LMFDB - Embedded newform 2394.2.b.i.1709.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1709,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.1709");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 10x^{6} + 30x^{4} + 24x^{2} + 1 $ x^8 + 10x^6 + 30x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1709.7
Root			$2.03530i$ of defining polynomial
Character	$\chi$	$=$	2394.1709
Dual form			2394.2.b.i.1709.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.02986i q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.02986i q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.02986i q^{10} -4.70358i q^{11} +6.26603i q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.70358i q^{17} +(4.28832 + 0.781225i) q^{19} +3.02986i q^{20} +4.70358i q^{22} +0.460942i q^{23} -4.18005 q^{25} -6.26603i q^{26} -1.00000 q^{28} +3.18005 q^{29} +7.73344i q^{31} -1.00000 q^{32} +4.70358i q^{34} -3.02986i q^{35} -1.81708i q^{37} +(-4.28832 - 0.781225i) q^{38} -3.02986i q^{40} +8.40856 q^{41} +4.86151 q^{43} -4.70358i q^{44} -0.460942i q^{46} +12.5321i q^{47} +1.00000 q^{49} +4.18005 q^{50} +6.26603i q^{52} -11.3897 q^{53} +14.2512 q^{55} +1.00000 q^{56} -3.18005 q^{58} +5.46492 q^{61} -7.73344i q^{62} +1.00000 q^{64} -18.9852 q^{65} +12.0711i q^{67} -4.70358i q^{68} +3.02986i q^{70} -2.20964 q^{71} -13.2216 q^{73} +1.81708i q^{74} +(4.28832 + 0.781225i) q^{76} +4.70358i q^{77} +8.69159i q^{79} +3.02986i q^{80} -8.40856 q^{82} +8.37401i q^{83} +14.2512 q^{85} -4.86151 q^{86} +4.70358i q^{88} -6.19892 q^{89} -6.26603i q^{91} +0.460942i q^{92} -12.5321i q^{94} +(-2.36700 + 12.9930i) q^{95} +9.50221i q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8} + 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} + 16 q^{29} - 8 q^{32} - 12 q^{38} - 24 q^{41} - 32 q^{43} + 8 q^{49} + 24 q^{50} - 48 q^{53} + 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} - 16 q^{65} + 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} + 32 q^{86} + 8 q^{89} - 8 q^{95} - 8 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^7 - 8 * q^8 + 8 * q^14 + 8 * q^16 + 12 * q^19 - 24 * q^25 - 8 * q^28 + 16 * q^29 - 8 * q^32 - 12 * q^38 - 24 * q^41 - 32 * q^43 + 8 * q^49 + 24 * q^50 - 48 * q^53 + 8 * q^56 - 16 * q^58 + 8 * q^61 + 8 * q^64 - 16 * q^65 + 16 * q^71 - 16 * q^73 + 12 * q^76 + 24 * q^82 + 32 * q^86 + 8 * q^89 - 8 * q^95 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$			3.02986i			1.35499i	0.735525	+	0.677497i	$0.236935\pi$
$5$			3.02986i			1.35499i	−0.735525	+	0.677497i	$0.763065\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$	−1.00000			−0.353553
$9$		0			0
$10$		−	3.02986i		−	0.958126i
$11$		−	4.70358i		−	1.41818i	−0.705116	−	0.709092i	$-0.749105\pi$
$11$		−	4.70358i		−	1.41818i	0.705116	−	0.709092i	$-0.250895\pi$
$12$		0			0
$13$			6.26603i			1.73788i	0.494913	+	0.868942i	$0.335200\pi$
$13$			6.26603i			1.73788i	−0.494913	+	0.868942i	$0.664800\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		−	4.70358i		−	1.14079i	−0.821372	−	0.570393i	$-0.806791\pi$
$17$		−	4.70358i		−	1.14079i	0.821372	−	0.570393i	$-0.193209\pi$
$18$		0			0
$19$	4.28832	+	0.781225i	0.983808	+	0.179225i
$19$	4.28832	+	0.781225i	0.983808	+	0.179225i
$20$			3.02986i			0.677497i
$21$		0			0
$22$			4.70358i			1.00281i
$23$			0.460942i			0.0961131i	0.998845	+	0.0480565i	$0.0153028\pi$
$23$			0.460942i			0.0961131i	−0.998845	+	0.0480565i	$0.984697\pi$
$24$		0			0
$25$	−4.18005			−0.836010
$26$		−	6.26603i		−	1.22887i
$27$		0			0
$28$	−1.00000			−0.188982
$29$	3.18005			0.590520			0.295260	−	0.955417i	$-0.404594\pi$
$29$	3.18005			0.590520			0.295260	+	0.955417i	$0.404594\pi$
$30$		0			0
$31$			7.73344i			1.38897i	0.719508	+	0.694484i	$0.244367\pi$
$31$			7.73344i			1.38897i	−0.719508	+	0.694484i	$0.755633\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			4.70358i			0.806658i
$35$		−	3.02986i		−	0.512140i
$36$		0			0
$37$		−	1.81708i		−	0.298726i	−0.988782	−	0.149363i	$-0.952278\pi$
$37$		−	1.81708i		−	0.298726i	0.988782	−	0.149363i	$-0.0477223\pi$
$38$	−4.28832	−	0.781225i	−0.695657	−	0.126731i
$39$		0			0
$40$		−	3.02986i		−	0.479063i
$41$	8.40856			1.31320			0.656598	−	0.754241i	$-0.271995\pi$
$41$	8.40856			1.31320			0.656598	+	0.754241i	$0.271995\pi$
$42$		0			0
$43$	4.86151			0.741373			0.370687	−	0.928758i	$-0.379122\pi$
$43$	4.86151			0.741373			0.370687	+	0.928758i	$0.379122\pi$
$44$		−	4.70358i		−	0.709092i
$45$		0			0
$46$		−	0.460942i		−	0.0679622i
$47$			12.5321i			1.82799i	0.405726	+	0.913995i	$0.367019\pi$
$47$			12.5321i			1.82799i	−0.405726	+	0.913995i	$0.632981\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$	4.18005			0.591148
$51$		0			0
$52$			6.26603i			0.868942i
$53$	−11.3897			−1.56449			−0.782247	−	0.622968i	$-0.785926\pi$
$53$	−11.3897			−1.56449			−0.782247	+	0.622968i	$0.785926\pi$
$54$		0			0
$55$	14.2512			1.92163
$56$	1.00000			0.133631
$57$		0			0
$58$	−3.18005			−0.417561
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	5.46492			0.699711			0.349856	−	0.936804i	$-0.386231\pi$
$61$	5.46492			0.699711			0.349856	+	0.936804i	$0.386231\pi$
$62$		−	7.73344i		−	0.982148i
$63$		0			0
$64$	1.00000			0.125000
$65$	−18.9852			−2.35482
$66$		0			0
$67$			12.0711i			1.47472i	0.675499	+	0.737361i	$0.263928\pi$
$67$			12.0711i			1.47472i	−0.675499	+	0.737361i	$0.736072\pi$
$68$		−	4.70358i		−	0.570393i
$69$		0			0
$70$			3.02986i			0.362137i
$71$	−2.20964			−0.262236			−0.131118	−	0.991367i	$-0.541857\pi$
$71$	−2.20964			−0.262236			−0.131118	+	0.991367i	$0.541857\pi$
$72$		0			0
$73$	−13.2216			−1.54747			−0.773736	−	0.633508i	$-0.781614\pi$
$73$	−13.2216			−1.54747			−0.773736	+	0.633508i	$0.781614\pi$
