LMFDB - Embedded newform 4140.2.f.d.829.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.16
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.d.829.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.405664 + 2.19896i) q^{5} -3.23277i q^{7} -1.54118 q^{11} -6.47137i q^{13} +7.55463i q^{17} +1.20957 q^{19} +1.00000i q^{23} +(-4.67087 + 1.78408i) q^{25} -1.14088 q^{29} -6.97027 q^{31} +(7.10874 - 1.31142i) q^{35} -5.33750i q^{37} -7.35747 q^{41} -3.63307i q^{43} -10.3307i q^{47} -3.45079 q^{49} +3.16191i q^{53} +(-0.625203 - 3.38901i) q^{55} -8.15340 q^{59} +0.160021 q^{61} +(14.2303 - 2.62520i) q^{65} +15.1069i q^{67} -3.24232 q^{71} +9.51829i q^{73} +4.98229i q^{77} +10.9150 q^{79} +3.26243i q^{83} +(-16.6124 + 3.06464i) q^{85} -15.5146 q^{89} -20.9205 q^{91} +(0.490678 + 2.65979i) q^{95} +9.21865i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) $ 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.405664	+	2.19896i	0.181418	+	0.983406i
$5$	0.405664	+	2.19896i	0.181418	+	0.983406i
$6$		0			0
$7$		−	3.23277i		−	1.22187i	−0.791680	−	0.610936i	$-0.790793\pi$
$7$		−	3.23277i		−	1.22187i	0.791680	−	0.610936i	$-0.209207\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.54118			−0.464685			−0.232342	−	0.972634i	$-0.574639\pi$
$11$	−1.54118			−0.464685			−0.232342	+	0.972634i	$0.574639\pi$
$12$		0			0
$13$		−	6.47137i		−	1.79484i	−0.441181	−	0.897418i	$-0.645440\pi$
$13$		−	6.47137i		−	1.79484i	0.441181	−	0.897418i	$-0.354560\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.55463i			1.83227i	0.400873	+	0.916134i	$0.368707\pi$
$17$			7.55463i			1.83227i	−0.400873	+	0.916134i	$0.631293\pi$
$18$		0			0
$19$	1.20957			0.277494			0.138747	−	0.990328i	$-0.455693\pi$
$19$	1.20957			0.277494			0.138747	+	0.990328i	$0.455693\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−4.67087	+	1.78408i	−0.934175	+	0.356816i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.14088			−0.211856			−0.105928	−	0.994374i	$-0.533781\pi$
$29$	−1.14088			−0.211856			−0.105928	+	0.994374i	$0.533781\pi$
$30$		0			0
$31$	−6.97027			−1.25190			−0.625948	−	0.779865i	$-0.715288\pi$
$31$	−6.97027			−1.25190			−0.625948	+	0.779865i	$0.715288\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.10874	−	1.31142i	1.20160	−	0.221670i
$36$		0			0
$37$		−	5.33750i		−	0.877479i	−0.898614	−	0.438740i	$-0.855425\pi$
$37$		−	5.33750i		−	0.877479i	0.898614	−	0.438740i	$-0.144575\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.35747			−1.14904			−0.574522	−	0.818489i	$-0.694812\pi$
$41$	−7.35747			−1.14904			−0.574522	+	0.818489i	$0.694812\pi$
$42$		0			0
$43$		−	3.63307i		−	0.554038i	−0.960864	−	0.277019i	$-0.910653\pi$
$43$		−	3.63307i		−	0.554038i	0.960864	−	0.277019i	$-0.0893466\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.3307i		−	1.50688i	−0.657515	−	0.753441i	$-0.728392\pi$
$47$		−	10.3307i		−	1.50688i	0.657515	−	0.753441i	$-0.271608\pi$
$48$		0			0
$49$	−3.45079			−0.492970
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.16191i			0.434322i	0.976136	+	0.217161i	$0.0696797\pi$
$53$			3.16191i			0.434322i	−0.976136	+	0.217161i	$0.930320\pi$
$54$		0			0
$55$	−0.625203	−	3.38901i	−0.0843023	−	0.456974i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.15340			−1.06148			−0.530741	−	0.847534i	$-0.678086\pi$
$59$	−8.15340			−1.06148			−0.530741	+	0.847534i	$0.678086\pi$
$60$		0			0
$61$	0.160021			0.0204885			0.0102443	−	0.999948i	$-0.496739\pi$
$61$	0.160021			0.0204885			0.0102443	+	0.999948i	$0.496739\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14.2303	−	2.62520i	1.76505	−	0.325616i
$66$		0			0
$67$			15.1069i			1.84560i	0.385285	+	0.922798i	$0.374103\pi$
$67$			15.1069i			1.84560i	−0.385285	+	0.922798i	$0.625897\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.24232			−0.384793			−0.192397	−	0.981317i	$-0.561626\pi$
$71$	−3.24232			−0.384793			−0.192397	+	0.981317i	$0.561626\pi$
$72$		0			0
$73$			9.51829i			1.11403i	0.830502	+	0.557016i	$0.188054\pi$
$73$			9.51829i			1.11403i	−0.830502	+	0.557016i	$0.811946\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.98229i			0.567785i
$78$		0			0
$79$	10.9150			1.22804			0.614019	−	0.789292i	$-0.289552\pi$
$79$	10.9150			1.22804			0.614019	+	0.789292i	$0.289552\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.26243i			0.358098i	0.983840	+	0.179049i	$0.0573021\pi$
$83$			3.26243i			0.358098i	−0.983840	+	0.179049i	$0.942698\pi$
$84$		0			0
$85$	−16.6124	+	3.06464i	−1.80186	+	0.332407i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.5146			−1.64454			−0.822272	−	0.569095i	$-0.807294\pi$
$89$	−15.5146			−1.64454			−0.822272	+	0.569095i	$0.807294\pi$
$90$		0			0
$91$	−20.9205			−2.19306
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.490678	+	2.65979i	0.0503425	+	0.272889i
$96$		0			0
$97$			9.21865i			0.936012i	0.883725	+	0.468006i	$0.155027\pi$
