LMFDB - Embedded newform 2070.3.c.b.91.21

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			91.21
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +3.49916i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +3.49916i q^{7} +2.82843 q^{8} -3.16228i q^{10} +11.9699i q^{11} -19.9639 q^{13} +4.94856i q^{14} +4.00000 q^{16} -26.2299i q^{17} -0.0297478i q^{19} -4.47214i q^{20} +16.9280i q^{22} +(9.54233 - 20.9271i) q^{23} -5.00000 q^{25} -28.2332 q^{26} +6.99832i q^{28} +41.0391 q^{29} +29.7926 q^{31} +5.65685 q^{32} -37.0947i q^{34} +7.82436 q^{35} -51.3489i q^{37} -0.0420698i q^{38} -6.32456i q^{40} +74.6506 q^{41} +51.7973i q^{43} +23.9398i q^{44} +(13.4949 - 29.5954i) q^{46} +27.5439 q^{47} +36.7559 q^{49} -7.07107 q^{50} -39.9278 q^{52} -15.0332i q^{53} +26.7655 q^{55} +9.89712i q^{56} +58.0381 q^{58} -1.55359 q^{59} +12.9254i q^{61} +42.1330 q^{62} +8.00000 q^{64} +44.6406i q^{65} -28.2878i q^{67} -52.4598i q^{68} +11.0653 q^{70} +94.7727 q^{71} -25.3868 q^{73} -72.6184i q^{74} -0.0594957i q^{76} -41.8845 q^{77} +80.5613i q^{79} -8.94427i q^{80} +105.572 q^{82} +55.8397i q^{83} -58.6518 q^{85} +73.2525i q^{86} +33.8559i q^{88} -81.9490i q^{89} -69.8568i q^{91} +(19.0847 - 41.8542i) q^{92} +38.9530 q^{94} -0.0665182 q^{95} -4.03039i q^{97} +51.9807 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			3.49916i			0.499880i	0.968261	+	0.249940i	$0.0804109\pi$
$7$			3.49916i			0.499880i	−0.968261	+	0.249940i	$0.919589\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$			11.9699i			1.08817i	0.839030	+	0.544086i	$0.183123\pi$
$11$			11.9699i			1.08817i	−0.839030	+	0.544086i	$0.816877\pi$
$12$		0			0
$13$	−19.9639			−1.53568			−0.767842	−	0.640640i	$-0.778669\pi$
$13$	−19.9639			−1.53568			−0.767842	+	0.640640i	$0.778669\pi$
$14$			4.94856i			0.353468i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	26.2299i		−	1.54293i	−0.636269	−	0.771467i	$-0.719523\pi$
$17$		−	26.2299i		−	1.54293i	0.636269	−	0.771467i	$-0.280477\pi$
$18$		0			0
$19$		−	0.0297478i		−	0.00156568i	−1.00000		0.000782838i	$-0.999751\pi$
$19$		−	0.0297478i		−	0.00156568i	1.00000		0.000782838i	$-0.000249185\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$			16.9280i			0.769453i
$23$	9.54233	−	20.9271i	0.414884	−	0.909874i
$23$	9.54233	−	20.9271i	0.414884	−	0.909874i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−28.2332			−1.08589
$27$		0			0
$28$			6.99832i			0.249940i
$29$	41.0391			1.41514			0.707571	−	0.706642i	$-0.249791\pi$
$29$	41.0391			1.41514			0.707571	+	0.706642i	$0.249791\pi$
$30$		0			0
$31$	29.7926			0.961050			0.480525	−	0.876981i	$-0.340446\pi$
$31$	29.7926			0.961050			0.480525	+	0.876981i	$0.340446\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$		−	37.0947i		−	1.09102i
$35$	7.82436			0.223553
$36$		0			0
$37$		−	51.3489i		−	1.38781i	−0.720067	−	0.693905i	$-0.755889\pi$
$37$		−	51.3489i		−	1.38781i	0.720067	−	0.693905i	$-0.244111\pi$
$38$		−	0.0420698i		−	0.00110710i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	74.6506			1.82075			0.910373	−	0.413789i	$-0.135795\pi$
$41$	74.6506			1.82075			0.910373	+	0.413789i	$0.135795\pi$
$42$		0			0
$43$			51.7973i			1.20459i	0.798274	+	0.602294i	$0.205747\pi$
$43$			51.7973i			1.20459i	−0.798274	+	0.602294i	$0.794253\pi$
$44$			23.9398i			0.544086i
$45$		0			0
$46$	13.4949	−	29.5954i	0.293367	−	0.643378i
$47$	27.5439			0.586040			0.293020	−	0.956106i	$-0.405340\pi$
$47$	27.5439			0.586040			0.293020	+	0.956106i	$0.405340\pi$
$48$		0			0
$49$	36.7559			0.750120
$50$	−7.07107			−0.141421
$51$		0			0
$52$	−39.9278			−0.767842
$53$		−	15.0332i		−	0.283645i	−0.989892	−	0.141823i	$-0.954704\pi$
$53$		−	15.0332i		−	0.283645i	0.989892	−	0.141823i	$-0.0452963\pi$
$54$		0			0
$55$	26.7655			0.486645
$56$			9.89712i			0.176734i
$57$		0			0
$58$	58.0381			1.00066
$59$	−1.55359			−0.0263320			−0.0131660	−	0.999913i	$-0.504191\pi$
$59$	−1.55359			−0.0263320			−0.0131660	+	0.999913i	$0.504191\pi$
$60$		0			0
$61$			12.9254i			0.211892i	0.994372	+	0.105946i	$0.0337871\pi$
$61$			12.9254i			0.211892i	−0.994372	+	0.105946i	$0.966213\pi$
$62$	42.1330			0.679565
$63$		0			0
$64$	8.00000			0.125000
$65$			44.6406i			0.686778i
$66$		0			0
$67$		−	28.2878i		−	0.422206i	−0.977464	−	0.211103i	$-0.932294\pi$
$67$		−	28.2878i		−	0.422206i	0.977464	−	0.211103i	$-0.0677055\pi$
$68$		−	52.4598i		−	0.771467i
$69$		0			0
$70$	11.0653			0.158076
$71$	94.7727			1.33483			0.667413	−	0.744688i	$-0.267401\pi$
$71$	94.7727			1.33483			0.667413	+	0.744688i	$0.267401\pi$
$72$		0			0
$73$	−25.3868			−0.347764			−0.173882	−	0.984766i	$-0.555631\pi$
$73$	−25.3868			−0.347764			−0.173882	+	0.984766i	$0.555631\pi$
$74$		−	72.6184i		−	0.981329i
$75$		0			0
$76$		−	0.0594957i		−	0.000782838i
$77$	−41.8845			−0.543955
$78$		0			0
$79$			80.5613i			1.01976i	0.860245	+	0.509882i	$0.170311\pi$
$79$			80.5613i			1.01976i	−0.860245	+	0.509882i	$0.829689\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	105.572			1.28746
$83$			55.8397i			0.672767i	0.941725	+	0.336384i	$0.109204\pi$
$83$			55.8397i			0.672767i	−0.941725	+	0.336384i	$0.890796\pi$
$84$		0			0
$85$	−58.6518			−0.690021
$86$			73.2525i			0.851773i
$87$		0			0
$88$			33.8559i			0.384727i
$89$		−	81.9490i		−	0.920775i	−0.887718	−	0.460387i	$-0.847710\pi$
$89$		−	81.9490i		−	0.920775i	0.887718	−	0.460387i	$-0.152290\pi$
$90$		0			0
$91$		−	69.8568i		−	0.767657i
$92$	19.0847	−	41.8542i	0.207442	−	0.454937i
$93$		0			0
