LMFDB - Embedded newform 2070.3.c.b.91.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.11
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.54509i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.54509i q^{7} -2.82843 q^{8} -3.16228i q^{10} +8.65106i q^{11} -19.9643 q^{13} +10.6704i q^{14} +4.00000 q^{16} -9.27071i q^{17} -35.9882i q^{19} +4.47214i q^{20} -12.2344i q^{22} +(-22.9761 + 1.04751i) q^{23} -5.00000 q^{25} +28.2338 q^{26} -15.0902i q^{28} +27.4794 q^{29} +22.3027 q^{31} -5.65685 q^{32} +13.1108i q^{34} +16.8713 q^{35} +42.9329i q^{37} +50.8949i q^{38} -6.32456i q^{40} +39.2309 q^{41} +20.8570i q^{43} +17.3021i q^{44} +(32.4932 - 1.48140i) q^{46} -66.5989 q^{47} -7.92839 q^{49} +7.07107 q^{50} -39.9286 q^{52} +64.5799i q^{53} -19.3444 q^{55} +21.3407i q^{56} -38.8617 q^{58} -90.0416 q^{59} +10.7927i q^{61} -31.5407 q^{62} +8.00000 q^{64} -44.6416i q^{65} -58.7797i q^{67} -18.5414i q^{68} -23.8597 q^{70} +29.3496 q^{71} +95.8335 q^{73} -60.7163i q^{74} -71.9763i q^{76} +65.2730 q^{77} +72.0229i q^{79} +8.94427i q^{80} -55.4809 q^{82} +106.235i q^{83} +20.7299 q^{85} -29.4963i q^{86} -24.4689i q^{88} -80.0068i q^{89} +150.633i q^{91} +(-45.9523 + 2.09502i) q^{92} +94.1850 q^{94} +80.4720 q^{95} +103.236i q^{97} +11.2124 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4}+O(q^{10}) $ 32 * q + 64 * q^4	$ 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	7.54509i		−	1.07787i	−0.842347	−	0.538935i	$-0.818827\pi$
$7$		−	7.54509i		−	1.07787i	0.842347	−	0.538935i	$-0.181173\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$			8.65106i			0.786460i	0.919440	+	0.393230i	$0.128642\pi$
$11$			8.65106i			0.786460i	−0.919440	+	0.393230i	$0.871358\pi$
$12$		0			0
$13$	−19.9643			−1.53572			−0.767858	−	0.640620i	$-0.778678\pi$
$13$	−19.9643			−1.53572			−0.767858	+	0.640620i	$0.778678\pi$
$14$			10.6704i			0.762169i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	9.27071i		−	0.545336i	−0.962108	−	0.272668i	$-0.912094\pi$
$17$		−	9.27071i		−	0.545336i	0.962108	−	0.272668i	$-0.0879061\pi$
$18$		0			0
$19$		−	35.9882i		−	1.89411i	−0.321066	−	0.947057i	$-0.604041\pi$
$19$		−	35.9882i		−	1.89411i	0.321066	−	0.947057i	$-0.395959\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$		−	12.2344i		−	0.556111i
$23$	−22.9761	+	1.04751i	−0.998962	+	0.0455439i
$23$	−22.9761	+	1.04751i	−0.998962	+	0.0455439i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	28.2338			1.08592
$27$		0			0
$28$		−	15.0902i		−	0.538935i
$29$	27.4794			0.947564			0.473782	−	0.880642i	$-0.342888\pi$
$29$	27.4794			0.947564			0.473782	+	0.880642i	$0.342888\pi$
$30$		0			0
$31$	22.3027			0.719441			0.359720	−	0.933060i	$-0.382872\pi$
$31$	22.3027			0.719441			0.359720	+	0.933060i	$0.382872\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			13.1108i			0.385611i
$35$	16.8713			0.482038
$36$		0			0
$37$			42.9329i			1.16035i	0.814493	+	0.580174i	$0.197015\pi$
$37$			42.9329i			1.16035i	−0.814493	+	0.580174i	$0.802985\pi$
$38$			50.8949i			1.33934i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	39.2309			0.956851			0.478426	−	0.878128i	$-0.341208\pi$
$41$	39.2309			0.956851			0.478426	+	0.878128i	$0.341208\pi$
$42$		0			0
$43$			20.8570i			0.485047i	0.970146	+	0.242523i	$0.0779751\pi$
$43$			20.8570i			0.485047i	−0.970146	+	0.242523i	$0.922025\pi$
$44$			17.3021i			0.393230i
$45$		0			0
$46$	32.4932	−	1.48140i	0.706373	−	0.0322044i
$47$	−66.5989			−1.41700			−0.708499	−	0.705712i	$-0.750627\pi$
$47$	−66.5989			−1.41700			−0.708499	+	0.705712i	$0.750627\pi$
$48$		0			0
$49$	−7.92839			−0.161804
$50$	7.07107			0.141421
$51$		0			0
$52$	−39.9286			−0.767858
$53$			64.5799i			1.21849i	0.792982	+	0.609245i	$0.208527\pi$
$53$			64.5799i			1.21849i	−0.792982	+	0.609245i	$0.791473\pi$
$54$		0			0
$55$	−19.3444			−0.351716
$56$			21.3407i			0.381085i
$57$		0			0
$58$	−38.8617			−0.670029
$59$	−90.0416			−1.52613			−0.763064	−	0.646323i	$-0.776306\pi$
$59$	−90.0416			−1.52613			−0.763064	+	0.646323i	$0.776306\pi$
$60$		0			0
$61$			10.7927i			0.176930i	0.996079	+	0.0884650i	$0.0281961\pi$
$61$			10.7927i			0.176930i	−0.996079	+	0.0884650i	$0.971804\pi$
$62$	−31.5407			−0.508722
$63$		0			0
$64$	8.00000			0.125000
$65$		−	44.6416i		−	0.686793i
$66$		0			0
$67$		−	58.7797i		−	0.877309i	−0.898656	−	0.438655i	$-0.855455\pi$
$67$		−	58.7797i		−	0.877309i	0.898656	−	0.438655i	$-0.144545\pi$
$68$		−	18.5414i		−	0.272668i
$69$		0			0
$70$	−23.8597			−0.340852
$71$	29.3496			0.413375			0.206688	−	0.978407i	$-0.433732\pi$
$71$	29.3496			0.413375			0.206688	+	0.978407i	$0.433732\pi$
$72$		0			0
$73$	95.8335			1.31279			0.656394	−	0.754418i	$-0.272081\pi$
$73$	95.8335			1.31279			0.656394	+	0.754418i	$0.272081\pi$
$74$		−	60.7163i		−	0.820490i
$75$		0			0
$76$		−	71.9763i		−	0.947057i
$77$	65.2730			0.847701
$78$		0			0
$79$			72.0229i			0.911682i	0.890061	+	0.455841i	$0.150661\pi$
$79$			72.0229i			0.911682i	−0.890061	+	0.455841i	$0.849339\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	−55.4809			−0.676596
$83$			106.235i			1.27994i	0.768401	+	0.639969i	$0.221053\pi$
$83$			106.235i			1.27994i	−0.768401	+	0.639969i	$0.778947\pi$
$84$		0			0
$85$	20.7299			0.243882
$86$		−	29.4963i		−	0.342980i
$87$		0			0
$88$		−	24.4689i		−	0.278056i
$89$		−	80.0068i		−	0.898952i	−0.893292	−	0.449476i	$-0.851611\pi$
$89$		−	80.0068i		−	0.898952i	0.893292	−	0.449476i	$-0.148389\pi$
$90$		0			0
$91$			150.633i			1.65530i
