LMFDB - Embedded newform 112.7.c.b.97.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,7,Mod(97,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("112.97");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	112.c (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:	$25.7660573654$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-510}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 510 $ x^2 + 510
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.2
Root			$22.5832i$ of defining polynomial
Character	$\chi$	$=$	112.97
Dual form			112.7.c.b.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+45.1664i q^{3} +45.1664i q^{5} +(-133.000 + 316.165i) q^{7} -1311.00 q^{9} +O(q^{10})$	$q+45.1664i q^{3} +45.1664i q^{5} +(-133.000 + 316.165i) q^{7} -1311.00 q^{9} -874.000 q^{11} -2213.15i q^{13} -2040.00 q^{15} +5961.96i q^{17} +3116.48i q^{19} +(-14280.0 - 6007.13i) q^{21} -4738.00 q^{23} +13585.0 q^{25} -26286.8i q^{27} +11146.0 q^{29} -27461.1i q^{31} -39475.4i q^{33} +(-14280.0 - 6007.13i) q^{35} +3002.00 q^{37} +99960.0 q^{39} -57541.9i q^{41} -31418.0 q^{43} -59213.1i q^{45} -72446.8i q^{47} +(-82271.0 - 84099.8i) q^{49} -269280. q^{51} -76406.0 q^{53} -39475.4i q^{55} -140760. q^{57} +113232. i q^{59} +275108. i q^{61} +(174363. - 414492. i) q^{63} +99960.0 q^{65} -495242. q^{67} -213998. i q^{69} +184406. q^{71} +60974.6i q^{73} +613585. i q^{75} +(116242. - 276328. i) q^{77} +534934. q^{79} +231561. q^{81} -714848. i q^{83} -269280. q^{85} +503424. i q^{87} +629529. i q^{89} +(699720. + 294349. i) q^{91} +1.24032e6 q^{93} -140760. q^{95} -814440. i q^{97} +1.14581e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 266 q^{7} - 2622 q^{9}+O(q^{10}) $ 2 * q - 266 * q^7 - 2622 * q^9	$ 2 q - 266 q^{7} - 2622 q^{9} - 1748 q^{11} - 4080 q^{15} - 28560 q^{21} - 9476 q^{23} + 27170 q^{25} + 22292 q^{29} - 28560 q^{35} + 6004 q^{37} + 199920 q^{39} - 62836 q^{43} - 164542 q^{49} - 538560 q^{51} - 152812 q^{53} - 281520 q^{57} + 348726 q^{63} + 199920 q^{65} - 990484 q^{67} + 368812 q^{71} + 232484 q^{77} + 1069868 q^{79} + 463122 q^{81} - 538560 q^{85} + 1399440 q^{91} + 2480640 q^{93} - 281520 q^{95} + 2291628 q^{99}+O(q^{100}) $ 2 * q - 266 * q^7 - 2622 * q^9 - 1748 * q^11 - 4080 * q^15 - 28560 * q^21 - 9476 * q^23 + 27170 * q^25 + 22292 * q^29 - 28560 * q^35 + 6004 * q^37 + 199920 * q^39 - 62836 * q^43 - 164542 * q^49 - 538560 * q^51 - 152812 * q^53 - 281520 * q^57 + 348726 * q^63 + 199920 * q^65 - 990484 * q^67 + 368812 * q^71 + 232484 * q^77 + 1069868 * q^79 + 463122 * q^81 - 538560 * q^85 + 1399440 * q^91 + 2480640 * q^93 - 281520 * q^95 + 2291628 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			45.1664i			1.67283i	0.548098	+	0.836414i	$0.315352\pi$
$3$			45.1664i			1.67283i	−0.548098	+	0.836414i	$0.684648\pi$
$4$		0			0
$5$			45.1664i			0.361331i	0.983545	+	0.180665i	$0.0578251\pi$
$5$			45.1664i			0.361331i	−0.983545	+	0.180665i	$0.942175\pi$
$6$		0			0
$7$	−133.000	+	316.165i	−0.387755	+	0.921762i
$7$	−133.000	+	316.165i	−0.387755	+	0.921762i
$8$		0			0
$9$	−1311.00			−1.79835
$10$		0			0
$11$	−874.000			−0.656649			−0.328325	−	0.944565i	$-0.606484\pi$
$11$	−874.000			−0.656649			−0.328325	+	0.944565i	$0.606484\pi$
$12$		0			0
$13$		−	2213.15i		−	1.00735i	−0.863893	−	0.503676i	$-0.831981\pi$
$13$		−	2213.15i		−	1.00735i	0.863893	−	0.503676i	$-0.168019\pi$
$14$		0			0
$15$	−2040.00			−0.604444
$16$		0			0
$17$			5961.96i			1.21351i	0.794890	+	0.606753i	$0.207528\pi$
$17$			5961.96i			1.21351i	−0.794890	+	0.606753i	$0.792472\pi$
$18$		0			0
$19$			3116.48i			0.454363i	0.973852	+	0.227182i	$0.0729511\pi$
$19$			3116.48i			0.454363i	−0.973852	+	0.227182i	$0.927049\pi$
$20$		0			0
$21$	−14280.0	−	6007.13i	−1.54195	−	0.648648i
$22$		0			0
$23$	−4738.00			−0.389414			−0.194707	−	0.980861i	$-0.562376\pi$
$23$	−4738.00			−0.389414			−0.194707	+	0.980861i	$0.562376\pi$
$24$		0			0
$25$	13585.0			0.869440
$26$		0			0
$27$		−	26286.8i		−	1.33551i
$28$		0			0
$29$	11146.0			0.457009			0.228505	−	0.973543i	$-0.426616\pi$
$29$	11146.0			0.457009			0.228505	+	0.973543i	$0.426616\pi$
$30$		0			0
$31$		−	27461.1i		−	0.921793i	−0.887454	−	0.460897i	$-0.847528\pi$
$31$		−	27461.1i		−	0.921793i	0.887454	−	0.460897i	$-0.152472\pi$
$32$		0			0
$33$		−	39475.4i		−	1.09846i
$34$		0			0
$35$	−14280.0	−	6007.13i	−0.333061	−	0.140108i
$36$		0			0
$37$	3002.00			0.0592660			0.0296330	−	0.999561i	$-0.490566\pi$
$37$	3002.00			0.0592660			0.0296330	+	0.999561i	$0.490566\pi$
$38$		0			0
$39$	99960.0			1.68513
$40$		0			0
$41$		−	57541.9i		−	0.834897i	−0.908701	−	0.417449i	$-0.862924\pi$
$41$		−	57541.9i		−	0.834897i	0.908701	−	0.417449i	$-0.137076\pi$
$42$		0			0
$43$	−31418.0			−0.395160			−0.197580	−	0.980287i	$-0.563308\pi$
$43$	−31418.0			−0.395160			−0.197580	+	0.980287i	$0.563308\pi$
$44$		0			0
$45$		−	59213.1i		−	0.649801i
$46$		0			0
$47$		−	72446.8i		−	0.697792i	−0.937161	−	0.348896i	$-0.886557\pi$
$47$		−	72446.8i		−	0.697792i	0.937161	−	0.348896i	$-0.113443\pi$
$48$		0			0
$49$	−82271.0	−	84099.8i	−0.699292	−	0.714836i
$50$		0			0
$51$	−269280.			−2.02999
$52$		0			0
$53$	−76406.0			−0.513216			−0.256608	−	0.966516i	$-0.582605\pi$
$53$	−76406.0			−0.513216			−0.256608	+	0.966516i	$0.582605\pi$
$54$		0			0
$55$		−	39475.4i		−	0.237268i
$56$		0			0
$57$	−140760.			−0.760072
$58$		0			0
$59$			113232.i			0.551332i	0.961253	+	0.275666i	$0.0888984\pi$
$59$			113232.i			0.551332i	−0.961253	+	0.275666i	$0.911102\pi$
$60$		0			0
$61$			275108.i			1.21203i	0.795452	+	0.606016i	$0.207233\pi$
$61$			275108.i			1.21203i	−0.795452	+	0.606016i	$0.792767\pi$
$62$		0			0
$63$	174363.	−	414492.i	0.697321	−	1.65766i
$64$		0			0
$65$	99960.0			0.363987
$66$		0			0
$67$	−495242.			−1.64662			−0.823309	−	0.567593i	$-0.807875\pi$
