LMFDB - Embedded newform 1840.2.i.b.1471.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561 $ x^16 - 8x^14 + 33x^12 - 98x^10 + 272x^8 - 882x^6 + 2673x^4 - 5832*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.6
Root			$-1.59950 - 0.664536i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.b.1471.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32907i q^{3} +1.00000i q^{5} -3.37473 q^{7} +1.23357 q^{9} +O(q^{10})$	$q-1.32907i q^{3} +1.00000i q^{5} -3.37473 q^{7} +1.23357 q^{9} +1.86992 q^{11} -4.25170 q^{13} +1.32907 q^{15} +3.38881i q^{17} +4.52807 q^{19} +4.48526i q^{21} +(3.47789 - 3.30216i) q^{23} -1.00000 q^{25} -5.62672i q^{27} +3.15524 q^{29} -2.04566i q^{31} -2.48526i q^{33} -3.37473i q^{35} +5.40694i q^{37} +5.65081i q^{39} +10.1258 q^{41} +2.07625 q^{43} +1.23357i q^{45} -5.06892i q^{47} +4.38881 q^{49} +4.50397 q^{51} -11.8922i q^{53} +1.86992i q^{55} -6.01813i q^{57} -11.1187i q^{59} +4.01813i q^{61} -4.16295 q^{63} -4.25170i q^{65} +5.68140 q^{67} +(-4.38881 - 4.62237i) q^{69} +1.69419i q^{71} +7.02931 q^{73} +1.32907i q^{75} -6.31048 q^{77} +4.52807 q^{79} -3.77762 q^{81} +3.81148 q^{83} -3.38881 q^{85} -4.19355i q^{87} +4.32861i q^{89} +14.3483 q^{91} -2.71883 q^{93} +4.52807i q^{95} +8.95239i q^{97} +2.30667 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.32907i		−	0.767340i	−0.923470	−	0.383670i	$-0.874660\pi$
$3$		−	1.32907i		−	0.767340i	0.923470	−	0.383670i	$-0.125340\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−3.37473			−1.27553			−0.637764	−	0.770232i	$-0.720141\pi$
$7$	−3.37473			−1.27553			−0.637764	+	0.770232i	$0.720141\pi$
$8$		0			0
$9$	1.23357			0.411189
$10$		0			0
$11$	1.86992			0.563803			0.281901	−	0.959443i	$-0.409035\pi$
$11$	1.86992			0.563803			0.281901	+	0.959443i	$0.409035\pi$
$12$		0			0
$13$	−4.25170			−1.17921			−0.589604	−	0.807692i	$-0.700716\pi$
$13$	−4.25170			−1.17921			−0.589604	+	0.807692i	$0.700716\pi$
$14$		0			0
$15$	1.32907			0.343165
$16$		0			0
$17$			3.38881i			0.821907i	0.911656	+	0.410953i	$0.134804\pi$
$17$			3.38881i			0.821907i	−0.911656	+	0.410953i	$0.865196\pi$
$18$		0			0
$19$	4.52807			1.03881			0.519405	−	0.854528i	$-0.326154\pi$
$19$	4.52807			1.03881			0.519405	+	0.854528i	$0.326154\pi$
$20$		0			0
$21$			4.48526i			0.978764i
$22$		0			0
$23$	3.47789	−	3.30216i	0.725191	−	0.688548i
$23$	3.47789	−	3.30216i	0.725191	−	0.688548i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	5.62672i		−	1.08286i
$28$		0			0
$29$	3.15524			0.585914			0.292957	−	0.956126i	$-0.405361\pi$
$29$	3.15524			0.585914			0.292957	+	0.956126i	$0.405361\pi$
$30$		0			0
$31$		−	2.04566i		−	0.367411i	−0.982981	−	0.183706i	$-0.941191\pi$
$31$		−	2.04566i		−	0.367411i	0.982981	−	0.183706i	$-0.0588093\pi$
$32$		0			0
$33$		−	2.48526i		−	0.432629i
$34$		0			0
$35$		−	3.37473i		−	0.570434i
$36$		0			0
$37$			5.40694i			0.888895i	0.895805	+	0.444448i	$0.146600\pi$
$37$			5.40694i			0.888895i	−0.895805	+	0.444448i	$0.853400\pi$
$38$		0			0
$39$			5.65081i			0.904854i
$40$		0			0
$41$	10.1258			1.58138			0.790689	−	0.612217i	$-0.209722\pi$
$41$	10.1258			1.58138			0.790689	+	0.612217i	$0.209722\pi$
$42$		0			0
$43$	2.07625			0.316625			0.158313	−	0.987389i	$-0.449395\pi$
$43$	2.07625			0.316625			0.158313	+	0.987389i	$0.449395\pi$
$44$		0			0
$45$			1.23357i			0.183889i
$46$		0			0
$47$		−	5.06892i		−	0.739377i	−0.929156	−	0.369689i	$-0.879464\pi$
$47$		−	5.06892i		−	0.739377i	0.929156	−	0.369689i	$-0.120536\pi$
$48$		0			0
$49$	4.38881			0.626973
$50$		0			0
$51$	4.50397			0.630682
$52$		0			0
$53$		−	11.8922i		−	1.63352i	−0.576978	−	0.816760i	$-0.695768\pi$
$53$		−	11.8922i		−	1.63352i	0.576978	−	0.816760i	$-0.304232\pi$
$54$		0			0
$55$			1.86992i			0.252140i
$56$		0			0
$57$		−	6.01813i		−	0.797121i
$58$		0			0
$59$		−	11.1187i		−	1.44754i	−0.690043	−	0.723769i	$-0.742408\pi$
$59$		−	11.1187i		−	1.44754i	0.690043	−	0.723769i	$-0.257592\pi$
$60$		0			0
$61$			4.01813i			0.514469i	0.966349	+	0.257234i	$0.0828112\pi$
$61$			4.01813i			0.514469i	−0.966349	+	0.257234i	$0.917189\pi$
$62$		0			0
$63$	−4.16295			−0.524483
$64$		0			0
$65$		−	4.25170i		−	0.527358i
$66$		0			0
$67$	5.68140			0.694094			0.347047	−	0.937848i	$-0.387184\pi$
$67$	5.68140			0.694094			0.347047	+	0.937848i	$0.387184\pi$
$68$		0			0
$69$	−4.38881	−	4.62237i	−0.528350	−	0.556468i
$70$		0			0
$71$			1.69419i			0.201063i	0.994934	+	0.100531i	$0.0320543\pi$
$71$			1.69419i			0.201063i	−0.994934	+	0.100531i	$0.967946\pi$
$72$		0			0
$73$	7.02931			0.822719			0.411359	−	0.911473i	$-0.365054\pi$
$73$	7.02931			0.822719			0.411359	+	0.911473i	$0.365054\pi$
$74$		0			0
$75$			1.32907i			0.153468i
$76$		0			0
$77$	−6.31048			−0.719146
$78$		0			0
$79$	4.52807			0.509447			0.254724	−	0.967014i	$-0.418015\pi$
$79$	4.52807			0.509447			0.254724	+	0.967014i	$0.418015\pi$
$80$		0			0
$81$	−3.77762			−0.419735
$82$		0			0
$83$	3.81148			0.418364			0.209182	−	0.977877i	$-0.432920\pi$
$83$	3.81148			0.418364			0.209182	+	0.977877i	$0.432920\pi$
$84$		0			0
$85$	−3.38881			−0.367568
$86$		0			0
$87$		−	4.19355i		−	0.449595i
