LMFDB - Embedded newform 315.8.a.g.1.1

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			1.1
Root			$15.4362$ 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)

$f(q)$	$=$	\(q-19.0104 q^{2} +233.393 q^{4} -125.000 q^{5} +343.000 q^{7} -2003.57 q^{8} +2376.29 q^{10} -4502.83 q^{11} -4274.51 q^{13} -6520.55 q^{14} +8214.16 q^{16} -18401.1 q^{17} +18009.6 q^{19} -29174.2 q^{20} +85600.4 q^{22} -60066.0 q^{23} +15625.0 q^{25} +81260.0 q^{26} +80054.0 q^{28} -120639. q^{29} -69029.6 q^{31} +100303. q^{32} +349811. q^{34} -42875.0 q^{35} +145432. q^{37} -342368. q^{38} +250446. q^{40} -195031. q^{41} +497258. q^{43} -1.05093e6 q^{44} +1.14188e6 q^{46} +878940. q^{47} +117649. q^{49} -297037. q^{50} -997643. q^{52} -1.14353e6 q^{53} +562854. q^{55} -687224. q^{56} +2.29339e6 q^{58} +1.77473e6 q^{59} -3.05405e6 q^{61} +1.31228e6 q^{62} -2.95820e6 q^{64} +534314. q^{65} -2.33226e6 q^{67} -4.29470e6 q^{68} +815069. q^{70} +462659. q^{71} -2.30754e6 q^{73} -2.76471e6 q^{74} +4.20332e6 q^{76} -1.54447e6 q^{77} -6.33467e6 q^{79} -1.02677e6 q^{80} +3.70761e6 q^{82} -7.03050e6 q^{83} +2.30014e6 q^{85} -9.45305e6 q^{86} +9.02172e6 q^{88} -5.92781e6 q^{89} -1.46616e6 q^{91} -1.40190e7 q^{92} -1.67090e7 q^{94} -2.25120e6 q^{95} -4.28990e6 q^{97} -2.23655e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−19.0104	−1.68029	−0.840147	−	0.542359i	$-0.817531\pi$
$2$	−19.0104	−1.68029	−0.840147	+	0.542359i	$0.817531\pi$
$3$	0	0
$3$	0	0
$4$	233.393	1.82339
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−2003.57	−1.38353
$9$	0	0
$10$	2376.29	0.751450
$11$	−4502.83	−1.02003	−0.510013	−	0.860167i	$-0.670359\pi$
$11$	−4502.83	−1.02003	−0.510013	+	0.860167i	$0.670359\pi$
$12$	0	0
$13$	−4274.51	−0.539616	−0.269808	−	0.962914i	$-0.586960\pi$
$13$	−4274.51	−0.539616	−0.269808	+	0.962914i	$0.586960\pi$
$14$	−6520.55	−0.635091
$15$	0	0
$16$	8214.16	0.501352
$17$	−18401.1	−0.908390	−0.454195	−	0.890902i	$-0.650073\pi$
$17$	−18401.1	−0.908390	−0.454195	+	0.890902i	$0.650073\pi$
$18$	0	0
$19$	18009.6	0.602373	0.301187	−	0.953565i	$-0.402617\pi$
$19$	18009.6	0.602373	0.301187	+	0.953565i	$0.402617\pi$
$20$	−29174.2	−0.815443
$21$	0	0
$22$	85600.4	1.71394
$23$	−60066.0	−1.02939	−0.514697	−	0.857372i	$-0.672096\pi$
$23$	−60066.0	−1.02939	−0.514697	+	0.857372i	$0.672096\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	81260.0	0.906713
$27$	0	0
$28$	80054.0	0.689175
$29$	−120639.	−0.918533	−0.459266	−	0.888299i	$-0.651888\pi$
$29$	−120639.	−0.918533	−0.459266	+	0.888299i	$0.651888\pi$
$30$	0	0
$31$	−69029.6	−0.416169	−0.208084	−	0.978111i	$-0.566723\pi$
$31$	−69029.6	−0.416169	−0.208084	+	0.978111i	$0.566723\pi$
$32$	100303.	0.541112
$33$	0	0
$34$	349811.	1.52636
$35$	−42875.0	−0.169031
$36$	0	0
$37$	145432.	0.472013	0.236007	−	0.971751i	$-0.424161\pi$
$37$	145432.	0.472013	0.236007	+	0.971751i	$0.424161\pi$
$38$	−342368.	−1.01216
$39$	0	0
$40$	250446.	0.618734
$41$	−195031.	−0.441937	−0.220969	−	0.975281i	$-0.570922\pi$
$41$	−195031.	−0.441937	−0.220969	+	0.975281i	$0.570922\pi$
$42$	0	0
$43$	497258.	0.953767	0.476884	−	0.878966i	$-0.341766\pi$
$43$	497258.	0.953767	0.476884	+	0.878966i	$0.341766\pi$
$44$	−1.05093e6	−1.85990
$45$	0	0
$46$	1.14188e6	1.72968
$47$	878940.	1.23486	0.617428	−	0.786627i	$-0.288175\pi$
$47$	878940.	1.23486	0.617428	+	0.786627i	$0.288175\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−297037.	−0.336059
$51$	0	0
$52$	−997643.	−0.983929
$53$	−1.14353e6	−1.05507	−0.527537	−	0.849532i	$-0.676884\pi$
$53$	−1.14353e6	−1.05507	−0.527537	+	0.849532i	$0.676884\pi$
$54$	0	0
$55$	562854.	0.456169
$56$	−687224.	−0.522926
$57$	0	0
$58$	2.29339e6	1.54341
$59$	1.77473e6	1.12499	0.562497	−	0.826799i	$-0.309841\pi$
$59$	1.77473e6	1.12499	0.562497	+	0.826799i	$0.309841\pi$
$60$	0	0
$61$	−3.05405e6	−1.72275	−0.861374	−	0.507971i	$-0.830396\pi$
$61$	−3.05405e6	−1.72275	−0.861374	+	0.507971i	$0.830396\pi$
$62$	1.31228e6	0.699286
$63$	0	0
$64$	−2.95820e6	−1.41058
$65$	534314.	0.241324
$66$	0	0
$67$	−2.33226e6	−0.947360	−0.473680	−	0.880697i	$-0.657075\pi$
$67$	−2.33226e6	−0.947360	−0.473680	+	0.880697i	$0.657075\pi$
$68$	−4.29470e6	−1.65635
$69$	0	0
$70$	815069.	0.284021
$71$	462659.	0.153411	0.0767056	−	0.997054i	$-0.475560\pi$
$71$	462659.	0.153411	0.0767056	+	0.997054i	$0.475560\pi$
$72$	0	0
$73$	−2.30754e6	−0.694255	−0.347128	−	0.937818i	$-0.612843\pi$
$73$	−2.30754e6	−0.694255	−0.347128	+	0.937818i	$0.612843\pi$
$74$	−2.76471e6	−0.793121
$75$	0	0
$76$	4.20332e6	1.09836
$77$	−1.54447e6	−0.385533
$78$	0	0
$79$	−6.33467e6	−1.44554	−0.722769	−	0.691090i	$-0.757131\pi$
$79$	−6.33467e6	−1.44554	−0.722769	+	0.691090i	$0.757131\pi$
$80$	−1.02677e6	−0.224212
$81$	0	0
$82$	3.70761e6	0.742585
$83$	−7.03050e6	−1.34963	−0.674813	−	0.737989i	$-0.735776\pi$
$83$	−7.03050e6	−1.34963	−0.674813	+	0.737989i	$0.735776\pi$
$84$	0	0
$85$	2.30014e6	0.406244
$86$	−9.45305e6	−1.60261
$87$	0	0
$88$	9.02172e6	1.41124
$89$	−5.92781e6	−0.891312	−0.445656	−	0.895204i	$-0.647029\pi$