$67$	−495242.			−1.64662			−0.823309	+	0.567593i	$0.807875\pi$
$68$		0			0
$69$		−	213998.i		−	0.651423i
$70$		0			0
$71$	184406.			0.515229			0.257614	−	0.966248i	$-0.417064\pi$
$71$	184406.			0.515229			0.257614	+	0.966248i	$0.417064\pi$
$72$		0			0
$73$			60974.6i			0.156740i	0.996924	+	0.0783701i	$0.0249716\pi$
$73$			60974.6i			0.156740i	−0.996924	+	0.0783701i	$0.975028\pi$
$74$		0			0
$75$			613585.i			1.45442i
$76$		0			0
$77$	116242.	−	276328.i	0.254619	−	0.605275i
$78$		0			0
$79$	534934.			1.08497			0.542486	−	0.840065i	$-0.317483\pi$
$79$	534934.			1.08497			0.542486	+	0.840065i	$0.317483\pi$
$80$		0			0
$81$	231561.			0.435723
$82$		0			0
$83$		−	714848.i		−	1.25020i	−0.780545	−	0.625100i	$-0.785058\pi$
$83$		−	714848.i		−	1.25020i	0.780545	−	0.625100i	$-0.214942\pi$
$84$		0			0
$85$	−269280.			−0.438478
$86$		0			0
$87$			503424.i			0.764498i
$88$		0			0
$89$			629529.i			0.892988i	0.894787	+	0.446494i	$0.147328\pi$
$89$			629529.i			0.892988i	−0.894787	+	0.446494i	$0.852672\pi$
$90$		0			0
$91$	699720.	+	294349.i	0.928539	+	0.390606i
$92$		0			0
$93$	1.24032e6			1.54200
$94$		0			0
$95$	−140760.			−0.164176
$96$		0			0
$97$		−	814440.i		−	0.892368i	−0.894941	−	0.446184i	$-0.852783\pi$
$97$		−	814440.i		−	0.892368i	0.894941	−	0.446184i	$-0.147217\pi$
$98$		0			0
$99$	1.14581e6			1.18089
$100$		0			0
$101$			1.95195e6i			1.89455i	0.320425	+	0.947274i	$0.396174\pi$
$101$			1.95195e6i			1.89455i	−0.320425	+	0.947274i	$0.603826\pi$
$102$		0			0
$103$			1.69744e6i			1.55340i	0.629871	+	0.776700i	$0.283108\pi$
$103$			1.69744e6i			1.55340i	−0.629871	+	0.776700i	$0.716892\pi$
$104$		0			0
$105$	271320.	−	644976.i	0.234376	−	0.557154i
$106$		0			0
$107$	−1.61603e6			−1.31916			−0.659579	−	0.751635i	$-0.729266\pi$
$107$	−1.61603e6			−1.31916			−0.659579	+	0.751635i	$0.729266\pi$
$108$		0			0
$109$	199226.			0.153839			0.0769195	−	0.997037i	$-0.475492\pi$
$109$	199226.			0.153839			0.0769195	+	0.997037i	$0.475492\pi$
$110$		0			0
$111$			135589.i			0.0991418i
$112$		0			0
$113$	−1.80762e6			−1.25277			−0.626386	−	0.779513i	$-0.715467\pi$
$113$	−1.80762e6			−1.25277			−0.626386	+	0.779513i	$0.715467\pi$
$114$		0			0
$115$		−	213998.i		−	0.140707i
$116$		0			0
$117$			2.90144e6i			1.81157i
$118$		0			0
$119$	−1.88496e6	−	792941.i	−1.11857	−	0.470543i
$120$		0			0
$121$	−1.00768e6			−0.568812
$122$		0			0
$123$	2.59896e6			1.39664
$124$		0			0
$125$			1.31931e6i			0.675486i
$126$		0			0
$127$	−3.32472e6			−1.62310			−0.811548	−	0.584286i	$-0.801375\pi$
$127$	−3.32472e6			−1.62310			−0.811548	+	0.584286i	$0.801375\pi$
$128$		0			0
$129$		−	1.41904e6i		−	0.661035i
$130$		0			0
$131$			3.13567e6i			1.39482i	0.716674	+	0.697408i	$0.245664\pi$
$131$			3.13567e6i			1.39482i	−0.716674	+	0.697408i	$0.754336\pi$
$132$		0			0
$133$	−985320.	−	414492.i	−0.418815	−	0.176182i
$134$		0			0
$135$	1.18728e6			0.482561
$136$		0			0
$137$	2.12927e6			0.828072			0.414036	−	0.910260i	$-0.364119\pi$
$137$	2.12927e6			0.828072			0.414036	+	0.910260i	$0.364119\pi$
$138$		0			0
$139$		−	1.68421e6i		−	0.627121i	−0.949568	−	0.313561i	$-0.898478\pi$
$139$		−	1.68421e6i		−	0.627121i	0.949568	−	0.313561i	$-0.101522\pi$
$140$		0			0
$141$	3.27216e6			1.16729
$142$		0			0
$143$			1.93429e6i			0.661477i
$144$		0			0
$145$			503424.i			0.165132i
$146$		0			0
$147$	3.79848e6	−	3.71588e6i	1.19580	−	1.16980i
$148$		0			0
$149$	−2.59573e6			−0.784696			−0.392348	−	0.919817i	$-0.628337\pi$
$149$	−2.59573e6			−0.784696			−0.392348	+	0.919817i	$0.628337\pi$
$150$		0			0
$151$	1.68557e6			0.489570			0.244785	−	0.969577i	$-0.421283\pi$
$151$	1.68557e6			0.489570			0.244785	+	0.969577i	$0.421283\pi$
$152$		0			0
$153$		−	7.81613e6i		−	2.18231i
$154$		0			0
$155$	1.24032e6			0.333072
$156$		0			0
$157$		−	3.67641e6i		−	0.950002i	−0.879985	−	0.475001i	$-0.842448\pi$
$157$		−	3.67641e6i		−	0.950002i	0.879985	−	0.475001i	$-0.157552\pi$
$158$		0			0
$159$		−	3.45098e6i		−	0.858521i
$160$		0			0
$161$	630154.	−	1.49799e6i	0.150997	−	0.358947i
$162$		0			0
$163$	−1.88191e6			−0.434547			−0.217274	−	0.976111i	$-0.569716\pi$
$163$	−1.88191e6			−0.434547			−0.217274	+	0.976111i	$0.569716\pi$
$164$		0			0
$165$	1.78296e6			0.396908
$166$		0			0
$167$		−	3.15595e6i		−	0.677612i	−0.940856	−	0.338806i	$-0.889977\pi$
$167$		−	3.15595e6i		−	0.677612i	0.940856	−	0.338806i	$-0.110023\pi$
$168$		0			0
$169$	−71231.0			−0.0147574
$170$		0			0
$171$		−	4.08570e6i		−	0.817106i
$172$		0			0
$173$			2.29477e6i			0.443201i	0.975138	+	0.221600i	$0.0711280\pi$
$173$			2.29477e6i			0.443201i	−0.975138	+	0.221600i	$0.928872\pi$
$174$		0			0
$175$	−1.80680e6	+	4.29509e6i	−0.337130	+	0.801417i
$176$		0			0
$177$	−5.11428e6			−0.922284
$178$		0			0
$179$	−3.51846e6			−0.613470			−0.306735	−	0.951795i	$-0.599237\pi$
$179$	−3.51846e6			−0.613470			−0.306735	+	0.951795i	$0.599237\pi$
$180$		0			0
$181$			7.48267e6i			1.26189i	0.775829	+	0.630944i	$0.217332\pi$
$181$			7.48267e6i			1.26189i	−0.775829	+	0.630944i	$0.782668\pi$
$182$		0			0
$183$	−1.24256e7			−2.02752
$184$		0			0
$185$			135589.i			0.0214146i
$186$		0			0
$187$		−	5.21075e6i		−	0.796848i
$188$		0			0
$189$	8.31096e6	+	3.49615e6i	1.23102	+	0.517850i
$190$		0			0
$191$	7.20028e6			1.03335			0.516677	−	0.856180i	$-0.327169\pi$
$191$	7.20028e6			1.03335			0.516677	+	0.856180i	$0.327169\pi$
$192$		0			0
$193$	−1.30889e7			−1.82067			−0.910335	−	0.413872i	$-0.864176\pi$
$193$	−1.30889e7			−1.82067			−0.910335	+	0.413872i	$0.864176\pi$
$194$		0			0
$195$			4.51483e6i			0.608888i
$196$		0			0
$197$	−9.17476e6			−1.20004			−0.600020	−	0.799985i	$-0.704841\pi$
$197$	−9.17476e6			−1.20004			−0.600020	+	0.799985i	$0.704841\pi$
$198$		0			0
$199$		−	6.78769e6i		−	0.861317i	−0.902515	−	0.430658i	$-0.858281\pi$
$199$		−	6.78769e6i		−	0.861317i	0.902515	−	0.430658i	$-0.141719\pi$
