LMFDB - Embedded newform 98.7.b.c.97.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,7,Mod(97,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.97"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	98.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,256,0,0,0,0,-1512] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$22.5453001947$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 $ x^8 - 2x^7 + 285x^6 + 282x^5 + 62091x^4 + 29260x^3 + 4838750x^2 + 2401000*x + 294122500
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$-6.30576 + 10.9219i$ of defining polynomial
Character	$\chi$	$=$	98.97
Dual form			98.7.b.c.97.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.65685 q^{2} +42.4218i q^{3} +32.0000 q^{4} -187.462i q^{5} -239.974i q^{6} -181.019 q^{8} -1070.61 q^{9} +1060.44i q^{10} +1111.85 q^{11} +1357.50i q^{12} +706.517i q^{13} +7952.48 q^{15} +1024.00 q^{16} +8469.41i q^{17} +6056.28 q^{18} +25.4105i q^{19} -5998.78i q^{20} -6289.56 q^{22} -849.476 q^{23} -7679.17i q^{24} -19517.0 q^{25} -3996.67i q^{26} -14491.7i q^{27} -15109.4 q^{29} -44986.0 q^{30} +1392.35i q^{31} -5792.62 q^{32} +47166.6i q^{33} -47910.2i q^{34} -34259.5 q^{36} -9292.94 q^{37} -143.743i q^{38} -29971.7 q^{39} +33934.2i q^{40} +109829. i q^{41} -45569.8 q^{43} +35579.1 q^{44} +200699. i q^{45} +4805.36 q^{46} +138408. i q^{47} +43439.9i q^{48} +110405. q^{50} -359288. q^{51} +22608.6i q^{52} +61875.9 q^{53} +81977.5i q^{54} -208429. i q^{55} -1077.96 q^{57} +85471.8 q^{58} -78537.2i q^{59} +254479. q^{60} -174701. i q^{61} -7876.29i q^{62} +32768.0 q^{64} +132445. q^{65} -266815. i q^{66} -435949. q^{67} +271021. i q^{68} -36036.3i q^{69} -561559. q^{71} +193801. q^{72} +221899. i q^{73} +52568.8 q^{74} -827946. i q^{75} +813.135i q^{76} +169546. q^{78} +666478. q^{79} -191961. i q^{80} -165710. q^{81} -621289. i q^{82} +653635. i q^{83} +1.58769e6 q^{85} +257782. q^{86} -640969. i q^{87} -201266. q^{88} +132332. i q^{89} -1.13532e6i q^{90} -27183.2 q^{92} -59065.8 q^{93} -782956. i q^{94} +4763.49 q^{95} -245733. i q^{96} +1.59480e6i q^{97} -1.19036e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 256 q^{4} - 1512 q^{9} + 2712 q^{11} + 27144 q^{15} + 8192 q^{16} + 13632 q^{18} + 25248 q^{22} + 8256 q^{23} - 9328 q^{25} - 30312 q^{29} - 19296 q^{30} - 48384 q^{36} + 12248 q^{37} - 201528 q^{39}+ \cdots - 4625928 q^{99}+O(q^{100}) $ 8 * q + 256 * q^4 - 1512 * q^9 + 2712 * q^11 + 27144 * q^15 + 8192 * q^16 + 13632 * q^18 + 25248 * q^22 + 8256 * q^23 - 9328 * q^25 - 30312 * q^29 - 19296 * q^30 - 48384 * q^36 + 12248 * q^37 - 201528 * q^39 - 297376 * q^43 + 86784 * q^44 + 388128 * q^46 + 38784 * q^50 - 507384 * q^51 - 556968 * q^53 - 81288 * q^57 + 339648 * q^58 + 868608 * q^60 + 262144 * q^64 - 16632 * q^65 - 1297616 * q^67 + 190128 * q^71 + 436224 * q^72 + 2316288 * q^74 + 1122432 * q^78 - 140192 * q^79 - 2355840 * q^81 + 2190984 * q^85 + 1146048 * q^86 + 807936 * q^88 + 264192 * q^92 + 46728 * q^93 - 1451280 * q^95 - 4625928 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−5.65685	−0.707107
$2$	−5.65685	−0.707107
$3$	42.4218i	1.57118i	0.618749	+	0.785589i	$0.287640\pi$
$3$	42.4218i	1.57118i	−0.618749	+	0.785589i	$0.712360\pi$
$4$	32.0000	0.500000
$5$	− 187.462i	− 1.49970i	−0.661610	−	0.749848i	$-0.730127\pi$
$5$	− 187.462i	− 1.49970i	0.661610	−	0.749848i	$-0.269873\pi$
$6$	− 239.974i	− 1.11099i
$7$	0	0
$7$	0	0
$8$	−181.019	−0.353553
$9$	−1070.61	−1.46860
$10$	1060.44i	1.06044i
$11$	1111.85	0.835348	0.417674	−	0.908597i	$-0.362845\pi$
$11$	1111.85	0.835348	0.417674	+	0.908597i	$0.362845\pi$
$12$	1357.50i	0.785589i
$13$	706.517i	0.321583i	0.986988	+	0.160791i	$0.0514046\pi$
$13$	706.517i	0.321583i	−0.986988	+	0.160791i	$0.948595\pi$
$14$	0	0
$15$	7952.48	2.35629
$16$	1024.00	0.250000
$17$	8469.41i	1.72388i	0.507012	+	0.861939i	$0.330750\pi$
$17$	8469.41i	1.72388i	−0.507012	+	0.861939i	$0.669250\pi$
$18$	6056.28	1.03846
$19$	25.4105i	0.00370469i	0.999998	+	0.00185234i	$0.000589620\pi$
$19$	25.4105i	0.00370469i	−0.999998	+	0.00185234i	$0.999410\pi$
$20$	− 5998.78i	− 0.749848i
$21$	0	0
$22$	−6289.56	−0.590680
$23$	−849.476	−0.0698180	−0.0349090	−	0.999390i	$-0.511114\pi$
$23$	−849.476	−0.0698180	−0.0349090	+	0.999390i	$0.511114\pi$
$24$	− 7679.17i	− 0.555495i
$25$	−19517.0	−1.24909
$26$	− 3996.67i	− 0.227393i
$27$	− 14491.7i	− 0.736255i
$28$	0	0
$29$	−15109.4	−0.619518	−0.309759	−	0.950815i	$-0.600248\pi$
$29$	−15109.4	−0.619518	−0.309759	+	0.950815i	$0.600248\pi$
$30$	−44986.0	−1.66615
$31$	1392.35i	0.0467371i	0.999727	+	0.0233686i	$0.00743912\pi$
$31$	1392.35i	0.0467371i	−0.999727	+	0.0233686i	$0.992561\pi$
$32$	−5792.62	−0.176777
$33$	47166.6i	1.31248i
$34$	− 47910.2i	− 1.21897i
$35$	0	0
$36$	−34259.5	−0.734300
$37$	−9292.94	−0.183463	−0.0917313	−	0.995784i	$-0.529240\pi$
$37$	−9292.94	−0.183463	−0.0917313	+	0.995784i	$0.529240\pi$
$38$	− 143.743i	− 0.00261961i
$39$	−29971.7	−0.505264
$40$	33934.2i	0.530222i
$41$	109829.i	1.59356i	0.604272	+	0.796778i	$0.293464\pi$
$41$	109829.i	1.59356i	−0.604272	+	0.796778i	$0.706536\pi$
$42$	0	0
$43$	−45569.8	−0.573155	−0.286577	−	0.958057i	$-0.592518\pi$
$43$	−45569.8	−0.573155	−0.286577	+	0.958057i	$0.592518\pi$
$44$	35579.1	0.417674
$45$	200699.i	2.20245i
$46$	4805.36	0.0493688
$47$	138408.i	1.33312i	0.745452	+	0.666559i	$0.232234\pi$
$47$	138408.i	1.33312i	−0.745452	+	0.666559i	$0.767766\pi$
$48$	43439.9i	0.392795i
$49$	0	0
$50$	110405.	0.883238
$51$	−359288.	−2.70852
$52$	22608.6i	0.160791i
$53$	61875.9	0.415617	0.207809	−	0.978170i	$-0.433367\pi$
$53$	61875.9	0.415617	0.207809	+	0.978170i	$0.433367\pi$
$54$	81977.5i	0.520611i
$55$	− 208429.i	− 1.25277i
$56$	0	0
$57$	−1077.96	−0.00582073
$58$	85471.8	0.438066
$59$	− 78537.2i	− 0.382402i	−0.981551	−	0.191201i	$-0.938762\pi$
$59$	− 78537.2i	− 0.382402i	0.981551	−	0.191201i	$-0.0612382\pi$
$60$	254479.	1.17814
$61$	− 174701.i	− 0.769673i	−0.922985	−	0.384836i	$-0.874258\pi$