$97$			9.21865i			0.936012i	−0.883725	+	0.468006i	$0.844973\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.17208			−0.813153			−0.406576	−	0.913617i	$-0.633277\pi$
$101$	−8.17208			−0.813153			−0.406576	+	0.913617i	$0.633277\pi$
$102$		0			0
$103$			4.60465i			0.453710i	0.973929	+	0.226855i	$0.0728443\pi$
$103$			4.60465i			0.453710i	−0.973929	+	0.226855i	$0.927156\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.0143941i		−	0.00139153i	−1.00000		0.000695764i	$-0.999779\pi$
$107$		−	0.0143941i		−	0.00139153i	1.00000		0.000695764i	$-0.000221468\pi$
$108$		0			0
$109$	−3.81119			−0.365046			−0.182523	−	0.983202i	$-0.558426\pi$
$109$	−3.81119			−0.365046			−0.182523	+	0.983202i	$0.558426\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2.31322i			0.217610i	0.994063	+	0.108805i	$0.0347024\pi$
$113$			2.31322i			0.217610i	−0.994063	+	0.108805i	$0.965298\pi$
$114$		0			0
$115$	−2.19896	+	0.405664i	−0.205054	+	0.0378283i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	24.4224			2.23879
$120$		0			0
$121$	−8.62475			−0.784068
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.81793	−	9.54734i	−0.520371	−	0.853940i
$126$		0			0
$127$		−	15.0379i		−	1.33440i	−0.744879	−	0.667200i	$-0.767493\pi$
$127$		−	15.0379i		−	1.33440i	0.744879	−	0.667200i	$-0.232507\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	16.2573			1.42041			0.710203	−	0.703997i	$-0.248603\pi$
$131$	16.2573			1.42041			0.710203	+	0.703997i	$0.248603\pi$
$132$		0			0
$133$		−	3.91025i		−	0.339062i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.3715i			0.886098i	0.896498	+	0.443049i	$0.146103\pi$
$137$			10.3715i			0.886098i	−0.896498	+	0.443049i	$0.853897\pi$
$138$		0			0
$139$	−17.7637			−1.50670			−0.753351	−	0.657619i	$-0.771564\pi$
$139$	−17.7637			−1.50670			−0.753351	+	0.657619i	$0.771564\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			9.97358i			0.834033i
$144$		0			0
$145$	−0.462813	−	2.50875i	−0.0384346	−	0.208340i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.00344			−0.655667			−0.327833	−	0.944736i	$-0.606318\pi$
$149$	−8.00344			−0.655667			−0.327833	+	0.944736i	$0.606318\pi$
$150$		0			0
$151$	3.24804			0.264321			0.132161	−	0.991228i	$-0.457808\pi$
$151$	3.24804			0.264321			0.132161	+	0.991228i	$0.457808\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.82758	−	15.3274i	−0.227117	−	1.23112i
$156$		0			0
$157$		−	10.0944i		−	0.805619i	−0.915284	−	0.402809i	$-0.868034\pi$
$157$		−	10.0944i		−	0.805619i	0.915284	−	0.402809i	$-0.131966\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.23277			0.254778
$162$		0			0
$163$		−	15.5370i		−	1.21695i	−0.793573	−	0.608475i	$-0.791781\pi$
$163$		−	15.5370i		−	1.21695i	0.793573	−	0.608475i	$-0.208219\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.9516i		−	1.46652i	−0.679949	−	0.733260i	$-0.737998\pi$
$167$		−	18.9516i		−	1.46652i	0.679949	−	0.733260i	$-0.262002\pi$
$168$		0			0
$169$	−28.8787			−2.22144
$170$		0			0
$171$		0			0
$172$		0			0
$173$			3.90203i			0.296666i	0.988937	+	0.148333i	$0.0473907\pi$
$173$			3.90203i			0.296666i	−0.988937	+	0.148333i	$0.952609\pi$
$174$		0			0
$175$	5.76751	+	15.0999i	0.435983	+	1.14144i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.65049			−0.422338			−0.211169	−	0.977450i	$-0.567727\pi$
$179$	−5.65049			−0.422338			−0.211169	+	0.977450i	$0.567727\pi$
$180$		0			0
$181$	−14.7671			−1.09763			−0.548814	−	0.835945i	$-0.684920\pi$
$181$	−14.7671			−1.09763			−0.548814	+	0.835945i	$0.684920\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.7370	−	2.16523i	0.862918	−	0.159191i
$186$		0			0
$187$		−	11.6431i		−	0.851426i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−21.2428			−1.53708			−0.768539	−	0.639803i	$-0.779016\pi$
$191$	−21.2428			−1.53708			−0.768539	+	0.639803i	$0.779016\pi$
$192$		0			0
$193$			10.1832i			0.733005i	0.930417	+	0.366502i	$0.119445\pi$
$193$			10.1832i			0.733005i	−0.930417	+	0.366502i	$0.880555\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	24.5346i		−	1.74802i	−0.485911	−	0.874008i	$-0.661512\pi$
$197$		−	24.5346i		−	1.74802i	0.485911	−	0.874008i	$-0.338488\pi$
$198$		0			0
$199$	6.17207			0.437526			0.218763	−	0.975778i	$-0.429798\pi$
$199$	6.17207			0.437526			0.218763	+	0.975778i	$0.429798\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			3.68820i			0.258861i
$204$		0			0
$205$	−2.98466	−	16.1788i	−0.208458	−	1.12998i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.86417			−0.128947
$210$		0			0
$211$	10.4430			0.718928			0.359464	−	0.933159i	$-0.382960\pi$
$211$	10.4430			0.718928			0.359464	+	0.933159i	$0.382960\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.98899	−	1.47381i	0.544845	−	0.100513i
$216$		0			0
$217$			22.5332i			1.52966i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	48.8888			3.28862
$222$		0			0
$223$		−	12.4215i		−	0.831805i	−0.909409	−	0.415903i	$-0.863466\pi$