$74$			1.81708i			0.211231i
$75$		0			0
$76$	4.28832	+	0.781225i	0.491904	+	0.0896126i
$77$			4.70358i			0.536023i
$78$		0			0
$79$			8.69159i			0.977881i	0.872317	+	0.488940i	$0.162616\pi$
$79$			8.69159i			0.977881i	−0.872317	+	0.488940i	$0.837384\pi$
$80$			3.02986i			0.338749i
$81$		0			0
$82$	−8.40856			−0.928570
$83$			8.37401i			0.919167i	0.888135	+	0.459583i	$0.152001\pi$
$83$			8.37401i			0.919167i	−0.888135	+	0.459583i	$0.847999\pi$
$84$		0			0
$85$	14.2512			1.54576
$86$	−4.86151			−0.524230
$87$		0			0
$88$			4.70358i			0.501404i
$89$	−6.19892			−0.657084			−0.328542	−	0.944489i	$-0.606557\pi$
$89$	−6.19892			−0.657084			−0.328542	+	0.944489i	$0.606557\pi$
$90$		0			0
$91$		−	6.26603i		−	0.656859i
$92$			0.460942i			0.0480565i
$93$		0			0
$94$		−	12.5321i		−	1.29258i
$95$	−2.36700	+	12.9930i	−0.242849	+	1.33305i
$96$		0			0
$97$			9.50221i			0.964803i	0.875950	+	0.482401i	$0.160235\pi$
$97$			9.50221i			0.964803i	−0.875950	+	0.482401i	$0.839765\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	−4.18005			−0.418005
$101$		−	15.5619i		−	1.54847i	−0.632899	−	0.774235i	$-0.718135\pi$
$101$		−	15.5619i		−	1.54847i	0.632899	−	0.774235i	$-0.281865\pi$
$102$		0			0
$103$		−	12.2307i		−	1.20513i	−0.798071	−	0.602564i	$-0.794146\pi$
$103$		−	12.2307i		−	1.20513i	0.798071	−	0.602564i	$-0.205854\pi$
$104$		−	6.26603i		−	0.614435i
$105$		0			0
$106$	11.3897			1.10626
$107$	−6.29177			−0.608248			−0.304124	−	0.952632i	$-0.598364\pi$
$107$	−6.29177			−0.608248			−0.304124	+	0.952632i	$0.598364\pi$
$108$		0			0
$109$		−	17.9245i		−	1.71686i	−0.512932	−	0.858429i	$-0.671441\pi$
$109$		−	17.9245i		−	1.71686i	0.512932	−	0.858429i	$-0.328559\pi$
$110$	−14.2512			−1.35880
$111$		0			0
$112$	−1.00000			−0.0944911
$113$	−18.0831			−1.70112			−0.850558	−	0.525881i	$-0.823736\pi$
$113$	−18.0831			−1.70112			−0.850558	+	0.525881i	$0.823736\pi$
$114$		0			0
$115$	−1.39659			−0.130233
$116$	3.18005			0.295260
$117$		0			0
$118$		0			0
$119$			4.70358i			0.431177i
$120$		0			0
$121$	−11.1237			−1.01124
$122$	−5.46492			−0.494770
$123$		0			0
$124$			7.73344i			0.694484i
$125$			2.48433i			0.222206i
$126$		0			0
$127$		−	7.18792i		−	0.637824i	−0.947784	−	0.318912i	$-0.896682\pi$
$127$		−	7.18792i		−	0.637824i	0.947784	−	0.318912i	$-0.103318\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	18.9852			1.66511
$131$			16.6314i			1.45309i	0.687120	+	0.726544i	$0.258875\pi$
$131$			16.6314i			1.45309i	−0.687120	+	0.726544i	$0.741125\pi$
$132$		0			0
$133$	−4.28832	−	0.781225i	−0.371844	−	0.0677408i
$134$		−	12.0711i		−	1.04279i
$135$		0			0
$136$			4.70358i			0.403329i
$137$		−	5.16452i		−	0.441235i	−0.975360	−	0.220618i	$-0.929193\pi$
$137$		−	5.16452i		−	0.441235i	0.975360	−	0.220618i	$-0.0708073\pi$
$138$		0			0
$139$	−20.9890			−1.78026			−0.890132	−	0.455702i	$-0.849388\pi$
$139$	−20.9890			−1.78026			−0.890132	+	0.455702i	$0.849388\pi$
$140$		−	3.02986i		−	0.256070i
$141$		0			0
$142$	2.20964			0.185429
$143$	29.4728			2.46464
$144$		0			0
$145$			9.63510i			0.800152i
$146$	13.2216			1.09423
$147$		0			0
$148$		−	1.81708i		−	0.149363i
$149$		−	0.921884i		−	0.0755237i	−0.999287	−	0.0377619i	$-0.987977\pi$
$149$		−	0.921884i		−	0.0755237i	0.999287	−	0.0377619i	$-0.0120228\pi$
$150$		0			0
$151$			16.3725i			1.33238i	0.745783	+	0.666189i	$0.232076\pi$
$151$			16.3725i			1.33238i	−0.745783	+	0.666189i	$0.767924\pi$
$152$	−4.28832	−	0.781225i	−0.347829	−	0.0633657i
$153$		0			0
$154$		−	4.70358i		−	0.379026i
$155$	−23.4312			−1.88204
$156$		0			0
$157$	−8.69343			−0.693811			−0.346906	−	0.937900i	$-0.612768\pi$
$157$	−8.69343			−0.693811			−0.346906	+	0.937900i	$0.612768\pi$
$158$		−	8.69159i		−	0.691466i
$159$		0			0
$160$		−	3.02986i		−	0.239531i
$161$		−	0.460942i		−	0.0363273i
$162$		0			0
$163$	8.12750			0.636595			0.318298	−	0.947991i	$-0.396889\pi$
$163$	8.12750			0.636595			0.318298	+	0.947991i	$0.396889\pi$
$164$	8.40856			0.656598
$165$		0			0
$166$		−	8.37401i		−	0.649949i
$167$	23.1127			1.78851			0.894257	−	0.447553i	$-0.147704\pi$
$167$	23.1127			1.78851			0.894257	+	0.447553i	$0.147704\pi$
$168$		0			0
$169$	−26.2632			−2.02024
$170$	−14.2512			−1.09302
$171$		0			0
$172$	4.86151			0.370687
$173$	−6.29177			−0.478355			−0.239177	−	0.970976i	$-0.576878\pi$
$173$	−6.29177			−0.478355			−0.239177	+	0.970976i	$0.576878\pi$
$174$		0			0
$175$	4.18005			0.315982
$176$		−	4.70358i		−	0.354546i
$177$		0			0
$178$	6.19892			0.464629
$179$	−4.15046			−0.310220			−0.155110	−	0.987897i	$-0.549573\pi$
$179$	−4.15046			−0.310220			−0.155110	+	0.987897i	$0.549573\pi$
$180$		0			0
$181$			24.6677i			1.83354i	0.399417	+	0.916769i	$0.369212\pi$
$181$			24.6677i			1.83354i	−0.399417	+	0.916769i	$0.630788\pi$
$182$			6.26603i			0.464469i
$183$		0			0
$184$		−	0.460942i		−	0.0339811i
$185$	5.50549			0.404772
$186$		0			0
$187$	−22.1237			−1.61784
$188$			12.5321i			0.913995i
$189$		0			0
$190$	2.36700	−	12.9930i	0.171720	−	0.942612i
$191$			1.51413i			0.109559i	0.998498	+	0.0547795i	$0.0174456\pi$
$191$			1.51413i			0.109559i	−0.998498	+	0.0547795i	$0.982554\pi$
$192$		0			0
$193$			13.3952i			0.964206i	0.876115	+	0.482103i	$0.160127\pi$
$193$			13.3952i			0.964206i	−0.876115	+	0.482103i	$0.839873\pi$
$194$		−	9.50221i		−	0.682219i
$195$		0			0
$196$	1.00000			0.0714286
$197$		−	12.5321i		−	0.892873i	−0.894815	−	0.446436i	$-0.852693\pi$
$197$		−	12.5321i		−	0.892873i	0.894815	−	0.446436i	$-0.147307\pi$
$198$		0			0
$199$	21.3453			1.51313			0.756564	−	0.653920i	$-0.226877\pi$
$199$	21.3453			1.51313			0.756564	+	0.653920i	$0.226877\pi$
$200$	4.18005			0.295574
$201$		0			0