$94$	38.9530			0.414393
$95$	−0.0665182			−0.000700191
$96$		0			0
$97$		−	4.03039i		−	0.0415504i	−0.999784	−	0.0207752i	$-0.993387\pi$
$97$		−	4.03039i		−	0.0415504i	0.999784	−	0.0207752i	$-0.00661343\pi$
$98$	51.9807			0.530415
$99$		0			0
$100$	−10.0000			−0.100000
$101$	162.974			1.61360			0.806801	−	0.590823i	$-0.201197\pi$
$101$	162.974			1.61360			0.806801	+	0.590823i	$0.201197\pi$
$102$		0			0
$103$			137.541i			1.33535i	0.744453	+	0.667675i	$0.232711\pi$
$103$			137.541i			1.33535i	−0.744453	+	0.667675i	$0.767289\pi$
$104$	−56.4664			−0.542946
$105$		0			0
$106$		−	21.2602i		−	0.200567i
$107$			66.5945i			0.622379i	0.950348	+	0.311189i	$0.100727\pi$
$107$			66.5945i			0.622379i	−0.950348	+	0.311189i	$0.899273\pi$
$108$		0			0
$109$		−	95.0049i		−	0.871605i	−0.900042	−	0.435802i	$-0.856465\pi$
$109$		−	95.0049i		−	0.871605i	0.900042	−	0.435802i	$-0.143535\pi$
$110$	37.8521			0.344110
$111$		0			0
$112$			13.9966i			0.124970i
$113$		−	194.493i		−	1.72118i	−0.509300	−	0.860589i	$-0.670096\pi$
$113$		−	194.493i		−	1.72118i	0.509300	−	0.860589i	$-0.329904\pi$
$114$		0			0
$115$	−46.7944	−	21.3373i	−0.406908	−	0.185542i
$116$	82.0782			0.707571
$117$		0			0
$118$	−2.19711			−0.0186195
$119$	91.7825			0.771282
$120$		0			0
$121$	−22.2782			−0.184117
$122$			18.2793i			0.149830i
$123$		0			0
$124$	59.5851			0.480525
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−139.254			−1.09649			−0.548244	−	0.836318i	$-0.684704\pi$
$127$	−139.254			−1.09649			−0.548244	+	0.836318i	$0.684704\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$			63.1313i			0.485626i
$131$	−87.8593			−0.670682			−0.335341	−	0.942097i	$-0.608851\pi$
$131$	−87.8593			−0.670682			−0.335341	+	0.942097i	$0.608851\pi$
$132$		0			0
$133$	0.104092			0.000782650
$134$		−	40.0050i		−	0.298545i
$135$		0			0
$136$		−	74.1893i		−	0.545510i
$137$		−	181.394i		−	1.32405i	−0.749484	−	0.662023i	$-0.769698\pi$
$137$		−	181.394i		−	1.32405i	0.749484	−	0.662023i	$-0.230302\pi$
$138$		0			0
$139$	−135.089			−0.971860			−0.485930	−	0.873998i	$-0.661519\pi$
$139$	−135.089			−0.971860			−0.485930	+	0.873998i	$0.661519\pi$
$140$	15.6487			0.111777
$141$		0			0
$142$	134.029			0.943865
$143$		−	238.965i		−	1.67109i
$144$		0			0
$145$		−	91.7662i		−	0.632871i
$146$	−35.9023			−0.245906
$147$		0			0
$148$		−	102.698i		−	0.693905i
$149$			56.7676i			0.380991i	0.981688	+	0.190495i	$0.0610094\pi$
$149$			56.7676i			0.380991i	−0.981688	+	0.190495i	$0.938991\pi$
$150$		0			0
$151$	36.4693			0.241519			0.120759	−	0.992682i	$-0.461467\pi$
$151$	36.4693			0.241519			0.120759	+	0.992682i	$0.461467\pi$
$152$		−	0.0841396i		−	0.000553550i
$153$		0			0
$154$	−59.2337			−0.384634
$155$		−	66.6182i		−	0.429795i
$156$		0			0
$157$		−	47.7035i		−	0.303844i	−0.988392	−	0.151922i	$-0.951454\pi$
$157$		−	47.7035i		−	0.303844i	0.988392	−	0.151922i	$-0.0485463\pi$
$158$			113.931i			0.721081i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	73.2273	+	33.3901i	0.454828	+	0.207392i
$162$		0			0
$163$	118.851			0.729148			0.364574	−	0.931174i	$-0.381215\pi$
$163$	118.851			0.729148			0.364574	+	0.931174i	$0.381215\pi$
$164$	149.301			0.910373
$165$		0			0
$166$			78.9692i			0.475718i
$167$	108.318			0.648610			0.324305	−	0.945952i	$-0.394870\pi$
$167$	108.318			0.648610			0.324305	+	0.945952i	$0.394870\pi$
$168$		0			0
$169$	229.557			1.35832
$170$	−82.9462			−0.487919
$171$		0			0
$172$			103.595i			0.602294i
$173$	−78.7417			−0.455154			−0.227577	−	0.973760i	$-0.573080\pi$
$173$	−78.7417			−0.455154			−0.227577	+	0.973760i	$0.573080\pi$
$174$		0			0
$175$		−	17.4958i		−	0.0999760i
$176$			47.8795i			0.272043i
$177$		0			0
$178$		−	115.893i		−	0.651086i
$179$	−225.677			−1.26077			−0.630383	−	0.776285i	$-0.717102\pi$
$179$	−225.677			−1.26077			−0.630383	+	0.776285i	$0.717102\pi$
$180$		0			0
$181$		−	219.280i		−	1.21149i	−0.795659	−	0.605745i	$-0.792875\pi$
$181$		−	219.280i		−	1.21149i	0.795659	−	0.605745i	$-0.207125\pi$
$182$		−	98.7924i		−	0.542816i
$183$		0			0
$184$	26.9898	−	59.1908i	0.146684	−	0.321689i
$185$	−114.820			−0.620647
$186$		0			0
$187$	313.969			1.67898
$188$	55.0878			0.293020
$189$		0			0
$190$	−0.0940709			−0.000495110
$191$			240.006i			1.25658i	0.777981	+	0.628288i	$0.216244\pi$
$191$			240.006i			1.25658i	−0.777981	+	0.628288i	$0.783756\pi$
$192$		0			0
$193$	368.019			1.90683			0.953417	−	0.301654i	$-0.0975388\pi$
$193$	368.019			1.90683			0.953417	+	0.301654i	$0.0975388\pi$
$194$		−	5.69984i		−	0.0293806i
$195$		0			0
$196$	73.5118			0.375060
$197$	1.84737			0.00937754			0.00468877	−	0.999989i	$-0.498508\pi$
$197$	1.84737			0.00937754			0.00468877	+	0.999989i	$0.498508\pi$
$198$		0			0
$199$			56.5030i			0.283934i	0.989871	+	0.141967i	$0.0453428\pi$
$199$			56.5030i			0.283934i	−0.989871	+	0.141967i	$0.954657\pi$
$200$	−14.1421			−0.0707107
$201$		0			0
$202$	230.480			1.14099
$203$			143.602i			0.707401i
$204$		0			0
$205$		−	166.924i		−	0.814262i
$206$			194.513i			0.944236i
$207$		0			0
$208$	−79.8555			−0.383921
$209$	0.356078			0.00170372
$210$		0			0
$211$	−88.2861			−0.418418			−0.209209	−	0.977871i	$-0.567089\pi$
$211$	−88.2861			−0.418418			−0.209209	+	0.977871i	$0.567089\pi$
$212$		−	30.0664i		−	0.141823i
$213$		0			0
$214$			94.1788i			0.440088i
$215$	115.822			0.538708
$216$		0			0
$217$			104.249i			0.480410i
$218$		−	134.357i		−	0.616317i
$219$		0			0
$220$	53.5310			0.243323
$221$			523.650i			2.36946i
$222$		0			0
