LMFDB - Embedded newform 3762.2.b.b.989.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(989,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3762.989");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.b (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:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			989.12
Character	$\chi$	$=$	3762.989
Dual form			3762.2.b.b.989.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.60040i q^{5} +1.65399i q^{7} +1.00000 q^{8} -1.60040i q^{10} +(-1.87038 - 2.73892i) q^{11} +1.23956i q^{13} +1.65399i q^{14} +1.00000 q^{16} +1.74559 q^{17} +1.00000i q^{19} -1.60040i q^{20} +(-1.87038 - 2.73892i) q^{22} +5.99268i q^{23} +2.43873 q^{25} +1.23956i q^{26} +1.65399i q^{28} -7.25108 q^{29} +10.5409 q^{31} +1.00000 q^{32} +1.74559 q^{34} +2.64705 q^{35} +7.98892 q^{37} +1.00000i q^{38} -1.60040i q^{40} +8.60750 q^{41} -5.30873i q^{43} +(-1.87038 - 2.73892i) q^{44} +5.99268i q^{46} -2.24687i q^{47} +4.26430 q^{49} +2.43873 q^{50} +1.23956i q^{52} -4.01550i q^{53} +(-4.38336 + 2.99335i) q^{55} +1.65399i q^{56} -7.25108 q^{58} -10.9235i q^{59} -0.355889i q^{61} +10.5409 q^{62} +1.00000 q^{64} +1.98378 q^{65} +9.96365 q^{67} +1.74559 q^{68} +2.64705 q^{70} -8.44330i q^{71} +1.24571i q^{73} +7.98892 q^{74} +1.00000i q^{76} +(4.53015 - 3.09360i) q^{77} -1.72159i q^{79} -1.60040i q^{80} +8.60750 q^{82} -8.04031 q^{83} -2.79364i q^{85} -5.30873i q^{86} +(-1.87038 - 2.73892i) q^{88} +4.10143i q^{89} -2.05022 q^{91} +5.99268i q^{92} -2.24687i q^{94} +1.60040 q^{95} -3.14641 q^{97} +4.26430 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{2} + 36 q^{4} + 36 q^{8} - 4 q^{11} + 36 q^{16} + 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} + 36 q^{32} + 16 q^{34} + 16 q^{35} + 16 q^{37} + 24 q^{41} - 4 q^{44} - 68 q^{49} - 28 q^{50}+ \cdots - 68 q^{98}+O(q^{100}) $ 36 * q + 36 * q^2 + 36 * q^4 + 36 * q^8 - 4 * q^11 + 36 * q^16 + 16 * q^17 - 4 * q^22 - 28 * q^25 - 8 * q^31 + 36 * q^32 + 16 * q^34 + 16 * q^35 + 16 * q^37 + 24 * q^41 - 4 * q^44 - 68 * q^49 - 28 * q^50 + 28 * q^55 - 8 * q^62 + 36 * q^64 + 32 * q^65 + 16 * q^67 + 16 * q^68 + 16 * q^70 + 16 * q^74 - 32 * q^77 + 24 * q^82 - 56 * q^83 - 4 * q^88 - 48 * q^91 + 8 * q^97 - 68 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3762\mathbb{Z}\right)^\times$.

$n$	$343$	$2377$	$2927$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	1.60040i		−	0.715719i	−0.933775	−	0.357860i	$-0.883507\pi$
$5$		−	1.60040i		−	0.715719i	0.933775	−	0.357860i	$-0.116493\pi$
$6$		0			0
$7$			1.65399i			0.625151i	0.949893	+	0.312575i	$0.101192\pi$
$7$			1.65399i			0.625151i	−0.949893	+	0.312575i	$0.898808\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$		−	1.60040i		−	0.506090i
$11$	−1.87038	−	2.73892i	−0.563941	−	0.825815i
$11$	−1.87038	−	2.73892i	−0.563941	−	0.825815i
$12$		0			0
$13$			1.23956i			0.343791i	0.985115	+	0.171896i	$0.0549891\pi$
$13$			1.23956i			0.343791i	−0.985115	+	0.171896i	$0.945011\pi$
$14$			1.65399i			0.442048i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.74559			0.423368			0.211684	−	0.977338i	$-0.432105\pi$
$17$	1.74559			0.423368			0.211684	+	0.977338i	$0.432105\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		−	1.60040i		−	0.357860i
$21$		0			0
$22$	−1.87038	−	2.73892i	−0.398767	−	0.583939i
$23$			5.99268i			1.24956i	0.780800	+	0.624781i	$0.214812\pi$
$23$			5.99268i			1.24956i	−0.780800	+	0.624781i	$0.785188\pi$
$24$		0			0
$25$	2.43873			0.487746
$26$			1.23956i			0.243097i
$27$		0			0
$28$			1.65399i			0.312575i
$29$	−7.25108			−1.34649			−0.673246	−	0.739419i	$-0.735101\pi$
$29$	−7.25108			−1.34649			−0.673246	+	0.739419i	$0.735101\pi$
$30$		0			0
$31$	10.5409			1.89320			0.946602	−	0.322403i	$-0.104491\pi$
$31$	10.5409			1.89320			0.946602	+	0.322403i	$0.104491\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	1.74559			0.299367
$35$	2.64705			0.447433
$36$		0			0
$37$	7.98892			1.31337			0.656685	−	0.754165i	$-0.271958\pi$
$37$	7.98892			1.31337			0.656685	+	0.754165i	$0.271958\pi$
$38$			1.00000i			0.162221i
$39$		0			0
$40$		−	1.60040i		−	0.253045i
$41$	8.60750			1.34427			0.672133	−	0.740430i	$-0.265378\pi$
$41$	8.60750			1.34427			0.672133	+	0.740430i	$0.265378\pi$
$42$		0			0
$43$		−	5.30873i		−	0.809573i	−0.914411	−	0.404787i	$-0.867346\pi$
$43$		−	5.30873i		−	0.809573i	0.914411	−	0.404787i	$-0.132654\pi$
$44$	−1.87038	−	2.73892i	−0.281971	−	0.412907i
$45$		0			0
$46$			5.99268i			0.883573i
$47$		−	2.24687i		−	0.327739i	−0.986482	−	0.163870i	$-0.947602\pi$
$47$		−	2.24687i		−	0.327739i	0.986482	−	0.163870i	$-0.0523976\pi$
$48$		0			0
$49$	4.26430			0.609186
$50$	2.43873			0.344888
$51$		0			0
$52$			1.23956i			0.171896i
$53$		−	4.01550i		−	0.551572i	−0.961219	−	0.275786i	$-0.911062\pi$
$53$		−	4.01550i		−	0.551572i	0.961219	−	0.275786i	$-0.0889381\pi$
$54$		0			0
$55$	−4.38336	+	2.99335i	−0.591052	+	0.403624i
$56$			1.65399i			0.221024i
$57$		0			0
$58$	−7.25108			−0.952114
$59$		−	10.9235i		−	1.42212i	−0.703132	−	0.711059i	$-0.748216\pi$
$59$		−	10.9235i		−	1.42212i	0.703132	−	0.711059i	$-0.251784\pi$
$60$		0			0
$61$		−	0.355889i		−	0.0455669i	−0.999740	−	0.0227834i	$-0.992747\pi$
$61$		−	0.355889i		−	0.0455669i	0.999740	−	0.0227834i	$-0.00725282\pi$
$62$	10.5409			1.33870
$63$		0			0
$64$	1.00000			0.125000
$65$	1.98378			0.246058
$66$		0			0
$67$	9.96365			1.21725			0.608627	−	0.793457i	$-0.291721\pi$
$67$	9.96365			1.21725			0.608627	+	0.793457i	$0.291721\pi$
$68$	1.74559			0.211684
$69$		0			0
$70$	2.64705			0.316383
$71$		−	8.44330i		−	1.00204i	−0.865437	−	0.501018i	$-0.832959\pi$
$71$		−	8.44330i		−	1.00204i	0.865437	−	0.501018i	$-0.167041\pi$
$72$		0			0
$73$			1.24571i			0.145799i	0.997339	+	0.0728997i	$0.0232253\pi$