$92$	−45.9523	+	2.09502i	−0.499481	+	0.0227720i
$93$		0			0
$94$	94.1850			1.00197
$95$	80.4720			0.847073
$96$		0			0
$97$			103.236i			1.06428i	0.846655	+	0.532142i	$0.178613\pi$
$97$			103.236i			1.06428i	−0.846655	+	0.532142i	$0.821387\pi$
$98$	11.2124			0.114413
$99$		0			0
$100$	−10.0000			−0.100000
$101$	−85.8682			−0.850180			−0.425090	−	0.905151i	$-0.639758\pi$
$101$	−85.8682			−0.850180			−0.425090	+	0.905151i	$0.639758\pi$
$102$		0			0
$103$			178.631i			1.73429i	0.498060	+	0.867143i	$0.334046\pi$
$103$			178.631i			1.73429i	−0.498060	+	0.867143i	$0.665954\pi$
$104$	56.4676			0.542958
$105$		0			0
$106$		−	91.3298i		−	0.861602i
$107$			21.2445i			0.198547i	0.995060	+	0.0992734i	$0.0316519\pi$
$107$			21.2445i			0.198547i	−0.995060	+	0.0992734i	$0.968348\pi$
$108$		0			0
$109$		−	187.493i		−	1.72012i	−0.510195	−	0.860059i	$-0.670427\pi$
$109$		−	187.493i		−	1.72012i	0.510195	−	0.860059i	$-0.329573\pi$
$110$	27.3570			0.248700
$111$		0			0
$112$		−	30.1804i		−	0.269468i
$113$			5.20875i			0.0460952i	0.999734	+	0.0230476i	$0.00733692\pi$
$113$			5.20875i			0.0460952i	−0.999734	+	0.0230476i	$0.992663\pi$
$114$		0			0
$115$	−2.34231	−	51.3762i	−0.0203679	−	0.446750i
$116$	54.9587			0.473782
$117$		0			0
$118$	127.338			1.07914
$119$	−69.9484			−0.587801
$120$		0			0
$121$	46.1592			0.381481
$122$		−	15.2632i		−	0.125108i
$123$		0			0
$124$	44.6053			0.359720
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	213.909			1.68432			0.842161	−	0.539227i	$-0.181283\pi$
$127$	213.909			1.68432			0.842161	+	0.539227i	$0.181283\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$			63.1327i			0.485636i
$131$	209.839			1.60183			0.800913	−	0.598780i	$-0.204348\pi$
$131$	209.839			1.60183			0.800913	+	0.598780i	$0.204348\pi$
$132$		0			0
$133$	−271.534			−2.04161
$134$			83.1271i			0.620351i
$135$		0			0
$136$			26.2215i			0.192805i
$137$			37.7195i			0.275325i	0.990479	+	0.137662i	$0.0439589\pi$
$137$			37.7195i			0.275325i	−0.990479	+	0.137662i	$0.956041\pi$
$138$		0			0
$139$	−8.08981			−0.0582001			−0.0291000	−	0.999577i	$-0.509264\pi$
$139$	−8.08981			−0.0582001			−0.0291000	+	0.999577i	$0.509264\pi$
$140$	33.7427			0.241019
$141$		0			0
$142$	−41.5067			−0.292300
$143$		−	172.712i		−	1.20778i
$144$		0			0
$145$			61.4457i			0.423764i
$146$	−135.529			−0.928281
$147$		0			0
$148$			85.8658i			0.580174i
$149$		−	112.471i		−	0.754837i	−0.926043	−	0.377419i	$-0.876812\pi$
$149$		−	112.471i		−	0.754837i	0.926043	−	0.377419i	$-0.123188\pi$
$150$		0			0
$151$	−214.567			−1.42098			−0.710488	−	0.703709i	$-0.751526\pi$
$151$	−214.567			−1.42098			−0.710488	+	0.703709i	$0.751526\pi$
$152$			101.790i			0.669670i
$153$		0			0
$154$	−92.3100			−0.599415
$155$			49.8703i			0.321744i
$156$		0			0
$157$			57.5819i			0.366764i	0.983042	+	0.183382i	$0.0587045\pi$
$157$			57.5819i			0.366764i	−0.983042	+	0.183382i	$0.941295\pi$
$158$		−	101.856i		−	0.644657i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	7.90356	+	173.357i	0.0490905	+	1.07675i
$162$		0			0
$163$	−208.382			−1.27842			−0.639208	−	0.769034i	$-0.720738\pi$
$163$	−208.382			−1.27842			−0.639208	+	0.769034i	$0.720738\pi$
$164$	78.4618			0.478426
$165$		0			0
$166$		−	150.239i		−	0.905053i
$167$	−253.727			−1.51932			−0.759661	−	0.650319i	$-0.774635\pi$
$167$	−253.727			−1.51932			−0.759661	+	0.650319i	$0.774635\pi$
$168$		0			0
$169$	229.574			1.35843
$170$	−29.3166			−0.172450
$171$		0			0
$172$			41.7140i			0.242523i
$173$	90.9046			0.525460			0.262730	−	0.964869i	$-0.415377\pi$
$173$	90.9046			0.525460			0.262730	+	0.964869i	$0.415377\pi$
$174$		0			0
$175$			37.7255i			0.215574i
$176$			34.6042i			0.196615i
$177$		0			0
$178$			113.147i			0.635655i
$179$	−27.7233			−0.154879			−0.0774393	−	0.996997i	$-0.524674\pi$
$179$	−27.7233			−0.154879			−0.0774393	+	0.996997i	$0.524674\pi$
$180$		0			0
$181$			321.337i			1.77534i	0.460479	+	0.887671i	$0.347678\pi$
$181$			321.337i			1.77534i	−0.460479	+	0.887671i	$0.652322\pi$
$182$		−	213.027i		−	1.17048i
$183$		0			0
$184$	64.9863	−	2.96281i	0.353187	−	0.0161022i
$185$	−96.0008			−0.518923
$186$		0			0
$187$	80.2015			0.428885
$188$	−133.198			−0.708499
$189$		0			0
$190$	−113.805			−0.598971
$191$			32.2385i			0.168788i	0.996432	+	0.0843940i	$0.0268954\pi$
$191$			32.2385i			0.168788i	−0.996432	+	0.0843940i	$0.973105\pi$
$192$		0			0
$193$	−30.8524			−0.159857			−0.0799285	−	0.996801i	$-0.525469\pi$
$193$	−30.8524			−0.159857			−0.0799285	+	0.996801i	$0.525469\pi$
$194$		−	145.997i		−	0.752562i
$195$		0			0
$196$	−15.8568			−0.0809019
$197$	−284.323			−1.44326			−0.721632	−	0.692277i	$-0.756608\pi$
$197$	−284.323			−1.44326			−0.721632	+	0.692277i	$0.756608\pi$
$198$		0			0
$199$			296.034i			1.48761i	0.668397	+	0.743805i	$0.266981\pi$
$199$			296.034i			1.48761i	−0.668397	+	0.743805i	$0.733019\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	121.436			0.601168
$203$		−	207.334i		−	1.02135i
$204$		0			0
$205$			87.7230i			0.427917i
$206$		−	252.623i		−	1.22633i
$207$		0			0
$208$	−79.8573			−0.383929
$209$	311.336			1.48964
$210$		0			0
$211$	−232.465			−1.10173			−0.550866	−	0.834594i	$-0.685703\pi$
$211$	−232.465			−1.10173			−0.550866	+	0.834594i	$0.685703\pi$
$212$			129.160i			0.609245i
$213$		0			0
$214$		−	30.0443i		−	0.140394i
$215$	−46.6377			−0.216920
$216$		0			0
$217$		−	168.276i		−	0.775464i
$218$			265.155i			1.21631i
$219$		0			0
$220$	−38.6887			−0.175858
$221$			185.084i			0.837482i