$202$			15.5619i			1.09493i
$203$	−3.18005			−0.223196
$204$		0			0
$205$			25.4768i			1.77937i
$206$			12.2307i			0.852154i
$207$		0			0
$208$			6.26603i			0.434471i
$209$	3.67456	−	20.1705i	0.254174	−	1.39522i
$210$		0			0
$211$			2.02339i			0.139296i	0.997572	+	0.0696480i	$0.0221876\pi$
$211$			2.02339i			0.139296i	−0.997572	+	0.0696480i	$0.977812\pi$
$212$	−11.3897			−0.782247
$213$		0			0
$214$	6.29177			0.430097
$215$			14.7297i			1.00456i
$216$		0			0
$217$		−	7.73344i		−	0.524980i
$218$			17.9245i			1.21400i
$219$		0			0
$220$	14.2512			0.960815
$221$	29.4728			1.98256
$222$		0			0
$223$		−	3.04609i		−	0.203981i	−0.994785	−	0.101991i	$-0.967479\pi$
$223$		−	3.04609i		−	0.203981i	0.994785	−	0.101991i	$-0.0325212\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	18.0831			1.20287
$227$	7.89110			0.523750			0.261875	−	0.965102i	$-0.415659\pi$
$227$	7.89110			0.523750			0.261875	+	0.965102i	$0.415659\pi$
$228$		0			0
$229$	8.12369			0.536829			0.268414	−	0.963304i	$-0.413500\pi$
$229$	8.12369			0.536829			0.268414	+	0.963304i	$0.413500\pi$
$230$	1.39659			0.0920884
$231$		0			0
$232$	−3.18005			−0.208780
$233$			12.2774i			0.804322i	0.915569	+	0.402161i	$0.131741\pi$
$233$			12.2774i			0.804322i	−0.915569	+	0.402161i	$0.868259\pi$
$234$		0			0
$235$	−37.9704			−2.47692
$236$		0			0
$237$		0			0
$238$		−	4.70358i		−	0.304888i
$239$			15.0647i			0.974455i	0.873275	+	0.487228i	$0.161992\pi$
$239$			15.0647i			0.974455i	−0.873275	+	0.487228i	$0.838008\pi$
$240$		0			0
$241$		−	4.53354i		−	0.292031i	−0.989282	−	0.146015i	$-0.953355\pi$
$241$		−	4.53354i		−	0.292031i	0.989282	−	0.146015i	$-0.0466449\pi$
$242$	11.1237			0.715058
$243$		0			0
$244$	5.46492			0.349856
$245$			3.02986i			0.193571i
$246$		0			0
$247$	−4.89518	+	26.8708i	−0.311473	+	1.70975i
$248$		−	7.73344i		−	0.491074i
$249$		0			0
$250$		−	2.48433i		−	0.157123i
$251$			25.4033i			1.60344i	0.597697	+	0.801722i	$0.296083\pi$
$251$			25.4033i			1.60344i	−0.597697	+	0.801722i	$0.703917\pi$
$252$		0			0
$253$	2.16808			0.136306
$254$			7.18792i			0.451010i
$255$		0			0
$256$	1.00000			0.0625000
$257$	23.9219			1.49221			0.746105	−	0.665829i	$-0.231922\pi$
$257$	23.9219			1.49221			0.746105	+	0.665829i	$0.231922\pi$
$258$		0			0
$259$			1.81708i			0.112908i
$260$	−18.9852			−1.17741
$261$		0			0
$262$		−	16.6314i		−	1.02749i
$263$		−	11.9398i		−	0.736241i	−0.929778	−	0.368120i	$-0.880001\pi$
$263$		−	11.9398i		−	0.736241i	0.929778	−	0.368120i	$-0.119999\pi$
$264$		0			0
$265$		−	34.5092i		−	2.11988i
$266$	4.28832	+	0.781225i	0.262934	+	0.0479000i
$267$		0			0
$268$			12.0711i			0.737361i
$269$	−7.91787			−0.482761			−0.241380	−	0.970431i	$-0.577600\pi$
$269$	−7.91787			−0.482761			−0.241380	+	0.970431i	$0.577600\pi$
$270$		0			0
$271$	5.87348			0.356788			0.178394	−	0.983959i	$-0.442910\pi$
$271$	5.87348			0.356788			0.178394	+	0.983959i	$0.442910\pi$
$272$		−	4.70358i		−	0.285197i
$273$		0			0
$274$			5.16452i			0.312000i
$275$			19.6612i			1.18562i
$276$		0			0
$277$	15.4542			0.928553			0.464277	−	0.885690i	$-0.346314\pi$
$277$	15.4542			0.928553			0.464277	+	0.885690i	$0.346314\pi$
$278$	20.9890			1.25884
$279$		0			0
$280$			3.02986i			0.181069i
$281$	16.7794			1.00097			0.500487	−	0.865744i	$-0.333154\pi$
$281$	16.7794			1.00097			0.500487	+	0.865744i	$0.333154\pi$
$282$		0			0
$283$	−1.63300			−0.0970717			−0.0485358	−	0.998821i	$-0.515456\pi$
$283$	−1.63300			−0.0970717			−0.0485358	+	0.998821i	$0.515456\pi$
$284$	−2.20964			−0.131118
$285$		0			0
$286$	−29.4728			−1.74276
$287$	−8.40856			−0.496342
$288$		0			0
$289$	−5.12369			−0.301394
$290$		−	9.63510i		−	0.565793i
$291$		0			0
$292$	−13.2216			−0.773736
$293$	−29.7913			−1.74043			−0.870214	−	0.492673i	$-0.836020\pi$
$293$	−29.7913			−1.74043			−0.870214	+	0.492673i	$0.836020\pi$
$294$		0			0
$295$		0			0
$296$			1.81708i			0.105616i
$297$		0			0
$298$			0.921884i			0.0534033i
$299$	−2.88828			−0.167033
$300$		0			0
$301$	−4.86151			−0.280213
$302$		−	16.3725i		−	0.942134i
$303$		0			0
$304$	4.28832	+	0.781225i	0.245952	+	0.0448063i
$305$			16.5579i			0.948105i
$306$		0			0
$307$			12.2454i			0.698879i	0.936959	+	0.349440i	$0.113628\pi$
$307$			12.2454i			0.698879i	−0.936959	+	0.349440i	$0.886372\pi$
$308$			4.70358i			0.268012i
$309$		0			0
$310$	23.4312			1.33081
$311$			17.4798i			0.991190i	0.868554	+	0.495595i	$0.165050\pi$
$311$			17.4798i			0.991190i	−0.868554	+	0.495595i	$0.834950\pi$
$312$		0			0
$313$	−11.1533			−0.630421			−0.315210	−	0.949022i	$-0.602075\pi$
$313$	−11.1533			−0.630421			−0.315210	+	0.949022i	$0.602075\pi$
$314$	8.69343			0.490599
$315$		0			0
$316$			8.69159i			0.488940i
$317$	−4.01380			−0.225438			−0.112719	−	0.993627i	$-0.535956\pi$
$317$	−4.01380			−0.225438			−0.112719	+	0.993627i	$0.535956\pi$
$318$		0			0
$319$		−	14.9576i		−	0.837466i
$320$			3.02986i			0.169374i
$321$		0			0
$322$			0.460942i			0.0256873i
$323$	3.67456	−	20.1705i	0.204458	−	1.12231i
$324$		0			0
$325$		−	26.1923i		−	1.45289i
$326$	−8.12750			−0.450141
$327$		0			0
$328$	−8.40856			−0.464285
$329$		−	12.5321i		−	0.690915i
$330$		0			0
$331$		−	9.45548i		−	0.519720i	−0.965646	−	0.259860i	$-0.916324\pi$
$331$		−	9.45548i		−	0.519720i	0.965646	−	0.259860i	$-0.0836765\pi$
$332$			8.37401i			0.459583i
$333$		0			0
$334$	−23.1127			−1.26467
$335$	−36.5738			−1.99824
$336$		0			0
$337$			30.2932i			1.65018i	0.565004	+	0.825088i	$0.308875\pi$
$337$			30.2932i			1.65018i	−0.565004	+	0.825088i	$0.691125\pi$
$338$	26.2632			1.42853
$339$		0			0
$340$	14.2512			0.772880
$341$	36.3749			1.96981
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	−4.86151			−0.262115
$345$		0			0