$223$		−	12.4215i		−	0.831805i	0.909409	−	0.415903i	$-0.136534\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	16.3837i		−	1.08743i	−0.839271	−	0.543713i	$-0.817018\pi$
$227$		−	16.3837i		−	1.08743i	0.839271	−	0.543713i	$-0.182982\pi$
$228$		0			0
$229$	−19.4860			−1.28767			−0.643835	−	0.765165i	$-0.722658\pi$
$229$	−19.4860			−1.28767			−0.643835	+	0.765165i	$0.722658\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.1683i			1.19024i	0.803636	+	0.595122i	$0.202896\pi$
$233$			18.1683i			1.19024i	−0.803636	+	0.595122i	$0.797104\pi$
$234$		0			0
$235$	22.7168	−	4.19078i	1.48188	−	0.273376i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.8349			−0.700855			−0.350427	−	0.936590i	$-0.613964\pi$
$239$	−10.8349			−0.700855			−0.350427	+	0.936590i	$0.613964\pi$
$240$		0			0
$241$	12.7685			0.822490			0.411245	−	0.911525i	$-0.365094\pi$
$241$	12.7685			0.822490			0.411245	+	0.911525i	$0.365094\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.39986	−	7.58815i	−0.0894338	−	0.484789i
$246$		0			0
$247$		−	7.82757i		−	0.498056i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.3084			−0.713781			−0.356890	−	0.934146i	$-0.616163\pi$
$251$	−11.3084			−0.713781			−0.356890	+	0.934146i	$0.616163\pi$
$252$		0			0
$253$		−	1.54118i		−	0.0968934i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.1895i		−	1.50890i	−0.656357	−	0.754451i	$-0.727903\pi$
$257$		−	24.1895i		−	1.50890i	0.656357	−	0.754451i	$-0.272097\pi$
$258$		0			0
$259$	−17.2549			−1.07217
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4.78146i		−	0.294837i	−0.989074	−	0.147419i	$-0.952904\pi$
$263$		−	4.78146i		−	0.294837i	0.989074	−	0.147419i	$-0.0470965\pi$
$264$		0			0
$265$	−6.95293	+	1.28267i	−0.427115	+	0.0787940i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.0836			−0.858694			−0.429347	−	0.903140i	$-0.641256\pi$
$269$	−14.0836			−0.858694			−0.429347	+	0.903140i	$0.641256\pi$
$270$		0			0
$271$	−17.6868			−1.07440			−0.537199	−	0.843456i	$-0.680517\pi$
$271$	−17.6868			−1.07440			−0.537199	+	0.843456i	$0.680517\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.19868	−	2.74959i	0.434097	−	0.165807i
$276$		0			0
$277$			5.66917i			0.340627i	0.985390	+	0.170314i	$0.0544781\pi$
$277$			5.66917i			0.340627i	−0.985390	+	0.170314i	$0.945522\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.1521			1.20218			0.601088	−	0.799183i	$-0.294734\pi$
$281$	20.1521			1.20218			0.601088	+	0.799183i	$0.294734\pi$
$282$		0			0
$283$			17.6352i			1.04831i	0.851624	+	0.524153i	$0.175618\pi$
$283$			17.6352i			1.04831i	−0.851624	+	0.524153i	$0.824382\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			23.7850i			1.40398i
$288$		0			0
$289$	−40.0724			−2.35720
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.77625i		−	0.279031i	−0.990220	−	0.139516i	$-0.955445\pi$
$293$		−	4.77625i		−	0.279031i	0.990220	−	0.139516i	$-0.0445546\pi$
$294$		0			0
$295$	−3.30754	−	17.9290i	−0.192572	−	1.04387i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.47137			0.374249
$300$		0			0
$301$	−11.7449			−0.676964
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.0649146	+	0.351879i	0.00371700	+	0.0201485i
$306$		0			0
$307$			18.7219i			1.06851i	0.845322	+	0.534257i	$0.179409\pi$
$307$			18.7219i			1.06851i	−0.845322	+	0.534257i	$0.820591\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.11393			−0.460099			−0.230049	−	0.973179i	$-0.573889\pi$
$311$	−8.11393			−0.460099			−0.230049	+	0.973179i	$0.573889\pi$
$312$		0			0
$313$		−	21.6494i		−	1.22370i	−0.790975	−	0.611849i	$-0.790426\pi$
$313$		−	21.6494i		−	1.22370i	0.790975	−	0.611849i	$-0.209574\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			29.3342i			1.64757i	0.566900	+	0.823787i	$0.308143\pi$
$317$			29.3342i			1.64757i	−0.566900	+	0.823787i	$0.691857\pi$
$318$		0			0
$319$	1.75830			0.0984462
$320$		0			0
$321$		0			0
$322$		0			0
$323$			9.13784i			0.508443i
$324$		0			0
$325$	11.5454	+	30.2270i	0.640426	+	1.67669i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−33.3967			−1.84122
$330$		0			0
$331$	11.4885			0.631465			0.315732	−	0.948848i	$-0.397750\pi$
$331$	11.4885			0.631465			0.315732	+	0.948848i	$0.397750\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−33.2194	+	6.12830i	−1.81497	+	0.334825i
$336$		0			0
$337$		−	21.7844i		−	1.18667i	−0.804956	−	0.593335i	$-0.797811\pi$
$337$		−	21.7844i		−	1.18667i	0.804956	−	0.593335i	$-0.202189\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.7425			0.581737
$342$		0			0
$343$		−	11.4738i		−	0.619526i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			0.999549i			0.0536586i	0.999640	+	0.0268293i	$0.00854105\pi$
$347$			0.999549i			0.0536586i	−0.999640	+	0.0268293i	$0.991459\pi$
$348$		0			0
$349$	9.48501			0.507721			0.253860	−	0.967241i	$-0.418300\pi$