$222$		0			0
$223$	−196.516			−0.881237			−0.440618	−	0.897694i	$-0.645241\pi$
$223$	−196.516			−0.881237			−0.440618	+	0.897694i	$0.645241\pi$
$224$			42.6815i			0.190542i
$225$		0			0
$226$		−	7.36629i		−	0.0325942i
$227$			302.870i			1.33423i	0.744955	+	0.667115i	$0.232471\pi$
$227$			302.870i			1.33423i	−0.744955	+	0.667115i	$0.767529\pi$
$228$		0			0
$229$			416.009i			1.81663i	0.418285	+	0.908316i	$0.362631\pi$
$229$			416.009i			1.81663i	−0.418285	+	0.908316i	$0.637369\pi$
$230$	3.31252	+	72.6569i	0.0144023	+	0.315900i
$231$		0			0
$232$	−77.7234			−0.335014
$233$	98.6862			0.423546			0.211773	−	0.977319i	$-0.432076\pi$
$233$	98.6862			0.423546			0.211773	+	0.977319i	$0.432076\pi$
$234$		0			0
$235$		−	148.920i		−	0.633701i
$236$	−180.083			−0.763064
$237$		0			0
$238$	98.9219			0.415638
$239$	24.2213			0.101344			0.0506722	−	0.998715i	$-0.483864\pi$
$239$	24.2213			0.101344			0.0506722	+	0.998715i	$0.483864\pi$
$240$		0			0
$241$			44.0207i			0.182658i	0.995821	+	0.0913292i	$0.0291115\pi$
$241$			44.0207i			0.182658i	−0.995821	+	0.0913292i	$0.970888\pi$
$242$	−65.2790			−0.269748
$243$		0			0
$244$			21.5855i			0.0884650i
$245$		−	17.7284i		−	0.0723609i
$246$		0			0
$247$			718.479i			2.90882i
$248$	−63.0815			−0.254361
$249$		0			0
$250$			15.8114i			0.0632456i
$251$		−	18.0472i		−	0.0719013i	−0.999354	−	0.0359507i	$-0.988554\pi$
$251$		−	18.0472i		−	0.0719013i	0.999354	−	0.0359507i	$-0.0114459\pi$
$252$		0			0
$253$	−9.06208	−	198.768i	−0.0358185	−	0.785644i
$254$	−302.513			−1.19100
$255$		0			0
$256$	16.0000			0.0625000
$257$	348.964			1.35784			0.678919	−	0.734213i	$-0.262449\pi$
$257$	348.964			1.35784			0.678919	+	0.734213i	$0.262449\pi$
$258$		0			0
$259$	323.932			1.25070
$260$		−	89.2832i		−	0.343397i
$261$		0			0
$262$	−296.758			−1.13266
$263$			138.601i			0.526999i	0.964660	+	0.263500i	$0.0848768\pi$
$263$			138.601i			0.526999i	−0.964660	+	0.263500i	$0.915123\pi$
$264$		0			0
$265$	−144.405			−0.544925
$266$	384.007			1.44364
$267$		0			0
$268$		−	117.559i		−	0.438655i
$269$	−175.666			−0.653034			−0.326517	−	0.945191i	$-0.605875\pi$
$269$	−175.666			−0.653034			−0.326517	+	0.945191i	$0.605875\pi$
$270$		0			0
$271$	245.596			0.906259			0.453130	−	0.891445i	$-0.350307\pi$
$271$	245.596			0.906259			0.453130	+	0.891445i	$0.350307\pi$
$272$		−	37.0829i		−	0.136334i
$273$		0			0
$274$		−	53.3434i		−	0.194684i
$275$		−	43.2553i		−	0.157292i
$276$		0			0
$277$	319.260			1.15256			0.576281	−	0.817251i	$-0.304503\pi$
$277$	319.260			1.15256			0.576281	+	0.817251i	$0.304503\pi$
$278$	11.4407			0.0411537
$279$		0			0
$280$	−47.7193			−0.170426
$281$			84.2955i			0.299984i	0.988687	+	0.149992i	$0.0479248\pi$
$281$			84.2955i			0.299984i	−0.988687	+	0.149992i	$0.952075\pi$
$282$		0			0
$283$			140.725i			0.497260i	0.968598	+	0.248630i	$0.0799804\pi$
$283$			140.725i			0.497260i	−0.968598	+	0.248630i	$0.920020\pi$
$284$	58.6993			0.206688
$285$		0			0
$286$			244.252i			0.854029i
$287$		−	296.001i		−	1.03136i
$288$		0			0
$289$	203.054			0.702608
$290$		−	86.8974i		−	0.299646i
$291$		0			0
$292$	191.667			0.656394
$293$			271.891i			0.927956i	0.885847	+	0.463978i	$0.153578\pi$
$293$			271.891i			0.927956i	−0.885847	+	0.463978i	$0.846422\pi$
$294$		0			0
$295$		−	201.339i		−	0.682505i
$296$		−	121.433i		−	0.410245i
$297$		0			0
$298$			159.058i			0.533751i
$299$	458.703	−	20.9128i	1.53412	−	0.0699426i
$300$		0			0
$301$	157.368			0.522817
$302$	303.444			1.00478
$303$		0			0
$304$		−	143.953i		−	0.473528i
$305$	−24.1333			−0.0791255
$306$		0			0
$307$	249.583			0.812975			0.406487	−	0.913656i	$-0.366753\pi$
$307$	249.583			0.812975			0.406487	+	0.913656i	$0.366753\pi$
$308$	130.546			0.423851
$309$		0			0
$310$		−	70.5272i		−	0.227507i
$311$	266.239			0.856074			0.428037	−	0.903761i	$-0.359205\pi$
$311$	266.239			0.856074			0.428037	+	0.903761i	$0.359205\pi$
$312$		0			0
$313$			252.371i			0.806297i	0.915135	+	0.403148i	$0.132084\pi$
$313$			252.371i			0.806297i	−0.915135	+	0.403148i	$0.867916\pi$
$314$		−	81.4331i		−	0.259341i
$315$		0			0
$316$			144.046i			0.455841i
$317$	−9.32174			−0.0294061			−0.0147031	−	0.999892i	$-0.504680\pi$
$317$	−9.32174			−0.0294061			−0.0147031	+	0.999892i	$0.504680\pi$
$318$		0			0
$319$			237.726i			0.745221i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−11.1773	−	245.164i	−0.0347122	−	0.761378i
$323$	−333.636			−1.03293
$324$		0			0
$325$	99.8216			0.307143
$326$	294.696			0.903976
$327$		0			0
$328$	−110.962			−0.338298
$329$			502.495i			1.52734i
$330$		0			0
$331$	−545.022			−1.64659			−0.823297	−	0.567611i	$-0.807868\pi$
$331$	−545.022			−1.64659			−0.823297	+	0.567611i	$0.807868\pi$
$332$			212.470i			0.639969i
$333$		0			0
$334$	358.824			1.07432
$335$	131.435			0.392345
$336$		0			0
$337$		−	9.80857i		−	0.0291055i	−0.999894	−	0.0145528i	$-0.995368\pi$
$337$		−	9.80857i		−	0.0291055i	0.999894	−	0.0145528i	$-0.00463245\pi$
$338$	−324.667			−0.960552
$339$		0			0
$340$	41.4599			0.121941
$341$			192.942i			0.565811i
$342$		0			0
$343$		−	309.889i		−	0.903467i
$344$		−	58.9925i		−	0.171490i
$345$		0			0
$346$	−128.559			−0.371557
$347$	−507.867			−1.46359			−0.731797	−	0.681522i	$-0.761318\pi$
$347$	−507.867			−1.46359			−0.731797	+	0.681522i	$0.761318\pi$
$348$		0			0
$349$	182.157			0.521941			0.260971	−	0.965347i	$-0.415957\pi$
$349$	182.157			0.521941			0.260971	+	0.965347i	$0.415957\pi$
$350$		−	53.3518i		−	0.152434i
$351$		0			0