$89$	−5.92781e6	−0.891312	−0.445656	+	0.895204i	$0.647029\pi$
$90$	0	0
$91$	−1.46616e6	−0.203956
$92$	−1.40190e7	−1.87698
$93$	0	0
$94$	−1.67090e7	−2.07492
$95$	−2.25120e6	−0.269390
$96$	0	0
$97$	−4.28990e6	−0.477251	−0.238625	−	0.971112i	$-0.576697\pi$
$97$	−4.28990e6	−0.477251	−0.238625	+	0.971112i	$0.576697\pi$
$98$	−2.23655e6	−0.240042
$99$	0	0
$100$	3.64677e6	0.364677
$101$	1.29025e7	1.24609	0.623046	−	0.782185i	$-0.285895\pi$
$101$	1.29025e7	1.24609	0.623046	+	0.782185i	$0.285895\pi$
$102$	0	0
$103$	−8.04379e6	−0.725321	−0.362660	−	0.931921i	$-0.618132\pi$
$103$	−8.04379e6	−0.725321	−0.362660	+	0.931921i	$0.618132\pi$
$104$	8.56427e6	0.746576
$105$	0	0
$106$	2.17389e7	1.77283
$107$	1.08308e7	0.854709	0.427354	−	0.904084i	$-0.359446\pi$
$107$	1.08308e7	0.854709	0.427354	+	0.904084i	$0.359446\pi$
$108$	0	0
$109$	3.03726e6	0.224641	0.112320	−	0.993672i	$-0.464172\pi$
$109$	3.03726e6	0.224641	0.112320	+	0.993672i	$0.464172\pi$
$110$	−1.07000e7	−0.766498
$111$	0	0
$112$	2.81746e6	0.189493
$113$	9.37656e6	0.611321	0.305660	−	0.952141i	$-0.401123\pi$
$113$	9.37656e6	0.611321	0.305660	+	0.952141i	$0.401123\pi$
$114$	0	0
$115$	7.50825e6	0.460359
$116$	−2.81564e7	−1.67484
$117$	0	0
$118$	−3.37383e7	−1.89032
$119$	−6.31158e6	−0.343339
$120$	0	0
$121$	788302.	0.0404524
$122$	5.80586e7	2.89472
$123$	0	0
$124$	−1.61111e7	−0.758837
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	1.86143e7	0.806368	0.403184	−	0.915119i	$-0.367903\pi$
$127$	1.86143e7	0.806368	0.403184	+	0.915119i	$0.367903\pi$
$128$	4.33977e7	1.82908
$129$	0	0
$130$	−1.01575e7	−0.405495
$131$	−9.02536e6	−0.350764	−0.175382	−	0.984500i	$-0.556116\pi$
$131$	−9.02536e6	−0.350764	−0.175382	+	0.984500i	$0.556116\pi$
$132$	0	0
$133$	6.17728e6	0.227676
$134$	4.43371e7	1.59184
$135$	0	0
$136$	3.68678e7	1.25679
$137$	3.54149e7	1.17669	0.588347	−	0.808608i	$-0.299779\pi$
$137$	3.54149e7	1.17669	0.588347	+	0.808608i	$0.299779\pi$
$138$	0	0
$139$	−4.05413e7	−1.28040	−0.640200	−	0.768208i	$-0.721149\pi$
$139$	−4.05413e7	−1.28040	−0.640200	+	0.768208i	$0.721149\pi$
$140$	−1.00067e7	−0.308209
$141$	0	0
$142$	−8.79531e6	−0.257776
$143$	1.92474e7	0.550422
$144$	0	0
$145$	1.50799e7	0.410780
$146$	4.38671e7	1.16655
$147$	0	0
$148$	3.39429e7	0.860662
$149$	−4.26530e7	−1.05633	−0.528163	−	0.849143i	$-0.677119\pi$
$149$	−4.26530e7	−1.05633	−0.528163	+	0.849143i	$0.677119\pi$
$150$	0	0
$151$	7.75258e7	1.83243	0.916214	−	0.400690i	$-0.131230\pi$
$151$	7.75258e7	1.83243	0.916214	+	0.400690i	$0.131230\pi$
$152$	−3.60834e7	−0.833402
$153$	0	0
$154$	2.93609e7	0.647809
$155$	8.62870e6	0.186116
$156$	0	0
$157$	1.80815e7	0.372895	0.186447	−	0.982465i	$-0.440303\pi$
$157$	1.80815e7	0.372895	0.186447	+	0.982465i	$0.440303\pi$
$158$	1.20424e8	2.42893
$159$	0	0
$160$	−1.25378e7	−0.241993
$161$	−2.06026e7	−0.389074
$162$	0	0
$163$	9.91147e7	1.79259	0.896296	−	0.443456i	$-0.146248\pi$
$163$	9.91147e7	1.79259	0.896296	+	0.443456i	$0.146248\pi$
$164$	−4.55190e7	−0.805823
$165$	0	0
$166$	1.33652e8	2.26777
$167$	4.49506e7	0.746840	0.373420	−	0.927662i	$-0.378185\pi$
$167$	4.49506e7	0.746840	0.373420	+	0.927662i	$0.378185\pi$
$168$	0	0
$169$	−4.44771e7	−0.708815
$170$	−4.37264e7	−0.682610
$171$	0	0
$172$	1.16057e8	1.73909
$173$	9.32139e7	1.36873	0.684367	−	0.729138i	$-0.260079\pi$
$173$	9.32139e7	1.36873	0.684367	+	0.729138i	$0.260079\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	−3.69869e7	−0.511392
$177$	0	0
$178$	1.12690e8	1.49767
$179$	5.82961e7	0.759720	0.379860	−	0.925044i	$-0.375972\pi$
$179$	5.82961e7	0.759720	0.379860	+	0.925044i	$0.375972\pi$
$180$	0	0
$181$	−9.20040e7	−1.15327	−0.576636	−	0.817001i	$-0.695635\pi$
$181$	−9.20040e7	−1.15327	−0.576636	+	0.817001i	$0.695635\pi$
$182$	2.78722e7	0.342705
$183$	0	0
$184$	1.20346e8	1.42420
$185$	−1.81790e7	−0.211091
$186$	0	0
$187$	8.28570e7	0.926581
$188$	2.05139e8	2.25162
$189$	0	0
$190$	4.27960e7	0.452654
$191$	−1.87816e8	−1.95037	−0.975184	−	0.221395i	$-0.928939\pi$
$191$	−1.87816e8	−1.95037	−0.975184	+	0.221395i	$0.928939\pi$
$192$	0	0
$193$	−6.42221e6	−0.0643034	−0.0321517	−	0.999483i	$-0.510236\pi$
$193$	−6.42221e6	−0.0643034	−0.0321517	+	0.999483i	$0.510236\pi$
$194$	8.15526e7	0.801921
$195$	0	0
$196$	2.74585e7	0.260484
$197$	1.14197e7	0.106420	0.0532102	−	0.998583i	$-0.483055\pi$
$197$	1.14197e7	0.106420	0.0532102	+	0.998583i	$0.483055\pi$
$198$	0	0
$199$	1.26893e8	1.14144	0.570719	−	0.821145i	$-0.306665\pi$
$199$	1.26893e8	1.14144	0.570719	+	0.821145i	$0.306665\pi$
$200$	−3.13057e7	−0.276706
$201$	0	0
$202$	−2.45282e8	−2.09380
$203$	−4.13792e7	−0.347173
$204$	0	0
$205$	2.43789e7	0.197640
$206$	1.52915e8	1.21875
$207$	0	0
$208$	−3.51115e7	−0.270538
$209$	−8.10940e7	−0.614436
$210$	0	0
$211$	−1.42587e8	−1.04494	−0.522470	−	0.852657i	$-0.674989\pi$
$211$	−1.42587e8	−1.04494	−0.522470	+	0.852657i	$0.674989\pi$
$212$	−2.66893e8	−1.92381
$213$	0	0
$214$	−2.05898e8	−1.43616
$215$	−6.21573e7	−0.426538
$216$	0	0
$217$	−2.36772e7	−0.157297
$218$	−5.77393e7	−0.377463
