LMFDB - Embedded newform 315.8.a.g.1.2

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			1.2
Root			$-9.12888$ 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)

$f(q)$	$=$	\(q-9.30777 q^{2} -41.3655 q^{4} -125.000 q^{5} +343.000 q^{7} +1576.41 q^{8} +1163.47 q^{10} +6789.89 q^{11} +14036.5 q^{13} -3192.56 q^{14} -9378.12 q^{16} +17740.4 q^{17} +47718.8 q^{19} +5170.68 q^{20} -63198.8 q^{22} +2469.43 q^{23} +15625.0 q^{25} -130649. q^{26} -14188.4 q^{28} -72802.1 q^{29} +260065. q^{31} -114492. q^{32} -165124. q^{34} -42875.0 q^{35} +118438. q^{37} -444156. q^{38} -197052. q^{40} -434543. q^{41} +599531. q^{43} -280867. q^{44} -22984.9 q^{46} -947778. q^{47} +117649. q^{49} -145434. q^{50} -580627. q^{52} +1.40065e6 q^{53} -848737. q^{55} +540710. q^{56} +677625. q^{58} +1.96604e6 q^{59} +726465. q^{61} -2.42063e6 q^{62} +2.26606e6 q^{64} -1.75457e6 q^{65} +756763. q^{67} -733841. q^{68} +399071. q^{70} -1.50605e6 q^{71} -6.38338e6 q^{73} -1.10240e6 q^{74} -1.97391e6 q^{76} +2.32893e6 q^{77} +4.43201e6 q^{79} +1.17227e6 q^{80} +4.04463e6 q^{82} -5.38121e6 q^{83} -2.21755e6 q^{85} -5.58030e6 q^{86} +1.07037e7 q^{88} +7.59795e6 q^{89} +4.81453e6 q^{91} -102149. q^{92} +8.82170e6 q^{94} -5.96485e6 q^{95} -7.91961e6 q^{97} -1.09505e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.30777	−0.822698	−0.411349	−	0.911478i	$-0.634942\pi$
$2$	−9.30777	−0.822698	−0.411349	+	0.911478i	$0.634942\pi$
$3$	0	0
$3$	0	0
$4$	−41.3655	−0.323168
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	1576.41	1.08857
$9$	0	0
$10$	1163.47	0.367922
$11$	6789.89	1.53811	0.769057	−	0.639180i	$-0.220726\pi$
$11$	6789.89	1.53811	0.769057	+	0.639180i	$0.220726\pi$
$12$	0	0
$13$	14036.5	1.77198	0.885988	−	0.463708i	$-0.153481\pi$
$13$	14036.5	1.77198	0.885988	+	0.463708i	$0.153481\pi$
$14$	−3192.56	−0.310951
$15$	0	0
$16$	−9378.12	−0.572395
$17$	17740.4	0.875775	0.437888	−	0.899030i	$-0.355727\pi$
$17$	17740.4	0.875775	0.437888	+	0.899030i	$0.355727\pi$
$18$	0	0
$19$	47718.8	1.59607	0.798036	−	0.602610i	$-0.205873\pi$
$19$	47718.8	1.59607	0.798036	+	0.602610i	$0.205873\pi$
$20$	5170.68	0.144525
$21$	0	0
$22$	−63198.8	−1.26540
$23$	2469.43	0.0423204	0.0211602	−	0.999776i	$-0.493264\pi$
$23$	2469.43	0.0423204	0.0211602	+	0.999776i	$0.493264\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−130649.	−1.45780
$27$	0	0
$28$	−14188.4	−0.122146
$29$	−72802.1	−0.554307	−0.277154	−	0.960826i	$-0.589391\pi$
$29$	−72802.1	−0.554307	−0.277154	+	0.960826i	$0.589391\pi$
$30$	0	0
$31$	260065.	1.56789	0.783947	−	0.620828i	$-0.213203\pi$
$31$	260065.	1.56789	0.783947	+	0.620828i	$0.213203\pi$
$32$	−114492.	−0.617659
$33$	0	0
$34$	−165124.	−0.720499
$35$	−42875.0	−0.169031
$36$	0	0
$37$	118438.	0.384403	0.192201	−	0.981356i	$-0.438437\pi$
$37$	118438.	0.384403	0.192201	+	0.981356i	$0.438437\pi$
$38$	−444156.	−1.31308
$39$	0	0
$40$	−197052.	−0.486822
$41$	−434543.	−0.984667	−0.492334	−	0.870407i	$-0.663856\pi$
$41$	−434543.	−0.984667	−0.492334	+	0.870407i	$0.663856\pi$
$42$	0	0
$43$	599531.	1.14993	0.574966	−	0.818177i	$-0.305015\pi$
$43$	599531.	1.14993	0.574966	+	0.818177i	$0.305015\pi$
$44$	−280867.	−0.497069
$45$	0	0
$46$	−22984.9	−0.0348169
$47$	−947778.	−1.33157	−0.665785	−	0.746143i	$-0.731903\pi$
$47$	−947778.	−1.33157	−0.665785	+	0.746143i	$0.731903\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−145434.	−0.164540
$51$	0	0
$52$	−580627.	−0.572646
$53$	1.40065e6	1.29230	0.646151	−	0.763210i	$-0.276378\pi$
$53$	1.40065e6	1.29230	0.646151	+	0.763210i	$0.276378\pi$
$54$	0	0
$55$	−848737.	−0.687866
$56$	540710.	0.411440
$57$	0	0
$58$	677625.	0.456028
$59$	1.96604e6	1.24626	0.623132	−	0.782117i	$-0.285860\pi$
$59$	1.96604e6	1.24626	0.623132	+	0.782117i	$0.285860\pi$
$60$	0	0
$61$	726465.	0.409789	0.204894	−	0.978784i	$-0.434315\pi$
$61$	726465.	0.409789	0.204894	+	0.978784i	$0.434315\pi$
$62$	−2.42063e6	−1.28990
$63$	0	0
$64$	2.26606e6	1.08054
$65$	−1.75457e6	−0.792452
$66$	0	0
$67$	756763.	0.307396	0.153698	−	0.988118i	$-0.450882\pi$
$67$	756763.	0.307396	0.153698	+	0.988118i	$0.450882\pi$
$68$	−733841.	−0.283022
$69$	0	0
$70$	399071.	0.139061
$71$	−1.50605e6	−0.499386	−0.249693	−	0.968325i	$-0.580330\pi$
$71$	−1.50605e6	−0.499386	−0.249693	+	0.968325i	$0.580330\pi$
$72$	0	0
$73$	−6.38338e6	−1.92053	−0.960265	−	0.279091i	$-0.909967\pi$
$73$	−6.38338e6	−1.92053	−0.960265	+	0.279091i	$0.909967\pi$
$74$	−1.10240e6	−0.316247
$75$	0	0
$76$	−1.97391e6	−0.515799
$77$	2.32893e6	0.581353
$78$	0	0
$79$	4.43201e6	1.01136	0.505680	−	0.862721i	$-0.331241\pi$
$79$	4.43201e6	1.01136	0.505680	+	0.862721i	$0.331241\pi$
$80$	1.17227e6	0.255983
$81$	0	0
$82$	4.04463e6	0.810084
$83$	−5.38121e6	−1.03301	−0.516507	−	0.856283i	$-0.672768\pi$
$83$	−5.38121e6	−1.03301	−0.516507	+	0.856283i	$0.672768\pi$
$84$	0	0
$85$	−2.21755e6	−0.391659
$86$	−5.58030e6	−0.946047
$87$	0	0
$88$	1.07037e7	1.67434
$89$	7.59795e6	1.14243	0.571217	−	0.820799i	$-0.306471\pi$
