LMFDB - Embedded newform 4012.2.b.b.237.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.32
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.45665i q^{3} +3.69601i q^{5} -3.33013i q^{7} +0.878183 q^{9} +O(q^{10})$	$q+1.45665i q^{3} +3.69601i q^{5} -3.33013i q^{7} +0.878183 q^{9} -4.75976i q^{11} +2.51524 q^{13} -5.38377 q^{15} +(-3.76296 + 1.68527i) q^{17} -0.982947 q^{19} +4.85082 q^{21} -4.02346i q^{23} -8.66046 q^{25} +5.64914i q^{27} -0.375643i q^{29} -3.17061i q^{31} +6.93328 q^{33} +12.3082 q^{35} +1.30716i q^{37} +3.66382i q^{39} -7.88667i q^{41} +9.82393 q^{43} +3.24577i q^{45} +13.2756 q^{47} -4.08976 q^{49} +(-2.45484 - 5.48130i) q^{51} +1.89912 q^{53} +17.5921 q^{55} -1.43181i q^{57} +1.00000 q^{59} -0.836969i q^{61} -2.92446i q^{63} +9.29635i q^{65} +1.30621 q^{67} +5.86076 q^{69} -10.8431i q^{71} +0.00541582i q^{73} -12.6152i q^{75} -15.8506 q^{77} +10.6850i q^{79} -5.59424 q^{81} +7.42159 q^{83} +(-6.22878 - 13.9079i) q^{85} +0.547179 q^{87} -4.58338 q^{89} -8.37608i q^{91} +4.61846 q^{93} -3.63298i q^{95} -10.4572i q^{97} -4.17994i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.45665i			0.840995i	0.907294	+	0.420497i	$0.138144\pi$
$3$			1.45665i			0.840995i	−0.907294	+	0.420497i	$0.861856\pi$
$4$		0			0
$5$			3.69601i			1.65290i	0.563007	+	0.826452i	$0.309644\pi$
$5$			3.69601i			1.65290i	−0.563007	+	0.826452i	$0.690356\pi$
$6$		0			0
$7$		−	3.33013i		−	1.25867i	−0.777134	−	0.629335i	$-0.783327\pi$
$7$		−	3.33013i		−	1.25867i	0.777134	−	0.629335i	$-0.216673\pi$
$8$		0			0
$9$	0.878183			0.292728
$10$		0			0
$11$		−	4.75976i		−	1.43512i	−0.696496	−	0.717561i	$-0.745259\pi$
$11$		−	4.75976i		−	1.43512i	0.696496	−	0.717561i	$-0.254741\pi$
$12$		0			0
$13$	2.51524			0.697603			0.348801	−	0.937197i	$-0.386589\pi$
$13$	2.51524			0.697603			0.348801	+	0.937197i	$0.386589\pi$
$14$		0			0
$15$	−5.38377			−1.39008
$16$		0			0
$17$	−3.76296	+	1.68527i	−0.912651	+	0.408739i
$17$	−3.76296	+	1.68527i	−0.912651	+	0.408739i
$18$		0			0
$19$	−0.982947			−0.225504			−0.112752	−	0.993623i	$-0.535966\pi$
$19$	−0.982947			−0.225504			−0.112752	+	0.993623i	$0.535966\pi$
$20$		0			0
$21$	4.85082			1.05854
$22$		0			0
$23$		−	4.02346i		−	0.838950i	−0.907767	−	0.419475i	$-0.862214\pi$
$23$		−	4.02346i		−	0.838950i	0.907767	−	0.419475i	$-0.137786\pi$
$24$		0			0
$25$	−8.66046			−1.73209
$26$		0			0
$27$			5.64914i			1.08718i
$28$		0			0
$29$		−	0.375643i		−	0.0697551i	−0.999392	−	0.0348776i	$-0.988896\pi$
$29$		−	0.375643i		−	0.0697551i	0.999392	−	0.0348776i	$-0.0111041\pi$
$30$		0			0
$31$		−	3.17061i		−	0.569459i	−0.958608	−	0.284729i	$-0.908096\pi$
$31$		−	3.17061i		−	0.569459i	0.958608	−	0.284729i	$-0.0919037\pi$
$32$		0			0
$33$	6.93328			1.20693
$34$		0			0
$35$	12.3082			2.08046
$36$		0			0
$37$			1.30716i			0.214896i	0.994211	+	0.107448i	$0.0342679\pi$