$346$	6.29177			0.338248
$347$			13.6015i			0.730166i	0.930975	+	0.365083i	$0.118959\pi$
$347$			13.6015i			0.730166i	−0.930975	+	0.365083i	$0.881041\pi$
$348$		0			0
$349$	6.61820			0.354264			0.177132	−	0.984187i	$-0.443318\pi$
$349$	6.61820			0.354264			0.177132	+	0.984187i	$0.443318\pi$
$350$	−4.18005			−0.223433
$351$		0			0
$352$			4.70358i			0.250702i
$353$		−	6.20726i		−	0.330379i	−0.986262	−	0.165190i	$-0.947176\pi$
$353$		−	6.20726i		−	0.330379i	0.986262	−	0.165190i	$-0.0528236\pi$
$354$		0			0
$355$		−	6.69489i		−	0.355328i
$356$	−6.19892			−0.328542
$357$		0			0
$358$	4.15046			0.219359
$359$			24.1906i			1.27673i	0.769734	+	0.638365i	$0.220389\pi$
$359$			24.1906i			1.27673i	−0.769734	+	0.638365i	$0.779611\pi$
$360$		0			0
$361$	17.7794	+	6.70028i	0.935757	+	0.352647i
$362$		−	24.6677i		−	1.29651i
$363$		0			0
$364$		−	6.26603i		−	0.328429i
$365$		−	40.0596i		−	2.09682i
$366$		0			0
$367$	4.88447			0.254967			0.127484	−	0.991841i	$-0.459310\pi$
$367$	4.88447			0.254967			0.127484	+	0.991841i	$0.459310\pi$
$368$			0.460942i			0.0240283i
$369$		0			0
$370$	−5.50549			−0.286217
$371$	11.3897			0.591323
$372$		0			0
$373$			7.08083i			0.366631i	0.983054	+	0.183316i	$0.0586831\pi$
$373$			7.08083i			0.366631i	−0.983054	+	0.183316i	$0.941317\pi$
$374$	22.1237			1.14399
$375$		0			0
$376$		−	12.5321i		−	0.646292i
$377$			19.9263i			1.02626i
$378$		0			0
$379$		−	31.1134i		−	1.59819i	−0.601206	−	0.799094i	$-0.705313\pi$
$379$		−	31.1134i		−	1.59819i	0.601206	−	0.799094i	$-0.294687\pi$
$380$	−2.36700	+	12.9930i	−0.121425	+	0.666527i
$381$		0			0
$382$		−	1.51413i		−	0.0774699i
$383$	−23.6569			−1.20881			−0.604407	−	0.796676i	$-0.706590\pi$
$383$	−23.6569			−1.20881			−0.604407	+	0.796676i	$0.706590\pi$
$384$		0			0
$385$	−14.2512			−0.726308
$386$		−	13.3952i		−	0.681797i
$387$		0			0
$388$			9.50221i			0.482401i
$389$			0.731804i			0.0371039i	0.999828	+	0.0185520i	$0.00590561\pi$
$389$			0.731804i			0.0371039i	−0.999828	+	0.0185520i	$0.994094\pi$
$390$		0			0
$391$	2.16808			0.109644
$392$	−1.00000			−0.0505076
$393$		0			0
$394$			12.5321i			0.631356i
$395$	−26.3343			−1.32502
$396$		0			0
$397$	−5.59933			−0.281022			−0.140511	−	0.990079i	$-0.544875\pi$
$397$	−5.59933			−0.281022			−0.140511	+	0.990079i	$0.544875\pi$
$398$	−21.3453			−1.06994
$399$		0			0
$400$	−4.18005			−0.209002
$401$	−5.77274			−0.288277			−0.144139	−	0.989558i	$-0.546041\pi$
$401$	−5.77274			−0.288277			−0.144139	+	0.989558i	$0.546041\pi$
$402$		0			0
$403$	−48.4580			−2.41387
$404$		−	15.5619i		−	0.774235i
$405$		0			0
$406$	3.18005			0.157823
$407$	−8.54678			−0.423648
$408$		0			0
$409$		−	10.7779i		−	0.532935i	−0.963844	−	0.266468i	$-0.914143\pi$
$409$		−	10.7779i		−	0.532935i	0.963844	−	0.266468i	$-0.0858565\pi$
$410$		−	25.4768i		−	1.25821i
$411$		0			0
$412$		−	12.2307i		−	0.602564i
$413$		0			0
$414$		0			0
$415$	−25.3721			−1.24547
$416$		−	6.26603i		−	0.307218i
$417$		0			0
$418$	−3.67456	+	20.1705i	−0.179728	+	0.986570i
$419$			2.53684i			0.123933i	0.998078	+	0.0619663i	$0.0197371\pi$
$419$			2.53684i			0.123933i	−0.998078	+	0.0619663i	$0.980263\pi$
$420$		0			0
$421$		−	11.6956i		−	0.570011i	−0.958526	−	0.285005i	$-0.908005\pi$
$421$		−	11.6956i		−	0.570011i	0.958526	−	0.285005i	$-0.0919954\pi$
$422$		−	2.02339i		−	0.0984972i
$423$		0			0
$424$	11.3897			0.553132
$425$			19.6612i			0.953709i
$426$		0			0
$427$	−5.46492			−0.264466
$428$	−6.29177			−0.304124
$429$		0			0
$430$		−	14.7297i		−	0.710329i
$431$	35.4690			1.70848			0.854241	−	0.519878i	$-0.174023\pi$
$431$	35.4690			1.70848			0.854241	+	0.519878i	$0.174023\pi$
$432$		0			0
$433$			15.1493i			0.728029i	0.931393	+	0.364014i	$0.118594\pi$
$433$			15.1493i			0.728029i	−0.931393	+	0.364014i	$0.881406\pi$
$434$			7.73344i			0.371217i
$435$		0			0
$436$		−	17.9245i		−	0.858429i
$437$	−0.360100	+	1.97667i	−0.0172259	+	0.0945568i
$438$		0			0
$439$		−	25.5293i		−	1.21845i	−0.792999	−	0.609223i	$-0.791482\pi$
$439$		−	25.5293i		−	1.21845i	0.792999	−	0.609223i	$-0.208518\pi$
$440$	−14.2512			−0.679399
$441$		0			0
$442$	−29.4728			−1.40188
$443$		−	4.29096i		−	0.203869i	−0.994791	−	0.101935i	$-0.967497\pi$
$443$		−	4.29096i		−	0.203869i	0.994791	−	0.101935i	$-0.0325033\pi$
$444$		0			0
$445$		−	18.7819i		−	0.890346i
$446$			3.04609i			0.144237i
$447$		0			0
$448$	−1.00000			−0.0472456
$449$	11.4542			0.540557			0.270279	−	0.962782i	$-0.412884\pi$
$449$	11.4542			0.540557			0.270279	+	0.962782i	$0.412884\pi$
$450$		0			0
$451$		−	39.5504i		−	1.86235i
$452$	−18.0831			−0.850558
$453$		0			0
$454$	−7.89110			−0.370348
$455$	18.9852			0.890040
$456$		0			0
$457$	−2.04058			−0.0954541			−0.0477270	−	0.998860i	$-0.515198\pi$
$457$	−2.04058			−0.0954541			−0.0477270	+	0.998860i	$0.515198\pi$
$458$	−8.12369			−0.379595
$459$		0			0
$460$	−1.39659			−0.0651163
$461$			12.0244i			0.560032i	0.959995	+	0.280016i	$0.0903398\pi$
$461$			12.0244i			0.560032i	−0.959995	+	0.280016i	$0.909660\pi$
$462$		0			0
$463$	36.2484			1.68460			0.842302	−	0.539006i	$-0.181200\pi$
$463$	36.2484			1.68460			0.842302	+	0.539006i	$0.181200\pi$
$464$	3.18005			0.147630
$465$		0			0
$466$		−	12.2774i		−	0.568741i
$467$		−	25.4621i		−	1.17825i	−0.808043	−	0.589123i	$-0.799473\pi$
$467$		−	25.4621i		−	1.17825i	0.808043	−	0.589123i	$-0.200527\pi$
$468$		0			0
$469$		−	12.0711i		−	0.557393i
$470$	37.9704			1.75144
$471$		0			0
$472$		0			0
$473$		−	22.8665i		−	1.05140i
$474$		0			0
$475$	−17.9254	−	3.26556i	−0.822473	−	0.149834i
$476$			4.70358i			0.215588i
$477$		0			0
$478$		−	15.0647i		−	0.689044i
$479$		−	11.6102i		−	0.530483i	−0.964182	−	0.265241i	$-0.914548\pi$