$88$		0			0
$89$			4.32861i			0.458832i	0.973328	+	0.229416i	$0.0736816\pi$
$89$			4.32861i			0.458832i	−0.973328	+	0.229416i	$0.926318\pi$
$90$		0			0
$91$	14.3483			1.50411
$92$		0			0
$93$	−2.71883			−0.281929
$94$		0			0
$95$			4.52807i			0.464570i
$96$		0			0
$97$			8.95239i			0.908978i	0.890752	+	0.454489i	$0.150178\pi$
$97$			8.95239i			0.908978i	−0.890752	+	0.454489i	$0.849822\pi$
$98$		0			0
$99$	2.30667			0.231829
$100$		0			0
$101$	11.8922			1.18332			0.591659	−	0.806188i	$-0.298473\pi$
$101$	11.8922			1.18332			0.591659	+	0.806188i	$0.298473\pi$
$102$		0			0
$103$	4.29764			0.423459			0.211730	−	0.977328i	$-0.432090\pi$
$103$	4.29764			0.423459			0.211730	+	0.977328i	$0.432090\pi$
$104$		0			0
$105$	−4.48526			−0.437717
$106$		0			0
$107$	−2.29303			−0.221676			−0.110838	−	0.993839i	$-0.535353\pi$
$107$	−2.29303			−0.221676			−0.110838	+	0.993839i	$0.535353\pi$
$108$		0			0
$109$		−	6.79575i		−	0.650914i	−0.945557	−	0.325457i	$-0.894482\pi$
$109$		−	6.79575i		−	0.650914i	0.945557	−	0.325457i	$-0.105518\pi$
$110$		0			0
$111$	7.18621			0.682085
$112$		0			0
$113$			3.56359i			0.335234i	0.985852	+	0.167617i	$0.0536072\pi$
$113$			3.56359i			0.335234i	−0.985852	+	0.167617i	$0.946393\pi$
$114$		0			0
$115$	3.30216	+	3.47789i	0.307928	+	0.324315i
$116$		0			0
$117$	−5.24475			−0.484877
$118$		0			0
$119$		−	11.4363i		−	1.04837i
$120$		0			0
$121$	−7.50339			−0.682127
$122$		0			0
$123$		−	13.4579i		−	1.21346i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	19.7928i		−	1.75633i	−0.478360	−	0.878164i	$-0.658768\pi$
$127$		−	19.7928i		−	1.75633i	0.478360	−	0.878164i	$-0.341232\pi$
$128$		0			0
$129$		−	2.75949i		−	0.242959i
$130$		0			0
$131$			13.8946i			1.21398i	0.794710	+	0.606990i	$0.207623\pi$
$131$			13.8946i			1.21398i	−0.794710	+	0.606990i	$0.792377\pi$
$132$		0			0
$133$	−15.2810			−1.32503
$134$		0			0
$135$	5.62672			0.484271
$136$		0			0
$137$		−	2.50339i		−	0.213879i	−0.994266	−	0.106940i	$-0.965895\pi$
$137$		−	2.50339i		−	0.213879i	0.994266	−	0.106940i	$-0.0341051\pi$
$138$		0			0
$139$		−	15.6058i		−	1.32366i	−0.749652	−	0.661832i	$-0.769779\pi$
$139$		−	15.6058i		−	1.32366i	0.749652	−	0.661832i	$-0.230221\pi$
$140$		0			0
$141$	−6.73696			−0.567354
$142$		0			0
$143$	−7.95034			−0.664841
$144$		0			0
$145$			3.15524i			0.262029i
$146$		0			0
$147$		−	5.83304i		−	0.481101i
$148$		0			0
$149$		−	21.2810i		−	1.74341i	−0.490033	−	0.871704i	$-0.663015\pi$
$149$		−	21.2810i		−	1.74341i	0.490033	−	0.871704i	$-0.336985\pi$
$150$		0			0
$151$			0.395705i			0.0322020i	0.999870	+	0.0161010i	$0.00512533\pi$
$151$			0.395705i			0.0322020i	−0.999870	+	0.0161010i	$0.994875\pi$
$152$		0			0
$153$			4.18032i			0.337959i
$154$		0			0
$155$	2.04566			0.164311
$156$		0			0
$157$			14.8446i			1.18473i	0.805671	+	0.592364i	$0.201805\pi$
$157$			14.8446i			1.18473i	−0.805671	+	0.592364i	$0.798195\pi$
$158$		0			0
$159$	−15.8056			−1.25347
$160$		0			0
$161$	−11.7370	+	11.1439i	−0.925002	+	0.878262i
$162$		0			0
$163$		−	2.44467i		−	0.191482i	−0.995406	−	0.0957408i	$-0.969478\pi$
$163$		−	2.44467i		−	0.191482i	0.995406	−	0.0957408i	$-0.0305220\pi$
$164$		0			0
$165$	2.48526			0.193477
$166$		0			0
$167$			14.8690i			1.15060i	0.817942	+	0.575300i	$0.195115\pi$
$167$			14.8690i			1.15060i	−0.817942	+	0.575300i	$0.804885\pi$
$168$		0			0
$169$	5.07692			0.390532
$170$		0			0
$171$	5.58567			0.427147
$172$		0			0
$173$	6.97052			0.529959			0.264980	−	0.964254i	$-0.414635\pi$
$173$	6.97052			0.529959			0.264980	+	0.964254i	$0.414635\pi$
$174$		0			0
$175$	3.37473			0.255106
$176$		0			0
$177$	−14.7776			−1.11075
$178$		0			0
$179$			8.19958i			0.612866i	0.951892	+	0.306433i	$0.0991355\pi$
$179$			8.19958i			0.612866i	−0.951892	+	0.306433i	$0.900865\pi$
$180$		0			0
$181$		−	25.7663i		−	1.91519i	−0.288115	−	0.957596i	$-0.593028\pi$
$181$		−	25.7663i		−	1.91519i	0.288115	−	0.957596i	$-0.406972\pi$
$182$		0			0
$183$	5.34039			0.394773
$184$		0			0
$185$	−5.40694			−0.397526
$186$		0			0
$187$			6.33681i			0.463393i
$188$		0			0
$189$			18.9887i			1.38122i
$190$		0			0
$191$	8.67738			0.627873			0.313937	−	0.949444i	$-0.398352\pi$
$191$	8.67738			0.627873			0.313937	+	0.949444i	$0.398352\pi$
$192$		0			0
$193$	3.74830			0.269809			0.134904	−	0.990859i	$-0.456927\pi$
$193$	3.74830			0.269809			0.134904	+	0.990859i	$0.456927\pi$
$194$		0			0
$195$	−5.65081			−0.404663
$196$		0			0
$197$	10.5984			0.755107			0.377554	−	0.925988i	$-0.376765\pi$
$197$	10.5984			0.755107			0.377554	+	0.925988i	$0.376765\pi$
$198$		0			0
$199$	−19.2520			−1.36474			−0.682368	−	0.731009i	$-0.739050\pi$
$199$	−19.2520			−1.36474			−0.682368	+	0.731009i	$0.739050\pi$
$200$		0			0
$201$		−	7.55100i		−	0.532606i
$202$		0			0
$203$	−10.6481			−0.747350
$204$		0			0
$205$			10.1258i			0.707214i
$206$		0			0
$207$	4.29021	−	4.07343i	0.298190	−	0.283123i
$208$		0			0
$209$	8.46713			0.585684
$210$		0			0
$211$		−	5.74127i		−	0.395245i	−0.980278	−	0.197623i	$-0.936678\pi$
$211$		−	5.74127i		−	0.395245i	0.980278	−	0.197623i	$-0.0633221\pi$
$212$		0			0
$213$	2.25170			0.154284