$219$	0	0
$220$	1.31366e8	0.831773
$221$	7.86557e7	0.490182
$222$	0	0
$223$	2.12588e8	1.28373	0.641863	−	0.766819i	$-0.278162\pi$
$223$	2.12588e8	1.28373	0.641863	+	0.766819i	$0.278162\pi$
$224$	3.44038e7	0.204521
$225$	0	0
$226$	−1.78252e8	−1.02720
$227$	−8.22201e7	−0.466538	−0.233269	−	0.972412i	$-0.574942\pi$
$227$	−8.22201e7	−0.466538	−0.233269	+	0.972412i	$0.574942\pi$
$228$	0	0
$229$	1.95654e8	1.07663	0.538314	−	0.842745i	$-0.319062\pi$
$229$	1.95654e8	1.07663	0.538314	+	0.842745i	$0.319062\pi$
$230$	−1.42735e8	−0.773538
$231$	0	0
$232$	2.41708e8	1.27082
$233$	2.59097e6	0.0134189	0.00670946	−	0.999977i	$-0.497864\pi$
$233$	2.59097e6	0.0134189	0.00670946	+	0.999977i	$0.497864\pi$
$234$	0	0
$235$	−1.09867e8	−0.552245
$236$	4.14211e8	2.05130
$237$	0	0
$238$	1.19985e8	0.576911
$239$	−2.68850e8	−1.27385	−0.636924	−	0.770927i	$-0.719793\pi$
$239$	−2.68850e8	−1.27385	−0.636924	+	0.770927i	$0.719793\pi$
$240$	0	0
$241$	−1.76419e8	−0.811867	−0.405933	−	0.913903i	$-0.633053\pi$
$241$	−1.76419e8	−0.811867	−0.405933	+	0.913903i	$0.633053\pi$
$242$	−1.49859e7	−0.0679718
$243$	0	0
$244$	−7.12796e8	−3.14124
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	−7.69821e7	−0.325050
$248$	1.38305e8	0.575783
$249$	0	0
$250$	3.71296e7	0.150290
$251$	2.77871e8	1.10914	0.554569	−	0.832138i	$-0.312883\pi$
$251$	2.77871e8	1.10914	0.554569	+	0.832138i	$0.312883\pi$
$252$	0	0
$253$	2.70467e8	1.05001
$254$	−3.53864e8	−1.35493
$255$	0	0
$256$	−4.46356e8	−1.66280
$257$	−1.92943e8	−0.709029	−0.354515	−	0.935050i	$-0.615354\pi$
$257$	−1.92943e8	−0.709029	−0.354515	+	0.935050i	$0.615354\pi$
$258$	0	0
$259$	4.98832e7	0.178404
$260$	1.24705e8	0.440026
$261$	0	0
$262$	1.71575e8	0.589386
$263$	3.57922e8	1.21323	0.606615	−	0.794995i	$-0.292527\pi$
$263$	3.57922e8	1.21323	0.606615	+	0.794995i	$0.292527\pi$
$264$	0	0
$265$	1.42941e8	0.471843
$266$	−1.17432e8	−0.382562
$267$	0	0
$268$	−5.44334e8	−1.72740
$269$	2.45154e8	0.767904	0.383952	−	0.923353i	$-0.374563\pi$
$269$	2.45154e8	0.767904	0.383952	+	0.923353i	$0.374563\pi$
$270$	0	0
$271$	3.89849e8	1.18988	0.594942	−	0.803769i	$-0.297175\pi$
$271$	3.89849e8	1.18988	0.594942	+	0.803769i	$0.297175\pi$
$272$	−1.51149e8	−0.455423
$273$	0	0
$274$	−6.73249e8	−1.97719
$275$	−7.03567e7	−0.204005
$276$	0	0
$277$	5.23694e8	1.48047	0.740233	−	0.672351i	$-0.234715\pi$
$277$	5.23694e8	1.48047	0.740233	+	0.672351i	$0.234715\pi$
$278$	7.70704e8	2.15145
$279$	0	0
$280$	8.59030e7	0.233859
$281$	5.48100e8	1.47363	0.736813	−	0.676096i	$-0.236330\pi$
$281$	5.48100e8	1.47363	0.736813	+	0.676096i	$0.236330\pi$
$282$	0	0
$283$	5.31402e8	1.39370	0.696852	−	0.717214i	$-0.254583\pi$
$283$	5.31402e8	1.39370	0.696852	+	0.717214i	$0.254583\pi$
$284$	1.07982e8	0.279728
$285$	0	0
$286$	−3.65900e8	−0.924871
$287$	−6.68957e7	−0.167037
$288$	0	0
$289$	−7.17384e7	−0.174827
$290$	−2.86674e8	−0.690232
$291$	0	0
$292$	−5.38565e8	−1.26590
$293$	−5.12957e8	−1.19136	−0.595682	−	0.803220i	$-0.703118\pi$
$293$	−5.12957e8	−1.19136	−0.595682	+	0.803220i	$0.703118\pi$
$294$	0	0
$295$	−2.21841e8	−0.503113
$296$	−2.91383e8	−0.653045
$297$	0	0
$298$	8.10849e8	1.77494
$299$	2.56753e8	0.555477
$300$	0	0
$301$	1.70560e8	0.360490
$302$	−1.47379e9	−3.07902
$303$	0	0
$304$	1.47933e8	0.302001
$305$	3.81756e8	0.770437
$306$	0	0
$307$	1.13872e8	0.224612	0.112306	−	0.993674i	$-0.464176\pi$
$307$	1.13872e8	0.224612	0.112306	+	0.993674i	$0.464176\pi$
$308$	−3.60469e8	−0.702977
$309$	0	0
$310$	−1.64035e8	−0.312730
$311$	−4.08851e8	−0.770732	−0.385366	−	0.922764i	$-0.625925\pi$
$311$	−4.08851e8	−0.770732	−0.385366	+	0.922764i	$0.625925\pi$
$312$	0	0
$313$	3.16669e7	0.0583715	0.0291858	−	0.999574i	$-0.490709\pi$
$313$	3.16669e7	0.0583715	0.0291858	+	0.999574i	$0.490709\pi$
$314$	−3.43736e8	−0.626573
$315$	0	0
$316$	−1.47847e9	−2.63577
$317$	−8.69607e7	−0.153326	−0.0766630	−	0.997057i	$-0.524427\pi$
$317$	−8.69607e7	−0.153326	−0.0766630	+	0.997057i	$0.524427\pi$
$318$	0	0
$319$	5.43217e8	0.936927
$320$	3.69775e8	0.630831
$321$	0	0
$322$	3.91664e8	0.653759
$323$	−3.31396e8	−0.547190
$324$	0	0
$325$	−6.67893e7	−0.107923
$326$	−1.88421e9	−3.01208
$327$	0	0
$328$	3.90758e8	0.611434
$329$	3.01476e8	0.466732
$330$	0	0
$331$	−6.22774e8	−0.943913	−0.471957	−	0.881622i	$-0.656452\pi$
$331$	−6.22774e8	−0.943913	−0.471957	+	0.881622i	$0.656452\pi$
$332$	−1.64087e9	−2.46089
$333$	0	0
$334$	−8.54526e8	−1.25491
$335$	2.91533e8	0.423672
$336$	0	0
$337$	−9.29789e7	−0.132336	−0.0661682	−	0.997808i	$-0.521077\pi$
$337$	−9.29789e7	−0.132336	−0.0661682	+	0.997808i	$0.521077\pi$
$338$	8.45525e8	1.19102
$339$	0	0
$340$	5.36837e8	0.740741
$341$	3.10829e8	0.424503
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	−9.96290e8	−1.31957
$345$	0	0
$346$	−1.77203e9	−2.29988
$347$	−2.89123e8	−0.371474	−0.185737	−	0.982599i	$-0.559467\pi$
$347$	−2.89123e8	−0.371474	−0.185737	+	0.982599i	$0.559467\pi$
$348$	0	0
$349$	−1.14349e9	−1.43994	−0.719971	−	0.694005i	$-0.755845\pi$
$349$	−1.14349e9	−1.43994	−0.719971	+	0.694005i	$0.755845\pi$