$61$	− 174701.i	− 0.769673i	0.922985	−	0.384836i	$-0.125742\pi$
$62$	− 7876.29i	− 0.0330481i
$63$	0	0
$64$	32768.0	0.125000
$65$	132445.	0.482276
$66$	− 266815.i	− 0.928064i
$67$	−435949.	−1.44948	−0.724738	−	0.689025i	$-0.758039\pi$
$67$	−435949.	−1.44948	−0.724738	+	0.689025i	$0.758039\pi$
$68$	271021.i	0.861939i
$69$	− 36036.3i	− 0.109697i
$70$	0	0
$71$	−561559.	−1.56899	−0.784495	−	0.620135i	$-0.787078\pi$
$71$	−561559.	−1.56899	−0.784495	+	0.620135i	$0.787078\pi$
$72$	193801.	0.519229
$73$	221899.i	0.570408i	0.958467	+	0.285204i	$0.0920614\pi$
$73$	221899.i	0.570408i	−0.958467	+	0.285204i	$0.907939\pi$
$74$	52568.8	0.129728
$75$	− 827946.i	− 1.96254i
$76$	813.135i	0.00185234i
$77$	0	0
$78$	169546.	0.357275
$79$	666478.	1.35178	0.675888	−	0.737004i	$-0.263760\pi$
$79$	666478.	1.35178	0.675888	+	0.737004i	$0.263760\pi$
$80$	− 191961.i	− 0.374924i
$81$	−165710.	−0.311812
$82$	− 621289.i	− 1.12681i
$83$	653635.i	1.14314i	0.820552	+	0.571572i	$0.193666\pi$
$83$	653635.i	1.14314i	−0.820552	+	0.571572i	$0.806334\pi$
$84$	0	0
$85$	1.58769e6	2.58529
$86$	257782.	0.405281
$87$	− 640969.i	− 0.973374i
$88$	−201266.	−0.295340
$89$	132332.i	0.187713i	0.995586	+	0.0938565i	$0.0299195\pi$
$89$	132332.i	0.187713i	−0.995586	+	0.0938565i	$0.970081\pi$
$90$	− 1.13532e6i	− 1.55737i
$91$	0	0
$92$	−27183.2	−0.0349090
$93$	−59065.8	−0.0734323
$94$	− 782956.i	− 0.942657i
$95$	4763.49	0.00555591
$96$	− 245733.i	− 0.277748i
$97$	1.59480e6i	1.74739i	0.486470	+	0.873697i	$0.338284\pi$
$97$	1.59480e6i	1.74739i	−0.486470	+	0.873697i	$0.661716\pi$
$98$	0	0
$99$	−1.19036e6	−1.22679
$100$	−624543.	−0.624543
$101$	1.30345e6i	1.26512i	0.774513	+	0.632559i	$0.217995\pi$
$101$	1.30345e6i	1.26512i	−0.774513	+	0.632559i	$0.782005\pi$
$102$	2.03244e6	1.91521
$103$	− 6637.07i	− 0.00607386i	−0.999995	−	0.00303693i	$-0.999033\pi$
$103$	− 6637.07i	− 0.00607386i	0.999995	−	0.00303693i	$-0.000966686\pi$
$104$	− 127893.i	− 0.113697i
$105$	0	0
$106$	−350023.	−0.293886
$107$	1.31198e6	1.07096	0.535482	−	0.844547i	$-0.320130\pi$
$107$	1.31198e6	1.07096	0.535482	+	0.844547i	$0.320130\pi$
$108$	− 463735.i	− 0.368128i
$109$	1.54261e6	1.19118	0.595589	−	0.803290i	$-0.296919\pi$
$109$	1.54261e6	1.19118	0.595589	+	0.803290i	$0.296919\pi$
$110$	1.17905e6i	0.885840i
$111$	− 394223.i	− 0.288253i
$112$	0	0
$113$	1.02212e6	0.708382	0.354191	−	0.935173i	$-0.384756\pi$
$113$	1.02212e6	0.708382	0.354191	+	0.935173i	$0.384756\pi$
$114$	6097.85	0.00411588
$115$	159244.i	0.104706i
$116$	−483502.	−0.309759
$117$	− 756405.i	− 0.472277i
$118$	444274.i	0.270399i
$119$	0	0
$120$	−1.43955e6	−0.833074
$121$	−535355.	−0.302194
$122$	988259.i	0.544241i
$123$	−4.65916e6	−2.50376
$124$	44555.0i	0.0233686i
$125$	729599.i	0.373554i
$126$	0	0
$127$	752429.	0.367329	0.183664	−	0.982989i	$-0.441204\pi$
$127$	752429.	0.367329	0.183664	+	0.982989i	$0.441204\pi$
$128$	−185364.	−0.0883883
$129$	− 1.93315e6i	− 0.900528i
$130$	−749223.	−0.341021
$131$	261503.i	0.116322i	0.998307	+	0.0581611i	$0.0185237\pi$
$131$	261503.i	0.116322i	−0.998307	+	0.0581611i	$0.981476\pi$
$132$	1.50933e6i	0.656240i
$133$	0	0
$134$	2.46610e6	1.02493
$135$	−2.71665e6	−1.10416
$136$	− 1.53313e6i	− 0.609483i
$137$	−698943.	−0.271819	−0.135910	−	0.990721i	$-0.543396\pi$
$137$	−698943.	−0.271819	−0.135910	+	0.990721i	$0.543396\pi$
$138$	203852.i	0.0775672i
$139$	− 1.60298e6i	− 0.596875i	−0.954429	−	0.298438i	$-0.903535\pi$
$139$	− 1.60298e6i	− 0.596875i	0.954429	−	0.298438i	$-0.0964655\pi$
$140$	0	0
$141$	−5.87153e6	−2.09457
$142$	3.17666e6	1.10944
$143$	785540.i	0.268633i
$144$	−1.09630e6	−0.367150
$145$	2.83244e6i	0.929089i
$146$	− 1.25525e6i	− 0.403340i
$147$	0	0
$148$	−297374.	−0.0917313
$149$	−4.17203e6	−1.26121	−0.630607	−	0.776103i	$-0.717194\pi$
$149$	−4.17203e6	−1.26121	−0.630607	+	0.776103i	$0.717194\pi$
$150$	4.68357e6i	1.38772i
$151$	6.20466e6	1.80213	0.901067	−	0.433680i	$-0.142785\pi$
$151$	6.20466e6	1.80213	0.901067	+	0.433680i	$0.142785\pi$
$152$	− 4599.79i	− 0.00130981i
$153$	− 9.06744e6i	− 2.53169i
$154$	0	0
$155$	261012.	0.0700914
$156$	−959096.	−0.252632
$157$	1.96839e6i	0.508641i	0.967120	+	0.254320i	$0.0818518\pi$
$157$	1.96839e6i	0.508641i	−0.967120	+	0.254320i	$0.918148\pi$
$158$	−3.77017e6	−0.955850
$159$	2.62489e6i	0.653009i
$160$	1.08590e6i	0.265111i
$161$	0	0
$162$	937397.	0.220485
$163$	1.21870e6	0.281407	0.140704	−	0.990052i	$-0.455064\pi$
$163$	1.21870e6	0.281407	0.140704	+	0.990052i	$0.455064\pi$
$164$	3.51454e6i	0.796778i
$165$	8.84194e6	1.96832
$166$	− 3.69752e6i	− 0.808325i
$167$	− 638229.i	− 0.137034i	−0.997650	−	0.0685168i	$-0.978173\pi$
$167$	− 638229.i	− 0.137034i	0.997650	−	0.0685168i	$-0.0218267\pi$
$168$	0	0
$169$	4.32764e6	0.896585
$170$	−8.98134e6	−1.82808
$171$	− 27204.7i	− 0.00544071i
$172$	−1.45823e6	−0.286577
$173$	621939.i	0.120118i	0.998195	+	0.0600592i	$0.0191289\pi$
$173$	621939.i	0.120118i	−0.998195	+	0.0600592i	$0.980871\pi$
$174$	3.62587e6i	0.688279i
$175$	0	0
$176$	1.13853e6	0.208837
$177$	3.33169e6	0.600821
$178$	− 748582.i	− 0.132733i
$179$	−5.52123e6	−0.962668	−0.481334	−	0.876537i	$-0.659848\pi$
$179$	−5.52123e6	−0.962668	−0.481334	+	0.876537i	$0.659848\pi$
$180$	6.42236e6i	1.10123i
$181$	3.79950e6i	0.640753i	0.947290	+	0.320377i	$0.103809\pi$
$181$	3.79950e6i	0.640753i	−0.947290	+	0.320377i	$0.896191\pi$
$182$	0	0
$183$	7.41114e6	1.20929
$184$	153772.	0.0246844
$185$	1.74207e6i	0.275138i
$186$	334127.	0.0519245
$187$	9.41670e6i	1.44004i
$188$	4.42907e6i	0.666559i
$189$	0	0
$190$	−26946.4	−0.00392862
$191$	−5.21125e6	−0.747896	−0.373948	−	0.927450i	$-0.621996\pi$
$191$	−5.21125e6	−0.747896	−0.373948	+	0.927450i	$0.621996\pi$
$192$	1.39008e6i	0.196397i
$193$	−3.62527e6	−0.504277	−0.252138	−	0.967691i	$-0.581134\pi$
$193$	−3.62527e6	−0.504277	−0.252138	+	0.967691i	$0.581134\pi$
$194$	− 9.02155e6i	− 1.23559i
$195$	5.61856e6i	0.757742i
$196$	0	0
$197$	−6.36761e6	−0.832871	−0.416435	−	0.909165i	$-0.636721\pi$
$197$	−6.36761e6	−0.832871	−0.416435	+	0.909165i	$0.636721\pi$