$214$		0			0
$215$			2.07625i			0.141599i
$216$		0			0
$217$			6.90355i			0.468643i
$218$		0			0
$219$		−	9.34247i		−	0.631305i
$220$		0			0
$221$		−	14.4082i		−	0.969199i
$222$		0			0
$223$			27.2042i			1.82173i	0.412709	+	0.910863i	$0.364583\pi$
$223$			27.2042i			1.82173i	−0.412709	+	0.910863i	$0.635417\pi$
$224$		0			0
$225$	−1.23357			−0.0822377
$226$		0			0
$227$	−18.6460			−1.23758			−0.618788	−	0.785558i	$-0.712376\pi$
$227$	−18.6460			−1.23758			−0.618788	+	0.785558i	$0.712376\pi$
$228$		0			0
$229$		−	2.52152i		−	0.166627i	−0.996523	−	0.0833134i	$-0.973450\pi$
$229$		−	2.52152i		−	0.166627i	0.996523	−	0.0833134i	$-0.0265503\pi$
$230$		0			0
$231$			8.38709i			0.551830i
$232$		0			0
$233$	23.9998			1.57228			0.786141	−	0.618048i	$-0.212076\pi$
$233$	23.9998			1.57228			0.786141	+	0.618048i	$0.212076\pi$
$234$		0			0
$235$	5.06892			0.330660
$236$		0			0
$237$		−	6.01813i		−	0.390920i
$238$		0			0
$239$			15.9572i			1.03219i	0.856532	+	0.516094i	$0.172614\pi$
$239$			15.9572i			1.03219i	−0.856532	+	0.516094i	$0.827386\pi$
$240$		0			0
$241$		−	4.19291i		−	0.270089i	−0.990840	−	0.135044i	$-0.956882\pi$
$241$		−	4.19291i		−	0.270089i	0.990840	−	0.135044i	$-0.0431177\pi$
$242$		0			0
$243$		−	11.8594i		−	0.760782i
$244$		0			0
$245$			4.38881i			0.280391i
$246$		0			0
$247$	−19.2520			−1.22497
$248$		0			0
$249$		−	5.06574i		−	0.321028i
$250$		0			0
$251$	16.1571			1.01983			0.509913	−	0.860226i	$-0.329678\pi$
$251$	16.1571			1.01983			0.509913	+	0.860226i	$0.329678\pi$
$252$		0			0
$253$	6.50339	−	6.17478i	0.408865	−	0.388205i
$254$		0			0
$255$			4.50397i			0.282050i
$256$		0			0
$257$	−19.0656			−1.18928			−0.594639	−	0.803993i	$-0.702705\pi$
$257$	−19.0656			−1.18928			−0.594639	+	0.803993i	$0.702705\pi$
$258$		0			0
$259$		−	18.2470i		−	1.13381i
$260$		0			0
$261$	3.89220			0.240921
$262$		0			0
$263$	−15.3584			−0.947039			−0.473519	−	0.880783i	$-0.657017\pi$
$263$	−15.3584			−0.947039			−0.473519	+	0.880783i	$0.657017\pi$
$264$		0			0
$265$	11.8922			0.730532
$266$		0			0
$267$	5.75304			0.352080
$268$		0			0
$269$	8.39999			0.512156			0.256078	−	0.966656i	$-0.417570\pi$
$269$	8.39999			0.512156			0.256078	+	0.966656i	$0.417570\pi$
$270$		0			0
$271$			30.3127i			1.84137i	0.390311	+	0.920683i	$0.372367\pi$
$271$			30.3127i			1.84137i	−0.390311	+	0.920683i	$0.627633\pi$
$272$		0			0
$273$		−	19.0700i		−	1.15417i
$274$		0			0
$275$	−1.86992			−0.112761
$276$		0			0
$277$	22.1565			1.33125			0.665627	−	0.746285i	$-0.268164\pi$
$277$	22.1565			1.33125			0.665627	+	0.746285i	$0.268164\pi$
$278$		0			0
$279$		−	2.52345i		−	0.151075i
$280$		0			0
$281$			12.4490i			0.742645i	0.928504	+	0.371323i	$0.121096\pi$
$281$			12.4490i			0.742645i	−0.928504	+	0.371323i	$0.878904\pi$
$282$		0			0
$283$	24.7901			1.47362			0.736809	−	0.676101i	$-0.236332\pi$
$283$	24.7901			1.47362			0.736809	+	0.676101i	$0.236332\pi$
$284$		0			0
$285$	6.01813			0.356483
$286$		0			0
$287$	−34.1717			−2.01709
$288$		0			0
$289$	5.51598			0.324469
$290$		0			0
$291$	11.8984			0.697495
$292$		0			0
$293$			14.6698i			0.857020i	0.903537	+	0.428510i	$0.140961\pi$
$293$			14.6698i			0.857020i	−0.903537	+	0.428510i	$0.859039\pi$
$294$		0			0
$295$	11.1187			0.647358
$296$		0			0
$297$		−	10.5215i		−	0.610521i
$298$		0			0
$299$	−14.7869	+	14.0398i	−0.855151	+	0.811941i
$300$		0			0
$301$	−7.00678			−0.403864
$302$		0			0
$303$		−	15.8056i		−	0.908008i
$304$		0			0
$305$	−4.01813			−0.230077
$306$		0			0
$307$		−	30.2411i		−	1.72595i	−0.505247	−	0.862975i	$-0.668599\pi$
$307$		−	30.2411i		−	1.72595i	0.505247	−	0.862975i	$-0.331401\pi$
$308$		0			0
$309$		−	5.71188i		−	0.324938i
$310$		0			0
$311$			16.9654i			0.962021i	0.876715	+	0.481011i	$0.159730\pi$
$311$			16.9654i			0.962021i	−0.876715	+	0.481011i	$0.840270\pi$
$312$		0			0
$313$			27.6403i			1.56232i	0.624328	+	0.781162i	$0.285373\pi$
$313$			27.6403i			1.56232i	−0.624328	+	0.781162i	$0.714627\pi$
$314$		0			0
$315$		−	4.16295i		−	0.234556i
$316$		0			0
$317$	−4.19291			−0.235497			−0.117749	−	0.993043i	$-0.537568\pi$
$317$	−4.19291			−0.235497			−0.117749	+	0.993043i	$0.537568\pi$
$318$		0			0
$319$	5.90006			0.330340
$320$		0			0
$321$			3.04761i			0.170101i
$322$		0			0
$323$			15.3447i			0.853805i
$324$		0			0
$325$	4.25170			0.235842
$326$		0			0
$327$	−9.03204			−0.499473
$328$		0			0
$329$			17.1062i			0.943097i
$330$		0			0
$331$		−	0.551304i		−	0.0303024i	−0.999885	−	0.0151512i	$-0.995177\pi$
$331$		−	0.551304i		−	0.0303024i	0.999885	−	0.0151512i	$-0.00482296\pi$
$332$		0			0
$333$			6.66982i			0.365504i
$334$		0			0
$335$			5.68140i			0.310408i
$336$		0			0
$337$			7.55100i			0.411329i	0.978623	+	0.205665i	$0.0659356\pi$
$337$			7.55100i			0.411329i	−0.978623	+	0.205665i	$0.934064\pi$
$338$		0			0
$339$	4.73626			0.257239
$340$		0			0
$341$		−	3.82522i		−	0.207147i
$342$		0			0
$343$	8.81207			0.475807
$344$		0			0
$345$	4.62237	−	4.38881i	0.248860	−	0.236285i
$346$		0			0
$347$		−	12.5897i		−	0.675848i	−0.941173	−	0.337924i	$-0.890275\pi$
$347$		−	12.5897i		−	0.675848i	0.941173	−	0.337924i	$-0.109725\pi$