$349$	9.48501			0.507721			0.253860	+	0.967241i	$0.418300\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6.86216i			0.365236i	0.983184	+	0.182618i	$0.0584571\pi$
$353$			6.86216i			0.365236i	−0.983184	+	0.182618i	$0.941543\pi$
$354$		0			0
$355$	−1.31529	−	7.12975i	−0.0698085	−	0.378408i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.0202			1.26774			0.633868	−	0.773441i	$-0.281466\pi$
$359$	24.0202			1.26774			0.633868	+	0.773441i	$0.281466\pi$
$360$		0			0
$361$	−17.5369			−0.922997
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.9304	+	3.86122i	−1.09555	+	0.202106i
$366$		0			0
$367$		−	27.8941i		−	1.45606i	−0.685545	−	0.728030i	$-0.740436\pi$
$367$		−	27.8941i		−	1.45606i	0.685545	−	0.728030i	$-0.259564\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.2217			0.530686
$372$		0			0
$373$		−	20.3434i		−	1.05334i	−0.850070	−	0.526670i	$-0.823440\pi$
$373$		−	20.3434i		−	1.05334i	0.850070	−	0.526670i	$-0.176560\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7.38306i			0.380247i
$378$		0			0
$379$	19.3139			0.992089			0.496044	−	0.868297i	$-0.334785\pi$
$379$	19.3139			0.992089			0.496044	+	0.868297i	$0.334785\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			21.5575i			1.10154i	0.834659	+	0.550768i	$0.185665\pi$
$383$			21.5575i			1.10154i	−0.834659	+	0.550768i	$0.814335\pi$
$384$		0			0
$385$	−10.9559	+	2.02114i	−0.558363	+	0.103007i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−34.2504			−1.73656			−0.868281	−	0.496072i	$-0.834775\pi$
$389$	−34.2504			−1.73656			−0.868281	+	0.496072i	$0.834775\pi$
$390$		0			0
$391$	−7.55463			−0.382054
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.42783	+	24.0018i	0.222789	+	1.20766i
$396$		0			0
$397$		−	10.6493i		−	0.534474i	−0.963631	−	0.267237i	$-0.913889\pi$
$397$		−	10.6493i		−	0.534474i	0.963631	−	0.267237i	$-0.0861106\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.70174			0.234794			0.117397	−	0.993085i	$-0.462545\pi$
$401$	4.70174			0.234794			0.117397	+	0.993085i	$0.462545\pi$
$402$		0			0
$403$			45.1072i			2.24695i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.22607i			0.407751i
$408$		0			0
$409$	−11.1631			−0.551978			−0.275989	−	0.961161i	$-0.589005\pi$
$409$	−11.1631			−0.551978			−0.275989	+	0.961161i	$0.589005\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			26.3581i			1.29700i
$414$		0			0
$415$	−7.17396	+	1.32345i	−0.352156	+	0.0649656i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.5495			0.808496			0.404248	−	0.914650i	$-0.367533\pi$
$419$	16.5495			0.808496			0.404248	+	0.914650i	$0.367533\pi$
$420$		0			0
$421$	27.8404			1.35686			0.678429	−	0.734666i	$-0.262661\pi$
$421$	27.8404			1.35686			0.678429	+	0.734666i	$0.262661\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.4781	−	35.2867i	−0.653782	−	1.71166i
$426$		0			0
$427$		−	0.517309i		−	0.0250344i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.9103			−1.19989			−0.599943	−	0.800043i	$-0.704810\pi$
$431$	−24.9103			−1.19989			−0.599943	+	0.800043i	$0.704810\pi$
$432$		0			0
$433$			27.4163i			1.31754i	0.752344	+	0.658771i	$0.228923\pi$
$433$			27.4163i			1.31754i	−0.752344	+	0.658771i	$0.771077\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.20957i			0.0578615i
$438$		0			0
$439$	12.1256			0.578721			0.289361	−	0.957220i	$-0.406557\pi$
$439$	12.1256			0.578721			0.289361	+	0.957220i	$0.406557\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.0760i			0.573747i	0.957968	+	0.286873i	$0.0926159\pi$
$443$			12.0760i			0.573747i	−0.957968	+	0.286873i	$0.907384\pi$
$444$		0			0
$445$	−6.29371	−	34.1160i	−0.298350	−	1.61725i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.7602			0.838154			0.419077	−	0.907951i	$-0.362354\pi$
$449$	17.7602			0.838154			0.419077	+	0.907951i	$0.362354\pi$
$450$		0			0
$451$	11.3392			0.533943
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.48667	−	46.0033i	−0.397861	−	2.15667i
$456$		0			0
$457$			2.60686i			0.121944i	0.998139	+	0.0609720i	$0.0194200\pi$
$457$			2.60686i			0.121944i	−0.998139	+	0.0609720i	$0.980580\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	22.7840			1.06116			0.530579	−	0.847636i	$-0.321975\pi$
$461$	22.7840			1.06116			0.530579	+	0.847636i	$0.321975\pi$
$462$		0			0
$463$			5.47834i			0.254600i	0.991864	+	0.127300i	$0.0406311\pi$
$463$			5.47834i			0.254600i	−0.991864	+	0.127300i	$0.959369\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	16.3930i		−	0.758578i	−0.925278	−	0.379289i	$-0.876169\pi$
$467$		−	16.3930i		−	0.758578i	0.925278	−	0.379289i	$-0.123831\pi$
$468$		0			0
$469$	48.8369			2.25508
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.59923i			0.257453i
$474$		0			0
$475$	−5.64974	+	2.15797i	−0.259228	+	0.0990142i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	42.2621			1.93100			0.965502	−	0.260395i	$-0.0838527\pi$
$479$	42.2621			1.93100			0.965502	+	0.260395i	$0.0838527\pi$