$200$		0			0
$201$		−	2.23683e7i		−	2.75451i
$202$		0			0
$203$	−1.48242e6	+	3.52397e6i	−0.177208	+	0.421254i
$204$		0			0
$205$	2.59896e6			0.301674
$206$		0			0
$207$	6.21152e6			0.700304
$208$		0			0
$209$		−	2.72380e6i		−	0.298357i
$210$		0			0
$211$	−8.40084e6			−0.894284			−0.447142	−	0.894463i	$-0.647558\pi$
$211$	−8.40084e6			−0.894284			−0.447142	+	0.894463i	$0.647558\pi$
$212$		0			0
$213$			8.32895e6i			0.861889i
$214$		0			0
$215$		−	1.41904e6i		−	0.142784i
$216$		0			0
$217$	8.68224e6	+	3.65233e6i	0.849675	+	0.357430i
$218$		0			0
$219$	−2.75400e6			−0.262199
$220$		0			0
$221$	1.31947e7			1.22243
$222$		0			0
$223$			4.90434e6i			0.442248i	0.975246	+	0.221124i	$0.0709726\pi$
$223$			4.90434e6i			0.442248i	−0.975246	+	0.221124i	$0.929027\pi$
$224$		0			0
$225$	−1.78099e7			−1.56356
$226$		0			0
$227$			1.32183e7i			1.13005i	0.825075	+	0.565023i	$0.191133\pi$
$227$			1.32183e7i			1.13005i	−0.825075	+	0.565023i	$0.808867\pi$
$228$		0			0
$229$			338974.i			0.0282266i	0.999900	+	0.0141133i	$0.00449256\pi$
$229$			338974.i			0.0282266i	−0.999900	+	0.0141133i	$0.995507\pi$
$230$		0			0
$231$	1.24807e7	+	5.25023e6i	1.01252	+	0.425934i
$232$		0			0
$233$	4.84146e6			0.382744			0.191372	−	0.981518i	$-0.438706\pi$
$233$	4.84146e6			0.382744			0.191372	+	0.981518i	$0.438706\pi$
$234$		0			0
$235$	3.27216e6			0.252134
$236$		0			0
$237$			2.41610e7i			1.81497i
$238$		0			0
$239$	1.37297e7			1.00570			0.502850	−	0.864374i	$-0.332285\pi$
$239$	1.37297e7			1.00570			0.502850	+	0.864374i	$0.332285\pi$
$240$		0			0
$241$		−	3.66913e6i		−	0.262127i	−0.991374	−	0.131064i	$-0.958161\pi$
$241$		−	3.66913e6i		−	0.262127i	0.991374	−	0.131064i	$-0.0418392\pi$
$242$		0			0
$243$		−	8.70433e6i		−	0.606619i
$244$		0			0
$245$	3.79848e6	−	3.71588e6i	0.258292	−	0.252676i
$246$		0			0
$247$	6.89724e6			0.457704
$248$		0			0
$249$	3.22871e7			2.09137
$250$		0			0
$251$			1.57289e7i			0.994664i	0.867560	+	0.497332i	$0.165687\pi$
$251$			1.57289e7i			0.994664i	−0.867560	+	0.497332i	$0.834313\pi$
$252$		0			0
$253$	4.14101e6			0.255708
$254$		0			0
$255$		−	1.21624e7i		−	0.733498i
$256$		0			0
$257$			7.54531e6i			0.444506i	0.974989	+	0.222253i	$0.0713411\pi$
$257$			7.54531e6i			0.444506i	−0.974989	+	0.222253i	$0.928659\pi$
$258$		0			0
$259$	−399266.	+	949126.i	−0.0229807	+	0.0546292i
$260$		0			0
$261$	−1.46124e7			−0.821864
$262$		0			0
$263$	1.32059e7			0.725942			0.362971	−	0.931800i	$-0.381762\pi$
$263$	1.32059e7			0.725942			0.362971	+	0.931800i	$0.381762\pi$
$264$		0			0
$265$		−	3.45098e6i		−	0.185441i
$266$		0			0
$267$	−2.84335e7			−1.49382
$268$		0			0
$269$		−	1.59600e7i		−	0.819930i	−0.912101	−	0.409965i	$-0.865541\pi$
$269$		−	1.59600e7i		−	0.819930i	0.912101	−	0.409965i	$-0.134459\pi$
$270$		0			0
$271$		−	2.48446e7i		−	1.24831i	−0.781299	−	0.624157i	$-0.785443\pi$
$271$		−	2.48446e7i		−	1.24831i	0.781299	−	0.624157i	$-0.214557\pi$
$272$		0			0
$273$	−1.32947e7	+	3.16038e7i	−0.653416	+	1.55329i
$274$		0			0
$275$	−1.18733e7			−0.570917
$276$		0			0
$277$	1.60013e7			0.752863			0.376432	−	0.926444i	$-0.377151\pi$
$277$	1.60013e7			0.752863			0.376432	+	0.926444i	$0.377151\pi$
$278$		0			0
$279$			3.60016e7i			1.65771i
$280$		0			0
$281$	−603566.			−0.0272023			−0.0136012	−	0.999908i	$-0.504330\pi$
$281$	−603566.			−0.0272023			−0.0136012	+	0.999908i	$0.504330\pi$
$282$		0			0
$283$			2.22195e7i			0.980334i	0.871629	+	0.490167i	$0.163064\pi$
$283$			2.22195e7i			0.980334i	−0.871629	+	0.490167i	$0.836936\pi$
$284$		0			0
$285$		−	6.35762e6i		−	0.274637i
$286$		0			0
$287$	1.81927e7	+	7.65308e6i	0.769577	+	0.323736i
$288$		0			0
$289$	−1.14074e7			−0.472599
$290$		0			0
$291$	3.67853e7			1.49278
$292$		0			0
$293$			4.28134e7i			1.70207i	0.525110	+	0.851034i	$0.324024\pi$
$293$			4.28134e7i			1.70207i	−0.525110	+	0.851034i	$0.675976\pi$
$294$		0			0
$295$	−5.11428e6			−0.199213
$296$		0			0
$297$			2.29747e7i			0.876961i
$298$		0			0
$299$			1.04859e7i			0.392277i
$300$		0			0
$301$	4.17859e6	−	9.93326e6i	0.153225	−	0.364244i
$302$		0			0
$303$	−8.81627e7			−3.16925
$304$		0			0
$305$	−1.24256e7			−0.437945
$306$		0			0
$307$		−	4.02152e7i		−	1.38987i	−0.719072	−	0.694936i	$-0.755433\pi$
$307$		−	4.02152e7i		−	1.38987i	0.719072	−	0.694936i	$-0.244567\pi$
$308$		0			0
$309$	−7.66673e7			−2.59857
$310$		0			0
$311$			1.47381e7i			0.489958i	0.969528	+	0.244979i	$0.0787811\pi$
$311$			1.47381e7i			0.489958i	−0.969528	+	0.244979i	$0.921219\pi$
$312$		0			0
$313$		−	4.19490e7i		−	1.36801i	−0.729479	−	0.684004i	$-0.760237\pi$
$313$		−	4.19490e7i		−	1.36801i	0.729479	−	0.684004i	$-0.239763\pi$
$314$		0			0
$315$	1.87211e7	+	7.87534e6i	0.598962	+	0.251964i
$316$		0			0
$317$	4.30922e7			1.35276			0.676380	−	0.736553i	$-0.263548\pi$
$317$	4.30922e7			1.35276			0.676380	+	0.736553i	$0.263548\pi$
$318$		0			0
$319$	−9.74160e6			−0.300095
$320$		0			0
$321$		−	7.29900e7i		−	2.20673i
$322$		0			0
$323$	−1.85803e7			−0.551373
$324$		0			0
$325$		−	3.00657e7i		−	0.875832i
$326$		0			0
$327$			8.99831e6i			0.257346i
$328$		0			0
$329$	2.29051e7	+	9.63543e6i	0.643198	+	0.270572i
$330$		0			0
$331$	5.32204e7			1.46755			0.733777	−	0.679390i	$-0.237756\pi$
$331$	5.32204e7			1.46755			0.733777	+	0.679390i	$0.237756\pi$
$332$		0			0
$333$	−3.93562e6			−0.106581
$334$		0			0
$335$		−	2.23683e7i		−	0.594974i
$336$		0			0
$337$	2.34579e6			0.0612913			0.0306456	−	0.999530i	$-0.490244\pi$
$337$	2.34579e6			0.0612913			0.0306456	+	0.999530i	$0.490244\pi$
$338$		0			0
$339$		−	8.16437e7i		−	2.09567i
$340$		0			0
$341$			2.40010e7i			0.605295i
$342$		0			0
$343$	3.75314e7	−	1.48259e7i	0.930063	−	0.367400i
$344$		0			0
$345$	9.66552e6			0.235379
$346$		0			0
$347$	4.80596e7			1.15025			0.575124	−	0.818066i	$-0.304954\pi$
$347$	4.80596e7			1.15025			0.575124	+	0.818066i	$0.304954\pi$