$198$	6.73367e6	0.867473
$199$	− 4.20350e6i	− 0.533399i	−0.963780	−	0.266699i	$-0.914067\pi$
$199$	− 4.20350e6i	− 0.533399i	0.963780	−	0.266699i	$-0.0859331\pi$
$200$	3.53295e6	0.441619
$201$	− 1.84937e7i	− 2.27739i
$202$	− 7.37343e6i	− 0.894573i
$203$	0	0
$204$	−1.14972e7	−1.35426
$205$	2.05888e7	2.38985
$206$	37544.9i	0.00429487i
$207$	909457.	0.102535
$208$	723474.i	0.0803957i
$209$	28252.6i	0.00309470i
$210$	0	0
$211$	−1.33548e7	−1.42164	−0.710822	−	0.703372i	$-0.751677\pi$
$211$	−1.33548e7	−1.42164	−0.710822	+	0.703372i	$0.751677\pi$
$212$	1.98003e6	0.207809
$213$	− 2.38223e7i	− 2.46516i
$214$	−7.42166e6	−0.757285
$215$	8.54260e6i	0.859557i
$216$	2.62328e6i	0.260306i
$217$	0	0
$218$	−8.72631e6	−0.842289
$219$	−9.41334e6	−0.896213
$220$	− 6.66973e6i	− 0.626384i
$221$	−5.98379e6	−0.554369
$222$	2.23006e6i	0.203825i
$223$	− 6.98597e6i	− 0.629959i	−0.949098	−	0.314980i	$-0.898002\pi$
$223$	− 6.98597e6i	− 0.629959i	0.949098	−	0.314980i	$-0.101998\pi$
$224$	0	0
$225$	2.08951e7	1.83441
$226$	−5.78200e6	−0.500902
$227$	480522.i	0.0410805i	0.999789	+	0.0205403i	$0.00653863\pi$
$227$	480522.i	0.0410805i	−0.999789	+	0.0205403i	$0.993461\pi$
$228$	−34494.7	−0.00291036
$229$	− 1.14561e7i	− 0.953959i	−0.878914	−	0.476980i	$-0.841732\pi$
$229$	− 1.14561e7i	− 0.953959i	0.878914	−	0.476980i	$-0.158268\pi$
$230$	− 900822.i	− 0.0740382i
$231$	0	0
$232$	2.73510e6	0.219033
$233$	1.43203e7	1.13210	0.566048	−	0.824372i	$-0.308472\pi$
$233$	1.43203e7	1.13210	0.566048	+	0.824372i	$0.308472\pi$
$234$	4.27887e6i	0.333950i
$235$	2.59463e7	1.99927
$236$	− 2.51319e6i	− 0.191201i
$237$	2.82732e7i	2.12388i
$238$	0	0
$239$	9.15640e6	0.670704	0.335352	−	0.942093i	$-0.391145\pi$
$239$	9.15640e6	0.670704	0.335352	+	0.942093i	$0.391145\pi$
$240$	8.14333e6	0.589072
$241$	− 2.49142e7i	− 1.77990i	−0.456056	−	0.889951i	$-0.650738\pi$
$241$	− 2.49142e7i	− 1.77990i	0.456056	−	0.889951i	$-0.349262\pi$
$242$	3.02843e6	0.213683
$243$	− 1.75942e7i	− 1.22617i
$244$	− 5.59043e6i	− 0.384836i
$245$	0	0
$246$	2.63562e7	1.77043
$247$	−17952.9	−0.00119136
$248$	− 252041.i	− 0.0165241i
$249$	−2.77284e7	−1.79608
$250$	− 4.12723e6i	− 0.264143i
$251$	− 2.06023e7i	− 1.30285i	−0.758714	−	0.651424i	$-0.774172\pi$
$251$	− 2.06023e7i	− 1.30285i	0.758714	−	0.651424i	$-0.225828\pi$
$252$	0	0
$253$	−944488.	−0.0583223
$254$	−4.25638e6	−0.259740
$255$	6.73528e7i	4.06195i
$256$	1.04858e6	0.0625000
$257$	1.24381e7i	0.732747i	0.930468	+	0.366373i	$0.119401\pi$
$257$	1.24381e7i	0.732747i	−0.930468	+	0.366373i	$0.880599\pi$
$258$	1.09356e7i	0.636769i
$259$	0	0
$260$	4.23824e6	0.241138
$261$	1.61763e7	0.909825
$262$	− 1.47928e6i	− 0.0822523i
$263$	2.84736e7	1.56522	0.782609	−	0.622514i	$-0.213888\pi$
$263$	2.84736e7	1.56522	0.782609	+	0.622514i	$0.213888\pi$
$264$	− 8.53807e6i	− 0.464032i
$265$	− 1.15994e7i	− 0.623299i
$266$	0	0
$267$	−5.61375e6	−0.294930
$268$	−1.39504e7	−0.724738
$269$	1.46502e7i	0.752640i	0.926490	+	0.376320i	$0.122811\pi$
$269$	1.46502e7i	0.752640i	−0.926490	+	0.376320i	$0.877189\pi$
$270$	1.53677e7	0.780758
$271$	1.48469e7i	0.745982i	0.927835	+	0.372991i	$0.121668\pi$
$271$	1.48469e7i	0.745982i	−0.927835	+	0.372991i	$0.878332\pi$
$272$	8.67268e6i	0.430969i
$273$	0	0
$274$	3.95382e6	0.192205
$275$	−2.16999e7	−1.04342
$276$	− 1.15316e6i	− 0.0548483i
$277$	2.26304e7	1.06476	0.532381	−	0.846505i	$-0.321297\pi$
$277$	2.26304e7	1.06476	0.532381	+	0.846505i	$0.321297\pi$
$278$	9.06782e6i	0.422055i
$279$	− 1.49066e6i	− 0.0686382i
$280$	0	0
$281$	−2.66513e7	−1.20116	−0.600579	−	0.799565i	$-0.705063\pi$
$281$	−2.66513e7	−1.20116	−0.600579	+	0.799565i	$0.705063\pi$
$282$	3.32144e7	1.48108
$283$	− 2.60337e7i	− 1.14862i	−0.818638	−	0.574310i	$-0.805270\pi$
$283$	− 2.60337e7i	− 1.14862i	0.818638	−	0.574310i	$-0.194730\pi$
$284$	−1.79699e7	−0.784495
$285$	202076.i	0.00872932i
$286$	− 4.44368e6i	− 0.189953i
$287$	0	0
$288$	6.20164e6	0.259614
$289$	−4.75934e7	−1.97175
$290$	− 1.60227e7i	− 0.656965i
$291$	−6.76543e7	−2.74547
$292$	7.10075e6i	0.285204i
$293$	− 2.40799e6i	− 0.0957310i	−0.998854	−	0.0478655i	$-0.984758\pi$
$293$	− 2.40799e6i	− 0.0957310i	0.998854	−	0.0478655i	$-0.0152419\pi$
$294$	0	0
$295$	−1.47227e7	−0.573486
$296$	1.68220e6	0.0648639
$297$	− 1.61126e7i	− 0.615029i
$298$	2.36006e7	0.891813
$299$	− 600169.i	− 0.0224523i
$300$	− 2.64943e7i	− 0.981269i
$301$	0	0
$302$	−3.50989e7	−1.27430
$303$	−5.52948e7	−1.98772
$304$	26020.3i	0 0.000926172i
$305$	−3.27498e7	−1.15427
$306$	5.12932e7i	1.79017i
$307$	− 1.92678e7i	− 0.665911i	−0.942942	−	0.332956i	$-0.891954\pi$
$307$	− 1.92678e7i	− 0.665911i	0.942942	−	0.332956i	$-0.108046\pi$
$308$	0	0
$309$	281556.	0.00954311
$310$	−1.47651e6	−0.0495621
$311$	3.76870e7i	1.25288i	0.779469	+	0.626441i	$0.215489\pi$
$311$	3.76870e7i	1.25288i	−0.779469	+	0.626441i	$0.784511\pi$
$312$	5.42547e6	0.178638
$313$	− 4.01419e7i	− 1.30908i	−0.756028	−	0.654539i	$-0.772863\pi$
$313$	− 4.01419e7i	− 1.30908i	0.756028	−	0.654539i	$-0.227137\pi$
$314$	− 1.11349e7i	− 0.359663i
$315$	0	0
$316$	2.13273e7	0.675888
$317$	5.34295e7	1.67727	0.838636	−	0.544693i	$-0.183354\pi$
$317$	5.34295e7	1.67727	0.838636	+	0.544693i	$0.183354\pi$
$318$	− 1.48486e7i	− 0.461747i
$319$	−1.67994e7	−0.517513
$320$	− 6.14275e6i	− 0.187462i
$321$	5.56564e7i	1.68267i
$322$	0	0
$323$	−215212.	−0.00638643
$324$	−5.30272e6	−0.155906
$325$	− 1.37891e7i	− 0.401685i
$326$	−6.89403e6	−0.198985
$327$	6.54403e7i	1.87155i
$328$	− 1.98813e7i	− 0.563407i
$329$	0	0
$330$	−5.00176e7	−1.39181
$331$	2.09302e7	0.577150	0.288575	−	0.957457i	$-0.406819\pi$
$331$	2.09302e7	0.577150	0.288575	+	0.957457i	$0.406819\pi$
$332$	2.09163e7i	0.571572i
$333$	9.94911e6	0.269433
$334$	3.61037e6i	0.0968973i
$335$	8.17238e7i	2.17377i
$336$	0	0
$337$	5.12346e6	0.133867	0.0669335	−	0.997757i	$-0.478678\pi$
$337$	5.12346e6	0.133867	0.0669335	+	0.997757i	$0.478678\pi$
$338$	−2.44808e7	−0.633981
$339$	4.33603e7i	1.11299i
$340$	5.08062e7	1.29265
$341$	1.54808e6i	0.0390417i
$342$	153893.i	0.00384716i