$352$		−	48.9378i		−	0.139028i
$353$	−148.258			−0.419994			−0.209997	−	0.977702i	$-0.567345\pi$
$353$	−148.258			−0.419994			−0.209997	+	0.977702i	$0.567345\pi$
$354$		0			0
$355$			65.6278i			0.184867i
$356$		−	160.014i		−	0.449476i
$357$		0			0
$358$	39.2066			0.109516
$359$			473.762i			1.31967i	0.751410	+	0.659836i	$0.229374\pi$
$359$			473.762i			1.31967i	−0.751410	+	0.659836i	$0.770626\pi$
$360$		0			0
$361$	−934.147			−2.58767
$362$		−	454.439i		−	1.25536i
$363$		0			0
$364$			301.265i			0.827652i
$365$			214.290i			0.587096i
$366$		0			0
$367$		−	25.7019i		−	0.0700324i	−0.999387	−	0.0350162i	$-0.988852\pi$
$367$		−	25.7019i		−	0.0700324i	0.999387	−	0.0350162i	$-0.0111483\pi$
$368$	−91.9045	+	4.19004i	−0.249741	+	0.0113860i
$369$		0			0
$370$	135.766			0.366934
$371$	487.261			1.31337
$372$		0			0
$373$			103.774i			0.278213i	0.990277	+	0.139107i	$0.0444231\pi$
$373$			103.774i			0.278213i	−0.990277	+	0.139107i	$0.955577\pi$
$374$	−113.422			−0.303267
$375$		0			0
$376$	188.370			0.500984
$377$	−548.607			−1.45519
$378$		0			0
$379$		−	60.4401i		−	0.159473i	−0.996816	−	0.0797363i	$-0.974592\pi$
$379$		−	60.4401i		−	0.159473i	0.996816	−	0.0797363i	$-0.0254078\pi$
$380$	160.944			0.423537
$381$		0			0
$382$		−	45.5921i		−	0.119351i
$383$			228.703i			0.597135i	0.954389	+	0.298567i	$0.0965087\pi$
$383$			228.703i			0.597135i	−0.954389	+	0.298567i	$0.903491\pi$
$384$		0			0
$385$			145.955i			0.379104i
$386$	43.6319			0.113036
$387$		0			0
$388$			206.471i			0.532142i
$389$			88.9467i			0.228655i	0.993443	+	0.114327i	$0.0364713\pi$
$389$			88.9467i			0.228655i	−0.993443	+	0.114327i	$0.963529\pi$
$390$		0			0
$391$	9.71117	+	213.005i	0.0248368	+	0.544770i
$392$	22.4249			0.0572063
$393$		0			0
$394$	402.093			1.02054
$395$	−161.048			−0.407717
$396$		0			0
$397$	480.726			1.21090			0.605449	−	0.795884i	$-0.292994\pi$
$397$	480.726			1.21090			0.605449	+	0.795884i	$0.292994\pi$
$398$		−	418.656i		−	1.05190i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			619.704i			1.54540i	0.634773	+	0.772699i	$0.281094\pi$
$401$			619.704i			1.54540i	−0.634773	+	0.772699i	$0.718906\pi$
$402$		0			0
$403$	−445.258			−1.10486
$404$	−171.736			−0.425090
$405$		0			0
$406$			293.215i			0.722204i
$407$	−371.415			−0.912567
$408$		0			0
$409$	−79.9943			−0.195585			−0.0977925	−	0.995207i	$-0.531178\pi$
$409$	−79.9943			−0.195585			−0.0977925	+	0.995207i	$0.531178\pi$
$410$		−	124.059i		−	0.302583i
$411$		0			0
$412$			357.263i			0.867143i
$413$			679.372i			1.64497i
$414$		0			0
$415$	−237.548			−0.572406
$416$	112.935			0.271479
$417$		0			0
$418$	−440.295			−1.05334
$419$			255.496i			0.609777i	0.952388	+	0.304888i	$0.0986192\pi$
$419$			255.496i			0.609777i	−0.952388	+	0.304888i	$0.901381\pi$
$420$		0			0
$421$			306.663i			0.728416i	0.931318	+	0.364208i	$0.118660\pi$
$421$			306.663i			0.728416i	−0.931318	+	0.364208i	$0.881340\pi$
$422$	328.756			0.779042
$423$		0			0
$424$		−	182.660i		−	0.430801i
$425$			46.3536i			0.109067i
$426$		0			0
$427$	81.4321			0.190708
$428$			42.4890i			0.0992734i
$429$		0			0
$430$	65.9557			0.153385
$431$		−	580.271i		−	1.34634i	−0.739489	−	0.673168i	$-0.764933\pi$
$431$		−	580.271i		−	1.34634i	0.739489	−	0.673168i	$-0.235067\pi$
$432$		0			0
$433$		−	695.400i		−	1.60600i	−0.595976	−	0.803002i	$-0.703235\pi$
$433$		−	695.400i		−	1.60600i	0.595976	−	0.803002i	$-0.296765\pi$
$434$			237.978i			0.548336i
$435$		0			0
$436$		−	374.986i		−	0.860059i
$437$	37.6980	+	826.869i	0.0862654	+	1.89215i
$438$		0			0
$439$	−315.512			−0.718705			−0.359353	−	0.933202i	$-0.617002\pi$
$439$	−315.512			−0.718705			−0.359353	+	0.933202i	$0.617002\pi$
$440$	54.7141			0.124350
$441$		0			0
$442$		−	261.748i		−	0.592189i
$443$	−536.137			−1.21024			−0.605121	−	0.796134i	$-0.706875\pi$
$443$	−536.137			−1.21024			−0.605121	+	0.796134i	$0.706875\pi$
$444$		0			0
$445$	178.901			0.402024
$446$	277.915			0.623129
$447$		0			0
$448$		−	60.3607i		−	0.134734i
$449$	676.452			1.50658			0.753288	−	0.657691i	$-0.228467\pi$
$449$	676.452			1.50658			0.753288	+	0.657691i	$0.228467\pi$
$450$		0			0
$451$			339.389i			0.752525i
$452$			10.4175i			0.0230476i
$453$		0			0
$454$		−	428.323i		−	0.943443i
$455$	−336.825			−0.740274
$456$		0			0
$457$		−	171.828i		−	0.375992i	−0.982170	−	0.187996i	$-0.939801\pi$
$457$		−	171.828i		−	0.375992i	0.982170	−	0.187996i	$-0.0601992\pi$
$458$		−	588.325i		−	1.28455i
$459$		0			0
$460$	−4.68461	−	102.752i	−0.0101839	−	0.223375i
$461$	744.326			1.61459			0.807295	−	0.590148i	$-0.200930\pi$
$461$	744.326			1.61459			0.807295	+	0.590148i	$0.200930\pi$
$462$		0			0
$463$	−115.174			−0.248757			−0.124378	−	0.992235i	$-0.539694\pi$
$463$	−115.174			−0.248757			−0.124378	+	0.992235i	$0.539694\pi$
$464$	109.917			0.236891
$465$		0			0
$466$	−139.563			−0.299492
$467$		−	563.253i		−	1.20611i	−0.797700	−	0.603055i	$-0.793950\pi$
$467$		−	563.253i		−	1.20611i	0.797700	−	0.603055i	$-0.206050\pi$
$468$		0			0
$469$	−443.498			−0.945626
$470$			210.604i			0.448094i
$471$		0			0
$472$	254.676			0.539568
$473$	−180.435			−0.381470
$474$		0			0
$475$			179.941i			0.378823i
$476$	−139.897			−0.293901
$477$		0			0
$478$	−34.2541			−0.0716613
$479$			422.812i			0.882698i	0.897336	+	0.441349i	$0.145500\pi$
$479$			422.812i			0.882698i	−0.897336	+	0.441349i	$0.854500\pi$
$480$		0			0
$481$		−	857.126i		−	1.78197i
$482$		−	62.2546i		−	0.129159i
$483$		0			0
$484$	92.3184			0.190740