$348$		0			0
$349$		−	1.00499e7i		−	0.236421i	−0.992989	−	0.118211i	$-0.962284\pi$
$349$		−	1.00499e7i		−	0.236421i	0.992989	−	0.118211i	$-0.0377158\pi$
$350$		0			0
$351$	−5.81767e7			−1.34533
$352$		0			0
$353$			7.65216e7i			1.73964i	0.493368	+	0.869821i	$0.335766\pi$
$353$			7.65216e7i			1.73964i	−0.493368	+	0.869821i	$0.664234\pi$
$354$		0			0
$355$			8.32895e6i			0.186168i
$356$		0			0
$357$	3.58142e7	−	8.51368e7i	0.787138	−	1.87117i
$358$		0			0
$359$	−8.39735e6			−0.181493			−0.0907463	−	0.995874i	$-0.528925\pi$
$359$	−8.39735e6			−0.181493			−0.0907463	+	0.995874i	$0.528925\pi$
$360$		0			0
$361$	3.73334e7			0.793554
$362$		0			0
$363$		−	4.55135e7i		−	0.951525i
$364$		0			0
$365$	−2.75400e6			−0.0566351
$366$		0			0
$367$		−	3.82776e6i		−	0.0774366i	−0.999250	−	0.0387183i	$-0.987672\pi$
$367$		−	3.82776e6i		−	0.0774366i	0.999250	−	0.0387183i	$-0.0123275\pi$
$368$		0			0
$369$			7.54375e7i			1.50144i
$370$		0			0
$371$	1.01620e7	−	2.41569e7i	0.199002	−	0.473063i
$372$		0			0
$373$	−4.93836e7			−0.951604			−0.475802	−	0.879552i	$-0.657842\pi$
$373$	−4.93836e7			−0.951604			−0.475802	+	0.879552i	$0.657842\pi$
$374$		0			0
$375$	−5.95884e7			−1.12997
$376$		0			0
$377$		−	2.46678e7i		−	0.460369i
$378$		0			0
$379$	−3.74561e7			−0.688026			−0.344013	−	0.938965i	$-0.611786\pi$
$379$	−3.74561e7			−0.688026			−0.344013	+	0.938965i	$0.611786\pi$
$380$		0			0
$381$		−	1.50166e8i		−	2.71516i
$382$		0			0
$383$			5.01003e7i			0.891752i	0.895095	+	0.445876i	$0.147108\pi$
$383$			5.01003e7i			0.891752i	−0.895095	+	0.445876i	$0.852892\pi$
$384$		0			0
$385$	1.24807e7	+	5.25023e6i	0.218704	+	0.0920017i
$386$		0			0
$387$	4.11890e7			0.710638
$388$		0			0
$389$	224986.			0.00382214			0.00191107	−	0.999998i	$-0.499392\pi$
$389$	224986.			0.00382214			0.00191107	+	0.999998i	$0.499392\pi$
$390$		0			0
$391$		−	2.82478e7i		−	0.472557i
$392$		0			0
$393$	−1.41627e8			−2.33329
$394$		0			0
$395$			2.41610e7i			0.392034i
$396$		0			0
$397$		−	871937.i		−	0.0139352i	−0.999976	−	0.00696760i	$-0.997782\pi$
$397$		−	871937.i		−	0.0139352i	0.999976	−	0.00696760i	$-0.00221787\pi$
$398$		0			0
$399$	1.87211e7	−	4.45033e7i	0.294722	−	0.700606i
$400$		0			0
$401$	1.44909e7			0.224730			0.112365	−	0.993667i	$-0.464157\pi$
$401$	1.44909e7			0.224730			0.112365	+	0.993667i	$0.464157\pi$
$402$		0			0
$403$	−6.07757e7			−0.928570
$404$		0			0
$405$			1.04588e7i			0.157440i
$406$		0			0
$407$	−2.62375e6			−0.0389170
$408$		0			0
$409$			1.04303e8i			1.52450i	0.647284	+	0.762249i	$0.275905\pi$
$409$			1.04303e8i			1.52450i	−0.647284	+	0.762249i	$0.724095\pi$
$410$		0			0
$411$			9.61712e7i			1.38522i
$412$		0			0
$413$	−3.58000e7	−	1.50599e7i	−0.508197	−	0.213782i
$414$		0			0
$415$	3.22871e7			0.451736
$416$		0			0
$417$	7.60696e7			1.04907
$418$		0			0
$419$			8.22075e7i			1.11756i	0.829317	+	0.558778i	$0.188730\pi$
$419$			8.22075e7i			1.11756i	−0.829317	+	0.558778i	$0.811270\pi$
$420$		0			0
$421$	−1.33780e7			−0.179285			−0.0896427	−	0.995974i	$-0.528573\pi$
$421$	−1.33780e7			−0.179285			−0.0896427	+	0.995974i	$0.528573\pi$
$422$		0			0
$423$			9.49778e7i			1.25488i
$424$		0			0
$425$			8.09932e7i			1.05507i
$426$		0			0
$427$	−8.69795e7	−	3.65894e7i	−1.11721	−	0.469972i
$428$		0			0
$429$	−8.73650e7			−1.10654
$430$		0			0
$431$	1.34244e8			1.67673			0.838367	−	0.545106i	$-0.183510\pi$
$431$	1.34244e8			1.67673			0.838367	+	0.545106i	$0.183510\pi$
$432$		0			0
$433$		−	1.03230e8i		−	1.27158i	−0.771863	−	0.635789i	$-0.780675\pi$
$433$		−	1.03230e8i		−	1.27158i	0.771863	−	0.635789i	$-0.219325\pi$
$434$		0			0
$435$	−2.27378e7			−0.276237
$436$		0			0
$437$		−	1.47659e7i		−	0.176935i
$438$		0			0
$439$		−	2.65816e7i		−	0.314186i	−0.987584	−	0.157093i	$-0.949788\pi$
$439$		−	2.65816e7i		−	0.314186i	0.987584	−	0.157093i	$-0.0502123\pi$
$440$		0			0
$441$	1.07857e8	+	1.10255e8i	1.25757	+	1.28553i
$442$		0			0
$443$	−1.31972e8			−1.51799			−0.758996	−	0.651096i	$-0.774310\pi$
$443$	−1.31972e8			−1.51799			−0.758996	+	0.651096i	$0.774310\pi$
$444$		0			0
$445$	−2.84335e7			−0.322664
$446$		0			0
$447$		−	1.17240e8i		−	1.31266i
$448$		0			0
$449$	1.47766e8			1.63244			0.816218	−	0.577743i	$-0.196067\pi$
$449$	1.47766e8			1.63244			0.816218	+	0.577743i	$0.196067\pi$
$450$		0			0
$451$			5.02917e7i			0.548234i
$452$		0			0
$453$			7.61309e7i			0.818967i
$454$		0			0
$455$	−1.32947e7	+	3.16038e7i	−0.141138	+	0.335510i
$456$		0			0
$457$	−8.22868e7			−0.862147			−0.431074	−	0.902317i	$-0.641865\pi$
$457$	−8.22868e7			−0.862147			−0.431074	+	0.902317i	$0.641865\pi$
$458$		0			0
$459$	1.56721e8			1.62065
$460$		0			0
$461$			1.31884e8i			1.34614i	0.739580	+	0.673068i	$0.235024\pi$
$461$			1.31884e8i			1.34614i	−0.739580	+	0.673068i	$0.764976\pi$
$462$		0			0
$463$	−1.39927e6			−0.0140981			−0.00704904	−	0.999975i	$-0.502244\pi$
$463$	−1.39927e6			−0.0140981			−0.00704904	+	0.999975i	$0.502244\pi$
$464$		0			0
$465$			5.60207e7i			0.557173i
$466$		0			0
$467$		−	1.81321e7i		−	0.178032i	−0.996030	−	0.0890158i	$-0.971628\pi$
$467$		−	1.81321e7i		−	0.178032i	0.996030	−	0.0890158i	$-0.0283722\pi$
$468$		0			0
$469$	6.58672e7	−	1.56578e8i	0.638485	−	1.51779i
$470$		0			0
$471$	1.66050e8			1.58919
$472$		0			0
$473$	2.74593e7			0.259482
$474$		0			0
$475$			4.23374e7i			0.395042i
$476$		0			0
$477$	1.00168e8			0.922943
$478$		0			0
$479$		−	2.89375e7i		−	0.263303i	−0.991296	−	0.131651i	$-0.957972\pi$
$479$		−	2.89375e7i		−	0.263303i	0.991296	−	0.131651i	$-0.0420279\pi$
$480$		0			0
$481$		−	6.64388e6i		−	0.0597017i
$482$		0			0
$483$	6.76586e7	+	2.84618e7i	0.600457	+	0.252592i
$484$		0			0
$485$	3.67853e7			0.322440
$486$		0			0
$487$	9.47515e7			0.820350			0.410175	−	0.912007i	$-0.365468\pi$
$487$	9.47515e7			0.820350			0.410175	+	0.912007i	$0.365468\pi$
$488$		0			0