$89$	7.59795e6	1.14243	0.571217	+	0.820799i	$0.306471\pi$
$90$	0	0
$91$	4.81453e6	0.669744
$92$	−102149.	−0.0136766
$93$	0	0
$94$	8.82170e6	1.09548
$95$	−5.96485e6	−0.713785
$96$	0	0
$97$	−7.91961e6	−0.881055	−0.440527	−	0.897739i	$-0.645209\pi$
$97$	−7.91961e6	−0.881055	−0.440527	+	0.897739i	$0.645209\pi$
$98$	−1.09505e6	−0.117528
$99$	0	0
$100$	−646335.	−0.0646335
$101$	−1.09106e7	−1.05371	−0.526856	−	0.849954i	$-0.676629\pi$
$101$	−1.09106e7	−1.05371	−0.526856	+	0.849954i	$0.676629\pi$
$102$	0	0
$103$	1.28843e7	1.16180	0.580898	−	0.813977i	$-0.302702\pi$
$103$	1.28843e7	1.16180	0.580898	+	0.813977i	$0.302702\pi$
$104$	2.21274e7	1.92892
$105$	0	0
$106$	−1.30369e7	−1.06317
$107$	1.28614e6	0.101495	0.0507475	−	0.998712i	$-0.483840\pi$
$107$	1.28614e6	0.101495	0.0507475	+	0.998712i	$0.483840\pi$
$108$	0	0
$109$	−2.51631e7	−1.86111	−0.930554	−	0.366155i	$-0.880674\pi$
$109$	−2.51631e7	−1.86111	−0.930554	+	0.366155i	$0.880674\pi$
$110$	7.89984e6	0.565906
$111$	0	0
$112$	−3.21670e6	−0.216345
$113$	−2.88841e6	−0.188315	−0.0941575	−	0.995557i	$-0.530016\pi$
$113$	−2.88841e6	−0.188315	−0.0941575	+	0.995557i	$0.530016\pi$
$114$	0	0
$115$	−308679.	−0.0189263
$116$	3.01149e6	0.179134
$117$	0	0
$118$	−1.82994e7	−1.02530
$119$	6.08497e6	0.331012
$120$	0	0
$121$	2.66155e7	1.36580
$122$	−6.76176e6	−0.337132
$123$	0	0
$124$	−1.07577e7	−0.506693
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−2.03929e7	−0.883418	−0.441709	−	0.897158i	$-0.645628\pi$
$127$	−2.03929e7	−0.883418	−0.441709	+	0.897158i	$0.645628\pi$
$128$	−6.43704e6	−0.271301
$129$	0	0
$130$	1.63311e7	0.651949
$131$	−4.77587e6	−0.185611	−0.0928054	−	0.995684i	$-0.529583\pi$
$131$	−4.77587e6	−0.185611	−0.0928054	+	0.995684i	$0.529583\pi$
$132$	0	0
$133$	1.63676e7	0.603258
$134$	−7.04377e6	−0.252894
$135$	0	0
$136$	2.79663e7	0.953341
$137$	2.50604e7	0.832657	0.416328	−	0.909214i	$-0.363317\pi$
$137$	2.50604e7	0.832657	0.416328	+	0.909214i	$0.363317\pi$
$138$	0	0
$139$	−1.35839e7	−0.429014	−0.214507	−	0.976722i	$-0.568814\pi$
$139$	−1.35839e7	−0.429014	−0.214507	+	0.976722i	$0.568814\pi$
$140$	1.77354e6	0.0546253
$141$	0	0
$142$	1.40180e7	0.410844
$143$	9.53065e7	2.72550
$144$	0	0
$145$	9.10026e6	0.247894
$146$	5.94151e7	1.58002
$147$	0	0
$148$	−4.89926e6	−0.124226
$149$	2.97393e6	0.0736509	0.0368255	−	0.999322i	$-0.488275\pi$
$149$	2.97393e6	0.0736509	0.0368255	+	0.999322i	$0.488275\pi$
$150$	0	0
$151$	2.41110e7	0.569897	0.284949	−	0.958543i	$-0.408023\pi$
$151$	2.41110e7	0.569897	0.284949	+	0.958543i	$0.408023\pi$
$152$	7.52247e7	1.73743
$153$	0	0
$154$	−2.16772e7	−0.478278
$155$	−3.25082e7	−0.701184
$156$	0	0
$157$	3.90767e7	0.805878	0.402939	−	0.915227i	$-0.367989\pi$
$157$	3.90767e7	0.805878	0.402939	+	0.915227i	$0.367989\pi$
$158$	−4.12521e7	−0.832045
$159$	0	0
$160$	1.43115e7	0.276226
$161$	847015.	0.0159956
$162$	0	0
$163$	3.63985e7	0.658305	0.329152	−	0.944277i	$-0.393237\pi$
$163$	3.63985e7	0.658305	0.329152	+	0.944277i	$0.393237\pi$
$164$	1.79751e7	0.318213
$165$	0	0
$166$	5.00870e7	0.849859
$167$	1.42610e7	0.236942	0.118471	−	0.992958i	$-0.462201\pi$
$167$	1.42610e7	0.236942	0.118471	+	0.992958i	$0.462201\pi$
$168$	0	0
$169$	1.34276e8	2.13990
$170$	2.06405e7	0.322217
$171$	0	0
$172$	−2.47999e7	−0.371621
$173$	−4.80295e7	−0.705257	−0.352628	−	0.935763i	$-0.614712\pi$
$173$	−4.80295e7	−0.705257	−0.352628	+	0.935763i	$0.614712\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	−6.36764e7	−0.880409
$177$	0	0
$178$	−7.07199e7	−0.939879
$179$	−5.97259e7	−0.778354	−0.389177	−	0.921163i	$-0.627241\pi$
$179$	−5.97259e7	−0.778354	−0.389177	+	0.921163i	$0.627241\pi$
$180$	0	0
$181$	7.26855e7	0.911115	0.455557	−	0.890206i	$-0.349440\pi$
$181$	7.26855e7	0.911115	0.455557	+	0.890206i	$0.349440\pi$
$182$	−4.48125e7	−0.550997
$183$	0	0
$184$	3.89285e6	0.0460686
$185$	−1.48048e7	−0.171910
$186$	0	0
$187$	1.20456e8	1.34704
$188$	3.92053e7	0.430320
$189$	0	0
$190$	5.55195e7	0.587229
$191$	7.16696e7	0.744249	0.372124	−	0.928183i	$-0.378629\pi$
$191$	7.16696e7	0.744249	0.372124	+	0.928183i	$0.378629\pi$
$192$	0	0
$193$	1.68899e8	1.69113	0.845566	−	0.533871i	$-0.179263\pi$
$193$	1.68899e8	1.69113	0.845566	+	0.533871i	$0.179263\pi$
$194$	7.37139e7	0.724842
$195$	0	0
$196$	−4.86660e6	−0.0461668
$197$	−1.43002e8	−1.33263	−0.666316	−	0.745669i	$-0.732130\pi$
$197$	−1.43002e8	−1.33263	−0.666316	+	0.745669i	$0.732130\pi$
$198$	0	0
$199$	5.31848e7	0.478411	0.239206	−	0.970969i	$-0.423113\pi$
$199$	5.31848e7	0.478411	0.239206	+	0.970969i	$0.423113\pi$
$200$	2.46315e7	0.217714
$201$	0	0
$202$	1.01553e8	0.866887
$203$	−2.49711e7	−0.209508
$204$	0	0
$205$	5.43179e7	0.440357
$206$	−1.19924e8	−0.955807
$207$	0	0
$208$	−1.31636e8	−1.01427
$209$	3.24006e8	2.45494
$210$	0	0
$211$	−3.30646e7	−0.242312	−0.121156	−	0.992633i	$-0.538660\pi$
$211$	−3.30646e7	−0.242312	−0.121156	+	0.992633i	$0.538660\pi$
$212$	−5.79385e7	−0.417630
$213$	0	0
$214$	−1.19711e7	−0.0834998
$215$	−7.49414e7	−0.514265
$216$	0	0