$343$	0	0
$344$	8.24901e6	0.202641
$345$	−6.75544e6	−0.164511
$346$	− 3.51822e6i	− 0.0849365i
$347$	1.33552e6	0.0319640	0.0159820	−	0.999872i	$-0.494913\pi$
$347$	1.33552e6	0.0319640	0.0159820	+	0.999872i	$0.494913\pi$
$348$	− 2.05110e7i	− 0.486687i
$349$	− 5.85989e7i	− 1.37852i	−0.724514	−	0.689260i	$-0.757936\pi$
$349$	− 5.85989e7i	− 1.37852i	0.724514	−	0.689260i	$-0.242064\pi$
$350$	0	0
$351$	1.02386e7	0.236767
$352$	−6.44051e6	−0.147670
$353$	5.81711e7i	1.32246i	0.750182	+	0.661231i	$0.229966\pi$
$353$	5.81711e7i	1.32246i	−0.750182	+	0.661231i	$0.770034\pi$
$354$	−1.88469e7	−0.424845
$355$	1.05271e8i	2.35301i
$356$	4.23462e6i	0.0938565i
$357$	0	0
$358$	3.12328e7	0.680709
$359$	−5.58319e7	−1.20670	−0.603350	−	0.797477i	$-0.706168\pi$
$359$	−5.58319e7	−1.20670	−0.603350	+	0.797477i	$0.706168\pi$
$360$	− 3.63303e7i	− 0.778685i
$361$	4.70452e7	0.999986
$362$	− 2.14932e7i	− 0.453081i
$363$	− 2.27107e7i	− 0.474801i
$364$	0	0
$365$	4.15975e7	0.855439
$366$	−4.19237e7	−0.855099
$367$	− 8.68077e6i	− 0.175614i	−0.996137	−	0.0878072i	$-0.972014\pi$
$367$	− 8.68077e6i	− 0.175614i	0.996137	−	0.0878072i	$-0.0279859\pi$
$368$	−869863.	−0.0174545
$369$	− 1.17585e8i	− 2.34030i
$370$	− 9.85465e6i	− 0.194552i
$371$	0	0
$372$	−1.89011e6	−0.0367162
$373$	6.88948e7	1.32758	0.663789	−	0.747920i	$-0.268947\pi$
$373$	6.88948e7	1.32758	0.663789	+	0.747920i	$0.268947\pi$
$374$	− 5.32689e7i	− 1.01826i
$375$	−3.09509e7	−0.586921
$376$	− 2.50546e7i	− 0.471329i
$377$	− 1.06751e7i	− 0.199226i
$378$	0	0
$379$	2.82976e7	0.519795	0.259897	−	0.965636i	$-0.416311\pi$
$379$	2.82976e7	0.519795	0.259897	+	0.965636i	$0.416311\pi$
$380$	152432.	0.00277795
$381$	3.19194e7i	0.577139i
$382$	2.94793e7	0.528843
$383$	− 1.07897e8i	− 1.92050i	−0.279149	−	0.960248i	$-0.590052\pi$
$383$	− 1.07897e8i	− 1.92050i	0.279149	−	0.960248i	$-0.409948\pi$
$384$	− 7.86347e6i	− 0.138874i
$385$	0	0
$386$	2.05076e7	0.356578
$387$	4.87875e7	0.841735
$388$	5.10336e7i	0.873697i
$389$	6.47117e7	1.09935	0.549673	−	0.835380i	$-0.314752\pi$
$389$	6.47117e7	1.09935	0.549673	+	0.835380i	$0.314752\pi$
$390$	− 3.17834e7i	− 0.535804i
$391$	− 7.19456e6i	− 0.120358i
$392$	0	0
$393$	−1.10934e7	−0.182763
$394$	3.60206e7	0.588928
$395$	− 1.24939e8i	− 2.02725i
$396$	−3.80914e7	−0.613396
$397$	7.56776e7i	1.20947i	0.796426	+	0.604736i	$0.206721\pi$
$397$	7.56776e7i	1.20947i	−0.796426	+	0.604736i	$0.793279\pi$
$398$	2.37786e7i	0.377170i
$399$	0	0
$400$	−1.99854e7	−0.312272
$401$	3.11432e6	0.0482981	0.0241490	−	0.999708i	$-0.492312\pi$
$401$	3.11432e6	0.0482981	0.0241490	+	0.999708i	$0.492312\pi$
$402$	1.04616e8i	1.61035i
$403$	−983716.	−0.0150298
$404$	4.17104e7i	0.632559i
$405$	3.10643e7i	0.467624i
$406$	0	0
$407$	−1.03323e7	−0.153255
$408$	6.50380e7	0.957606
$409$	− 1.21267e8i	− 1.77245i	−0.463256	−	0.886225i	$-0.653319\pi$
$409$	− 1.21267e8i	− 1.77245i	0.463256	−	0.886225i	$-0.346681\pi$
$410$	−1.16468e8	−1.68988
$411$	− 2.96504e7i	− 0.427076i
$412$	− 212386.i	− 0.00303693i
$413$	0	0
$414$	−5.14467e6	−0.0725030
$415$	1.22532e8	1.71437
$416$	− 4.09259e6i	− 0.0568483i
$417$	6.80013e7	0.937797
$418$	− 159821.i	− 0.00218829i
$419$	− 1.07947e7i	− 0.146746i	−0.997305	−	0.0733732i	$-0.976624\pi$
$419$	− 1.07947e7i	− 0.146746i	0.997305	−	0.0733732i	$-0.0233764\pi$
$420$	0	0
$421$	−5.16567e6	−0.0692278	−0.0346139	−	0.999401i	$-0.511020\pi$
$421$	−5.16567e6	−0.0692278	−0.0346139	+	0.999401i	$0.511020\pi$
$422$	7.55463e7	1.00525
$423$	− 1.48181e8i	− 1.95782i
$424$	−1.12007e7	−0.146943
$425$	− 1.65297e8i	− 2.15327i
$426$	1.34759e8i	1.74313i
$427$	0	0
$428$	4.19832e7	0.535482
$429$	−3.33240e7	−0.422071
$430$	− 4.83243e7i	− 0.607799i
$431$	−1.14858e8	−1.43460	−0.717301	−	0.696764i	$-0.754623\pi$
$431$	−1.14858e8	−1.43460	−0.717301	+	0.696764i	$0.754623\pi$
$432$	− 1.48395e7i	− 0.184064i
$433$	− 2.10551e7i	− 0.259355i	−0.991556	−	0.129677i	$-0.958606\pi$
$433$	− 2.10551e7i	− 0.259355i	0.991556	−	0.129677i	$-0.0413942\pi$
$434$	0	0
$435$	−1.20157e8	−1.45976
$436$	4.93635e7	0.595589
$437$	− 21585.6i	0 0.000258654i
$438$	5.32499e7	0.633719
$439$	6.57735e6i	0.0777423i	0.999244	+	0.0388711i	$0.0123762\pi$
$439$	6.57735e6i	0.0777423i	−0.999244	+	0.0388711i	$0.987624\pi$
$440$	3.77297e7i	0.442920i
$441$	0	0
$442$	3.38494e7	0.391998
$443$	−7.79214e7	−0.896284	−0.448142	−	0.893962i	$-0.647914\pi$
$443$	−7.79214e7	−0.896284	−0.448142	+	0.893962i	$0.647914\pi$
$444$	− 1.26151e7i	− 0.144126i
$445$	2.48072e7	0.281512
$446$	3.95186e7i	0.445448i
$447$	− 1.76985e8i	− 1.98159i
$448$	0	0
$449$	1.49956e8	1.65663	0.828316	−	0.560261i	$-0.189299\pi$
$449$	1.49956e8	1.65663	0.828316	+	0.560261i	$0.189299\pi$
$450$	−1.18200e8	−1.29712
$451$	1.22114e8i	1.33117i
$452$	3.27079e7	0.354191
$453$	2.63213e8i	2.83147i
$454$	− 2.71825e6i	− 0.0290483i
$455$	0	0
$456$	195131.	0.00205794
$457$	−1.26875e8	−1.32932	−0.664658	−	0.747147i	$-0.731423\pi$
$457$	−1.26875e8	−1.32932	−0.664658	+	0.747147i	$0.731423\pi$
$458$	6.48054e7i	0.674551i
$459$	1.22736e8	1.26921
$460$	5.09582e6i	0.0523529i
$461$	8.34728e7i	0.852005i	0.904722	+	0.426002i	$0.140078\pi$
$461$	8.34728e7i	0.852005i	−0.904722	+	0.426002i	$0.859922\pi$
$462$	0	0
$463$	−1.31072e8	−1.32058	−0.660292	−	0.751009i	$-0.729567\pi$
$463$	−1.31072e8	−1.32058	−0.660292	+	0.751009i	$0.729567\pi$
$464$	−1.54721e7	−0.154880
$465$	1.10726e7i	0.110126i
$466$	−8.10077e7	−0.800513
$467$	− 8.69972e7i	− 0.854190i	−0.904207	−	0.427095i	$-0.859537\pi$
$467$	− 8.69972e7i	− 0.854190i	0.904207	−	0.427095i	$-0.140463\pi$
$468$	− 2.42049e7i	− 0.236138i
$469$	0	0
$470$	−1.46774e8	−1.41370
$471$	−8.35025e7	−0.799165
$472$	1.42168e7i	0.135199i
$473$	−5.06667e7	−0.478783
$474$	− 1.59937e8i	− 1.50181i
$475$	− 495936.i	− 0.00462748i
$476$	0	0
$477$	−6.62449e7	−0.610376
$478$	−5.17964e7	−0.474259
$479$	1.40986e8i	1.28283i	0.767195	+	0.641413i	$0.221652\pi$
$479$	1.40986e8i	1.28283i	−0.767195	+	0.641413i	$0.778348\pi$
$480$	−4.60657e7	−0.416537
$481$	− 6.56562e6i	− 0.0589984i
$482$	1.40936e8i	1.25858i