$489$		−	8.49992e7i		−	0.726923i
$490$		0			0
$491$	2.58834e7			0.218663			0.109332	−	0.994005i	$-0.465129\pi$
$491$	2.58834e7			0.218663			0.109332	+	0.994005i	$0.465129\pi$
$492$		0			0
$493$			6.64520e7i			0.554584i
$494$		0			0
$495$			5.17522e7i			0.426691i
$496$		0			0
$497$	−2.45260e7	+	5.83026e7i	−0.199783	+	0.474918i
$498$		0			0
$499$	−1.56023e8			−1.25571			−0.627853	−	0.778332i	$-0.716066\pi$
$499$	−1.56023e8			−1.25571			−0.627853	+	0.778332i	$0.716066\pi$
$500$		0			0
$501$	1.42543e8			1.13353
$502$		0			0
$503$			1.11476e8i			0.875950i	0.898987	+	0.437975i	$0.144304\pi$
$503$			1.11476e8i			0.875950i	−0.898987	+	0.437975i	$0.855696\pi$
$504$		0			0
$505$	−8.81627e7			−0.684559
$506$		0			0
$507$		−	3.21724e6i		−	0.0246865i
$508$		0			0
$509$		−	7.45421e7i		−	0.565260i	−0.959229	−	0.282630i	$-0.908793\pi$
$509$		−	7.45421e7i		−	0.565260i	0.959229	−	0.282630i	$-0.0912068\pi$
$510$		0			0
$511$	−1.92780e7	−	8.10962e6i	−0.144477	−	0.0607768i
$512$		0			0
$513$	8.19223e7			0.606806
$514$		0			0
$515$	−7.66673e7			−0.561291
$516$		0			0
$517$			6.33185e7i			0.458204i
$518$		0			0
$519$	−1.03646e8			−0.741398
$520$		0			0
$521$			1.96232e8i			1.38758i	0.720179	+	0.693789i	$0.244060\pi$
$521$			1.96232e8i			1.38758i	−0.720179	+	0.693789i	$0.755940\pi$
$522$		0			0
$523$			4.62080e7i			0.323007i	0.986872	+	0.161504i	$0.0516344\pi$
$523$			4.62080e7i			0.323007i	−0.986872	+	0.161504i	$0.948366\pi$
$524$		0			0
$525$	−1.93994e8	−	8.16068e7i	−1.34063	−	0.563960i
$526$		0			0
$527$	1.63722e8			1.11860
$528$		0			0
$529$	−1.25587e8			−0.848357
$530$		0			0
$531$		−	1.48447e8i		−	0.991490i
$532$		0			0
$533$	−1.27349e8			−0.841035
$534$		0			0
$535$		−	7.29900e7i		−	0.476653i
$536$		0			0
$537$		−	1.58916e8i		−	1.02623i
$538$		0			0
$539$	7.19049e7	+	7.35032e7i	0.459189	+	0.469397i
$540$		0			0
$541$	−7.52906e7			−0.475498			−0.237749	−	0.971327i	$-0.576410\pi$
$541$	−7.52906e7			−0.475498			−0.237749	+	0.971327i	$0.576410\pi$
$542$		0			0
$543$	−3.37965e8			−2.11092
$544$		0			0
$545$			8.99831e6i			0.0555868i
$546$		0			0
$547$	−7.26760e7			−0.444047			−0.222023	−	0.975041i	$-0.571266\pi$
$547$	−7.26760e7			−0.444047			−0.222023	+	0.975041i	$0.571266\pi$
$548$		0			0
$549$		−	3.60667e8i		−	2.17966i
$550$		0			0
$551$			3.47363e7i			0.207648i
$552$		0			0
$553$	−7.11462e7	+	1.69127e8i	−0.420704	+	1.00009i
$554$		0			0
$555$	−6.12408e6			−0.0358230
$556$		0			0
$557$	−3.10741e8			−1.79818			−0.899090	−	0.437765i	$-0.855770\pi$
$557$	−3.10741e8			−1.79818			−0.899090	+	0.437765i	$0.855770\pi$
$558$		0			0
$559$			6.95328e7i			0.398065i
$560$		0			0
$561$	2.35351e8			1.33299
$562$		0			0
$563$			5.66378e7i			0.317381i	0.987328	+	0.158690i	$0.0507272\pi$
$563$			5.66378e7i			0.317381i	−0.987328	+	0.158690i	$0.949273\pi$
$564$		0			0
$565$		−	8.16437e7i		−	0.452665i
$566$		0			0
$567$	−3.07976e7	+	7.32114e7i	−0.168954	+	0.401633i
$568$		0			0
$569$	5.17304e7			0.280808			0.140404	−	0.990094i	$-0.455160\pi$
$569$	5.17304e7			0.280808			0.140404	+	0.990094i	$0.455160\pi$
$570$		0			0
$571$	−8.68765e7			−0.466653			−0.233326	−	0.972398i	$-0.574961\pi$
$571$	−8.68765e7			−0.466653			−0.233326	+	0.972398i	$0.574961\pi$
$572$		0			0
$573$			3.25210e8i			1.72862i
$574$		0			0
$575$	−6.43657e7			−0.338572
$576$		0			0
$577$		−	5.89865e7i		−	0.307062i	−0.988144	−	0.153531i	$-0.950936\pi$
$577$		−	5.89865e7i		−	0.307062i	0.988144	−	0.153531i	$-0.0490644\pi$
$578$		0			0
$579$		−	5.91178e8i		−	3.04567i
$580$		0			0
$581$	2.26010e8	+	9.50748e7i	1.15239	+	0.484771i
$582$		0			0
$583$	6.67788e7			0.337003
$584$		0			0
$585$	−1.31048e8			−0.654578
$586$		0			0
$587$			3.10848e8i			1.53686i	0.639934	+	0.768430i	$0.278962\pi$
$587$			3.10848e8i			1.53686i	−0.639934	+	0.768430i	$0.721038\pi$
$588$		0			0
$589$	8.55821e7			0.418829
$590$		0			0
$591$		−	4.14390e8i		−	2.00746i
$592$		0			0
$593$		−	3.46714e8i		−	1.66268i	−0.555766	−	0.831339i	$-0.687575\pi$
$593$		−	3.46714e8i		−	1.66268i	0.555766	−	0.831339i	$-0.312425\pi$
$594$		0			0
$595$	3.58142e7	−	8.51368e7i	0.170022	−	0.404172i
$596$		0			0
$597$	3.06575e8			1.44083
$598$		0			0
$599$	−9.47771e7			−0.440984			−0.220492	−	0.975389i	$-0.570766\pi$
$599$	−9.47771e7			−0.440984			−0.220492	+	0.975389i	$0.570766\pi$
$600$		0			0
$601$		−	2.04951e8i		−	0.944119i	−0.881567	−	0.472060i	$-0.843511\pi$
$601$		−	2.04951e8i		−	0.944119i	0.881567	−	0.472060i	$-0.156489\pi$
$602$		0			0
$603$	6.49262e8			2.96120
$604$		0			0
$605$		−	4.55135e7i		−	0.205529i
$606$		0			0
$607$		−	3.28634e8i		−	1.46942i	−0.678379	−	0.734712i	$-0.737317\pi$
$607$		−	3.28634e8i		−	1.46942i	0.678379	−	0.734712i	$-0.262683\pi$
$608$		0			0
$609$	−1.59165e8	−	6.69554e7i	−0.704686	−	0.296438i
$610$		0			0
$611$	−1.60336e8			−0.702922
$612$		0			0
$613$	2.67967e8			1.16332			0.581660	−	0.813432i	$-0.302403\pi$
$613$	2.67967e8			1.16332			0.581660	+	0.813432i	$0.302403\pi$
$614$		0			0
$615$			1.17386e8i			0.504649i
$616$		0			0
$617$	−3.88909e8			−1.65574			−0.827870	−	0.560921i	$-0.810447\pi$
$617$	−3.88909e8			−1.65574			−0.827870	+	0.560921i	$0.810447\pi$
$618$		0			0
$619$			6.30894e7i			0.266002i	0.991116	+	0.133001i	$0.0424613\pi$
$619$			6.30894e7i			0.266002i	−0.991116	+	0.133001i	$0.957539\pi$
$620$		0			0
$621$			1.24547e8i			0.520066i
$622$		0			0
$623$	−1.99035e8	−	8.37273e7i	−0.823123	−	0.346261i
$624$		0			0
$625$	1.52677e8			0.625366
$626$		0			0
$627$	1.23024e8			0.499101
$628$		0			0
$629$			1.78978e7i			0.0719197i
$630$		0			0
$631$	1.30827e8			0.520725			0.260363	−	0.965511i	$-0.416158\pi$
$631$	1.30827e8			0.520725			0.260363	+	0.965511i	$0.416158\pi$
$632$		0			0
$633$		−	3.79435e8i		−	1.49598i
$634$		0			0
$635$		−	1.50166e8i		−	0.586475i
$636$		0			0
$637$	−1.86126e8	+	1.82078e8i	−0.720091	+	0.704433i
$638$		0			0
$639$	−2.41756e8			−0.926563