$223$	129.548			0.580934			0.290467	−	0.956885i	$-0.406189\pi$
$223$	129.548			0.580934			0.290467	+	0.956885i	$0.406189\pi$
$224$			19.7942i			0.0883671i
$225$		0			0
$226$		−	275.055i		−	1.21706i
$227$		−	76.1848i		−	0.335616i	−0.985820	−	0.167808i	$-0.946331\pi$
$227$		−	76.1848i		−	0.335616i	0.985820	−	0.167808i	$-0.0536688\pi$
$228$		0			0
$229$			81.3519i			0.355248i	0.984098	+	0.177624i	$0.0568411\pi$
$229$			81.3519i			0.355248i	−0.984098	+	0.177624i	$0.943159\pi$
$230$	−66.1773	−	30.1755i	−0.287728	−	0.131198i
$231$		0			0
$232$	116.076			0.500328
$233$	−229.590			−0.985366			−0.492683	−	0.870209i	$-0.663984\pi$
$233$	−229.590			−0.985366			−0.492683	+	0.870209i	$0.663984\pi$
$234$		0			0
$235$		−	61.5900i		−	0.262085i
$236$	−3.10718			−0.0131660
$237$		0			0
$238$	129.800			0.545379
$239$	156.740			0.655818			0.327909	−	0.944709i	$-0.393656\pi$
$239$	156.740			0.655818			0.327909	+	0.944709i	$0.393656\pi$
$240$		0			0
$241$		−	191.086i		−	0.792887i	−0.918059	−	0.396444i	$-0.870244\pi$
$241$		−	191.086i		−	0.792887i	0.918059	−	0.396444i	$-0.129756\pi$
$242$	−31.5061			−0.130190
$243$		0			0
$244$			25.8508i			0.105946i
$245$		−	82.1887i		−	0.335464i
$246$		0			0
$247$			0.593882i			0.00240438i
$248$	84.2661			0.339783
$249$		0			0
$250$			15.8114i			0.0632456i
$251$		−	456.878i		−	1.82023i	−0.414354	−	0.910116i	$-0.635992\pi$
$251$		−	456.878i		−	1.82023i	0.414354	−	0.910116i	$-0.364008\pi$
$252$		0			0
$253$	250.495	+	114.221i	0.990099	+	0.451465i
$254$	−196.935			−0.775335
$255$		0			0
$256$	16.0000			0.0625000
$257$	−212.501			−0.826851			−0.413425	−	0.910538i	$-0.635668\pi$
$257$	−212.501			−0.826851			−0.413425	+	0.910538i	$0.635668\pi$
$258$		0			0
$259$	179.678			0.693738
$260$			89.2812i			0.343389i
$261$		0			0
$262$	−124.252			−0.474243
$263$			478.231i			1.81837i	0.416392	+	0.909185i	$0.363294\pi$
$263$			478.231i			1.81837i	−0.416392	+	0.909185i	$0.636706\pi$
$264$		0			0
$265$	−33.6153			−0.126850
$266$	0.147209			0.000553417
$267$		0			0
$268$		−	56.5756i		−	0.211103i
$269$	−525.346			−1.95296			−0.976479	−	0.215613i	$-0.930825\pi$
$269$	−525.346			−1.95296			−0.976479	+	0.215613i	$0.930825\pi$
$270$		0			0
$271$	374.732			1.38278			0.691388	−	0.722484i	$-0.257000\pi$
$271$	374.732			1.38278			0.691388	+	0.722484i	$0.257000\pi$
$272$		−	104.920i		−	0.385734i
$273$		0			0
$274$		−	256.530i		−	0.936241i
$275$		−	59.8494i		−	0.217634i
$276$		0			0
$277$	−406.934			−1.46907			−0.734537	−	0.678569i	$-0.762601\pi$
$277$	−406.934			−1.46907			−0.734537	+	0.678569i	$0.762601\pi$
$278$	−191.044			−0.687209
$279$		0			0
$280$	22.1306			0.0790379
$281$			189.257i			0.673511i	0.941592	+	0.336755i	$0.109330\pi$
$281$			189.257i			0.673511i	−0.941592	+	0.336755i	$0.890670\pi$
$282$		0			0
$283$			157.312i			0.555872i	0.960600	+	0.277936i	$0.0896503\pi$
$283$			157.312i			0.555872i	−0.960600	+	0.277936i	$0.910350\pi$
$284$	189.545			0.667413
$285$		0			0
$286$		−	337.948i		−	1.18164i
$287$			261.214i			0.910154i
$288$		0			0
$289$	−399.007			−1.38065
$290$		−	129.777i		−	0.447507i
$291$		0			0
$292$	−50.7736			−0.173882
$293$		−	463.682i		−	1.58253i	−0.611472	−	0.791266i	$-0.709422\pi$
$293$		−	463.682i		−	1.58253i	0.611472	−	0.791266i	$-0.290578\pi$
$294$		0			0
$295$			3.47393i			0.0117760i
$296$		−	145.237i		−	0.490665i
$297$		0			0
$298$			80.2815i			0.269401i
$299$	−190.502	+	417.786i	−0.637130	+	1.39728i
$300$		0			0
$301$	−181.247			−0.602150
$302$	51.5754			0.170780
$303$		0			0
$304$		−	0.118991i		−	0.000391419i
$305$	28.9021			0.0947609
$306$		0			0
$307$	350.361			1.14124			0.570620	−	0.821214i	$-0.306703\pi$
$307$	350.361			1.14124			0.570620	+	0.821214i	$0.306703\pi$
$308$	−83.7691			−0.271977
$309$		0			0
$310$		−	94.2123i		−	0.303911i
$311$	−152.396			−0.490018			−0.245009	−	0.969521i	$-0.578791\pi$
$311$	−152.396			−0.490018			−0.245009	+	0.969521i	$0.578791\pi$
$312$		0			0
$313$		−	47.6313i		−	0.152177i	−0.997101	−	0.0760884i	$-0.975757\pi$
$313$		−	47.6313i		−	0.152177i	0.997101	−	0.0760884i	$-0.0242431\pi$
$314$		−	67.4629i		−	0.214850i
$315$		0			0
$316$			161.123i			0.509882i
$317$	−410.929			−1.29631			−0.648153	−	0.761511i	$-0.724458\pi$
$317$	−410.929			−1.29631			−0.648153	+	0.761511i	$0.724458\pi$
$318$		0			0
$319$			491.233i			1.53992i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	103.559	+	47.2208i	0.321612	+	0.146648i
$323$	−0.780282			−0.00241573
$324$		0			0
$325$	99.8194			0.307137
$326$	168.081			0.515585
$327$		0			0
$328$	211.144			0.643731
$329$			96.3805i			0.292950i
$330$		0			0
$331$	−392.356			−1.18537			−0.592683	−	0.805436i	$-0.701931\pi$
$331$	−392.356			−1.18537			−0.592683	+	0.805436i	$0.701931\pi$
$332$			111.679i			0.336384i
$333$		0			0
$334$	153.185			0.458637
$335$	−63.2534			−0.188816
$336$		0			0
$337$			458.368i			1.36014i	0.733146	+	0.680071i	$0.238051\pi$
$337$			458.368i			1.36014i	−0.733146	+	0.680071i	$0.761949\pi$
$338$	324.642			0.960480
$339$		0			0
$340$	−117.304			−0.345011
$341$			356.614i			1.04579i
$342$		0			0
$343$			300.073i			0.874850i
$344$			146.505i			0.425886i
$345$		0			0
$346$	−111.358			−0.321843
$347$	178.744			0.515112			0.257556	−	0.966263i	$-0.417083\pi$
$347$	178.744			0.515112			0.257556	+	0.966263i	$0.417083\pi$
$348$		0			0
$349$	−24.0318			−0.0688591			−0.0344295	−	0.999407i	$-0.510961\pi$
$349$	−24.0318			−0.0688591			−0.0344295	+	0.999407i	$0.510961\pi$
$350$		−	24.7428i		−	0.0706937i
$351$		0			0