$348$		0			0
$349$	−13.8153			−0.739515			−0.369758	−	0.929128i	$-0.620559\pi$
$349$	−13.8153			−0.739515			−0.369758	+	0.929128i	$0.620559\pi$
$350$		0			0
$351$			23.9231i			1.27692i
$352$		0			0
$353$	−5.68935			−0.302814			−0.151407	−	0.988472i	$-0.548380\pi$
$353$	−5.68935			−0.302814			−0.151407	+	0.988472i	$0.548380\pi$
$354$		0			0
$355$	−1.69419			−0.0899180
$356$		0			0
$357$	−15.1997			−0.804453
$358$		0			0
$359$	10.5473			0.556665			0.278333	−	0.960485i	$-0.410218\pi$
$359$	10.5473			0.556665			0.278333	+	0.960485i	$0.410218\pi$
$360$		0			0
$361$	1.50339			0.0791259
$362$		0			0
$363$			9.97255i			0.523423i
$364$		0			0
$365$			7.02931i			0.367931i
$366$		0			0
$367$	−19.8865			−1.03806			−0.519032	−	0.854755i	$-0.673708\pi$
$367$	−19.8865			−1.03806			−0.519032	+	0.854755i	$0.673708\pi$
$368$		0			0
$369$	12.4908			0.650245
$370$		0			0
$371$			40.1330i			2.08360i
$372$		0			0
$373$			9.96374i			0.515903i	0.966158	+	0.257951i	$0.0830475\pi$
$373$			9.96374i			0.515903i	−0.966158	+	0.257951i	$0.916953\pi$
$374$		0			0
$375$	−1.32907			−0.0686330
$376$		0			0
$377$	−13.4151			−0.690914
$378$		0			0
$379$	−26.6803			−1.37047			−0.685237	−	0.728320i	$-0.740301\pi$
$379$	−26.6803			−1.37047			−0.685237	+	0.728320i	$0.740301\pi$
$380$		0			0
$381$	−26.3061			−1.34770
$382$		0			0
$383$	−30.6043			−1.56381			−0.781904	−	0.623399i	$-0.785751\pi$
$383$	−30.6043			−1.56381			−0.781904	+	0.623399i	$0.785751\pi$
$384$		0			0
$385$		−	6.31048i		−	0.321612i
$386$		0			0
$387$	2.56119			0.130193
$388$		0			0
$389$			5.28101i			0.267758i	0.990998	+	0.133879i	$0.0427433\pi$
$389$			5.28101i			0.267758i	−0.990998	+	0.133879i	$0.957257\pi$
$390$		0			0
$391$	11.1904	+	11.7859i	0.565922	+	0.596039i
$392$		0			0
$393$	18.4670			0.931535
$394$		0			0
$395$			4.52807i			0.227832i
$396$		0			0
$397$	30.6599			1.53878			0.769388	−	0.638782i	$-0.220562\pi$
$397$	30.6599			1.53878			0.769388	+	0.638782i	$0.220562\pi$
$398$		0			0
$399$			20.3096i			1.01675i
$400$		0			0
$401$			26.4082i			1.31876i	0.751809	+	0.659381i	$0.229181\pi$
$401$			26.4082i			1.31876i	−0.751809	+	0.659381i	$0.770819\pi$
$402$		0			0
$403$			8.69752i			0.433254i
$404$		0			0
$405$		−	3.77762i		−	0.187711i
$406$		0			0
$407$			10.1106i			0.501161i
$408$		0			0
$409$	−13.1915			−0.652278			−0.326139	−	0.945322i	$-0.605748\pi$
$409$	−13.1915			−0.652278			−0.326139	+	0.945322i	$0.605748\pi$
$410$		0			0
$411$	−3.32719			−0.164118
$412$		0			0
$413$			37.5228i			1.84637i
$414$		0			0
$415$			3.81148i			0.187098i
$416$		0			0
$417$	−20.7412			−1.01570
$418$		0			0
$419$	−35.1187			−1.71566			−0.857831	−	0.513931i	$-0.828189\pi$
$419$	−35.1187			−1.71566			−0.857831	+	0.513931i	$0.828189\pi$
$420$		0			0
$421$			10.6782i			0.520422i	0.965552	+	0.260211i	$0.0837922\pi$
$421$			10.6782i			0.520422i	−0.965552	+	0.260211i	$0.916208\pi$
$422$		0			0
$423$		−	6.25284i		−	0.304024i
$424$		0			0
$425$		−	3.38881i		−	0.164381i
$426$		0			0
$427$		−	13.5601i		−	0.656220i
$428$		0			0
$429$			10.5666i			0.510159i
$430$		0			0
$431$	−24.9470			−1.20166			−0.600828	−	0.799379i	$-0.705162\pi$
$431$	−24.9470			−1.20166			−0.600828	+	0.799379i	$0.705162\pi$
$432$		0			0
$433$		−	33.3299i		−	1.60173i	−0.598844	−	0.800865i	$-0.704373\pi$
$433$		−	33.3299i		−	1.60173i	0.598844	−	0.800865i	$-0.295627\pi$
$434$		0			0
$435$	4.19355			0.201065
$436$		0			0
$437$	15.7481	−	14.9524i	0.753336	−	0.715270i
$438$		0			0
$439$			17.2557i			0.823570i	0.911281	+	0.411785i	$0.135095\pi$
$439$			17.2557i			0.823570i	−0.911281	+	0.411785i	$0.864905\pi$
$440$		0			0
$441$	5.41389			0.257804
$442$		0			0
$443$			4.75135i			0.225743i	0.993610	+	0.112872i	$0.0360049\pi$
$443$			4.75135i			0.225743i	−0.993610	+	0.112872i	$0.963995\pi$
$444$		0			0
$445$	−4.32861			−0.205196
$446$		0			0
$447$	−28.2840			−1.33779
$448$		0			0
$449$	−25.1732			−1.18800			−0.593999	−	0.804466i	$-0.702452\pi$
$449$	−25.1732			−1.18800			−0.593999	+	0.804466i	$0.702452\pi$
$450$		0			0
$451$	18.9344			0.891586
$452$		0			0
$453$	0.525920			0.0247099
$454$		0			0
$455$			14.3483i			0.672660i
$456$		0			0
$457$		−	21.5455i		−	1.00785i	−0.863746	−	0.503927i	$-0.831888\pi$
$457$		−	21.5455i		−	1.00785i	0.863746	−	0.503927i	$-0.168112\pi$
$458$		0			0
$459$	19.0679			0.890012
$460$		0			0
$461$	13.0112			0.605991			0.302996	−	0.952992i	$-0.402013\pi$
$461$	13.0112			0.605991			0.302996	+	0.952992i	$0.402013\pi$
$462$		0			0
$463$			3.21595i			0.149458i	0.997204	+	0.0747288i	$0.0238091\pi$
$463$			3.21595i			0.149458i	−0.997204	+	0.0747288i	$0.976191\pi$
$464$		0			0
$465$		−	2.71883i		−	0.126083i
$466$		0			0
$467$	−26.7453			−1.23762			−0.618812	−	0.785539i	$-0.712386\pi$
$467$	−26.7453			−1.23762			−0.618812	+	0.785539i	$0.712386\pi$
$468$		0			0
$469$	−19.1732			−0.885336
$470$		0			0
$471$	19.7295			0.909089
$472$		0			0
$473$	3.88242			0.178514
$474$		0			0
$475$	−4.52807			−0.207762
$476$		0			0
$477$		−	14.6698i		−	0.671685i
$478$		0			0
$479$	27.0799			1.23731			0.618657	−	0.785661i	$-0.287677\pi$
$479$	27.0799			1.23731			0.618657	+	0.785661i	$0.287677\pi$