$485$	−230.842			−0.475962
$486$		0			0
$487$	813.888			1.67123			0.835614	−	0.549317i	$-0.185112\pi$
$487$	813.888			1.67123			0.835614	+	0.549317i	$0.185112\pi$
$488$		−	30.5264i		−	0.0625542i
$489$		0			0
$490$			25.0718i			0.0511669i
$491$	−162.937			−0.331848			−0.165924	−	0.986139i	$-0.553061\pi$
$491$	−162.937			−0.331848			−0.165924	+	0.986139i	$0.553061\pi$
$492$		0			0
$493$		−	254.753i		−	0.516741i
$494$		−	1016.08i		−	2.05685i
$495$		0			0
$496$	89.2107			0.179860
$497$		−	221.446i		−	0.445565i
$498$		0			0
$499$	410.699			0.823044			0.411522	−	0.911400i	$-0.364997\pi$
$499$	410.699			0.823044			0.411522	+	0.911400i	$0.364997\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$			25.5226i			0.0508419i
$503$		−	817.224i		−	1.62470i	−0.583170	−	0.812350i	$-0.698188\pi$
$503$		−	817.224i		−	1.62470i	0.583170	−	0.812350i	$-0.301812\pi$
$504$		0			0
$505$		−	192.007i		−	0.380212i
$506$	12.8157	+	281.100i	0.0253275	+	0.555534i
$507$		0			0
$508$	427.818			0.842161
$509$	−411.147			−0.807755			−0.403878	−	0.914813i	$-0.632338\pi$
$509$	−411.147			−0.807755			−0.403878	+	0.914813i	$0.632338\pi$
$510$		0			0
$511$		−	723.072i		−	1.41501i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−493.510			−0.960136
$515$	−399.432			−0.775596
$516$		0			0
$517$		−	576.151i		−	1.11441i
$518$	−458.110			−0.884382
$519$		0			0
$520$			126.265i			0.242818i
$521$			848.413i			1.62843i	0.580562	+	0.814216i	$0.302833\pi$
$521$			848.413i			1.62843i	−0.580562	+	0.814216i	$0.697167\pi$
$522$		0			0
$523$		−	527.585i		−	1.00877i	−0.863480	−	0.504383i	$-0.831720\pi$
$523$		−	527.585i		−	1.00877i	0.863480	−	0.504383i	$-0.168280\pi$
$524$	419.679			0.800913
$525$		0			0
$526$		−	196.011i		−	0.372645i
$527$		−	206.762i		−	0.392337i
$528$		0			0
$529$	526.805	−	48.1355i	0.995851	−	0.0909934i
$530$	204.220			0.385320
$531$		0			0
$532$	−543.068			−1.02080
$533$	−783.218			−1.46945
$534$		0			0
$535$	−47.5042			−0.0887929
$536$			166.254i			0.310176i
$537$		0			0
$538$	248.429			0.461765
$539$		−	68.5889i		−	0.127252i
$540$		0			0
$541$	−693.810			−1.28246			−0.641230	−	0.767349i	$-0.721575\pi$
$541$	−693.810			−1.28246			−0.641230	+	0.767349i	$0.721575\pi$
$542$	−347.326			−0.640822
$543$		0			0
$544$			52.4431i			0.0964027i
$545$	419.247			0.769260
$546$		0			0
$547$	−235.004			−0.429624			−0.214812	−	0.976655i	$-0.568914\pi$
$547$	−235.004			−0.429624			−0.214812	+	0.976655i	$0.568914\pi$
$548$			75.4389i			0.137662i
$549$		0			0
$550$			61.1722i			0.111222i
$551$		−	988.931i		−	1.79479i
$552$		0			0
$553$	543.419			0.982675
$554$	−451.502			−0.814985
$555$		0			0
$556$	−16.1796			−0.0291000
$557$			894.637i			1.60617i	0.595864	+	0.803085i	$0.296810\pi$
$557$			894.637i			1.60617i	−0.595864	+	0.803085i	$0.703190\pi$
$558$		0			0
$559$		−	416.396i		−	0.744895i
$560$	67.4853			0.120510
$561$		0			0
$562$		−	119.212i		−	0.212121i
$563$		−	718.198i		−	1.27566i	−0.770176	−	0.637831i	$-0.779832\pi$
$563$		−	718.198i		−	1.27566i	0.770176	−	0.637831i	$-0.220168\pi$
$564$		0			0
$565$	−11.6471			−0.0206144
$566$		−	199.015i		−	0.351616i
$567$		0			0
$568$	−83.0133			−0.146150
$569$			644.903i			1.13340i	0.823925	+	0.566699i	$0.191780\pi$
$569$			644.903i			1.13340i	−0.823925	+	0.566699i	$0.808220\pi$
$570$		0			0
$571$		−	908.958i		−	1.59187i	−0.605381	−	0.795935i	$-0.706979\pi$
$571$		−	908.958i		−	1.59187i	0.605381	−	0.795935i	$-0.293021\pi$
$572$		−	345.425i		−	0.603890i
$573$		0			0
$574$			418.608i			0.729283i
$575$	114.881	−	5.23755i	0.199792	−	0.00910879i
$576$		0			0
$577$	−156.784			−0.271723			−0.135861	−	0.990728i	$-0.543380\pi$
$577$	−156.784			−0.271723			−0.135861	+	0.990728i	$0.543380\pi$
$578$	−287.162			−0.496819
$579$		0			0
$580$			122.891i			0.211882i
$581$	801.552			1.37961
$582$		0			0
$583$	−558.685			−0.958293
$584$	−271.058			−0.464140
$585$		0			0
$586$		−	384.512i		−	0.656164i
$587$	68.2770			0.116315			0.0581576	−	0.998307i	$-0.481477\pi$
$587$	68.2770			0.116315			0.0581576	+	0.998307i	$0.481477\pi$
$588$		0			0
$589$		−	802.632i		−	1.36270i
$590$			284.736i			0.482604i
$591$		0			0
$592$			171.732i			0.290087i
$593$	167.137			0.281849			0.140925	−	0.990020i	$-0.454992\pi$
$593$	167.137			0.281849			0.140925	+	0.990020i	$0.454992\pi$
$594$		0			0
$595$		−	156.409i		−	0.262873i
$596$		−	224.941i		−	0.377419i
$597$		0			0
$598$	−648.704	+	29.5752i	−1.08479	+	0.0494569i
$599$	603.116			1.00687			0.503436	−	0.864033i	$-0.332069\pi$
$599$	603.116			1.00687			0.503436	+	0.864033i	$0.332069\pi$
$600$		0			0
$601$	270.772			0.450536			0.225268	−	0.974297i	$-0.427674\pi$
$601$	270.772			0.450536			0.225268	+	0.974297i	$0.427674\pi$
$602$	−222.552			−0.369688
$603$		0			0
$604$	−429.135			−0.710488
$605$			103.215i			0.170603i
$606$		0			0
$607$	279.365			0.460239			0.230119	−	0.973162i	$-0.426088\pi$
$607$	279.365			0.460239			0.230119	+	0.973162i	$0.426088\pi$
$608$			203.580i			0.334835i
$609$		0			0
$610$	34.1296			0.0559502
$611$	1329.60			2.17611
$612$		0			0
$613$		−	361.882i		−	0.590345i	−0.955444	−	0.295173i	$-0.904623\pi$
$613$		−	361.882i		−	0.590345i	0.955444	−	0.295173i	$-0.0953771\pi$
$614$	−352.964			−0.574860
$615$		0			0
$616$	−184.620			−0.299708
$617$			132.515i			0.214773i	0.994217	+	0.107387i	$0.0342483\pi$
$617$			132.515i			0.214773i	−0.994217	+	0.107387i	$0.965752\pi$
$618$		0			0
$619$		−	76.7590i		−	0.124005i	−0.998076	−	0.0620024i	$-0.980251\pi$
