LMFDB - Embedded newform 1936.2.e.d.1935.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,2,Mod(1935,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, 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("1936.1935");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1936.e (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:	$15.4590378313$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1935.4
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1936.1935
Dual form			1936.2.e.d.1935.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +1.00000 q^{5} +2.44949 q^{7} +O(q^{10})$	$q+1.73205i q^{3} +1.00000 q^{5} +2.44949 q^{7} +5.65685i q^{13} +1.73205i q^{15} -4.24264i q^{17} +4.24264i q^{21} +8.66025i q^{23} -4.00000 q^{25} +5.19615i q^{27} -5.65685i q^{29} -1.73205i q^{31} +2.44949 q^{35} +5.00000 q^{37} -9.79796 q^{39} -5.65685i q^{41} +12.2474 q^{43} +6.92820i q^{47} -1.00000 q^{49} +7.34847 q^{51} +4.00000 q^{53} +12.1244i q^{59} -5.65685i q^{61} +5.65685i q^{65} +5.19615i q^{67} -15.0000 q^{69} -12.1244i q^{71} +11.3137i q^{73} -6.92820i q^{75} +7.34847 q^{79} -9.00000 q^{81} -2.44949 q^{83} -4.24264i q^{85} +9.79796 q^{87} -17.0000 q^{89} +13.8564i q^{91} +3.00000 q^{93} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} - 16 q^{25} + 20 q^{37} - 4 q^{49} + 16 q^{53} - 60 q^{69} - 36 q^{81} - 68 q^{89} + 12 q^{93} + 28 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 16 * q^25 + 20 * q^37 - 4 * q^49 + 16 * q^53 - 60 * q^69 - 36 * q^81 - 68 * q^89 + 12 * q^93 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1936\mathbb{Z}\right)^\times$.

$n$	$485$	$849$	$1695$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.73205i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$3$	1.73205i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	1.73205i	0.447214i
$16$	0	0
$17$	− 4.24264i	− 1.02899i	−0.857493	−	0.514496i	$-0.827979\pi$
$17$	− 4.24264i	− 1.02899i	0.857493	−	0.514496i	$-0.172021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.24264i	0.925820i
$22$	0	0
$23$	8.66025i	1.80579i	0.429863	+	0.902894i	$0.358562\pi$
$23$	8.66025i	1.80579i	−0.429863	+	0.902894i	$0.641438\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.19615i	1.00000i
$28$	0	0
$29$	− 5.65685i	− 1.05045i	−0.850963	−	0.525226i	$-0.823981\pi$
$29$	− 5.65685i	− 1.05045i	0.850963	−	0.525226i	$-0.176019\pi$
$30$	0	0
$31$	− 1.73205i	− 0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	− 1.73205i	− 0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.44949	0.414039
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	−9.79796	−1.56893
$40$	0	0
$41$	− 5.65685i	− 0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	− 5.65685i	− 0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$	0	0
$43$	12.2474	1.86772	0.933859	−	0.357641i	$-0.116419\pi$
$43$	12.2474	1.86772	0.933859	+	0.357641i	$0.116419\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.92820i	1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$	6.92820i	1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	7.34847	1.02899
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.1244i	1.57846i	0.614100	+	0.789228i	$0.289519\pi$
$59$	12.1244i	1.57846i	−0.614100	+	0.789228i	$0.710481\pi$
$60$	0	0
$61$	− 5.65685i	− 0.724286i	−0.932123	−	0.362143i	$-0.882045\pi$
$61$	− 5.65685i	− 0.724286i	0.932123	−	0.362143i	$-0.117955\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.65685i	0.701646i
$66$	0	0
$67$	5.19615i	0.634811i	0.948290	+	0.317406i	$0.102812\pi$
$67$	5.19615i	0.634811i	−0.948290	+	0.317406i	$0.897188\pi$
$68$	0	0
$69$	−15.0000	−1.80579
$70$	0	0
$71$	− 12.1244i	− 1.43890i	−0.694546	−	0.719448i	$-0.744395\pi$
$71$	− 12.1244i	− 1.43890i	0.694546	−	0.719448i	$-0.255605\pi$
$72$	0	0
$73$	11.3137i	1.32417i	0.749429	+	0.662085i	$0.230328\pi$
$73$	11.3137i	1.32417i	−0.749429	+	0.662085i	$0.769672\pi$
$74$	0	0
$75$	− 6.92820i	− 0.800000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.34847	0.826767	0.413384	−	0.910557i	$-0.364347\pi$