$480$		0			0
$481$		−	22.9887i		−	1.04819i
$482$		0			0
$483$	14.8110	+	15.5993i	0.673926	+	0.709791i
$484$		0			0
$485$	−8.95239			−0.406507
$486$		0			0
$487$		−	9.57289i		−	0.433789i	−0.976195	−	0.216894i	$-0.930407\pi$
$487$		−	9.57289i		−	0.433789i	0.976195	−	0.216894i	$-0.0695928\pi$
$488$		0			0
$489$	−3.24915			−0.146932
$490$		0			0
$491$		−	2.12775i		−	0.0960240i	−0.998847	−	0.0480120i	$-0.984711\pi$
$491$		−	2.12775i		−	0.0960240i	0.998847	−	0.0480120i	$-0.0152886\pi$
$492$		0			0
$493$			10.6925i			0.481566i
$494$		0			0
$495$			2.30667i			0.103677i
$496$		0			0
$497$		−	5.71742i		−	0.256461i
$498$		0			0
$499$		−	17.1001i		−	0.765506i	−0.923851	−	0.382753i	$-0.874976\pi$
$499$		−	17.1001i		−	0.765506i	0.923851	−	0.382753i	$-0.125024\pi$
$500$		0			0
$501$	19.7620			0.882903
$502$		0			0
$503$	−42.2060			−1.88187			−0.940937	−	0.338582i	$-0.890053\pi$
$503$	−42.2060			−1.88187			−0.940937	+	0.338582i	$0.890053\pi$
$504$		0			0
$505$			11.8922i			0.529196i
$506$		0			0
$507$		−	6.74759i		−	0.299671i
$508$		0			0
$509$	3.92308			0.173888			0.0869438	−	0.996213i	$-0.472290\pi$
$509$	3.92308			0.173888			0.0869438	+	0.996213i	$0.472290\pi$
$510$		0			0
$511$	−23.7220			−1.04940
$512$		0			0
$513$		−	25.4782i		−	1.12489i
$514$		0			0
$515$			4.29764i			0.189377i
$516$		0			0
$517$		−	9.47848i		−	0.416863i
$518$		0			0
$519$		−	9.26433i		−	0.406659i
$520$		0			0
$521$			5.22238i			0.228797i	0.993435	+	0.114398i	$0.0364940\pi$
$521$			5.22238i			0.228797i	−0.993435	+	0.114398i	$0.963506\pi$
$522$		0			0
$523$	−17.2337			−0.753578			−0.376789	−	0.926299i	$-0.622972\pi$
$523$	−17.2337			−0.753578			−0.376789	+	0.926299i	$0.622972\pi$
$524$		0			0
$525$		−	4.48526i		−	0.195753i
$526$		0			0
$527$	6.93234			0.301978
$528$		0			0
$529$	1.19150	−	22.9691i	0.0518044	−	0.998657i
$530$		0			0
$531$		−	13.7157i		−	0.595211i
$532$		0			0
$533$	−43.0517			−1.86478
$534$		0			0
$535$		−	2.29303i		−	0.0991364i
$536$		0			0
$537$	10.8978			0.470277
$538$		0			0
$539$	8.20673			0.353489
$540$		0			0
$541$	43.1423			1.85483			0.927417	−	0.374030i	$-0.122024\pi$
$541$	43.1423			1.85483			0.927417	+	0.374030i	$0.122024\pi$
$542$		0			0
$543$	−34.2452			−1.46960
$544$		0			0
$545$	6.79575			0.291098
$546$		0			0
$547$			0.807044i			0.0345067i	0.999851	+	0.0172534i	$0.00549218\pi$
$547$			0.807044i			0.0345067i	−0.999851	+	0.0172534i	$0.994508\pi$
$548$		0			0
$549$			4.95663i			0.211544i
$550$		0			0
$551$	14.2871			0.608653
$552$		0			0
$553$	−15.2810			−0.649815
$554$		0			0
$555$			7.18621i			0.305038i
$556$		0			0
$557$			20.9398i			0.887248i	0.896213	+	0.443624i	$0.146307\pi$
$557$			20.9398i			0.887248i	−0.896213	+	0.443624i	$0.853693\pi$
$558$		0			0
$559$	−8.82758			−0.373367
$560$		0			0
$561$	8.42208			0.355580
$562$		0			0
$563$	11.2027			0.472138			0.236069	−	0.971736i	$-0.424141\pi$
$563$	11.2027			0.472138			0.236069	+	0.971736i	$0.424141\pi$
$564$		0			0
$565$	−3.56359			−0.149921
$566$		0			0
$567$	12.7484			0.535384
$568$		0			0
$569$		−	10.0000i		−	0.419222i	−0.977785	−	0.209611i	$-0.932780\pi$
$569$		−	10.0000i		−	0.419222i	0.977785	−	0.209611i	$-0.0672197\pi$
$570$		0			0
$571$	21.0398			0.880488			0.440244	−	0.897878i	$-0.354892\pi$
$571$	21.0398			0.880488			0.440244	+	0.897878i	$0.354892\pi$
$572$		0			0
$573$		−	11.5329i		−	0.481792i
$574$		0			0
$575$	−3.47789	+	3.30216i	−0.145038	+	0.137710i
$576$		0			0
$577$	44.6933			1.86061			0.930304	−	0.366790i	$-0.119543\pi$
$577$	44.6933			1.86061			0.930304	+	0.366790i	$0.119543\pi$
$578$		0			0
$579$		−	4.98177i		−	0.207035i
$580$		0			0
$581$	−12.8627			−0.533636
$582$		0			0
$583$		−	22.2375i		−	0.920983i
$584$		0			0
$585$		−	5.24475i		−	0.216844i
$586$		0			0
$587$			8.43001i			0.347944i	0.984751	+	0.173972i	$0.0556602\pi$
$587$			8.43001i			0.347944i	−0.984751	+	0.173972i	$0.944340\pi$
$588$		0			0
$589$		−	9.26288i		−	0.381670i
$590$		0			0
$591$		−	14.0861i		−	0.579424i
$592$		0			0
$593$	−8.91190			−0.365968			−0.182984	−	0.983116i	$-0.558576\pi$
$593$	−8.91190			−0.365968			−0.182984	+	0.983116i	$0.558576\pi$
$594$		0			0
$595$	11.4363			0.468843
$596$		0			0
$597$			25.5873i			1.04722i
$598$		0			0
$599$			2.30667i			0.0942481i	0.998889	+	0.0471240i	$0.0150056\pi$
$599$			2.30667i			0.0942481i	−0.998889	+	0.0471240i	$0.984994\pi$
$600$		0			0
$601$	−15.4657			−0.630860			−0.315430	−	0.948949i	$-0.602149\pi$
$601$	−15.4657			−0.630860			−0.315430	+	0.948949i	$0.602149\pi$
$602$		0			0
$603$	7.00839			0.285404
$604$		0			0
$605$		−	7.50339i		−	0.305056i
$606$		0			0
$607$			19.8678i			0.806408i	0.915110	+	0.403204i	$0.132103\pi$
$607$			19.8678i			0.806408i	−0.915110	+	0.403204i	$0.867897\pi$
$608$		0			0
$609$			14.1521i			0.573471i
$610$		0			0
$611$			21.5515i			0.871880i
$612$		0			0
$613$		−	23.6640i		−	0.955780i	−0.878420	−	0.477890i	$-0.841402\pi$
$613$		−	23.6640i		−	0.955780i	0.878420	−	0.477890i	$-0.158598\pi$
$614$		0			0
$615$	13.4579			0.542674
$616$		0			0
$617$		−	45.6222i		−	1.83668i	−0.395791	−	0.918341i	$-0.629530\pi$