$350$	−1.01884e8	−0.127018
$351$	0	0
$352$	−4.51646e8	−0.551948
$353$	1.43830e9	1.74036	0.870179	−	0.492736i	$-0.164003\pi$
$353$	1.43830e9	1.74036	0.870179	+	0.492736i	$0.164003\pi$
$354$	0	0
$355$	−5.78324e7	−0.0686076
$356$	−1.38351e9	−1.62521
$357$	0	0
$358$	−1.10823e9	−1.27655
$359$	1.31227e9	1.49690	0.748451	−	0.663190i	$-0.230798\pi$
$359$	1.31227e9	1.49690	0.748451	+	0.663190i	$0.230798\pi$
$360$	0	0
$361$	−5.69527e8	−0.637146
$362$	1.74903e9	1.93784
$363$	0	0
$364$	−3.42192e8	−0.371890
$365$	2.88442e8	0.310480
$366$	0	0
$367$	6.60076e8	0.697048	0.348524	−	0.937300i	$-0.386683\pi$
$367$	6.60076e8	0.697048	0.348524	+	0.937300i	$0.386683\pi$
$368$	−4.93392e8	−0.516089
$369$	0	0
$370$	3.45589e8	0.354694
$371$	−3.92231e8	−0.398780
$372$	0	0
$373$	−3.17296e8	−0.316580	−0.158290	−	0.987393i	$-0.550598\pi$
$373$	−3.17296e8	−0.316580	−0.158290	+	0.987393i	$0.550598\pi$
$374$	−1.57514e9	−1.55693
$375$	0	0
$376$	−1.76102e9	−1.70846
$377$	5.15673e8	0.495655
$378$	0	0
$379$	−3.39230e8	−0.320078	−0.160039	−	0.987111i	$-0.551162\pi$
$379$	−3.39230e8	−0.320078	−0.160039	+	0.987111i	$0.551162\pi$
$380$	−5.25415e8	−0.491201
$381$	0	0
$382$	3.57046e9	3.27719
$383$	7.33467e8	0.667091	0.333546	−	0.942734i	$-0.391755\pi$
$383$	7.33467e8	0.667091	0.333546	+	0.942734i	$0.391755\pi$
$384$	0	0
$385$	1.93059e8	0.172416
$386$	1.22089e8	0.108049
$387$	0	0
$388$	−1.00124e9	−0.870212
$389$	1.92717e9	1.65996	0.829978	−	0.557796i	$-0.188353\pi$
$389$	1.92717e9	1.65996	0.829978	+	0.557796i	$0.188353\pi$
$390$	0	0
$391$	1.10528e9	0.935091
$392$	−2.35718e8	−0.197647
$393$	0	0
$394$	−2.17093e8	−0.178817
$395$	7.91834e8	0.646464
$396$	0	0
$397$	1.61772e8	0.129759	0.0648794	−	0.997893i	$-0.479334\pi$
$397$	1.61772e8	0.129759	0.0648794	+	0.997893i	$0.479334\pi$
$398$	−2.41228e9	−1.91795
$399$	0	0
$400$	1.28346e8	0.100270
$401$	−7.37769e8	−0.571367	−0.285683	−	0.958324i	$-0.592221\pi$
$401$	−7.37769e8	−0.571367	−0.285683	+	0.958324i	$0.592221\pi$
$402$	0	0
$403$	2.95068e8	0.224571
$404$	3.01137e9	2.27211
$405$	0	0
$406$	7.86633e8	0.583352
$407$	−6.54856e8	−0.481465
$408$	0	0
$409$	2.25702e9	1.63119	0.815595	−	0.578623i	$-0.196410\pi$
$409$	2.25702e9	1.63119	0.815595	+	0.578623i	$0.196410\pi$
$410$	−4.63452e8	−0.332094
$411$	0	0
$412$	−1.87737e9	−1.32254
$413$	6.08733e8	0.425208
$414$	0	0
$415$	8.78813e8	0.603571
$416$	−4.28745e8	−0.291993
$417$	0	0
$418$	1.54163e9	1.03243
$419$	−9.19680e7	−0.0610784	−0.0305392	−	0.999534i	$-0.509722\pi$
$419$	−9.19680e7	−0.0610784	−0.0305392	+	0.999534i	$0.509722\pi$
$420$	0	0
$421$	−2.02409e9	−1.32203	−0.661015	−	0.750373i	$-0.729874\pi$
$421$	−2.02409e9	−1.32203	−0.661015	+	0.750373i	$0.729874\pi$
$422$	2.71063e9	1.75581
$423$	0	0
$424$	2.29114e9	1.45973
$425$	−2.87517e8	−0.181678
$426$	0	0
$427$	−1.04754e9	−0.651138
$428$	2.52784e9	1.55846
$429$	0	0
$430$	1.18163e9	0.716708
$431$	3.17852e9	1.91229	0.956146	−	0.292890i	$-0.0946168\pi$
$431$	3.17852e9	1.91229	0.956146	+	0.292890i	$0.0946168\pi$
$432$	0	0
$433$	1.76089e9	1.04238	0.521189	−	0.853442i	$-0.325489\pi$
$433$	1.76089e9	1.04238	0.521189	+	0.853442i	$0.325489\pi$
$434$	4.50111e8	0.264305
$435$	0	0
$436$	7.08876e8	0.409607
$437$	−1.08176e9	−0.620079
$438$	0	0
$439$	−2.44879e9	−1.38142	−0.690710	−	0.723132i	$-0.742702\pi$
$439$	−2.44879e9	−1.38142	−0.690710	+	0.723132i	$0.742702\pi$
$440$	−1.12772e9	−0.631125
$441$	0	0
$442$	−1.49527e9	−0.823649
$443$	−2.38357e9	−1.30261	−0.651305	−	0.758816i	$-0.725778\pi$
$443$	−2.38357e9	−1.30261	−0.651305	+	0.758816i	$0.725778\pi$
$444$	0	0
$445$	7.40977e8	0.398607
$446$	−4.04138e9	−2.15704
$447$	0	0
$448$	−1.01466e9	−0.533149
$449$	3.69146e9	1.92458	0.962290	−	0.272024i	$-0.0876931\pi$
$449$	3.69146e9	1.92458	0.962290	+	0.272024i	$0.0876931\pi$
$450$	0	0
$451$	8.78193e8	0.450788
$452$	2.18843e9	1.11467
$453$	0	0
$454$	1.56303e9	0.783922
$455$	1.83270e8	0.0912118
$456$	0	0
$457$	−2.03116e9	−0.995490	−0.497745	−	0.867323i	$-0.665839\pi$
$457$	−2.03116e9	−0.995490	−0.497745	+	0.867323i	$0.665839\pi$
$458$	−3.71946e9	−1.80905
$459$	0	0
$460$	1.75238e9	0.839412
$461$	−3.41432e9	−1.62312	−0.811560	−	0.584269i	$-0.801381\pi$
$461$	−3.41432e9	−1.62312	−0.811560	+	0.584269i	$0.801381\pi$
$462$	0	0
$463$	2.40382e9	1.12556	0.562779	−	0.826608i	$-0.309732\pi$
$463$	2.40382e9	1.12556	0.562779	+	0.826608i	$0.309732\pi$
$464$	−9.90948e8	−0.460509
$465$	0	0
$466$	−4.92553e7	−0.0225477
$467$	2.43203e9	1.10499	0.552496	−	0.833516i	$-0.313675\pi$
$467$	2.43203e9	1.10499	0.552496	+	0.833516i	$0.313675\pi$
$468$	0	0
$469$	−7.99965e8	−0.358069
$470$	2.08862e9	0.927933
$471$	0	0
$472$	−3.55579e9	−1.55647
$473$	−2.23907e9	−0.972867
$474$	0	0
$475$	2.81400e8	0.120475
$476$	−1.47308e9	−0.626040
$477$	0	0
$478$	5.11093e9	2.14044
$479$	3.57965e8	0.148822	0.0744109	−	0.997228i	$-0.476292\pi$
$479$	3.57965e8	0.148822	0.0744109	+	0.997228i	$0.476292\pi$
$480$	0	0
$481$	−6.21651e8	−0.254706
$482$	3.35378e9	1.36417
$483$	0	0
$484$	1.83985e8	0.0737603
$485$	5.36238e8	0.213433
