LMFDB - Embedded newform 912.2.f.g.113.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(113,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("912.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.f (of order $2$, degree $1$, not 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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.3
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	912.113
Dual form			912.2.f.g.113.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 - 0.707107i) q^{3} -2.23607i q^{5} +1.00000 q^{7} +(2.00000 - 2.23607i) q^{9} +O(q^{10})$	$q+(1.58114 - 0.707107i) q^{3} -2.23607i q^{5} +1.00000 q^{7} +(2.00000 - 2.23607i) q^{9} +2.23607i q^{11} -4.24264i q^{13} +(-1.58114 - 3.53553i) q^{15} -2.23607i q^{17} +(-1.00000 + 4.24264i) q^{19} +(1.58114 - 0.707107i) q^{21} -4.47214i q^{23} +(1.58114 - 4.94975i) q^{27} +4.24264i q^{31} +(1.58114 + 3.53553i) q^{33} -2.23607i q^{35} +4.24264i q^{37} +(-3.00000 - 6.70820i) q^{39} -9.48683 q^{41} +7.00000 q^{43} +(-5.00000 - 4.47214i) q^{45} -11.1803i q^{47} -6.00000 q^{49} +(-1.58114 - 3.53553i) q^{51} +9.48683 q^{53} +5.00000 q^{55} +(1.41886 + 7.41531i) q^{57} -9.48683 q^{59} +11.0000 q^{61} +(2.00000 - 2.23607i) q^{63} -9.48683 q^{65} +8.48528i q^{67} +(-3.16228 - 7.07107i) q^{69} -9.48683 q^{71} +5.00000 q^{73} +2.23607i q^{77} +(-1.00000 - 8.94427i) q^{81} +8.94427i q^{83} -5.00000 q^{85} +9.48683 q^{89} -4.24264i q^{91} +(3.00000 + 6.70820i) q^{93} +(9.48683 + 2.23607i) q^{95} +12.7279i q^{97} +(5.00000 + 4.47214i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 + 8 * q^9	$ 4 q + 4 q^{7} + 8 q^{9} - 4 q^{19} - 12 q^{39} + 28 q^{43} - 20 q^{45} - 24 q^{49} + 20 q^{55} + 12 q^{57} + 44 q^{61} + 8 q^{63} + 20 q^{73} - 4 q^{81} - 20 q^{85} + 12 q^{93} + 20 q^{99}+O(q^{100}) $ 4 * q + 4 * q^7 + 8 * q^9 - 4 * q^19 - 12 * q^39 + 28 * q^43 - 20 * q^45 - 24 * q^49 + 20 * q^55 + 12 * q^57 + 44 * q^61 + 8 * q^63 + 20 * q^73 - 4 * q^81 - 20 * q^85 + 12 * q^93 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	1.58114	−	0.707107i	0.912871	−	0.408248i
$3$	1.58114	−	0.707107i	0.912871	−	0.408248i
$4$		0			0
$5$		−	2.23607i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$5$		−	2.23607i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$		0			0
$7$	1.00000			0.377964			0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000			0.377964			0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	2.00000	−	2.23607i	0.666667	−	0.745356i
$10$		0			0
$11$			2.23607i			0.674200i	0.941469	+	0.337100i	$0.109446\pi$
$11$			2.23607i			0.674200i	−0.941469	+	0.337100i	$0.890554\pi$
$12$		0			0
$13$		−	4.24264i		−	1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$		−	4.24264i		−	1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$		0			0
$15$	−1.58114	−	3.53553i	−0.408248	−	0.912871i
$16$		0			0
$17$		−	2.23607i		−	0.542326i	−0.962533	−	0.271163i	$-0.912592\pi$
$17$		−	2.23607i		−	0.542326i	0.962533	−	0.271163i	$-0.0874083\pi$
$18$		0			0
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$20$		0			0
$21$	1.58114	−	0.707107i	0.345033	−	0.154303i
$22$		0			0
$23$		−	4.47214i		−	0.932505i	−0.884652	−	0.466252i	$-0.845604\pi$
$23$		−	4.47214i		−	0.932505i	0.884652	−	0.466252i	$-0.154396\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.58114	−	4.94975i	0.304290	−	0.952579i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			4.24264i			0.762001i	0.924575	+	0.381000i	$0.124420\pi$
$31$			4.24264i			0.762001i	−0.924575	+	0.381000i	$0.875580\pi$
$32$		0			0
$33$	1.58114	+	3.53553i	0.275241	+	0.615457i
$34$		0			0
$35$		−	2.23607i		−	0.377964i
$36$		0			0
$37$			4.24264i			0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$			4.24264i			0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$		0			0
$39$	−3.00000	−	6.70820i	−0.480384	−	1.07417i
$40$		0			0
$41$	−9.48683			−1.48159			−0.740797	−	0.671729i	$-0.765552\pi$
$41$	−9.48683			−1.48159			−0.740797	+	0.671729i	$0.765552\pi$
$42$		0			0
$43$	7.00000			1.06749			0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000			1.06749			0.533745	+	0.845645i	$0.320784\pi$
$44$		0			0
$45$	−5.00000	−	4.47214i	−0.745356	−	0.666667i
$46$		0			0
$47$		−	11.1803i		−	1.63082i	−0.578884	−	0.815410i	$-0.696511\pi$
$47$		−	11.1803i		−	1.63082i	0.578884	−	0.815410i	$-0.303489\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$	−1.58114	−	3.53553i	−0.221404	−	0.495074i
$52$		0			0
$53$	9.48683			1.30312			0.651558	−	0.758599i	$-0.274116\pi$
$53$	9.48683			1.30312			0.651558	+	0.758599i	$0.274116\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		0			0
$57$	1.41886	+	7.41531i	0.187933	+	0.982182i
$58$		0			0
$59$	−9.48683			−1.23508			−0.617540	−	0.786539i	$-0.711871\pi$
$59$	−9.48683			−1.23508			−0.617540	+	0.786539i	$0.711871\pi$
$60$		0			0
$61$	11.0000			1.40841			0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000			1.40841			0.704203	+	0.709999i	$0.251305\pi$
$62$		0			0
$63$	2.00000	−	2.23607i	0.251976	−	0.281718i
$64$		0			0
$65$	−9.48683			−1.17670
$66$		0			0
$67$			8.48528i			1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$67$			8.48528i			1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$	−3.16228	−	7.07107i	−0.380693	−	0.851257i
$70$		0			0
$71$	−9.48683			−1.12588			−0.562940	−	0.826498i	$-0.690330\pi$
$71$	−9.48683			−1.12588			−0.562940	+	0.826498i	$0.690330\pi$
$72$		0			0