$79$	7.34847	0.826767	0.413384	+	0.910557i	$0.364347\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−2.44949	−0.268866	−0.134433	−	0.990923i	$-0.542921\pi$
$83$	−2.44949	−0.268866	−0.134433	+	0.990923i	$0.542921\pi$
$84$	0	0
$85$	− 4.24264i	− 0.460179i
$86$	0	0
$87$	9.79796	1.05045
$88$	0	0
$89$	−17.0000	−1.80200	−0.900998	−	0.433823i	$-0.857164\pi$
$89$	−17.0000	−1.80200	−0.900998	+	0.433823i	$0.857164\pi$
$90$	0	0
$91$	13.8564i	1.45255i
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.41421i	− 0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$	− 1.41421i	− 0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$	0	0
$103$	− 6.92820i	− 0.682656i	−0.939944	−	0.341328i	$-0.889123\pi$
$103$	− 6.92820i	− 0.682656i	0.939944	−	0.341328i	$-0.110877\pi$
$104$	0	0
$105$	4.24264i	0.414039i
$106$	0	0
$107$	−12.2474	−1.18401	−0.592003	−	0.805936i	$-0.701663\pi$
$107$	−12.2474	−1.18401	−0.592003	+	0.805936i	$0.701663\pi$
$108$	0	0
$109$	7.07107i	0.677285i	0.940915	+	0.338643i	$0.109968\pi$
$109$	7.07107i	0.677285i	−0.940915	+	0.338643i	$0.890032\pi$
$110$	0	0
$111$	8.66025i	0.821995i
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	0	0
$115$	8.66025i	0.807573i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 10.3923i	− 0.952661i
$120$	0	0
$121$	0	0
$122$	0	0
$123$	9.79796	0.883452
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−19.5959	−1.73886	−0.869428	−	0.494059i	$-0.835513\pi$
$127$	−19.5959	−1.73886	−0.869428	+	0.494059i	$0.835513\pi$
$128$	0	0
$129$	21.2132i	1.86772i
$130$	0	0
$131$	17.1464	1.49809	0.749045	−	0.662519i	$-0.230513\pi$
$131$	17.1464	1.49809	0.749045	+	0.662519i	$0.230513\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.19615i	0.447214i
$136$	0	0
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	0	0
$139$	−12.2474	−1.03882	−0.519408	−	0.854527i	$-0.673847\pi$
$139$	−12.2474	−1.03882	−0.519408	+	0.854527i	$0.673847\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 5.65685i	− 0.469776i
$146$	0	0
$147$	− 1.73205i	− 0.142857i
$148$	0	0
$149$	− 12.7279i	− 1.04271i	−0.853339	−	0.521356i	$-0.825426\pi$
$149$	− 12.7279i	− 1.04271i	0.853339	−	0.521356i	$-0.174574\pi$
$150$	0	0
$151$	2.44949	0.199337	0.0996683	−	0.995021i	$-0.468222\pi$
$151$	2.44949	0.199337	0.0996683	+	0.995021i	$0.468222\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.73205i	− 0.139122i
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	0	0
$159$	6.92820i	0.549442i
$160$	0	0
$161$	21.2132i	1.67183i
$162$	0	0
$163$	13.8564i	1.08532i	0.839953	+	0.542659i	$0.182582\pi$
$163$	13.8564i	1.08532i	−0.839953	+	0.542659i	$0.817418\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.79796	−0.758189	−0.379094	−	0.925358i	$-0.623764\pi$
$167$	−9.79796	−0.758189	−0.379094	+	0.925358i	$0.623764\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.07107i	− 0.537603i	−0.963196	−	0.268802i	$-0.913372\pi$
$173$	− 7.07107i	− 0.537603i	0.963196	−	0.268802i	$-0.0866276\pi$
$174$	0	0
$175$	−9.79796	−0.740656
$176$	0	0
$177$	−21.0000	−1.57846
$178$	0	0
$179$	5.19615i	0.388379i	0.980964	+	0.194189i	$0.0622076\pi$
$179$	5.19615i	0.388379i	−0.980964	+	0.194189i	$0.937792\pi$
$180$	0	0
$181$	3.00000	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$182$	0	0
$183$	9.79796	0.724286
$184$	0	0
$185$	5.00000	0.367607
$186$	0	0
$187$	0	0
$188$	0	0
$189$	12.7279i	0.925820i
$190$	0	0
$191$	− 5.19615i	− 0.375980i	−0.982171	−	0.187990i	$-0.939803\pi$
$191$	− 5.19615i	− 0.375980i	0.982171	−	0.187990i	$-0.0601973\pi$
$192$	0	0
$193$	5.65685i	0.407189i	0.979055	+	0.203595i	$0.0652625\pi$
$193$	5.65685i	0.407189i	−0.979055	+	0.203595i	$0.934738\pi$
$194$	0	0
$195$	−9.79796	−0.701646
$196$	0	0
$197$	1.41421i	0.100759i	0.998730	+	0.0503793i	$0.0160430\pi$
$197$	1.41421i	0.100759i	−0.998730	+	0.0503793i	$0.983957\pi$
$198$	0	0
$199$	− 6.92820i	− 0.491127i	−0.969380	−	0.245564i	$-0.921027\pi$
$199$	− 6.92820i	− 0.491127i	0.969380	−	0.245564i	$-0.0789730\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	0	0
$203$	− 13.8564i	− 0.972529i
$204$	0	0
$205$	− 5.65685i	− 0.395092i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	19.5959	1.34904	0.674519	−	0.738257i	$-0.264351\pi$