$486$	0	0
$487$	−2.09507e9	−0.821952	−0.410976	−	0.911646i	$-0.634812\pi$
$487$	−2.09507e9	−0.821952	−0.410976	+	0.911646i	$0.634812\pi$
$488$	6.11900e9	2.38348
$489$	0	0
$490$	2.79569e8	0.107350
$491$	3.01356e9	1.14893	0.574467	−	0.818528i	$-0.305209\pi$
$491$	3.01356e9	1.14893	0.574467	+	0.818528i	$0.305209\pi$
$492$	0	0
$493$	2.21989e9	0.834386
$494$	1.46346e9	0.546180
$495$	0	0
$496$	−5.67020e8	−0.208647
$497$	1.58692e8	0.0579840
$498$	0	0
$499$	3.82232e9	1.37713	0.688566	−	0.725174i	$-0.258241\pi$
$499$	3.82232e9	1.37713	0.688566	+	0.725174i	$0.258241\pi$
$500$	−4.55847e8	−0.163089
$501$	0	0
$502$	−5.28243e9	−1.86368
$503$	8.29703e8	0.290693	0.145347	−	0.989381i	$-0.453570\pi$
$503$	8.29703e8	0.290693	0.145347	+	0.989381i	$0.453570\pi$
$504$	0	0
$505$	−1.61282e9	−0.557270
$506$	−5.14167e9	−1.76432
$507$	0	0
$508$	4.34445e9	1.47032
$509$	4.75397e9	1.59788	0.798940	−	0.601411i	$-0.205394\pi$
$509$	4.75397e9	1.59788	0.798940	+	0.601411i	$0.205394\pi$
$510$	0	0
$511$	−7.91486e8	−0.262404
$512$	2.93048e9	0.964925
$513$	0	0
$514$	3.66792e9	1.19138
$515$	1.00547e9	0.324373
$516$	0	0
$517$	−3.95772e9	−1.25959
$518$	−9.48297e8	−0.299771
$519$	0	0
$520$	−1.07053e9	−0.333879
$521$	−4.66521e9	−1.44524	−0.722619	−	0.691246i	$-0.757062\pi$
$521$	−4.66521e9	−1.44524	−0.722619	+	0.691246i	$0.757062\pi$
$522$	0	0
$523$	2.94800e9	0.901098	0.450549	−	0.892752i	$-0.351228\pi$
$523$	2.94800e9	0.901098	0.450549	+	0.892752i	$0.351228\pi$
$524$	−2.10646e9	−0.639578
$525$	0	0
$526$	−6.80423e9	−2.03858
$527$	1.27022e9	0.378044
$528$	0	0
$529$	2.03103e8	0.0596514
$530$	−2.71737e9	−0.792835
$531$	0	0
$532$	1.44174e9	0.415141
$533$	8.33664e8	0.238477
$534$	0	0
$535$	−1.35385e9	−0.382237
$536$	4.67284e9	1.31070
$537$	0	0
$538$	−4.66047e9	−1.29030
$539$	−5.29753e8	−0.145718
$540$	0	0
$541$	4.94883e9	1.34373	0.671865	−	0.740673i	$-0.265493\pi$
$541$	4.94883e9	1.34373	0.671865	+	0.740673i	$0.265493\pi$
$542$	−7.41117e9	−1.99935
$543$	0	0
$544$	−1.84568e9	−0.491541
$545$	−3.79657e8	−0.100462
$546$	0	0
$547$	2.66452e9	0.696088	0.348044	−	0.937478i	$-0.386846\pi$
$547$	2.66452e9	0.696088	0.348044	+	0.937478i	$0.386846\pi$
$548$	8.26560e9	2.14557
$549$	0	0
$550$	1.33751e9	0.342789
$551$	−2.17266e9	−0.553300
$552$	0	0
$553$	−2.17279e9	−0.546362
$554$	−9.95560e9	−2.48762
$555$	0	0
$556$	−9.46207e9	−2.33466
$557$	2.24000e9	0.549230	0.274615	−	0.961554i	$-0.411450\pi$
$557$	2.24000e9	0.549230	0.274615	+	0.961554i	$0.411450\pi$
$558$	0	0
$559$	−2.12554e9	−0.514668
$560$	−3.52182e8	−0.0847440
$561$	0	0
$562$	−1.04196e10	−2.47612
$563$	−3.40259e9	−0.803582	−0.401791	−	0.915731i	$-0.631612\pi$
$563$	−3.40259e9	−0.803582	−0.401791	+	0.915731i	$0.631612\pi$
$564$	0	0
$565$	−1.17207e9	−0.273391
$566$	−1.01021e10	−2.34183
$567$	0	0
$568$	−9.26969e8	−0.212249
$569$	−4.63265e9	−1.05423	−0.527116	−	0.849793i	$-0.676727\pi$
$569$	−4.63265e9	−1.05423	−0.527116	+	0.849793i	$0.676727\pi$
$570$	0	0
$571$	−2.60224e9	−0.584954	−0.292477	−	0.956273i	$-0.594479\pi$
$571$	−2.60224e9	−0.584954	−0.292477	+	0.956273i	$0.594479\pi$
$572$	4.49222e9	1.00363
$573$	0	0
$574$	1.27171e9	0.280671
$575$	−9.38532e8	−0.205879
$576$	0	0
$577$	5.02957e8	0.108997	0.0544986	−	0.998514i	$-0.482644\pi$
$577$	5.02957e8	0.108997	0.0544986	+	0.998514i	$0.482644\pi$
$578$	1.36377e9	0.293761
$579$	0	0
$580$	3.51954e9	0.749012
$581$	−2.41146e9	−0.510110
$582$	0	0
$583$	5.14913e9	1.07620
$584$	4.62331e9	0.960524
$585$	0	0
$586$	9.75150e9	2.00184
$587$	1.96930e9	0.401863	0.200932	−	0.979605i	$-0.435603\pi$
$587$	1.96930e9	0.401863	0.200932	+	0.979605i	$0.435603\pi$
$588$	0	0
$589$	−1.24319e9	−0.250689
$590$	4.21728e9	0.845378
$591$	0	0
$592$	1.19460e9	0.236645
$593$	−5.02109e9	−0.988797	−0.494398	−	0.869236i	$-0.664612\pi$
$593$	−5.02109e9	−0.988797	−0.494398	+	0.869236i	$0.664612\pi$
$594$	0	0
$595$	7.88947e8	0.153546
$596$	−9.95494e9	−1.92609
$597$	0	0
$598$	−4.88096e9	−0.933365
$599$	3.93975e9	0.748989	0.374494	−	0.927229i	$-0.377816\pi$
$599$	3.93975e9	0.748989	0.374494	+	0.927229i	$0.377816\pi$
$600$	0	0
$601$	2.82069e9	0.530024	0.265012	−	0.964245i	$-0.414624\pi$
$601$	2.82069e9	0.530024	0.265012	+	0.964245i	$0.414624\pi$
$602$	−3.24240e9	−0.605729
$603$	0	0
$604$	1.80940e10	3.34122
$605$	−9.85377e7	−0.0180908
$606$	0	0
$607$	1.08437e10	1.96796	0.983979	−	0.178287i	$-0.0570555\pi$
$607$	1.08437e10	1.96796	0.983979	+	0.178287i	$0.0570555\pi$
$608$	1.80641e9	0.325952
$609$	0	0
$610$	−7.25733e9	−1.29456
$611$	−3.75704e9	−0.666349
$612$	0	0
$613$	3.16454e9	0.554881	0.277440	−	0.960743i	$-0.410514\pi$
$613$	3.16454e9	0.554881	0.277440	+	0.960743i	$0.410514\pi$
$614$	−2.16475e9	−0.377414
$615$	0	0
$616$	3.09445e9	0.533398
$617$	−6.68640e9	−1.14603	−0.573013	−	0.819546i	$-0.694226\pi$
$617$	−6.68640e9	−1.14603	−0.573013	+	0.819546i	$0.694226\pi$
$618$	0	0
$619$	−6.31362e9	−1.06994	−0.534972	−	0.844870i	$-0.679678\pi$
$619$	−6.31362e9	−1.06994	−0.534972	+	0.844870i	$0.679678\pi$
$620$	2.01388e9	0.339362
$621$	0	0