$217$	8.92025e7	0.592608
$218$	2.34212e8	1.53113
$219$	0	0
$220$	3.51084e7	0.222296
$221$	2.49014e8	1.55185
$222$	0	0
$223$	8.22325e6	0.0496565	0.0248283	−	0.999692i	$-0.492096\pi$
$223$	8.22325e6	0.0496565	0.0248283	+	0.999692i	$0.492096\pi$
$224$	−3.92706e7	−0.233453
$225$	0	0
$226$	2.68847e7	0.154926
$227$	−2.49659e8	−1.41663	−0.708314	−	0.705897i	$-0.750544\pi$
$227$	−2.49659e8	−1.41663	−0.708314	+	0.705897i	$0.750544\pi$
$228$	0	0
$229$	−3.17343e8	−1.74624	−0.873122	−	0.487501i	$-0.837908\pi$
$229$	−3.17343e8	−1.74624	−0.873122	+	0.487501i	$0.837908\pi$
$230$	2.87311e6	0.0155706
$231$	0	0
$232$	−1.14766e8	−0.603401
$233$	−1.67569e8	−0.867856	−0.433928	−	0.900948i	$-0.642873\pi$
$233$	−1.67569e8	−0.867856	−0.433928	+	0.900948i	$0.642873\pi$
$234$	0	0
$235$	1.18472e8	0.595496
$236$	−8.13261e7	−0.402752
$237$	0	0
$238$	−5.66374e7	−0.272323
$239$	2.17743e8	1.03169	0.515847	−	0.856681i	$-0.327477\pi$
$239$	2.17743e8	1.03169	0.515847	+	0.856681i	$0.327477\pi$
$240$	0	0
$241$	−3.09126e8	−1.42258	−0.711288	−	0.702901i	$-0.751888\pi$
$241$	−3.09126e8	−1.42258	−0.711288	+	0.702901i	$0.751888\pi$
$242$	−2.47731e8	−1.12364
$243$	0	0
$244$	−3.00505e7	−0.132430
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	6.69807e8	2.82820
$248$	4.09971e8	1.70676
$249$	0	0
$250$	1.81792e7	0.0735844
$251$	−2.51050e8	−1.00208	−0.501040	−	0.865424i	$-0.667049\pi$
$251$	−2.51050e8	−1.00208	−0.501040	+	0.865424i	$0.667049\pi$
$252$	0	0
$253$	1.67672e7	0.0650936
$254$	1.89812e8	0.726786
$255$	0	0
$256$	−2.30141e8	−0.857343
$257$	3.91200e8	1.43758	0.718791	−	0.695226i	$-0.244696\pi$
$257$	3.91200e8	1.43758	0.718791	+	0.695226i	$0.244696\pi$
$258$	0	0
$259$	4.06244e7	0.145291
$260$	7.25784e7	0.256095
$261$	0	0
$262$	4.44527e7	0.152702
$263$	1.12277e8	0.380581	0.190291	−	0.981728i	$-0.439057\pi$
$263$	1.12277e8	0.380581	0.190291	+	0.981728i	$0.439057\pi$
$264$	0	0
$265$	−1.75081e8	−0.577935
$266$	−1.52345e8	−0.496299
$267$	0	0
$268$	−3.13038e7	−0.0993404
$269$	−2.96616e8	−0.929097	−0.464548	−	0.885548i	$-0.653783\pi$
$269$	−2.96616e8	−0.929097	−0.464548	+	0.885548i	$0.653783\pi$
$270$	0	0
$271$	−5.01060e8	−1.52932	−0.764658	−	0.644436i	$-0.777092\pi$
$271$	−5.01060e8	−1.52932	−0.764658	+	0.644436i	$0.777092\pi$
$272$	−1.66372e8	−0.501290
$273$	0	0
$274$	−2.33256e8	−0.685025
$275$	1.06092e8	0.307623
$276$	0	0
$277$	−2.72956e8	−0.771638	−0.385819	−	0.922575i	$-0.626081\pi$
$277$	−2.72956e8	−0.771638	−0.385819	+	0.922575i	$0.626081\pi$
$278$	1.26435e8	0.352949
$279$	0	0
$280$	−6.75888e7	−0.184002
$281$	−5.96358e8	−1.60338	−0.801688	−	0.597743i	$-0.796064\pi$
$281$	−5.96358e8	−1.60338	−0.801688	+	0.597743i	$0.796064\pi$
$282$	0	0
$283$	4.28880e8	1.12482	0.562410	−	0.826858i	$-0.309874\pi$
$283$	4.28880e8	1.12482	0.562410	+	0.826858i	$0.309874\pi$
$284$	6.22986e7	0.161385
$285$	0	0
$286$	−8.87091e8	−2.24227
$287$	−1.49048e8	−0.372169
$288$	0	0
$289$	−9.56161e7	−0.233017
$290$	−8.47031e7	−0.203942
$291$	0	0
$292$	2.64052e8	0.620653
$293$	−2.59401e8	−0.602470	−0.301235	−	0.953550i	$-0.597399\pi$
$293$	−2.59401e8	−0.602470	−0.301235	+	0.953550i	$0.597399\pi$
$294$	0	0
$295$	−2.45755e8	−0.557346
$296$	1.86708e8	0.418448
$297$	0	0
$298$	−2.76806e7	−0.0605925
$299$	3.46623e7	0.0749907
$300$	0	0
$301$	2.05639e8	0.434633
$302$	−2.24420e8	−0.468853
$303$	0	0
$304$	−4.47513e8	−0.913583
$305$	−9.08081e7	−0.183263
$306$	0	0
$307$	1.96834e8	0.388254	0.194127	−	0.980976i	$-0.437813\pi$
$307$	1.96834e8	0.388254	0.194127	+	0.980976i	$0.437813\pi$
$308$	−9.63374e7	−0.187874
$309$	0	0
$310$	3.02579e8	0.576862
$311$	2.98512e8	0.562730	0.281365	−	0.959601i	$-0.409213\pi$
$311$	2.98512e8	0.562730	0.281365	+	0.959601i	$0.409213\pi$
$312$	0	0
$313$	−2.65037e7	−0.0488541	−0.0244271	−	0.999702i	$-0.507776\pi$
$313$	−2.65037e7	−0.0488541	−0.0244271	+	0.999702i	$0.507776\pi$
$314$	−3.63717e8	−0.662994
$315$	0	0
$316$	−1.83332e8	−0.326839
$317$	−2.07006e8	−0.364986	−0.182493	−	0.983207i	$-0.558417\pi$
$317$	−2.07006e8	−0.364986	−0.182493	+	0.983207i	$0.558417\pi$
$318$	0	0
$319$	−4.94318e8	−0.852588
$320$	−2.83258e8	−0.483233
$321$	0	0
$322$	−7.88382e6	−0.0131596
$323$	8.46552e8	1.39780
$324$	0	0
$325$	2.19321e8	0.354395
$326$	−3.38789e8	−0.541586
$327$	0	0
$328$	−6.85020e8	−1.07188
$329$	−3.25088e8	−0.503286
$330$	0	0
$331$	−4.69055e8	−0.710929	−0.355464	−	0.934690i	$-0.615677\pi$
$331$	−4.69055e8	−0.710929	−0.355464	+	0.934690i	$0.615677\pi$
$332$	2.22596e8	0.333837
$333$	0	0
$334$	−1.32738e8	−0.194932
$335$	−9.45954e7	−0.137472
$336$	0	0
$337$	8.39783e7	0.119526	0.0597630	−	0.998213i	$-0.480965\pi$
$337$	8.39783e7	0.119526	0.0597630	+	0.998213i	$0.480965\pi$
$338$	−1.24981e9	−1.76049
$339$	0	0
$340$	9.17301e7	0.126571
$341$	1.76582e9	2.41160
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	9.45109e8	1.25178
$345$	0	0
$346$	4.47048e8	0.580213
$347$	−1.28097e9	−1.64583	−0.822916	−	0.568163i	$-0.807654\pi$
$347$	−1.28097e9	−1.64583	−0.822916	+	0.568163i	$0.807654\pi$