$211$	19.5959	1.34904	0.674519	+	0.738257i	$0.264351\pi$
$212$	0	0
$213$	21.0000	1.43890
$214$	0	0
$215$	12.2474	0.835269
$216$	0	0
$217$	− 4.24264i	− 0.288009i
$218$	0	0
$219$	−19.5959	−1.32417
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	5.19615i	0.347960i	0.984749	+	0.173980i	$0.0556628\pi$
$223$	5.19615i	0.347960i	−0.984749	+	0.173980i	$0.944337\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.44949	0.162578	0.0812892	−	0.996691i	$-0.474096\pi$
$227$	2.44949	0.162578	0.0812892	+	0.996691i	$0.474096\pi$
$228$	0	0
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 22.6274i	− 1.48237i	−0.671300	−	0.741186i	$-0.734264\pi$
$233$	− 22.6274i	− 1.48237i	0.671300	−	0.741186i	$-0.265736\pi$
$234$	0	0
$235$	6.92820i	0.451946i
$236$	0	0
$237$	12.7279i	0.826767i
$238$	0	0
$239$	22.0454	1.42600	0.712999	−	0.701165i	$-0.247336\pi$
$239$	22.0454	1.42600	0.712999	+	0.701165i	$0.247336\pi$
$240$	0	0
$241$	− 22.6274i	− 1.45756i	−0.684748	−	0.728780i	$-0.740088\pi$
$241$	− 22.6274i	− 1.45756i	0.684748	−	0.728780i	$-0.259912\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 4.24264i	− 0.268866i
$250$	0	0
$251$	− 8.66025i	− 0.546630i	−0.961925	−	0.273315i	$-0.911880\pi$
$251$	− 8.66025i	− 0.546630i	0.961925	−	0.273315i	$-0.0881202\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	7.34847	0.460179
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	12.2474	0.761019
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.1464	1.05729	0.528647	−	0.848842i	$-0.322700\pi$
$263$	17.1464	1.05729	0.528647	+	0.848842i	$0.322700\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	− 29.4449i	− 1.80200i
$268$	0	0
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.0416i	1.44452i	0.691621	+	0.722261i	$0.256897\pi$
$277$	24.0416i	1.44452i	−0.691621	+	0.722261i	$0.743103\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.89949i	0.590554i	0.955412	+	0.295277i	$0.0954120\pi$
$281$	9.89949i	0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$282$	0	0
$283$	9.79796	0.582428	0.291214	−	0.956658i	$-0.405941\pi$
$283$	9.79796	0.582428	0.291214	+	0.956658i	$0.405941\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 13.8564i	− 0.817918i
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	12.1244i	0.710742i
$292$	0	0
$293$	− 33.9411i	− 1.98286i	−0.130632	−	0.991431i	$-0.541701\pi$
$293$	− 33.9411i	− 1.98286i	0.130632	−	0.991431i	$-0.458299\pi$
$294$	0	0
$295$	12.1244i	0.705907i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−48.9898	−2.83315
$300$	0	0
$301$	30.0000	1.72917
$302$	0	0
$303$	2.44949	0.140720
$304$	0	0
$305$	− 5.65685i	− 0.323911i
$306$	0	0
$307$	−29.3939	−1.67760	−0.838799	−	0.544442i	$-0.816741\pi$
$307$	−29.3939	−1.67760	−0.838799	+	0.544442i	$0.816741\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	− 31.1769i	− 1.76788i	−0.467600	−	0.883940i	$-0.654881\pi$
$311$	− 31.1769i	− 1.76788i	0.467600	−	0.883940i	$-0.345119\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.0000	1.29181	0.645904	−	0.763418i	$-0.276480\pi$
$317$	23.0000	1.29181	0.645904	+	0.763418i	$0.276480\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	− 21.2132i	− 1.18401i
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 22.6274i	− 1.25514i
$326$	0	0
$327$	−12.2474	−0.677285
$328$	0	0
$329$	16.9706i	0.935617i
$330$	0	0
$331$	5.19615i	0.285606i	0.989751	+	0.142803i	$0.0456116\pi$
$331$	5.19615i	0.285606i	−0.989751	+	0.142803i	$0.954388\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.19615i	0.283896i
$336$	0	0
$337$	1.41421i	0.0770371i	0.999258	+	0.0385186i	$0.0122639\pi$
$337$	1.41421i	0.0770371i	−0.999258	+	0.0385186i	$0.987736\pi$
$338$	0	0
$339$	22.5167i	1.22294i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.5959	−1.05808
$344$	0	0
$345$	−15.0000	−0.807573
$346$	0	0
$347$	9.79796	0.525982	0.262991	−	0.964798i	$-0.415291\pi$
$347$	9.79796	0.525982	0.262991	+	0.964798i	$0.415291\pi$
$348$	0	0
$349$	− 24.0416i	− 1.28692i	−0.765480	−	0.643459i	$-0.777498\pi$
$349$	− 24.0416i	− 1.28692i	0.765480	−	0.643459i	$-0.222502\pi$
$350$	0	0
$351$	−29.3939	−1.56893
$352$	0	0