$622$	7.77240e9	1.29506
$623$	−2.03324e9	−0.336884
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−6.02000e8	−0.0980813
$627$	0	0
$628$	4.22011e9	0.679932
$629$	−2.67611e9	−0.428772
$630$	0	0
$631$	5.96493e9	0.945154	0.472577	−	0.881289i	$-0.343324\pi$
$631$	5.96493e9	0.945154	0.472577	+	0.881289i	$0.343324\pi$
$632$	1.26919e10	1.99995
$633$	0	0
$634$	1.65315e9	0.257633
$635$	−2.32678e9	−0.360619
$636$	0	0
$637$	−5.02892e8	−0.0770880
$638$	−1.03267e10	−1.57431
$639$	0	0
$640$	−5.42471e9	−0.817988
$641$	7.69364e9	1.15379	0.576897	−	0.816817i	$-0.304263\pi$
$641$	7.69364e9	1.15379	0.576897	+	0.816817i	$0.304263\pi$
$642$	0	0
$643$	8.95380e9	1.32822	0.664109	−	0.747636i	$-0.268811\pi$
$643$	8.95380e9	1.32822	0.664109	+	0.747636i	$0.268811\pi$
$644$	−4.80852e9	−0.709433
$645$	0	0
$646$	6.29995e9	0.919440
$647$	−1.13171e10	−1.64275	−0.821376	−	0.570388i	$-0.806793\pi$
$647$	−1.13171e10	−1.64275	−0.821376	+	0.570388i	$0.806793\pi$
$648$	0	0
$649$	−7.99131e9	−1.14752
$650$	1.26969e9	0.181343
$651$	0	0
$652$	2.31327e10	3.26859
$653$	3.31017e8	0.0465216	0.0232608	−	0.999729i	$-0.492595\pi$
$653$	3.31017e8	0.0465216	0.0232608	+	0.999729i	$0.492595\pi$
$654$	0	0
$655$	1.12817e9	0.156866
$656$	−1.60202e9	−0.221566
$657$	0	0
$658$	−5.73117e9	−0.784247
$659$	−7.69647e9	−1.04759	−0.523796	−	0.851844i	$-0.675485\pi$
$659$	−7.69647e9	−1.04759	−0.523796	+	0.851844i	$0.675485\pi$
$660$	0	0
$661$	−9.01041e9	−1.21350	−0.606749	−	0.794893i	$-0.707527\pi$
$661$	−9.01041e9	−1.21350	−0.606749	+	0.794893i	$0.707527\pi$
$662$	1.18391e10	1.58605
$663$	0	0
$664$	1.40861e10	1.86725
$665$	−7.72160e8	−0.101820
$666$	0	0
$667$	7.24631e9	0.945532
$668$	1.04912e10	1.36178
$669$	0	0
$670$	−5.54214e9	−0.711894
$671$	1.37519e10	1.75725
$672$	0	0
$673$	5.44057e9	0.688005	0.344002	−	0.938969i	$-0.388217\pi$
$673$	5.44057e9	0.688005	0.344002	+	0.938969i	$0.388217\pi$
$674$	1.76756e9	0.222364
$675$	0	0
$676$	−1.03807e10	−1.29244
$677$	4.22948e9	0.523873	0.261937	−	0.965085i	$-0.415639\pi$
$677$	4.22948e9	0.523873	0.261937	+	0.965085i	$0.415639\pi$
$678$	0	0
$679$	−1.47144e9	−0.180384
$680$	−4.60848e9	−0.562052
$681$	0	0
$682$	−5.90896e9	−0.713289
$683$	3.23839e9	0.388917	0.194459	−	0.980911i	$-0.437705\pi$
$683$	3.23839e9	0.388917	0.194459	+	0.980911i	$0.437705\pi$
$684$	0	0
$685$	−4.42686e9	−0.526234
$686$	−7.67136e8	−0.0907273
$687$	0	0
$688$	4.08456e9	0.478173
$689$	4.88804e9	0.569334
$690$	0	0
$691$	2.28886e7	0.00263904	0.00131952	−	0.999999i	$-0.499580\pi$
$691$	2.28886e7	0.00263904	0.00131952	+	0.999999i	$0.499580\pi$
$692$	2.17555e10	2.49573
$693$	0	0
$694$	5.49632e9	0.624186
$695$	5.06766e9	0.572612
$696$	0	0
$697$	3.58879e9	0.401452
$698$	2.17382e10	2.41952
$699$	0	0
$700$	1.25084e9	0.137835
$701$	9.45855e9	1.03708	0.518539	−	0.855054i	$-0.326476\pi$
$701$	9.45855e9	1.03708	0.518539	+	0.855054i	$0.326476\pi$
$702$	0	0
$703$	2.61917e9	0.284328
$704$	1.33203e10	1.43883
$705$	0	0
$706$	−2.73426e10	−2.92431
$707$	4.42557e9	0.470979
$708$	0	0
$709$	3.82006e9	0.402539	0.201270	−	0.979536i	$-0.435493\pi$
$709$	3.82006e9	0.402539	0.201270	+	0.979536i	$0.435493\pi$
$710$	1.09941e9	0.115281
$711$	0	0
$712$	1.18768e10	1.23316
$713$	4.14634e9	0.428402
$714$	0	0
$715$	−2.40592e9	−0.246156
$716$	1.36059e10	1.38526
$717$	0	0
$718$	−2.49468e10	−2.51524
$719$	−7.33673e9	−0.736125	−0.368062	−	0.929801i	$-0.619979\pi$
$719$	−7.33673e9	−0.736125	−0.368062	+	0.929801i	$0.619979\pi$
$720$	0	0
$721$	−2.75902e9	−0.274146
$722$	1.08269e10	1.07059
$723$	0	0
$724$	−2.14731e10	−2.10286
$725$	−1.88498e9	−0.183707
$726$	0	0
$727$	−1.82474e10	−1.76129	−0.880644	−	0.473778i	$-0.842890\pi$
$727$	−1.82474e10	−1.76129	−0.880644	+	0.473778i	$0.842890\pi$
$728$	2.93755e9	0.282179
$729$	0	0
$730$	−5.48339e9	−0.521698
$731$	−9.15010e9	−0.866393
$732$	0	0
$733$	1.15884e10	1.08682	0.543412	−	0.839466i	$-0.317132\pi$
$733$	1.15884e10	1.08682	0.543412	+	0.839466i	$0.317132\pi$
$734$	−1.25483e10	−1.17125
$735$	0	0
$736$	−6.02478e9	−0.557018
$737$	1.05018e10	0.966332
$738$	0	0
$739$	4.70724e9	0.429053	0.214527	−	0.976718i	$-0.431179\pi$
$739$	4.70724e9	0.429053	0.214527	+	0.976718i	$0.431179\pi$
$740$	−4.24286e9	−0.384900
$741$	0	0
$742$	7.45646e9	0.670068
$743$	−1.20000e10	−1.07330	−0.536648	−	0.843806i	$-0.680310\pi$
$743$	−1.20000e10	−1.07330	−0.536648	+	0.843806i	$0.680310\pi$
$744$	0	0
$745$	5.33163e9	0.472403
$746$	6.03191e9	0.531947
$747$	0	0
$748$	1.93383e10	1.68952
$749$	3.71497e9	0.323050
$750$	0	0
$751$	8.66014e9	0.746080	0.373040	−	0.927815i	$-0.378315\pi$
$751$	8.66014e9	0.746080	0.373040	+	0.927815i	$0.378315\pi$
$752$	7.21975e9	0.619098
$753$	0	0
$754$	−9.80312e9	−0.832846
$755$	−9.69072e9	−0.819486
$756$	0	0
$757$	−9.64571e9	−0.808163	−0.404081	−	0.914723i	$-0.632409\pi$
$757$	−9.64571e9	−0.808163	−0.404081	+	0.914723i	$0.632409\pi$
$758$	6.44887e9	0.537826
$759$	0	0
$760$	4.51042e9	0.372709
$761$	9.41824e9	0.774682	0.387341	−	0.921937i	$-0.373394\pi$
$761$	9.41824e9	0.774682	0.387341	+	0.921937i	$0.373394\pi$