$348$	0	0
$349$	1.71031e8	0.215370	0.107685	−	0.994185i	$-0.465656\pi$
$349$	1.71031e8	0.215370	0.107685	+	0.994185i	$0.465656\pi$
$350$	−4.98838e7	−0.0621901
$351$	0	0
$352$	−7.77386e8	−0.950030
$353$	−9.88908e8	−1.19659	−0.598294	−	0.801277i	$-0.704154\pi$
$353$	−9.88908e8	−1.19659	−0.598294	+	0.801277i	$0.704154\pi$
$354$	0	0
$355$	1.88257e8	0.223332
$356$	−3.14293e8	−0.369198
$357$	0	0
$358$	5.55915e8	0.640351
$359$	−2.20546e8	−0.251576	−0.125788	−	0.992057i	$-0.540146\pi$
$359$	−2.20546e8	−0.251576	−0.125788	+	0.992057i	$0.540146\pi$
$360$	0	0
$361$	1.38322e9	1.54744
$362$	−6.76540e8	−0.749572
$363$	0	0
$364$	−1.99155e8	−0.216440
$365$	7.97923e8	0.858887
$366$	0	0
$367$	9.89258e8	1.04467	0.522334	−	0.852741i	$-0.325061\pi$
$367$	9.89258e8	1.04467	0.522334	+	0.852741i	$0.325061\pi$
$368$	−2.31586e7	−0.0242240
$369$	0	0
$370$	1.37800e8	0.141430
$371$	4.80423e8	0.488444
$372$	0	0
$373$	−1.37864e8	−0.137553	−0.0687764	−	0.997632i	$-0.521910\pi$
$373$	−1.37864e8	−0.137553	−0.0687764	+	0.997632i	$0.521910\pi$
$374$	−1.12117e9	−1.10821
$375$	0	0
$376$	−1.49409e9	−1.44950
$377$	−1.02189e9	−0.982220
$378$	0	0
$379$	−8.30994e8	−0.784080	−0.392040	−	0.919948i	$-0.628231\pi$
$379$	−8.30994e8	−0.784080	−0.392040	+	0.919948i	$0.628231\pi$
$380$	2.46739e8	0.230672
$381$	0	0
$382$	−6.67084e8	−0.612292
$383$	1.89312e9	1.72180	0.860901	−	0.508772i	$-0.169901\pi$
$383$	1.89312e9	1.72180	0.860901	+	0.508772i	$0.169901\pi$
$384$	0	0
$385$	−2.91117e8	−0.259989
$386$	−1.57208e9	−1.39129
$387$	0	0
$388$	3.27598e8	0.284728
$389$	2.11450e9	1.82131	0.910655	−	0.413169i	$-0.135578\pi$
$389$	2.11450e9	1.82131	0.910655	+	0.413169i	$0.135578\pi$
$390$	0	0
$391$	4.38088e7	0.0370632
$392$	1.85464e8	0.155510
$393$	0	0
$394$	1.33103e9	1.09635
$395$	−5.54002e8	−0.452294
$396$	0	0
$397$	−1.96954e9	−1.57978	−0.789891	−	0.613247i	$-0.789863\pi$
$397$	−1.96954e9	−1.57978	−0.789891	+	0.613247i	$0.789863\pi$
$398$	−4.95031e8	−0.393588
$399$	0	0
$400$	−1.46533e8	−0.114479
$401$	−2.12538e9	−1.64601	−0.823003	−	0.568037i	$-0.807703\pi$
$401$	−2.12538e9	−1.64601	−0.823003	+	0.568037i	$0.807703\pi$
$402$	0	0
$403$	3.65042e9	2.77827
$404$	4.51320e8	0.340526
$405$	0	0
$406$	2.32425e8	0.172362
$407$	8.04184e8	0.591255
$408$	0	0
$409$	−2.47760e9	−1.79061	−0.895303	−	0.445457i	$-0.853041\pi$
$409$	−2.47760e9	−1.79061	−0.895303	+	0.445457i	$0.853041\pi$
$410$	−5.05579e8	−0.362281
$411$	0	0
$412$	−5.32964e8	−0.375455
$413$	6.74351e8	0.471044
$414$	0	0
$415$	6.72651e8	0.461978
$416$	−1.60707e9	−1.09448
$417$	0	0
$418$	−3.01577e9	−2.01967
$419$	2.06521e9	1.37156	0.685781	−	0.727808i	$-0.259461\pi$
$419$	2.06521e9	1.37156	0.685781	+	0.727808i	$0.259461\pi$
$420$	0	0
$421$	−2.63000e9	−1.71778	−0.858890	−	0.512160i	$-0.828846\pi$
$421$	−2.63000e9	−1.71778	−0.858890	+	0.512160i	$0.828846\pi$
$422$	3.07757e8	0.199349
$423$	0	0
$424$	2.20800e9	1.40676
$425$	2.77194e8	0.175155
$426$	0	0
$427$	2.49177e8	0.154886
$428$	−5.32017e7	−0.0327999
$429$	0	0
$430$	6.97537e8	0.423085
$431$	−3.92684e8	−0.236251	−0.118125	−	0.992999i	$-0.537688\pi$
$431$	−3.92684e8	−0.236251	−0.118125	+	0.992999i	$0.537688\pi$
$432$	0	0
$433$	2.79693e9	1.65567	0.827836	−	0.560970i	$-0.189572\pi$
$433$	2.79693e9	1.65567	0.827836	+	0.560970i	$0.189572\pi$
$434$	−8.30276e8	−0.487538
$435$	0	0
$436$	1.04088e9	0.601450
$437$	1.17838e8	0.0675463
$438$	0	0
$439$	3.20853e8	0.181001	0.0905003	−	0.995896i	$-0.471153\pi$
$439$	3.20853e8	0.181001	0.0905003	+	0.995896i	$0.471153\pi$
$440$	−1.33796e9	−0.748788
$441$	0	0
$442$	−2.31776e9	−1.27671
$443$	1.82780e9	0.998887	0.499443	−	0.866346i	$-0.333538\pi$
$443$	1.82780e9	0.998887	0.499443	+	0.866346i	$0.333538\pi$
$444$	0	0
$445$	−9.49743e8	−0.510912
$446$	−7.65401e7	−0.0408523
$447$	0	0
$448$	7.77259e8	0.408407
$449$	6.98256e7	0.0364043	0.0182021	−	0.999834i	$-0.494206\pi$
$449$	6.98256e7	0.0364043	0.0182021	+	0.999834i	$0.494206\pi$
$450$	0	0
$451$	−2.95050e9	−1.51453
$452$	1.19481e8	0.0608573
$453$	0	0
$454$	2.32376e9	1.16546
$455$	−6.01816e8	−0.299519
$456$	0	0
$457$	−7.05463e7	−0.0345754	−0.0172877	−	0.999851i	$-0.505503\pi$
$457$	−7.05463e7	−0.0345754	−0.0172877	+	0.999851i	$0.505503\pi$
$458$	2.95376e9	1.43663
$459$	0	0
$460$	1.27686e7	0.00611635
$461$	1.27040e9	0.603929	0.301965	−	0.953319i	$-0.402358\pi$
$461$	1.27040e9	0.603929	0.301965	+	0.953319i	$0.402358\pi$
$462$	0	0
$463$	−1.50939e9	−0.706754	−0.353377	−	0.935481i	$-0.614967\pi$
$463$	−1.50939e9	−0.706754	−0.353377	+	0.935481i	$0.614967\pi$
$464$	6.82746e8	0.317283
$465$	0	0
$466$	1.55969e9	0.713983
$467$	4.10525e9	1.86522	0.932611	−	0.360883i	$-0.117524\pi$
$467$	4.10525e9	1.86522	0.932611	+	0.360883i	$0.117524\pi$
$468$	0	0
$469$	2.59570e8	0.116185
$470$	−1.10271e9	−0.489914
$471$	0	0
$472$	3.09929e9	1.35664
$473$	4.07075e9	1.76873
$474$	0	0
$475$	7.45607e8	0.319214
$476$	−2.51707e8	−0.106972
$477$	0	0
$478$	−2.02670e9	−0.848773
$479$	−3.40485e9	−1.41555	−0.707773	−	0.706440i	$-0.750300\pi$