$73$			1.24571i			0.145799i	−0.997339	+	0.0728997i	$0.976775\pi$
$74$	7.98892			0.928693
$75$		0			0
$76$			1.00000i			0.114708i
$77$	4.53015	−	3.09360i	0.516259	−	0.352548i
$78$		0			0
$79$		−	1.72159i		−	0.193694i	−0.995299	−	0.0968472i	$-0.969124\pi$
$79$		−	1.72159i		−	0.193694i	0.995299	−	0.0968472i	$-0.0308758\pi$
$80$		−	1.60040i		−	0.178930i
$81$		0			0
$82$	8.60750			0.950540
$83$	−8.04031			−0.882538			−0.441269	−	0.897375i	$-0.645472\pi$
$83$	−8.04031			−0.882538			−0.441269	+	0.897375i	$0.645472\pi$
$84$		0			0
$85$		−	2.79364i		−	0.303013i
$86$		−	5.30873i		−	0.572455i
$87$		0			0
$88$	−1.87038	−	2.73892i	−0.199383	−	0.291970i
$89$			4.10143i			0.434750i	0.976088	+	0.217375i	$0.0697495\pi$
$89$			4.10143i			0.434750i	−0.976088	+	0.217375i	$0.930251\pi$
$90$		0			0
$91$	−2.05022			−0.214921
$92$			5.99268i			0.624781i
$93$		0			0
$94$		−	2.24687i		−	0.231747i
$95$	1.60040			0.164197
$96$		0			0
$97$	−3.14641			−0.319470			−0.159735	−	0.987160i	$-0.551064\pi$
$97$	−3.14641			−0.319470			−0.159735	+	0.987160i	$0.551064\pi$
$98$	4.26430			0.430760
$99$		0			0
$100$	2.43873			0.243873
$101$	2.63345			0.262038			0.131019	−	0.991380i	$-0.458175\pi$
$101$	2.63345			0.262038			0.131019	+	0.991380i	$0.458175\pi$
$102$		0			0
$103$	−17.0803			−1.68298			−0.841488	−	0.540276i	$-0.818320\pi$
$103$	−17.0803			−1.68298			−0.841488	+	0.540276i	$0.818320\pi$
$104$			1.23956i			0.121548i
$105$		0			0
$106$		−	4.01550i		−	0.390020i
$107$	6.36868			0.615683			0.307842	−	0.951438i	$-0.400393\pi$
$107$	6.36868			0.615683			0.307842	+	0.951438i	$0.400393\pi$
$108$		0			0
$109$			0.457818i			0.0438510i	0.999760	+	0.0219255i	$0.00697966\pi$
$109$			0.457818i			0.0438510i	−0.999760	+	0.0219255i	$0.993020\pi$
$110$	−4.38336	+	2.99335i	−0.417937	+	0.285405i
$111$		0			0
$112$			1.65399i			0.156288i
$113$			4.14054i			0.389510i	0.980852	+	0.194755i	$0.0623911\pi$
$113$			4.14054i			0.389510i	−0.980852	+	0.194755i	$0.937609\pi$
$114$		0			0
$115$	9.59068			0.894335
$116$	−7.25108			−0.673246
$117$		0			0
$118$		−	10.9235i		−	1.00559i
$119$			2.88720i			0.264669i
$120$		0			0
$121$	−4.00335	+	10.2456i	−0.363940	+	0.931422i
$122$		−	0.355889i		−	0.0322206i
$123$		0			0
$124$	10.5409			0.946602
$125$		−	11.9049i		−	1.06481i
$126$		0			0
$127$		−	18.7747i		−	1.66599i	−0.553282	−	0.832994i	$-0.686625\pi$
$127$		−	18.7747i		−	1.66599i	0.553282	−	0.832994i	$-0.313375\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	1.98378			0.173989
$131$	6.13297			0.535840			0.267920	−	0.963441i	$-0.413664\pi$
$131$	6.13297			0.535840			0.267920	+	0.963441i	$0.413664\pi$
$132$		0			0
$133$	−1.65399			−0.143419
$134$	9.96365			0.860728
$135$		0			0
$136$	1.74559			0.149683
$137$		−	2.54878i		−	0.217757i	−0.994055	−	0.108879i	$-0.965274\pi$
$137$		−	2.54878i		−	0.217757i	0.994055	−	0.108879i	$-0.0347260\pi$
$138$		0			0
$139$			18.9662i			1.60870i	0.594159	+	0.804348i	$0.297485\pi$
$139$			18.9662i			1.60870i	−0.594159	+	0.804348i	$0.702515\pi$
$140$	2.64705			0.223716
$141$		0			0
$142$		−	8.44330i		−	0.708546i
$143$	3.39504	−	2.31844i	0.283908	−	0.193878i
$144$		0			0
$145$			11.6046i			0.963711i
$146$			1.24571i			0.103096i
$147$		0			0
$148$	7.98892			0.656685
$149$	20.8646			1.70929			0.854647	−	0.519209i	$-0.173773\pi$
$149$	20.8646			1.70929			0.854647	+	0.519209i	$0.173773\pi$
$150$		0			0
$151$			15.2282i			1.23926i	0.784896	+	0.619628i	$0.212717\pi$
$151$			15.2282i			1.23926i	−0.784896	+	0.619628i	$0.787283\pi$
$152$			1.00000i			0.0811107i
$153$		0			0
$154$	4.53015	−	3.09360i	0.365050	−	0.249289i
$155$		−	16.8697i		−	1.35500i
$156$		0			0
$157$	9.10926			0.726998			0.363499	−	0.931595i	$-0.381582\pi$
$157$	9.10926			0.726998			0.363499	+	0.931595i	$0.381582\pi$
$158$		−	1.72159i		−	0.136963i
$159$		0			0
$160$		−	1.60040i		−	0.126523i
$161$	−9.91186			−0.781164
$162$		0			0
$163$	2.84605			0.222920			0.111460	−	0.993769i	$-0.464447\pi$
$163$	2.84605			0.222920			0.111460	+	0.993769i	$0.464447\pi$
$164$	8.60750			0.672133
$165$		0			0
$166$	−8.04031			−0.624049
$167$	−5.22557			−0.404367			−0.202183	−	0.979348i	$-0.564804\pi$
$167$	−5.22557			−0.404367			−0.202183	+	0.979348i	$0.564804\pi$
$168$		0			0
$169$	11.4635			0.881808
$170$		−	2.79364i		−	0.214262i
$171$		0			0
$172$		−	5.30873i		−	0.404787i
$173$	6.83864			0.519932			0.259966	−	0.965618i	$-0.416289\pi$
$173$	6.83864			0.519932			0.259966	+	0.965618i	$0.416289\pi$
$174$		0			0
$175$			4.03364i			0.304915i
$176$	−1.87038	−	2.73892i	−0.140985	−	0.206454i
$177$		0			0
$178$			4.10143i			0.307415i
$179$		−	8.39539i		−	0.627501i	−0.949506	−	0.313750i	$-0.898415\pi$
$179$		−	8.39539i		−	0.627501i	0.949506	−	0.313750i	$-0.101585\pi$
$180$		0			0
$181$	−9.21653			−0.685060			−0.342530	−	0.939507i	$-0.611284\pi$
$181$	−9.21653			−0.685060			−0.342530	+	0.939507i	$0.611284\pi$
$182$	−2.05022			−0.151972
$183$		0			0
$184$			5.99268i			0.441787i
$185$		−	12.7854i		−	0.940005i
$186$		0			0
$187$	−3.26492	−	4.78103i	−0.238755	−	0.349624i
$188$		−	2.24687i		−	0.163870i
$189$		0			0
$190$	1.60040			0.116105
$191$			23.5308i			1.70263i	0.524658	+	0.851313i	$0.324193\pi$
$191$			23.5308i			1.70263i	−0.524658	+	0.851313i	$0.675807\pi$
$192$		0			0
$193$			0.604960i			0.0435460i	0.999763	+	0.0217730i	$0.00693111\pi$
$193$			0.604960i			0.0435460i	−0.999763	+	0.0217730i	$0.993069\pi$
$194$	−3.14641			−0.225899
$195$		0			0
$196$	4.26430			0.304593
$197$	−4.85995			−0.346257			−0.173129	−	0.984899i	$-0.555388\pi$
$197$	−4.85995			−0.346257			−0.173129	+	0.984899i	$0.555388\pi$
$198$		0			0
$199$	−15.6837			−1.11179			−0.555895	−	0.831253i	$-0.687624\pi$
$199$	−15.6837			−1.11179			−0.555895	+	0.831253i	$0.687624\pi$
$200$	2.43873			0.172444