$483$	0	0
$484$	−1.71314e7	−0.151097
$485$	2.98964e8	2.62056
$486$	9.95277e7i	0.867032i
$487$	1.43030e8	1.23834	0.619171	−	0.785256i	$-0.287469\pi$
$487$	1.43030e8	1.23834	0.619171	+	0.785256i	$0.287469\pi$
$488$	3.16243e7i	0.272120i
$489$	5.16996e7i	0.442141i
$490$	0	0
$491$	9.10418e7	0.769124	0.384562	−	0.923099i	$-0.374353\pi$
$491$	9.10418e7	0.769124	0.384562	+	0.923099i	$0.374353\pi$
$492$	−1.49093e8	−1.25188
$493$	− 1.27968e8i	− 1.06797i
$494$	101557.	0.000842422 0
$495$	2.23146e8i	1.83982i
$496$	1.42576e6i	0.0116843i
$497$	0	0
$498$	1.56855e8	1.27002
$499$	−5.66145e7	−0.455645	−0.227822	−	0.973703i	$-0.573161\pi$
$499$	−5.66145e7	−0.455645	−0.227822	+	0.973703i	$0.573161\pi$
$500$	2.33472e7i	0.186777i
$501$	2.70748e7	0.215304
$502$	1.16544e8i	0.921253i
$503$	− 5.77667e7i	− 0.453914i	−0.973905	−	0.226957i	$-0.927122\pi$
$503$	− 5.77667e7i	− 0.453914i	0.973905	−	0.226957i	$-0.0728777\pi$
$504$	0	0
$505$	2.44348e8	1.89729
$506$	5.34283e6	0.0412401
$507$	1.83586e8i	1.40869i
$508$	2.40777e7	0.183664
$509$	1.65313e8i	1.25359i	0.779186	+	0.626793i	$0.215633\pi$
$509$	1.65313e8i	1.25359i	−0.779186	+	0.626793i	$0.784367\pi$
$510$	− 3.81005e8i	− 2.87224i
$511$	0	0
$512$	−5.93164e6	−0.0441942
$513$	368241.	0.00272760
$514$	− 7.03604e7i	− 0.518130i
$515$	−1.24420e6	−0.00910894
$516$	− 6.18609e7i	− 0.450264i
$517$	1.53889e8i	1.11362i
$518$	0	0
$519$	−2.63838e7	−0.188727
$520$	−2.39751e7	−0.170510
$521$	− 1.24209e6i	− 0.00878297i	−0.999990	−	0.00439148i	$-0.998602\pi$
$521$	− 1.24209e6i	− 0.00878297i	0.999990	−	0.00439148i	$-0.00139786\pi$
$522$	−9.15070e7	−0.643343
$523$	4.06365e7i	0.284061i	0.989862	+	0.142030i	$0.0453631\pi$
$523$	4.06365e7i	0.284061i	−0.989862	+	0.142030i	$0.954637\pi$
$524$	8.36810e6i	0.0581611i
$525$	0	0
$526$	−1.61071e8	−1.10678
$527$	−1.17923e7	−0.0805691
$528$	4.82986e7i	0.328120i
$529$	−1.47314e8	−0.995125
$530$	6.56159e7i	0.440739i
$531$	8.40828e7i	0.561595i
$532$	0	0
$533$	−7.75964e7	−0.512460
$534$	3.17562e7	0.208547
$535$	− 2.45946e8i	− 1.60612i
$536$	7.89152e7	0.512467
$537$	− 2.34221e8i	− 1.51252i
$538$	− 8.28742e7i	− 0.532197i
$539$	0	0
$540$	−8.69326e7	−0.552080
$541$	−1.95600e8	−1.23532	−0.617658	−	0.786447i	$-0.711918\pi$
$541$	−1.95600e8	−1.23532	−0.617658	+	0.786447i	$0.711918\pi$
$542$	− 8.39868e7i	− 0.527489i
$543$	−1.61182e8	−1.00674
$544$	− 4.90601e7i	− 0.304741i
$545$	− 2.89180e8i	− 1.78640i
$546$	0	0
$547$	9.54532e7	0.583215	0.291607	−	0.956538i	$-0.405810\pi$
$547$	9.54532e7	0.583215	0.291607	+	0.956538i	$0.405810\pi$
$548$	−2.23662e7	−0.135910
$549$	1.87037e8i	1.13034i
$550$	1.22753e8	0.737811
$551$	− 383938.i	− 0.00229512i
$552$	6.52327e6i	0.0387836i
$553$	0	0
$554$	−1.28017e8	−0.752900
$555$	−7.39018e7	−0.432291
$556$	− 5.12953e7i	− 0.298438i
$557$	−1.20225e8	−0.695714	−0.347857	−	0.937548i	$-0.613091\pi$
$557$	−1.20225e8	−0.695714	−0.347857	+	0.937548i	$0.613091\pi$
$558$	8.43244e6i	0.0485345i
$559$	− 3.21959e7i	− 0.184317i
$560$	0	0
$561$	−3.99473e8	−2.26256
$562$	1.50763e8	0.849347
$563$	1.70936e7i	0.0957872i	0.998852	+	0.0478936i	$0.0152508\pi$
$563$	1.70936e7i	0.0957872i	−0.998852	+	0.0478936i	$0.984749\pi$
$564$	−1.87889e8	−1.04728
$565$	− 1.91609e8i	− 1.06236i
$566$	1.47269e8i	0.812197i
$567$	0	0
$568$	1.01653e8	0.554722
$569$	−8.25688e7	−0.448208	−0.224104	−	0.974565i	$-0.571945\pi$
$569$	−8.25688e7	−0.448208	−0.224104	+	0.974565i	$0.571945\pi$
$570$	− 1.14311e6i	− 0.00617256i
$571$	2.98470e7	0.160322	0.0801609	−	0.996782i	$-0.474457\pi$
$571$	2.98470e7	0.160322	0.0801609	+	0.996782i	$0.474457\pi$
$572$	2.51373e7i	0.134317i
$573$	− 2.21070e8i	− 1.17508i
$574$	0	0
$575$	1.65792e7	0.0872088
$576$	−3.50817e7	−0.183575
$577$	2.97159e8i	1.54690i	0.633860	+	0.773448i	$0.281470\pi$
$577$	2.97159e8i	1.54690i	−0.633860	+	0.773448i	$0.718530\pi$
$578$	2.69229e8	1.39424
$579$	− 1.53791e8i	− 0.792309i
$580$	9.06382e7i	0.464544i
$581$	0	0
$582$	3.82711e8	1.94134
$583$	6.87965e7	0.347185
$584$	− 4.01679e7i	− 0.201670i
$585$	−1.41797e8	−0.708271
$586$	1.36217e7i	0.0676920i
$587$	− 3.09361e7i	− 0.152950i	−0.997071	−	0.0764752i	$-0.975633\pi$
$587$	− 3.09361e7i	− 0.152950i	0.997071	−	0.0764752i	$-0.0243666\pi$
$588$	0	0
$589$	−35380.1	−0.000173146 0
$590$	8.32844e7	0.405516
$591$	− 2.70125e8i	− 1.30859i
$592$	−9.51597e6	−0.0458657
$593$	2.23573e8i	1.07215i	0.844171	+	0.536075i	$0.180093\pi$
$593$	2.23573e8i	1.07215i	−0.844171	+	0.536075i	$0.819907\pi$
$594$	9.11465e7i	0.434891i
$595$	0	0
$596$	−1.33505e8	−0.630607
$597$	1.78320e8	0.838065
$598$	3.39507e6i	0.0158762i
$599$	2.29575e8	1.06818	0.534090	−	0.845427i	$-0.320654\pi$
$599$	2.29575e8	1.06818	0.534090	+	0.845427i	$0.320654\pi$
$600$	1.49874e8i	0.693862i
$601$	2.45821e8i	1.13239i	0.824272	+	0.566194i	$0.191585\pi$
$601$	2.45821e8i	1.13239i	−0.824272	+	0.566194i	$0.808415\pi$
$602$	0	0
$603$	4.66731e8	2.12870
$604$	1.98549e8	0.901067
$605$	1.00359e8i	0.453199i
$606$	3.12794e8	1.40553
$607$	− 2.25750e7i	− 0.100940i	−0.998726	−	0.0504699i	$-0.983928\pi$
$607$	− 2.25750e7i	− 0.100940i	0.998726	−	0.0504699i	$-0.0160719\pi$
$608$	− 147193.i	0 0.000654903i
$609$	0	0
$610$	1.85261e8	0.816196
$611$	−9.77879e7	−0.428708
$612$	− 2.90158e8i	− 1.26584i
$613$	9.79984e7	0.425439	0.212720	−	0.977113i	$-0.431768\pi$
$613$	9.79984e7	0.425439	0.212720	+	0.977113i	$0.431768\pi$
$614$	1.08995e8i	0.470870i
$615$	8.73416e8i	3.75488i
$616$	0	0
$617$	1.51949e7	0.0646907	0.0323454	−	0.999477i	$-0.489702\pi$
$617$	1.51949e7	0.0646907	0.0323454	+	0.999477i	$0.489702\pi$
$618$	−1.59272e6	−0.00674800
$619$	4.11478e8i	1.73490i	0.497524	+	0.867450i	$0.334243\pi$
$619$	4.11478e8i	1.73490i	−0.497524	+	0.867450i	$0.665757\pi$
$620$	8.35238e6	0.0350457
$621$	1.23104e7i	0.0514039i
$622$	− 2.13190e8i	− 0.885922i
$623$	0	0
$624$	−3.06911e7	−0.126316
$625$	−1.68181e8	−0.688869
$626$	2.27077e8i	0.925658i
$627$	−1.19853e6	−0.00486233
$628$	6.29883e7i	0.254320i
$629$	− 7.87057e7i	− 0.316267i
$630$	0	0
$631$	−3.87782e8	−1.54348	−0.771738	−	0.635941i	$-0.780612\pi$