$479$	−3.40485e9	−1.41555	−0.707773	+	0.706440i	$0.750300\pi$
$480$	0	0
$481$	1.66246e9	0.681153
$482$	2.87727e9	1.17035
$483$	0	0
$484$	−1.10096e9	−0.441381
$485$	9.89952e8	0.394020
$486$	0	0
$487$	2.93163e9	1.15016	0.575079	−	0.818098i	$-0.304971\pi$
$487$	2.93163e9	1.15016	0.575079	+	0.818098i	$0.304971\pi$
$488$	1.14521e9	0.446083
$489$	0	0
$490$	1.36881e8	0.0525603
$491$	−1.54232e9	−0.588014	−0.294007	−	0.955803i	$-0.594989\pi$
$491$	−1.54232e9	−0.588014	−0.294007	+	0.955803i	$0.594989\pi$
$492$	0	0
$493$	−1.29154e9	−0.485449
$494$	−6.23441e9	−2.32676
$495$	0	0
$496$	−2.43893e9	−0.897455
$497$	−5.16577e8	−0.188750
$498$	0	0
$499$	1.16624e9	0.420181	0.210091	−	0.977682i	$-0.432624\pi$
$499$	1.16624e9	0.420181	0.210091	+	0.977682i	$0.432624\pi$
$500$	8.07919e7	0.0289050
$501$	0	0
$502$	2.33672e9	0.824409
$503$	−2.81637e9	−0.986739	−0.493369	−	0.869820i	$-0.664235\pi$
$503$	−2.81637e9	−0.986739	−0.493369	+	0.869820i	$0.664235\pi$
$504$	0	0
$505$	1.36382e9	0.471235
$506$	−1.56065e8	−0.0535524
$507$	0	0
$508$	8.43562e8	0.285492
$509$	1.97626e9	0.664252	0.332126	−	0.943235i	$-0.392234\pi$
$509$	1.97626e9	0.664252	0.332126	+	0.943235i	$0.392234\pi$
$510$	0	0
$511$	−2.18950e9	−0.725892
$512$	2.96604e9	0.976636
$513$	0	0
$514$	−3.64120e9	−1.18270
$515$	−1.61053e9	−0.519571
$516$	0	0
$517$	−6.43531e9	−2.04811
$518$	−3.78122e8	−0.119530
$519$	0	0
$520$	−2.76592e9	−0.862638
$521$	−3.49564e9	−1.08292	−0.541458	−	0.840728i	$-0.682127\pi$
$521$	−3.49564e9	−1.08292	−0.541458	+	0.840728i	$0.682127\pi$
$522$	0	0
$523$	4.42047e9	1.35118	0.675589	−	0.737278i	$-0.263889\pi$
$523$	4.42047e9	1.35118	0.675589	+	0.737278i	$0.263889\pi$
$524$	1.97556e8	0.0599834
$525$	0	0
$526$	−1.04505e9	−0.313103
$527$	4.61367e9	1.37312
$528$	0	0
$529$	−3.39873e9	−0.998209
$530$	1.62961e9	0.475466
$531$	0	0
$532$	−6.77052e8	−0.194954
$533$	−6.09948e9	−1.74481
$534$	0	0
$535$	−1.60767e8	−0.0453900
$536$	1.19297e9	0.334621
$537$	0	0
$538$	2.76083e9	0.764366
$539$	7.98824e8	0.219731
$540$	0	0
$541$	3.20008e9	0.868901	0.434451	−	0.900696i	$-0.356943\pi$
$541$	3.20008e9	0.868901	0.434451	+	0.900696i	$0.356943\pi$
$542$	4.66375e9	1.25817
$543$	0	0
$544$	−2.03113e9	−0.540931
$545$	3.14539e9	0.832313
$546$	0	0
$547$	−5.82743e9	−1.52237	−0.761187	−	0.648533i	$-0.775383\pi$
$547$	−5.82743e9	−1.52237	−0.761187	+	0.648533i	$0.775383\pi$
$548$	−1.03663e9	−0.269088
$549$	0	0
$550$	−9.87481e8	−0.253081
$551$	−3.47403e9	−0.884714
$552$	0	0
$553$	1.52018e9	0.382258
$554$	2.54061e9	0.634825
$555$	0	0
$556$	5.61903e8	0.138643
$557$	6.64579e9	1.62950	0.814748	−	0.579815i	$-0.196875\pi$
$557$	6.64579e9	1.62950	0.814748	+	0.579815i	$0.196875\pi$
$558$	0	0
$559$	8.41534e9	2.03765
$560$	4.02087e8	0.0967524
$561$	0	0
$562$	5.55077e9	1.31909
$563$	−5.27415e9	−1.24558	−0.622792	−	0.782387i	$-0.714002\pi$
$563$	−5.27415e9	−1.24558	−0.622792	+	0.782387i	$0.714002\pi$
$564$	0	0
$565$	3.61052e8	0.0842170
$566$	−3.99191e9	−0.925388
$567$	0	0
$568$	−2.37417e9	−0.543616
$569$	2.71359e9	0.617520	0.308760	−	0.951140i	$-0.400086\pi$
$569$	2.71359e9	0.617520	0.308760	+	0.951140i	$0.400086\pi$
$570$	0	0
$571$	−1.11076e9	−0.249686	−0.124843	−	0.992177i	$-0.539843\pi$
$571$	−1.11076e9	−0.249686	−0.124843	+	0.992177i	$0.539843\pi$
$572$	−3.94240e9	−0.880794
$573$	0	0
$574$	1.38731e9	0.306183
$575$	3.85849e7	0.00846408
$576$	0	0
$577$	7.08441e9	1.53528	0.767642	−	0.640879i	$-0.221430\pi$
$577$	7.08441e9	1.53528	0.767642	+	0.640879i	$0.221430\pi$
$578$	8.89972e8	0.191703
$579$	0	0
$580$	−3.76436e8	−0.0801113
$581$	−1.84575e9	−0.390443
$582$	0	0
$583$	9.51026e9	1.98771
$584$	−1.00629e10	−2.09063
$585$	0	0
$586$	2.41445e9	0.495651
$587$	3.01875e8	0.0616019	0.0308010	−	0.999526i	$-0.490194\pi$
$587$	3.01875e8	0.0616019	0.0308010	+	0.999526i	$0.490194\pi$
$588$	0	0
$589$	1.24100e10	2.50247
$590$	2.28743e9	0.458528
$591$	0	0
$592$	−1.11073e9	−0.220030
$593$	7.32796e9	1.44308	0.721542	−	0.692371i	$-0.243434\pi$
$593$	7.32796e9	1.44308	0.721542	+	0.692371i	$0.243434\pi$
$594$	0	0
$595$	−7.60621e8	−0.148033
$596$	−1.23018e8	−0.0238016
$597$	0	0
$598$	−3.22628e8	−0.0616947
$599$	5.96777e9	1.13454	0.567268	−	0.823533i	$-0.308000\pi$
$599$	5.96777e9	1.13454	0.567268	+	0.823533i	$0.308000\pi$
$600$	0	0
$601$	−9.24419e9	−1.73703	−0.868517	−	0.495660i	$-0.834926\pi$
$601$	−9.24419e9	−1.73703	−0.868517	+	0.495660i	$0.834926\pi$
$602$	−1.91404e9	−0.357572
$603$	0	0
$604$	−9.97364e8	−0.184172
$605$	−3.32694e9	−0.610802
$606$	0	0
$607$	4.53221e9	0.822526	0.411263	−	0.911517i	$-0.365088\pi$
$607$	4.53221e9	0.822526	0.411263	+	0.911517i	$0.365088\pi$
$608$	−5.46341e9	−0.985828
$609$	0	0
$610$	8.45221e8	0.150770
$611$	−1.33035e10	−2.35951
$612$	0	0
$613$	4.90735e9	0.860469	0.430234	−	0.902717i	$-0.358431\pi$
$613$	4.90735e9	0.860469	0.430234	+	0.902717i	$0.358431\pi$
$614$	−1.83209e9	−0.319416
$615$	0	0
$616$	3.67136e9	0.632842
$617$	−1.56563e9	−0.268344	−0.134172	−	0.990958i	$-0.542837\pi$