$617$		−	45.6222i		−	1.83668i	0.395791	−	0.918341i	$-0.370470\pi$
$618$		0			0
$619$	−42.3426			−1.70189			−0.850946	−	0.525253i	$-0.823971\pi$
$619$	−42.3426			−1.70189			−0.850946	+	0.525253i	$0.823971\pi$
$620$		0			0
$621$	−18.5803	−	19.5691i	−0.745602	−	0.785282i
$622$		0			0
$623$		−	14.6079i		−	0.585253i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	11.2534i		−	0.449419i
$628$		0			0
$629$	−18.3231			−0.730589
$630$		0			0
$631$	38.4459			1.53051			0.765254	−	0.643728i	$-0.222613\pi$
$631$	38.4459			1.53051			0.765254	+	0.643728i	$0.222613\pi$
$632$		0			0
$633$	−7.63056			−0.303288
$634$		0			0
$635$	19.7928			0.785454
$636$		0			0
$637$	−18.6599			−0.739331
$638$		0			0
$639$			2.08989i			0.0826748i
$640$		0			0
$641$		−	11.9456i		−	0.471823i	−0.971775	−	0.235912i	$-0.924192\pi$
$641$		−	11.9456i		−	0.471823i	0.971775	−	0.235912i	$-0.0758076\pi$
$642$		0			0
$643$	33.2111			1.30972			0.654859	−	0.755751i	$-0.272728\pi$
$643$	33.2111			1.30972			0.654859	+	0.755751i	$0.272728\pi$
$644$		0			0
$645$	2.75949			0.108655
$646$		0			0
$647$		−	27.7672i		−	1.09164i	−0.837901	−	0.545822i	$-0.816218\pi$
$647$		−	27.7672i		−	1.09164i	0.837901	−	0.545822i	$-0.183782\pi$
$648$		0			0
$649$		−	20.7912i		−	0.816125i
$650$		0			0
$651$	9.17531			0.359609
$652$		0			0
$653$	−5.33980			−0.208962			−0.104481	−	0.994527i	$-0.533318\pi$
$653$	−5.33980			−0.208962			−0.104481	+	0.994527i	$0.533318\pi$
$654$		0			0
$655$	−13.8946			−0.542908
$656$		0			0
$657$	8.67112			0.338293
$658$		0			0
$659$	−49.7574			−1.93827			−0.969136	−	0.246528i	$-0.920710\pi$
$659$	−49.7574			−1.93827			−0.969136	+	0.246528i	$0.920710\pi$
$660$		0			0
$661$			40.1674i			1.56233i	0.624324	+	0.781165i	$0.285374\pi$
$661$			40.1674i			1.56233i	−0.624324	+	0.781165i	$0.714626\pi$
$662$		0			0
$663$	−19.1495			−0.743706
$664$		0			0
$665$		−	15.2810i		−	0.592572i
$666$		0			0
$667$	10.9736	−	10.4191i	0.424899	−	0.403429i
$668$		0			0
$669$	36.1563			1.39788
$670$		0			0
$671$			7.51359i			0.290059i
$672$		0			0
$673$	−25.6531			−0.988854			−0.494427	−	0.869219i	$-0.664622\pi$
$673$	−25.6531			−0.988854			−0.494427	+	0.869219i	$0.664622\pi$
$674$		0			0
$675$			5.62672i			0.216572i
$676$		0			0
$677$			14.3370i			0.551014i	0.961299	+	0.275507i	$0.0888458\pi$
$677$			14.3370i			0.551014i	−0.961299	+	0.275507i	$0.911154\pi$
$678$		0			0
$679$		−	30.2119i		−	1.15943i
$680$		0			0
$681$			24.7819i			0.949643i
$682$		0			0
$683$		−	22.3898i		−	0.856721i	−0.903608	−	0.428360i	$-0.859091\pi$
$683$		−	22.3898i		−	0.856721i	0.903608	−	0.428360i	$-0.140909\pi$
$684$		0			0
$685$	2.50339			0.0956497
$686$		0			0
$687$	−3.35128			−0.127859
$688$		0			0
$689$			50.5620i			1.92626i
$690$		0			0
$691$			34.2966i			1.30470i	0.757916	+	0.652352i	$0.226218\pi$
$691$			34.2966i			1.30470i	−0.757916	+	0.652352i	$0.773782\pi$
$692$		0			0
$693$	−7.78440			−0.295705
$694$		0			0
$695$	15.6058			0.591960
$696$		0			0
$697$			34.3143i			1.29975i
$698$		0			0
$699$		−	31.8975i		−	1.20648i
$700$		0			0
$701$		−	31.2266i		−	1.17941i	−0.807618	−	0.589707i	$-0.799243\pi$
$701$		−	31.2266i		−	1.17941i	0.807618	−	0.589707i	$-0.200757\pi$
$702$		0			0
$703$			24.4830i			0.923393i
$704$		0			0
$705$		−	6.73696i		−	0.253729i
$706$		0			0
$707$	−40.1330			−1.50936
$708$		0			0
$709$			10.6391i			0.399560i	0.979841	+	0.199780i	$0.0640227\pi$
$709$			10.6391i			0.399560i	−0.979841	+	0.199780i	$0.935977\pi$
$710$		0			0
$711$	5.58567			0.209479
$712$		0			0
$713$	−6.75509	−	7.11458i	−0.252980	−	0.266443i
$714$		0			0
$715$		−	7.95034i		−	0.297326i
$716$		0			0
$717$	21.2083			0.792039
$718$		0			0
$719$			46.1183i			1.71992i	0.510359	+	0.859961i	$0.329512\pi$
$719$			46.1183i			1.71992i	−0.510359	+	0.859961i	$0.670488\pi$
$720$		0			0
$721$	−14.5034			−0.540135
$722$		0			0
$723$	−5.57268			−0.207250
$724$		0			0
$725$	−3.15524			−0.117183
$726$		0			0
$727$	9.44534			0.350308			0.175154	−	0.984541i	$-0.443958\pi$
$727$	9.44534			0.350308			0.175154	+	0.984541i	$0.443958\pi$
$728$		0			0
$729$	−27.0949			−1.00351
$730$		0			0
$731$			7.03601i			0.260236i
$732$		0			0
$733$		−	14.4545i		−	0.533891i	−0.963712	−	0.266945i	$-0.913986\pi$
$733$		−	14.4545i		−	0.533891i	0.963712	−	0.266945i	$-0.0860143\pi$
$734$		0			0
$735$	5.83304			0.215155
$736$		0			0
$737$	10.6238			0.391332
$738$		0			0
$739$		−	16.3972i		−	0.603180i	−0.953438	−	0.301590i	$-0.902483\pi$
$739$		−	16.3972i		−	0.603180i	0.953438	−	0.301590i	$-0.0975173\pi$
$740$		0			0
$741$			25.5873i			0.939971i
$742$		0			0
$743$	43.8494			1.60868			0.804339	−	0.594171i	$-0.202520\pi$
$743$	43.8494			1.60868			0.804339	+	0.594171i	$0.202520\pi$
$744$		0			0
$745$	21.2810			0.779676
$746$		0			0
$747$	4.70171			0.172027
$748$		0			0
$749$	7.73837			0.282754
$750$		0			0
$751$	2.10540			0.0768272			0.0384136	−	0.999262i	$-0.487770\pi$
$751$	2.10540			0.0768272			0.0384136	+	0.999262i	$0.487770\pi$
$752$		0			0
$753$		−	21.4739i		−	0.782553i
$754$		0			0
$755$	−0.395705			−0.0144012
$756$		0			0
$757$		−	29.0826i		−	1.05702i	−0.848926	−	0.528512i	$-0.822750\pi$