$762$	0	0
$763$	1.04178e9	0.0849063
$764$	−4.38351e10	−3.55628
$765$	0	0
$766$	−1.39435e10	−1.12091
$767$	−7.58611e9	−0.607065
$768$	0	0
$769$	−7.36510e9	−0.584032	−0.292016	−	0.956414i	$-0.594326\pi$
$769$	−7.36510e9	−0.584032	−0.292016	+	0.956414i	$0.594326\pi$
$770$	−3.67012e9	−0.289709
$771$	0	0
$772$	−1.49890e9	−0.117250
$773$	−1.33080e10	−1.03630	−0.518150	−	0.855290i	$-0.673379\pi$
$773$	−1.33080e10	−1.03630	−0.518150	+	0.855290i	$0.673379\pi$
$774$	0	0
$775$	−1.07859e9	−0.0832338
$776$	8.59511e9	0.660291
$777$	0	0
$778$	−3.66362e10	−2.78921
$779$	−3.51243e9	−0.266211
$780$	0	0
$781$	−2.08328e9	−0.156483
$782$	−2.10118e10	−1.57123
$783$	0	0
$784$	9.66387e8	0.0716218
$785$	−2.26019e9	−0.166764
$786$	0	0
$787$	7.47085e9	0.546334	0.273167	−	0.961967i	$-0.411929\pi$
$787$	7.47085e9	0.546334	0.273167	+	0.961967i	$0.411929\pi$
$788$	2.66530e9	0.194045
$789$	0	0
$790$	−1.50530e10	−1.08625
$791$	3.21616e9	0.231058
$792$	0	0
$793$	1.30546e10	0.929623
$794$	−3.07534e9	−0.218033
$795$	0	0
$796$	2.96160e10	2.08128
$797$	1.26160e10	0.882706	0.441353	−	0.897333i	$-0.354499\pi$
$797$	1.26160e10	0.882706	0.441353	+	0.897333i	$0.354499\pi$
$798$	0	0
$799$	−1.61735e10	−1.12173
$800$	1.56723e9	0.108222
$801$	0	0
$802$	1.40252e10	0.960064
$803$	1.03905e10	0.708158
$804$	0	0
$805$	2.57533e9	0.173999
$806$	−5.60935e9	−0.377346
$807$	0	0
$808$	−2.58511e10	−1.72401
$809$	1.66973e10	1.10873	0.554367	−	0.832272i	$-0.312960\pi$
$809$	1.66973e10	1.10873	0.554367	+	0.832272i	$0.312960\pi$
$810$	0	0
$811$	1.32145e10	0.869915	0.434957	−	0.900451i	$-0.356763\pi$
$811$	1.32145e10	0.869915	0.434957	+	0.900451i	$0.356763\pi$
$812$	−9.65763e9	−0.633030
$813$	0	0
$814$	1.24490e10	0.809003
$815$	−1.23893e10	−0.801672
$816$	0	0
$817$	8.95541e9	0.574524
$818$	−4.29068e10	−2.74088
$819$	0	0
$820$	5.68988e9	0.360375
$821$	−1.98844e10	−1.25404	−0.627020	−	0.779003i	$-0.715726\pi$
$821$	−1.98844e10	−1.25404	−0.627020	+	0.779003i	$0.715726\pi$
$822$	0	0
$823$	−2.59040e10	−1.61982	−0.809910	−	0.586554i	$-0.800484\pi$
$823$	−2.59040e10	−1.61982	−0.809910	+	0.586554i	$0.800484\pi$
$824$	1.61163e10	1.00350
$825$	0	0
$826$	−1.15722e10	−0.714474
$827$	1.41678e10	0.871027	0.435514	−	0.900182i	$-0.356567\pi$
$827$	1.41678e10	0.871027	0.435514	+	0.900182i	$0.356567\pi$
$828$	0	0
$829$	1.00028e10	0.609791	0.304895	−	0.952386i	$-0.401378\pi$
$829$	1.00028e10	0.609791	0.304895	+	0.952386i	$0.401378\pi$
$830$	−1.67065e10	−1.01418
$831$	0	0
$832$	1.26449e10	0.761171
$833$	−2.16487e9	−0.129770
$834$	0	0
$835$	−5.61882e9	−0.333997
$836$	−1.89268e10	−1.12035
$837$	0	0
$838$	1.74834e9	0.102630
$839$	−7.40779e9	−0.433034	−0.216517	−	0.976279i	$-0.569470\pi$
$839$	−7.40779e9	−0.433034	−0.216517	+	0.976279i	$0.569470\pi$
$840$	0	0
$841$	−2.69611e9	−0.156297
$842$	3.84786e10	2.22140
$843$	0	0
$844$	−3.32789e10	−1.90533
$845$	5.55963e9	0.316992
$846$	0	0
$847$	2.70388e8	0.0152896
$848$	−9.39315e9	−0.528963
$849$	0	0
$850$	5.46580e9	0.305272
$851$	−8.73552e9	−0.485887
$852$	0	0
$853$	−1.30062e10	−0.717511	−0.358755	−	0.933432i	$-0.616799\pi$
$853$	−1.30062e10	−0.717511	−0.358755	+	0.933432i	$0.616799\pi$
$854$	1.99141e10	1.09410
$855$	0	0
$856$	−2.17003e10	−1.18252
$857$	−1.39742e10	−0.758394	−0.379197	−	0.925316i	$-0.623800\pi$
$857$	−1.39742e10	−0.758394	−0.379197	+	0.925316i	$0.623800\pi$
$858$	0	0
$859$	1.39136e10	0.748967	0.374484	−	0.927234i	$-0.377820\pi$
$859$	1.39136e10	0.748967	0.374484	+	0.927234i	$0.377820\pi$
$860$	−1.45071e10	−0.777743
$861$	0	0
$862$	−6.04248e10	−3.21321
$863$	−3.55415e10	−1.88234	−0.941169	−	0.337937i	$-0.890271\pi$
$863$	−3.55415e10	−1.88234	−0.941169	+	0.337937i	$0.890271\pi$
$864$	0	0
$865$	−1.16517e10	−0.612117
$866$	−3.34752e10	−1.75150
$867$	0	0
$868$	−5.52609e9	−0.286813
$869$	2.85240e10	1.47449
$870$	0	0
$871$	9.96927e9	0.511211
$872$	−6.08535e9	−0.310798
$873$	0	0
$874$	2.05647e10	1.04192
$875$	−6.69922e8	−0.0338062
$876$	0	0
$877$	−2.17698e10	−1.08982	−0.544911	−	0.838494i	$-0.683437\pi$
$877$	−2.17698e10	−1.08982	−0.544911	+	0.838494i	$0.683437\pi$
$878$	4.65524e10	2.32119
$879$	0	0
$880$	4.62337e9	0.228702
$881$	5.94782e9	0.293050	0.146525	−	0.989207i	$-0.453191\pi$
$881$	5.94782e9	0.293050	0.146525	+	0.989207i	$0.453191\pi$
$882$	0	0
$883$	−1.67373e10	−0.818130	−0.409065	−	0.912505i	$-0.634145\pi$
$883$	−1.67373e10	−0.818130	−0.409065	+	0.912505i	$0.634145\pi$
$884$	1.83577e10	0.893791
$885$	0	0
$886$	4.53124e10	2.18877
$887$	−3.58618e10	−1.72544	−0.862720	−	0.505682i	$-0.831241\pi$
$887$	−3.58618e10	−1.72544	−0.862720	+	0.505682i	$0.831241\pi$
$888$	0	0
$889$	6.38470e9	0.304778
$890$	−1.40862e10	−0.669776
$891$	0	0
$892$	4.96168e10	2.34073
$893$	1.58293e10	0.743845
$894$	0	0
$895$	−7.28701e9	−0.339757
$896$	1.48854e10	0.691326
$897$	0	0
$898$	−7.01760e10	−3.23386
$899$	8.32766e9	0.382265
$900$	0	0
$901$	2.10422e10	0.958418
$902$	−1.66948e10	−0.757455
$903$	0	0
$904$	−1.87866e10	−0.845781
$905$	1.15005e10	0.515759
$906$	0	0
$907$	−1.57578e9	−0.0701247	−0.0350623	−	0.999385i	$-0.511163\pi$