$631$	−3.87782e8	−1.54348	−0.771738	+	0.635941i	$0.780612\pi$
$632$	−1.20645e8	−0.477925
$633$	− 5.66536e8i	− 2.23366i
$634$	−3.02243e8	−1.18601
$635$	− 1.41052e8i	− 0.550881i
$636$	8.39963e7i	0.326504i
$637$	0	0
$638$	9.50317e7	0.365937
$639$	6.01210e8	2.30422
$640$	3.47487e7i	0.132556i
$641$	−1.40298e8	−0.532694	−0.266347	−	0.963877i	$-0.585817\pi$
$641$	−1.40298e8	−0.532694	−0.266347	+	0.963877i	$0.585817\pi$
$642$	− 3.14840e8i	− 1.18983i
$643$	− 2.07832e8i	− 0.781772i	−0.920439	−	0.390886i	$-0.872169\pi$
$643$	− 2.07832e8i	− 0.781772i	0.920439	−	0.390886i	$-0.127831\pi$
$644$	0	0
$645$	−3.62393e8	−1.35052
$646$	1.21742e6	0.00451589
$647$	− 6.95528e7i	− 0.256804i	−0.991722	−	0.128402i	$-0.959015\pi$
$647$	− 6.95528e7i	− 0.256804i	0.991722	−	0.128402i	$-0.0409848\pi$
$648$	2.99967e7	0.110242
$649$	− 8.73215e7i	− 0.319438i
$650$	7.80029e7i	0.284034i
$651$	0	0
$652$	3.89985e7	0.140704
$653$	−9.96144e7	−0.357753	−0.178876	−	0.983872i	$-0.557246\pi$
$653$	−9.96144e7	−0.357753	−0.178876	+	0.983872i	$0.557246\pi$
$654$	− 3.70186e8i	− 1.32339i
$655$	4.90219e7	0.174448
$656$	1.12465e8i	0.398389i
$657$	− 2.37567e8i	− 0.837702i
$658$	0	0
$659$	3.37346e8	1.17874	0.589372	−	0.807862i	$-0.299375\pi$
$659$	3.37346e8	1.17874	0.589372	+	0.807862i	$0.299375\pi$
$660$	2.82942e8	0.984160
$661$	− 3.65502e8i	− 1.26557i	−0.774329	−	0.632784i	$-0.781912\pi$
$661$	− 3.65502e8i	− 1.26557i	0.774329	−	0.632784i	$-0.218088\pi$
$662$	−1.18399e8	−0.408107
$663$	− 2.53843e8i	− 0.871013i
$664$	− 1.18321e8i	− 0.404163i
$665$	0	0
$666$	−5.62807e7	−0.190518
$667$	1.28351e7	0.0432535
$668$	− 2.04233e7i	− 0.0685168i
$669$	2.96358e8	0.989778
$670$	− 4.62300e8i	− 1.53709i
$671$	− 1.94241e8i	− 0.642944i
$672$	0	0
$673$	−4.77356e8	−1.56602	−0.783010	−	0.622008i	$-0.786317\pi$
$673$	−4.77356e8	−1.56602	−0.783010	+	0.622008i	$0.786317\pi$
$674$	−2.89827e7	−0.0946582
$675$	2.82835e8i	0.919647i
$676$	1.38485e8	0.448292
$677$	− 5.64706e8i	− 1.81994i	−0.414676	−	0.909969i	$-0.636105\pi$
$677$	− 5.64706e8i	− 1.81994i	0.414676	−	0.909969i	$-0.363895\pi$
$678$	− 2.45283e8i	− 0.787006i
$679$	0	0
$680$	−2.87403e8	−0.914039
$681$	−2.03846e7	−0.0645448
$682$	− 8.75724e6i	− 0.0276067i
$683$	5.07749e7	0.159363	0.0796814	−	0.996820i	$-0.474610\pi$
$683$	5.07749e7	0.159363	0.0796814	+	0.996820i	$0.474610\pi$
$684$	− 870550.i	− 0.00272035i
$685$	1.31025e8i	0.407646i
$686$	0	0
$687$	4.85988e8	1.49884
$688$	−4.66635e7	−0.143289
$689$	4.37164e7i	0.133655i
$690$	3.82145e7	0.116327
$691$	1.73814e8i	0.526805i	0.964686	+	0.263402i	$0.0848447\pi$
$691$	1.73814e8i	0.526805i	−0.964686	+	0.263402i	$0.915155\pi$
$692$	1.99020e7i	0.0600592i
$693$	0	0
$694$	−7.55483e6	−0.0226020
$695$	−3.00498e8	−0.895131
$696$	1.16028e8i	0.344140i
$697$	−9.30191e8	−2.74710
$698$	3.31485e8i	0.974760i
$699$	6.07492e8i	1.77873i
$700$	0	0
$701$	1.13297e7	0.0328899	0.0164450	−	0.999865i	$-0.494765\pi$
$701$	1.13297e7	0.0328899	0.0164450	+	0.999865i	$0.494765\pi$
$702$	−5.79185e7	−0.167420
$703$	− 236138.i	0 0.000679672i
$704$	3.64330e7	0.104418
$705$	1.10069e9i	3.14121i
$706$	− 3.29066e8i	− 0.935123i
$707$	0	0
$708$	1.06614e8	0.300410
$709$	1.91238e8	0.536580	0.268290	−	0.963338i	$-0.413541\pi$
$709$	1.91238e8	0.536580	0.268290	+	0.963338i	$0.413541\pi$
$710$	− 5.95502e8i	− 1.66383i
$711$	−7.13538e8	−1.98522
$712$	− 2.39546e7i	− 0.0663665i
$713$	− 1.18276e6i	− 0.00326309i
$714$	0	0
$715$	1.47259e8	0.402868
$716$	−1.76679e8	−0.481334
$717$	3.88431e8i	1.05380i
$718$	3.15833e8	0.853265
$719$	− 3.23542e8i	− 0.870450i	−0.900322	−	0.435225i	$-0.856669\pi$
$719$	− 3.23542e8i	− 0.870450i	0.900322	−	0.435225i	$-0.143331\pi$
$720$	2.05515e8i	0.550614i
$721$	0	0
$722$	−2.66128e8	−0.707097
$723$	1.05691e9	2.79654
$724$	1.21584e8i	0.320377i
$725$	2.94890e8	0.773832
$726$	1.28471e8i	0.335735i
$727$	− 1.14232e8i	− 0.297293i	−0.988890	−	0.148646i	$-0.952508\pi$
$727$	− 1.14232e8i	− 0.297293i	0.988890	−	0.148646i	$-0.0474916\pi$
$728$	0	0
$729$	6.25574e8	1.61472
$730$	−2.35311e8	−0.604887
$731$	− 3.85949e8i	− 0.988048i
$732$	2.37156e8	0.604646
$733$	− 1.35572e8i	− 0.344238i	−0.985076	−	0.172119i	$-0.944939\pi$
$733$	− 1.35572e8i	− 0.344238i	0.985076	−	0.172119i	$-0.0550613\pi$
$734$	4.91059e7i	0.124178i
$735$	0	0
$736$	4.92069e6	0.0123422
$737$	−4.84709e8	−1.21082
$738$	6.65158e8i	1.65484i
$739$	6.32855e8	1.56809	0.784044	−	0.620705i	$-0.213153\pi$
$739$	6.32855e8	1.56809	0.784044	+	0.620705i	$0.213153\pi$
$740$	5.57463e7i	0.137569i
$741$	− 761596.i	− 0.00187185i
$742$	0	0
$743$	−2.81712e8	−0.686814	−0.343407	−	0.939187i	$-0.611581\pi$
$743$	−2.81712e8	−0.686814	−0.343407	+	0.939187i	$0.611581\pi$
$744$	1.06921e7	0.0259622
$745$	7.82097e8i	1.89144i
$746$	−3.89728e8	−0.938739
$747$	− 6.99788e8i	− 1.67882i
$748$	3.01334e8i	0.720019i
$749$	0	0
$750$	1.75085e8	0.415016
$751$	6.08354e7	0.143627	0.0718136	−	0.997418i	$-0.477121\pi$
$751$	6.08354e7	0.143627	0.0718136	+	0.997418i	$0.477121\pi$
$752$	1.41730e8i	0.333280i
$753$	8.73985e8	2.04701
$754$	6.03873e7i	0.140874i
$755$	− 1.16314e9i	− 2.70265i
$756$	0	0
$757$	6.73226e8	1.55193	0.775967	−	0.630774i	$-0.217262\pi$
$757$	6.73226e8	1.55193	0.775967	+	0.630774i	$0.217262\pi$
$758$	−1.60075e8	−0.367551
$759$	− 4.00669e7i	− 0.0916348i
$760$	−862285.	−0.00196431
$761$	7.94474e8i	1.80271i	0.433082	+	0.901355i	$0.357426\pi$
$761$	7.94474e8i	1.80271i	−0.433082	+	0.901355i	$0.642574\pi$
$762$	− 1.80564e8i	− 0.408099i
$763$	0	0
$764$	−1.66760e8	−0.373948
$765$	−1.69980e9	−3.79676
$766$	6.10358e8i	1.35800i
$767$	5.54879e7	0.122974
$768$	4.44825e7i	0.0981986i
$769$	− 8.61161e7i	− 0.189367i	−0.995507	−	0.0946837i	$-0.969816\pi$
$769$	− 8.61161e7i	− 0.189367i	0.995507	−	0.0946837i	$-0.0301840\pi$
$770$	0	0
$771$	−5.27646e8	−1.15128
$772$	−1.16009e8	−0.252138
$773$	5.62940e8i	1.21877i	0.792873	+	0.609387i	$0.208584\pi$
$773$	5.62940e8i	1.21877i	−0.792873	+	0.609387i	$0.791416\pi$
$774$	−2.75984e8	−0.595197
$775$	− 2.71744e7i	− 0.0583787i