$757$		−	29.0826i		−	1.05702i	0.848926	−	0.528512i	$-0.177250\pi$
$758$		0			0
$759$	−8.20673	−	8.64348i	−0.297885	−	0.313738i
$760$		0			0
$761$	−0.162025			−0.00587340			−0.00293670	−	0.999996i	$-0.500935\pi$
$761$	−0.162025			−0.00587340			−0.00293670	+	0.999996i	$0.500935\pi$
$762$		0			0
$763$			22.9338i			0.830260i
$764$		0			0
$765$	−4.18032			−0.151140
$766$		0			0
$767$			47.2735i			1.70695i
$768$		0			0
$769$			43.6306i			1.57336i	0.617362	+	0.786679i	$0.288201\pi$
$769$			43.6306i			1.57336i	−0.617362	+	0.786679i	$0.711799\pi$
$770$		0			0
$771$			25.3395i			0.912581i
$772$		0			0
$773$			26.0042i			0.935307i	0.883912	+	0.467654i	$0.154901\pi$
$773$			26.0042i			0.935307i	−0.883912	+	0.467654i	$0.845099\pi$
$774$		0			0
$775$			2.04566i			0.0734822i
$776$		0			0
$777$	−24.2515			−0.870019
$778$		0			0
$779$	45.8501			1.64275
$780$		0			0
$781$			3.16800i			0.113360i
$782$		0			0
$783$		−	17.7537i		−	0.634464i
$784$		0			0
$785$	−14.8446			−0.529826
$786$		0			0
$787$	−3.43273			−0.122364			−0.0611818	−	0.998127i	$-0.519487\pi$
$787$	−3.43273			−0.122364			−0.0611818	+	0.998127i	$0.519487\pi$
$788$		0			0
$789$			20.4124i			0.726701i
$790$		0			0
$791$		−	12.0261i		−	0.427600i
$792$		0			0
$793$		−	17.0839i		−	0.606666i
$794$		0			0
$795$		−	15.8056i		−	0.560567i
$796$		0			0
$797$		−	2.81942i		−	0.0998688i	−0.998752	−	0.0499344i	$-0.984099\pi$
$797$		−	2.81942i		−	0.0998688i	0.998752	−	0.0499344i	$-0.0159012\pi$
$798$		0			0
$799$	17.1776			0.607699
$800$		0			0
$801$			5.33963i			0.188667i
$802$		0			0
$803$	13.1443			0.463851
$804$		0			0
$805$	−11.1439	−	11.7370i	−0.392771	−	0.413673i
$806$		0			0
$807$		−	11.1642i		−	0.392998i
$808$		0			0
$809$	4.18032			0.146972			0.0734861	−	0.997296i	$-0.476588\pi$
$809$	4.18032			0.146972			0.0734861	+	0.997296i	$0.476588\pi$
$810$		0			0
$811$		−	52.5060i		−	1.84373i	−0.387507	−	0.921867i	$-0.626664\pi$
$811$		−	52.5060i		−	1.84373i	0.387507	−	0.921867i	$-0.373336\pi$
$812$		0			0
$813$	40.2878			1.41295
$814$		0			0
$815$	2.44467			0.0856331
$816$		0			0
$817$	9.40140			0.328913
$818$		0			0
$819$	17.6996			0.618475
$820$		0			0
$821$	15.7481			0.549614			0.274807	−	0.961499i	$-0.411386\pi$
$821$	15.7481			0.549614			0.274807	+	0.961499i	$0.411386\pi$
$822$		0			0
$823$		−	51.7555i		−	1.80408i	−0.431650	−	0.902041i	$-0.642068\pi$
$823$		−	51.7555i		−	1.80408i	0.431650	−	0.902041i	$-0.357932\pi$
$824$		0			0
$825$			2.48526i			0.0865257i
$826$		0			0
$827$	27.5915			0.959451			0.479725	−	0.877419i	$-0.340736\pi$
$827$	27.5915			0.959451			0.479725	+	0.877419i	$0.340736\pi$
$828$		0			0
$829$	−12.5160			−0.434698			−0.217349	−	0.976094i	$-0.569741\pi$
$829$	−12.5160			−0.434698			−0.217349	+	0.976094i	$0.569741\pi$
$830$		0			0
$831$		−	29.4476i		−	1.02153i
$832$		0			0
$833$			14.8728i			0.515313i
$834$		0			0
$835$	−14.8690			−0.514564
$836$		0			0
$837$	−11.5103			−0.397856
$838$		0			0
$839$	−19.1296			−0.660427			−0.330214	−	0.943906i	$-0.607121\pi$
$839$	−19.1296			−0.660427			−0.330214	+	0.943906i	$0.607121\pi$
$840$		0			0
$841$	−19.0444			−0.656705
$842$		0			0
$843$	16.5456			0.569862
$844$		0			0
$845$			5.07692i			0.174651i
$846$		0			0
$847$	25.3219			0.870072
$848$		0			0
$849$		−	32.9478i		−	1.13077i
$850$		0			0
$851$	17.8546	+	18.8048i	0.612047	+	0.644619i
$852$		0			0
$853$	−48.5620			−1.66273			−0.831366	−	0.555725i	$-0.812441\pi$
$853$	−48.5620			−1.66273			−0.831366	+	0.555725i	$0.812441\pi$
$854$		0			0
$855$			5.58567i			0.191026i
$856$		0			0
$857$	32.8978			1.12377			0.561884	−	0.827216i	$-0.310077\pi$
$857$	32.8978			1.12377			0.561884	+	0.827216i	$0.310077\pi$
$858$		0			0
$859$		−	46.5140i		−	1.58704i	−0.608546	−	0.793519i	$-0.708247\pi$
$859$		−	46.5140i		−	1.58704i	0.608546	−	0.793519i	$-0.291753\pi$
$860$		0			0
$861$			45.4167i			1.54780i
$862$		0			0
$863$		−	8.90384i		−	0.303090i	−0.988450	−	0.151545i	$-0.951575\pi$
$863$		−	8.90384i		−	0.303090i	0.988450	−	0.151545i	$-0.0484249\pi$
$864$		0			0
$865$			6.97052i			0.237005i
$866$		0			0
$867$		−	7.33114i		−	0.248979i
$868$		0			0
$869$	8.46713			0.287228
$870$		0			0
$871$	−24.1556			−0.818481
$872$		0			0
$873$			11.0434i			0.373761i
$874$		0			0
$875$			3.37473i			0.114087i
$876$		0			0
$877$	23.6640			0.799077			0.399538	−	0.916716i	$-0.369170\pi$
$877$	23.6640			0.799077			0.399538	+	0.916716i	$0.369170\pi$
$878$		0			0
$879$	19.4972			0.657626
$880$		0			0
$881$		−	2.00000i		−	0.0673817i	−0.999432	−	0.0336909i	$-0.989274\pi$
$881$		−	2.00000i		−	0.0673817i	0.999432	−	0.0336909i	$-0.0107262\pi$
$882$		0			0
$883$			3.70407i			0.124652i	0.998056	+	0.0623260i	$0.0198518\pi$
$883$			3.70407i			0.124652i	−0.998056	+	0.0623260i	$0.980148\pi$
$884$		0			0
$885$		−	14.7776i		−	0.496744i
$886$		0			0
$887$		−	28.8489i		−	0.968653i	−0.874887	−	0.484326i	$-0.839065\pi$
$887$		−	28.8489i		−	0.968653i	0.874887	−	0.484326i	$-0.160935\pi$
$888$		0			0
$889$			66.7954i			2.24025i
$890$		0			0
$891$	−7.06385			−0.236648
$892$		0			0
$893$		−	22.9524i		−	0.768073i
$894$		0			0
$895$	−8.19958			−0.274082
$896$		0			0