$201$		0			0
$202$	2.63345			0.185289
$203$		−	11.9932i		−	0.841761i
$204$		0			0
$205$		−	13.7754i		−	0.962117i
$206$	−17.0803			−1.19004
$207$		0			0
$208$			1.23956i			0.0859478i
$209$	2.73892	−	1.87038i	0.189455	−	0.129377i
$210$		0			0
$211$			10.2138i			0.703144i	0.936161	+	0.351572i	$0.114353\pi$
$211$			10.2138i			0.703144i	−0.936161	+	0.351572i	$0.885647\pi$
$212$		−	4.01550i		−	0.275786i
$213$		0			0
$214$	6.36868			0.435354
$215$	−8.49607			−0.579427
$216$		0			0
$217$			17.4346i			1.18354i
$218$			0.457818i			0.0310073i
$219$		0			0
$220$	−4.38336	+	2.99335i	−0.295526	+	0.201812i
$221$			2.16376i			0.145550i
$222$		0			0
$223$	−22.1003			−1.47995			−0.739973	−	0.672636i	$-0.765162\pi$
$223$	−22.1003			−1.47995			−0.739973	+	0.672636i	$0.765162\pi$
$224$			1.65399i			0.110512i
$225$		0			0
$226$			4.14054i			0.275425i
$227$	17.1078			1.13549			0.567743	−	0.823206i	$-0.307817\pi$
$227$	17.1078			1.13549			0.567743	+	0.823206i	$0.307817\pi$
$228$		0			0
$229$	9.64766			0.637535			0.318768	−	0.947833i	$-0.396731\pi$
$229$	9.64766			0.637535			0.318768	+	0.947833i	$0.396731\pi$
$230$	9.59068			0.632390
$231$		0			0
$232$	−7.25108			−0.476057
$233$	−8.90102			−0.583125			−0.291563	−	0.956552i	$-0.594175\pi$
$233$	−8.90102			−0.583125			−0.291563	+	0.956552i	$0.594175\pi$
$234$		0			0
$235$	−3.59588			−0.234569
$236$		−	10.9235i		−	0.711059i
$237$		0			0
$238$			2.88720i			0.187149i
$239$	−16.7800			−1.08541			−0.542703	−	0.839925i	$-0.682599\pi$
$239$	−16.7800			−1.08541			−0.542703	+	0.839925i	$0.682599\pi$
$240$		0			0
$241$			5.80518i			0.373944i	0.982365	+	0.186972i	$0.0598674\pi$
$241$			5.80518i			0.373944i	−0.982365	+	0.186972i	$0.940133\pi$
$242$	−4.00335	+	10.2456i	−0.257345	+	0.658615i
$243$		0			0
$244$		−	0.355889i		−	0.0227834i
$245$		−	6.82458i		−	0.436007i
$246$		0			0
$247$	−1.23956			−0.0788711
$248$	10.5409			0.669349
$249$		0			0
$250$		−	11.9049i		−	0.752933i
$251$			9.00520i			0.568403i	0.958765	+	0.284202i	$0.0917285\pi$
$251$			9.00520i			0.568403i	−0.958765	+	0.284202i	$0.908271\pi$
$252$		0			0
$253$	16.4135	−	11.2086i	1.03191	−	0.704679i
$254$		−	18.7747i		−	1.17803i
$255$		0			0
$256$	1.00000			0.0625000
$257$			14.5525i			0.907761i	0.891063	+	0.453880i	$0.149961\pi$
$257$			14.5525i			0.907761i	−0.891063	+	0.453880i	$0.850039\pi$
$258$		0			0
$259$			13.2136i			0.821055i
$260$	1.98378			0.123029
$261$		0			0
$262$	6.13297			0.378896
$263$	11.7893			0.726960			0.363480	−	0.931602i	$-0.381589\pi$
$263$	11.7893			0.726960			0.363480	+	0.931602i	$0.381589\pi$
$264$		0			0
$265$	−6.42640			−0.394771
$266$	−1.65399			−0.101413
$267$		0			0
$268$	9.96365			0.608627
$269$		−	9.54840i		−	0.582177i	−0.956696	−	0.291088i	$-0.905983\pi$
$269$		−	9.54840i		−	0.582177i	0.956696	−	0.291088i	$-0.0940173\pi$
$270$		0			0
$271$		−	16.4893i		−	1.00166i	−0.865547	−	0.500828i	$-0.833029\pi$
$271$		−	16.4893i		−	1.00166i	0.865547	−	0.500828i	$-0.166971\pi$
$272$	1.74559			0.105842
$273$		0			0
$274$		−	2.54878i		−	0.153978i
$275$	−4.56135	−	6.67948i	−0.275060	−	0.402788i
$276$		0			0
$277$		−	10.4004i		−	0.624901i	−0.949934	−	0.312450i	$-0.898850\pi$
$277$		−	10.4004i		−	0.624901i	0.949934	−	0.312450i	$-0.101150\pi$
$278$			18.9662i			1.13752i
$279$		0			0
$280$	2.64705			0.158191
$281$	29.8925			1.78324			0.891619	−	0.452786i	$-0.149570\pi$
$281$	29.8925			1.78324			0.891619	+	0.452786i	$0.149570\pi$
$282$		0			0
$283$		−	20.4726i		−	1.21697i	−0.793565	−	0.608486i	$-0.791777\pi$
$283$		−	20.4726i		−	1.21697i	0.793565	−	0.608486i	$-0.208223\pi$
$284$		−	8.44330i		−	0.501018i
$285$		0			0
$286$	3.39504	−	2.31844i	0.200753	−	0.137092i
$287$			14.2368i			0.840369i
$288$		0			0
$289$	−13.9529			−0.820759
$290$			11.6046i			0.681446i
$291$		0			0
$292$			1.24571i			0.0728997i
$293$	−23.3628			−1.36487			−0.682434	−	0.730947i	$-0.739079\pi$
$293$	−23.3628			−1.36487			−0.682434	+	0.730947i	$0.739079\pi$
$294$		0			0
$295$	−17.4819			−1.01784
$296$	7.98892			0.464347
$297$		0			0
$298$	20.8646			1.20865
$299$	−7.42827			−0.429588
$300$		0			0
$301$	8.78060			0.506105
$302$			15.2282i			0.876286i
$303$		0			0
$304$			1.00000i			0.0573539i
$305$	−0.569563			−0.0326131
$306$		0			0
$307$			26.5375i			1.51457i	0.653082	+	0.757287i	$0.273476\pi$
$307$			26.5375i			1.51457i	−0.653082	+	0.757287i	$0.726524\pi$
$308$	4.53015	−	3.09360i	0.258129	−	0.176274i
$309$		0			0
$310$		−	16.8697i		−	0.958132i
$311$		−	28.2467i		−	1.60172i	−0.598851	−	0.800860i	$-0.704376\pi$
$311$		−	28.2467i		−	1.60172i	0.598851	−	0.800860i	$-0.295624\pi$
$312$		0			0
$313$	10.7742			0.608995			0.304497	−	0.952513i	$-0.401512\pi$
$313$	10.7742			0.608995			0.304497	+	0.952513i	$0.401512\pi$
$314$	9.10926			0.514065
$315$		0			0
$316$		−	1.72159i		−	0.0968472i
$317$			3.30204i			0.185461i	0.995691	+	0.0927306i	$0.0295595\pi$
$317$			3.30204i			0.185461i	−0.995691	+	0.0927306i	$0.970440\pi$
$318$		0			0
$319$	13.5623	+	19.8601i	0.759342	+	1.11195i
$320$		−	1.60040i		−	0.0894649i
$321$		0			0
$322$	−9.91186			−0.552366
$323$			1.74559i			0.0971273i
$324$		0			0
$325$			3.02294i			0.167683i
$326$	2.84605			0.157628
$327$		0			0
$328$	8.60750			0.475270
$329$	3.71630			0.204886
$330$		0			0
$331$	−0.420921			−0.0231359			−0.0115680	−	0.999933i	$-0.503682\pi$
$331$	−0.420921			−0.0231359			−0.0115680	+	0.999933i	$0.503682\pi$
$332$	−8.04031			−0.441269
$333$		0			0
$334$	−5.22557			−0.285930
$335$		−	15.9458i		−	0.871212i
$336$		0			0
$337$			21.5763i			1.17534i	0.809101	+	0.587669i	$0.199954\pi$
$337$			21.5763i			1.17534i	−0.809101	+	0.587669i	$0.800046\pi$
$338$	11.4635			0.623532
$339$		0			0
$340$		−	2.79364i		−	0.151506i
$341$	−19.7155	−	28.8707i	−1.06766	−	1.56344i
$342$		0			0
$343$			18.6311i			1.00598i