$353$	17.0000	0.904819	0.452409	−	0.891810i	$-0.350565\pi$
$353$	17.0000	0.904819	0.452409	+	0.891810i	$0.350565\pi$
$354$	0	0
$355$	− 12.1244i	− 0.643494i
$356$	0	0
$357$	18.0000	0.952661
$358$	0	0
$359$	−29.3939	−1.55135	−0.775675	−	0.631133i	$-0.782590\pi$
$359$	−29.3939	−1.55135	−0.775675	+	0.631133i	$0.782590\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.3137i	0.592187i
$366$	0	0
$367$	− 1.73205i	− 0.0904123i	−0.998978	−	0.0452062i	$-0.985606\pi$
$367$	− 1.73205i	− 0.0904123i	0.998978	−	0.0452062i	$-0.0143945\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.79796	0.508685
$372$	0	0
$373$	− 18.3848i	− 0.951928i	−0.879465	−	0.475964i	$-0.842099\pi$
$373$	− 18.3848i	− 0.951928i	0.879465	−	0.475964i	$-0.157901\pi$
$374$	0	0
$375$	− 15.5885i	− 0.804984i
$376$	0	0
$377$	32.0000	1.64808
$378$	0	0
$379$	− 15.5885i	− 0.800725i	−0.916357	−	0.400363i	$-0.868884\pi$
$379$	− 15.5885i	− 0.800725i	0.916357	−	0.400363i	$-0.131116\pi$
$380$	0	0
$381$	− 33.9411i	− 1.73886i
$382$	0	0
$383$	− 5.19615i	− 0.265511i	−0.991149	−	0.132755i	$-0.957617\pi$
$383$	− 5.19615i	− 0.265511i	0.991149	−	0.132755i	$-0.0423825\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.0000	0.557722	0.278861	−	0.960331i	$-0.410043\pi$
$389$	11.0000	0.557722	0.278861	+	0.960331i	$0.410043\pi$
$390$	0	0
$391$	36.7423	1.85814
$392$	0	0
$393$	29.6985i	1.49809i
$394$	0	0
$395$	7.34847	0.369742
$396$	0	0
$397$	36.0000	1.80679	0.903394	−	0.428811i	$-0.141067\pi$
$397$	36.0000	1.80679	0.903394	+	0.428811i	$0.141067\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	0	0
$403$	9.79796	0.488071
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.5563i	0.769212i	0.923081	+	0.384606i	$0.125663\pi$
$409$	15.5563i	0.769212i	−0.923081	+	0.384606i	$0.874337\pi$
$410$	0	0
$411$	− 19.0526i	− 0.939793i
$412$	0	0
$413$	29.6985i	1.46137i
$414$	0	0
$415$	−2.44949	−0.120241
$416$	0	0
$417$	− 21.2132i	− 1.03882i
$418$	0	0
$419$	− 13.8564i	− 0.676930i	−0.940979	−	0.338465i	$-0.890092\pi$
$419$	− 13.8564i	− 0.676930i	0.940979	−	0.338465i	$-0.109908\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	16.9706i	0.823193i
$426$	0	0
$427$	− 13.8564i	− 0.670559i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.1464	−0.825914	−0.412957	−	0.910750i	$-0.635504\pi$
$431$	−17.1464	−0.825914	−0.412957	+	0.910750i	$0.635504\pi$
$432$	0	0
$433$	3.00000	0.144171	0.0720854	−	0.997398i	$-0.477035\pi$
$433$	3.00000	0.144171	0.0720854	+	0.997398i	$0.477035\pi$
$434$	0	0
$435$	9.79796	0.469776
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−19.5959	−0.935262	−0.467631	−	0.883924i	$-0.654892\pi$
$439$	−19.5959	−0.935262	−0.467631	+	0.883924i	$0.654892\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.19615i	− 0.246877i	−0.992352	−	0.123438i	$-0.960608\pi$
$443$	− 5.19615i	− 0.246877i	0.992352	−	0.123438i	$-0.0393921\pi$
$444$	0	0
$445$	−17.0000	−0.805877
$446$	0	0
$447$	22.0454	1.04271
$448$	0	0
$449$	25.0000	1.17982	0.589911	−	0.807468i	$-0.299163\pi$
$449$	25.0000	1.17982	0.589911	+	0.807468i	$0.299163\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	4.24264i	0.199337i
$454$	0	0
$455$	13.8564i	0.649598i
$456$	0	0
$457$	22.6274i	1.05847i	0.848477	+	0.529233i	$0.177520\pi$
$457$	22.6274i	1.05847i	−0.848477	+	0.529233i	$0.822480\pi$
$458$	0	0
$459$	22.0454	1.02899
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 8.66025i	− 0.402476i	−0.979542	−	0.201238i	$-0.935504\pi$
$463$	− 8.66025i	− 0.402476i	0.979542	−	0.201238i	$-0.0644965\pi$
$464$	0	0
$465$	3.00000	0.139122
$466$	0	0
$467$	− 19.0526i	− 0.881647i	−0.897594	−	0.440824i	$-0.854686\pi$
$467$	− 19.0526i	− 0.881647i	0.897594	−	0.440824i	$-0.145314\pi$
$468$	0	0
$469$	12.7279i	0.587721i
$470$	0	0
$471$	5.19615i	0.239426i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.1918	1.79072	0.895360	−	0.445342i	$-0.146918\pi$
$479$	39.1918	1.79072	0.895360	+	0.445342i	$0.146918\pi$
$480$	0	0
$481$	28.2843i	1.28965i
$482$	0	0
$483$	−36.7423	−1.67183
$484$	0	0
$485$	7.00000	0.317854
$486$	0	0
$487$	− 15.5885i	− 0.706380i	−0.935552	−	0.353190i	$-0.885097\pi$
