0.709623\pi$
$751$	−1.42070e10	−1.22394	−0.611972	+	0.790880i	$0.709623\pi$
$752$	1.44192e9	0.123646
$753$	0	0
$754$	2.89071e9	0.245587
$755$	−7.99967e9	−0.676485
$756$	0	0
$757$	3.78427e9	0.317064	0.158532	−	0.987354i	$-0.449324\pi$
$757$	3.78427e9	0.317064	0.158532	+	0.987354i	$0.449324\pi$
$758$	−5.60911e9	−0.467791
$759$	0	0
$760$	−3.97311e9	−0.328309
$761$	−1.01511e10	−0.834966	−0.417483	−	0.908685i	$-0.637088\pi$
$761$	−1.01511e10	−0.834966	−0.417483	+	0.908685i	$0.637088\pi$
$762$	0	0
$763$	1.60414e8	0.0130740
$764$	−6.22314e9	−0.504874
$765$	0	0
$766$	6.47993e9	0.520919
$767$	−5.79954e9	−0.464098
$768$	0	0
$769$	−1.31076e10	−1.03939	−0.519696	−	0.854351i	$-0.673955\pi$
$769$	−1.31076e10	−1.03939	−0.519696	+	0.854351i	$0.673955\pi$
$770$	−4.94750e8	−0.0390543
$771$	0	0
$772$	9.59751e9	0.750755
$773$	1.40370e10	1.09307	0.546533	−	0.837438i	$-0.315947\pi$
$773$	1.40370e10	1.09307	0.546533	+	0.837438i	$0.315947\pi$
$774$	0	0
$775$	1.06797e9	0.0824142
$776$	−1.05994e10	−0.814265
$777$	0	0
$778$	−3.59985e9	−0.274066
$779$	2.49713e10	1.89261
$780$	0	0
$781$	−7.35474e9	−0.552445
$782$	−6.90524e9	−0.516363
$783$	0	0
$784$	1.41687e9	0.105009
$785$	−7.50622e8	−0.0553831
$786$	0	0
$787$	−3.87288e9	−0.283219	−0.141610	−	0.989923i	$-0.545228\pi$
$787$	−3.87288e9	−0.283219	−0.141610	+	0.989923i	$0.545228\pi$
$788$	8.78764e9	0.639780
$789$	0	0
$790$	−6.96987e8	−0.0502956
$791$	4.99145e9	0.358599
$792$	0	0
$793$	−2.01445e10	−1.43450
$794$	1.01355e9	0.0718578
$795$	0	0
$796$	1.22794e8	0.00862942
$797$	8.56688e9	0.599403	0.299701	−	0.954033i	$-0.403113\pi$
$797$	8.56688e9	0.599403	0.299701	+	0.954033i	$0.403113\pi$
$798$	0	0
$799$	2.33926e9	0.162243
$800$	2.31106e9	0.159586
$801$	0	0
$802$	2.66302e9	0.182290
$803$	1.50026e10	1.02250
$804$	0	0
$805$	4.43803e9	0.299850
$806$	−3.30431e9	−0.222284
$807$	0	0
$808$	−1.55684e10	−1.03825
$809$	1.49503e10	0.992731	0.496365	−	0.868114i	$-0.334668\pi$
$809$	1.49503e10	0.992731	0.496365	+	0.868114i	$0.334668\pi$
$810$	0	0
$811$	−8.61940e9	−0.567419	−0.283710	−	0.958910i	$-0.591565\pi$
$811$	−8.61940e9	−0.567419	−0.283710	+	0.958910i	$0.591565\pi$
$812$	2.38612e9	0.156403
$813$	0	0
$814$	3.07538e9	0.199854
$815$	−1.22878e10	−0.795104
$816$	0	0
$817$	2.95350e10	1.89478
$818$	2.98810e9	0.190879
$819$	0	0
$820$	−9.53186e9	−0.603711
$821$	−2.53165e10	−1.59663	−0.798313	−	0.602243i	$-0.794274\pi$
$821$	−2.53165e10	−1.59663	−0.798313	+	0.602243i	$0.794274\pi$
$822$	0	0
$823$	1.51790e9	0.0949171	0.0474586	−	0.998873i	$-0.484888\pi$
$823$	1.51790e9	0.0949171	0.0474586	+	0.998873i	$0.484888\pi$
$824$	−4.42811e9	−0.275723
$825$	0	0
$826$	4.79699e8	0.0296168
$827$	−1.68322e10	−1.03484	−0.517418	−	0.855733i	$-0.673107\pi$
$827$	−1.68322e10	−1.03484	−0.517418	+	0.855733i	$0.673107\pi$
$828$	0	0
$829$	−1.54259e10	−0.940396	−0.470198	−	0.882561i	$-0.655818\pi$