$480$		0			0
$481$	−34.5409			−1.57493
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.2715	+	3.73967i	−0.920480	+	0.169810i
$486$		0			0
$487$			6.62454i			0.300187i	0.988672	+	0.150093i	$0.0479574\pi$
$487$			6.62454i			0.300187i	−0.988672	+	0.150093i	$0.952043\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.07235			0.0935236			0.0467618	−	0.998906i	$-0.485110\pi$
$491$	2.07235			0.0935236			0.0467618	+	0.998906i	$0.485110\pi$
$492$		0			0
$493$		−	8.61892i		−	0.388177i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			10.4817i			0.470168i
$498$		0			0
$499$	24.4746			1.09563			0.547817	−	0.836598i	$-0.315459\pi$
$499$	24.4746			1.09563			0.547817	+	0.836598i	$0.315459\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			25.1260i			1.12031i	0.828386	+	0.560157i	$0.189259\pi$
$503$			25.1260i			1.12031i	−0.828386	+	0.560157i	$0.810741\pi$
$504$		0			0
$505$	−3.31512	−	17.9701i	−0.147521	−	0.799659i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.88306			0.216438			0.108219	−	0.994127i	$-0.465485\pi$
$509$	4.88306			0.216438			0.108219	+	0.994127i	$0.465485\pi$
$510$		0			0
$511$	30.7704			1.36120
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.1255	+	1.86794i	−0.446181	+	0.0823113i
$516$		0			0
$517$			15.9215i			0.700225i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.70144			0.0745413			0.0372706	−	0.999305i	$-0.488134\pi$
$521$	1.70144			0.0745413			0.0372706	+	0.999305i	$0.488134\pi$
$522$		0			0
$523$			16.5240i			0.722543i	0.932461	+	0.361271i	$0.117657\pi$
$523$			16.5240i			0.722543i	−0.932461	+	0.361271i	$0.882343\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	52.6578i		−	2.29381i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			47.6129i			2.06235i
$534$		0			0
$535$	0.0316520	−	0.00583915i	0.00136844	−	0.000252449i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.31830			0.229075
$540$		0			0
$541$	−12.5024			−0.537521			−0.268761	−	0.963207i	$-0.586614\pi$
$541$	−12.5024			−0.537521			−0.268761	+	0.963207i	$0.586614\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.54606	−	8.38067i	−0.0662260	−	0.358988i
$546$		0			0
$547$			0.0134071i			0.000573248i	1.00000		0.000286624i	$9.12352e-5\pi$
$547$			0.0134071i			0.000573248i	−1.00000		0.000286624i	$0.999909\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.37997			−0.0587887
$552$		0			0
$553$		−	35.2858i		−	1.50050i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			21.9167i			0.928639i	0.885668	+	0.464319i	$0.153701\pi$
$557$			21.9167i			0.928639i	−0.885668	+	0.464319i	$0.846299\pi$
$558$		0			0
$559$	−23.5110			−0.994408
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.26360i		−	0.179689i	−0.995956	−	0.0898446i	$-0.971363\pi$
$563$		−	4.26360i		−	0.179689i	0.995956	−	0.0898446i	$-0.0286370\pi$
$564$		0			0
$565$	−5.08669	+	0.938392i	−0.213999	+	0.0394784i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	33.5323			1.40575			0.702875	−	0.711314i	$-0.251900\pi$
$569$	33.5323			1.40575			0.702875	+	0.711314i	$0.251900\pi$
$570$		0			0
$571$	−2.45524			−0.102748			−0.0513742	−	0.998679i	$-0.516360\pi$
$571$	−2.45524			−0.102748			−0.0513742	+	0.998679i	$0.516360\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.78408	−	4.67087i	−0.0744012	−	0.194789i
$576$		0			0
$577$			0.303142i			0.0126200i	0.999980	+	0.00630998i	$0.00200854\pi$
$577$			0.303142i			0.0126200i	−0.999980	+	0.00630998i	$0.997991\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.5467			0.437550
$582$		0			0
$583$		−	4.87309i		−	0.201823i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.59450i		−	0.0658119i	−0.999458	−	0.0329060i	$-0.989524\pi$
$587$		−	1.59450i		−	0.0658119i	0.999458	−	0.0329060i	$-0.0104762\pi$
$588$		0			0
$589$	−8.43101			−0.347394
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	44.9962i		−	1.84777i	−0.382667	−	0.923886i	$-0.624994\pi$
$593$		−	44.9962i		−	1.84777i	0.382667	−	0.923886i	$-0.375006\pi$
$594$		0			0
$595$	9.90727	+	53.7039i	0.406159	+	2.20164i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−37.4163			−1.52879			−0.764395	−	0.644748i	$-0.776962\pi$
$599$	−37.4163			−1.52879			−0.764395	+	0.644748i	$0.776962\pi$
$600$		0			0
$601$	−0.430999			−0.0175808			−0.00879041	−	0.999961i	$-0.502798\pi$
$601$	−0.430999			−0.0175808			−0.00879041	+	0.999961i	$0.502798\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.49875	−	18.9655i	−0.142244	−	0.771057i
$606$		0			0
$607$			29.1493i			1.18314i	0.806255	+	0.591568i	$0.201491\pi$
$607$			29.1493i			1.18314i	−0.806255	+	0.591568i	$0.798509\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−66.8536			−2.70461
$612$		0			0
$613$			24.4753i			0.988548i	0.869306	+	0.494274i	$0.164566\pi$
$613$			24.4753i			0.988548i	−0.869306	+	0.494274i	$0.835434\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	40.6677i		−	1.63722i	−0.574349	−	0.818611i	$-0.694745\pi$