$344$		−	5.30873i		−	0.286227i
$345$		0			0
$346$	6.83864			0.367647
$347$	−23.4062			−1.25651			−0.628254	−	0.778008i	$-0.716230\pi$
$347$	−23.4062			−1.25651			−0.628254	+	0.778008i	$0.716230\pi$
$348$		0			0
$349$			1.17120i			0.0626928i	0.999509	+	0.0313464i	$0.00997950\pi$
$349$			1.17120i			0.0626928i	−0.999509	+	0.0313464i	$0.990020\pi$
$350$			4.03364i			0.215607i
$351$		0			0
$352$	−1.87038	−	2.73892i	−0.0996917	−	0.145985i
$353$		−	6.72587i		−	0.357982i	−0.983851	−	0.178991i	$-0.942717\pi$
$353$		−	6.72587i		−	0.357982i	0.983851	−	0.178991i	$-0.0572833\pi$
$354$		0			0
$355$	−13.5126			−0.717176
$356$			4.10143i			0.217375i
$357$		0			0
$358$		−	8.39539i		−	0.443710i
$359$	−25.2102			−1.33054			−0.665272	−	0.746601i	$-0.731685\pi$
$359$	−25.2102			−1.33054			−0.665272	+	0.746601i	$0.731685\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−9.21653			−0.484410
$363$		0			0
$364$	−2.05022			−0.107461
$365$	1.99363			0.104351
$366$		0			0
$367$	20.9049			1.09123			0.545614	−	0.838037i	$-0.316296\pi$
$367$	20.9049			1.09123			0.545614	+	0.838037i	$0.316296\pi$
$368$			5.99268i			0.312390i
$369$		0			0
$370$		−	12.7854i		−	0.664684i
$371$	6.64162			0.344816
$372$		0			0
$373$		−	14.9317i		−	0.773133i	−0.922261	−	0.386567i	$-0.873661\pi$
$373$		−	14.9317i		−	0.773133i	0.922261	−	0.386567i	$-0.126339\pi$
$374$	−3.26492	−	4.78103i	−0.168825	−	0.247221i
$375$		0			0
$376$		−	2.24687i		−	0.115873i
$377$		−	8.98812i		−	0.462912i
$378$		0			0
$379$	22.9383			1.17826			0.589131	−	0.808038i	$-0.299470\pi$
$379$	22.9383			1.17826			0.589131	+	0.808038i	$0.299470\pi$
$380$	1.60040			0.0820986
$381$		0			0
$382$			23.5308i			1.20394i
$383$		−	26.7584i		−	1.36729i	−0.729814	−	0.683646i	$-0.760393\pi$
$383$		−	26.7584i		−	1.36729i	0.729814	−	0.683646i	$-0.239607\pi$
$384$		0			0
$385$	−4.95099	−	7.25005i	−0.252326	−	0.369497i
$386$			0.604960i			0.0307917i
$387$		0			0
$388$	−3.14641			−0.159735
$389$			18.5342i			0.939723i	0.882740	+	0.469861i	$0.155696\pi$
$389$			18.5342i			0.939723i	−0.882740	+	0.469861i	$0.844304\pi$
$390$		0			0
$391$			10.4608i			0.529024i
$392$	4.26430			0.215380
$393$		0			0
$394$	−4.85995			−0.244841
$395$	−2.75523			−0.138631
$396$		0			0
$397$	8.18631			0.410859			0.205430	−	0.978672i	$-0.434141\pi$
$397$	8.18631			0.410859			0.205430	+	0.978672i	$0.434141\pi$
$398$	−15.6837			−0.786154
$399$		0			0
$400$	2.43873			0.121936
$401$			18.5674i			0.927209i	0.886042	+	0.463605i	$0.153444\pi$
$401$			18.5674i			0.927209i	−0.886042	+	0.463605i	$0.846556\pi$
$402$		0			0
$403$			13.0661i			0.650867i
$404$	2.63345			0.131019
$405$		0			0
$406$		−	11.9932i		−	0.595215i
$407$	−14.9423	−	21.8810i	−0.740664	−	1.08460i
$408$		0			0
$409$			37.0679i			1.83289i	0.400160	+	0.916445i	$0.368955\pi$
$409$			37.0679i			1.83289i	−0.400160	+	0.916445i	$0.631045\pi$
$410$		−	13.7754i		−	0.680320i
$411$		0			0
$412$	−17.0803			−0.841488
$413$	18.0674			0.889039
$414$		0			0
$415$			12.8677i			0.631650i
$416$			1.23956i			0.0607742i
$417$		0			0
$418$	2.73892	−	1.87038i	0.133965	−	0.0914834i
$419$		−	8.88277i		−	0.433952i	−0.976177	−	0.216976i	$-0.930381\pi$
$419$		−	8.88277i		−	0.433952i	0.976177	−	0.216976i	$-0.0696193\pi$
$420$		0			0
$421$	−15.0103			−0.731559			−0.365779	−	0.930702i	$-0.619197\pi$
$421$	−15.0103			−0.731559			−0.365779	+	0.930702i	$0.619197\pi$
$422$			10.2138i			0.497198i
$423$		0			0
$424$		−	4.01550i		−	0.195010i
$425$	4.25703			0.206496
$426$		0			0
$427$	0.588638			0.0284862
$428$	6.36868			0.307842
$429$		0			0
$430$	−8.49607			−0.409717
$431$	11.5164			0.554728			0.277364	−	0.960765i	$-0.410539\pi$
$431$	11.5164			0.554728			0.277364	+	0.960765i	$0.410539\pi$
$432$		0			0
$433$	−24.3150			−1.16850			−0.584252	−	0.811573i	$-0.698612\pi$
$433$	−24.3150			−1.16850			−0.584252	+	0.811573i	$0.698612\pi$
$434$			17.4346i			0.836888i
$435$		0			0
$436$			0.457818i			0.0219255i
$437$	−5.99268			−0.286669
$438$		0			0
$439$			36.9879i			1.76534i	0.469998	+	0.882668i	$0.344255\pi$
$439$			36.9879i			1.76534i	−0.469998	+	0.882668i	$0.655745\pi$
$440$	−4.38336	+	2.99335i	−0.208968	+	0.142703i
$441$		0			0
$442$			2.16376i			0.102920i
$443$		−	26.6046i		−	1.26402i	−0.774959	−	0.632011i	$-0.782230\pi$
$443$		−	26.6046i		−	1.26402i	0.774959	−	0.632011i	$-0.217770\pi$
$444$		0			0
$445$	6.56391			0.311159
$446$	−22.1003			−1.04648
$447$		0			0
$448$			1.65399i			0.0781439i
$449$			5.55083i			0.261960i	0.991385	+	0.130980i	$0.0418124\pi$
$449$			5.55083i			0.261960i	−0.991385	+	0.130980i	$0.958188\pi$
$450$		0			0
$451$	−16.0993	−	23.5752i	−0.758087	−	1.11011i
$452$			4.14054i			0.194755i
$453$		0			0
$454$	17.1078			0.802910
$455$			3.28116i			0.153823i
$456$		0			0
$457$		−	20.2718i		−	0.948274i	−0.880451	−	0.474137i	$-0.842760\pi$
$457$		−	20.2718i		−	0.948274i	0.880451	−	0.474137i	$-0.157240\pi$
$458$	9.64766			0.450805
$459$		0			0
$460$	9.59068			0.447168
$461$	23.4834			1.09373			0.546866	−	0.837220i	$-0.315821\pi$
$461$	23.4834			1.09373			0.546866	+	0.837220i	$0.315821\pi$
$462$		0			0
$463$	−5.10399			−0.237203			−0.118601	−	0.992942i	$-0.537841\pi$
$463$	−5.10399			−0.237203			−0.118601	+	0.992942i	$0.537841\pi$
$464$	−7.25108			−0.336623
$465$		0			0
$466$	−8.90102			−0.412332
$467$			24.1746i			1.11867i	0.828942	+	0.559334i	$0.188943\pi$
$467$			24.1746i			1.11867i	−0.828942	+	0.559334i	$0.811057\pi$
$468$		0			0
$469$			16.4798i			0.760967i
$470$	−3.59588			−0.165866
$471$		0			0
$472$		−	10.9235i		−	0.502795i
$473$	−14.5402	+	9.92935i	−0.668558	+	0.456552i
$474$		0			0
$475$			2.43873i			0.111897i
$476$			2.88720i			0.132335i
$477$		0			0
$478$	−16.7800			−0.767498
$479$	−4.36286			−0.199344			−0.0996722	−	0.995020i	$-0.531779\pi$