$640$		0			0
$641$	−7.17536e7			−0.272439			−0.136220	−	0.990679i	$-0.543495\pi$
$641$	−7.17536e7			−0.272439			−0.136220	+	0.990679i	$0.543495\pi$
$642$		0			0
$643$		−	2.56068e8i		−	0.963214i	−0.876387	−	0.481607i	$-0.840053\pi$
$643$		−	2.56068e8i		−	0.963214i	0.876387	−	0.481607i	$-0.159947\pi$
$644$		0			0
$645$	6.40927e7			0.238852
$646$		0			0
$647$			4.93122e8i			1.82071i	0.413827	+	0.910356i	$0.364192\pi$
$647$			4.93122e8i			1.82071i	−0.413827	+	0.910356i	$0.635808\pi$
$648$		0			0
$649$		−	9.89648e7i		−	0.362032i
$650$		0			0
$651$	−1.64963e8	+	3.92145e8i	−0.597919	+	1.42136i
$652$		0			0
$653$	1.55036e8			0.556793			0.278397	−	0.960466i	$-0.410197\pi$
$653$	1.55036e8			0.556793			0.278397	+	0.960466i	$0.410197\pi$
$654$		0			0
$655$	−1.41627e8			−0.503990
$656$		0			0
$657$		−	7.99377e7i		−	0.281874i
$658$		0			0
$659$	3.01683e8			1.05413			0.527065	−	0.849825i	$-0.323293\pi$
$659$	3.01683e8			1.05413			0.527065	+	0.849825i	$0.323293\pi$
$660$		0			0
$661$			1.80227e8i			0.624044i	0.950075	+	0.312022i	$0.101006\pi$
$661$			1.80227e8i			0.624044i	−0.950075	+	0.312022i	$0.898994\pi$
$662$		0			0
$663$			5.95957e8i			2.04491i
$664$		0			0
$665$	1.87211e7	−	4.45033e7i	0.0636599	−	0.151331i
$666$		0			0
$667$	−5.28097e7			−0.177966
$668$		0			0
$669$	−2.21511e8			−0.739806
$670$		0			0
$671$		−	2.40445e8i		−	0.795880i
$672$		0			0
$673$	4.06265e8			1.33280			0.666399	−	0.745595i	$-0.267835\pi$
$673$	4.06265e8			1.33280			0.666399	+	0.745595i	$0.267835\pi$
$674$		0			0
$675$		−	3.57106e8i		−	1.16114i
$676$		0			0
$677$			1.77837e7i			0.0573133i	0.999589	+	0.0286566i	$0.00912294\pi$
$677$			1.77837e7i			0.0573133i	−0.999589	+	0.0286566i	$0.990877\pi$
$678$		0			0
$679$	2.57497e8	+	1.08320e8i	0.822551	+	0.346020i
$680$		0			0
$681$	−5.97020e8			−1.89037
$682$		0			0
$683$	2.66054e8			0.835042			0.417521	−	0.908667i	$-0.362899\pi$
$683$	2.66054e8			0.835042			0.417521	+	0.908667i	$0.362899\pi$
$684$		0			0
$685$			9.61712e7i			0.299208i
$686$		0			0
$687$	−1.53102e7			−0.0472183
$688$		0			0
$689$			1.69098e8i			0.516989i
$690$		0			0
$691$			2.44451e8i			0.740898i	0.928853	+	0.370449i	$0.120796\pi$
$691$			2.44451e8i			0.740898i	−0.928853	+	0.370449i	$0.879204\pi$
$692$		0			0
$693$	−1.52393e8	+	3.62266e8i	−0.457895	+	1.08850i
$694$		0			0
$695$	7.60696e7			0.226598
$696$		0			0
$697$	3.43063e8			1.01315
$698$		0			0
$699$			2.18671e8i			0.640265i
$700$		0			0
$701$	2.31727e8			0.672702			0.336351	−	0.941737i	$-0.390807\pi$
$701$	2.31727e8			0.672702			0.336351	+	0.941737i	$0.390807\pi$
$702$		0			0
$703$			9.35567e6i			0.0269283i
$704$		0			0
$705$			1.47792e8i			0.421776i
$706$		0			0
$707$	−6.17139e8	−	2.59610e8i	−1.74632	−	0.734621i
$708$		0			0
$709$	−3.09705e8			−0.868979			−0.434489	−	0.900677i	$-0.643071\pi$
$709$	−3.09705e8			−0.868979			−0.434489	+	0.900677i	$0.643071\pi$
$710$		0			0
$711$	−7.01298e8			−1.95117
$712$		0			0
$713$			1.30111e8i			0.358959i
$714$		0			0
$715$	−8.73650e7			−0.239012
$716$		0			0
$717$			6.20122e8i			1.68236i
$718$		0			0
$719$		−	3.85416e8i		−	1.03692i	−0.855103	−	0.518458i	$-0.826506\pi$
$719$		−	3.85416e8i		−	1.03692i	0.855103	−	0.518458i	$-0.173494\pi$
$720$		0			0
$721$	−5.36671e8	−	2.25760e8i	−1.43187	−	0.602339i
$722$		0			0
$723$	1.65721e8			0.438494
$724$		0			0
$725$	1.51418e8			0.397342
$726$		0			0
$727$			3.13918e8i			0.816983i	0.912762	+	0.408491i	$0.133945\pi$
$727$			3.13918e8i			0.816983i	−0.912762	+	0.408491i	$0.866055\pi$
$728$		0			0
$729$	5.61951e8			1.45049
$730$		0			0
$731$		−	1.87313e8i		−	0.479530i
$732$		0			0
$733$			7.18118e7i			0.182341i	0.995835	+	0.0911704i	$0.0290608\pi$
$733$			7.18118e7i			0.182341i	−0.995835	+	0.0911704i	$0.970939\pi$
$734$		0			0
$735$	1.67833e8	+	1.71564e8i	0.422683	+	0.432079i
$736$		0			0
$737$	4.32842e8			1.08125
$738$		0			0
$739$	−2.77815e7			−0.0688370			−0.0344185	−	0.999408i	$-0.510958\pi$
$739$	−2.77815e7			−0.0688370			−0.0344185	+	0.999408i	$0.510958\pi$
$740$		0			0
$741$			3.11523e8i			0.765660i
$742$		0			0
$743$	7.03366e8			1.71481			0.857403	−	0.514646i	$-0.172077\pi$
$743$	7.03366e8			1.71481			0.857403	+	0.514646i	$0.172077\pi$
$744$		0			0
$745$		−	1.17240e8i		−	0.283535i
$746$		0			0
$747$			9.37166e8i			2.24830i
$748$		0			0
$749$	2.14931e8	−	5.10930e8i	0.511510	−	1.21595i
$750$		0			0
$751$	3.00617e8			0.709731			0.354866	−	0.934917i	$-0.384527\pi$
$751$	3.00617e8			0.709731			0.354866	+	0.934917i	$0.384527\pi$
$752$		0			0
$753$	−7.10416e8			−1.66390
$754$		0			0
$755$			7.61309e7i			0.176897i
$756$		0			0
$757$	1.17057e8			0.269841			0.134921	−	0.990856i	$-0.456922\pi$
$757$	1.17057e8			0.269841			0.134921	+	0.990856i	$0.456922\pi$
$758$		0			0
$759$			1.87034e8i			0.427756i
$760$		0			0
$761$			2.63542e8i			0.597992i	0.954254	+	0.298996i	$0.0966518\pi$
$761$			2.63542e8i			0.597992i	−0.954254	+	0.298996i	$0.903348\pi$
$762$		0			0
$763$	−2.64971e7	+	6.29882e7i	−0.0596519	+	0.141803i
$764$		0			0
$765$	3.53026e8			0.788538
$766$		0			0
$767$	2.50600e8			0.555385
$768$		0			0
$769$			1.25689e8i			0.276388i	0.990405	+	0.138194i	$0.0441298\pi$
$769$			1.25689e8i			0.276388i	−0.990405	+	0.138194i	$0.955870\pi$
$770$		0			0
$771$	−3.40794e8			−0.743582
$772$		0			0
$773$			1.58329e8i			0.342786i	0.985203	+	0.171393i	$0.0548268\pi$
$773$			1.58329e8i			0.342786i	−0.985203	+	0.171393i	$0.945173\pi$
$774$		0			0
$775$		−	3.73060e8i		−	0.801444i
$776$		0			0
$777$	−4.28686e7	−	1.80334e7i	−0.0913852	−	0.0384427i
$778$		0			0
$779$	1.79328e8			0.379347
$780$		0			0
$781$	−1.61171e8			−0.338324
$782$		0			0
$783$		−	2.92993e8i		−	0.610340i
$784$		0			0
$785$	1.66050e8			0.343265
$786$		0			0
$787$			5.15726e8i			1.05802i	0.848615	+	0.529012i	$0.177437\pi$
$787$			5.15726e8i			1.05802i	−0.848615	+	0.529012i	$0.822563\pi$