$617$	−1.56563e9	−0.268344	−0.134172	+	0.990958i	$0.542837\pi$
$618$	0	0
$619$	−3.72881e9	−0.631906	−0.315953	−	0.948775i	$-0.602324\pi$
$619$	−3.72881e9	−0.631906	−0.315953	+	0.948775i	$0.602324\pi$
$620$	1.34472e9	0.226600
$621$	0	0
$622$	−2.77848e9	−0.462957
$623$	2.60610e9	0.431800
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	2.46690e8	0.0401922
$627$	0	0
$628$	−1.61643e9	−0.260434
$629$	2.10115e9	0.336650
$630$	0	0
$631$	7.08510e9	1.12265	0.561324	−	0.827596i	$-0.310292\pi$
$631$	7.08510e9	1.12265	0.561324	+	0.827596i	$0.310292\pi$
$632$	6.98669e9	1.10093
$633$	0	0
$634$	1.92677e9	0.300273
$635$	2.54911e9	0.395076
$636$	0	0
$637$	1.65138e9	0.253140
$638$	4.60100e9	0.701423
$639$	0	0
$640$	8.04630e8	0.121329
$641$	−1.82759e9	−0.274079	−0.137039	−	0.990566i	$-0.543759\pi$
$641$	−1.82759e9	−0.274079	−0.137039	+	0.990566i	$0.543759\pi$
$642$	0	0
$643$	5.18343e9	0.768916	0.384458	−	0.923143i	$-0.374388\pi$
$643$	5.18343e9	0.768916	0.384458	+	0.923143i	$0.374388\pi$
$644$	−3.50372e7	−0.00516926
$645$	0	0
$646$	−7.87951e9	−1.14997
$647$	−1.50002e9	−0.217736	−0.108868	−	0.994056i	$-0.534723\pi$
$647$	−1.50002e9	−0.217736	−0.108868	+	0.994056i	$0.534723\pi$
$648$	0	0
$649$	1.33492e10	1.91690
$650$	−2.04139e9	−0.291560
$651$	0	0
$652$	−1.50564e9	−0.212743
$653$	−7.11169e9	−0.999486	−0.499743	−	0.866174i	$-0.666572\pi$
$653$	−7.11169e9	−0.999486	−0.499743	+	0.866174i	$0.666572\pi$
$654$	0	0
$655$	5.96984e8	0.0830076
$656$	4.07520e9	0.563619
$657$	0	0
$658$	3.02584e9	0.414053
$659$	−1.25636e10	−1.71007	−0.855036	−	0.518568i	$-0.826465\pi$
$659$	−1.25636e10	−1.71007	−0.855036	+	0.518568i	$0.826465\pi$
$660$	0	0
$661$	2.01267e9	0.271061	0.135530	−	0.990773i	$-0.456726\pi$
$661$	2.01267e9	0.271061	0.135530	+	0.990773i	$0.456726\pi$
$662$	4.36586e9	0.584880
$663$	0	0
$664$	−8.48301e9	−1.12451
$665$	−2.04595e9	−0.269785
$666$	0	0
$667$	−1.79780e8	−0.0234585
$668$	−5.89912e8	−0.0765719
$669$	0	0
$670$	8.80472e8	0.113098
$671$	4.93262e9	0.630302
$672$	0	0
$673$	−8.16453e9	−1.03247	−0.516236	−	0.856446i	$-0.672667\pi$
$673$	−8.16453e9	−1.03247	−0.516236	+	0.856446i	$0.672667\pi$
$674$	−7.81651e8	−0.0983339
$675$	0	0
$676$	−5.55437e9	−0.691547
$677$	1.73643e9	0.215079	0.107539	−	0.994201i	$-0.465703\pi$
$677$	1.73643e9	0.215079	0.107539	+	0.994201i	$0.465703\pi$
$678$	0	0
$679$	−2.71643e9	−0.333007
$680$	−3.49578e9	−0.426347
$681$	0	0
$682$	−1.64358e10	−1.98402
$683$	4.08957e9	0.491140	0.245570	−	0.969379i	$-0.421025\pi$
$683$	4.08957e9	0.491140	0.245570	+	0.969379i	$0.421025\pi$
$684$	0	0
$685$	−3.13255e9	−0.372375
$686$	−3.75602e8	−0.0444215
$687$	0	0
$688$	−5.62247e9	−0.658215
$689$	1.96603e10	2.28993
$690$	0	0
$691$	−6.97574e9	−0.804298	−0.402149	−	0.915574i	$-0.631737\pi$
$691$	−6.97574e9	−0.804298	−0.402149	+	0.915574i	$0.631737\pi$
$692$	1.98676e9	0.227916
$693$	0	0
$694$	1.19230e10	1.35402
$695$	1.69798e9	0.191861
$696$	0	0
$697$	−7.70898e9	−0.862348
$698$	−1.59191e9	−0.177185
$699$	0	0
$700$	−2.21693e8	−0.0244292
$701$	5.98140e9	0.655827	0.327914	−	0.944708i	$-0.393654\pi$
$701$	5.98140e9	0.655827	0.327914	+	0.944708i	$0.393654\pi$
$702$	0	0
$703$	5.65174e9	0.613534
$704$	1.53863e10	1.66200
$705$	0	0
$706$	9.20453e9	0.984430
$707$	−3.74232e9	−0.398266
$708$	0	0
$709$	3.47164e9	0.365825	0.182912	−	0.983129i	$-0.441448\pi$
$709$	3.47164e9	0.365825	0.182912	+	0.983129i	$0.441448\pi$
$710$	−1.75225e9	−0.183735
$711$	0	0
$712$	1.19775e10	1.24362
$713$	6.42214e8	0.0663539
$714$	0	0
$715$	−1.19133e10	−1.21888
$716$	2.47059e9	0.251539
$717$	0	0
$718$	2.05280e9	0.206971
$719$	9.20843e9	0.923921	0.461960	−	0.886901i	$-0.347146\pi$
$719$	9.20843e9	0.923921	0.461960	+	0.886901i	$0.347146\pi$
$720$	0	0
$721$	4.41931e9	0.439117
$722$	−1.28746e10	−1.27308
$723$	0	0
$724$	−3.00667e9	−0.294443
$725$	−1.13753e9	−0.110861
$726$	0	0
$727$	−2.27106e9	−0.219209	−0.109605	−	0.993975i	$-0.534958\pi$
$727$	−2.27106e9	−0.219209	−0.109605	+	0.993975i	$0.534958\pi$
$728$	7.58969e9	0.729062
$729$	0	0
$730$	−7.42688e9	−0.706605
$731$	1.06359e10	1.00708
$732$	0	0
$733$	1.15821e9	0.108623	0.0543116	−	0.998524i	$-0.482704\pi$
$733$	1.15821e9	0.108623	0.0543116	+	0.998524i	$0.482704\pi$
$734$	−9.20778e9	−0.859447
$735$	0	0
$736$	−2.82729e8	−0.0261396
$737$	5.13834e9	0.472810
$738$	0	0
$739$	−2.29733e9	−0.209396	−0.104698	−	0.994504i	$-0.533388\pi$
$739$	−2.29733e9	−0.209396	−0.104698	+	0.994504i	$0.533388\pi$
$740$	6.12407e8	0.0555558
$741$	0	0
$742$	−4.47166e9	−0.401842
$743$	−2.57569e9	−0.230374	−0.115187	−	0.993344i	$-0.536747\pi$
$743$	−2.57569e9	−0.230374	−0.115187	+	0.993344i	$0.536747\pi$
$744$	0	0
$745$	−3.71741e8	−0.0329377
$746$	1.28320e9	0.113164
$747$	0	0
$748$	−4.98270e9	−0.435321
$749$	4.41146e8	0.0383615
$750$	0	0
$751$	3.45645e9	0.297777	0.148888	−	0.988854i	$-0.452430\pi$
$751$	3.45645e9	0.297777	0.148888	+	0.988854i	$0.452430\pi$
$752$	8.88838e9	0.762184
$753$	0	0
$754$	9.51150e9	0.808070
$755$	−3.01388e9	−0.254866
$756$	0	0