$73$	5.00000			0.585206			0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000			0.585206			0.292603	+	0.956234i	$0.405479\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.23607i			0.254824i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$		0			0
$83$			8.94427i			0.981761i	0.871227	+	0.490881i	$0.163325\pi$
$83$			8.94427i			0.981761i	−0.871227	+	0.490881i	$0.836675\pi$
$84$		0			0
$85$	−5.00000			−0.542326
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.48683			1.00560			0.502801	−	0.864402i	$-0.332303\pi$
$89$	9.48683			1.00560			0.502801	+	0.864402i	$0.332303\pi$
$90$		0			0
$91$		−	4.24264i		−	0.444750i
$92$		0			0
$93$	3.00000	+	6.70820i	0.311086	+	0.695608i
$94$		0			0
$95$	9.48683	+	2.23607i	0.973329	+	0.229416i
$96$		0			0
$97$			12.7279i			1.29232i	0.763200	+	0.646162i	$0.223627\pi$
$97$			12.7279i			1.29232i	−0.763200	+	0.646162i	$0.776373\pi$
$98$		0			0
$99$	5.00000	+	4.47214i	0.502519	+	0.449467i
$100$		0			0
$101$		−	8.94427i		−	0.889988i	−0.895533	−	0.444994i	$-0.853206\pi$
$101$		−	8.94427i		−	0.889988i	0.895533	−	0.444994i	$-0.146794\pi$
$102$		0			0
$103$			8.48528i			0.836080i	0.908429	+	0.418040i	$0.137283\pi$
$103$			8.48528i			0.836080i	−0.908429	+	0.418040i	$0.862717\pi$
$104$		0			0
$105$	−1.58114	−	3.53553i	−0.154303	−	0.345033i
$106$		0			0
$107$	18.9737			1.83425			0.917127	−	0.398596i	$-0.130502\pi$
$107$	18.9737			1.83425			0.917127	+	0.398596i	$0.130502\pi$
$108$		0			0
$109$			8.48528i			0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$			8.48528i			0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$		0			0
$111$	3.00000	+	6.70820i	0.284747	+	0.636715i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−10.0000			−0.932505
$116$		0			0
$117$	−9.48683	−	8.48528i	−0.877058	−	0.784465i
$118$		0			0
$119$		−	2.23607i		−	0.204980i
$120$		0			0
$121$	6.00000			0.545455
$122$		0			0
$123$	−15.0000	+	6.70820i	−1.35250	+	0.604858i
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$		−	16.9706i		−	1.50589i	−0.658081	−	0.752947i	$-0.728632\pi$
$127$		−	16.9706i		−	1.50589i	0.658081	−	0.752947i	$-0.271368\pi$
$128$		0			0
$129$	11.0680	−	4.94975i	0.974481	−	0.435801i
$130$		0			0
$131$			15.6525i			1.36756i	0.729687	+	0.683782i	$0.239666\pi$
$131$			15.6525i			1.36756i	−0.729687	+	0.683782i	$0.760334\pi$
$132$		0			0
$133$	−1.00000	+	4.24264i	−0.0867110	+	0.367884i
$134$		0			0
$135$	−11.0680	−	3.53553i	−0.952579	−	0.304290i
$136$		0			0
$137$			11.1803i			0.955201i	0.878577	+	0.477600i	$0.158493\pi$
$137$			11.1803i			0.955201i	−0.878577	+	0.477600i	$0.841507\pi$
$138$		0			0
$139$	−11.0000			−0.933008			−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000			−0.933008			−0.466504	+	0.884519i	$0.654487\pi$
$140$		0			0
$141$	−7.90569	−	17.6777i	−0.665780	−	1.48873i
$142$		0			0
$143$	9.48683			0.793329
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−9.48683	+	4.24264i	−0.782461	+	0.349927i
$148$		0			0
$149$			11.1803i			0.915929i	0.888970	+	0.457965i	$0.151421\pi$
$149$			11.1803i			0.915929i	−0.888970	+	0.457965i	$0.848579\pi$
$150$		0			0
$151$			4.24264i			0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$			4.24264i			0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$		0			0
$153$	−5.00000	−	4.47214i	−0.404226	−	0.361551i
$154$		0			0
$155$	9.48683			0.762001
$156$		0			0
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$		0			0
$159$	15.0000	−	6.70820i	1.18958	−	0.531995i
$160$		0			0
$161$		−	4.47214i		−	0.352454i
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$	7.90569	−	3.53553i	0.615457	−	0.275241i
$166$		0			0
$167$	−18.9737			−1.46823			−0.734113	−	0.679027i	$-0.762402\pi$
$167$	−18.9737			−1.46823			−0.734113	+	0.679027i	$0.762402\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	7.48683	+	10.7213i	0.572532	+	0.819882i
$172$		0			0
$173$	18.9737			1.44254			0.721271	−	0.692653i	$-0.243558\pi$
$173$	18.9737			1.44254			0.721271	+	0.692653i	$0.243558\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−15.0000	+	6.70820i	−1.12747	+	0.504219i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$			25.4558i			1.89212i	0.323994	+	0.946059i	$0.394974\pi$
$181$			25.4558i			1.89212i	−0.323994	+	0.946059i	$0.605026\pi$
$182$		0			0
$183$	17.3925	−	7.77817i	1.28569	−	0.574979i
$184$		0			0
$185$	9.48683			0.697486
$186$		0			0
$187$	5.00000			0.365636
$188$		0			0
$189$	1.58114	−	4.94975i	0.115011	−	0.360041i
$190$		0			0
$191$			2.23607i			0.161796i	0.996722	+	0.0808981i	$0.0257788\pi$
$191$			2.23607i			0.161796i	−0.996722	+	0.0808981i	$0.974221\pi$
$192$		0			0
$193$		−	25.4558i		−	1.83235i	−0.400776	−	0.916176i	$-0.631260\pi$
$193$		−	25.4558i		−	1.83235i	0.400776	−	0.916176i	$-0.368740\pi$
$194$		0			0
$195$	−15.0000	+	6.70820i	−1.07417	+	0.480384i
$196$		0			0
$197$			17.8885i			1.27451i	0.770655	+	0.637253i	$0.219929\pi$
$197$			17.8885i			1.27451i	−0.770655	+	0.637253i	$0.780071\pi$
$198$		0			0
$199$	13.0000			0.921546			0.460773	−	0.887518i	$-0.347572\pi$
$199$	13.0000			0.921546			0.460773	+	0.887518i	$0.347572\pi$
$200$		0			0
$201$	6.00000	+	13.4164i	0.423207	+	0.946320i
$202$		0			0
$203$		0			0
$204$		0			0
$205$			21.2132i			1.48159i
$206$		0			0
$207$	−10.0000	−	8.94427i	−0.695048	−	0.621670i
$208$		0			0
$209$	−9.48683	−	2.23607i	−0.656218	−	0.154672i
$210$		0			0
$211$			4.24264i			0.292075i	0.989279	+	0.146038i	$0.0466521\pi$