$617$		−	40.6677i		−	1.63722i	0.574349	−	0.818611i	$-0.305255\pi$
$618$		0			0
$619$	25.7331			1.03430			0.517150	−	0.855895i	$-0.326993\pi$
$619$	25.7331			1.03430			0.517150	+	0.855895i	$0.326993\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			50.1551i			2.00942i
$624$		0			0
$625$	18.6341	−	16.6664i	0.745365	−	0.666657i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.3228			1.60778
$630$		0			0
$631$	37.5617			1.49531			0.747654	−	0.664089i	$-0.231180\pi$
$631$	37.5617			1.49531			0.747654	+	0.664089i	$0.231180\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	33.0678	−	6.10034i	1.31226	−	0.242085i
$636$		0			0
$637$			22.3313i			0.884800i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−39.5324			−1.56144			−0.780718	−	0.624883i	$-0.785147\pi$
$641$	−39.5324			−1.56144			−0.780718	+	0.624883i	$0.785147\pi$
$642$		0			0
$643$			42.6224i			1.68086i	0.541916	+	0.840432i	$0.317699\pi$
$643$			42.6224i			1.68086i	−0.541916	+	0.840432i	$0.682301\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	22.5601i		−	0.886931i	−0.896292	−	0.443465i	$-0.853749\pi$
$647$		−	22.5601i		−	0.886931i	0.896292	−	0.443465i	$-0.146251\pi$
$648$		0			0
$649$	12.5659			0.493255
$650$		0			0
$651$		0			0
$652$		0			0
$653$			33.6226i			1.31575i	0.753126	+	0.657877i	$0.228545\pi$
$653$			33.6226i			1.31575i	−0.753126	+	0.657877i	$0.771455\pi$
$654$		0			0
$655$	6.59499	+	35.7492i	0.257688	+	1.39684i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.90805			0.230145			0.115072	−	0.993357i	$-0.463290\pi$
$659$	5.90805			0.230145			0.115072	+	0.993357i	$0.463290\pi$
$660$		0			0
$661$	−1.15131			−0.0447808			−0.0223904	−	0.999749i	$-0.507128\pi$
$661$	−1.15131			−0.0447808			−0.0223904	+	0.999749i	$0.507128\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.59850	−	1.58625i	0.333436	−	0.0615121i
$666$		0			0
$667$		−	1.14088i		−	0.0441750i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.246621			−0.00952070
$672$		0			0
$673$			21.6959i			0.836315i	0.908375	+	0.418157i	$0.137324\pi$
$673$			21.6959i			0.836315i	−0.908375	+	0.418157i	$0.862676\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	17.7024i		−	0.680357i	−0.940361	−	0.340178i	$-0.889513\pi$
$677$		−	17.7024i		−	0.680357i	0.940361	−	0.340178i	$-0.110487\pi$
$678$		0			0
$679$	29.8018			1.14369
$680$		0			0
$681$		0			0
$682$		0			0
$683$			26.6160i			1.01843i	0.860638	+	0.509217i	$0.170065\pi$
$683$			26.6160i			1.01843i	−0.860638	+	0.509217i	$0.829935\pi$
$684$		0			0
$685$	−22.8066	+	4.20734i	−0.871394	+	0.160754i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.4619			0.779538
$690$		0			0
$691$	18.1006			0.688578			0.344289	−	0.938864i	$-0.388120\pi$
$691$	18.1006			0.688578			0.344289	+	0.938864i	$0.388120\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.20611	−	39.0618i	−0.273343	−	1.48170i
$696$		0			0
$697$		−	55.5830i		−	2.10536i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.5844			−0.588614			−0.294307	−	0.955711i	$-0.595089\pi$
$701$	−15.5844			−0.588614			−0.294307	+	0.955711i	$0.595089\pi$
$702$		0			0
$703$		−	6.45607i		−	0.243495i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			26.4184i			0.993568i
$708$		0			0
$709$	9.78051			0.367315			0.183657	−	0.982990i	$-0.441206\pi$
$709$	9.78051			0.367315			0.183657	+	0.982990i	$0.441206\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.97027i		−	0.261038i
$714$		0			0
$715$	−21.9315	+	4.04592i	−0.820193	+	0.151309i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	40.9542			1.52733			0.763667	−	0.645611i	$-0.223397\pi$
$719$	40.9542			1.52733			0.763667	+	0.645611i	$0.223397\pi$
$720$		0			0
$721$	14.8858			0.554375
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.32890	−	2.03542i	0.197910	−	0.0755935i
$726$		0			0
$727$			37.3620i			1.38568i	0.721092	+	0.692839i	$0.243640\pi$
$727$			37.3620i			1.38568i	−0.721092	+	0.692839i	$0.756360\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	27.4465			1.01515
$732$		0			0
$733$		−	27.0326i		−	0.998474i	−0.866466	−	0.499237i	$-0.833614\pi$
$733$		−	27.0326i		−	0.998474i	0.866466	−	0.499237i	$-0.166386\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	23.2824i		−	0.857620i
$738$		0			0
$739$	14.1381			0.520078			0.260039	−	0.965598i	$-0.416265\pi$
$739$	14.1381			0.520078			0.260039	+	0.965598i	$0.416265\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			15.1617i			0.556230i	0.960548	+	0.278115i	$0.0897096\pi$
$743$			15.1617i			0.556230i	−0.960548	+	0.278115i	$0.910290\pi$
$744$		0			0
$745$	−3.24670	−	17.5993i	−0.118950	−	0.644787i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.0465327			−0.00170027
$750$		0			0
$751$	−39.3087			−1.43440			−0.717198	−	0.696869i	$-0.754576\pi$
$751$	−39.3087			−1.43440			−0.717198	+	0.696869i	$0.754576\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.31761	+	7.14231i	0.0479528	+	0.259935i