$479$		−	11.6102i		−	0.530483i	0.964182	−	0.265241i	$-0.0854516\pi$
$480$		0			0
$481$	11.3859			0.519151
$482$			4.53354i			0.206497i
$483$		0			0
$484$	−11.1237			−0.505622
$485$	−28.7904			−1.30730
$486$		0			0
$487$		−	9.10422i		−	0.412552i	−0.978494	−	0.206276i	$-0.933866\pi$
$487$		−	9.10422i		−	0.412552i	0.978494	−	0.206276i	$-0.0661344\pi$
$488$	−5.46492			−0.247385
$489$		0			0
$490$		−	3.02986i		−	0.136875i
$491$		−	10.5408i		−	0.475698i	−0.971302	−	0.237849i	$-0.923558\pi$
$491$		−	10.5408i		−	0.475698i	0.971302	−	0.237849i	$-0.0764423\pi$
$492$		0			0
$493$		−	14.9576i		−	0.673658i
$494$	4.89518	−	26.8708i	0.220245	−	1.20897i
$495$		0			0
$496$			7.73344i			0.347242i
$497$	2.20964			0.0991158
$498$		0			0
$499$	34.4432			1.54189			0.770945	−	0.636902i	$-0.219784\pi$
$499$	34.4432			1.54189			0.770945	+	0.636902i	$0.219784\pi$
$500$			2.48433i			0.111103i
$501$		0			0
$502$		−	25.4033i		−	1.13381i
$503$		−	15.9174i		−	0.709720i	−0.934919	−	0.354860i	$-0.884528\pi$
$503$		−	15.9174i		−	0.709720i	0.934919	−	0.354860i	$-0.115472\pi$
$504$		0			0
$505$	47.1504			2.09817
$506$	−2.16808			−0.0963829
$507$		0			0
$508$		−	7.18792i		−	0.318912i
$509$	36.9436			1.63750			0.818749	−	0.574152i	$-0.194668\pi$
$509$	36.9436			1.63750			0.818749	+	0.574152i	$0.194668\pi$
$510$		0			0
$511$	13.2216			0.584890
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−23.9219			−1.05515
$515$	37.0573			1.63294
$516$		0			0
$517$	58.9456			2.59242
$518$		−	1.81708i		−	0.0798379i
$519$		0			0
$520$	18.9852			0.832556
$521$	−23.7952			−1.04248			−0.521242	−	0.853409i	$-0.674531\pi$
$521$	−23.7952			−1.04248			−0.521242	+	0.853409i	$0.674531\pi$
$522$		0			0
$523$			0.980657i			0.0428811i	0.999770	+	0.0214406i	$0.00682526\pi$
$523$			0.980657i			0.0428811i	−0.999770	+	0.0214406i	$0.993175\pi$
$524$			16.6314i			0.726544i
$525$		0			0
$526$			11.9398i			0.520601i
$527$	36.3749			1.58451
$528$		0			0
$529$	22.7875			0.990762
$530$			34.5092i			1.49898i
$531$		0			0
$532$	−4.28832	−	0.781225i	−0.185922	−	0.0338704i
$533$			52.6883i			2.28218i
$534$		0			0
$535$		−	19.0632i		−	0.824173i
$536$		−	12.0711i		−	0.521393i
$537$		0			0
$538$	7.91787			0.341363
$539$		−	4.70358i		−	0.202598i
$540$		0			0
$541$	−16.0869			−0.691631			−0.345816	−	0.938303i	$-0.612398\pi$
$541$	−16.0869			−0.691631			−0.345816	+	0.938303i	$0.612398\pi$
$542$	−5.87348			−0.252288
$543$		0			0
$544$			4.70358i			0.201664i
$545$	54.3088			2.32633
$546$		0			0
$547$		−	18.3534i		−	0.784734i	−0.919809	−	0.392367i	$-0.871656\pi$
$547$		−	18.3534i		−	0.784734i	0.919809	−	0.392367i	$-0.128344\pi$
$548$		−	5.16452i		−	0.220618i
$549$		0			0
$550$		−	19.6612i		−	0.838357i
$551$	13.6371	+	2.48433i	0.580959	+	0.105836i
$552$		0			0
$553$		−	8.69159i		−	0.369604i
$554$	−15.4542			−0.656586
$555$		0			0
$556$	−20.9890			−0.890132
$557$			29.1109i			1.23347i	0.787171	+	0.616735i	$0.211545\pi$
$557$			29.1109i			1.23347i	−0.787171	+	0.616735i	$0.788455\pi$
$558$		0			0
$559$			30.4624i			1.28842i
$560$		−	3.02986i		−	0.128035i
$561$		0			0
$562$	−16.7794			−0.707795
$563$	34.8625			1.46928			0.734639	−	0.678458i	$-0.237351\pi$
$563$	34.8625			1.46928			0.734639	+	0.678458i	$0.237351\pi$
$564$		0			0
$565$		−	54.7893i		−	2.30500i
$566$	1.63300			0.0686400
$567$		0			0
$568$	2.20964			0.0927143
$569$	−20.5736			−0.862488			−0.431244	−	0.902235i	$-0.641925\pi$
$569$	−20.5736			−0.862488			−0.431244	+	0.902235i	$0.641925\pi$
$570$		0			0
$571$	−18.5014			−0.774260			−0.387130	−	0.922025i	$-0.626534\pi$
$571$	−18.5014			−0.774260			−0.387130	+	0.922025i	$0.626534\pi$
$572$	29.4728			1.23232
$573$		0			0
$574$	8.40856			0.350967
$575$		−	1.92676i		−	0.0803515i
$576$		0			0
$577$	30.5392			1.27136			0.635681	−	0.771952i	$-0.280719\pi$
$577$	30.5392			1.27136			0.635681	+	0.771952i	$0.280719\pi$
$578$	5.12369			0.213117
$579$		0			0
$580$			9.63510i			0.400076i
$581$		−	8.37401i		−	0.347412i
$582$		0			0
$583$			53.5723i			2.21874i
$584$	13.2216			0.547114
$585$		0			0
$586$	29.7913			1.23067
$587$		−	13.3427i		−	0.550711i	−0.961342	−	0.275356i	$-0.911204\pi$
$587$		−	13.3427i		−	0.550711i	0.961342	−	0.275356i	$-0.0887956\pi$
$588$		0			0
$589$	−6.04156	+	33.1635i	−0.248938	+	1.36648i
$590$		0			0
$591$		0			0
$592$		−	1.81708i		−	0.0746815i
$593$			12.8938i			0.529484i	0.964319	+	0.264742i	$0.0852868\pi$
$593$			12.8938i			0.529484i	−0.964319	+	0.264742i	$0.914713\pi$
$594$		0			0
$595$	−14.2512			−0.584242
$596$		−	0.921884i		−	0.0377619i
$597$		0			0
$598$	2.88828			0.118110
$599$	1.48944			0.0608568			0.0304284	−	0.999537i	$-0.490313\pi$
$599$	1.48944			0.0608568			0.0304284	+	0.999537i	$0.490313\pi$
$600$		0			0
$601$		−	30.1603i		−	1.23026i	−0.788424	−	0.615132i	$-0.789103\pi$
$601$		−	30.1603i		−	1.23026i	0.788424	−	0.615132i	$-0.210897\pi$
$602$	4.86151			0.198140
$603$		0			0
$604$			16.3725i			0.666189i
$605$		−	33.7032i		−	1.37023i
$606$		0			0
$607$		−	9.64975i		−	0.391671i	−0.980637	−	0.195836i	$-0.937258\pi$
$607$		−	9.64975i		−	0.391671i	0.980637	−	0.195836i	$-0.0627419\pi$
$608$	−4.28832	−	0.781225i	−0.173914	−	0.0316829i
$609$		0			0
$610$		−	16.5579i		−	0.670411i
$611$	−78.5263			−3.17684
$612$		0			0
$613$	44.5401			1.79896			0.899480	−	0.436963i	$-0.143946\pi$
$613$	44.5401			1.79896			0.899480	+	0.436963i	$0.143946\pi$
$614$		−	12.2454i		−	0.494182i
$615$		0			0
$616$		−	4.70358i		−	0.189513i
$617$		−	22.9603i		−	0.924349i	−0.886789	−	0.462174i	$-0.847069\pi$
$617$		−	22.9603i		−	0.924349i	0.886789	−	0.462174i	$-0.152931\pi$
$618$		0			0
$619$	2.36293			0.0949741			0.0474871	−	0.998872i	$-0.484879\pi$
$619$	2.36293			0.0949741			0.0474871	+	0.998872i	$0.484879\pi$