$619$		−	76.7590i		−	0.124005i	0.998076	−	0.0620024i	$-0.0197487\pi$
$620$			99.7406i			0.160872i
$621$		0			0
$622$	−376.519			−0.605336
$623$	−603.658			−0.968954
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	356.906i		−	0.570138i
$627$		0			0
$628$			115.164i			0.183382i
$629$	398.018			0.632780
$630$		0			0
$631$		−	832.399i		−	1.31917i	−0.751628	−	0.659587i	$-0.770731\pi$
$631$		−	832.399i		−	1.31917i	0.751628	−	0.659587i	$-0.229269\pi$
$632$		−	203.712i		−	0.322328i
$633$		0			0
$634$	13.1829			0.0207933
$635$			478.315i			0.753251i
$636$		0			0
$637$	158.285			0.248485
$638$		−	336.195i		−	0.526951i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$		−	139.943i		−	0.218319i	−0.994024	−	0.109160i	$-0.965184\pi$
$641$		−	139.943i		−	0.218319i	0.994024	−	0.109160i	$-0.0348160\pi$
$642$		0			0
$643$			973.488i			1.51398i	0.653428	+	0.756989i	$0.273330\pi$
$643$			973.488i			1.51398i	−0.653428	+	0.756989i	$0.726670\pi$
$644$	15.8071	+	346.714i	0.0245452	+	0.538376i
$645$		0			0
$646$	471.832			0.730391
$647$	465.146			0.718928			0.359464	−	0.933159i	$-0.382960\pi$
$647$	465.146			0.718928			0.359464	+	0.933159i	$0.382960\pi$
$648$		0			0
$649$		−	778.955i		−	1.20024i
$650$	−141.169			−0.217183
$651$		0			0
$652$	−416.763			−0.639208
$653$	−333.454			−0.510649			−0.255325	−	0.966855i	$-0.582182\pi$
$653$	−333.454			−0.510649			−0.255325	+	0.966855i	$0.582182\pi$
$654$		0			0
$655$			469.215i			0.716359i
$656$	156.924			0.239213
$657$		0			0
$658$		−	710.635i		−	1.07999i
$659$		−	608.381i		−	0.923188i	−0.887091	−	0.461594i	$-0.847278\pi$
$659$		−	608.381i		−	0.923188i	0.887091	−	0.461594i	$-0.152722\pi$
$660$		0			0
$661$			62.9560i			0.0952435i	0.998865	+	0.0476218i	$0.0151642\pi$
$661$			62.9560i			0.0952435i	−0.998865	+	0.0476218i	$0.984836\pi$
$662$	770.778			1.16432
$663$		0			0
$664$		−	300.478i		−	0.452527i
$665$		−	607.168i		−	0.913035i
$666$		0			0
$667$	−631.369	+	28.7849i	−0.946581	+	0.0431558i
$668$	−507.453			−0.759661
$669$		0			0
$670$	−185.878			−0.277430
$671$	−93.3685			−0.139148
$672$		0			0
$673$	−910.479			−1.35287			−0.676433	−	0.736504i	$-0.736475\pi$
$673$	−910.479			−1.35287			−0.676433	+	0.736504i	$0.736475\pi$
$674$			13.8714i			0.0205807i
$675$		0			0
$676$	459.148			0.679213
$677$			470.376i			0.694794i	0.937718	+	0.347397i	$0.112934\pi$
$677$			470.376i			0.694794i	−0.937718	+	0.347397i	$0.887066\pi$
$678$		0			0
$679$	778.921			1.14716
$680$	−58.6331			−0.0862252
$681$		0			0
$682$		−	272.861i		−	0.400089i
$683$	−764.068			−1.11869			−0.559347	−	0.828933i	$-0.688948\pi$
$683$	−764.068			−1.11869			−0.559347	+	0.828933i	$0.688948\pi$
$684$		0			0
$685$	−84.3433			−0.123129
$686$			438.249i			0.638847i
$687$		0			0
$688$			83.4281i			0.121262i
$689$		−	1289.29i		−	1.87125i
$690$		0			0
$691$	106.591			0.154257			0.0771284	−	0.997021i	$-0.475425\pi$
$691$	106.591			0.154257			0.0771284	+	0.997021i	$0.475425\pi$
$692$	181.809			0.262730
$693$		0			0
$694$	718.233			1.03492
$695$		−	18.0894i		−	0.0260279i
$696$		0			0
$697$		−	363.698i		−	0.521806i
$698$	−257.610			−0.369068
$699$		0			0
$700$			75.4509i			0.107787i
$701$		−	792.708i		−	1.13082i	−0.824808	−	0.565412i	$-0.808717\pi$
$701$		−	792.708i		−	1.13082i	0.824808	−	0.565412i	$-0.191283\pi$
$702$		0			0
$703$	1545.08			2.19783
$704$			69.2085i			0.0983075i
$705$		0			0
$706$	209.668			0.296981
$707$			647.883i			0.916383i
$708$		0			0
$709$			89.5195i			0.126262i	0.998005	+	0.0631309i	$0.0201085\pi$
$709$			89.5195i			0.126262i	−0.998005	+	0.0631309i	$0.979891\pi$
$710$		−	92.8117i		−	0.130721i
$711$		0			0
$712$			226.293i			0.317828i
$713$	−512.429	+	23.3623i	−0.718694	+	0.0327662i
$714$		0			0
$715$	386.197			0.540135
$716$	−55.4465			−0.0774393
$717$		0			0
$718$		−	670.001i		−	0.933149i
$719$	277.614			0.386112			0.193056	−	0.981188i	$-0.438160\pi$
$719$	277.614			0.386112			0.193056	+	0.981188i	$0.438160\pi$
$720$		0			0
$721$	1347.79			1.86933
$722$	1321.08			1.82976
$723$		0			0
$724$			642.674i			0.887671i
$725$	−137.397			−0.189513
$726$		0			0
$727$		−	1240.96i		−	1.70696i	−0.521123	−	0.853481i	$-0.674487\pi$
$727$		−	1240.96i		−	1.70696i	0.521123	−	0.853481i	$-0.325513\pi$
$728$		−	426.053i		−	0.585238i
$729$		0			0
$730$		−	303.052i		−	0.415140i
$731$	193.359			0.264514
$732$		0			0
$733$			996.086i			1.35892i	0.733714	+	0.679459i	$0.237785\pi$
$733$			996.086i			1.35892i	−0.733714	+	0.679459i	$0.762215\pi$
$734$			36.3480i			0.0495204i
$735$		0			0
$736$	129.973	−	5.92562i	0.176593	−	0.00805111i
$737$	508.507			0.689969
$738$		0			0
$739$	−413.675			−0.559776			−0.279888	−	0.960033i	$-0.590297\pi$
$739$	−413.675			−0.559776			−0.279888	+	0.960033i	$0.590297\pi$
$740$	−192.002			−0.259462
$741$		0			0
$742$	−689.092			−0.928695
$743$		−	704.731i		−	0.948494i	−0.880392	−	0.474247i	$-0.842720\pi$
$743$		−	704.731i		−	0.948494i	0.880392	−	0.474247i	$-0.157280\pi$
$744$		0			0
$745$	251.492			0.337573
$746$		−	146.758i		−	0.196726i
$747$		0			0
$748$	160.403			0.214442
$749$	160.292			0.214008
$750$		0			0
$751$			996.202i			1.32650i	0.748398	+	0.663250i	$0.230824\pi$
$751$			996.202i			1.32650i	−0.748398	+	0.663250i	$0.769176\pi$
$752$	−266.396			−0.354249
$753$		0			0
$754$	775.847			1.02897
$755$		−	479.787i		−	0.635480i
$756$		0			0
$757$			532.958i			0.704040i	0.935992	+	0.352020i	$0.114505\pi$
$757$			532.958i			0.704040i	−0.935992	+	0.352020i	$0.885495\pi$
$758$			85.4752i			0.112764i
$759$		0			0
$760$	−227.609			−0.299486