$352$			67.7119i			0.192363i
$353$	568.260			1.60980			0.804901	−	0.593409i	$-0.202218\pi$
$353$	568.260			1.60980			0.804901	+	0.593409i	$0.202218\pi$
$354$		0			0
$355$		−	211.918i		−	0.596953i
$356$		−	163.898i		−	0.460387i
$357$		0			0
$358$	−319.155			−0.891496
$359$		−	540.074i		−	1.50438i	−0.658945	−	0.752192i	$-0.728997\pi$
$359$		−	540.074i		−	1.50438i	0.658945	−	0.752192i	$-0.271003\pi$
$360$		0			0
$361$	360.999			0.999998
$362$		−	310.108i		−	0.856653i
$363$		0			0
$364$		−	139.714i		−	0.383829i
$365$			56.7666i			0.155525i
$366$		0			0
$367$		−	404.550i		−	1.10232i	−0.834401	−	0.551158i	$-0.814186\pi$
$367$		−	404.550i		−	1.10232i	0.834401	−	0.551158i	$-0.185814\pi$
$368$	38.1693	−	83.7084i	0.103721	−	0.227469i
$369$		0			0
$370$	−162.380			−0.438864
$371$	52.6035			0.141789
$372$		0			0
$373$		−	192.435i		−	0.515911i	−0.966157	−	0.257955i	$-0.916951\pi$
$373$		−	192.435i		−	0.515911i	0.966157	−	0.257955i	$-0.0830487\pi$
$374$	444.019			1.18722
$375$		0			0
$376$	77.9059			0.207197
$377$	−819.300			−2.17321
$378$		0			0
$379$		−	711.208i		−	1.87654i	−0.345908	−	0.938269i	$-0.612429\pi$
$379$		−	711.208i		−	1.87654i	0.345908	−	0.938269i	$-0.387571\pi$
$380$	−0.133036			−0.000350096
$381$		0			0
$382$			339.420i			0.888533i
$383$		−	308.566i		−	0.805655i	−0.915276	−	0.402827i	$-0.868027\pi$
$383$		−	308.566i		−	0.805655i	0.915276	−	0.402827i	$-0.131973\pi$
$384$		0			0
$385$			93.6567i			0.243264i
$386$	520.458			1.34834
$387$		0			0
$388$		−	8.06079i		−	0.0207752i
$389$		−	197.980i		−	0.508947i	−0.967080	−	0.254474i	$-0.918098\pi$
$389$		−	197.980i		−	0.508947i	0.967080	−	0.254474i	$-0.0819022\pi$
$390$		0			0
$391$	−548.916	−	250.294i	−1.40388	−	0.640139i
$392$	103.961			0.265208
$393$		0			0
$394$	2.61258			0.00663092
$395$	180.141			0.456052
$396$		0			0
$397$	267.667			0.674224			0.337112	−	0.941465i	$-0.390550\pi$
$397$	267.667			0.674224			0.337112	+	0.941465i	$0.390550\pi$
$398$			79.9073i			0.200772i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			749.856i			1.86997i	0.354694	+	0.934983i	$0.384585\pi$
$401$			749.856i			1.86997i	−0.354694	+	0.934983i	$0.615415\pi$
$402$		0			0
$403$	−594.775			−1.47587
$404$	325.948			0.806801
$405$		0			0
$406$			203.084i			0.500208i
$407$	614.641			1.51017
$408$		0			0
$409$	291.243			0.712086			0.356043	−	0.934469i	$-0.384126\pi$
$409$	291.243			0.712086			0.356043	+	0.934469i	$0.384126\pi$
$410$		−	236.066i		−	0.575770i
$411$		0			0
$412$			275.082i			0.667675i
$413$		−	5.43625i		−	0.0131628i
$414$		0			0
$415$	124.861			0.300871
$416$	−112.933			−0.271473
$417$		0			0
$418$	0.503571			0.00120471
$419$		−	334.184i		−	0.797574i	−0.917044	−	0.398787i	$-0.869431\pi$
$419$		−	334.184i		−	0.797574i	0.917044	−	0.398787i	$-0.130569\pi$
$420$		0			0
$421$			414.441i			0.984422i	0.870476	+	0.492211i	$0.163811\pi$
$421$			414.441i			0.984422i	−0.870476	+	0.492211i	$0.836189\pi$
$422$	−124.855			−0.295866
$423$		0			0
$424$		−	42.5203i		−	0.100284i
$425$			131.149i			0.308587i
$426$		0			0
$427$	−45.2280			−0.105920
$428$			133.189i			0.311189i
$429$		0			0
$430$	163.797			0.380924
$431$			721.242i			1.67342i	0.547649	+	0.836708i	$0.315523\pi$
$431$			721.242i			1.67342i	−0.547649	+	0.836708i	$0.684477\pi$
$432$		0			0
$433$		−	231.360i		−	0.534319i	−0.963652	−	0.267160i	$-0.913915\pi$
$433$		−	231.360i		−	0.534319i	0.963652	−	0.267160i	$-0.0860851\pi$
$434$			147.430i			0.339701i
$435$		0			0
$436$		−	190.010i		−	0.435802i
$437$	−0.622536	−	0.283864i	−0.00142457	−	0.000649574i
$438$		0			0
$439$	110.160			0.250935			0.125468	−	0.992098i	$-0.459957\pi$
$439$	110.160			0.250935			0.125468	+	0.992098i	$0.459957\pi$
$440$	75.7042			0.172055
$441$		0			0
$442$			740.553i			1.67546i
$443$	−140.713			−0.317636			−0.158818	−	0.987308i	$-0.550768\pi$
$443$	−140.713			−0.317636			−0.158818	+	0.987308i	$0.550768\pi$
$444$		0			0
$445$	−183.243			−0.411783
$446$	183.209			0.410782
$447$		0			0
$448$			27.9933i			0.0624850i
$449$	−222.010			−0.494455			−0.247227	−	0.968958i	$-0.579519\pi$
$449$	−222.010			−0.494455			−0.247227	+	0.968958i	$0.579519\pi$
$450$		0			0
$451$			893.559i			1.98128i
$452$		−	388.986i		−	0.860589i
$453$		0			0
$454$		−	107.742i		−	0.237316i
$455$	−156.205			−0.343307
$456$		0			0
$457$			448.070i			0.980460i	0.871593	+	0.490230i	$0.163087\pi$
$457$			448.070i			0.980460i	−0.871593	+	0.490230i	$0.836913\pi$
$458$			115.049i			0.251199i
$459$		0			0
$460$	−93.5889	−	42.6746i	−0.203454	−	0.0927709i
$461$	−819.214			−1.77704			−0.888519	−	0.458841i	$-0.848265\pi$
$461$	−819.214			−1.77704			−0.888519	+	0.458841i	$0.848265\pi$
$462$		0			0
$463$	−793.838			−1.71455			−0.857276	−	0.514857i	$-0.827845\pi$
$463$	−793.838			−1.71455			−0.857276	+	0.514857i	$0.827845\pi$
$464$	164.156			0.353785
$465$		0			0
$466$	−324.690			−0.696759
$467$		−	335.892i		−	0.719254i	−0.933096	−	0.359627i	$-0.882904\pi$
$467$		−	335.892i		−	0.719254i	0.933096	−	0.359627i	$-0.117096\pi$
$468$		0			0
$469$	98.9835			0.211052
$470$		−	87.1015i		−	0.185322i
$471$		0			0
$472$	−4.39421			−0.00930977
$473$	−620.008			−1.31080
$474$		0			0
$475$			0.148739i			0.000313135i
$476$	183.565			0.385641
$477$		0			0
$478$	221.664			0.463733
$479$		−	578.078i		−	1.20684i	−0.797422	−	0.603422i	$-0.793804\pi$
$479$		−	578.078i		−	1.20684i	0.797422	−	0.603422i	$-0.206196\pi$
$480$		0			0
$481$			1025.12i			2.13124i
$482$		−	270.236i		−	0.560656i
$483$		0			0
$484$	−44.5563			−0.0920585