$829$	−1.54259e10	−0.940396	−0.470198	+	0.882561i	$0.655818\pi$
$830$	−1.49338e9	−0.0906559
$831$	0	0
$832$	1.46761e10	0.883442
$833$	2.29863e9	0.137788
$834$	0	0
$835$	7.25988e9	0.431546
$836$	1.49802e10	0.886737
$837$	0	0
$838$	−3.65536e9	−0.214574
$839$	1.06044e9	0.0619894	0.0309947	−	0.999520i	$-0.490132\pi$
$839$	1.06044e9	0.0619894	0.0309947	+	0.999520i	$0.490132\pi$
$840$	0	0
$841$	−1.36745e10	−0.792730
$842$	8.67880e9	0.501034
$843$	0	0
$844$	−1.33181e9	−0.0762509
$845$	−1.72159e10	−0.981595
$846$	0	0
$847$	−2.76638e9	−0.156430
$848$	−2.69892e9	−0.151986
$849$	0	0
$850$	1.04235e9	0.0582165
$851$	−2.75869e10	−1.53444
$852$	0	0
$853$	3.03756e10	1.67573	0.837864	−	0.545878i	$-0.183804\pi$
$853$	3.03756e10	1.67573	0.837864	+	0.545878i	$0.183804\pi$
$854$	1.66622e9	0.0915438
$855$	0	0
$856$	1.45847e10	0.794764
$857$	1.27509e10	0.692001	0.346001	−	0.938234i	$-0.387539\pi$
$857$	1.27509e10	0.692001	0.346001	+	0.938234i	$0.387539\pi$
$858$	0	0
$859$	1.08205e10	0.582467	0.291233	−	0.956652i	$-0.405934\pi$
$859$	1.08205e10	0.582467	0.291233	+	0.956652i	$0.405934\pi$
$860$	−1.12739e10	−0.604405
$861$	0	0
$862$	−3.04476e9	−0.161912
$863$	−6.05794e9	−0.320839	−0.160419	−	0.987049i	$-0.551285\pi$
$863$	−6.05794e9	−0.320839	−0.160419	+	0.987049i	$0.551285\pi$
$864$	0	0
$865$	−2.89682e9	−0.152183
$866$	−3.54981e9	−0.185734
$867$	0	0
$868$	−2.72752e9	−0.141563
$869$	5.51915e9	0.285301
$870$	0	0
$871$	−5.97560e10	−3.06421
$872$	−3.90175e8	−0.0199274
$873$	0	0
$874$	1.34651e10	0.682211
$875$	−6.69922e8	−0.0338062
$876$	0	0
$877$	5.64172e9	0.282431	0.141216	−	0.989979i	$-0.454899\pi$
$877$	5.64172e9	0.282431	0.141216	+	0.989979i	$0.454899\pi$
$878$	−5.32407e9	−0.265468
$879$	0	0
$880$	−5.08772e9	−0.251671
$881$	1.41128e9	0.0695339	0.0347669	−	0.999395i	$-0.488931\pi$
$881$	1.41128e9	0.0695339	0.0347669	+	0.999395i	$0.488931\pi$
$882$	0	0
$883$	1.83461e10	0.896773	0.448386	−	0.893840i	$-0.351999\pi$
$883$	1.83461e10	0.896773	0.448386	+	0.893840i	$0.351999\pi$
$884$	3.21846e10	1.56699
$885$	0	0
$886$	9.55923e8	0.0461748
$887$	−6.77821e9	−0.326124	−0.163062	−	0.986616i	$-0.552137\pi$
$887$	−6.77821e9	−0.326124	−0.163062	+	0.986616i	$0.552137\pi$
$888$	0	0
$889$	7.00065e8	0.0334182
$890$	8.68889e8	0.0413142
$891$	0	0
$892$	−3.27617e10	−1.54557
$893$	−4.56151e9	−0.214353
$894$	0	0
$895$	4.72378e9	0.220246
$896$	−7.70764e9	−0.357967
$897$	0	0
$898$	−2.11163e9	−0.0973085
$899$	−4.08695e9	−0.187603
$900$	0	0
$901$	−4.37852e9	−0.199430
$902$	−7.56332e9	−0.343155
$903$	0	0
$904$	−1.21407e10	−0.546579
$905$	1.13666e9	0.0509752
$906$	0	0
$907$	−3.32813e9	−0.148107	−0.0740535	−	0.997254i	$-0.523594\pi$
$907$	−3.32813e9	−0.148107	−0.0740535	+	0.997254i	$0.523594\pi$
$908$	−4.65718e9	−0.206454
$909$	0	0
$910$	2.07275e9	0.0911806
$911$	−7.48533e9	−0.328017	−0.164009	−	0.986459i	$-0.552442\pi$