$907$	−1.57578e9	−0.0701247	−0.0350623	+	0.999385i	$0.511163\pi$
$908$	−1.91896e10	−0.850680
$909$	0	0
$910$	−3.48402e9	−0.153263
$911$	−2.34451e10	−1.02740	−0.513699	−	0.857971i	$-0.671725\pi$
$911$	−2.34451e10	−1.02740	−0.513699	+	0.857971i	$0.671725\pi$
$912$	0	0
$913$	3.16572e10	1.37665
$914$	3.86130e10	1.67272
$915$	0	0
$916$	4.56645e10	1.96311
$917$	−3.09570e9	−0.132576
$918$	0	0
$919$	3.95547e10	1.68110	0.840551	−	0.541732i	$-0.182231\pi$
$919$	3.95547e10	1.68110	0.840551	+	0.541732i	$0.182231\pi$
$920$	−1.50433e10	−0.636921
$921$	0	0
$922$	6.49074e10	2.72732
$923$	−1.97764e9	−0.0827831
$924$	0	0
$925$	2.27238e9	0.0944026
$926$	−4.56974e10	−1.89127
$927$	0	0
$928$	−1.21004e10	−0.497030
$929$	−1.81939e10	−0.744512	−0.372256	−	0.928130i	$-0.621416\pi$
$929$	−1.81939e10	−0.744512	−0.372256	+	0.928130i	$0.621416\pi$
$930$	0	0
$931$	2.11881e9	0.0860533
$932$	6.04717e8	0.0244679
$933$	0	0
$934$	−4.62337e10	−1.85671
$935$	−1.03571e10	−0.414380
$936$	0	0
$937$	−4.78803e8	−0.0190138	−0.00950688	−	0.999955i	$-0.503026\pi$
$937$	−4.78803e8	−0.0190138	−0.00950688	+	0.999955i	$0.503026\pi$
$938$	1.52076e10	0.601660
$939$	0	0
$940$	−2.56424e10	−1.00696
$941$	−2.86712e10	−1.12171	−0.560857	−	0.827912i	$-0.689528\pi$
$941$	−2.86712e10	−1.12171	−0.560857	+	0.827912i	$0.689528\pi$
$942$	0	0
$943$	1.17148e10	0.454928
$944$	1.45779e10	0.564019
$945$	0	0
$946$	4.25655e10	1.63470
$947$	−8.72415e9	−0.333809	−0.166905	−	0.985973i	$-0.553377\pi$
$947$	−8.72415e9	−0.333809	−0.166905	+	0.985973i	$0.553377\pi$
$948$	0	0
$949$	9.86360e9	0.374631
$950$	−5.34950e9	−0.202433
$951$	0	0
$952$	1.26457e10	0.475021
$953$	−9.64039e9	−0.360802	−0.180401	−	0.983593i	$-0.557740\pi$
$953$	−9.64039e9	−0.360802	−0.180401	+	0.983593i	$0.557740\pi$
$954$	0	0
$955$	2.34770e10	0.872231
$956$	−6.27478e10	−2.32272
$957$	0	0
$958$	−6.80504e9	−0.250064
$959$	1.21473e10	0.444749
$960$	0	0
$961$	−2.27475e10	−0.826804
$962$	1.18178e10	0.427981
$963$	0	0
$964$	−4.11750e10	−1.48035
$965$	8.02777e8	0.0287574
$966$	0	0
$967$	−2.56468e10	−0.912094	−0.456047	−	0.889956i	$-0.650735\pi$
$967$	−2.56468e10	−0.912094	−0.456047	+	0.889956i	$0.650735\pi$
$968$	−1.57942e9	−0.0559671
$969$	0	0
$970$	−1.01941e10	−0.358630
$971$	3.19514e10	1.12001	0.560007	−	0.828488i	$-0.310799\pi$
$971$	3.19514e10	1.12001	0.560007	+	0.828488i	$0.310799\pi$
$972$	0	0
$973$	−1.39057e10	−0.483946
$974$	3.98279e10	1.38112
$975$	0	0
$976$	−2.50865e10	−0.863704
$977$	4.30520e10	1.47694	0.738470	−	0.674287i	$-0.235549\pi$
$977$	4.30520e10	1.47694	0.738470	+	0.674287i	$0.235549\pi$
$978$	0	0
$979$	2.66919e10	0.909161
$980$	−3.43231e9	−0.116492
$981$	0	0
$982$	−5.72889e10	−1.93055
$983$	5.12333e10	1.72034	0.860171	−	0.510006i	$-0.170357\pi$
$983$	5.12333e10	1.72034	0.860171	+	0.510006i	$0.170357\pi$
$984$	0	0
$985$	−1.42747e9	−0.0475926
$986$	−4.22009e10	−1.40201
$987$	0	0
$988$	−1.79671e10	−0.592692
$989$	−2.98683e10	−0.981802
$990$	0	0
$991$	2.98088e10	0.972940	0.486470	−	0.873697i	$-0.338284\pi$
$991$	2.98088e10	0.972940	0.486470	+	0.873697i	$0.338284\pi$
$992$	−6.92385e9	−0.225194
$993$	0	0
$994$	−3.01679e9	−0.0974301
$995$	−1.58616e10	−0.510467
$996$	0	0
$997$	−6.10695e9	−0.195160	−0.0975801	−	0.995228i	$-0.531110\pi$
$997$	−6.10695e9	−0.195160	−0.0975801	+	0.995228i	$0.531110\pi$
$998$	−7.26637e10	−2.31399
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.g.1.1		4
3.2	odd	2		105.8.a.h.1.4	✓	4
15.14	odd	2		525.8.a.i.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.h.1.4	✓	4	3.2	odd	2
315.8.a.g.1.1		4	1.1	even	1	trivial
525.8.a.i.1.1		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)

\(f(q)\)	\(=\)	\(q-19.0104 q^{2} +233.393 q^{4} -125.000 q^{5} +343.000 q^{7} -2003.57 q^{8} +2376.29 q^{10} -4502.83 q^{11} -4274.51 q^{13} -6520.55 q^{14} +8214.16 q^{16} -18401.1 q^{17} +18009.6 q^{19} -29174.2 q^{20} +85600.4 q^{22} -60066.0 q^{23} +15625.0 q^{25} +81260.0 q^{26} +80054.0 q^{28} -120639. q^{29} -69029.6 q^{31} +100303. q^{32} +349811. q^{34} -42875.0 q^{35} +145432. q^{37} -342368. q^{38} +250446. q^{40} -195031. q^{41} +497258. q^{43} -1.05093e6 q^{44} +1.14188e6 q^{46} +878940. q^{47} +117649. q^{49} -297037. q^{50} -997643. q^{52} -1.14353e6 q^{53} +562854. q^{55} -687224. q^{56} +2.29339e6 q^{58} +1.77473e6 q^{59} -3.05405e6 q^{61} +1.31228e6 q^{62} -2.95820e6 q^{64} +534314. q^{65} -2.33226e6 q^{67} -4.29470e6 q^{68} +815069. q^{70} +462659. q^{71} -2.30754e6 q^{73} -2.76471e6 q^{74} +4.20332e6 q^{76} -1.54447e6 q^{77} -6.33467e6 q^{79} -1.02677e6 q^{80} +3.70761e6 q^{82} -7.03050e6 q^{83} +2.30014e6 q^{85} -9.45305e6 q^{86} +9.02172e6 q^{88} -5.92781e6 q^{89} -1.46616e6 q^{91} -1.40190e7 q^{92} -1.67090e7 q^{94} -2.25120e6 q^{95} -4.28990e6 q^{97} -2.23655e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) \) 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

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Database

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