$788$		0			0
$789$			5.96464e8i			1.21438i
$790$		0			0
$791$	2.40414e8	−	5.71506e8i	0.485769	−	1.15476i
$792$		0			0
$793$	6.08856e8			1.22094
$794$		0			0
$795$	1.55868e8			0.310210
$796$		0			0
$797$		−	9.29429e8i		−	1.83587i	−0.396735	−	0.917933i	$-0.629857\pi$
$797$		−	9.29429e8i		−	1.83587i	0.396735	−	0.917933i	$-0.370143\pi$
$798$		0			0
$799$	4.31925e8			0.846775
$800$		0			0
$801$		−	8.25312e8i		−	1.60591i
$802$		0			0
$803$		−	5.32918e7i		−	0.102923i
$804$		0			0
$805$	6.76586e7	+	2.84618e7i	0.129699	+	0.0545600i
$806$		0			0
$807$	7.20856e8			1.37160
$808$		0			0
$809$	−6.89731e8			−1.30267			−0.651335	−	0.758791i	$-0.725791\pi$
$809$	−6.89731e8			−1.30267			−0.651335	+	0.758791i	$0.725791\pi$
$810$		0			0
$811$		−	2.85961e7i		−	0.0536099i	−0.999641	−	0.0268049i	$-0.991467\pi$
$811$		−	2.85961e7i		−	0.0536099i	0.999641	−	0.0268049i	$-0.00853330\pi$
$812$		0			0
$813$	1.12214e9			2.08821
$814$		0			0
$815$		−	8.49992e7i		−	0.157015i
$816$		0			0
$817$		−	9.79135e7i		−	0.179546i
$818$		0			0
$819$	−9.17333e8	−	3.85892e8i	−1.66984	−	0.702447i
$820$		0			0
$821$	−2.51151e8			−0.453844			−0.226922	−	0.973913i	$-0.572866\pi$
$821$	−2.51151e8			−0.453844			−0.226922	+	0.973913i	$0.572866\pi$
$822$		0			0
$823$	−2.40085e8			−0.430691			−0.215346	−	0.976538i	$-0.569088\pi$
$823$	−2.40085e8			−0.430691			−0.215346	+	0.976538i	$0.569088\pi$
$824$		0			0
$825$		−	5.36273e8i		−	0.955046i
$826$		0			0
$827$	−9.14380e8			−1.61663			−0.808314	−	0.588751i	$-0.799620\pi$
$827$	−9.14380e8			−1.61663			−0.808314	+	0.588751i	$0.799620\pi$
$828$		0			0
$829$			5.22917e8i			0.917845i	0.888476	+	0.458922i	$0.151764\pi$
$829$			5.22917e8i			0.917845i	−0.888476	+	0.458922i	$0.848236\pi$
$830$		0			0
$831$			7.22721e8i			1.25941i
$832$		0			0
$833$	5.01399e8	−	4.90496e8i	0.867459	−	0.848596i
$834$		0			0
$835$	1.42543e8			0.244842
$836$		0			0
$837$	−7.21866e8			−1.23106
$838$		0			0
$839$		−	9.46684e8i		−	1.60295i	−0.598030	−	0.801474i	$-0.704050\pi$
$839$		−	9.46684e8i		−	1.60295i	0.598030	−	0.801474i	$-0.295950\pi$
$840$		0			0
$841$	−4.70590e8			−0.791142
$842$		0			0
$843$		−	2.72609e7i		−	0.0455048i
$844$		0			0
$845$		−	3.21724e6i		−	0.00533229i
$846$		0			0
$847$	1.34022e8	−	3.18594e8i	0.220560	−	0.524309i
$848$		0			0
$849$	−1.00357e9			−1.63993
$850$		0			0
$851$	−1.42235e7			−0.0230790
$852$		0			0
$853$		−	2.30334e8i		−	0.371117i	−0.982633	−	0.185558i	$-0.940591\pi$
$853$		−	2.30334e8i		−	0.371117i	0.982633	−	0.185558i	$-0.0594094\pi$
$854$		0			0
$855$	1.84536e8			0.295246
$856$		0			0
$857$		−	4.34496e7i		−	0.0690308i	−0.999404	−	0.0345154i	$-0.989011\pi$
$857$		−	4.34496e7i		−	0.0690308i	0.999404	−	0.0345154i	$-0.0109888\pi$
$858$		0			0
$859$			1.61129e8i			0.254211i	0.991889	+	0.127105i	$0.0405686\pi$
$859$			1.61129e8i			0.254211i	−0.991889	+	0.127105i	$0.959431\pi$
$860$		0			0
$861$	−3.45662e8	+	8.21699e8i	−0.541554	+	1.28737i
$862$		0			0
$863$	−2.56357e8			−0.398853			−0.199426	−	0.979913i	$-0.563908\pi$
$863$	−2.56357e8			−0.398853			−0.199426	+	0.979913i	$0.563908\pi$
$864$		0			0
$865$	−1.03646e8			−0.160142
$866$		0			0
$867$		−	5.15230e8i		−	0.790577i
$868$		0			0
$869$	−4.67532e8			−0.712447
$870$		0			0
$871$			1.09605e9i			1.65872i
$872$		0			0
$873$			1.06773e9i			1.60479i
$874$		0			0
$875$	−4.17119e8	−	1.75468e8i	−0.622638	−	0.261923i
$876$		0			0
$877$	2.47675e8			0.367184			0.183592	−	0.983003i	$-0.441228\pi$
$877$	2.47675e8			0.367184			0.183592	+	0.983003i	$0.441228\pi$
$878$		0			0
$879$	−1.93373e9			−2.84727
$880$		0			0
$881$			1.05612e9i			1.54450i	0.635320	+	0.772249i	$0.280868\pi$
$881$			1.05612e9i			1.54450i	−0.635320	+	0.772249i	$0.719132\pi$
$882$		0			0
$883$	5.38572e8			0.782278			0.391139	−	0.920332i	$-0.372081\pi$
$883$	5.38572e8			0.782278			0.391139	+	0.920332i	$0.372081\pi$
$884$		0			0
$885$		−	2.30993e8i		−	0.333250i
$886$		0			0
$887$		−	1.13395e8i		−	0.162489i	−0.996694	−	0.0812445i	$-0.974111\pi$
$887$		−	1.13395e8i		−	0.162489i	0.996694	−	0.0812445i	$-0.0258895\pi$
$888$		0			0
$889$	4.42188e8	−	1.05116e9i	0.629364	−	1.49611i
$890$		0			0
$891$	−2.02384e8			−0.286117
$892$		0			0
$893$	2.25779e8			0.317051
$894$		0			0
$895$		−	1.58916e8i		−	0.221666i
$896$		0			0
$897$	−4.73610e8			−0.656212
$898$		0			0
$899$		−	3.06082e8i		−	0.421268i
$900$		0			0
$901$		−	4.55529e8i		−	0.622791i
$902$		0			0
$903$	4.48649e8	+	1.88732e8i	0.609317	+	0.256320i
$904$		0			0
$905$	−3.37965e8			−0.455959
$906$		0			0
$907$	4.58651e8			0.614696			0.307348	−	0.951597i	$-0.400558\pi$
$907$	4.58651e8			0.614696			0.307348	+	0.951597i	$0.400558\pi$
$908$		0			0
$909$		−	2.55901e9i		−	3.40707i
$910$		0			0
$911$	−6.00692e8			−0.794505			−0.397253	−	0.917709i	$-0.630036\pi$
$911$	−6.00692e8			−0.794505			−0.397253	+	0.917709i	$0.630036\pi$
$912$		0			0
$913$			6.24777e8i			0.820943i
$914$		0			0
$915$		−	5.61221e8i		−	0.732606i
$916$		0			0
$917$	−9.91389e8	−	4.17045e8i	−1.28569	−	0.540847i
$918$		0			0
$919$	−7.19637e8			−0.927187			−0.463593	−	0.886048i	$-0.653440\pi$
$919$	−7.19637e8			−0.927187			−0.463593	+	0.886048i	$0.653440\pi$
$920$		0			0
$921$	1.81637e9			2.32502
$922$		0			0
$923$		−	4.08118e8i		−	0.519016i
$924$		0			0
$925$	4.07822e7			0.0515282
$926$		0			0
$927$		−	2.22535e9i		−	2.79356i
$928$		0			0
$929$		−	4.63919e8i		−	0.578622i	−0.957235	−	0.289311i	$-0.906574\pi$
$929$		−	4.63919e8i		−	0.578622i	0.957235	−	0.289311i	$-0.0934262\pi$
$930$		0			0
$931$	2.62095e8	−	2.56396e8i	0.324795	−	0.317733i
$932$		0			0
$933$	−6.65664e8			−0.819616
$934$		0			0
$935$	2.35351e8			0.287926
$936$		0			0
$937$			1.26911e9i			1.54269i	0.636417	+	0.771345i	$0.280416\pi$
$937$			1.26911e9i			1.54269i	−0.636417	+	0.771345i	$0.719584\pi$
$938$		0			0
$939$	1.89468e9			2.28844
$940$		0			0