$485$	−9.01223			−0.0185819
$486$		0			0
$487$	−921.262			−1.89171			−0.945854	−	0.324591i	$-0.894773\pi$
$487$	−921.262			−1.89171			−0.945854	+	0.324591i	$0.894773\pi$
$488$			36.5586i			0.0749151i
$489$		0			0
$490$		−	116.232i		−	0.237209i
$491$	830.923			1.69231			0.846153	−	0.532940i	$-0.178913\pi$
$491$	830.923			1.69231			0.846153	+	0.532940i	$0.178913\pi$
$492$		0			0
$493$		−	1076.45i		−	2.18347i
$494$			0.839876i			0.00170015i
$495$		0			0
$496$	119.170			0.240263
$497$			331.625i			0.667253i
$498$		0			0
$499$	−547.907			−1.09801			−0.549005	−	0.835819i	$-0.684993\pi$
$499$	−547.907			−1.09801			−0.549005	+	0.835819i	$0.684993\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$		−	646.123i		−	1.28710i
$503$		−	40.5138i		−	0.0805443i	−0.999189	−	0.0402722i	$-0.987178\pi$
$503$		−	40.5138i		−	0.0805443i	0.999189	−	0.0402722i	$-0.0128225\pi$
$504$		0			0
$505$		−	364.420i		−	0.721625i
$506$	354.254	+	161.532i	0.700106	+	0.319234i
$507$		0			0
$508$	−278.508			−0.548244
$509$	943.330			1.85330			0.926650	−	0.375925i	$-0.122675\pi$
$509$	943.330			1.85330			0.926650	+	0.375925i	$0.122675\pi$
$510$		0			0
$511$		−	88.8324i		−	0.173840i
$512$	22.6274			0.0441942
$513$		0			0
$514$	−300.521			−0.584672
$515$	307.551			0.597187
$516$		0			0
$517$			329.697i			0.637712i
$518$	254.103			0.490547
$519$		0			0
$520$			126.263i			0.242813i
$521$			409.118i			0.785255i	0.919698	+	0.392628i	$0.128434\pi$
$521$			409.118i			0.785255i	−0.919698	+	0.392628i	$0.871566\pi$
$522$		0			0
$523$			88.2598i			0.168757i	0.996434	+	0.0843784i	$0.0268904\pi$
$523$			88.2598i			0.168757i	−0.996434	+	0.0843784i	$0.973110\pi$
$524$	−175.719			−0.335341
$525$		0			0
$526$			676.321i			1.28578i
$527$		−	781.455i		−	1.48284i
$528$		0			0
$529$	−346.888	−	399.387i	−0.655742	−	0.754985i
$530$	−47.5391			−0.0896965
$531$		0			0
$532$	0.208185			0.000391325
$533$	−1490.31			−2.79609
$534$		0			0
$535$	148.910			0.278336
$536$		−	80.0099i		−	0.149272i
$537$		0			0
$538$	−742.951			−1.38095
$539$			439.964i			0.816259i
$540$		0			0
$541$	383.774			0.709380			0.354690	−	0.934984i	$-0.384586\pi$
$541$	383.774			0.709380			0.354690	+	0.934984i	$0.384586\pi$
$542$	529.951			0.977770
$543$		0			0
$544$		−	148.379i		−	0.272755i
$545$	−212.437			−0.389793
$546$		0			0
$547$	−143.193			−0.261780			−0.130890	−	0.991397i	$-0.541783\pi$
$547$	−143.193			−0.261780			−0.130890	+	0.991397i	$0.541783\pi$
$548$		−	362.788i		−	0.662023i
$549$		0			0
$550$		−	84.6399i		−	0.153891i
$551$		−	1.22082i		−	0.00221565i
$552$		0			0
$553$	−281.897			−0.509759
$554$	−575.491			−1.03879
$555$		0			0
$556$	−270.177			−0.485930
$557$			624.965i			1.12202i	0.827809	+	0.561010i	$0.189587\pi$
$557$			624.965i			1.12202i	−0.827809	+	0.561010i	$0.810413\pi$
$558$		0			0
$559$		−	1034.08i		−	1.84987i
$560$	31.2974			0.0558883
$561$		0			0
$562$			267.649i			0.476244i
$563$		−	697.193i		−	1.23835i	−0.785251	−	0.619177i	$-0.787466\pi$
$563$		−	697.193i		−	1.23835i	0.785251	−	0.619177i	$-0.212534\pi$
$564$		0			0
$565$	−434.900			−0.769734
$566$			222.472i			0.393061i
$567$		0			0
$568$	268.058			0.471932
$569$		−	943.778i		−	1.65866i	−0.558759	−	0.829330i	$-0.688722\pi$
$569$		−	943.778i		−	1.65866i	0.558759	−	0.829330i	$-0.311278\pi$
$570$		0			0
$571$		−	159.659i		−	0.279613i	−0.990179	−	0.139806i	$-0.955352\pi$
$571$		−	159.659i		−	0.279613i	0.990179	−	0.139806i	$-0.0446480\pi$
$572$		−	477.931i		−	0.835543i
$573$		0			0
$574$			369.413i			0.643576i
$575$	−47.7117	+	104.636i	−0.0829768	+	0.181975i
$576$		0			0
$577$	689.947			1.19575			0.597874	−	0.801590i	$-0.296012\pi$
$577$	689.947			1.19575			0.597874	+	0.801590i	$0.296012\pi$
$578$	−564.281			−0.976265
$579$		0			0
$580$		−	183.532i		−	0.316435i
$581$	−195.392			−0.336303
$582$		0			0
$583$	179.946			0.308655
$584$	−71.8047			−0.122953
$585$		0			0
$586$		−	655.745i		−	1.11902i
$587$	−46.9308			−0.0799502			−0.0399751	−	0.999201i	$-0.512728\pi$
$587$	−46.9308			−0.0799502			−0.0399751	+	0.999201i	$0.512728\pi$
$588$		0			0
$589$		−	0.886264i		−	0.00150469i
$590$			4.91288i			0.00832691i
$591$		0			0
$592$		−	205.396i		−	0.346952i
$593$	−52.6508			−0.0887871			−0.0443936	−	0.999014i	$-0.514136\pi$
$593$	−52.6508			−0.0887871			−0.0443936	+	0.999014i	$0.514136\pi$
$594$		0			0
$595$		−	205.232i		−	0.344928i
$596$			113.535i			0.190495i
$597$		0			0
$598$	−269.411	+	590.839i	−0.450519	+	0.988025i
$599$	208.189			0.347560			0.173780	−	0.984784i	$-0.444402\pi$
$599$	208.189			0.347560			0.173780	+	0.984784i	$0.444402\pi$
$600$		0			0
$601$	116.243			0.193417			0.0967083	−	0.995313i	$-0.469169\pi$
$601$	116.243			0.193417			0.0967083	+	0.995313i	$0.469169\pi$
$602$	−256.322			−0.425784
$603$		0			0
$604$	72.9387			0.120759
$605$			49.8155i			0.0823396i
$606$		0			0
$607$	−211.295			−0.348097			−0.174048	−	0.984737i	$-0.555685\pi$
$607$	−211.295			−0.348097			−0.174048	+	0.984737i	$0.555685\pi$
$608$		−	0.168279i		−	0.000276775i
$609$		0			0
$610$	40.8737			0.0670061
$611$	−549.883			−0.899973
$612$		0			0
$613$			939.938i			1.53334i	0.642040	+	0.766671i	$0.278088\pi$
$613$			939.938i			1.53334i	−0.642040	+	0.766671i	$0.721912\pi$
$614$	495.485			0.806979
$615$		0			0
$616$	−118.467			−0.192317
$617$			571.784i			0.926717i	0.886171	+	0.463359i	$0.153356\pi$
$617$			571.784i			0.926717i	−0.886171	+	0.463359i	$0.846644\pi$
$618$		0			0
$619$			944.586i			1.52599i	0.646406	+	0.762994i	$0.276271\pi$