$620$	−23.4312			−0.941021
$621$		0			0
$622$		−	17.4798i		−	0.700877i
$623$	6.19892			0.248355
$624$		0			0
$625$	−28.4274			−1.13710
$626$	11.1533			0.445775
$627$		0			0
$628$	−8.69343			−0.346906
$629$	−8.54678			−0.340782
$630$		0			0
$631$	10.2234			0.406989			0.203494	−	0.979076i	$-0.434770\pi$
$631$	10.2234			0.406989			0.203494	+	0.979076i	$0.434770\pi$
$632$		−	8.69159i		−	0.345733i
$633$		0			0
$634$	4.01380			0.159409
$635$	21.7784			0.864249
$636$		0			0
$637$			6.26603i			0.248269i
$638$			14.9576i			0.592178i
$639$		0			0
$640$		−	3.02986i		−	0.119766i
$641$	−41.2603			−1.62969			−0.814843	−	0.579682i	$-0.803177\pi$
$641$	−41.2603			−1.62969			−0.814843	+	0.579682i	$0.803177\pi$
$642$		0			0
$643$	−46.7765			−1.84469			−0.922343	−	0.386371i	$-0.873728\pi$
$643$	−46.7765			−1.84469			−0.922343	+	0.386371i	$0.873728\pi$
$644$		−	0.460942i		−	0.0181637i
$645$		0			0
$646$	−3.67456	+	20.1705i	−0.144573	+	0.793596i
$647$			10.1065i			0.397328i	0.980068	+	0.198664i	$0.0636602\pi$
$647$			10.1065i			0.397328i	−0.980068	+	0.198664i	$0.936340\pi$
$648$		0			0
$649$		0			0
$650$			26.1923i			1.02735i
$651$		0			0
$652$	8.12750			0.318298
$653$			5.13784i			0.201059i	0.994934	+	0.100530i	$0.0320537\pi$
$653$			5.13784i			0.201059i	−0.994934	+	0.100530i	$0.967946\pi$
$654$		0			0
$655$	−50.3907			−1.96893
$656$	8.40856			0.328299
$657$		0			0
$658$			12.5321i			0.488551i
$659$	43.3639			1.68922			0.844609	−	0.535384i	$-0.179833\pi$
$659$	43.3639			1.68922			0.844609	+	0.535384i	$0.179833\pi$
$660$		0			0
$661$		−	44.0314i		−	1.71262i	−0.516460	−	0.856311i	$-0.672750\pi$
$661$		−	44.0314i		−	1.71262i	0.516460	−	0.856311i	$-0.327250\pi$
$662$			9.45548i			0.367498i
$663$		0			0
$664$		−	8.37401i		−	0.324975i
$665$	2.36700	−	12.9930i	0.0917884	−	0.503847i
$666$		0			0
$667$			1.46582i			0.0567567i
$668$	23.1127			0.894257
$669$		0			0
$670$	36.5738			1.41297
$671$		−	25.7047i		−	0.992319i
$672$		0			0
$673$		−	21.8133i		−	0.840842i	−0.907329	−	0.420421i	$-0.861882\pi$
$673$		−	21.8133i		−	0.840842i	0.907329	−	0.420421i	$-0.138118\pi$
$674$		−	30.2932i		−	1.16685i
$675$		0			0
$676$	−26.2632			−1.01012
$677$	−8.81614			−0.338832			−0.169416	−	0.985545i	$-0.554188\pi$
$677$	−8.81614			−0.338832			−0.169416	+	0.985545i	$0.554188\pi$
$678$		0			0
$679$		−	9.50221i		−	0.364661i
$680$	−14.2512			−0.546508
$681$		0			0
$682$	−36.3749			−1.39287
$683$	0.254026			0.00972003			0.00486002	−	0.999988i	$-0.498453\pi$
$683$	0.254026			0.00972003			0.00486002	+	0.999988i	$0.498453\pi$
$684$		0			0
$685$	15.6478			0.597871
$686$	1.00000			0.0381802
$687$		0			0
$688$	4.86151			0.185343
$689$		−	71.3682i		−	2.71891i
$690$		0			0
$691$	−12.4331			−0.472977			−0.236488	−	0.971634i	$-0.575997\pi$
$691$	−12.4331			−0.472977			−0.236488	+	0.971634i	$0.575997\pi$
$692$	−6.29177			−0.239177
$693$		0			0
$694$		−	13.6015i		−	0.516306i
$695$		−	63.5938i		−	2.41225i
$696$		0			0
$697$		−	39.5504i		−	1.49808i
$698$	−6.61820			−0.250502
$699$		0			0
$700$	4.18005			0.157991
$701$		−	23.1215i		−	0.873288i	−0.899634	−	0.436644i	$-0.856167\pi$
$701$		−	23.1215i		−	0.873288i	0.899634	−	0.436644i	$-0.143833\pi$
$702$		0			0
$703$	1.41955	−	7.79221i	0.0535392	−	0.293889i
$704$		−	4.70358i		−	0.177273i
$705$		0			0
$706$			6.20726i			0.233613i
$707$			15.5619i			0.585266i
$708$		0			0
$709$	−18.3425			−0.688866			−0.344433	−	0.938811i	$-0.611929\pi$
$709$	−18.3425			−0.688866			−0.344433	+	0.938811i	$0.611929\pi$
$710$			6.69489i			0.251255i
$711$		0			0
$712$	6.19892			0.232314
$713$	−3.56467			−0.133498
$714$		0			0
$715$			89.2985i			3.33957i
$716$	−4.15046			−0.155110
$717$		0			0
$718$		−	24.1906i		−	0.902784i
$719$			35.7903i			1.33475i	0.744721	+	0.667376i	$0.232583\pi$
$719$			35.7903i			1.33475i	−0.744721	+	0.667376i	$0.767417\pi$
$720$		0			0
$721$			12.2307i			0.455496i
$722$	−17.7794	−	6.70028i	−0.661680	−	0.249359i
$723$		0			0
$724$			24.6677i			0.916769i
$725$	−13.2928			−0.493681
$726$		0			0
$727$	3.98620			0.147840			0.0739199	−	0.997264i	$-0.476449\pi$
$727$	3.98620			0.147840			0.0739199	+	0.997264i	$0.476449\pi$
$728$			6.26603i			0.232235i
$729$		0			0
$730$			40.0596i			1.48267i
$731$		−	22.8665i		−	0.845748i
$732$		0			0
$733$	−28.3274			−1.04630			−0.523148	−	0.852242i	$-0.675243\pi$
$733$	−28.3274			−1.04630			−0.523148	+	0.852242i	$0.675243\pi$
$734$	−4.88447			−0.180289
$735$		0			0
$736$		−	0.460942i		−	0.0169906i
$737$	56.7775			2.09143
$738$		0			0
$739$	−48.4794			−1.78334			−0.891672	−	0.452681i	$-0.850468\pi$
$739$	−48.4794			−1.78334			−0.891672	+	0.452681i	$0.850468\pi$
$740$	5.50549			0.202386
$741$		0			0
$742$	−11.3897			−0.418129
$743$	9.17624			0.336643			0.168322	−	0.985732i	$-0.446165\pi$
$743$	9.17624			0.336643			0.168322	+	0.985732i	$0.446165\pi$
$744$		0			0
$745$	2.79318			0.102334
$746$		−	7.08083i		−	0.259248i
$747$		0			0
$748$	−22.1237			−0.808922
$749$	6.29177			0.229896
$750$		0			0
$751$		−	33.8848i		−	1.23647i	−0.785991	−	0.618237i	$-0.787847\pi$
$751$		−	33.8848i		−	1.23647i	0.785991	−	0.618237i	$-0.212153\pi$
$752$			12.5321i			0.456997i
$753$		0			0
$754$		−	19.9263i		−	0.725673i
$755$	−49.6065			−1.80536
$756$		0			0
$757$	15.3175			0.556726			0.278363	−	0.960476i	$-0.410208\pi$
$757$	15.3175			0.556726			0.278363	+	0.960476i	$0.410208\pi$
$758$			31.1134i			1.13009i
$759$		0			0
$760$	2.36700	−	12.9930i	0.0858602	−	0.471306i
$761$			21.8967i			0.793754i	0.917872	+	0.396877i	$0.129906\pi$
$761$			21.8967i			0.793754i	−0.917872	+	0.396877i	$0.870094\pi$
$762$		0			0
$763$			17.9245i			0.648912i
$764$			1.51413i			0.0547795i
$765$		0			0