$941$		−	1.06092e8i		−	0.127325i	−0.997971	−	0.0636626i	$-0.979722\pi$
$941$		−	1.06092e8i		−	0.127325i	0.997971	−	0.0636626i	$-0.0202782\pi$
$942$		0			0
$943$			2.72634e8i			0.325121i
$944$		0			0
$945$	−1.57908e8	+	3.75376e8i	−0.187115	+	0.444806i
$946$		0			0
$947$	−3.60651e8			−0.424655			−0.212328	−	0.977199i	$-0.568104\pi$
$947$	−3.60651e8			−0.424655			−0.212328	+	0.977199i	$0.568104\pi$
$948$		0			0
$949$	1.34946e8			0.157892
$950$		0			0
$951$			1.94632e9i			2.26294i
$952$		0			0
$953$	1.35809e9			1.56910			0.784551	−	0.620064i	$-0.212893\pi$
$953$	1.35809e9			1.56910			0.784551	+	0.620064i	$0.212893\pi$
$954$		0			0
$955$			3.25210e8i			0.373383i
$956$		0			0
$957$		−	4.39993e8i		−	0.502007i
$958$		0			0
$959$	−2.83192e8	+	6.73198e8i	−0.321089	+	0.763286i
$960$		0			0
$961$	1.33389e8			0.150297
$962$		0			0
$963$	2.11861e9			2.37231
$964$		0			0
$965$		−	5.91178e8i		−	0.657864i
$966$		0			0
$967$	−4.83872e8			−0.535119			−0.267560	−	0.963541i	$-0.586217\pi$
$967$	−4.83872e8			−0.535119			−0.267560	+	0.963541i	$0.586217\pi$
$968$		0			0
$969$		−	8.39205e8i		−	0.922353i
$970$		0			0
$971$			3.07992e8i			0.336420i	0.985751	+	0.168210i	$0.0537987\pi$
$971$			3.07992e8i			0.336420i	−0.985751	+	0.168210i	$0.946201\pi$
$972$		0			0
$973$	5.32487e8	+	2.24000e8i	0.578057	+	0.243169i
$974$		0			0
$975$	1.35796e9			1.46512
$976$		0			0
$977$	−2.33410e8			−0.250285			−0.125143	−	0.992139i	$-0.539939\pi$
$977$	−2.33410e8			−0.250285			−0.125143	+	0.992139i	$0.539939\pi$
$978$		0			0
$979$		−	5.50208e8i		−	0.586380i
$980$		0			0
$981$	−2.61185e8			−0.276657
$982$		0			0
$983$			6.02617e8i			0.634426i	0.948354	+	0.317213i	$0.102747\pi$
$983$			6.02617e8i			0.634426i	−0.948354	+	0.317213i	$0.897253\pi$
$984$		0			0
$985$		−	4.14390e8i		−	0.433612i
$986$		0			0
$987$	−4.35197e8	+	1.03454e9i	−0.452621	+	1.07596i
$988$		0			0
$989$	1.48858e8			0.153881
$990$		0			0
$991$	−1.12772e9			−1.15873			−0.579364	−	0.815069i	$-0.696699\pi$
$991$	−1.12772e9			−1.15873			−0.579364	+	0.815069i	$0.696699\pi$
$992$		0			0
$993$			2.40377e9i			2.45497i
$994$		0			0
$995$	3.06575e8			0.311220
$996$		0			0
$997$			1.17224e9i			1.18286i	0.806357	+	0.591429i	$0.201436\pi$
$997$			1.17224e9i			1.18286i	−0.806357	+	0.591429i	$0.798564\pi$
$998$		0			0
$999$		−	7.89130e7i		−	0.0791503i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.7.c.b.97.2		2
4.3	odd	2		7.7.b.b.6.1	✓	2
7.6	odd	2	inner	112.7.c.b.97.1		2
8.3	odd	2		448.7.c.d.321.2		2
8.5	even	2		448.7.c.c.321.1		2
12.11	even	2		63.7.d.d.55.1		2
20.3	even	4		175.7.c.c.174.4		4
20.7	even	4		175.7.c.c.174.1		4
20.19	odd	2		175.7.d.e.76.2		2
28.3	even	6		49.7.d.d.19.2		4
28.11	odd	6		49.7.d.d.19.1		4
28.19	even	6		49.7.d.d.31.1		4
28.23	odd	6		49.7.d.d.31.2		4
28.27	even	2		7.7.b.b.6.2	yes	2
56.13	odd	2		448.7.c.c.321.2		2
56.27	even	2		448.7.c.d.321.1		2
84.83	odd	2		63.7.d.d.55.2		2
140.27	odd	4		175.7.c.c.174.2		4
140.83	odd	4		175.7.c.c.174.3		4
140.139	even	2		175.7.d.e.76.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.7.b.b.6.1	✓	2	4.3	odd	2
7.7.b.b.6.2	yes	2	28.27	even	2
49.7.d.d.19.1		4	28.11	odd	6
49.7.d.d.19.2		4	28.3	even	6
49.7.d.d.31.1		4	28.19	even	6
49.7.d.d.31.2		4	28.23	odd	6
63.7.d.d.55.1		2	12.11	even	2
63.7.d.d.55.2		2	84.83	odd	2
112.7.c.b.97.1		2	7.6	odd	2	inner
112.7.c.b.97.2		2	1.1	even	1	trivial
175.7.c.c.174.1		4	20.7	even	4
175.7.c.c.174.2		4	140.27	odd	4
175.7.c.c.174.3		4	140.83	odd	4
175.7.c.c.174.4		4	20.3	even	4
175.7.d.e.76.1		2	140.139	even	2
175.7.d.e.76.2		2	20.19	odd	2
448.7.c.c.321.1		2	8.5	even	2
448.7.c.c.321.2		2	56.13	odd	2
448.7.c.d.321.1		2	56.27	even	2
448.7.c.d.321.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	112.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+45.1664i q^{3} +45.1664i q^{5} +(-133.000 + 316.165i) q^{7} -1311.00 q^{9} +O(q^{10})\)	\(q+45.1664i q^{3} +45.1664i q^{5} +(-133.000 + 316.165i) q^{7} -1311.00 q^{9} -874.000 q^{11} -2213.15i q^{13} -2040.00 q^{15} +5961.96i q^{17} +3116.48i q^{19} +(-14280.0 - 6007.13i) q^{21} -4738.00 q^{23} +13585.0 q^{25} -26286.8i q^{27} +11146.0 q^{29} -27461.1i q^{31} -39475.4i q^{33} +(-14280.0 - 6007.13i) q^{35} +3002.00 q^{37} +99960.0 q^{39} -57541.9i q^{41} -31418.0 q^{43} -59213.1i q^{45} -72446.8i q^{47} +(-82271.0 - 84099.8i) q^{49} -269280. q^{51} -76406.0 q^{53} -39475.4i q^{55} -140760. q^{57} +113232. i q^{59} +275108. i q^{61} +(174363. - 414492. i) q^{63} +99960.0 q^{65} -495242. q^{67} -213998. i q^{69} +184406. q^{71} +60974.6i q^{73} +613585. i q^{75} +(116242. - 276328. i) q^{77} +534934. q^{79} +231561. q^{81} -714848. i q^{83} -269280. q^{85} +503424. i q^{87} +629529. i q^{89} +(699720. + 294349. i) q^{91} +1.24032e6 q^{93} -140760. q^{95} -814440. i q^{97} +1.14581e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 266 q^{7} - 2622 q^{9}+O(q^{10}) \) 2 * q - 266 * q^7 - 2622 * q^9	\( 2 q - 266 q^{7} - 2622 q^{9} - 1748 q^{11} - 4080 q^{15} - 28560 q^{21} - 9476 q^{23} + 27170 q^{25} + 22292 q^{29} - 28560 q^{35} + 6004 q^{37} + 199920 q^{39} - 62836 q^{43} - 164542 q^{49} - 538560 q^{51} - 152812 q^{53} - 281520 q^{57} + 348726 q^{63} + 199920 q^{65} - 990484 q^{67} + 368812 q^{71} + 232484 q^{77} + 1069868 q^{79} + 463122 q^{81} - 538560 q^{85} + 1399440 q^{91} + 2480640 q^{93} - 281520 q^{95} + 2291628 q^{99}+O(q^{100}) \) 2 * q - 266 * q^7 - 2622 * q^9 - 1748 * q^11 - 4080 * q^15 - 28560 * q^21 - 9476 * q^23 + 27170 * q^25 + 22292 * q^29 - 28560 * q^35 + 6004 * q^37 + 199920 * q^39 - 62836 * q^43 - 164542 * q^49 - 538560 * q^51 - 152812 * q^53 - 281520 * q^57 + 348726 * q^63 + 199920 * q^65 - 990484 * q^67 + 368812 * q^71 + 232484 * q^77 + 1069868 * q^79 + 463122 * q^81 - 538560 * q^85 + 1399440 * q^91 + 2480640 * q^93 - 281520 * q^95 + 2291628 * q^99

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