$619$			944.586i			1.52599i	−0.646406	+	0.762994i	$0.723729\pi$
$620$		−	133.236i		−	0.214897i
$621$		0			0
$622$	−215.520			−0.346495
$623$	286.752			0.460277
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	67.3608i		−	0.107605i
$627$		0			0
$628$		−	95.4070i		−	0.151922i
$629$	−1346.88			−2.14130
$630$		0			0
$631$			1035.90i			1.64169i	0.571154	+	0.820843i	$0.306496\pi$
$631$			1035.90i			1.64169i	−0.571154	+	0.820843i	$0.693504\pi$
$632$			227.862i			0.360541i
$633$		0			0
$634$	−581.141			−0.916626
$635$			311.382i			0.490365i
$636$		0			0
$637$	−733.790			−1.15195
$638$			694.709i			1.08889i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$			291.199i			0.454289i	0.973861	+	0.227144i	$0.0729389\pi$
$641$			291.199i			0.454289i	−0.973861	+	0.227144i	$0.927061\pi$
$642$		0			0
$643$			183.214i			0.284936i	0.989799	+	0.142468i	$0.0455038\pi$
$643$			183.214i			0.284936i	−0.989799	+	0.142468i	$0.954496\pi$
$644$	146.455	+	66.7803i	0.227414	+	0.103696i
$645$		0			0
$646$	−1.10349			−0.00170818
$647$	196.253			0.303328			0.151664	−	0.988432i	$-0.451537\pi$
$647$	196.253			0.303328			0.151664	+	0.988432i	$0.451537\pi$
$648$		0			0
$649$		−	18.5963i		−	0.0286537i
$650$	141.166			0.217178
$651$		0			0
$652$	237.702			0.364574
$653$	215.371			0.329818			0.164909	−	0.986309i	$-0.447267\pi$
$653$	215.371			0.329818			0.164909	+	0.986309i	$0.447267\pi$
$654$		0			0
$655$			196.459i			0.299938i
$656$	298.602			0.455186
$657$		0			0
$658$			136.303i			0.207147i
$659$		−	144.360i		−	0.219059i	−0.993984	−	0.109529i	$-0.965066\pi$
$659$		−	144.360i		−	0.219059i	0.993984	−	0.109529i	$-0.0349344\pi$
$660$		0			0
$661$		−	639.386i		−	0.967301i	−0.875261	−	0.483650i	$-0.839311\pi$
$661$		−	639.386i		−	0.967301i	0.875261	−	0.483650i	$-0.160689\pi$
$662$	−554.876			−0.838181
$663$		0			0
$664$			157.938i			0.237859i
$665$		−	0.232758i		−	0.000350012i
$666$		0			0
$667$	391.609	−	858.830i	0.587120	−	1.28760i
$668$	216.636			0.324305
$669$		0			0
$670$	−89.4538			−0.133513
$671$	−154.716			−0.230575
$672$		0			0
$673$	594.376			0.883174			0.441587	−	0.897218i	$-0.354416\pi$
$673$	594.376			0.883174			0.441587	+	0.897218i	$0.354416\pi$
$674$			648.230i			0.961766i
$675$		0			0
$676$	459.113			0.679162
$677$			514.242i			0.759590i	0.925071	+	0.379795i	$0.124005\pi$
$677$			514.242i			0.759590i	−0.925071	+	0.379795i	$0.875995\pi$
$678$		0			0
$679$	14.1030			0.0207702
$680$	−165.892			−0.243959
$681$		0			0
$682$			504.328i			0.739483i
$683$	−304.332			−0.445581			−0.222790	−	0.974866i	$-0.571517\pi$
$683$	−304.332			−0.445581			−0.222790	+	0.974866i	$0.571517\pi$
$684$		0			0
$685$	−405.610			−0.592131
$686$			424.368i			0.618612i
$687$		0			0
$688$			207.189i			0.301147i
$689$			300.121i			0.435589i
$690$		0			0
$691$	845.096			1.22300			0.611502	−	0.791243i	$-0.290566\pi$
$691$	845.096			1.22300			0.611502	+	0.791243i	$0.290566\pi$
$692$	−157.483			−0.227577
$693$		0			0
$694$	252.782			0.364239
$695$			302.067i			0.434629i
$696$		0			0
$697$		−	1958.08i		−	2.80929i
$698$	−33.9861			−0.0486907
$699$		0			0
$700$		−	34.9916i		−	0.0499880i
$701$		−	192.249i		−	0.274250i	−0.990554	−	0.137125i	$-0.956214\pi$
$701$		−	192.249i		−	0.274250i	0.990554	−	0.137125i	$-0.0437861\pi$
$702$		0			0
$703$	−1.52752			−0.00217286
$704$			95.7591i			0.136021i
$705$		0			0
$706$	803.641			1.13830
$707$			570.271i			0.806607i
$708$		0			0
$709$			1303.54i			1.83856i	0.393599	+	0.919282i	$0.371230\pi$
$709$			1303.54i			1.83856i	−0.393599	+	0.919282i	$0.628770\pi$
$710$		−	299.698i		−	0.422109i
$711$		0			0
$712$		−	231.787i		−	0.325543i
$713$	284.291	−	623.472i	0.398724	−	0.874435i
$714$		0			0
$715$	−534.343			−0.747333
$716$	−451.354			−0.630383
$717$		0			0
$718$		−	763.779i		−	1.06376i
$719$	−359.296			−0.499717			−0.249858	−	0.968282i	$-0.580384\pi$
$719$	−359.296			−0.499717			−0.249858	+	0.968282i	$0.580384\pi$
$720$		0			0
$721$	−481.278			−0.667515
$722$	510.530			0.707105
$723$		0			0
$724$		−	438.560i		−	0.605745i
$725$	−205.196			−0.283028
$726$		0			0
$727$			1033.74i			1.42193i	0.703227	+	0.710966i	$0.251742\pi$
$727$			1033.74i			1.42193i	−0.703227	+	0.710966i	$0.748258\pi$
$728$		−	197.585i		−	0.271408i
$729$		0			0
$730$			80.2801i			0.109973i
$731$	1358.64			1.85860
$732$		0			0
$733$			88.8871i			0.121265i	0.998160	+	0.0606324i	$0.0193117\pi$
$733$			88.8871i			0.121265i	−0.998160	+	0.0606324i	$0.980688\pi$
$734$		−	572.120i		−	0.779455i
$735$		0			0
$736$	53.9796	−	118.382i	0.0733418	−	0.160845i
$737$	338.602			0.459432
$738$		0			0
$739$	284.091			0.384426			0.192213	−	0.981353i	$-0.438434\pi$
$739$	284.091			0.384426			0.192213	+	0.981353i	$0.438434\pi$
$740$	−229.639			−0.310324
$741$		0			0
$742$	74.3927			0.100260
$743$			972.637i			1.30907i	0.756033	+	0.654534i	$0.227135\pi$
$743$			972.637i			1.30907i	−0.756033	+	0.654534i	$0.772865\pi$
$744$		0			0
$745$	126.936			0.170384
$746$		−	272.144i		−	0.364804i
$747$		0			0
$748$	627.937			0.839489
$749$	−233.025			−0.311114
$750$		0			0
$751$			85.8549i			0.114321i	0.998365	+	0.0571604i	$0.0182046\pi$
$751$			85.8549i			0.114321i	−0.998365	+	0.0571604i	$0.981795\pi$
$752$	110.176			0.146510
$753$		0			0
$754$	−1158.67			−1.53669
$755$		−	81.5479i		−	0.108010i
$756$		0			0
$757$			130.267i			0.172084i	0.996292	+	0.0860419i	$0.0274219\pi$
$757$			130.267i			0.172084i	−0.996292	+	0.0860419i	$0.972578\pi$
$758$		−	1005.80i		−	1.32691i
$759$		0			0