$766$	23.6569			0.854760
$767$		0			0
$768$		0			0
$769$	6.43308			0.231983			0.115991	−	0.993250i	$-0.462995\pi$
$769$	6.43308			0.231983			0.115991	+	0.993250i	$0.462995\pi$
$770$	14.2512			0.513577
$771$		0			0
$772$			13.3952i			0.482103i
$773$	−5.68527			−0.204485			−0.102243	−	0.994760i	$-0.532602\pi$
$773$	−5.68527			−0.204485			−0.102243	+	0.994760i	$0.532602\pi$
$774$		0			0
$775$		−	32.3262i		−	1.16119i
$776$		−	9.50221i		−	0.341109i
$777$		0			0
$778$		−	0.731804i		−	0.0262364i
$779$	36.0586	+	6.56898i	1.29193	+	0.235358i
$780$		0			0
$781$			10.3932i			0.371898i
$782$	−2.16808			−0.0775304
$783$		0			0
$784$	1.00000			0.0357143
$785$		−	26.3399i		−	0.940110i
$786$		0			0
$787$			1.08566i			0.0386995i	0.999813	+	0.0193498i	$0.00615961\pi$
$787$			1.08566i			0.0386995i	−0.999813	+	0.0193498i	$0.993840\pi$
$788$		−	12.5321i		−	0.446436i
$789$		0			0
$790$	26.3343			0.936933
$791$	18.0831			0.642962
$792$		0			0
$793$			34.2434i			1.21602i
$794$	5.59933			0.198713
$795$		0			0
$796$	21.3453			0.756564
$797$	−40.9456			−1.45037			−0.725184	−	0.688555i	$-0.758245\pi$
$797$	−40.9456			−1.45037			−0.725184	+	0.688555i	$0.758245\pi$
$798$		0			0
$799$	58.9456			2.08535
$800$	4.18005			0.147787
$801$		0			0
$802$	5.77274			0.203843
$803$			62.1889i			2.19460i
$804$		0			0
$805$	1.39659			0.0492233
$806$	48.4580			1.70686
$807$		0			0
$808$			15.5619i			0.547467i
$809$		−	18.5063i		−	0.650648i	−0.945603	−	0.325324i	$-0.894527\pi$
$809$		−	18.5063i		−	0.650648i	0.945603	−	0.325324i	$-0.105473\pi$
$810$		0			0
$811$			6.11310i			0.214660i	0.994223	+	0.107330i	$0.0342301\pi$
$811$			6.11310i			0.214660i	−0.994223	+	0.107330i	$0.965770\pi$
$812$	−3.18005			−0.111598
$813$		0			0
$814$	8.54678			0.299564
$815$			24.6252i			0.862583i
$816$		0			0
$817$	20.8477	+	3.79793i	0.729369	+	0.132873i
$818$			10.7779i			0.376842i
$819$		0			0
$820$			25.4768i			0.889687i
$821$		−	8.06726i		−	0.281549i	−0.990042	−	0.140775i	$-0.955041\pi$
$821$		−	8.06726i		−	0.281549i	0.990042	−	0.140775i	$-0.0449593\pi$
$822$		0			0
$823$	45.4231			1.58335			0.791674	−	0.610943i	$-0.209210\pi$
$823$	45.4231			1.58335			0.791674	+	0.610943i	$0.209210\pi$
$824$			12.2307i			0.426077i
$825$		0			0
$826$		0			0
$827$	−14.5244			−0.505062			−0.252531	−	0.967589i	$-0.581263\pi$
$827$	−14.5244			−0.505062			−0.252531	+	0.967589i	$0.581263\pi$
$828$		0			0
$829$			49.5418i			1.72066i	0.509740	+	0.860329i	$0.329742\pi$
$829$			49.5418i			1.72066i	−0.509740	+	0.860329i	$0.670258\pi$
$830$	25.3721			0.880677
$831$		0			0
$832$			6.26603i			0.217236i
$833$		−	4.70358i		−	0.162969i
$834$		0			0
$835$			70.0283i			2.42343i
$836$	3.67456	−	20.1705i	0.127087	−	0.697610i
$837$		0			0
$838$		−	2.53684i		−	0.0876335i
$839$	−8.02550			−0.277071			−0.138536	−	0.990357i	$-0.544240\pi$
$839$	−8.02550			−0.277071			−0.138536	+	0.990357i	$0.544240\pi$
$840$		0			0
$841$	−18.8873			−0.651286
$842$			11.6956i			0.403058i
$843$		0			0
$844$			2.02339i			0.0696480i
$845$		−	79.5737i		−	2.73742i
$846$		0			0
$847$	11.1237			0.382215
$848$	−11.3897			−0.391123
$849$		0			0
$850$		−	19.6612i		−	0.674374i
$851$	0.837568			0.0287115
$852$		0			0
$853$	−20.7608			−0.710835			−0.355418	−	0.934708i	$-0.615661\pi$
$853$	−20.7608			−0.710835			−0.355418	+	0.934708i	$0.615661\pi$
$854$	5.46492			0.187006
$855$		0			0
$856$	6.29177			0.215048
$857$	55.5858			1.89877			0.949387	−	0.314109i	$-0.101706\pi$
$857$	55.5858			1.89877			0.949387	+	0.314109i	$0.101706\pi$
$858$		0			0
$859$	11.2474			0.383756			0.191878	−	0.981419i	$-0.438542\pi$
$859$	11.2474			0.383756			0.191878	+	0.981419i	$0.438542\pi$
$860$			14.7297i			0.502278i
$861$		0			0
$862$	−35.4690			−1.20808
$863$	14.1376			0.481251			0.240625	−	0.970618i	$-0.422647\pi$
$863$	14.1376			0.481251			0.240625	+	0.970618i	$0.422647\pi$
$864$		0			0
$865$		−	19.0632i		−	0.648168i
$866$		−	15.1493i		−	0.514794i
$867$		0			0
$868$		−	7.73344i		−	0.262490i
$869$	40.8816			1.38681
$870$		0			0
$871$	−75.6381			−2.56290
$872$			17.9245i			0.607001i
$873$		0			0
$874$	0.360100	−	1.97667i	0.0121805	−	0.0668618i
$875$		−	2.48433i		−	0.0839858i
$876$		0			0
$877$			32.5954i			1.10067i	0.834944	+	0.550335i	$0.185500\pi$
$877$			32.5954i			1.10067i	−0.834944	+	0.550335i	$0.814500\pi$
$878$			25.5293i			0.861571i
$879$		0			0
$880$	14.2512			0.480408
$881$			9.16299i			0.308709i	0.988016	+	0.154355i	$0.0493298\pi$
$881$			9.16299i			0.308709i	−0.988016	+	0.154355i	$0.950670\pi$
$882$		0			0
$883$	23.3491			0.785760			0.392880	−	0.919590i	$-0.371479\pi$
$883$	23.3491			0.785760			0.392880	+	0.919590i	$0.371479\pi$
$884$	29.4728			0.991278
$885$		0			0
$886$			4.29096i			0.144157i
$887$	−28.4741			−0.956065			−0.478033	−	0.878342i	$-0.658650\pi$
$887$	−28.4741			−0.956065			−0.478033	+	0.878342i	$0.658650\pi$
$888$		0			0
$889$			7.18792i			0.241075i
$890$			18.7819i			0.629570i
$891$		0			0
$892$		−	3.04609i		−	0.101991i
$893$	−9.79036	+	53.7415i	−0.327622	+	1.79839i
$894$		0			0
$895$		−	12.5753i		−	0.420346i
$896$	1.00000			0.0334077
$897$		0			0
$898$	−11.4542			−0.382232
$899$			24.5927i			0.820214i
$900$		0			0
$901$			53.5723i			1.78475i
$902$			39.5504i			1.31688i
$903$		0			0
$904$	18.0831			0.601436
$905$	−74.7398			−2.48443
$906$		0			0
$907$		−	46.5749i		−	1.54649i	−0.634105	−	0.773247i	$-0.718631\pi$
$907$		−	46.5749i		−	1.54649i	0.634105	−	0.773247i	$-0.281369\pi$
$908$	7.89110			0.261875
$909$		0			0
$910$	−18.9852			−0.629353
$911$	−12.0831			−0.400332			−0.200166	−	0.979762i	$-0.564148\pi$
$911$	−12.0831			−0.400332			−0.200166	+	0.979762i	$0.564148\pi$
$912$		0			0
$913$	39.3878			1.30355
$914$	2.04058			0.0674962