$487$	− 15.5885i	− 0.706380i	0.935552	−	0.353190i	$-0.114903\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	−29.3939	−1.32653	−0.663264	−	0.748386i	$-0.730829\pi$
$491$	−29.3939	−1.32653	−0.663264	+	0.748386i	$0.730829\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 29.6985i	− 1.33216i
$498$	0	0
$499$	− 41.5692i	− 1.86089i	−0.366427	−	0.930447i	$-0.619419\pi$
$499$	− 41.5692i	− 1.86089i	0.366427	−	0.930447i	$-0.380581\pi$
$500$	0	0
$501$	− 16.9706i	− 0.758189i
$502$	0	0
$503$	−26.9444	−1.20139	−0.600695	−	0.799478i	$-0.705110\pi$
$503$	−26.9444	−1.20139	−0.600695	+	0.799478i	$0.705110\pi$
$504$	0	0
$505$	− 1.41421i	− 0.0629317i
$506$	0	0
$507$	− 32.9090i	− 1.46154i
$508$	0	0
$509$	−23.0000	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$509$	−23.0000	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$510$	0	0
$511$	27.7128i	1.22594i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6.92820i	− 0.305293i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.2474	0.537603
$520$	0	0
$521$	−31.0000	−1.35813	−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000	−1.35813	−0.679067	+	0.734076i	$0.737616\pi$
$522$	0	0
$523$	−19.5959	−0.856870	−0.428435	−	0.903573i	$-0.640935\pi$
$523$	−19.5959	−0.856870	−0.428435	+	0.903573i	$0.640935\pi$
$524$	0	0
$525$	− 16.9706i	− 0.740656i
$526$	0	0
$527$	−7.34847	−0.320104
$528$	0	0
$529$	−52.0000	−2.26087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.0000	1.38607
$534$	0	0
$535$	−12.2474	−0.529503
$536$	0	0
$537$	−9.00000	−0.388379
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 28.2843i	− 1.21604i	−0.793923	−	0.608018i	$-0.791965\pi$
$541$	− 28.2843i	− 1.21604i	0.793923	−	0.608018i	$-0.208035\pi$
$542$	0	0
$543$	5.19615i	0.222988i
$544$	0	0
$545$	7.07107i	0.302891i
$546$	0	0
$547$	19.5959	0.837861	0.418930	−	0.908018i	$-0.362405\pi$
$547$	19.5959	0.837861	0.418930	+	0.908018i	$0.362405\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	18.0000	0.765438
$554$	0	0
$555$	8.66025i	0.367607i
$556$	0	0
$557$	4.24264i	0.179766i	0.995952	+	0.0898832i	$0.0286494\pi$
$557$	4.24264i	0.179766i	−0.995952	+	0.0898832i	$0.971351\pi$
$558$	0	0
$559$	69.2820i	2.93032i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.79796	−0.412935	−0.206467	−	0.978453i	$-0.566197\pi$
$563$	−9.79796	−0.412935	−0.206467	+	0.978453i	$0.566197\pi$
$564$	0	0
$565$	13.0000	0.546914
$566$	0	0
$567$	−22.0454	−0.925820
$568$	0	0
$569$	− 16.9706i	− 0.711443i	−0.934592	−	0.355722i	$-0.884235\pi$
$569$	− 16.9706i	− 0.711443i	0.934592	−	0.355722i	$-0.115765\pi$
$570$	0	0
$571$	9.79796	0.410032	0.205016	−	0.978759i	$-0.434275\pi$
$571$	9.79796	0.410032	0.205016	+	0.978759i	$0.434275\pi$
$572$	0	0
$573$	9.00000	0.375980
$574$	0	0
$575$	− 34.6410i	− 1.44463i
$576$	0	0
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	0	0
$579$	−9.79796	−0.407189
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.44949	−0.100759
$592$	0	0
$593$	− 39.5980i	− 1.62609i	−0.582198	−	0.813047i	$-0.697807\pi$
$593$	− 39.5980i	− 1.62609i	0.582198	−	0.813047i	$-0.302193\pi$
$594$	0	0
$595$	− 10.3923i	− 0.426043i
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	− 34.6410i	− 1.41539i	−0.706516	−	0.707697i	$-0.749734\pi$
$599$	− 34.6410i	− 1.41539i	0.706516	−	0.707697i	$-0.250266\pi$
$600$	0	0
$601$	9.89949i	0.403809i	0.979405	+	0.201904i	$0.0647130\pi$
$601$	9.89949i	0.403809i	−0.979405	+	0.201904i	$0.935287\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.6413	1.69017	0.845085	−	0.534633i	$-0.179550\pi$
$607$	41.6413	1.69017	0.845085	+	0.534633i	$0.179550\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	−39.1918	−1.58553
$612$	0	0
$613$	28.2843i	1.14239i	0.820814	+	0.571195i	$0.193520\pi$
$613$	28.2843i	1.14239i	−0.820814	+	0.571195i	$0.806480\pi$
$614$	0	0
$615$	9.79796	0.395092
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	1.73205i	0.0696170i	0.999394	+	0.0348085i	$0.0110821\pi$
$619$	1.73205i	0.0696170i	−0.999394	+	0.0348085i	$0.988918\pi$
$620$	0	0
$621$	−45.0000	−1.80579
$622$	0	0
$623$	−41.6413	−1.66832
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 21.2132i	− 0.845826i