$911$	−7.48533e9	−0.328017	−0.164009	+	0.986459i	$0.552442\pi$
$912$	0	0
$913$	1.18254e10	0.514244
$914$	3.09797e9	0.134204
$915$	0	0
$916$	1.44953e10	0.623149
$917$	3.13615e9	0.134309
$918$	0	0
$919$	3.84326e10	1.63341	0.816705	−	0.577056i	$-0.195799\pi$
$919$	3.84326e10	1.63341	0.816705	+	0.577056i	$0.195799\pi$
$920$	−1.07946e10	−0.457034
$921$	0	0
$922$	8.25354e9	0.346802
$923$	3.08127e10	1.28980
$924$	0	0
$925$	4.16425e9	0.172998
$926$	−1.07326e9	−0.0444189
$927$	0	0
$928$	−8.84406e9	−0.363274
$929$	−2.74203e10	−1.12206	−0.561032	−	0.827794i	$-0.689596\pi$
$929$	−2.74203e10	−1.12206	−0.561032	+	0.827794i	$0.689596\pi$
$930$	0	0
$931$	−4.48228e9	−0.182043
$932$	−3.74022e10	−1.51336
$933$	0	0
$934$	−1.61955e8	−0.00650400
$935$	−8.25392e9	−0.330233
$936$	0	0
$937$	−2.33959e9	−0.0929077	−0.0464539	−	0.998920i	$-0.514792\pi$
$937$	−2.33959e9	−0.0929077	−0.0464539	+	0.998920i	$0.514792\pi$
$938$	4.94261e9	0.195545
$939$	0	0
$940$	1.74118e9	0.0683750
$941$	4.04198e10	1.58136	0.790680	−	0.612230i	$-0.209727\pi$
$941$	4.04198e10	1.58136	0.790680	+	0.612230i	$0.209727\pi$
$942$	0	0
$943$	6.78449e10	2.63467
$944$	4.93294e9	0.190855
$945$	0	0
$946$	−8.94557e9	−0.343549
$947$	−2.68103e10	−1.02583	−0.512917	−	0.858438i	$-0.671435\pi$
$947$	−2.68103e10	−1.02583	−0.512917	+	0.858438i	$0.671435\pi$
$948$	0	0
$949$	−6.28533e10	−2.38724
$950$	−2.03256e9	−0.0769148
$951$	0	0
$952$	−5.59094e9	−0.210018
$953$	6.72263e9	0.251602	0.125801	−	0.992056i	$-0.459850\pi$
$953$	6.72263e9	0.251602	0.125801	+	0.992056i	$0.459850\pi$
$954$	0	0
$955$	−6.68625e9	−0.248411
$956$	−3.43297e10	−1.27077
$957$	0	0
$958$	7.12902e9	0.261970
$959$	−1.24415e10	−0.455520
$960$	0	0
$961$	−2.28409e10	−0.830198
$962$	−1.28843e10	−0.466602
$963$	0	0
$964$	−1.17373e10	−0.421988
$965$	1.03117e10	0.369391
$966$	0	0
$967$	2.39023e10	0.850055	0.425027	−	0.905180i	$-0.360264\pi$
$967$	2.39023e10	0.850055	0.425027	+	0.905180i	$0.360264\pi$
$968$	6.72864e9	0.238431
$969$	0	0
$970$	−5.42242e9	−0.190762
$971$	−1.65771e10	−0.581088	−0.290544	−	0.956862i	$-0.593836\pi$
$971$	−1.65771e10	−0.581088	−0.290544	+	0.956862i	$0.593836\pi$
$972$	0	0
$973$	−7.84862e9	−0.273148
$974$	−7.48681e9	−0.259621
$975$	0	0
$976$	1.71344e10	0.589921
$977$	2.03177e10	0.697018	0.348509	−	0.937305i	$-0.386688\pi$
$977$	2.03177e10	0.697018	0.348509	+	0.937305i	$0.386688\pi$
$978$	0	0
$979$	−6.88038e9	−0.234354
$980$	1.71094e9	0.0580689
$981$	0	0
$982$	−7.47841e9	−0.252011
$983$	−4.78793e10	−1.60772	−0.803861	−	0.594818i	$-0.797224\pi$
$983$	−4.78793e10	−1.60772	−0.803861	+	0.594818i	$0.797224\pi$
$984$	0	0
$985$	9.44160e9	0.314788
$986$	−3.98891e9	−0.132521
$987$	0	0
$988$	−6.27594e10	−2.07028
$989$	8.02440e10	2.63770
$990$	0	0
$991$	−1.99787e10	−0.652092	−0.326046	−	0.945354i	$-0.605716\pi$