$760$	−0.188142			−0.000247555
$761$	255.675			0.335973			0.167987	−	0.985789i	$-0.446273\pi$
$761$	255.675			0.335973			0.167987	+	0.985789i	$0.446273\pi$
$762$		0			0
$763$	332.437			0.435698
$764$			480.012i			0.628288i
$765$		0			0
$766$		−	436.378i		−	0.569684i
$767$	31.0156			0.0404376
$768$		0			0
$769$		−	254.418i		−	0.330843i	−0.986223	−	0.165421i	$-0.947102\pi$
$769$		−	254.418i		−	0.330843i	0.986223	−	0.165421i	$-0.0528984\pi$
$770$			132.451i			0.172014i
$771$		0			0
$772$	736.038			0.953417
$773$		−	849.508i		−	1.09898i	−0.835502	−	0.549488i	$-0.814823\pi$
$773$		−	849.508i		−	1.09898i	0.835502	−	0.549488i	$-0.185177\pi$
$774$		0			0
$775$	−148.963			−0.192210
$776$		−	11.3997i		−	0.0146903i
$777$		0			0
$778$		−	279.987i		−	0.359880i
$779$		−	2.22069i		−	0.00285070i
$780$		0			0
$781$			1134.42i			1.45252i
$782$	−776.284	−	353.970i	−0.992691	−	0.452647i
$783$		0			0
$784$	147.024			0.187530
$785$	−106.668			−0.135883
$786$		0			0
$787$			911.783i			1.15856i	0.815130	+	0.579278i	$0.196665\pi$
$787$			911.783i			1.15856i	−0.815130	+	0.579278i	$0.803335\pi$
$788$	3.69475			0.00468877
$789$		0			0
$790$	254.757			0.322477
$791$	680.562			0.860382
$792$		0			0
$793$		−	258.041i		−	0.325399i
$794$	378.538			0.476748
$795$		0			0
$796$			113.006i			0.141967i
$797$			1137.11i			1.42673i	0.700791	+	0.713366i	$0.252830\pi$
$797$			1137.11i			1.42673i	−0.700791	+	0.713366i	$0.747170\pi$
$798$		0			0
$799$		−	722.473i		−	0.904222i
$800$	−28.2843			−0.0353553
$801$		0			0
$802$			1060.46i			1.32226i
$803$		−	303.877i		−	0.378427i
$804$		0			0
$805$	74.6626	−	163.741i	0.0927486	−	0.203405i
$806$	−841.139			−1.04360
$807$		0			0
$808$	460.960			0.570494
$809$	−946.247			−1.16965			−0.584825	−	0.811160i	$-0.698837\pi$
$809$	−946.247			−1.16965			−0.584825	+	0.811160i	$0.698837\pi$
$810$		0			0
$811$	−499.192			−0.615526			−0.307763	−	0.951463i	$-0.599580\pi$
$811$	−499.192			−0.615526			−0.307763	+	0.951463i	$0.599580\pi$
$812$			287.205i			0.353700i
$813$		0			0
$814$	869.234			1.06785
$815$		−	265.759i		−	0.326085i
$816$		0			0
$817$	1.54086			0.00188600
$818$	411.880			0.503521
$819$		0			0
$820$		−	333.847i		−	0.407131i
$821$	495.979			0.604116			0.302058	−	0.953290i	$-0.402326\pi$
$821$	495.979			0.604116			0.302058	+	0.953290i	$0.402326\pi$
$822$		0			0
$823$	−527.562			−0.641023			−0.320512	−	0.947245i	$-0.603855\pi$
$823$	−527.562			−0.641023			−0.320512	+	0.947245i	$0.603855\pi$
$824$			389.025i			0.472118i
$825$		0			0
$826$		−	7.68802i		−	0.00930753i
$827$			871.134i			1.05337i	0.850062	+	0.526683i	$0.176565\pi$
$827$			871.134i			1.05337i	−0.850062	+	0.526683i	$0.823435\pi$
$828$		0			0
$829$	−406.931			−0.490869			−0.245435	−	0.969413i	$-0.578931\pi$
$829$	−406.931			−0.490869			−0.245435	+	0.969413i	$0.578931\pi$
$830$	176.581			0.212748
$831$		0			0
$832$	−159.711			−0.191960
$833$		−	964.103i		−	1.15739i
$834$		0			0
$835$		−	242.206i		−	0.290067i
$836$	0.712156			0.000851862
$837$		0			0
$838$		−	472.607i		−	0.563970i
$839$		−	994.738i		−	1.18562i	−0.805341	−	0.592812i	$-0.798018\pi$
$839$		−	994.738i		−	1.18562i	0.805341	−	0.592812i	$-0.201982\pi$
$840$		0			0
$841$	843.208			1.00263
$842$			586.109i			0.696091i
$843$		0			0
$844$	−176.572			−0.209209
$845$		−	513.304i		−	0.607461i
$846$		0			0
$847$		−	77.9548i		−	0.0920364i
$848$		−	60.1328i		−	0.0709113i
$849$		0			0
$850$			185.473i			0.218204i
$851$	−1074.58	−	489.989i	−1.26273	−	0.575780i
$852$		0			0
$853$	−1043.88			−1.22378			−0.611888	−	0.790945i	$-0.709590\pi$
$853$	−1043.88			−1.22378			−0.611888	+	0.790945i	$0.709590\pi$
$854$	−63.9621			−0.0748971
$855$		0			0
$856$			188.358i			0.220044i
$857$	1226.75			1.43144			0.715722	−	0.698385i	$-0.246098\pi$
$857$	1226.75			1.43144			0.715722	+	0.698385i	$0.246098\pi$
$858$		0			0
$859$	−1008.87			−1.17448			−0.587238	−	0.809414i	$-0.699785\pi$
$859$	−1008.87			−1.17448			−0.587238	+	0.809414i	$0.699785\pi$
$860$	231.645			0.269354
$861$		0			0
$862$			1019.99i			1.18328i
$863$	232.004			0.268834			0.134417	−	0.990925i	$-0.457084\pi$
$863$	232.004			0.268834			0.134417	+	0.990925i	$0.457084\pi$
$864$		0			0
$865$			176.072i			0.203551i
$866$		−	327.193i		−	0.377821i
$867$		0			0
$868$			208.498i			0.240205i
$869$	−964.309			−1.10968
$870$		0			0
$871$			564.734i			0.648374i
$872$		−	268.714i		−	0.308159i
$873$		0			0
$874$	−0.880399	−	0.401444i	−0.00100732	−	0.000459318i
$875$	−39.1218			−0.0447106
$876$		0			0
$877$	−90.6500			−0.103364			−0.0516819	−	0.998664i	$-0.516458\pi$
$877$	−90.6500			−0.103364			−0.0516819	+	0.998664i	$0.516458\pi$
$878$	155.790			0.177438
$879$		0			0
$880$	107.062			0.121661
$881$			870.146i			0.987680i	0.869553	+	0.493840i	$0.164407\pi$
$881$			870.146i			0.987680i	−0.869553	+	0.493840i	$0.835593\pi$
$882$		0			0
$883$	−841.486			−0.952986			−0.476493	−	0.879178i	$-0.658092\pi$
$883$	−841.486			−0.952986			−0.476493	+	0.879178i	$0.658092\pi$
$884$			1047.30i			1.18473i
$885$		0			0
$886$	−198.998			−0.224602
$887$	138.461			0.156100			0.0780500	−	0.996949i	$-0.475131\pi$
$887$	138.461			0.156100			0.0780500	+	0.996949i	$0.475131\pi$
$888$		0			0
$889$		−	487.272i		−	0.548113i
$890$	−259.145			−0.291175
$891$		0			0
$892$	259.097			0.290467
$893$		−	0.819371i		−	0.000917549i
$894$		0			0
$895$			504.629i			0.563831i
$896$			39.5885i			0.0441836i
$897$		0			0
$898$	−313.970			−0.349632
$899$	1222.66			1.36002
$900$		0			0
$901$	−394.319			−0.437646