$630$	0	0
$631$	8.66025i	0.344759i	0.985031	+	0.172380i	$0.0551456\pi$
$631$	8.66025i	0.344759i	−0.985031	+	0.172380i	$0.944854\pi$
$632$	0	0
$633$	33.9411i	1.34904i
$634$	0	0
$635$	−19.5959	−0.777640
$636$	0	0
$637$	− 5.65685i	− 0.224133i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	0	0
$643$	25.9808i	1.02458i	0.858812	+	0.512291i	$0.171203\pi$
$643$	25.9808i	1.02458i	−0.858812	+	0.512291i	$0.828797\pi$
$644$	0	0
$645$	21.2132i	0.835269i
$646$	0	0
$647$	− 43.3013i	− 1.70235i	−0.524883	−	0.851174i	$-0.675891\pi$
$647$	− 43.3013i	− 1.70235i	0.524883	−	0.851174i	$-0.324109\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	7.34847	0.288009
$652$	0	0
$653$	−23.0000	−0.900060	−0.450030	−	0.893014i	$-0.648587\pi$
$653$	−23.0000	−0.900060	−0.450030	+	0.893014i	$0.648587\pi$
$654$	0	0
$655$	17.1464	0.669966
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.2474	0.477093	0.238546	−	0.971131i	$-0.423329\pi$
$659$	12.2474	0.477093	0.238546	+	0.971131i	$0.423329\pi$
$660$	0	0
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	0	0
$663$	41.5692i	1.61441i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.9898	1.89689
$668$	0	0
$669$	−9.00000	−0.347960
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 26.8701i	− 1.03576i	−0.855452	−	0.517882i	$-0.826721\pi$
$673$	− 26.8701i	− 1.03576i	0.855452	−	0.517882i	$-0.173279\pi$
$674$	0	0
$675$	− 20.7846i	− 0.800000i
$676$	0	0
$677$	− 7.07107i	− 0.271763i	−0.990725	−	0.135882i	$-0.956613\pi$
$677$	− 7.07107i	− 0.271763i	0.990725	−	0.135882i	$-0.0433867\pi$
$678$	0	0
$679$	17.1464	0.658020
$680$	0	0
$681$	4.24264i	0.162578i
$682$	0	0
$683$	17.3205i	0.662751i	0.943499	+	0.331375i	$0.107513\pi$
$683$	17.3205i	0.662751i	−0.943499	+	0.331375i	$0.892487\pi$
$684$	0	0
$685$	−11.0000	−0.420288
$686$	0	0
$687$	15.5885i	0.594737i
$688$	0	0
$689$	22.6274i	0.862036i
$690$	0	0
$691$	− 12.1244i	− 0.461232i	−0.973045	−	0.230616i	$-0.925926\pi$
$691$	− 12.1244i	− 0.461232i	0.973045	−	0.230616i	$-0.0740742\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.2474	−0.464572
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	39.1918	1.48237
$700$	0	0
$701$	− 4.24264i	− 0.160242i	−0.996785	−	0.0801212i	$-0.974469\pi$
$701$	− 4.24264i	− 0.160242i	0.996785	−	0.0801212i	$-0.0255307\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	− 3.46410i	− 0.130281i
$708$	0	0
$709$	33.0000	1.23934	0.619671	−	0.784862i	$-0.287266\pi$
$709$	33.0000	1.23934	0.619671	+	0.784862i	$0.287266\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.0000	0.561754
$714$	0	0
$715$	0	0
$716$	0	0
$717$	38.1838i	1.42600i
$718$	0	0
$719$	− 1.73205i	− 0.0645946i	−0.999478	−	0.0322973i	$-0.989718\pi$
$719$	− 1.73205i	− 0.0645946i	0.999478	−	0.0322973i	$-0.0102823\pi$
$720$	0	0
$721$	− 16.9706i	− 0.632017i
$722$	0	0
$723$	39.1918	1.45756
$724$	0	0
$725$	22.6274i	0.840361i
$726$	0	0
$727$	− 12.1244i	− 0.449667i	−0.974397	−	0.224834i	$-0.927816\pi$
$727$	− 12.1244i	− 0.449667i	0.974397	−	0.224834i	$-0.0721839\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	− 51.9615i	− 1.92187i
$732$	0	0
$733$	− 11.3137i	− 0.417881i	−0.977928	−	0.208941i	$-0.932998\pi$
$733$	− 11.3137i	− 0.417881i	0.977928	−	0.208941i	$-0.0670016\pi$
$734$	0	0
$735$	− 1.73205i	− 0.0638877i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	7.34847	0.270318	0.135159	−	0.990824i	$-0.456846\pi$
$739$	7.34847	0.270318	0.135159	+	0.990824i	$0.456846\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	− 12.7279i	− 0.466315i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−30.0000	−1.09618
$750$	0	0
$751$	36.3731i	1.32727i	0.748056	+	0.663636i	$0.230988\pi$
$751$	36.3731i	1.32727i	−0.748056	+	0.663636i	$0.769012\pi$
$752$	0	0
$753$	15.0000	0.546630
$754$	0	0
$755$	2.44949	0.0891461
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.9706i	0.615182i	0.951519	+	0.307591i	$0.0995229\pi$
$761$	16.9706i	0.615182i	−0.951519	+	0.307591i	$0.900477\pi$
$762$	0	0
$763$	17.3205i	0.627044i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−68.5857	−2.47649
$768$	0	0
$769$	45.2548i	1.63193i	0.578101	+	0.815966i	$0.303794\pi$