$761$	−1294.59			−1.70117			−0.850586	−	0.525835i	$-0.823753\pi$
$761$	−1294.59			−1.70117			−0.850586	+	0.525835i	$0.823753\pi$
$762$		0			0
$763$	−1414.65			−1.85406
$764$			64.4770i			0.0843940i
$765$		0			0
$766$		−	323.434i		−	0.422238i
$767$	1797.62			2.34370
$768$		0			0
$769$			691.918i			0.899764i	0.893088	+	0.449882i	$0.148534\pi$
$769$			691.918i			0.899764i	−0.893088	+	0.449882i	$0.851466\pi$
$770$		−	206.411i		−	0.268067i
$771$		0			0
$772$	−61.7048			−0.0799285
$773$		−	721.253i		−	0.933057i	−0.884506	−	0.466529i	$-0.845504\pi$
$773$		−	721.253i		−	0.933057i	0.884506	−	0.466529i	$-0.154496\pi$
$774$		0			0
$775$	−111.513			−0.143888
$776$		−	291.994i		−	0.376281i
$777$		0			0
$778$		−	125.790i		−	0.161683i
$779$		−	1411.85i		−	1.81238i
$780$		0			0
$781$			253.905i			0.325103i
$782$	−13.7337	−	301.235i	−0.0175622	−	0.385211i
$783$		0			0
$784$	−31.7135			−0.0404510
$785$	−128.757			−0.164022
$786$		0			0
$787$			1085.29i			1.37903i	0.724274	+	0.689513i	$0.242175\pi$
$787$			1085.29i			1.37903i	−0.724274	+	0.689513i	$0.757825\pi$
$788$	−568.646			−0.721632
$789$		0			0
$790$	227.756			0.288299
$791$	39.3005			0.0496846
$792$		0			0
$793$		−	215.470i		−	0.271714i
$794$	−679.849			−0.856234
$795$		0			0
$796$			592.069i			0.743805i
$797$		−	518.555i		−	0.650634i	−0.945605	−	0.325317i	$-0.894529\pi$
$797$		−	518.555i		−	0.650634i	0.945605	−	0.325317i	$-0.105471\pi$
$798$		0			0
$799$			617.419i			0.772740i
$800$	28.2843			0.0353553
$801$		0			0
$802$		−	876.394i		−	1.09276i
$803$			829.061i			1.03245i
$804$		0			0
$805$	−387.638	+	17.6729i	−0.481538	+	0.0219539i
$806$	629.689			0.781252
$807$		0			0
$808$	242.872			0.300584
$809$	890.746			1.10105			0.550523	−	0.834820i	$-0.314428\pi$
$809$	890.746			1.10105			0.550523	+	0.834820i	$0.314428\pi$
$810$		0			0
$811$	584.404			0.720597			0.360298	−	0.932837i	$-0.382675\pi$
$811$	584.404			0.720597			0.360298	+	0.932837i	$0.382675\pi$
$812$		−	414.668i		−	0.510675i
$813$		0			0
$814$	525.260			0.645282
$815$		−	465.956i		−	0.571725i
$816$		0			0
$817$	750.605			0.918734
$818$	113.129			0.138299
$819$		0			0
$820$			175.446i			0.213958i
$821$	−599.765			−0.730529			−0.365265	−	0.930904i	$-0.619022\pi$
$821$	−599.765			−0.730529			−0.365265	+	0.930904i	$0.619022\pi$
$822$		0			0
$823$	878.550			1.06750			0.533749	−	0.845643i	$-0.320783\pi$
$823$	878.550			1.06750			0.533749	+	0.845643i	$0.320783\pi$
$824$		−	505.246i		−	0.613163i
$825$		0			0
$826$		−	960.777i		−	1.16317i
$827$		−	1493.08i		−	1.80541i	−0.430258	−	0.902706i	$-0.641577\pi$
$827$		−	1493.08i		−	1.80541i	0.430258	−	0.902706i	$-0.358423\pi$
$828$		0			0
$829$	−614.423			−0.741161			−0.370581	−	0.928800i	$-0.620841\pi$
$829$	−614.423			−0.741161			−0.370581	+	0.928800i	$0.620841\pi$
$830$	335.944			0.404752
$831$		0			0
$832$	−159.715			−0.191965
$833$			73.5018i			0.0882375i
$834$		0			0
$835$		−	567.350i		−	0.679461i
$836$	622.671			0.744822
$837$		0			0
$838$		−	361.327i		−	0.431177i
$839$			20.0377i			0.0238829i	0.999929	+	0.0119414i	$0.00380117\pi$
$839$			20.0377i			0.0238829i	−0.999929	+	0.0119414i	$0.996199\pi$
$840$		0			0
$841$	−85.8849			−0.102122
$842$		−	433.687i		−	0.515068i
$843$		0			0
$844$	−464.931			−0.550866
$845$			513.343i			0.607507i
$846$		0			0
$847$		−	348.275i		−	0.411187i
$848$			258.320i			0.304622i
$849$		0			0
$850$		−	65.5539i		−	0.0771222i
$851$	−44.9727	−	986.432i	−0.0528468	−	1.15914i
$852$		0			0
$853$	−342.579			−0.401617			−0.200808	−	0.979631i	$-0.564357\pi$
$853$	−342.579			−0.401617			−0.200808	+	0.979631i	$0.564357\pi$
$854$	−115.162			−0.134851
$855$		0			0
$856$		−	60.0886i		−	0.0701969i
$857$	−206.677			−0.241163			−0.120582	−	0.992703i	$-0.538476\pi$
$857$	−206.677			−0.241163			−0.120582	+	0.992703i	$0.538476\pi$
$858$		0			0
$859$	−131.130			−0.152655			−0.0763274	−	0.997083i	$-0.524319\pi$
$859$	−131.130			−0.152655			−0.0763274	+	0.997083i	$0.524319\pi$
$860$	−93.2754			−0.108460
$861$		0			0
$862$			820.627i			0.952004i
$863$	−506.478			−0.586881			−0.293440	−	0.955977i	$-0.594800\pi$
$863$	−506.478			−0.586881			−0.293440	+	0.955977i	$0.594800\pi$
$864$		0			0
$865$			203.269i			0.234993i
$866$			983.444i			1.13562i
$867$		0			0
$868$		−	336.551i		−	0.387732i
$869$	−623.074			−0.717001
$870$		0			0
$871$			1173.50i			1.34730i
$872$			530.310i			0.608153i
$873$		0			0
$874$	−53.3130	−	1169.37i	−0.0609989	−	1.33795i
$875$	−84.3567			−0.0964076
$876$		0			0
$877$	605.650			0.690593			0.345297	−	0.938494i	$-0.387778\pi$
$877$	605.650			0.690593			0.345297	+	0.938494i	$0.387778\pi$
$878$	446.201			0.508201
$879$		0			0
$880$	−77.3774			−0.0879289
$881$		−	870.965i		−	0.988609i	−0.869289	−	0.494305i	$-0.835423\pi$
$881$		−	870.965i		−	0.988609i	0.869289	−	0.494305i	$-0.164577\pi$
$882$		0			0
$883$	−790.016			−0.894696			−0.447348	−	0.894360i	$-0.647631\pi$
$883$	−790.016			−0.894696			−0.447348	+	0.894360i	$0.647631\pi$
$884$			370.167i			0.418741i
$885$		0			0
$886$	758.212			0.855770
$887$	329.981			0.372020			0.186010	−	0.982548i	$-0.440444\pi$
$887$	329.981			0.372020			0.186010	+	0.982548i	$0.440444\pi$
$888$		0			0
$889$		−	1613.96i		−	1.81548i
$890$	−253.004			−0.284274
$891$		0			0
$892$	−393.032			−0.440618
$893$			2396.77i			2.68395i
$894$		0			0
$895$		−	61.9911i		−	0.0692638i
$896$			85.3630i			0.0952712i
$897$		0			0
$898$	−956.648			−1.06531
$899$	612.863			0.681716
$900$		0			0
$901$	598.702			0.664486