$756$		0			0
$757$		−	5.25823i		−	0.191114i	−0.995424	−	0.0955569i	$-0.969537\pi$
$757$		−	5.25823i		−	0.191114i	0.995424	−	0.0955569i	$-0.0304632\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.9905			−0.434655			−0.217328	−	0.976099i	$-0.569734\pi$
$761$	−11.9905			−0.434655			−0.217328	+	0.976099i	$0.569734\pi$
$762$		0			0
$763$			12.3207i			0.446039i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			52.7637i			1.90519i
$768$		0			0
$769$	−17.3925			−0.627189			−0.313594	−	0.949557i	$-0.601533\pi$
$769$	−17.3925			−0.627189			−0.313594	+	0.949557i	$0.601533\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			23.5445i			0.846837i	0.905934	+	0.423418i	$0.139170\pi$
$773$			23.5445i			0.846837i	−0.905934	+	0.423418i	$0.860830\pi$
$774$		0			0
$775$	32.5572	−	12.4355i	1.16949	−	0.446697i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.89936			−0.318853
$780$		0			0
$781$	4.99702			0.178807
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.1972	−	4.09492i	0.792250	−	0.146154i
$786$		0			0
$787$			33.4328i			1.19175i	0.803076	+	0.595876i	$0.203195\pi$
$787$			33.4328i			1.19175i	−0.803076	+	0.595876i	$0.796805\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.47812			0.265891
$792$		0			0
$793$		−	1.03555i		−	0.0367736i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			13.1794i			0.466838i	0.972376	+	0.233419i	$0.0749913\pi$
$797$			13.1794i			0.466838i	−0.972376	+	0.233419i	$0.925009\pi$
$798$		0			0
$799$	78.0444			2.76101
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	14.6694i		−	0.517673i
$804$		0			0
$805$	1.31142	+	7.10874i	0.0462214	+	0.250550i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.13443			−0.110201			−0.0551004	−	0.998481i	$-0.517548\pi$
$809$	−3.13443			−0.110201			−0.0551004	+	0.998481i	$0.517548\pi$
$810$		0			0
$811$	37.6825			1.32321			0.661606	−	0.749852i	$-0.269875\pi$
$811$	37.6825			1.32321			0.661606	+	0.749852i	$0.269875\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	34.1653	−	6.30279i	1.19676	−	0.220777i
$816$		0			0
$817$		−	4.39445i		−	0.153742i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−43.3829			−1.51407			−0.757037	−	0.653372i	$-0.773354\pi$
$821$	−43.3829			−1.51407			−0.757037	+	0.653372i	$0.773354\pi$
$822$		0			0
$823$		−	52.2672i		−	1.82192i	−0.412496	−	0.910959i	$-0.635343\pi$
$823$		−	52.2672i		−	1.82192i	0.412496	−	0.910959i	$-0.364657\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	5.95772i		−	0.207170i	−0.994621	−	0.103585i	$-0.966969\pi$
$827$		−	5.95772i		−	0.207170i	0.994621	−	0.103585i	$-0.0330314\pi$
$828$		0			0
$829$	45.5397			1.58166			0.790830	−	0.612036i	$-0.209649\pi$
$829$	45.5397			1.58166			0.790830	+	0.612036i	$0.209649\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	26.0694i		−	0.903252i
$834$		0			0
$835$	41.6739	−	7.68798i	1.44218	−	0.266054i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−42.5230			−1.46806			−0.734029	−	0.679118i	$-0.762362\pi$
$839$	−42.5230			−1.46806			−0.734029	+	0.679118i	$0.762362\pi$
$840$		0			0
$841$	−27.6984			−0.955117
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.7150	−	63.5032i	−0.403010	−	2.18458i
$846$		0			0
$847$			27.8818i			0.958031i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.33750			0.182967
$852$		0			0
$853$		−	46.5906i		−	1.59523i	−0.603167	−	0.797615i	$-0.706095\pi$
$853$		−	46.5906i		−	1.59523i	0.603167	−	0.797615i	$-0.293905\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	52.2679i		−	1.78544i	−0.450616	−	0.892718i	$-0.648795\pi$
$857$		−	52.2679i		−	1.78544i	0.450616	−	0.892718i	$-0.351205\pi$
$858$		0			0
$859$	−9.41393			−0.321199			−0.160600	−	0.987020i	$-0.551343\pi$
$859$	−9.41393			−0.321199			−0.160600	+	0.987020i	$0.551343\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.7353i			1.25048i	0.780431	+	0.625241i	$0.214999\pi$
$863$			36.7353i			1.25048i	−0.780431	+	0.625241i	$0.785001\pi$
$864$		0			0
$865$	−8.58041	+	1.58291i	−0.291743	+	0.0538206i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−16.8221			−0.570650
$870$		0			0
$871$	97.7621			3.31254
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−30.8643	+	18.8080i	−1.04340	+	0.635827i
$876$		0			0
$877$			11.8456i			0.399997i	0.979796	+	0.199999i	$0.0640937\pi$
$877$			11.8456i			0.399997i	−0.979796	+	0.199999i	$0.935906\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.69400			0.0570724			0.0285362	−	0.999593i	$-0.490915\pi$
$881$	1.69400			0.0570724			0.0285362	+	0.999593i	$0.490915\pi$
$882$		0			0
$883$		−	35.8233i		−	1.20555i	−0.797911	−	0.602776i	$-0.794061\pi$
$883$		−	35.8233i		−	1.20555i	0.797911	−	0.602776i	$-0.205939\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	30.5808i		−	1.02680i	−0.858149	−	0.513401i	$-0.828385\pi$
$887$		−	30.5808i		−	1.02680i	0.858149	−	0.513401i	$-0.171615\pi$
$888$		0			0