$897$	18.6599	+	19.6529i	0.623035	+	0.656192i
$898$		0			0
$899$		−	6.45455i		−	0.215271i
$900$		0			0
$901$	40.3004			1.34260
$902$		0			0
$903$			9.31252i			0.309901i
$904$		0			0
$905$	25.7663			0.856500
$906$		0			0
$907$	−0.129634			−0.00430442			−0.00215221	−	0.999998i	$-0.500685\pi$
$907$	−0.129634			−0.00430442			−0.00215221	+	0.999998i	$0.500685\pi$
$908$		0			0
$909$	14.6698			0.486567
$910$		0			0
$911$	−10.4040			−0.344701			−0.172350	−	0.985036i	$-0.555136\pi$
$911$	−10.4040			−0.344701			−0.172350	+	0.985036i	$0.555136\pi$
$912$		0			0
$913$	7.12717			0.235875
$914$		0			0
$915$			5.34039i			0.176548i
$916$		0			0
$917$		−	46.8906i		−	1.54846i
$918$		0			0
$919$	16.2214			0.535096			0.267548	−	0.963545i	$-0.413787\pi$
$919$	16.2214			0.535096			0.267548	+	0.963545i	$0.413787\pi$
$920$		0			0
$921$	−40.1926			−1.32439
$922$		0			0
$923$		−	7.20316i		−	0.237095i
$924$		0			0
$925$		−	5.40694i		−	0.177779i
$926$		0			0
$927$	5.30143			0.174122
$928$		0			0
$929$	0.903382			0.0296390			0.0148195	−	0.999890i	$-0.495283\pi$
$929$	0.903382			0.0296390			0.0148195	+	0.999890i	$0.495283\pi$
$930$		0			0
$931$	19.8728			0.651305
$932$		0			0
$933$	22.5483			0.738198
$934$		0			0
$935$	−6.33681			−0.207236
$936$		0			0
$937$			5.60962i			0.183258i	0.995793	+	0.0916292i	$0.0292074\pi$
$937$			5.60962i			0.183258i	−0.995793	+	0.0916292i	$0.970793\pi$
$938$		0			0
$939$	36.7360			1.19883
$940$		0			0
$941$		−	58.1926i		−	1.89702i	−0.316741	−	0.948512i	$-0.602589\pi$
$941$		−	58.1926i		−	1.89702i	0.316741	−	0.948512i	$-0.397411\pi$
$942$		0			0
$943$	35.2163	−	33.4369i	1.14680	−	1.08885i
$944$		0			0
$945$	−18.9887			−0.617701
$946$		0			0
$947$			11.0400i			0.358751i	0.983781	+	0.179375i	$0.0574076\pi$
$947$			11.0400i			0.358751i	−0.983781	+	0.179375i	$0.942592\pi$
$948$		0			0
$949$	−29.8865			−0.970157
$950$		0			0
$951$			5.57268i			0.180707i
$952$		0			0
$953$			41.9634i			1.35933i	0.733523	+	0.679664i	$0.237874\pi$
$953$			41.9634i			1.35933i	−0.733523	+	0.679664i	$0.762126\pi$
$954$		0			0
$955$			8.67738i			0.280793i
$956$		0			0
$957$		−	7.84160i		−	0.253483i
$958$		0			0
$959$			8.44827i			0.272809i
$960$		0			0
$961$	26.8153			0.865009
$962$		0			0
$963$	−2.82861			−0.0911506
$964$		0			0
$965$			3.74830i			0.120662i
$966$		0			0
$967$		−	46.4731i		−	1.49447i	−0.664558	−	0.747237i	$-0.731380\pi$
$967$		−	46.4731i		−	1.49447i	0.664558	−	0.747237i	$-0.268620\pi$
$968$		0			0
$969$	20.3943			0.655159
$970$		0			0
$971$	12.8169			0.411314			0.205657	−	0.978624i	$-0.434067\pi$
$971$	12.8169			0.411314			0.205657	+	0.978624i	$0.434067\pi$
$972$		0			0
$973$			52.6653i			1.68837i
$974$		0			0
$975$		−	5.65081i		−	0.180971i
$976$		0			0
$977$			1.44320i			0.0461720i	0.999733	+	0.0230860i	$0.00734915\pi$
$977$			1.44320i			0.0461720i	−0.999733	+	0.0230860i	$0.992651\pi$
$978$		0			0
$979$			8.09417i			0.258691i
$980$		0			0
$981$		−	8.38300i		−	0.267649i
$982$		0			0
$983$	−28.9492			−0.923336			−0.461668	−	0.887053i	$-0.652749\pi$
$983$	−28.9492			−0.923336			−0.461668	+	0.887053i	$0.652749\pi$
$984$		0			0
$985$			10.5984i			0.337694i
$986$		0			0
$987$	22.7354			0.723676
$988$		0			0
$989$	7.22098	−	6.85610i	0.229614	−	0.218011i
$990$		0			0
$991$			12.0299i			0.382142i	0.981576	+	0.191071i	$0.0611961\pi$
$991$			12.0299i			0.382142i	−0.981576	+	0.191071i	$0.938804\pi$
$992$		0			0
$993$	−0.732723			−0.0232523
$994$		0			0
$995$		−	19.2520i		−	0.610328i
$996$		0			0
$997$	−8.22917			−0.260620			−0.130310	−	0.991473i	$-0.541597\pi$
$997$	−8.22917			−0.260620			−0.130310	+	0.991473i	$0.541597\pi$
$998$		0			0
$999$	30.4233			0.962551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.b.1471.6	yes	16
4.3	odd	2	inner	1840.2.i.b.1471.12	yes	16
23.22	odd	2	inner	1840.2.i.b.1471.5	✓	16
92.91	even	2	inner	1840.2.i.b.1471.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.b.1471.5	✓	16	23.22	odd	2	inner
1840.2.i.b.1471.6	yes	16	1.1	even	1	trivial
1840.2.i.b.1471.11	yes	16	92.91	even	2	inner
1840.2.i.b.1471.12	yes	16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.32907i q^{3} +1.00000i q^{5} -3.37473 q^{7} +1.23357 q^{9} +O(q^{10})\)	\(q-1.32907i q^{3} +1.00000i q^{5} -3.37473 q^{7} +1.23357 q^{9} +1.86992 q^{11} -4.25170 q^{13} +1.32907 q^{15} +3.38881i q^{17} +4.52807 q^{19} +4.48526i q^{21} +(3.47789 - 3.30216i) q^{23} -1.00000 q^{25} -5.62672i q^{27} +3.15524 q^{29} -2.04566i q^{31} -2.48526i q^{33} -3.37473i q^{35} +5.40694i q^{37} +5.65081i q^{39} +10.1258 q^{41} +2.07625 q^{43} +1.23357i q^{45} -5.06892i q^{47} +4.38881 q^{49} +4.50397 q^{51} -11.8922i q^{53} +1.86992i q^{55} -6.01813i q^{57} -11.1187i q^{59} +4.01813i q^{61} -4.16295 q^{63} -4.25170i q^{65} +5.68140 q^{67} +(-4.38881 - 4.62237i) q^{69} +1.69419i q^{71} +7.02931 q^{73} +1.32907i q^{75} -6.31048 q^{77} +4.52807 q^{79} -3.77762 q^{81} +3.81148 q^{83} -3.38881 q^{85} -4.19355i q^{87} +4.32861i q^{89} +14.3483 q^{91} -2.71883 q^{93} +4.52807i q^{95} +8.95239i q^{97} +2.30667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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