$902$			1263.68i			1.40098i
$903$		0			0
$904$		−	550.110i		−	0.608528i
$905$	−490.325			−0.541795
$906$		0			0
$907$		−	533.764i		−	0.588494i	−0.955729	−	0.294247i	$-0.904931\pi$
$907$		−	533.764i		−	0.588494i	0.955729	−	0.294247i	$-0.0950689\pi$
$908$		−	152.370i		−	0.167808i
$909$		0			0
$910$	−220.907			−0.242754
$911$		−	953.921i		−	1.04711i	−0.851991	−	0.523557i	$-0.824605\pi$
$911$		−	953.921i		−	1.04711i	0.851991	−	0.523557i	$-0.175395\pi$
$912$		0			0
$913$	−668.395			−0.732086
$914$			633.667i			0.693290i
$915$		0			0
$916$			162.704i			0.177624i
$917$		−	307.434i		−	0.335260i
$918$		0			0
$919$			1229.95i			1.33835i	0.743104	+	0.669176i	$0.233353\pi$
$919$			1229.95i			1.33835i	−0.743104	+	0.669176i	$0.766647\pi$
$920$	−132.355	−	60.3510i	−0.143864	−	0.0655989i
$921$		0			0
$922$	−1158.54			−1.25655
$923$	−1892.03			−2.04987
$924$		0			0
$925$			256.745i			0.277562i
$926$	−1122.66			−1.21237
$927$		0			0
$928$	232.152			0.250164
$929$	−212.171			−0.228387			−0.114193	−	0.993459i	$-0.536428\pi$
$929$	−212.171			−0.228387			−0.114193	+	0.993459i	$0.536428\pi$
$930$		0			0
$931$		−	1.09341i		−	0.00117444i
$932$	−459.180			−0.492683
$933$		0			0
$934$		−	475.022i		−	0.508589i
$935$		−	702.055i		−	0.750861i
$936$		0			0
$937$		−	1127.07i		−	1.20285i	−0.798931	−	0.601423i	$-0.794601\pi$
$937$		−	1127.07i		−	1.20285i	0.798931	−	0.601423i	$-0.205399\pi$
$938$	139.984			0.149236
$939$		0			0
$940$		−	123.180i		−	0.131043i
$941$			117.572i			0.124943i	0.998047	+	0.0624717i	$0.0198983\pi$
$941$			117.572i			0.124943i	−0.998047	+	0.0624717i	$0.980102\pi$
$942$		0			0
$943$	712.340	−	1562.22i	0.755398	−	1.65665i
$944$	−6.21435			−0.00658300
$945$		0			0
$946$	−876.824			−0.926875
$947$	1763.85			1.86257			0.931283	−	0.364297i	$-0.118691\pi$
$947$	1763.85			1.86257			0.931283	+	0.364297i	$0.118691\pi$
$948$		0			0
$949$	506.819			0.534056
$950$			0.210349i			0.000221420i
$951$		0			0
$952$	259.600			0.272689
$953$			1497.38i			1.57123i	0.618716	+	0.785615i	$0.287653\pi$
$953$			1497.38i			1.57123i	−0.618716	+	0.785615i	$0.712347\pi$
$954$		0			0
$955$	536.670			0.561958
$956$	313.481			0.327909
$957$		0			0
$958$		−	817.526i		−	0.853367i
$959$	634.727			0.661863
$960$		0			0
$961$	−73.4034			−0.0763823
$962$			1449.74i			1.50701i
$963$		0			0
$964$		−	382.172i		−	0.396444i
$965$		−	822.916i		−	0.852762i
$966$		0			0
$967$	355.699			0.367837			0.183919	−	0.982941i	$-0.441122\pi$
$967$	355.699			0.367837			0.183919	+	0.982941i	$0.441122\pi$
$968$	−63.0122			−0.0650952
$969$		0			0
$970$	−12.7452			−0.0131394
$971$		−	398.185i		−	0.410077i	−0.978754	−	0.205039i	$-0.934268\pi$
$971$		−	398.185i		−	0.410077i	0.978754	−	0.205039i	$-0.0657320\pi$
$972$		0			0
$973$		−	472.696i		−	0.485813i
$974$	−1302.86			−1.33764
$975$		0			0
$976$			51.7016i			0.0529730i
$977$			1697.18i			1.73714i	0.495570	+	0.868568i	$0.334959\pi$
$977$			1697.18i			1.73714i	−0.495570	+	0.868568i	$0.665041\pi$
$978$		0			0
$979$	980.920			1.00196
$980$		−	164.377i		−	0.167732i
$981$		0			0
$982$	1175.10			1.19664
$983$		−	358.215i		−	0.364410i	−0.983261	−	0.182205i	$-0.941677\pi$
$983$		−	358.215i		−	0.364410i	0.983261	−	0.182205i	$-0.0583234\pi$
$984$		0			0
$985$		−	4.13086i		−	0.00419376i
$986$		−	1522.33i		−	1.54395i
$987$		0			0
$988$			1.18776i			0.00120219i
$989$	1083.97	+	494.267i	1.09602	+	0.499765i
$990$		0			0
$991$	786.854			0.794000			0.397000	−	0.917819i	$-0.370051\pi$
$991$	786.854			0.794000			0.397000	+	0.917819i	$0.370051\pi$
$992$	168.532			0.169891
$993$		0			0
$994$			468.988i			0.471819i
$995$	126.344			0.126979
$996$		0			0
$997$	−436.962			−0.438277			−0.219139	−	0.975694i	$-0.570325\pi$
$997$	−436.962			−0.438277			−0.219139	+	0.975694i	$0.570325\pi$
$998$	−774.858			−0.776411
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.21		32
3.2	odd	2		690.3.c.a.91.15	yes	32
23.22	odd	2	inner	2070.3.c.b.91.28		32
69.68	even	2		690.3.c.a.91.10	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.10	✓	32	69.68	even	2
690.3.c.a.91.15	yes	32	3.2	odd	2
2070.3.c.b.91.21		32	1.1	even	1	trivial
2070.3.c.b.91.28		32	23.22	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +3.49916i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +3.49916i q^{7} +2.82843 q^{8} -3.16228i q^{10} +11.9699i q^{11} -19.9639 q^{13} +4.94856i q^{14} +4.00000 q^{16} -26.2299i q^{17} -0.0297478i q^{19} -4.47214i q^{20} +16.9280i q^{22} +(9.54233 - 20.9271i) q^{23} -5.00000 q^{25} -28.2332 q^{26} +6.99832i q^{28} +41.0391 q^{29} +29.7926 q^{31} +5.65685 q^{32} -37.0947i q^{34} +7.82436 q^{35} -51.3489i q^{37} -0.0420698i q^{38} -6.32456i q^{40} +74.6506 q^{41} +51.7973i q^{43} +23.9398i q^{44} +(13.4949 - 29.5954i) q^{46} +27.5439 q^{47} +36.7559 q^{49} -7.07107 q^{50} -39.9278 q^{52} -15.0332i q^{53} +26.7655 q^{55} +9.89712i q^{56} +58.0381 q^{58} -1.55359 q^{59} +12.9254i q^{61} +42.1330 q^{62} +8.00000 q^{64} +44.6406i q^{65} -28.2878i q^{67} -52.4598i q^{68} +11.0653 q^{70} +94.7727 q^{71} -25.3868 q^{73} -72.6184i q^{74} -0.0594957i q^{76} -41.8845 q^{77} +80.5613i q^{79} -8.94427i q^{80} +105.572 q^{82} +55.8397i q^{83} -58.6518 q^{85} +73.2525i q^{86} +33.8559i q^{88} -81.9490i q^{89} -69.8568i q^{91} +(19.0847 - 41.8542i) q^{92} +38.9530 q^{94} -0.0665182 q^{95} -4.03039i q^{97} +51.9807 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 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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