$889$	−48.6141			−1.63047
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	12.4956i		−	0.418151i
$894$		0			0
$895$	−2.29220	−	12.4252i	−0.0766198	−	0.415329i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7.95223			0.265222
$900$		0			0
$901$	−23.8871			−0.795794
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.99046	−	32.4722i	−0.199130	−	1.07941i
$906$		0			0
$907$		−	31.4194i		−	1.04326i	−0.853171	−	0.521631i	$-0.825324\pi$
$907$		−	31.4194i		−	1.04326i	0.853171	−	0.521631i	$-0.174676\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−45.9786			−1.52334			−0.761669	−	0.647966i	$-0.775620\pi$
$911$	−45.9786			−1.52334			−0.761669	+	0.647966i	$0.775620\pi$
$912$		0			0
$913$		−	5.02801i		−	0.166403i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	52.5560i		−	1.73555i
$918$		0			0
$919$	13.9290			0.459474			0.229737	−	0.973253i	$-0.426213\pi$
$919$	13.9290			0.459474			0.229737	+	0.973253i	$0.426213\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			20.9823i			0.690641i
$924$		0			0
$925$	9.52252	+	24.9308i	0.313098	+	0.819719i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	57.5727			1.88890			0.944450	−	0.328656i	$-0.106596\pi$
$929$	57.5727			1.88890			0.944450	+	0.328656i	$0.106596\pi$
$930$		0			0
$931$	−4.17396			−0.136796
$932$		0			0
$933$		0			0
$934$		0			0
$935$	25.6027	−	4.72318i	0.837298	−	0.154464i
$936$		0			0
$937$		−	43.9696i		−	1.43642i	−0.695824	−	0.718212i	$-0.744961\pi$
$937$		−	43.9696i		−	1.43642i	0.695824	−	0.718212i	$-0.255039\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.8753			−0.387125			−0.193562	−	0.981088i	$-0.562004\pi$
$941$	−11.8753			−0.387125			−0.193562	+	0.981088i	$0.562004\pi$
$942$		0			0
$943$		−	7.35747i		−	0.239592i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	38.4406i		−	1.24915i	−0.780964	−	0.624576i	$-0.785272\pi$
$947$		−	38.4406i		−	1.24915i	0.780964	−	0.624576i	$-0.214728\pi$
$948$		0			0
$949$	61.5964			1.99950
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.22062i			0.0395398i	0.999805	+	0.0197699i	$0.00629337\pi$
$953$			1.22062i			0.0395398i	−0.999805	+	0.0197699i	$0.993707\pi$
$954$		0			0
$955$	−8.61745	−	46.7122i	−0.278854	−	1.51157i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	33.5287			1.08270
$960$		0			0
$961$	17.5846			0.567245
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.3925	+	4.13097i	−0.720841	+	0.132981i
$966$		0			0
$967$		−	24.7003i		−	0.794309i	−0.917752	−	0.397154i	$-0.869998\pi$
$967$		−	24.7003i		−	0.794309i	0.917752	−	0.397154i	$-0.130002\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41.0376			−1.31696			−0.658480	−	0.752598i	$-0.728800\pi$
$971$	−41.0376			−1.31696			−0.658480	+	0.752598i	$0.728800\pi$
$972$		0			0
$973$			57.4261i			1.84100i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			46.6025i			1.49095i	0.666535	+	0.745474i	$0.267777\pi$
$977$			46.6025i			1.49095i	−0.666535	+	0.745474i	$0.732223\pi$
$978$		0			0
$979$	23.9108			0.764194
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	42.6181i		−	1.35931i	−0.733534	−	0.679653i	$-0.762130\pi$
$983$		−	42.6181i		−	1.35931i	0.733534	−	0.679653i	$-0.237870\pi$
$984$		0			0
$985$	53.9506	−	9.95279i	1.71901	−	0.317122i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.63307			0.115525
$990$		0			0
$991$	−26.7258			−0.848972			−0.424486	−	0.905435i	$-0.639545\pi$
$991$	−26.7258			−0.848972			−0.424486	+	0.905435i	$0.639545\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.50379	+	13.5722i	0.0793753	+	0.430266i
$996$		0			0
$997$			39.6293i			1.25507i	0.778588	+	0.627536i	$0.215936\pi$
$997$			39.6293i			1.25507i	−0.778588	+	0.627536i	$0.784064\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.d.829.16	yes	24
3.2	odd	2	inner	4140.2.f.d.829.9	✓	24
5.4	even	2	inner	4140.2.f.d.829.15	yes	24
15.14	odd	2	inner	4140.2.f.d.829.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.f.d.829.9	✓	24	3.2	odd	2	inner
4140.2.f.d.829.10	yes	24	15.14	odd	2	inner
4140.2.f.d.829.15	yes	24	5.4	even	2	inner
4140.2.f.d.829.16	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.405664 + 2.19896i) q^{5} -3.23277i q^{7} -1.54118 q^{11} -6.47137i q^{13} +7.55463i q^{17} +1.20957 q^{19} +1.00000i q^{23} +(-4.67087 + 1.78408i) q^{25} -1.14088 q^{29} -6.97027 q^{31} +(7.10874 - 1.31142i) q^{35} -5.33750i q^{37} -7.35747 q^{41} -3.63307i q^{43} -10.3307i q^{47} -3.45079 q^{49} +3.16191i q^{53} +(-0.625203 - 3.38901i) q^{55} -8.15340 q^{59} +0.160021 q^{61} +(14.2303 - 2.62520i) q^{65} +15.1069i q^{67} -3.24232 q^{71} +9.51829i q^{73} +4.98229i q^{77} +10.9150 q^{79} +3.26243i q^{83} +(-16.6124 + 3.06464i) q^{85} -15.5146 q^{89} -20.9205 q^{91} +(0.490678 + 2.65979i) q^{95} +9.21865i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) \) 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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