$915$		0			0
$916$	8.12369			0.268414
$917$		−	16.6314i		−	0.549216i
$918$		0			0
$919$	49.5735			1.63528			0.817640	−	0.575730i	$-0.195282\pi$
$919$	49.5735			1.63528			0.817640	+	0.575730i	$0.195282\pi$
$920$	1.39659			0.0460442
$921$		0			0
$922$		−	12.0244i		−	0.396002i
$923$		−	13.8457i		−	0.455736i
$924$		0			0
$925$			7.59548i			0.249738i
$926$	−36.2484			−1.19120
$927$		0			0
$928$	−3.18005			−0.104390
$929$		−	46.3466i		−	1.52058i	−0.649583	−	0.760291i	$-0.725056\pi$
$929$		−	46.3466i		−	1.52058i	0.649583	−	0.760291i	$-0.274944\pi$
$930$		0			0
$931$	4.28832	+	0.781225i	0.140544	+	0.0256036i
$932$			12.2774i			0.402161i
$933$		0			0
$934$			25.4621i			0.833146i
$935$		−	67.0317i		−	2.19217i
$936$		0			0
$937$	19.6169			0.640858			0.320429	−	0.947273i	$-0.396173\pi$
$937$	19.6169			0.640858			0.320429	+	0.947273i	$0.396173\pi$
$938$			12.0711i			0.394136i
$939$		0			0
$940$	−37.9704			−1.23846
$941$	4.69626			0.153094			0.0765468	−	0.997066i	$-0.475611\pi$
$941$	4.69626			0.153094			0.0765468	+	0.997066i	$0.475611\pi$
$942$		0			0
$943$			3.87586i			0.126215i
$944$		0			0
$945$		0			0
$946$			22.8665i			0.743454i
$947$			44.3127i			1.43997i	0.693990	+	0.719985i	$0.255851\pi$
$947$			44.3127i			1.43997i	−0.693990	+	0.719985i	$0.744149\pi$
$948$		0			0
$949$		−	82.8470i		−	2.68933i
$950$	17.9254	+	3.26556i	0.581576	+	0.105949i
$951$		0			0
$952$		−	4.70358i		−	0.152444i
$953$	−59.4007			−1.92418			−0.962088	−	0.272739i	$-0.912071\pi$
$953$	−59.4007			−1.92418			−0.962088	+	0.272739i	$0.912071\pi$
$954$		0			0
$955$	−4.58761			−0.148452
$956$			15.0647i			0.487228i
$957$		0			0
$958$			11.6102i			0.375108i
$959$			5.16452i			0.166771i
$960$		0			0
$961$	−28.8061			−0.929230
$962$	−11.3859			−0.367095
$963$		0			0
$964$		−	4.53354i		−	0.146015i
$965$	−40.5855			−1.30649
$966$		0			0
$967$	−16.6381			−0.535044			−0.267522	−	0.963552i	$-0.586205\pi$
$967$	−16.6381			−0.535044			−0.267522	+	0.963552i	$0.586205\pi$
$968$	11.1237			0.357529
$969$		0			0
$970$	28.7904			0.924402
$971$	28.4608			0.913352			0.456676	−	0.889633i	$-0.349040\pi$
$971$	28.4608			0.913352			0.456676	+	0.889633i	$0.349040\pi$
$972$		0			0
$973$	20.9890			0.672877
$974$			9.10422i			0.291718i
$975$		0			0
$976$	5.46492			0.174928
$977$	2.84291			0.0909527			0.0454763	−	0.998965i	$-0.485519\pi$
$977$	2.84291			0.0909527			0.0454763	+	0.998965i	$0.485519\pi$
$978$		0			0
$979$			29.1571i			0.931866i
$980$			3.02986i			0.0967853i
$981$		0			0
$982$			10.5408i			0.336369i
$983$	−3.42053			−0.109098			−0.0545490	−	0.998511i	$-0.517372\pi$
$983$	−3.42053			−0.109098			−0.0545490	+	0.998511i	$0.517372\pi$
$984$		0			0
$985$	37.9704			1.20984
$986$			14.9576i			0.476348i
$987$		0			0
$988$	−4.89518	+	26.8708i	−0.155736	+	0.854873i
$989$			2.24087i			0.0712557i
$990$		0			0
$991$		−	11.2347i		−	0.356882i	−0.983951	−	0.178441i	$-0.942895\pi$
$991$		−	11.2347i		−	0.356882i	0.983951	−	0.178441i	$-0.0571054\pi$
$992$		−	7.73344i		−	0.245537i
$993$		0			0
$994$	−2.20964			−0.0700855
$995$			64.6733i			2.05028i
$996$		0			0
$997$	44.2068			1.40004			0.700022	−	0.714121i	$-0.253174\pi$
$997$	44.2068			1.40004			0.700022	+	0.714121i	$0.253174\pi$
$998$	−34.4432			−1.09028
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.b.i.1709.7	yes	8
3.2	odd	2		2394.2.b.j.1709.2	yes	8
19.18	odd	2		2394.2.b.j.1709.7	yes	8
57.56	even	2	inner	2394.2.b.i.1709.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.b.i.1709.2	✓	8	57.56	even	2	inner
2394.2.b.i.1709.7	yes	8	1.1	even	1	trivial
2394.2.b.j.1709.2	yes	8	3.2	odd	2
2394.2.b.j.1709.7	yes	8	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.02986i q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.02986i q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.02986i q^{10} -4.70358i q^{11} +6.26603i q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.70358i q^{17} +(4.28832 + 0.781225i) q^{19} +3.02986i q^{20} +4.70358i q^{22} +0.460942i q^{23} -4.18005 q^{25} -6.26603i q^{26} -1.00000 q^{28} +3.18005 q^{29} +7.73344i q^{31} -1.00000 q^{32} +4.70358i q^{34} -3.02986i q^{35} -1.81708i q^{37} +(-4.28832 - 0.781225i) q^{38} -3.02986i q^{40} +8.40856 q^{41} +4.86151 q^{43} -4.70358i q^{44} -0.460942i q^{46} +12.5321i q^{47} +1.00000 q^{49} +4.18005 q^{50} +6.26603i q^{52} -11.3897 q^{53} +14.2512 q^{55} +1.00000 q^{56} -3.18005 q^{58} +5.46492 q^{61} -7.73344i q^{62} +1.00000 q^{64} -18.9852 q^{65} +12.0711i q^{67} -4.70358i q^{68} +3.02986i q^{70} -2.20964 q^{71} -13.2216 q^{73} +1.81708i q^{74} +(4.28832 + 0.781225i) q^{76} +4.70358i q^{77} +8.69159i q^{79} +3.02986i q^{80} -8.40856 q^{82} +8.37401i q^{83} +14.2512 q^{85} -4.86151 q^{86} +4.70358i q^{88} -6.19892 q^{89} -6.26603i q^{91} +0.460942i q^{92} -12.5321i q^{94} +(-2.36700 + 12.9930i) q^{95} +9.50221i q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8} + 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} + 16 q^{29} - 8 q^{32} - 12 q^{38} - 24 q^{41} - 32 q^{43} + 8 q^{49} + 24 q^{50} - 48 q^{53} + 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} - 16 q^{65} + 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} + 32 q^{86} + 8 q^{89} - 8 q^{95} - 8 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^7 - 8 * q^8 + 8 * q^14 + 8 * q^16 + 12 * q^19 - 24 * q^25 - 8 * q^28 + 16 * q^29 - 8 * q^32 - 12 * q^38 - 24 * q^41 - 32 * q^43 + 8 * q^49 + 24 * q^50 - 48 * q^53 + 8 * q^56 - 16 * q^58 + 8 * q^61 + 8 * q^64 - 16 * q^65 + 16 * q^71 - 16 * q^73 + 12 * q^76 + 24 * q^82 + 32 * q^86 + 8 * q^89 - 8 * q^95 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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