$757$	−1.23571e10	−1.03534	−0.517669	−	0.855581i	$-0.673200\pi$
$757$	−1.23571e10	−1.03534	−0.517669	+	0.855581i	$0.673200\pi$
$758$	7.73470e9	0.645062
$759$	0	0
$760$	−9.40308e9	−0.777003
$761$	−1.74837e10	−1.43810	−0.719048	−	0.694960i	$-0.755422\pi$
$761$	−1.74837e10	−1.43810	−0.719048	+	0.694960i	$0.755422\pi$
$762$	0	0
$763$	−8.63094e9	−0.703433
$764$	−2.96465e9	−0.240517
$765$	0	0
$766$	−1.76208e10	−1.41652
$767$	2.75964e10	2.20835
$768$	0	0
$769$	1.28053e10	1.01542	0.507710	−	0.861528i	$-0.330492\pi$
$769$	1.28053e10	1.01542	0.507710	+	0.861528i	$0.330492\pi$
$770$	2.70965e9	0.213892
$771$	0	0
$772$	−6.98660e9	−0.546519
$773$	−5.44386e9	−0.423915	−0.211958	−	0.977279i	$-0.567984\pi$
$773$	−5.44386e9	−0.423915	−0.211958	+	0.977279i	$0.567984\pi$
$774$	0	0
$775$	4.06352e9	0.313579
$776$	−1.24846e10	−0.959088
$777$	0	0
$778$	−1.96813e10	−1.49839
$779$	−2.07359e10	−1.57160
$780$	0	0
$781$	−1.02259e10	−0.768113
$782$	−4.07762e8	−0.0304918
$783$	0	0
$784$	−1.10333e9	−0.0817707
$785$	−4.88459e9	−0.360400
$786$	0	0
$787$	2.46027e10	1.79916	0.899581	−	0.436753i	$-0.143872\pi$
$787$	2.46027e10	1.79916	0.899581	+	0.436753i	$0.143872\pi$
$788$	5.91535e9	0.430664
$789$	0	0
$790$	5.15652e9	0.372102
$791$	−9.90726e8	−0.0711764
$792$	0	0
$793$	1.01970e10	0.726136
$794$	1.83320e10	1.29968
$795$	0	0
$796$	−2.20001e9	−0.154607
$797$	6.71494e9	0.469827	0.234914	−	0.972016i	$-0.424519\pi$
$797$	6.71494e9	0.469827	0.234914	+	0.972016i	$0.424519\pi$
$798$	0	0
$799$	−1.68140e10	−1.16616
$800$	−1.78893e9	−0.123532
$801$	0	0
$802$	1.97826e10	1.35417
$803$	−4.33425e10	−2.95399
$804$	0	0
$805$	−1.05877e8	−0.00715345
$806$	−3.39772e10	−2.28568
$807$	0	0
$808$	−1.71996e10	−1.14704
$809$	5.89951e9	0.391738	0.195869	−	0.980630i	$-0.437247\pi$
$809$	5.89951e9	0.391738	0.195869	+	0.980630i	$0.437247\pi$
$810$	0	0
$811$	−9.53678e9	−0.627811	−0.313905	−	0.949454i	$-0.601637\pi$
$811$	−9.53678e9	−0.627811	−0.313905	+	0.949454i	$0.601637\pi$
$812$	1.03294e9	0.0677064
$813$	0	0
$814$	−7.48516e9	−0.486425
$815$	−4.54982e9	−0.294403
$816$	0	0
$817$	2.86089e10	1.83537
$818$	2.30610e10	1.47313
$819$	0	0
$820$	−2.24689e9	−0.142309
$821$	1.72163e10	1.08577	0.542887	−	0.839806i	$-0.317331\pi$
$821$	1.72163e10	1.08577	0.542887	+	0.839806i	$0.317331\pi$
$822$	0	0
$823$	−7.08048e9	−0.442755	−0.221377	−	0.975188i	$-0.571055\pi$
$823$	−7.08048e9	−0.442755	−0.221377	+	0.975188i	$0.571055\pi$
$824$	2.03110e10	1.26469
$825$	0	0
$826$	−6.27671e9	−0.387527
$827$	2.54934e9	0.156732	0.0783660	−	0.996925i	$-0.475030\pi$
$827$	2.54934e9	0.156732	0.0783660	+	0.996925i	$0.475030\pi$
$828$	0	0
$829$	−1.14560e10	−0.698378	−0.349189	−	0.937052i	$-0.613543\pi$
$829$	−1.14560e10	−0.698378	−0.349189	+	0.937052i	$0.613543\pi$
$830$	−6.26088e9	−0.380069
$831$	0	0
$832$	3.18076e10	1.91470
$833$	2.08714e9	0.125111
$834$	0	0
$835$	−1.78262e9	−0.105964
$836$	−1.34026e10	−0.793357
$837$	0	0
$838$	−1.92225e10	−1.12838
$839$	−2.17022e10	−1.26864	−0.634319	−	0.773071i	$-0.718719\pi$
$839$	−2.17022e10	−1.26864	−0.634319	+	0.773071i	$0.718719\pi$
$840$	0	0
$841$	−1.19497e10	−0.692743
$842$	2.44794e10	1.41321
$843$	0	0
$844$	1.36773e9	0.0783073
$845$	−1.67845e10	−0.956993
$846$	0	0
$847$	9.12911e9	0.516222
$848$	−1.31355e10	−0.739707
$849$	0	0
$850$	−2.58006e9	−0.144100
$851$	2.92475e8	0.0162681
$852$	0	0
$853$	−1.96358e10	−1.08324	−0.541622	−	0.840622i	$-0.682190\pi$
$853$	−1.96358e10	−1.08324	−0.541622	+	0.840622i	$0.682190\pi$
$854$	−2.31929e9	−0.127424
$855$	0	0
$856$	2.02749e9	0.110484
$857$	−1.17660e10	−0.638550	−0.319275	−	0.947662i	$-0.603439\pi$
$857$	−1.17660e10	−0.638550	−0.319275	+	0.947662i	$0.603439\pi$
$858$	0	0
$859$	9.94494e9	0.535335	0.267668	−	0.963511i	$-0.413747\pi$
$859$	9.94494e9	0.535335	0.267668	+	0.963511i	$0.413747\pi$
$860$	3.09998e9	0.166194
$861$	0	0
$862$	3.65502e9	0.194363
$863$	2.33995e10	1.23928	0.619640	−	0.784886i	$-0.287279\pi$
$863$	2.33995e10	1.23928	0.619640	+	0.784886i	$0.287279\pi$
$864$	0	0
$865$	6.00369e9	0.315400
$866$	−2.60332e10	−1.36212
$867$	0	0
$868$	−3.68990e9	−0.191512
$869$	3.00929e10	1.55559
$870$	0	0
$871$	1.06223e10	0.544698
$872$	−3.96675e10	−2.02594
$873$	0	0
$874$	−1.09681e9	−0.0555703
$875$	−6.69922e8	−0.0338062
$876$	0	0
$877$	−1.83836e10	−0.920306	−0.460153	−	0.887840i	$-0.652206\pi$
$877$	−1.83836e10	−0.920306	−0.460153	+	0.887840i	$0.652206\pi$
$878$	−2.98642e9	−0.148909
$879$	0	0
$880$	7.95956e9	0.393731
$881$	−1.42714e10	−0.703153	−0.351577	−	0.936159i	$-0.614354\pi$
$881$	−1.42714e10	−0.703153	−0.351577	+	0.936159i	$0.614354\pi$
$882$	0	0
$883$	8.73358e9	0.426904	0.213452	−	0.976954i	$-0.431529\pi$
$883$	8.73358e9	0.426904	0.213452	+	0.976954i	$0.431529\pi$
$884$	−1.03006e10	−0.501509
$885$	0	0
$886$	−1.70128e10	−0.821782
$887$	1.26531e10	0.608787	0.304394	−	0.952546i	$-0.401546\pi$
$887$	1.26531e10	0.608787	0.304394	+	0.952546i	$0.401546\pi$
$888$	0	0
$889$	−6.99477e9	−0.333901
$890$	8.83999e9	0.420326
$891$	0	0
$892$	−3.40158e8	−0.0160474