$991$	−1.99787e10	−0.652092	−0.326046	+	0.945354i	$0.605716\pi$
$992$	1.01095e10	0.328804
$993$	0	0
$994$	−2.54861e9	−0.0823098
$995$	1.31932e8	0.00424590
$996$	0	0
$997$	4.49634e10	1.43690	0.718450	−	0.695579i	$-0.244852\pi$
$997$	4.49634e10	1.43690	0.718450	+	0.695579i	$0.244852\pi$
$998$	2.24235e9	0.0714080
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.g.1.3		4
3.2	odd	2		105.8.a.h.1.2	✓	4
15.14	odd	2		525.8.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.h.1.2	✓	4	3.2	odd	2
315.8.a.g.1.3		4	1.1	even	1	trivial
525.8.a.i.1.3		4	15.14	odd	2

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41438 q^{2} -116.342 q^{4} -125.000 q^{5} +343.000 q^{7} -834.277 q^{8} -426.798 q^{10} +3379.63 q^{11} -14159.0 q^{13} +1171.13 q^{14} +12043.2 q^{16} +19538.0 q^{17} -38098.7 q^{19} +14542.8 q^{20} +11539.4 q^{22} -103511. q^{23} +15625.0 q^{25} -48344.1 q^{26} -39905.3 q^{28} -59794.5 q^{29} +68349.9 q^{31} +147908. q^{32} +66710.3 q^{34} -42875.0 q^{35} +266512. q^{37} -130084. q^{38} +104285. q^{40} -655437. q^{41} -775223. q^{43} -393193. q^{44} -353426. q^{46} +119729. q^{47} +117649. q^{49} +53349.7 q^{50} +1.64728e6 q^{52} -224102. q^{53} -422454. q^{55} -286157. q^{56} -204161. q^{58} +409602. q^{59} +1.42274e6 q^{61} +233373. q^{62} -1.03652e6 q^{64} +1.76987e6 q^{65} +4.22037e6 q^{67} -2.27309e6 q^{68} -146392. q^{70} -2.17619e6 q^{71} +4.43912e6 q^{73} +909973. q^{74} +4.43248e6 q^{76} +1.15921e6 q^{77} +1.63306e6 q^{79} -1.50540e6 q^{80} -2.23791e6 q^{82} +3.49903e6 q^{83} -2.44225e6 q^{85} -2.64691e6 q^{86} -2.81955e6 q^{88} -2.03583e6 q^{89} -4.85652e6 q^{91} +1.20427e7 q^{92} +408799. q^{94} +4.76234e6 q^{95} +1.27049e7 q^{97} +401698. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) \) 4 * q - 11 * q^2 + 141 * q^4 - 500 * q^5 + 1372 * q^7 - 2133 * q^8 + 1375 * q^10 + 2708 * q^11 - 2212 * q^13 - 3773 * q^14 - 9599 * q^16 + 17016 * q^17 + 32668 * q^19 - 17625 * q^20 - 7196 * q^22 - 87696 * q^23 + 62500 * q^25 - 67354 * q^26 + 48363 * q^28 - 200488 * q^29 + 258852 * q^31 - 39445 * q^32 + 225518 * q^34 - 171500 * q^35 + 666856 * q^37 - 846556 * q^38 + 266625 * q^40 - 833000 * q^41 + 1051896 * q^43 - 1918236 * q^44 + 1786160 * q^46 - 529824 * q^47 + 470596 * q^49 - 171875 * q^50 + 211718 * q^52 - 791764 * q^53 - 338500 * q^55 - 731619 * q^56 + 3500242 * q^58 + 3825904 * q^59 + 212432 * q^61 - 882400 * q^62 - 1515063 * q^64 + 276500 * q^65 + 5165272 * q^67 - 7423202 * q^68 + 471625 * q^70 + 623260 * q^71 - 2083228 * q^73 - 1059642 * q^74 + 6990964 * q^76 + 928844 * q^77 - 1325016 * q^79 + 1199875 * q^80 + 11798982 * q^82 - 11447616 * q^83 - 2127000 * q^85 - 7525948 * q^86 + 19484572 * q^88 + 5805808 * q^89 - 758716 * q^91 + 2716288 * q^92 - 15552560 * q^94 - 4083500 * q^95 + 5857516 * q^97 - 1294139 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more