$479$	−4.36286			−0.199344			−0.0996722	+	0.995020i	$0.531779\pi$
$480$		0			0
$481$			9.90271i			0.451525i
$482$			5.80518i			0.264419i
$483$		0			0
$484$	−4.00335	+	10.2456i	−0.181970	+	0.465711i
$485$			5.03551i			0.228651i
$486$		0			0
$487$	−21.1036			−0.956296			−0.478148	−	0.878279i	$-0.658692\pi$
$487$	−21.1036			−0.956296			−0.478148	+	0.878279i	$0.658692\pi$
$488$		−	0.355889i		−	0.0161103i
$489$		0			0
$490$		−	6.82458i		−	0.308303i
$491$	−34.5369			−1.55863			−0.779315	−	0.626632i	$-0.784433\pi$
$491$	−34.5369			−1.55863			−0.779315	+	0.626632i	$0.784433\pi$
$492$		0			0
$493$	−12.6574			−0.570062
$494$	−1.23956			−0.0557703
$495$		0			0
$496$	10.5409			0.473301
$497$	13.9652			0.626423
$498$		0			0
$499$	2.33181			0.104386			0.0521931	−	0.998637i	$-0.483379\pi$
$499$	2.33181			0.104386			0.0521931	+	0.998637i	$0.483379\pi$
$500$		−	11.9049i		−	0.532404i
$501$		0			0
$502$			9.00520i			0.401922i
$503$	−3.31385			−0.147757			−0.0738786	−	0.997267i	$-0.523538\pi$
$503$	−3.31385			−0.147757			−0.0738786	+	0.997267i	$0.523538\pi$
$504$		0			0
$505$		−	4.21456i		−	0.187545i
$506$	16.4135	−	11.2086i	0.729668	−	0.498283i
$507$		0			0
$508$		−	18.7747i		−	0.832994i
$509$			25.9710i			1.15114i	0.817751	+	0.575571i	$0.195220\pi$
$509$			25.9710i			1.15114i	−0.817751	+	0.575571i	$0.804780\pi$
$510$		0			0
$511$	−2.06040			−0.0911466
$512$	1.00000			0.0441942
$513$		0			0
$514$			14.5525i			0.641884i
$515$			27.3353i			1.20454i
$516$		0			0
$517$	−6.15398	+	4.20250i	−0.270652	+	0.184826i
$518$			13.2136i			0.580573i
$519$		0			0
$520$	1.98378			0.0869946
$521$			34.8826i			1.52823i	0.645078	+	0.764117i	$0.276825\pi$
$521$			34.8826i			1.52823i	−0.645078	+	0.764117i	$0.723175\pi$
$522$		0			0
$523$			33.1973i			1.45161i	0.687898	+	0.725807i	$0.258534\pi$
$523$			33.1973i			1.45161i	−0.687898	+	0.725807i	$0.741466\pi$
$524$	6.13297			0.267920
$525$		0			0
$526$	11.7893			0.514038
$527$	18.4001			0.801523
$528$		0			0
$529$	−12.9123			−0.561403
$530$	−6.42640			−0.279145
$531$		0			0
$532$	−1.65399			−0.0717097
$533$			10.6695i			0.462147i
$534$		0			0
$535$		−	10.1924i		−	0.440656i
$536$	9.96365			0.430364
$537$		0			0
$538$		−	9.54840i		−	0.411661i
$539$	−7.97588	−	11.6796i	−0.343545	−	0.503075i
$540$		0			0
$541$			24.6969i			1.06180i	0.847434	+	0.530900i	$0.178146\pi$
$541$			24.6969i			1.06180i	−0.847434	+	0.530900i	$0.821854\pi$
$542$		−	16.4893i		−	0.708277i
$543$		0			0
$544$	1.74559			0.0748416
$545$	0.732690			0.0313850
$546$		0			0
$547$		−	38.2759i		−	1.63656i	−0.574819	−	0.818281i	$-0.694927\pi$
$547$		−	38.2759i		−	1.63656i	0.574819	−	0.818281i	$-0.305073\pi$
$548$		−	2.54878i		−	0.108879i
$549$		0			0
$550$	−4.56135	−	6.67948i	−0.194497	−	0.284814i
$551$		−	7.25108i		−	0.308906i
$552$		0			0
$553$	2.84750			0.121088
$554$		−	10.4004i		−	0.441871i
$555$		0			0
$556$			18.9662i			0.804348i
$557$	−37.7397			−1.59908			−0.799541	−	0.600611i	$-0.794924\pi$
$557$	−37.7397			−1.59908			−0.799541	+	0.600611i	$0.794924\pi$
$558$		0			0
$559$	6.58047			0.278324
$560$	2.64705			0.111858
$561$		0			0
$562$	29.8925			1.26094
$563$	20.0944			0.846876			0.423438	−	0.905925i	$-0.360823\pi$
$563$	20.0944			0.846876			0.423438	+	0.905925i	$0.360823\pi$
$564$		0			0
$565$	6.62651			0.278780
$566$		−	20.4726i		−	0.860528i
$567$		0			0
$568$		−	8.44330i		−	0.354273i
$569$	2.04519			0.0857390			0.0428695	−	0.999081i	$-0.486350\pi$
$569$	2.04519			0.0857390			0.0428695	+	0.999081i	$0.486350\pi$
$570$		0			0
$571$			15.4377i			0.646047i	0.946391	+	0.323024i	$0.104699\pi$
$571$			15.4377i			0.646047i	−0.946391	+	0.323024i	$0.895301\pi$
$572$	3.39504	−	2.31844i	0.141954	−	0.0969390i
$573$		0			0
$574$			14.2368i			0.594231i
$575$			14.6145i			0.609468i
$576$		0			0
$577$	3.56099			0.148246			0.0741229	−	0.997249i	$-0.476384\pi$
$577$	3.56099			0.148246			0.0741229	+	0.997249i	$0.476384\pi$
$578$	−13.9529			−0.580364
$579$		0			0
$580$			11.6046i			0.481855i
$581$		−	13.2986i		−	0.551720i
$582$		0			0
$583$	−10.9981	+	7.51052i	−0.455496	+	0.311054i
$584$			1.24571i			0.0515479i
$585$		0			0
$586$	−23.3628			−0.965107
$587$		−	15.1317i		−	0.624553i	−0.949991	−	0.312276i	$-0.898908\pi$
$587$		−	15.1317i		−	0.624553i	0.949991	−	0.312276i	$-0.101092\pi$
$588$		0			0
$589$			10.5409i			0.434331i
$590$	−17.4819			−0.719720
$591$		0			0
$592$	7.98892			0.328343
$593$	−26.2323			−1.07723			−0.538615	−	0.842552i	$-0.681052\pi$
$593$	−26.2323			−1.07723			−0.538615	+	0.842552i	$0.681052\pi$
$594$		0			0
$595$	4.62066			0.189429
$596$	20.8646			0.854647
$597$		0			0
$598$	−7.42827			−0.303764
$599$			8.03822i			0.328433i	0.986424	+	0.164216i	$0.0525095\pi$
$599$			8.03822i			0.328433i	−0.986424	+	0.164216i	$0.947490\pi$
$600$		0			0
$601$		−	42.5220i		−	1.73451i	−0.497866	−	0.867254i	$-0.665883\pi$
$601$		−	42.5220i		−	1.73451i	0.497866	−	0.867254i	$-0.334117\pi$
$602$	8.78060			0.357871
$603$		0			0
$604$			15.2282i			0.619628i
$605$	16.3971	+	6.40694i	0.666637	+	0.260479i
$606$		0			0
$607$			16.7886i			0.681428i	0.940167	+	0.340714i	$0.110669\pi$
$607$			16.7886i			0.681428i	−0.940167	+	0.340714i	$0.889331\pi$
$608$			1.00000i			0.0405554i
$609$		0			0
$610$	−0.569563			−0.0230609
$611$	2.78512			0.112674
$612$		0			0
$613$			14.3919i			0.581285i	0.956832	+	0.290643i	$0.0938690\pi$
$613$			14.3919i			0.581285i	−0.956832	+	0.290643i	$0.906131\pi$
$614$			26.5375i			1.07097i
$615$		0			0
$616$	4.53015	−	3.09360i	0.182525	−	0.124645i
$617$		−	4.62417i		−	0.186162i	−0.995659	−	0.0930810i	$-0.970328\pi$
$617$		−	4.62417i		−	0.186162i	0.995659	−	0.0930810i	$-0.0296715\pi$
$618$		0			0
$619$	−28.8606			−1.16001			−0.580003	−	0.814614i	$-0.696949\pi$
$619$	−28.8606			−1.16001			−0.580003	+	0.814614i	$0.696949\pi$