$211$			4.24264i			0.292075i	−0.989279	+	0.146038i	$0.953348\pi$
$212$		0			0
$213$	−15.0000	+	6.70820i	−1.02778	+	0.459639i
$214$		0			0
$215$		−	15.6525i		−	1.06749i
$216$		0			0
$217$			4.24264i			0.288009i
$218$		0			0
$219$	7.90569	−	3.53553i	0.534217	−	0.238909i
$220$		0			0
$221$	−9.48683			−0.638153
$222$		0			0
$223$		−	4.24264i		−	0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$		−	4.24264i		−	0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.48683			−0.629663			−0.314832	−	0.949148i	$-0.601948\pi$
$227$	−9.48683			−0.629663			−0.314832	+	0.949148i	$0.601948\pi$
$228$		0			0
$229$	−13.0000			−0.859064			−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000			−0.859064			−0.429532	+	0.903052i	$0.641321\pi$
$230$		0			0
$231$	1.58114	+	3.53553i	0.104031	+	0.232621i
$232$		0			0
$233$			24.5967i			1.61139i	0.592333	+	0.805693i	$0.298207\pi$
$233$			24.5967i			1.61139i	−0.592333	+	0.805693i	$0.701793\pi$
$234$		0			0
$235$	−25.0000			−1.63082
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	11.1803i		−	0.723196i	−0.932334	−	0.361598i	$-0.882231\pi$
$239$		−	11.1803i		−	0.723196i	0.932334	−	0.361598i	$-0.117769\pi$
$240$		0			0
$241$			4.24264i			0.273293i	0.990620	+	0.136646i	$0.0436324\pi$
$241$			4.24264i			0.273293i	−0.990620	+	0.136646i	$0.956368\pi$
$242$		0			0
$243$	−7.90569	−	13.4350i	−0.507151	−	0.861858i
$244$		0			0
$245$			13.4164i			0.857143i
$246$		0			0
$247$	18.0000	+	4.24264i	1.14531	+	0.269953i
$248$		0			0
$249$	6.32456	+	14.1421i	0.400802	+	0.896221i
$250$		0			0
$251$			29.0689i			1.83481i	0.397953	+	0.917406i	$0.369721\pi$
$251$			29.0689i			1.83481i	−0.397953	+	0.917406i	$0.630279\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		0			0
$255$	−7.90569	+	3.53553i	−0.495074	+	0.221404i
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$			4.24264i			0.263625i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.6525i			0.965173i	0.875848	+	0.482587i	$0.160303\pi$
$263$			15.6525i			0.965173i	−0.875848	+	0.482587i	$0.839697\pi$
$264$		0			0
$265$		−	21.2132i		−	1.30312i
$266$		0			0
$267$	15.0000	−	6.70820i	0.917985	−	0.410535i
$268$		0			0
$269$	−28.4605			−1.73527			−0.867634	−	0.497204i	$-0.834360\pi$
$269$	−28.4605			−1.73527			−0.867634	+	0.497204i	$0.834360\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$		0			0
$273$	−3.00000	−	6.70820i	−0.181568	−	0.405999i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.00000			−0.0600842			−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000			−0.0600842			−0.0300421	+	0.999549i	$0.509564\pi$
$278$		0			0
$279$	9.48683	+	8.48528i	0.567962	+	0.508001i
$280$		0			0
$281$	18.9737			1.13187			0.565937	−	0.824448i	$-0.308515\pi$
$281$	18.9737			1.13187			0.565937	+	0.824448i	$0.308515\pi$
$282$		0			0
$283$	13.0000			0.772770			0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000			0.772770			0.386385	+	0.922338i	$0.373724\pi$
$284$		0			0
$285$	16.5811	−	3.17267i	0.982182	−	0.187933i
$286$		0			0
$287$	−9.48683			−0.559990
$288$		0			0
$289$	12.0000			0.705882
$290$		0			0
$291$	9.00000	+	20.1246i	0.527589	+	1.17973i
$292$		0			0
$293$	−18.9737			−1.10845			−0.554227	−	0.832366i	$-0.686986\pi$
$293$	−18.9737			−1.10845			−0.554227	+	0.832366i	$0.686986\pi$
$294$		0			0
$295$			21.2132i			1.23508i
$296$		0			0
$297$	11.0680	+	3.53553i	0.642229	+	0.205152i
$298$		0			0
$299$	−18.9737			−1.09728
$300$		0			0
$301$	7.00000			0.403473
$302$		0			0
$303$	−6.32456	−	14.1421i	−0.363336	−	0.812444i
$304$		0			0
$305$		−	24.5967i		−	1.40841i
$306$		0			0
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$	6.00000	+	13.4164i	0.341328	+	0.763233i
$310$		0			0
$311$			2.23607i			0.126796i	0.997988	+	0.0633979i	$0.0201937\pi$
$311$			2.23607i			0.126796i	−0.997988	+	0.0633979i	$0.979806\pi$
$312$		0			0
$313$	−28.0000			−1.58265			−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000			−1.58265			−0.791327	+	0.611393i	$0.790609\pi$
$314$		0			0
$315$	−5.00000	−	4.47214i	−0.281718	−	0.251976i
$316$		0			0
$317$	9.48683			0.532834			0.266417	−	0.963858i	$-0.414160\pi$
$317$	9.48683			0.532834			0.266417	+	0.963858i	$0.414160\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	30.0000	−	13.4164i	1.67444	−	0.748831i
$322$		0			0
$323$	9.48683	+	2.23607i	0.527862	+	0.124418i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	6.00000	+	13.4164i	0.331801	+	0.741929i
$328$		0			0
$329$		−	11.1803i		−	0.616392i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	9.48683	+	8.48528i	0.519875	+	0.464991i
$334$		0			0
$335$	18.9737			1.03664
$336$		0			0
$337$		−	12.7279i		−	0.693334i	−0.937988	−	0.346667i	$-0.887313\pi$
$337$		−	12.7279i		−	0.693334i	0.937988	−	0.346667i	$-0.112687\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9.48683			−0.513741
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$	−15.8114	+	7.07107i	−0.851257	+	0.380693i
$346$		0			0
$347$			2.23607i			0.120038i	0.998197	+	0.0600192i	$0.0191162\pi$
$347$			2.23607i			0.120038i	−0.998197	+	0.0600192i	$0.980884\pi$
$348$		0			0
$349$	29.0000			1.55233			0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000			1.55233			0.776167	+	0.630527i	$0.217161\pi$
$350$		0			0
$351$	−21.0000	−	6.70820i	−1.12090	−	0.358057i
$352$		0			0
$353$		−	8.94427i		−	0.476056i	−0.971258	−	0.238028i	$-0.923499\pi$