$769$	45.2548i	1.63193i	−0.578101	+	0.815966i	$0.696206\pi$
$770$	0	0
$771$	13.8564i	0.499026i
$772$	0	0
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	6.92820i	0.248868i
$776$	0	0
$777$	21.2132i	0.761019i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	29.3939	1.05045
$784$	0	0
$785$	3.00000	0.107075
$786$	0	0
$787$	−29.3939	−1.04778	−0.523889	−	0.851786i	$-0.675519\pi$
$787$	−29.3939	−1.04778	−0.523889	+	0.851786i	$0.675519\pi$
$788$	0	0
$789$	29.6985i	1.05729i
$790$	0	0
$791$	31.8434	1.13222
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	6.92820i	0.245718i
$796$	0	0
$797$	23.0000	0.814702	0.407351	−	0.913272i	$-0.366453\pi$
$797$	23.0000	0.814702	0.407351	+	0.913272i	$0.366453\pi$
$798$	0	0
$799$	29.3939	1.03988
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	21.2132i	0.747667i
$806$	0	0
$807$	6.92820i	0.243884i
$808$	0	0
$809$	− 16.9706i	− 0.596653i	−0.954464	−	0.298327i	$-0.903572\pi$
$809$	− 16.9706i	− 0.596653i	0.954464	−	0.298327i	$-0.0964285\pi$
$810$	0	0
$811$	−17.1464	−0.602093	−0.301046	−	0.953610i	$-0.597336\pi$
$811$	−17.1464	−0.602093	−0.301046	+	0.953610i	$0.597336\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.8564i	0.485369i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 43.8406i	− 1.53005i	−0.644002	−	0.765024i	$-0.722727\pi$
$821$	− 43.8406i	− 1.53005i	0.644002	−	0.765024i	$-0.277273\pi$
$822$	0	0
$823$	− 19.0526i	− 0.664130i	−0.943256	−	0.332065i	$-0.892255\pi$
$823$	− 19.0526i	− 0.664130i	0.943256	−	0.332065i	$-0.107745\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.3939	−1.02213	−0.511063	−	0.859543i	$-0.670748\pi$
$827$	−29.3939	−1.02213	−0.511063	+	0.859543i	$0.670748\pi$
$828$	0	0
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	−41.6413	−1.44452
$832$	0	0
$833$	4.24264i	0.146999i
$834$	0	0
$835$	−9.79796	−0.339072
$836$	0	0
$837$	9.00000	0.311086
$838$	0	0
$839$	1.73205i	0.0597970i	0.999553	+	0.0298985i	$0.00951841\pi$
$839$	1.73205i	0.0597970i	−0.999553	+	0.0298985i	$0.990482\pi$
$840$	0	0
$841$	−3.00000	−0.103448
$842$	0	0
$843$	−17.1464	−0.590554
$844$	0	0
$845$	−19.0000	−0.653620
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.9706i	0.582428i
$850$	0	0
$851$	43.3013i	1.48435i
$852$	0	0
$853$	− 7.07107i	− 0.242109i	−0.992646	−	0.121054i	$-0.961372\pi$
$853$	− 7.07107i	− 0.242109i	0.992646	−	0.121054i	$-0.0386275\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.9706i	0.579703i	0.957072	+	0.289852i	$0.0936060\pi$
$857$	16.9706i	0.579703i	−0.957072	+	0.289852i	$0.906394\pi$
$858$	0	0
$859$	43.3013i	1.47742i	0.674023	+	0.738710i	$0.264565\pi$
$859$	43.3013i	1.47742i	−0.674023	+	0.738710i	$0.735435\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	− 6.92820i	− 0.235839i	−0.993023	−	0.117919i	$-0.962378\pi$
$863$	− 6.92820i	− 0.235839i	0.993023	−	0.117919i	$-0.0376224\pi$
$864$	0	0
$865$	− 7.07107i	− 0.240424i
$866$	0	0
$867$	− 1.73205i	− 0.0588235i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−29.3939	−0.995974
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22.0454	−0.745271
$876$	0	0
$877$	− 22.6274i	− 0.764074i	−0.924147	−	0.382037i	$-0.875223\pi$
$877$	− 22.6274i	− 0.764074i	0.924147	−	0.382037i	$-0.124777\pi$
$878$	0	0
$879$	58.7878	1.98286
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	0	0
$883$	41.5692i	1.39892i	0.714674	+	0.699458i	$0.246575\pi$
$883$	41.5692i	1.39892i	−0.714674	+	0.699458i	$0.753425\pi$
$884$	0	0
$885$	−21.0000	−0.705907
$886$	0	0
$887$	17.1464	0.575721	0.287860	−	0.957672i	$-0.407056\pi$
$887$	17.1464	0.575721	0.287860	+	0.957672i	$0.407056\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	5.19615i	0.173688i
$896$	0	0
$897$	− 84.8528i	− 2.83315i
$898$	0	0
$899$	−9.79796	−0.326780
$900$	0	0
$901$	− 16.9706i	− 0.565371i
$902$	0	0
$903$	51.9615i	1.72917i
$904$	0	0
$905$	3.00000	0.0997234
$906$	0	0
$907$	41.5692i	1.38028i	0.723674	+	0.690142i	$0.242452\pi$
$907$	41.5692i	1.38028i	−0.723674	+	0.690142i	$0.757548\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 17.3205i	− 0.573854i	−0.957952	−	0.286927i	$-0.907366\pi$