$620$		−	16.8697i		−	0.677502i
$621$		0			0
$622$		−	28.2467i		−	1.13259i
$623$	−6.78373			−0.271784
$624$		0			0
$625$	−6.85896			−0.274358
$626$	10.7742			0.430624
$627$		0			0
$628$	9.10926			0.363499
$629$	13.9454			0.556039
$630$		0			0
$631$	−4.02870			−0.160380			−0.0801900	−	0.996780i	$-0.525553\pi$
$631$	−4.02870			−0.160380			−0.0801900	+	0.996780i	$0.525553\pi$
$632$		−	1.72159i		−	0.0684813i
$633$		0			0
$634$			3.30204i			0.131141i
$635$	−30.0470			−1.19238
$636$		0			0
$637$			5.28585i			0.209433i
$638$	13.5623	+	19.8601i	0.536936	+	0.786270i
$639$		0			0
$640$		−	1.60040i		−	0.0632613i
$641$			20.9610i			0.827909i	0.910298	+	0.413954i	$0.135853\pi$
$641$			20.9610i			0.827909i	−0.910298	+	0.413954i	$0.864147\pi$
$642$		0			0
$643$	27.1544			1.07086			0.535432	−	0.844578i	$-0.320149\pi$
$643$	27.1544			1.07086			0.535432	+	0.844578i	$0.320149\pi$
$644$	−9.91186			−0.390582
$645$		0			0
$646$			1.74559i			0.0686794i
$647$			20.6187i			0.810606i	0.914182	+	0.405303i	$0.132834\pi$
$647$			20.6187i			0.810606i	−0.914182	+	0.405303i	$0.867166\pi$
$648$		0			0
$649$	−29.9186	+	20.4311i	−1.17441	+	0.801991i
$650$			3.02294i			0.118569i
$651$		0			0
$652$	2.84605			0.111460
$653$			14.0720i			0.550680i	0.961347	+	0.275340i	$0.0887905\pi$
$653$			14.0720i			0.550680i	−0.961347	+	0.275340i	$0.911210\pi$
$654$		0			0
$655$		−	9.81519i		−	0.383511i
$656$	8.60750			0.336066
$657$		0			0
$658$	3.71630			0.144877
$659$	−42.4730			−1.65451			−0.827257	−	0.561823i	$-0.810100\pi$
$659$	−42.4730			−1.65451			−0.827257	+	0.561823i	$0.810100\pi$
$660$		0			0
$661$	−1.55006			−0.0602902			−0.0301451	−	0.999546i	$-0.509597\pi$
$661$	−1.55006			−0.0602902			−0.0301451	+	0.999546i	$0.509597\pi$
$662$	−0.420921			−0.0163596
$663$		0			0
$664$	−8.04031			−0.312024
$665$			2.64705i			0.102648i
$666$		0			0
$667$		−	43.4534i		−	1.68252i
$668$	−5.22557			−0.202183
$669$		0			0
$670$		−	15.9458i		−	0.616040i
$671$	−0.974750	+	0.665648i	−0.0376298	+	0.0256970i
$672$		0			0
$673$		−	27.4187i		−	1.05691i	−0.848960	−	0.528457i	$-0.822771\pi$
$673$		−	27.4187i		−	1.05691i	0.848960	−	0.528457i	$-0.177229\pi$
$674$			21.5763i			0.831090i
$675$		0			0
$676$	11.4635			0.440904
$677$	−10.8853			−0.418358			−0.209179	−	0.977877i	$-0.567079\pi$
$677$	−10.8853			−0.418358			−0.209179	+	0.977877i	$0.567079\pi$
$678$		0			0
$679$		−	5.20415i		−	0.199717i
$680$		−	2.79364i		−	0.107131i
$681$		0			0
$682$	−19.7155	−	28.8707i	−0.754947	−	1.10552i
$683$		−	9.11796i		−	0.348889i	−0.984667	−	0.174444i	$-0.944187\pi$
$683$		−	9.11796i		−	0.348889i	0.984667	−	0.174444i	$-0.0558129\pi$
$684$		0			0
$685$	−4.07906			−0.155853
$686$			18.6311i			0.711338i
$687$		0			0
$688$		−	5.30873i		−	0.202393i
$689$	4.97744			0.189625
$690$		0			0
$691$	18.5738			0.706580			0.353290	−	0.935514i	$-0.385063\pi$
$691$	18.5738			0.706580			0.353290	+	0.935514i	$0.385063\pi$
$692$	6.83864			0.259966
$693$		0			0
$694$	−23.4062			−0.888485
$695$	30.3535			1.15137
$696$		0			0
$697$	15.0252			0.569120
$698$			1.17120i			0.0443305i
$699$		0			0
$700$			4.03364i			0.152457i
$701$	5.85097			0.220988			0.110494	−	0.993877i	$-0.464757\pi$
$701$	5.85097			0.220988			0.110494	+	0.993877i	$0.464757\pi$
$702$		0			0
$703$			7.98892i			0.301308i
$704$	−1.87038	−	2.73892i	−0.0704927	−	0.103227i
$705$		0			0
$706$		−	6.72587i		−	0.253131i
$707$			4.35570i			0.163813i
$708$		0			0
$709$	−15.0947			−0.566894			−0.283447	−	0.958988i	$-0.591478\pi$
$709$	−15.0947			−0.566894			−0.283447	+	0.958988i	$0.591478\pi$
$710$	−13.5126			−0.507120
$711$		0			0
$712$			4.10143i			0.153707i
$713$			63.1684i			2.36567i
$714$		0			0
$715$	−3.71043	−	5.43342i	−0.138762	−	0.203198i
$716$		−	8.39539i		−	0.313750i
$717$		0			0
$718$	−25.2102			−0.940837
$719$		−	37.8816i		−	1.41274i	−0.707840	−	0.706372i	$-0.750330\pi$
$719$		−	37.8816i		−	1.41274i	0.707840	−	0.706372i	$-0.249670\pi$
$720$		0			0
$721$		−	28.2508i		−	1.05211i
$722$	−1.00000			−0.0372161
$723$		0			0
$724$	−9.21653			−0.342530
$725$	−17.6834			−0.656746
$726$		0			0
$727$	−5.68835			−0.210969			−0.105485	−	0.994421i	$-0.533639\pi$
$727$	−5.68835			−0.210969			−0.105485	+	0.994421i	$0.533639\pi$
$728$	−2.05022			−0.0759861
$729$		0			0
$730$	1.99363			0.0737876
$731$		−	9.26687i		−	0.342748i
$732$		0			0
$733$		−	42.5732i		−	1.57248i	−0.617924	−	0.786238i	$-0.712026\pi$
$733$		−	42.5732i		−	1.57248i	0.617924	−	0.786238i	$-0.287974\pi$
$734$	20.9049			0.771615
$735$		0			0
$736$			5.99268i			0.220893i
$737$	−18.6358	−	27.2896i	−0.686460	−	1.00523i
$738$		0			0
$739$		−	10.8151i		−	0.397842i	−0.980016	−	0.198921i	$-0.936256\pi$
$739$		−	10.8151i		−	0.397842i	0.980016	−	0.198921i	$-0.0637436\pi$
$740$		−	12.7854i		−	0.470002i
$741$		0			0
$742$	6.64162			0.243821
$743$	−40.9336			−1.50171			−0.750854	−	0.660469i	$-0.770358\pi$
$743$	−40.9336			−1.50171			−0.750854	+	0.660469i	$0.770358\pi$
$744$		0			0
$745$		−	33.3916i		−	1.22338i
$746$		−	14.9317i		−	0.546688i
$747$		0			0
$748$	−3.26492	−	4.78103i	−0.119377	−	0.174812i
$749$			10.5338i			0.384895i
$750$		0			0
$751$	−50.4354			−1.84041			−0.920207	−	0.391431i	$-0.871980\pi$
$751$	−50.4354			−1.84041			−0.920207	+	0.391431i	$0.871980\pi$
$752$		−	2.24687i		−	0.0819348i
$753$		0			0
$754$		−	8.98812i		−	0.327328i
$755$	24.3712			0.886960
$756$		0			0
$757$	−11.9447			−0.434136			−0.217068	−	0.976156i	$-0.569649\pi$
$757$	−11.9447			−0.434136			−0.217068	+	0.976156i	$0.569649\pi$
$758$	22.9383			0.833157
$759$		0			0
$760$	1.60040			0.0580525
$761$	33.6940			1.22141			0.610704	−	0.791859i	$-0.290887\pi$
$761$	33.6940			1.22141			0.610704	+	0.791859i	$0.290887\pi$
$762$		0			0
$763$	−0.757228			−0.0274135
$764$			23.5308i			0.851313i