$353$		−	8.94427i		−	0.476056i	0.971258	−	0.238028i	$-0.0765009\pi$
$354$		0			0
$355$			21.2132i			1.12588i
$356$		0			0
$357$	−1.58114	−	3.53553i	−0.0836827	−	0.187120i
$358$		0			0
$359$		−	11.1803i		−	0.590076i	−0.955486	−	0.295038i	$-0.904668\pi$
$359$		−	11.1803i		−	0.590076i	0.955486	−	0.295038i	$-0.0953323\pi$
$360$		0			0
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$		0			0
$363$	9.48683	−	4.24264i	0.497930	−	0.222681i
$364$		0			0
$365$		−	11.1803i		−	0.585206i
$366$		0			0
$367$	−26.0000			−1.35719			−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000			−1.35719			−0.678594	+	0.734513i	$0.737411\pi$
$368$		0			0
$369$	−18.9737	+	21.2132i	−0.987730	+	1.10432i
$370$		0			0
$371$	9.48683			0.492532
$372$		0			0
$373$		−	16.9706i		−	0.878702i	−0.898315	−	0.439351i	$-0.855208\pi$
$373$		−	16.9706i		−	0.878702i	0.898315	−	0.439351i	$-0.144792\pi$
$374$		0			0
$375$	−7.90569	−	17.6777i	−0.408248	−	0.912871i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	33.9411i		−	1.74344i	−0.490006	−	0.871719i	$-0.663005\pi$
$379$		−	33.9411i		−	1.74344i	0.490006	−	0.871719i	$-0.336995\pi$
$380$		0			0
$381$	−12.0000	−	26.8328i	−0.614779	−	1.37469i
$382$		0			0
$383$	−9.48683			−0.484755			−0.242377	−	0.970182i	$-0.577927\pi$
$383$	−9.48683			−0.484755			−0.242377	+	0.970182i	$0.577927\pi$
$384$		0			0
$385$	5.00000			0.254824
$386$		0			0
$387$	14.0000	−	15.6525i	0.711660	−	0.795660i
$388$		0			0
$389$		−	15.6525i		−	0.793612i	−0.917902	−	0.396806i	$-0.870119\pi$
$389$		−	15.6525i		−	0.793612i	0.917902	−	0.396806i	$-0.129881\pi$
$390$		0			0
$391$	−10.0000			−0.505722
$392$		0			0
$393$	11.0680	+	24.7487i	0.558305	+	1.24841i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−19.0000			−0.953583			−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000			−0.953583			−0.476791	+	0.879017i	$0.658200\pi$
$398$		0			0
$399$	1.41886	+	7.41531i	0.0710319	+	0.371230i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	18.0000			0.896644
$404$		0			0
$405$	−20.0000	+	2.23607i	−0.993808	+	0.111111i
$406$		0			0
$407$	−9.48683			−0.470245
$408$		0			0
$409$		−	29.6985i		−	1.46850i	−0.678882	−	0.734248i	$-0.737535\pi$
$409$		−	29.6985i		−	1.46850i	0.678882	−	0.734248i	$-0.262465\pi$
$410$		0			0
$411$	7.90569	+	17.6777i	0.389959	+	0.871975i
$412$		0			0
$413$	−9.48683			−0.466817
$414$		0			0
$415$	20.0000			0.981761
$416$		0			0
$417$	−17.3925	+	7.77817i	−0.851716	+	0.380899i
$418$		0			0
$419$		−	4.47214i		−	0.218478i	−0.994016	−	0.109239i	$-0.965159\pi$
$419$		−	4.47214i		−	0.218478i	0.994016	−	0.109239i	$-0.0348414\pi$
$420$		0			0
$421$			21.2132i			1.03387i	0.856025	+	0.516934i	$0.172927\pi$
$421$			21.2132i			1.03387i	−0.856025	+	0.516934i	$0.827073\pi$
$422$		0			0
$423$	−25.0000	−	22.3607i	−1.21554	−	1.08721i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	11.0000			0.532327
$428$		0			0
$429$	15.0000	−	6.70820i	0.724207	−	0.323875i
$430$		0			0
$431$	9.48683			0.456965			0.228482	−	0.973548i	$-0.426624\pi$
$431$	9.48683			0.456965			0.228482	+	0.973548i	$0.426624\pi$
$432$		0			0
$433$		−	25.4558i		−	1.22333i	−0.791117	−	0.611665i	$-0.790500\pi$
$433$		−	25.4558i		−	1.22333i	0.791117	−	0.611665i	$-0.209500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.9737	+	4.47214i	0.907634	+	0.213931i
$438$		0			0
$439$		−	8.48528i		−	0.404980i	−0.979284	−	0.202490i	$-0.935097\pi$
$439$		−	8.48528i		−	0.404980i	0.979284	−	0.202490i	$-0.0649034\pi$
$440$		0			0
$441$	−12.0000	+	13.4164i	−0.571429	+	0.638877i
$442$		0			0
$443$		−	24.5967i		−	1.16863i	−0.811528	−	0.584313i	$-0.801364\pi$
$443$		−	24.5967i		−	1.16863i	0.811528	−	0.584313i	$-0.198636\pi$
$444$		0			0
$445$		−	21.2132i		−	1.00560i
$446$		0			0
$447$	7.90569	+	17.6777i	0.373927	+	0.836125i
$448$		0			0
$449$	37.9473			1.79085			0.895423	−	0.445217i	$-0.146873\pi$
$449$	37.9473			1.79085			0.895423	+	0.445217i	$0.146873\pi$
$450$		0			0
$451$		−	21.2132i		−	0.998891i
$452$		0			0
$453$	3.00000	+	6.70820i	0.140952	+	0.315179i
$454$		0			0
$455$	−9.48683			−0.444750
$456$		0			0
$457$	5.00000			0.233890			0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000			0.233890			0.116945	+	0.993138i	$0.462690\pi$
$458$		0			0
$459$	−11.0680	−	3.53553i	−0.516609	−	0.165025i
$460$		0			0
$461$		−	29.0689i		−	1.35387i	−0.736041	−	0.676936i	$-0.763307\pi$
$461$		−	29.0689i		−	1.35387i	0.736041	−	0.676936i	$-0.236693\pi$
$462$		0			0
$463$	−5.00000			−0.232370			−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000			−0.232370			−0.116185	+	0.993228i	$0.537067\pi$
$464$		0			0
$465$	15.0000	−	6.70820i	0.695608	−	0.311086i
$466$		0			0
$467$		−	38.0132i		−	1.75904i	−0.475863	−	0.879520i	$-0.657864\pi$
$467$		−	38.0132i		−	1.75904i	0.475863	−	0.879520i	$-0.342136\pi$
$468$		0			0
$469$			8.48528i			0.391814i
$470$		0			0
$471$	−6.32456	+	2.82843i	−0.291420	+	0.130327i
$472$		0			0
$473$			15.6525i			0.719702i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	18.9737	−	21.2132i	0.868744	−	0.971286i
$478$		0			0
$479$		−	31.3050i		−	1.43036i	−0.698940	−	0.715180i	$-0.746345\pi$
$479$		−	31.3050i		−	1.43036i	0.698940	−	0.715180i	$-0.253655\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$	−3.16228	−	7.07107i	−0.143889	−	0.321745i
$484$		0			0
$485$	28.4605			1.29232
$486$		0			0
$487$			12.7279i			0.576757i	0.957516	+	0.288379i	$0.0931162\pi$
$487$			12.7279i			0.576757i	−0.957516	+	0.288379i	$0.906884\pi$