$776$	− 2.88690e8i	− 0.617797i
$777$	0	0
$778$	−3.66065e8	−0.777354
$779$	−2.79082e6	−0.00590363
$780$	1.79794e8i	0.378871i
$781$	−6.24368e8	−1.31065
$782$	4.06986e7i	0.0851058i
$783$	2.18962e8i	0.456124i
$784$	0	0
$785$	3.68997e8	0.762806
$786$	6.27539e7	0.129233
$787$	− 8.92284e8i	− 1.83054i	−0.402842	−	0.915269i	$-0.631978\pi$
$787$	− 8.92284e8i	− 1.83054i	0.402842	−	0.915269i	$-0.368022\pi$
$788$	−2.03763e8	−0.416435
$789$	1.20790e9i	2.45924i
$790$	7.06764e8i	1.43348i
$791$	0	0
$792$	2.15477e8	0.433737
$793$	1.23429e8	0.247513
$794$	− 4.28097e8i	− 0.855226i
$795$	4.92066e8	0.979314
$796$	− 1.34512e8i	− 0.266699i
$797$	− 9.48842e7i	− 0.187421i	−0.995599	−	0.0937106i	$-0.970127\pi$
$797$	− 9.48842e7i	− 0.187421i	0.995599	−	0.0937106i	$-0.0298729\pi$
$798$	0	0
$799$	−1.17224e9	−2.29813
$800$	1.13054e8	0.220809
$801$	− 1.41676e8i	− 0.275675i
$802$	−1.76172e7	−0.0341519
$803$	2.46717e8i	0.476489i
$804$	− 5.91800e8i	− 1.13869i
$805$	0	0
$806$	5.56474e6	0.0106277
$807$	−6.21489e8	−1.18253
$808$	− 2.35950e8i	− 0.447286i
$809$	8.27475e8	1.56282	0.781411	−	0.624017i	$-0.214500\pi$
$809$	8.27475e8	1.56282	0.781411	+	0.624017i	$0.214500\pi$
$810$	− 1.75726e8i	− 0.330660i
$811$	9.28491e8i	1.74066i	0.492465	+	0.870332i	$0.336096\pi$
$811$	9.28491e8i	1.74066i	−0.492465	+	0.870332i	$0.663904\pi$
$812$	0	0
$813$	−6.29833e8	−1.17207
$814$	5.84485e7	0.108368
$815$	− 2.28461e8i	− 0.422025i
$816$	−3.67911e8	−0.677130
$817$	− 1.15795e6i	− 0.00212336i
$818$	6.85991e8i	1.25331i
$819$	0	0
$820$	6.58843e8	1.19492
$821$	5.54691e8	1.00236	0.501178	−	0.865344i	$-0.332900\pi$
$821$	5.54691e8	1.00236	0.501178	+	0.865344i	$0.332900\pi$
$822$	1.67728e8i	0.301989i
$823$	−8.91822e8	−1.59985	−0.799924	−	0.600102i	$-0.795127\pi$
$823$	−8.91822e8	−1.59985	−0.799924	+	0.600102i	$0.795127\pi$
$824$	1.20144e6i	0.00214743i
$825$	− 9.20550e8i	− 1.63940i
$826$	0	0
$827$	−1.38059e8	−0.244089	−0.122044	−	0.992525i	$-0.538945\pi$
$827$	−1.38059e8	−0.244089	−0.122044	+	0.992525i	$0.538945\pi$
$828$	2.91026e7	0.0512674
$829$	1.39508e8i	0.244870i	0.992477	+	0.122435i	$0.0390704\pi$
$829$	1.39508e8i	0.244870i	−0.992477	+	0.122435i	$0.960930\pi$
$830$	−6.93144e8	−1.21224
$831$	9.60022e8i	1.67293i
$832$	2.31512e7i	0.0401978i
$833$	0	0
$834$	−3.84673e8	−0.663123
$835$	−1.19644e8	−0.205509
$836$	904082.i	0.00154735i
$837$	2.01775e7	0.0344105
$838$	6.10639e7i	0.103765i
$839$	− 2.67389e8i	− 0.452750i	−0.974040	−	0.226375i	$-0.927313\pi$
$839$	− 2.67389e8i	− 0.452750i	0.974040	−	0.226375i	$-0.0726874\pi$
$840$	0	0
$841$	−3.66528e8	−0.616197
$842$	2.92214e7	0.0489514
$843$	− 1.13060e9i	− 1.88723i
$844$	−4.27355e8	−0.710822
$845$	− 8.11268e8i	− 1.34460i
$846$	8.38240e8i	1.38439i
$847$	0	0
$848$	6.33609e7	0.103904
$849$	1.10440e9	1.80469
$850$	9.35063e8i	1.52259i
$851$	7.89412e6	0.0128090
$852$	− 7.62315e8i	− 1.23258i
$853$	6.69761e8i	1.07913i	0.841945	+	0.539564i	$0.181411\pi$
$853$	6.69761e8i	1.07913i	−0.841945	+	0.539564i	$0.818589\pi$
$854$	0	0
$855$	−5.09985e6	−0.00815941
$856$	−2.37493e8	−0.378643
$857$	4.04750e8i	0.643049i	0.946901	+	0.321524i	$0.104195\pi$
$857$	4.04750e8i	0.643049i	−0.946901	+	0.321524i	$0.895805\pi$
$858$	1.88509e8	0.298449
$859$	− 3.43703e8i	− 0.542255i	−0.962543	−	0.271128i	$-0.912603\pi$
$859$	− 3.43703e8i	− 0.542255i	0.962543	−	0.271128i	$-0.0873965\pi$
$860$	2.73363e8i	0.429779i
$861$	0	0
$862$	6.49737e8	1.01442
$863$	7.49906e8	1.16674	0.583371	−	0.812206i	$-0.301733\pi$
$863$	7.49906e8	1.16674	0.583371	+	0.812206i	$0.301733\pi$
$864$	8.39450e7i	0.130153i
$865$	1.16590e8	0.180141
$866$	1.19106e8i	0.183392i
$867$	− 2.01900e9i	− 3.09798i
$868$	0	0
$869$	7.41023e8	1.12920
$870$	6.79713e8	1.03221
$871$	− 3.08005e8i	− 0.466127i
$872$	−2.79242e8	−0.421145
$873$	− 1.70741e9i	− 2.56623i
$874$	122106.i	0 0.000182896i
$875$	0	0
$876$	−3.01227e8	−0.448107
$877$	−1.48951e8	−0.220824	−0.110412	−	0.993886i	$-0.535217\pi$
$877$	−1.48951e8	−0.220824	−0.110412	+	0.993886i	$0.535217\pi$
$878$	− 3.72071e7i	− 0.0549721i
$879$	1.02151e8	0.150410
$880$	− 2.13431e8i	− 0.313192i
$881$	1.17850e9i	1.72346i	0.507368	+	0.861729i	$0.330618\pi$
$881$	1.17850e9i	1.72346i	−0.507368	+	0.861729i	$0.669382\pi$
$882$	0	0
$883$	4.71777e8	0.685259	0.342630	−	0.939471i	$-0.388682\pi$
$883$	4.71777e8	0.685259	0.342630	+	0.939471i	$0.388682\pi$
$884$	−1.91481e8	−0.277185
$885$	− 6.24565e8i	− 0.901048i
$886$	4.40790e8	0.633768
$887$	− 3.08900e7i	− 0.0442636i	−0.999755	−	0.0221318i	$-0.992955\pi$
$887$	− 3.08900e7i	− 0.0442636i	0.999755	−	0.0221318i	$-0.00704534\pi$
$888$	7.13620e7i	0.101913i
$889$	0	0
$890$	−1.40331e8	−0.199059
$891$	−1.84244e8	−0.260472
$892$	− 2.23551e8i	− 0.314980i
$893$	−3.51702e6	−0.00493879
$894$	1.00118e9i	1.40120i
$895$	1.03502e9i	1.44371i
$896$	0	0
$897$	2.54603e7	0.0352765
$898$	−8.48281e8	−1.17142
$899$	− 2.10375e7i	− 0.0289545i
$900$	6.68642e8	0.917205
$901$	5.24052e8i	0.716473i
$902$	− 6.90779e8i	− 0.941282i
$903$	0	0
$904$	−1.85024e8	−0.250451
$905$	7.12262e8	0.960935
$906$	− 1.48896e9i	− 2.00215i
$907$	−7.06064e8	−0.946286	−0.473143	−	0.880986i	$-0.656881\pi$
$907$	−7.06064e8	−0.946286	−0.473143	+	0.880986i	$0.656881\pi$
$908$	1.53767e7i	0.0205403i
$909$	− 1.39549e9i	− 1.85795i
$910$	0	0
$911$	4.46529e8	0.590602	0.295301	−	0.955404i	$-0.404580\pi$
$911$	4.46529e8	0.590602	0.295301	+	0.955404i	$0.404580\pi$
$912$	−1.10383e6	−0.00145518
$913$	7.26743e8i	0.954923i
$914$	7.17715e8	0.939969
$915$	− 1.38931e9i	− 1.81357i
$916$	− 3.66595e8i	− 0.476980i
$917$	0	0
$918$	−6.94301e8	−0.897470
$919$	−6.01034e8	−0.774377	−0.387188	−	0.922001i	$-0.626554\pi$
$919$	−6.01034e8	−0.774377	−0.387188	+	0.922001i	$0.626554\pi$
$920$	− 2.88263e7i	− 0.0370191i
$921$	8.17374e8	1.04627
$922$	− 4.72193e8i	− 0.602458i
$923$	− 3.96751e8i	− 0.504560i
$924$	0	0
$925$	1.81370e8	0.229161
$926$	7.41454e8	0.933794
$927$	7.10571e6i	0.00892007i
$928$	8.75232e7	0.109516
$929$	9.77935e8i	1.21973i	0.792506	+	0.609864i	$0.208776\pi$
$929$	9.77935e8i	1.21973i	−0.792506	+	0.609864i	$0.791224\pi$