$765$		0			0
$766$		−	26.7584i		−	0.966821i
$767$	13.5403			0.488912
$768$		0			0
$769$		−	49.5991i		−	1.78859i	−0.447480	−	0.894294i	$-0.647678\pi$
$769$		−	49.5991i		−	1.78859i	0.447480	−	0.894294i	$-0.352322\pi$
$770$	−4.95099	−	7.25005i	−0.178421	−	0.261273i
$771$		0			0
$772$			0.604960i			0.0217730i
$773$		−	27.3879i		−	0.985073i	−0.870292	−	0.492536i	$-0.836070\pi$
$773$		−	27.3879i		−	0.985073i	0.870292	−	0.492536i	$-0.163930\pi$
$774$		0			0
$775$	25.7064			0.923402
$776$	−3.14641			−0.112950
$777$		0			0
$778$			18.5342i			0.664484i
$779$			8.60750i			0.308396i
$780$		0			0
$781$	−23.1255	+	15.7922i	−0.827496	+	0.565089i
$782$			10.4608i			0.374077i
$783$		0			0
$784$	4.26430			0.152297
$785$		−	14.5784i		−	0.520327i
$786$		0			0
$787$		−	4.39881i		−	0.156801i	−0.996922	−	0.0784003i	$-0.975019\pi$
$787$		−	4.39881i		−	0.156801i	0.996922	−	0.0784003i	$-0.0249812\pi$
$788$	−4.85995			−0.173129
$789$		0			0
$790$	−2.75523			−0.0980268
$791$	−6.84843			−0.243502
$792$		0			0
$793$	0.441144			0.0156655
$794$	8.18631			0.290521
$795$		0			0
$796$	−15.6837			−0.555895
$797$			0.513083i			0.0181743i	0.999959	+	0.00908716i	$0.00289257\pi$
$797$			0.513083i			0.0181743i	−0.999959	+	0.00908716i	$0.997107\pi$
$798$		0			0
$799$		−	3.92211i		−	0.138754i
$800$	2.43873			0.0862221
$801$		0			0
$802$			18.5674i			0.655636i
$803$	3.41190	−	2.32995i	0.120403	−	0.0822223i
$804$		0			0
$805$			15.8629i			0.559094i
$806$			13.0661i			0.460232i
$807$		0			0
$808$	2.63345			0.0926443
$809$	0.482347			0.0169584			0.00847922	−	0.999964i	$-0.497301\pi$
$809$	0.482347			0.0169584			0.00847922	+	0.999964i	$0.497301\pi$
$810$		0			0
$811$		−	36.9347i		−	1.29695i	−0.761234	−	0.648477i	$-0.775406\pi$
$811$		−	36.9347i		−	1.29695i	0.761234	−	0.648477i	$-0.224594\pi$
$812$		−	11.9932i		−	0.420880i
$813$		0			0
$814$	−14.9423	−	21.8810i	−0.523728	−	0.766929i
$815$		−	4.55482i		−	0.159548i
$816$		0			0
$817$	5.30873			0.185729
$818$			37.0679i			1.29605i
$819$		0			0
$820$		−	13.7754i		−	0.481059i
$821$	10.9394			0.381789			0.190894	−	0.981611i	$-0.438861\pi$
$821$	10.9394			0.381789			0.190894	+	0.981611i	$0.438861\pi$
$822$		0			0
$823$	34.6772			1.20877			0.604386	−	0.796692i	$-0.293419\pi$
$823$	34.6772			1.20877			0.604386	+	0.796692i	$0.293419\pi$
$824$	−17.0803			−0.595022
$825$		0			0
$826$	18.0674			0.628645
$827$	−46.3601			−1.61210			−0.806049	−	0.591849i	$-0.798398\pi$
$827$	−46.3601			−1.61210			−0.806049	+	0.591849i	$0.798398\pi$
$828$		0			0
$829$	−17.9018			−0.621756			−0.310878	−	0.950450i	$-0.600623\pi$
$829$	−17.9018			−0.621756			−0.310878	+	0.950450i	$0.600623\pi$
$830$			12.8677i			0.446644i
$831$		0			0
$832$			1.23956i			0.0429739i
$833$	7.44374			0.257910
$834$		0			0
$835$			8.36299i			0.289413i
$836$	2.73892	−	1.87038i	0.0947275	−	0.0646885i
$837$		0			0
$838$		−	8.88277i		−	0.306850i
$839$			7.82142i			0.270026i	0.990844	+	0.135013i	$0.0431075\pi$
$839$			7.82142i			0.270026i	−0.990844	+	0.135013i	$0.956892\pi$
$840$		0			0
$841$	23.5782			0.813041
$842$	−15.0103			−0.517290
$843$		0			0
$844$			10.2138i			0.351572i
$845$		−	18.3462i		−	0.631127i
$846$		0			0
$847$	−16.9462	−	6.62151i	−0.582279	−	0.227518i
$848$		−	4.01550i		−	0.137893i
$849$		0			0
$850$	4.25703			0.146015
$851$			47.8751i			1.64114i
$852$		0			0
$853$		−	39.7054i		−	1.35949i	−0.733449	−	0.679744i	$-0.762091\pi$
$853$		−	39.7054i		−	1.35949i	0.733449	−	0.679744i	$-0.237909\pi$
$854$	0.588638			0.0201428
$855$		0			0
$856$	6.36868			0.217677
$857$	32.5490			1.11185			0.555927	−	0.831231i	$-0.312363\pi$
$857$	32.5490			1.11185			0.555927	+	0.831231i	$0.312363\pi$
$858$		0			0
$859$	−25.9610			−0.885777			−0.442889	−	0.896577i	$-0.646046\pi$
$859$	−25.9610			−0.885777			−0.442889	+	0.896577i	$0.646046\pi$
$860$	−8.49607			−0.289714
$861$		0			0
$862$	11.5164			0.392252
$863$		−	31.2800i		−	1.06478i	−0.846498	−	0.532392i	$-0.821293\pi$
$863$		−	31.2800i		−	1.06478i	0.846498	−	0.532392i	$-0.178707\pi$
$864$		0			0
$865$		−	10.9445i		−	0.372125i
$866$	−24.3150			−0.826257
$867$		0			0
$868$			17.4346i			0.591769i
$869$	−4.71530	+	3.22004i	−0.159956	+	0.109232i
$870$		0			0
$871$			12.3505i			0.418481i
$872$			0.457818i			0.0155037i
$873$		0			0
$874$	−5.99268			−0.202706
$875$	19.6907			0.665666
$876$		0			0
$877$		−	45.5435i		−	1.53790i	−0.639311	−	0.768948i	$-0.720781\pi$
$877$		−	45.5435i		−	1.53790i	0.639311	−	0.768948i	$-0.279219\pi$
$878$			36.9879i			1.24828i
$879$		0			0
$880$	−4.38336	+	2.99335i	−0.147763	+	0.100906i
$881$			42.4479i			1.43011i	0.699070	+	0.715054i	$0.253598\pi$
$881$			42.4479i			1.43011i	−0.699070	+	0.715054i	$0.746402\pi$
$882$		0			0
$883$	−38.8422			−1.30714			−0.653572	−	0.756865i	$-0.726730\pi$
$883$	−38.8422			−1.30714			−0.653572	+	0.756865i	$0.726730\pi$
$884$			2.16376i			0.0727751i
$885$		0			0
$886$		−	26.6046i		−	0.893799i
$887$	−36.3100			−1.21917			−0.609586	−	0.792720i	$-0.708664\pi$
$887$	−36.3100			−1.21917			−0.609586	+	0.792720i	$0.708664\pi$
$888$		0			0
$889$	31.0533			1.04149
$890$	6.56391			0.220023
$891$		0			0
$892$	−22.1003			−0.739973
$893$	2.24687			0.0751885
$894$		0			0
$895$	−13.4360			−0.449114
$896$			1.65399i			0.0552561i
$897$		0			0
$898$			5.55083i			0.185234i
$899$	−76.4331			−2.54919
$900$		0			0
$901$		−	7.00943i		−	0.233518i
$902$	−16.0993	−	23.5752i	−0.536049	−	0.784970i
$903$		0			0
$904$			4.14054i			0.137712i
$905$			14.7501i			0.490310i
$906$		0			0
$907$	−50.4551			−1.67533			−0.837667	−	0.546181i	$-0.816081\pi$
$907$	−50.4551			−1.67533			−0.837667	+	0.546181i	$0.816081\pi$
$908$	17.1078			0.567743
$909$		0			0
$910$			3.28116i			0.108770i
$911$			7.53950i			0.249795i	0.992170	+	0.124897i	$0.0398602\pi$