$488$		0			0
$489$	−3.16228	+	1.41421i	−0.143003	+	0.0639529i
$490$		0			0
$491$		−	4.47214i		−	0.201825i	−0.994895	−	0.100912i	$-0.967824\pi$
$491$		−	4.47214i		−	0.201825i	0.994895	−	0.100912i	$-0.0321762\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	10.0000	−	11.1803i	0.449467	−	0.502519i
$496$		0			0
$497$	−9.48683			−0.425543
$498$		0			0
$499$	37.0000			1.65635			0.828174	−	0.560471i	$-0.189380\pi$
$499$	37.0000			1.65635			0.828174	+	0.560471i	$0.189380\pi$
$500$		0			0
$501$	−30.0000	+	13.4164i	−1.34030	+	0.599401i
$502$		0			0
$503$			22.3607i			0.997013i	0.866886	+	0.498507i	$0.166118\pi$
$503$			22.3607i			0.997013i	−0.866886	+	0.498507i	$0.833882\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$	−7.90569	+	3.53553i	−0.351104	+	0.157019i
$508$		0			0
$509$	−28.4605			−1.26149			−0.630745	−	0.775990i	$-0.717250\pi$
$509$	−28.4605			−1.26149			−0.630745	+	0.775990i	$0.717250\pi$
$510$		0			0
$511$	5.00000			0.221187
$512$		0			0
$513$	19.4189	+	11.6580i	0.857364	+	0.514711i
$514$		0			0
$515$	18.9737			0.836080
$516$		0			0
$517$	25.0000			1.09950
$518$		0			0
$519$	30.0000	−	13.4164i	1.31685	−	0.588915i
$520$		0			0
$521$	−18.9737			−0.831251			−0.415626	−	0.909536i	$-0.636437\pi$
$521$	−18.9737			−0.831251			−0.415626	+	0.909536i	$0.636437\pi$
$522$		0			0
$523$		−	16.9706i		−	0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$		−	16.9706i		−	0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.48683			0.413253
$528$		0			0
$529$	3.00000			0.130435
$530$		0			0
$531$	−18.9737	+	21.2132i	−0.823387	+	0.920575i
$532$		0			0
$533$			40.2492i			1.74339i
$534$		0			0
$535$		−	42.4264i		−	1.83425i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	13.4164i		−	0.577886i
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$	18.0000	+	40.2492i	0.772454	+	1.72726i
$544$		0			0
$545$	18.9737			0.812743
$546$		0			0
$547$		−	4.24264i		−	0.181402i	−0.995878	−	0.0907011i	$-0.971089\pi$
$547$		−	4.24264i		−	0.181402i	0.995878	−	0.0907011i	$-0.0289108\pi$
$548$		0			0
$549$	22.0000	−	24.5967i	0.938937	−	1.04976i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	15.0000	−	6.70820i	0.636715	−	0.284747i
$556$		0			0
$557$		−	15.6525i		−	0.663217i	−0.943417	−	0.331608i	$-0.892409\pi$
$557$		−	15.6525i		−	0.663217i	0.943417	−	0.331608i	$-0.107591\pi$
$558$		0			0
$559$		−	29.6985i		−	1.25611i
$560$		0			0
$561$	7.90569	−	3.53553i	0.333779	−	0.149270i
$562$		0			0
$563$	9.48683			0.399822			0.199911	−	0.979814i	$-0.435935\pi$
$563$	9.48683			0.399822			0.199911	+	0.979814i	$0.435935\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.00000	−	8.94427i	−0.0419961	−	0.375624i
$568$		0			0
$569$	−28.4605			−1.19313			−0.596563	−	0.802566i	$-0.703467\pi$
$569$	−28.4605			−1.19313			−0.596563	+	0.802566i	$0.703467\pi$
$570$		0			0
$571$	−14.0000			−0.585882			−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000			−0.585882			−0.292941	+	0.956131i	$0.594634\pi$
$572$		0			0
$573$	1.58114	+	3.53553i	0.0660530	+	0.147699i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$	−18.0000	−	40.2492i	−0.748054	−	1.67270i
$580$		0			0
$581$			8.94427i			0.371071i
$582$		0			0
$583$			21.2132i			0.878561i
$584$		0			0
$585$	−18.9737	+	21.2132i	−0.784465	+	0.877058i
$586$		0			0
$587$			15.6525i			0.646047i	0.946391	+	0.323023i	$0.104699\pi$
$587$			15.6525i			0.646047i	−0.946391	+	0.323023i	$0.895301\pi$
$588$		0			0
$589$	−18.0000	−	4.24264i	−0.741677	−	0.174815i
$590$		0			0
$591$	12.6491	+	28.2843i	0.520315	+	1.16346i
$592$		0			0
$593$		−	22.3607i		−	0.918243i	−0.888373	−	0.459122i	$-0.848164\pi$
$593$		−	22.3607i		−	0.918243i	0.888373	−	0.459122i	$-0.151836\pi$
$594$		0			0
$595$	−5.00000			−0.204980
$596$		0			0
$597$	20.5548	−	9.19239i	0.841252	−	0.376219i
$598$		0			0
$599$	18.9737			0.775243			0.387621	−	0.921819i	$-0.373297\pi$
$599$	18.9737			0.775243			0.387621	+	0.921819i	$0.373297\pi$
$600$		0			0
$601$			38.1838i			1.55755i	0.627304	+	0.778774i	$0.284158\pi$
$601$			38.1838i			1.55755i	−0.627304	+	0.778774i	$0.715842\pi$
$602$		0			0
$603$	18.9737	+	16.9706i	0.772667	+	0.691095i
$604$		0			0
$605$		−	13.4164i		−	0.545455i
$606$		0			0
$607$		−	4.24264i		−	0.172203i	−0.996286	−	0.0861017i	$-0.972559\pi$
$607$		−	4.24264i		−	0.172203i	0.996286	−	0.0861017i	$-0.0274410\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−47.4342			−1.91898
$612$		0			0
$613$	5.00000			0.201948			0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000			0.201948			0.100974	+	0.994889i	$0.467804\pi$
$614$		0			0
$615$	15.0000	+	33.5410i	0.604858	+	1.35250i
$616$		0			0
$617$			11.1803i			0.450104i	0.974347	+	0.225052i	$0.0722551\pi$
$617$			11.1803i			0.450104i	−0.974347	+	0.225052i	$0.927745\pi$
$618$		0			0
$619$	−8.00000			−0.321547			−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000			−0.321547			−0.160774	+	0.986991i	$0.551399\pi$
$620$		0			0
$621$	−22.1359	−	7.07107i	−0.888285	−	0.283752i
$622$		0			0
$623$	9.48683			0.380082
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$	−16.5811	+	3.17267i	−0.662187	+	0.126704i
$628$		0			0
$629$	9.48683			0.378265
$630$		0			0
$631$	−41.0000			−1.63218			−0.816092	−	0.577922i	$-0.803864\pi$
$631$	−41.0000			−1.63218			−0.816092	+	0.577922i	$0.803864\pi$
$632$		0			0
$633$	3.00000	+	6.70820i	0.119239	+	0.266627i
$634$		0			0
$635$	−37.9473			−1.50589