$930$	− 6.26360e7i	− 0.0778709i
$931$	0	0
$932$	4.58249e8	0.566048
$933$	−1.59875e9	−1.96850
$934$	4.92131e8i	0.604004i
$935$	1.76527e9	2.15962
$936$	1.36924e8i	0.166975i
$937$	1.61274e8i	0.196041i	0.995184	+	0.0980205i	$0.0312511\pi$
$937$	1.61274e8i	0.196041i	−0.995184	+	0.0980205i	$0.968749\pi$
$938$	0	0
$939$	1.70289e9	2.05679
$940$	8.30282e8	0.999636
$941$	− 2.46122e8i	− 0.295381i	−0.989034	−	0.147690i	$-0.952816\pi$
$941$	− 2.46122e8i	− 0.295381i	0.989034	−	0.147690i	$-0.0471839\pi$
$942$	4.72361e8	0.565095
$943$	− 9.32975e7i	− 0.111259i
$944$	− 8.04221e7i	− 0.0956004i
$945$	0	0
$946$	2.86614e8	0.338551
$947$	1.41653e9	1.66792	0.833960	−	0.551825i	$-0.186068\pi$
$947$	1.41653e9	1.66792	0.833960	+	0.551825i	$0.186068\pi$
$948$	9.04743e8i	1.06194i
$949$	−1.56775e8	−0.183434
$950$	2.80544e6i	0.00327212i
$951$	2.26658e9i	2.63529i
$952$	0	0
$953$	7.96546e8	0.920306	0.460153	−	0.887840i	$-0.347795\pi$
$953$	7.96546e8	0.920306	0.460153	+	0.887840i	$0.347795\pi$
$954$	3.74738e8	0.431601
$955$	9.76910e8i	1.12162i
$956$	2.93005e8	0.335352
$957$	− 7.12660e8i	− 0.813105i
$958$	− 7.97535e8i	− 0.907095i
$959$	0	0
$960$	2.60587e8	0.294536
$961$	8.85565e8	0.997816
$962$	3.71408e7i	0.0417182i
$963$	−1.40461e9	−1.57282
$964$	− 7.97255e8i	− 0.889951i
$965$	6.79601e8i	0.756262i
$966$	0	0
$967$	−3.27333e6	−0.00362002	−0.00181001	−	0.999998i	$-0.500576\pi$
$967$	−3.27333e6	−0.00362002	−0.00181001	+	0.999998i	$0.500576\pi$
$968$	9.69096e7	0.106842
$969$	− 9.12967e6i	− 0.0100342i
$970$	−1.69120e9	−1.85302
$971$	− 7.28505e8i	− 0.795747i	−0.917440	−	0.397873i	$-0.869748\pi$
$971$	− 7.28505e8i	− 0.795747i	0.917440	−	0.397873i	$-0.130252\pi$
$972$	− 5.63014e8i	− 0.613084i
$973$	0	0
$974$	−8.09100e8	−0.875640
$975$	5.84958e8	0.631118
$976$	− 1.78894e8i	− 0.192418i
$977$	4.76512e8	0.510964	0.255482	−	0.966814i	$-0.417766\pi$
$977$	4.76512e8	0.510964	0.255482	+	0.966814i	$0.417766\pi$
$978$	− 2.92457e8i	− 0.312641i
$979$	1.47133e8i	0.156806i
$980$	0	0
$981$	−1.65153e9	−1.74936
$982$	−5.15010e8	−0.543853
$983$	1.67235e8i	0.176063i	0.996118	+	0.0880313i	$0.0280575\pi$
$983$	1.67235e8i	0.176063i	−0.996118	+	0.0880313i	$0.971942\pi$
$984$	8.43399e8	0.885213
$985$	1.19368e9i	1.24905i
$986$	7.23896e8i	0.755171i
$987$	0	0
$988$	−574494.	−0.000595682 0
$989$	3.87104e7	0.0400165
$990$	− 1.26231e9i	− 1.30095i
$991$	−2.18138e8	−0.224135	−0.112068	−	0.993701i	$-0.535747\pi$
$991$	−2.18138e8	−0.224135	−0.112068	+	0.993701i	$0.535747\pi$
$992$	− 8.06533e6i	− 0.00826203i
$993$	8.87896e8i	0.906806i
$994$	0	0
$995$	−7.87997e8	−0.799936
$996$	−8.87308e8	−0.898042
$997$	1.62988e8i	0.164464i	0.996613	+	0.0822319i	$0.0262048\pi$
$997$	1.62988e8i	0.164464i	−0.996613	+	0.0822319i	$0.973795\pi$
$998$	3.20260e8	0.322189
$999$	1.34671e8i	0.135075i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.7.b.c.97.4		8
7.2	even	3		14.7.d.a.3.3	✓	8
7.3	odd	6		14.7.d.a.5.3	yes	8
7.4	even	3		98.7.d.c.19.4		8
7.5	odd	6		98.7.d.c.31.4		8
7.6	odd	2	inner	98.7.b.c.97.1		8
21.2	odd	6		126.7.n.c.73.2		8
21.17	even	6		126.7.n.c.19.2		8
28.3	even	6		112.7.s.c.33.4		8
28.23	odd	6		112.7.s.c.17.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.7.d.a.3.3	✓	8	7.2	even	3
14.7.d.a.5.3	yes	8	7.3	odd	6
98.7.b.c.97.1		8	7.6	odd	2	inner
98.7.b.c.97.4		8	1.1	even	1	trivial
98.7.d.c.19.4		8	7.4	even	3
98.7.d.c.31.4		8	7.5	odd	6
112.7.s.c.17.4		8	28.23	odd	6
112.7.s.c.33.4		8	28.3	even	6
126.7.n.c.19.2		8	21.17	even	6
126.7.n.c.73.2		8	21.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	98.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.65685 q^{2} +42.4218i q^{3} +32.0000 q^{4} -187.462i q^{5} -239.974i q^{6} -181.019 q^{8} -1070.61 q^{9} +1060.44i q^{10} +1111.85 q^{11} +1357.50i q^{12} +706.517i q^{13} +7952.48 q^{15} +1024.00 q^{16} +8469.41i q^{17} +6056.28 q^{18} +25.4105i q^{19} -5998.78i q^{20} -6289.56 q^{22} -849.476 q^{23} -7679.17i q^{24} -19517.0 q^{25} -3996.67i q^{26} -14491.7i q^{27} -15109.4 q^{29} -44986.0 q^{30} +1392.35i q^{31} -5792.62 q^{32} +47166.6i q^{33} -47910.2i q^{34} -34259.5 q^{36} -9292.94 q^{37} -143.743i q^{38} -29971.7 q^{39} +33934.2i q^{40} +109829. i q^{41} -45569.8 q^{43} +35579.1 q^{44} +200699. i q^{45} +4805.36 q^{46} +138408. i q^{47} +43439.9i q^{48} +110405. q^{50} -359288. q^{51} +22608.6i q^{52} +61875.9 q^{53} +81977.5i q^{54} -208429. i q^{55} -1077.96 q^{57} +85471.8 q^{58} -78537.2i q^{59} +254479. q^{60} -174701. i q^{61} -7876.29i q^{62} +32768.0 q^{64} +132445. q^{65} -266815. i q^{66} -435949. q^{67} +271021. i q^{68} -36036.3i q^{69} -561559. q^{71} +193801. q^{72} +221899. i q^{73} +52568.8 q^{74} -827946. i q^{75} +813.135i q^{76} +169546. q^{78} +666478. q^{79} -191961. i q^{80} -165710. q^{81} -621289. i q^{82} +653635. i q^{83} +1.58769e6 q^{85} +257782. q^{86} -640969. i q^{87} -201266. q^{88} +132332. i q^{89} -1.13532e6i q^{90} -27183.2 q^{92} -59065.8 q^{93} -782956. i q^{94} +4763.49 q^{95} -245733. i q^{96} +1.59480e6i q^{97} -1.19036e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 256 q^{4} - 1512 q^{9} + 2712 q^{11} + 27144 q^{15} + 8192 q^{16} + 13632 q^{18} + 25248 q^{22} + 8256 q^{23} - 9328 q^{25} - 30312 q^{29} - 19296 q^{30} - 48384 q^{36} + 12248 q^{37} - 201528 q^{39}+ \cdots - 4625928 q^{99}+O(q^{100}) \) 8 * q + 256 * q^4 - 1512 * q^9 + 2712 * q^11 + 27144 * q^15 + 8192 * q^16 + 13632 * q^18 + 25248 * q^22 + 8256 * q^23 - 9328 * q^25 - 30312 * q^29 - 19296 * q^30 - 48384 * q^36 + 12248 * q^37 - 201528 * q^39 - 297376 * q^43 + 86784 * q^44 + 388128 * q^46 + 38784 * q^50 - 507384 * q^51 - 556968 * q^53 - 81288 * q^57 + 339648 * q^58 + 868608 * q^60 + 262144 * q^64 - 16632 * q^65 - 1297616 * q^67 + 190128 * q^71 + 436224 * q^72 + 2316288 * q^74 + 1122432 * q^78 - 140192 * q^79 - 2355840 * q^81 + 2190984 * q^85 + 1146048 * q^86 + 807936 * q^88 + 264192 * q^92 + 46728 * q^93 - 1451280 * q^95 - 4625928 * q^99

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