$911$			7.53950i			0.249795i	−0.992170	+	0.124897i	$0.960140\pi$
$912$		0			0
$913$	15.0384	+	22.0217i	0.497700	+	0.728813i
$914$		−	20.2718i		−	0.670531i
$915$		0			0
$916$	9.64766			0.318768
$917$			10.1439i			0.334981i
$918$		0			0
$919$		−	2.36625i		−	0.0780554i	−0.999238	−	0.0390277i	$-0.987574\pi$
$919$		−	2.36625i		−	0.0780554i	0.999238	−	0.0390277i	$-0.0124261\pi$
$920$	9.59068			0.316195
$921$		0			0
$922$	23.4834			0.773385
$923$	10.4659			0.344491
$924$		0			0
$925$	19.4828			0.640591
$926$	−5.10399			−0.167728
$927$		0			0
$928$	−7.25108			−0.238028
$929$			56.8993i			1.86681i	0.358829	+	0.933403i	$0.383176\pi$
$929$			56.8993i			1.86681i	−0.358829	+	0.933403i	$0.616824\pi$
$930$		0			0
$931$			4.26430i			0.139757i
$932$	−8.90102			−0.291563
$933$		0			0
$934$			24.1746i			0.791018i
$935$	−7.65155	+	5.22517i	−0.250233	+	0.170881i
$936$		0			0
$937$		−	47.1121i		−	1.53908i	−0.638596	−	0.769542i	$-0.720485\pi$
$937$		−	47.1121i		−	1.53908i	0.638596	−	0.769542i	$-0.279515\pi$
$938$			16.4798i			0.538085i
$939$		0			0
$940$	−3.59588			−0.117285
$941$	−8.97930			−0.292717			−0.146358	−	0.989232i	$-0.546755\pi$
$941$	−8.97930			−0.292717			−0.146358	+	0.989232i	$0.546755\pi$
$942$		0			0
$943$			51.5820i			1.67974i
$944$		−	10.9235i		−	0.355530i
$945$		0			0
$946$	−14.5402	+	9.92935i	−0.472742	+	0.322831i
$947$			54.9193i			1.78464i	0.451406	+	0.892319i	$0.350923\pi$
$947$			54.9193i			1.78464i	−0.451406	+	0.892319i	$0.649077\pi$
$948$		0			0
$949$	−1.54413			−0.0501245
$950$			2.43873i			0.0791228i
$951$		0			0
$952$			2.88720i			0.0935746i
$953$	−41.7950			−1.35387			−0.676936	−	0.736042i	$-0.736693\pi$
$953$	−41.7950			−1.35387			−0.676936	+	0.736042i	$0.736693\pi$
$954$		0			0
$955$	37.6586			1.21860
$956$	−16.7800			−0.542703
$957$		0			0
$958$	−4.36286			−0.140958
$959$	4.21567			0.136131
$960$		0			0
$961$	80.1110			2.58422
$962$			9.90271i			0.319276i
$963$		0			0
$964$			5.80518i			0.186972i
$965$	0.968177			0.0311667
$966$		0			0
$967$			62.0406i			1.99509i	0.0700296	+	0.997545i	$0.477691\pi$
$967$			62.0406i			1.99509i	−0.0700296	+	0.997545i	$0.522309\pi$
$968$	−4.00335	+	10.2456i	−0.128672	+	0.329307i
$969$		0			0
$970$			5.03551i			0.161680i
$971$		−	19.0298i		−	0.610694i	−0.952241	−	0.305347i	$-0.901228\pi$
$971$		−	19.0298i		−	0.610694i	0.952241	−	0.305347i	$-0.0987724\pi$
$972$		0			0
$973$	−31.3700			−1.00568
$974$	−21.1036			−0.676203
$975$		0			0
$976$		−	0.355889i		−	0.0113917i
$977$		−	6.43660i		−	0.205925i	−0.994685	−	0.102962i	$-0.967168\pi$
$977$		−	6.43660i		−	0.205925i	0.994685	−	0.102962i	$-0.0328322\pi$
$978$		0			0
$979$	11.2335	−	7.67123i	0.359023	−	0.245174i
$980$		−	6.82458i		−	0.218003i
$981$		0			0
$982$	−34.5369			−1.10212
$983$			23.9053i			0.762462i	0.924480	+	0.381231i	$0.124500\pi$
$983$			23.9053i			0.762462i	−0.924480	+	0.381231i	$0.875500\pi$
$984$		0			0
$985$			7.77785i			0.247823i
$986$	−12.6574			−0.403095
$987$		0			0
$988$	−1.23956			−0.0394355
$989$	31.8135			1.01161
$990$		0			0
$991$	−57.8293			−1.83701			−0.918504	−	0.395412i	$-0.870602\pi$
$991$	−57.8293			−1.83701			−0.918504	+	0.395412i	$0.870602\pi$
$992$	10.5409			0.334674
$993$		0			0
$994$	13.9652			0.442948
$995$			25.1002i			0.795729i
$996$		0			0
$997$		−	42.9055i		−	1.35883i	−0.733753	−	0.679416i	$-0.762233\pi$
$997$		−	42.9055i		−	1.35883i	0.733753	−	0.679416i	$-0.237767\pi$
$998$	2.33181			0.0738121
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.b.b.989.12	yes	36
3.2	odd	2		3762.2.b.a.989.25	yes	36
11.10	odd	2		3762.2.b.a.989.12	✓	36
33.32	even	2	inner	3762.2.b.b.989.25	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3762.2.b.a.989.12	✓	36	11.10	odd	2
3762.2.b.a.989.25	yes	36	3.2	odd	2
3762.2.b.b.989.12	yes	36	1.1	even	1	trivial
3762.2.b.b.989.25	yes	36	33.32	even	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.60040i q^{5} +1.65399i q^{7} +1.00000 q^{8} -1.60040i q^{10} +(-1.87038 - 2.73892i) q^{11} +1.23956i q^{13} +1.65399i q^{14} +1.00000 q^{16} +1.74559 q^{17} +1.00000i q^{19} -1.60040i q^{20} +(-1.87038 - 2.73892i) q^{22} +5.99268i q^{23} +2.43873 q^{25} +1.23956i q^{26} +1.65399i q^{28} -7.25108 q^{29} +10.5409 q^{31} +1.00000 q^{32} +1.74559 q^{34} +2.64705 q^{35} +7.98892 q^{37} +1.00000i q^{38} -1.60040i q^{40} +8.60750 q^{41} -5.30873i q^{43} +(-1.87038 - 2.73892i) q^{44} +5.99268i q^{46} -2.24687i q^{47} +4.26430 q^{49} +2.43873 q^{50} +1.23956i q^{52} -4.01550i q^{53} +(-4.38336 + 2.99335i) q^{55} +1.65399i q^{56} -7.25108 q^{58} -10.9235i q^{59} -0.355889i q^{61} +10.5409 q^{62} +1.00000 q^{64} +1.98378 q^{65} +9.96365 q^{67} +1.74559 q^{68} +2.64705 q^{70} -8.44330i q^{71} +1.24571i q^{73} +7.98892 q^{74} +1.00000i q^{76} +(4.53015 - 3.09360i) q^{77} -1.72159i q^{79} -1.60040i q^{80} +8.60750 q^{82} -8.04031 q^{83} -2.79364i q^{85} -5.30873i q^{86} +(-1.87038 - 2.73892i) q^{88} +4.10143i q^{89} -2.05022 q^{91} +5.99268i q^{92} -2.24687i q^{94} +1.60040 q^{95} -3.14641 q^{97} +4.26430 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{2} + 36 q^{4} + 36 q^{8} - 4 q^{11} + 36 q^{16} + 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} + 36 q^{32} + 16 q^{34} + 16 q^{35} + 16 q^{37} + 24 q^{41} - 4 q^{44} - 68 q^{49} - 28 q^{50}+ \cdots - 68 q^{98}+O(q^{100}) \) 36 * q + 36 * q^2 + 36 * q^4 + 36 * q^8 - 4 * q^11 + 36 * q^16 + 16 * q^17 - 4 * q^22 - 28 * q^25 - 8 * q^31 + 36 * q^32 + 16 * q^34 + 16 * q^35 + 16 * q^37 + 24 * q^41 - 4 * q^44 - 68 * q^49 - 28 * q^50 + 28 * q^55 - 8 * q^62 + 36 * q^64 + 32 * q^65 + 16 * q^67 + 16 * q^68 + 16 * q^70 + 16 * q^74 - 32 * q^77 + 24 * q^82 - 56 * q^83 - 4 * q^88 - 48 * q^91 + 8 * q^97 - 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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