$636$		0			0
$637$			25.4558i			1.00860i
$638$		0			0
$639$	−18.9737	+	21.2132i	−0.750587	+	0.839181i
$640$		0			0
$641$	−28.4605			−1.12412			−0.562061	−	0.827096i	$-0.689991\pi$
$641$	−28.4605			−1.12412			−0.562061	+	0.827096i	$0.689991\pi$
$642$		0			0
$643$	−35.0000			−1.38027			−0.690133	−	0.723683i	$-0.742448\pi$
$643$	−35.0000			−1.38027			−0.690133	+	0.723683i	$0.742448\pi$
$644$		0			0
$645$	−11.0680	−	24.7487i	−0.435801	−	0.974481i
$646$		0			0
$647$			15.6525i			0.615362i	0.951490	+	0.307681i	$0.0995530\pi$
$647$			15.6525i			0.615362i	−0.951490	+	0.307681i	$0.900447\pi$
$648$		0			0
$649$		−	21.2132i		−	0.832691i
$650$		0			0
$651$	3.00000	+	6.70820i	0.117579	+	0.262915i
$652$		0			0
$653$		−	29.0689i		−	1.13755i	−0.822492	−	0.568777i	$-0.807417\pi$
$653$		−	29.0689i		−	1.13755i	0.822492	−	0.568777i	$-0.192583\pi$
$654$		0			0
$655$	35.0000			1.36756
$656$		0			0
$657$	10.0000	−	11.1803i	0.390137	−	0.436187i
$658$		0			0
$659$	18.9737			0.739109			0.369555	−	0.929209i	$-0.379510\pi$
$659$	18.9737			0.739109			0.369555	+	0.929209i	$0.379510\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	−15.0000	+	6.70820i	−0.582552	+	0.260525i
$664$		0			0
$665$	9.48683	+	2.23607i	0.367884	+	0.0867110i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.00000	−	6.70820i	−0.115987	−	0.259354i
$670$		0			0
$671$			24.5967i			0.949547i
$672$		0			0
$673$		−	12.7279i		−	0.490625i	−0.969444	−	0.245313i	$-0.921109\pi$
$673$		−	12.7279i		−	0.490625i	0.969444	−	0.245313i	$-0.0788906\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.9737			0.729217			0.364609	−	0.931161i	$-0.381203\pi$
$677$	18.9737			0.729217			0.364609	+	0.931161i	$0.381203\pi$
$678$		0			0
$679$			12.7279i			0.488453i
$680$		0			0
$681$	−15.0000	+	6.70820i	−0.574801	+	0.257059i
$682$		0			0
$683$	−28.4605			−1.08901			−0.544505	−	0.838757i	$-0.683283\pi$
$683$	−28.4605			−1.08901			−0.544505	+	0.838757i	$0.683283\pi$
$684$		0			0
$685$	25.0000			0.955201
$686$		0			0
$687$	−20.5548	+	9.19239i	−0.784215	+	0.350711i
$688$		0			0
$689$		−	40.2492i		−	1.53337i
$690$		0			0
$691$	−23.0000			−0.874961			−0.437481	−	0.899228i	$-0.644129\pi$
$691$	−23.0000			−0.874961			−0.437481	+	0.899228i	$0.644129\pi$
$692$		0			0
$693$	5.00000	+	4.47214i	0.189934	+	0.169882i
$694$		0			0
$695$			24.5967i			0.933008i
$696$		0			0
$697$			21.2132i			0.803507i
$698$		0			0
$699$	17.3925	+	38.8909i	0.657846	+	1.47099i
$700$		0			0
$701$			17.8885i			0.675641i	0.941211	+	0.337820i	$0.109690\pi$
$701$			17.8885i			0.675641i	−0.941211	+	0.337820i	$0.890310\pi$
$702$		0			0
$703$	−18.0000	−	4.24264i	−0.678883	−	0.160014i
$704$		0			0
$705$	−39.5285	+	17.6777i	−1.48873	+	0.665780i
$706$		0			0
$707$		−	8.94427i		−	0.336384i
$708$		0			0
$709$	8.00000			0.300446			0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000			0.300446			0.150223	+	0.988652i	$0.452001\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.9737			0.710569
$714$		0			0
$715$		−	21.2132i		−	0.793329i
$716$		0			0
$717$	−7.90569	−	17.6777i	−0.295244	−	0.660185i
$718$		0			0
$719$			29.0689i			1.08409i	0.840351	+	0.542043i	$0.182349\pi$
$719$			29.0689i			1.08409i	−0.840351	+	0.542043i	$0.817651\pi$
$720$		0			0
$721$			8.48528i			0.316008i
$722$		0			0
$723$	3.00000	+	6.70820i	0.111571	+	0.249481i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	31.0000			1.14973			0.574863	−	0.818250i	$-0.305055\pi$
$727$	31.0000			1.14973			0.574863	+	0.818250i	$0.305055\pi$
$728$		0			0
$729$	−22.0000	−	15.6525i	−0.814815	−	0.579721i
$730$		0			0
$731$		−	15.6525i		−	0.578928i
$732$		0			0
$733$	−40.0000			−1.47743			−0.738717	−	0.674016i	$-0.764568\pi$
$733$	−40.0000			−1.47743			−0.738717	+	0.674016i	$0.764568\pi$
$734$		0			0
$735$	9.48683	+	21.2132i	0.349927	+	0.782461i
$736$		0			0
$737$	−18.9737			−0.698904
$738$		0			0
$739$	19.0000			0.698926			0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000			0.698926			0.349463	+	0.936950i	$0.386364\pi$
$740$		0			0
$741$	31.4605	−	6.01972i	1.15573	−	0.221140i
$742$		0			0
$743$	37.9473			1.39215			0.696076	−	0.717968i	$-0.254928\pi$
$743$	37.9473			1.39215			0.696076	+	0.717968i	$0.254928\pi$
$744$		0			0
$745$	25.0000			0.915929
$746$		0			0
$747$	20.0000	+	17.8885i	0.731762	+	0.654508i
$748$		0			0
$749$	18.9737			0.693283
$750$		0			0
$751$		−	42.4264i		−	1.54816i	−0.633087	−	0.774081i	$-0.718212\pi$
$751$		−	42.4264i		−	1.54816i	0.633087	−	0.774081i	$-0.281788\pi$
$752$		0			0
$753$	20.5548	+	45.9619i	0.749059	+	1.67495i
$754$		0			0
$755$	9.48683			0.345261
$756$		0			0
$757$	−1.00000			−0.0363456			−0.0181728	−	0.999835i	$-0.505785\pi$
$757$	−1.00000			−0.0363456			−0.0181728	+	0.999835i	$0.505785\pi$
$758$		0			0
$759$	15.8114	−	7.07107i	0.573917	−	0.256664i
$760$		0			0
$761$			38.0132i			1.37798i	0.724773	+	0.688988i	$0.241945\pi$
$761$			38.0132i			1.37798i	−0.724773	+	0.688988i	$0.758055\pi$
$762$		0			0
$763$			8.48528i			0.307188i
$764$		0			0
$765$	−10.0000	+	11.1803i	−0.361551	+	0.404226i
$766$		0			0
$767$			40.2492i			1.45332i
$768$		0			0
$769$	23.0000			0.829401			0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000			0.829401			0.414701	+	0.909958i	$0.363886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28.4605			−1.02365			−0.511826	−	0.859089i	$-0.671031\pi$