$893$	−4.52269e10	−2.12528
$894$	0	0
$895$	7.46574e9	0.348091
$896$	−2.20790e9	−0.102542
$897$	0	0
$898$	−6.49920e8	−0.0299497
$899$	−1.89333e10	−0.869095
$900$	0	0
$901$	2.48481e10	1.13177
$902$	2.74626e10	1.24600
$903$	0	0
$904$	−4.55334e9	−0.204994
$905$	−9.08569e9	−0.407463
$906$	0	0
$907$	1.32930e10	0.591558	0.295779	−	0.955256i	$-0.404421\pi$
$907$	1.32930e10	0.591558	0.295779	+	0.955256i	$0.404421\pi$
$908$	1.03272e10	0.457809
$909$	0	0
$910$	5.60157e9	0.246414
$911$	−3.22600e10	−1.41368	−0.706839	−	0.707375i	$-0.749879\pi$
$911$	−3.22600e10	−1.41368	−0.706839	+	0.707375i	$0.749879\pi$
$912$	0	0
$913$	−3.65378e10	−1.58889
$914$	6.56629e8	0.0284452
$915$	0	0
$916$	1.31270e10	0.564330
$917$	−1.63812e9	−0.0701543
$918$	0	0
$919$	2.07823e9	0.0883262	0.0441631	−	0.999024i	$-0.485938\pi$
$919$	2.07823e9	0.0883262	0.0441631	+	0.999024i	$0.485938\pi$
$920$	−4.86606e8	−0.0206025
$921$	0	0
$922$	−1.18246e10	−0.496851
$923$	−2.11398e10	−0.884901
$924$	0	0
$925$	1.85060e9	0.0768805
$926$	1.40491e10	0.581445
$927$	0	0
$928$	8.33523e9	0.342373
$929$	2.99678e10	1.22631	0.613155	−	0.789962i	$-0.289900\pi$
$929$	2.99678e10	1.22631	0.613155	+	0.789962i	$0.289900\pi$
$930$	0	0
$931$	5.61407e9	0.228010
$932$	6.93156e9	0.280463
$933$	0	0
$934$	−3.82107e10	−1.53452
$935$	−1.50569e10	−0.602416
$936$	0	0
$937$	−6.83304e9	−0.271347	−0.135674	−	0.990754i	$-0.543320\pi$
$937$	−6.83304e9	−0.271347	−0.135674	+	0.990754i	$0.543320\pi$
$938$	−2.41601e9	−0.0955850
$939$	0	0
$940$	−4.90066e9	−0.192445
$941$	−1.47984e9	−0.0578962	−0.0289481	−	0.999581i	$-0.509216\pi$
$941$	−1.47984e9	−0.0578962	−0.0289481	+	0.999581i	$0.509216\pi$
$942$	0	0
$943$	−1.07308e9	−0.0416715
$944$	−1.84378e10	−0.713355
$945$	0	0
$946$	−3.78896e10	−1.45513
$947$	1.45643e10	0.557270	0.278635	−	0.960397i	$-0.410118\pi$
$947$	1.45643e10	0.557270	0.278635	+	0.960397i	$0.410118\pi$
$948$	0	0
$949$	−8.96006e10	−3.40313
$950$	−6.93993e9	−0.262617
$951$	0	0
$952$	9.59243e9	0.360329
$953$	7.86423e8	0.0294328	0.0147164	−	0.999892i	$-0.495315\pi$
$953$	7.86423e8	0.0294328	0.0147164	+	0.999892i	$0.495315\pi$
$954$	0	0
$955$	−8.95870e9	−0.332838
$956$	−9.00702e9	−0.333410
$957$	0	0
$958$	3.16916e10	1.16457
$959$	8.59572e9	0.314715
$960$	0	0
$961$	4.01214e10	1.45829
$962$	−1.54738e10	−0.560383
$963$	0	0
$964$	1.27871e10	0.459730
$965$	−2.11124e10	−0.756297
$966$	0	0
$967$	1.60421e10	0.570516	0.285258	−	0.958451i	$-0.407921\pi$
$967$	1.60421e10	0.570516	0.285258	+	0.958451i	$0.407921\pi$
$968$	4.19570e10	1.48676
$969$	0	0
$970$	−9.21424e9	−0.324159
$971$	−1.32951e10	−0.466040	−0.233020	−	0.972472i	$-0.574861\pi$
$971$	−1.32951e10	−0.466040	−0.233020	+	0.972472i	$0.574861\pi$
$972$	0	0
$973$	−4.65927e9	−0.162152
$974$	−2.72869e10	−0.946232
$975$	0	0
$976$	−6.81287e9	−0.234561
$977$	−1.82382e10	−0.625679	−0.312839	−	0.949806i	$-0.601280\pi$
$977$	−1.82382e10	−0.625679	−0.312839	+	0.949806i	$0.601280\pi$
$978$	0	0
$979$	5.15892e10	1.75719
$980$	6.08326e8	0.0206464
$981$	0	0
$982$	1.43555e10	0.483758
$983$	1.40430e9	0.0471544	0.0235772	−	0.999722i	$-0.492494\pi$
$983$	1.40430e9	0.0471544	0.0235772	+	0.999722i	$0.492494\pi$
$984$	0	0
$985$	1.78753e10	0.595971
$986$	1.20213e10	0.399378
$987$	0	0
$988$	−2.77069e10	−0.913983
$989$	1.48050e9	0.0486656
$990$	0	0
$991$	4.54958e9	0.148495	0.0742477	−	0.997240i	$-0.476344\pi$
$991$	4.54958e9	0.148495	0.0742477	+	0.997240i	$0.476344\pi$
$992$	−2.97753e10	−0.968424
$993$	0	0
$994$	4.80818e9	0.155284
$995$	−6.64810e9	−0.213952
$996$	0	0
$997$	−2.67218e10	−0.853950	−0.426975	−	0.904263i	$-0.640421\pi$
$997$	−2.67218e10	−0.853950	−0.426975	+	0.904263i	$0.640421\pi$
$998$	−1.08551e10	−0.345682
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-9.30777 q^{2} -41.3655 q^{4} -125.000 q^{5} +343.000 q^{7} +1576.41 q^{8} +1163.47 q^{10} +6789.89 q^{11} +14036.5 q^{13} -3192.56 q^{14} -9378.12 q^{16} +17740.4 q^{17} +47718.8 q^{19} +5170.68 q^{20} -63198.8 q^{22} +2469.43 q^{23} +15625.0 q^{25} -130649. q^{26} -14188.4 q^{28} -72802.1 q^{29} +260065. q^{31} -114492. q^{32} -165124. q^{34} -42875.0 q^{35} +118438. q^{37} -444156. q^{38} -197052. q^{40} -434543. q^{41} +599531. q^{43} -280867. q^{44} -22984.9 q^{46} -947778. q^{47} +117649. q^{49} -145434. q^{50} -580627. q^{52} +1.40065e6 q^{53} -848737. q^{55} +540710. q^{56} +677625. q^{58} +1.96604e6 q^{59} +726465. q^{61} -2.42063e6 q^{62} +2.26606e6 q^{64} -1.75457e6 q^{65} +756763. q^{67} -733841. q^{68} +399071. q^{70} -1.50605e6 q^{71} -6.38338e6 q^{73} -1.10240e6 q^{74} -1.97391e6 q^{76} +2.32893e6 q^{77} +4.43201e6 q^{79} +1.17227e6 q^{80} +4.04463e6 q^{82} -5.38121e6 q^{83} -2.21755e6 q^{85} -5.58030e6 q^{86} +1.07037e7 q^{88} +7.59795e6 q^{89} +4.81453e6 q^{91} -102149. q^{92} +8.82170e6 q^{94} -5.96485e6 q^{95} -7.91961e6 q^{97} -1.09505e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) \) 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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