$902$		−	479.968i		−	0.532116i
$903$		0			0
$904$		−	14.7326i		−	0.0162971i
$905$	−718.531			−0.793957
$906$		0			0
$907$		−	990.130i		−	1.09165i	−0.837898	−	0.545827i	$-0.816216\pi$
$907$		−	990.130i		−	1.09165i	0.837898	−	0.545827i	$-0.183784\pi$
$908$			605.740i			0.667115i
$909$		0			0
$910$	476.342			0.523453
$911$		−	399.780i		−	0.438836i	−0.975631	−	0.219418i	$-0.929584\pi$
$911$		−	399.780i		−	0.438836i	0.975631	−	0.219418i	$-0.0704158\pi$
$912$		0			0
$913$	−919.044			−1.00662
$914$			243.002i			0.265867i
$915$		0			0
$916$			832.017i			0.908316i
$917$		−	1583.26i		−	1.72656i
$918$		0			0
$919$			1471.91i			1.60164i	0.598903	+	0.800822i	$0.295604\pi$
$919$			1471.91i			1.60164i	−0.598903	+	0.800822i	$0.704396\pi$
$920$	6.62504	+	145.314i	0.00720113	+	0.157950i
$921$		0			0
$922$	−1052.64			−1.14169
$923$	−585.946			−0.634827
$924$		0			0
$925$		−	214.664i		−	0.232070i
$926$	162.881			0.175897
$927$		0			0
$928$	−155.447			−0.167507
$929$	−407.172			−0.438291			−0.219145	−	0.975692i	$-0.570327\pi$
$929$	−407.172			−0.438291			−0.219145	+	0.975692i	$0.570327\pi$
$930$		0			0
$931$			285.328i			0.306475i
$932$	197.372			0.211773
$933$		0			0
$934$			796.560i			0.852848i
$935$			179.336i			0.191803i
$936$		0			0
$937$			981.778i			1.04779i	0.851783	+	0.523895i	$0.175521\pi$
$937$			981.778i			1.04779i	−0.851783	+	0.523895i	$0.824479\pi$
$938$	627.201			0.668658
$939$		0			0
$940$		−	297.839i		−	0.316850i
$941$		−	1305.07i		−	1.38690i	−0.720506	−	0.693449i	$-0.756090\pi$
$941$		−	1305.07i		−	1.38690i	0.720506	−	0.693449i	$-0.243910\pi$
$942$		0			0
$943$	−901.374	+	41.0948i	−0.955858	+	0.0435788i
$944$	−360.166			−0.381532
$945$		0			0
$946$	255.174			0.269740
$947$	363.792			0.384152			0.192076	−	0.981380i	$-0.438478\pi$
$947$	363.792			0.384152			0.192076	+	0.981380i	$0.438478\pi$
$948$		0			0
$949$	−1913.25			−2.01607
$950$		−	254.475i		−	0.267868i
$951$		0			0
$952$	197.844			0.207819
$953$		−	995.226i		−	1.04431i	−0.852851	−	0.522154i	$-0.825128\pi$
$953$		−	995.226i		−	1.04431i	0.852851	−	0.522154i	$-0.174872\pi$
$954$		0			0
$955$	−72.0875			−0.0754843
$956$	48.4426			0.0506722
$957$		0			0
$958$		−	597.947i		−	0.624162i
$959$	284.597			0.296764
$960$		0			0
$961$	−463.591			−0.482405
$962$			1212.16i			1.26004i
$963$		0			0
$964$			88.0413i			0.0913292i
$965$		−	68.9881i		−	0.0714902i
$966$		0			0
$967$	1577.54			1.63138			0.815688	−	0.578493i	$-0.196359\pi$
$967$	1577.54			1.63138			0.815688	+	0.578493i	$0.196359\pi$
$968$	−130.558			−0.134874
$969$		0			0
$970$	326.459			0.336556
$971$			1397.09i			1.43882i	0.694586	+	0.719410i	$0.255588\pi$
$971$			1397.09i			1.43882i	−0.694586	+	0.719410i	$0.744412\pi$
$972$		0			0
$973$			61.0384i			0.0627321i
$974$	−1151.01			−1.18174
$975$		0			0
$976$			43.1709i			0.0442325i
$977$		−	1329.91i		−	1.36122i	−0.732648	−	0.680608i	$-0.761716\pi$
$977$		−	1329.91i		−	1.36122i	0.732648	−	0.680608i	$-0.238284\pi$
$978$		0			0
$979$	692.143			0.706990
$980$		−	35.4568i		−	0.0361804i
$981$		0			0
$982$	230.428			0.234652
$983$		−	1362.46i		−	1.38603i	−0.720925	−	0.693013i	$-0.756283\pi$
$983$		−	1362.46i		−	1.38603i	0.720925	−	0.693013i	$-0.243717\pi$
$984$		0			0
$985$		−	635.766i		−	0.645447i
$986$			360.276i			0.365391i
$987$		0			0
$988$			1436.96i			1.45441i
$989$	−21.8479	−	479.214i	−0.0220909	−	0.484543i
$990$		0			0
$991$	−1614.70			−1.62937			−0.814684	−	0.579905i	$-0.803090\pi$
$991$	−1614.70			−1.62937			−0.814684	+	0.579905i	$0.803090\pi$
$992$	−126.163			−0.127180
$993$		0			0
$994$			313.172i			0.315062i
$995$	−661.953			−0.665279
$996$		0			0
$997$	−931.722			−0.934525			−0.467263	−	0.884119i	$-0.654760\pi$
$997$	−931.722			−0.934525			−0.467263	+	0.884119i	$0.654760\pi$
$998$	−580.816			−0.581980
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.11		32
3.2	odd	2		690.3.c.a.91.25	✓	32
23.22	odd	2	inner	2070.3.c.b.91.6		32
69.68	even	2		690.3.c.a.91.32	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.25	✓	32	3.2	odd	2
690.3.c.a.91.32	yes	32	69.68	even	2
2070.3.c.b.91.6		32	23.22	odd	2	inner
2070.3.c.b.91.11		32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.54509i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.54509i q^{7} -2.82843 q^{8} -3.16228i q^{10} +8.65106i q^{11} -19.9643 q^{13} +10.6704i q^{14} +4.00000 q^{16} -9.27071i q^{17} -35.9882i q^{19} +4.47214i q^{20} -12.2344i q^{22} +(-22.9761 + 1.04751i) q^{23} -5.00000 q^{25} +28.2338 q^{26} -15.0902i q^{28} +27.4794 q^{29} +22.3027 q^{31} -5.65685 q^{32} +13.1108i q^{34} +16.8713 q^{35} +42.9329i q^{37} +50.8949i q^{38} -6.32456i q^{40} +39.2309 q^{41} +20.8570i q^{43} +17.3021i q^{44} +(32.4932 - 1.48140i) q^{46} -66.5989 q^{47} -7.92839 q^{49} +7.07107 q^{50} -39.9286 q^{52} +64.5799i q^{53} -19.3444 q^{55} +21.3407i q^{56} -38.8617 q^{58} -90.0416 q^{59} +10.7927i q^{61} -31.5407 q^{62} +8.00000 q^{64} -44.6416i q^{65} -58.7797i q^{67} -18.5414i q^{68} -23.8597 q^{70} +29.3496 q^{71} +95.8335 q^{73} -60.7163i q^{74} -71.9763i q^{76} +65.2730 q^{77} +72.0229i q^{79} +8.94427i q^{80} -55.4809 q^{82} +106.235i q^{83} +20.7299 q^{85} -29.4963i q^{86} -24.4689i q^{88} -80.0068i q^{89} +150.633i q^{91} +(-45.9523 + 2.09502i) q^{92} +94.1850 q^{94} +80.4720 q^{95} +103.236i q^{97} +11.2124 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 32 * q + 64 * q^4	\( 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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