$773$	−28.4605			−1.02365			−0.511826	+	0.859089i	$0.671031\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.00000	+	6.70820i	0.107624	+	0.240655i
$778$		0			0
$779$	9.48683	−	40.2492i	0.339901	−	1.44208i
$780$		0			0
$781$		−	21.2132i		−	0.759068i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			8.94427i			0.319235i
$786$		0			0
$787$		−	33.9411i		−	1.20987i	−0.796275	−	0.604935i	$-0.793199\pi$
$787$		−	33.9411i		−	1.20987i	0.796275	−	0.604935i	$-0.206801\pi$
$788$		0			0
$789$	11.0680	+	24.7487i	0.394030	+	0.881078i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	46.6690i		−	1.65727i
$794$		0			0
$795$	−15.0000	−	33.5410i	−0.531995	−	1.18958i
$796$		0			0
$797$	−18.9737			−0.672082			−0.336041	−	0.941847i	$-0.609088\pi$
$797$	−18.9737			−0.672082			−0.336041	+	0.941847i	$0.609088\pi$
$798$		0			0
$799$	−25.0000			−0.884436
$800$		0			0
$801$	18.9737	−	21.2132i	0.670402	−	0.749532i
$802$		0			0
$803$			11.1803i			0.394546i
$804$		0			0
$805$	−10.0000			−0.352454
$806$		0			0
$807$	−45.0000	+	20.1246i	−1.58408	+	0.708420i
$808$		0			0
$809$			11.1803i			0.393080i	0.980496	+	0.196540i	$0.0629705\pi$
$809$			11.1803i			0.393080i	−0.980496	+	0.196540i	$0.937029\pi$
$810$		0			0
$811$			4.24264i			0.148979i	0.997222	+	0.0744896i	$0.0237328\pi$
$811$			4.24264i			0.148979i	−0.997222	+	0.0744896i	$0.976267\pi$
$812$		0			0
$813$	−3.16228	+	1.41421i	−0.110906	+	0.0495986i
$814$		0			0
$815$			4.47214i			0.156652i
$816$		0			0
$817$	−7.00000	+	29.6985i	−0.244899	+	1.03902i
$818$		0			0
$819$	−9.48683	−	8.48528i	−0.331497	−	0.296500i
$820$		0			0
$821$			38.0132i			1.32667i	0.748323	+	0.663334i	$0.230859\pi$
$821$			38.0132i			1.32667i	−0.748323	+	0.663334i	$0.769141\pi$
$822$		0			0
$823$	−47.0000			−1.63832			−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−47.0000			−1.63832			−0.819159	+	0.573567i	$0.805559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.9737			0.659779			0.329890	−	0.944020i	$-0.392989\pi$
$827$	18.9737			0.659779			0.329890	+	0.944020i	$0.392989\pi$
$828$		0			0
$829$			42.4264i			1.47353i	0.676149	+	0.736765i	$0.263648\pi$
$829$			42.4264i			1.47353i	−0.676149	+	0.736765i	$0.736352\pi$
$830$		0			0
$831$	−1.58114	+	0.707107i	−0.0548491	+	0.0245293i
$832$		0			0
$833$			13.4164i			0.464851i
$834$		0			0
$835$			42.4264i			1.46823i
$836$		0			0
$837$	21.0000	+	6.70820i	0.725866	+	0.231869i
$838$		0			0
$839$	−56.9210			−1.96513			−0.982566	−	0.185917i	$-0.940475\pi$
$839$	−56.9210			−1.96513			−0.982566	+	0.185917i	$0.940475\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	30.0000	−	13.4164i	1.03325	−	0.462086i
$844$		0			0
$845$			11.1803i			0.384615i
$846$		0			0
$847$	6.00000			0.206162
$848$		0			0
$849$	20.5548	−	9.19239i	0.705439	−	0.315482i
$850$		0			0
$851$	18.9737			0.650409
$852$		0			0
$853$	−28.0000			−0.958702			−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000			−0.958702			−0.479351	+	0.877623i	$0.659128\pi$
$854$		0			0
$855$	23.9737	−	16.7411i	0.819882	−	0.572532i
$856$		0			0
$857$	−47.4342			−1.62032			−0.810160	−	0.586209i	$-0.800620\pi$
$857$	−47.4342			−1.62032			−0.810160	+	0.586209i	$0.800620\pi$
$858$		0			0
$859$	1.00000			0.0341196			0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000			0.0341196			0.0170598	+	0.999854i	$0.494569\pi$
$860$		0			0
$861$	−15.0000	+	6.70820i	−0.511199	+	0.228615i
$862$		0			0
$863$	−28.4605			−0.968807			−0.484403	−	0.874845i	$-0.660963\pi$
$863$	−28.4605			−0.968807			−0.484403	+	0.874845i	$0.660963\pi$
$864$		0			0
$865$		−	42.4264i		−	1.44254i
$866$		0			0
$867$	18.9737	−	8.48528i	0.644379	−	0.288175i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	36.0000			1.21981
$872$		0			0
$873$	28.4605	+	25.4558i	0.963242	+	0.861550i
$874$		0			0
$875$		−	11.1803i		−	0.377964i
$876$		0			0
$877$		−	8.48528i		−	0.286528i	−0.989685	−	0.143264i	$-0.954240\pi$
$877$		−	8.48528i		−	0.286528i	0.989685	−	0.143264i	$-0.0457597\pi$
$878$		0			0
$879$	−30.0000	+	13.4164i	−1.01187	+	0.452524i
$880$		0			0
$881$			38.0132i			1.28070i	0.768085	+	0.640348i	$0.221210\pi$
$881$			38.0132i			1.28070i	−0.768085	+	0.640348i	$0.778790\pi$
$882$		0			0
$883$	43.0000			1.44707			0.723533	−	0.690290i	$-0.242517\pi$
$883$	43.0000			1.44707			0.723533	+	0.690290i	$0.242517\pi$
$884$		0			0
$885$	15.0000	+	33.5410i	0.504219	+	1.12747i
$886$		0			0
$887$	28.4605			0.955610			0.477805	−	0.878466i	$-0.341433\pi$
$887$	28.4605			0.955610			0.477805	+	0.878466i	$0.341433\pi$
$888$		0			0
$889$		−	16.9706i		−	0.569174i
$890$		0			0
$891$	20.0000	−	2.23607i	0.670025	−	0.0749111i
$892$		0			0
$893$	47.4342	+	11.1803i	1.58732	+	0.374136i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−30.0000	+	13.4164i	−1.00167	+	0.447961i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	21.2132i		−	0.706714i
$902$		0			0
$903$	11.0680	−	4.94975i	0.368319	−	0.164717i
$904$		0			0
$905$	56.9210			1.89212
$906$		0			0
$907$			4.24264i			0.140875i	0.997516	+	0.0704373i	$0.0224395\pi$
$907$			4.24264i			0.140875i	−0.997516	+	0.0704373i	$0.977561\pi$
$908$		0			0
$909$	−20.0000	−	17.8885i	−0.663358	−	0.593326i
$910$		0			0
$911$	−18.9737			−0.628626			−0.314313	−	0.949319i	$-0.601774\pi$
$911$	−18.9737			−0.628626			−0.314313	+	0.949319i	$0.601774\pi$
$912$		0			0
$913$	−20.0000			−0.661903
$914$		0			0
$915$	−17.3925	−	38.8909i	−0.574979	−	1.28569i
$916$		0			0
$917$			15.6525i			0.516890i
$918$		0			0