$911$	− 17.3205i	− 0.573854i	0.957952	−	0.286927i	$-0.0926337\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	9.79796	0.323911
$916$	0	0
$917$	42.0000	1.38696
$918$	0	0
$919$	−17.1464	−0.565608	−0.282804	−	0.959178i	$-0.591265\pi$
$919$	−17.1464	−0.565608	−0.282804	+	0.959178i	$0.591265\pi$
$920$	0	0
$921$	− 50.9117i	− 1.67760i
$922$	0	0
$923$	68.5857	2.25753
$924$	0	0
$925$	−20.0000	−0.657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	54.0000	1.76788
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.6274i	0.739205i	0.929190	+	0.369603i	$0.120506\pi$
$937$	22.6274i	0.739205i	−0.929190	+	0.369603i	$0.879494\pi$
$938$	0	0
$939$	− 19.0526i	− 0.621757i
$940$	0	0
$941$	35.3553i	1.15255i	0.817255	+	0.576276i	$0.195494\pi$
$941$	35.3553i	1.15255i	−0.817255	+	0.576276i	$0.804506\pi$
$942$	0	0
$943$	48.9898	1.59533
$944$	0	0
$945$	12.7279i	0.414039i
$946$	0	0
$947$	25.9808i	0.844261i	0.906535	+	0.422131i	$0.138718\pi$
$947$	25.9808i	0.844261i	−0.906535	+	0.422131i	$0.861282\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	0	0
$951$	39.8372i	1.29181i
$952$	0	0
$953$	− 55.1543i	− 1.78662i	−0.449437	−	0.893312i	$-0.648375\pi$
$953$	− 55.1543i	− 1.78662i	0.449437	−	0.893312i	$-0.351625\pi$
$954$	0	0
$955$	− 5.19615i	− 0.168144i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26.9444	−0.870080
$960$	0	0
$961$	28.0000	0.903226
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.65685i	0.182101i
$966$	0	0
$967$	−19.5959	−0.630162	−0.315081	−	0.949065i	$-0.602032\pi$
$967$	−19.5959	−0.630162	−0.315081	+	0.949065i	$0.602032\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.19615i	0.166752i	0.996518	+	0.0833762i	$0.0265703\pi$
$971$	5.19615i	0.166752i	−0.996518	+	0.0833762i	$0.973430\pi$
$972$	0	0
$973$	−30.0000	−0.961756
$974$	0	0
$975$	39.1918	1.25514
$976$	0	0
$977$	43.0000	1.37569	0.687846	−	0.725857i	$-0.258556\pi$
$977$	43.0000	1.37569	0.687846	+	0.725857i	$0.258556\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.9808i	0.828658i	0.910127	+	0.414329i	$0.135984\pi$
$983$	25.9808i	0.828658i	−0.910127	+	0.414329i	$0.864016\pi$
$984$	0	0
$985$	1.41421i	0.0450606i
$986$	0	0
$987$	−29.3939	−0.935617
$988$	0	0
$989$	106.066i	3.37270i
$990$	0	0
$991$	20.7846i	0.660245i	0.943938	+	0.330122i	$0.107090\pi$
$991$	20.7846i	0.660245i	−0.943938	+	0.330122i	$0.892910\pi$
$992$	0	0
$993$	−9.00000	−0.285606
$994$	0	0
$995$	− 6.92820i	− 0.219639i
$996$	0	0
$997$	32.5269i	1.03014i	0.857149	+	0.515069i	$0.172234\pi$
$997$	32.5269i	1.03014i	−0.857149	+	0.515069i	$0.827766\pi$
$998$	0	0
$999$	25.9808i	0.821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.2.e.d.1935.4	yes	4
4.3	odd	2	inner	1936.2.e.d.1935.1	✓	4
11.10	odd	2	inner	1936.2.e.d.1935.3	yes	4
44.43	even	2	inner	1936.2.e.d.1935.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1936.2.e.d.1935.1	✓	4	4.3	odd	2	inner
1936.2.e.d.1935.2	yes	4	44.43	even	2	inner
1936.2.e.d.1935.3	yes	4	11.10	odd	2	inner
1936.2.e.d.1935.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1936.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +1.00000 q^{5} +2.44949 q^{7} +O(q^{10})\)	\(q+1.73205i q^{3} +1.00000 q^{5} +2.44949 q^{7} +5.65685i q^{13} +1.73205i q^{15} -4.24264i q^{17} +4.24264i q^{21} +8.66025i q^{23} -4.00000 q^{25} +5.19615i q^{27} -5.65685i q^{29} -1.73205i q^{31} +2.44949 q^{35} +5.00000 q^{37} -9.79796 q^{39} -5.65685i q^{41} +12.2474 q^{43} +6.92820i q^{47} -1.00000 q^{49} +7.34847 q^{51} +4.00000 q^{53} +12.1244i q^{59} -5.65685i q^{61} +5.65685i q^{65} +5.19615i q^{67} -15.0000 q^{69} -12.1244i q^{71} +11.3137i q^{73} -6.92820i q^{75} +7.34847 q^{79} -9.00000 q^{81} -2.44949 q^{83} -4.24264i q^{85} +9.79796 q^{87} -17.0000 q^{89} +13.8564i q^{91} +3.00000 q^{93} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} - 16 q^{25} + 20 q^{37} - 4 q^{49} + 16 q^{53} - 60 q^{69} - 36 q^{81} - 68 q^{89} + 12 q^{93} + 28 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 16 * q^25 + 20 * q^37 - 4 * q^49 + 16 * q^53 - 60 * q^69 - 36 * q^81 - 68 * q^89 + 12 * q^93 + 28 * q^97

Introduction

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