$919$	−2.00000			−0.0659739			−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000			−0.0659739			−0.0329870	+	0.999456i	$0.510502\pi$
$920$		0			0
$921$	6.00000	+	13.4164i	0.197707	+	0.442086i
$922$		0			0
$923$			40.2492i			1.32482i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	18.9737	+	16.9706i	0.623177	+	0.557386i
$928$		0			0
$929$		−	8.94427i		−	0.293452i	−0.989177	−	0.146726i	$-0.953126\pi$
$929$		−	8.94427i		−	0.293452i	0.989177	−	0.146726i	$-0.0468736\pi$
$930$		0			0
$931$	6.00000	−	25.4558i	0.196642	−	0.834282i
$932$		0			0
$933$	1.58114	+	3.53553i	0.0517642	+	0.115748i
$934$		0			0
$935$		−	11.1803i		−	0.365636i
$936$		0			0
$937$	−19.0000			−0.620703			−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000			−0.620703			−0.310351	+	0.950622i	$0.600447\pi$
$938$		0			0
$939$	−44.2719	+	19.7990i	−1.44476	+	0.646116i
$940$		0			0
$941$	37.9473			1.23705			0.618524	−	0.785766i	$-0.287731\pi$
$941$	37.9473			1.23705			0.618524	+	0.785766i	$0.287731\pi$
$942$		0			0
$943$			42.4264i			1.38159i
$944$		0			0
$945$	−11.0680	−	3.53553i	−0.360041	−	0.115011i
$946$		0			0
$947$		−	31.3050i		−	1.01727i	−0.860981	−	0.508637i	$-0.830149\pi$
$947$		−	31.3050i		−	1.01727i	0.860981	−	0.508637i	$-0.169851\pi$
$948$		0			0
$949$		−	21.2132i		−	0.688610i
$950$		0			0
$951$	15.0000	−	6.70820i	0.486408	−	0.217528i
$952$		0			0
$953$	18.9737			0.614617			0.307309	−	0.951610i	$-0.400572\pi$
$953$	18.9737			0.614617			0.307309	+	0.951610i	$0.400572\pi$
$954$		0			0
$955$	5.00000			0.161796
$956$		0			0
$957$		0			0
$958$		0			0
$959$			11.1803i			0.361032i
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	37.9473	−	42.4264i	1.22284	−	1.36717i
$964$		0			0
$965$	−56.9210			−1.83235
$966$		0			0
$967$	−50.0000			−1.60789			−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000			−1.60789			−0.803946	+	0.594703i	$0.797270\pi$
$968$		0			0
$969$	16.5811	−	3.17267i	0.532663	−	0.101921i
$970$		0			0
$971$	37.9473			1.21779			0.608894	−	0.793252i	$-0.291614\pi$
$971$	37.9473			1.21779			0.608894	+	0.793252i	$0.291614\pi$
$972$		0			0
$973$	−11.0000			−0.352644
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.48683			−0.303511			−0.151755	−	0.988418i	$-0.548493\pi$
$977$	−9.48683			−0.303511			−0.151755	+	0.988418i	$0.548493\pi$
$978$		0			0
$979$			21.2132i			0.677977i
$980$		0			0
$981$	18.9737	+	16.9706i	0.605783	+	0.541828i
$982$		0			0
$983$	18.9737			0.605166			0.302583	−	0.953123i	$-0.402151\pi$
$983$	18.9737			0.605166			0.302583	+	0.953123i	$0.402151\pi$
$984$		0			0
$985$	40.0000			1.27451
$986$		0			0
$987$	−7.90569	−	17.6777i	−0.251641	−	0.562686i
$988$		0			0
$989$		−	31.3050i		−	0.995440i
$990$		0			0
$991$		−	4.24264i		−	0.134772i	−0.997727	−	0.0673860i	$-0.978534\pi$
$991$		−	4.24264i		−	0.134772i	0.997727	−	0.0673860i	$-0.0214659\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	29.0689i		−	0.921546i
$996$		0			0
$997$	5.00000			0.158352			0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000			0.158352			0.0791758	+	0.996861i	$0.474771\pi$
$998$		0			0
$999$	21.0000	+	6.70820i	0.664411	+	0.212238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.f.g.113.3		4
3.2	odd	2	inner	912.2.f.g.113.1		4
4.3	odd	2		228.2.d.b.113.2	yes	4
12.11	even	2		228.2.d.b.113.4	yes	4
19.18	odd	2	inner	912.2.f.g.113.2		4
57.56	even	2	inner	912.2.f.g.113.4		4
76.75	even	2		228.2.d.b.113.3	yes	4
228.227	odd	2		228.2.d.b.113.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.d.b.113.1	✓	4	228.227	odd	2
228.2.d.b.113.2	yes	4	4.3	odd	2
228.2.d.b.113.3	yes	4	76.75	even	2
228.2.d.b.113.4	yes	4	12.11	even	2
912.2.f.g.113.1		4	3.2	odd	2	inner
912.2.f.g.113.2		4	19.18	odd	2	inner
912.2.f.g.113.3		4	1.1	even	1	trivial
912.2.f.g.113.4		4	57.56	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 - 0.707107i) q^{3} -2.23607i q^{5} +1.00000 q^{7} +(2.00000 - 2.23607i) q^{9} +O(q^{10})\)	\(q+(1.58114 - 0.707107i) q^{3} -2.23607i q^{5} +1.00000 q^{7} +(2.00000 - 2.23607i) q^{9} +2.23607i q^{11} -4.24264i q^{13} +(-1.58114 - 3.53553i) q^{15} -2.23607i q^{17} +(-1.00000 + 4.24264i) q^{19} +(1.58114 - 0.707107i) q^{21} -4.47214i q^{23} +(1.58114 - 4.94975i) q^{27} +4.24264i q^{31} +(1.58114 + 3.53553i) q^{33} -2.23607i q^{35} +4.24264i q^{37} +(-3.00000 - 6.70820i) q^{39} -9.48683 q^{41} +7.00000 q^{43} +(-5.00000 - 4.47214i) q^{45} -11.1803i q^{47} -6.00000 q^{49} +(-1.58114 - 3.53553i) q^{51} +9.48683 q^{53} +5.00000 q^{55} +(1.41886 + 7.41531i) q^{57} -9.48683 q^{59} +11.0000 q^{61} +(2.00000 - 2.23607i) q^{63} -9.48683 q^{65} +8.48528i q^{67} +(-3.16228 - 7.07107i) q^{69} -9.48683 q^{71} +5.00000 q^{73} +2.23607i q^{77} +(-1.00000 - 8.94427i) q^{81} +8.94427i q^{83} -5.00000 q^{85} +9.48683 q^{89} -4.24264i q^{91} +(3.00000 + 6.70820i) q^{93} +(9.48683 + 2.23607i) q^{95} +12.7279i q^{97} +(5.00000 + 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 + 8 * q^9	\( 4 q + 4 q^{7} + 8 q^{9} - 4 q^{19} - 12 q^{39} + 28 q^{43} - 20 q^{45} - 24 q^{49} + 20 q^{55} + 12 q^{57} + 44 q^{61} + 8 q^{63} + 20 q^{73} - 4 q^{81} - 20 q^{85} + 12 q^{93} + 20 q^{99}+O(q^{100}) \) 4 * q + 4 * q^7 + 8 * q^9 - 4 * q^19 - 12 * q^39 + 28 * q^43 - 20 * q^45 - 24 * q^49 + 20 * q^55 + 12 * q^57 + 44 * q^61 + 8 * q^63 + 20 * q^73 - 4 * q^81 - 20 * q^85 + 12 * q^93 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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