$37$			1.30716i			0.214896i	−0.994211	+	0.107448i	$0.965732\pi$
$38$		0			0
$39$			3.66382i			0.586680i
$40$		0			0
$41$		−	7.88667i		−	1.23169i	−0.787867	−	0.615845i	$-0.788815\pi$
$41$		−	7.88667i		−	1.23169i	0.787867	−	0.615845i	$-0.211185\pi$
$42$		0			0
$43$	9.82393			1.49814			0.749068	−	0.662494i	$-0.230502\pi$
$43$	9.82393			1.49814			0.749068	+	0.662494i	$0.230502\pi$
$44$		0			0
$45$			3.24577i			0.483851i
$46$		0			0
$47$	13.2756			1.93644			0.968221	−	0.250098i	$-0.0804628\pi$
$47$	13.2756			1.93644			0.968221	+	0.250098i	$0.0804628\pi$
$48$		0			0
$49$	−4.08976			−0.584251
$50$		0			0
$51$	−2.45484	−	5.48130i	−0.343747	−	0.767535i
$52$		0			0
$53$	1.89912			0.260864			0.130432	−	0.991457i	$-0.458364\pi$
$53$	1.89912			0.260864			0.130432	+	0.991457i	$0.458364\pi$
$54$		0			0
$55$	17.5921			2.37212
$56$		0			0
$57$		−	1.43181i		−	0.189647i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	0.836969i		−	0.107163i	−0.998563	−	0.0535814i	$-0.982936\pi$
$61$		−	0.836969i		−	0.107163i	0.998563	−	0.0535814i	$-0.0170637\pi$
$62$		0			0
$63$		−	2.92446i		−	0.368448i
$64$		0			0
$65$			9.29635i			1.15307i
$66$		0			0
$67$	1.30621			0.159579			0.0797894	−	0.996812i	$-0.474575\pi$
$67$	1.30621			0.159579			0.0797894	+	0.996812i	$0.474575\pi$
$68$		0			0
$69$	5.86076			0.705553
$70$		0			0
$71$		−	10.8431i		−	1.28683i	−0.765516	−	0.643417i	$-0.777516\pi$
$71$		−	10.8431i		−	1.28683i	0.765516	−	0.643417i	$-0.222484\pi$
$72$		0			0
$73$			0.00541582i			0.000633874i	1.00000		0.000316937i	$0.000100884\pi$
$73$			0.00541582i			0.000633874i	−1.00000		0.000316937i	$0.999899\pi$
$74$		0			0
$75$		−	12.6152i		−	1.45668i
$76$		0			0
$77$	−15.8506			−1.80634
$78$		0			0
$79$			10.6850i			1.20215i	0.799192	+	0.601076i	$0.205261\pi$
$79$			10.6850i			1.20215i	−0.799192	+	0.601076i	$0.794739\pi$
$80$		0			0
$81$	−5.59424			−0.621583
$82$		0			0
$83$	7.42159			0.814625			0.407313	−	0.913289i	$-0.366466\pi$
$83$	7.42159			0.814625			0.407313	+	0.913289i	$0.366466\pi$
$84$		0			0
$85$	−6.22878	−	13.9079i	−0.675606	−	1.50853i
$86$		0			0
$87$	0.547179			0.0586637
$88$		0			0
$89$	−4.58338			−0.485838			−0.242919	−	0.970047i	$-0.578105\pi$
$89$	−4.58338			−0.485838			−0.242919	+	0.970047i	$0.578105\pi$
$90$		0			0
$91$		−	8.37608i		−	0.878052i
$92$		0			0
$93$	4.61846			0.478912
$94$		0			0
$95$		−	3.63298i		−	0.372736i
$96$		0			0
$97$		−	10.4572i		−	1.06177i	−0.847445	−	0.530884i	$-0.821860\pi$
$97$		−	10.4572i		−	1.06177i	0.847445	−	0.530884i	$-0.178140\pi$
$98$		0			0
$99$		−	4.17994i		−	0.420100i
$100$		0			0
$101$	7.08784			0.705266			0.352633	−	0.935762i	$-0.385286\pi$
$101$	7.08784			0.705266			0.352633	+	0.935762i	$0.385286\pi$
$102$		0			0
$103$	9.40257			0.926462			0.463231	−	0.886238i	$-0.346690\pi$
$103$	9.40257			0.926462			0.463231	+	0.886238i	$0.346690\pi$
$104$		0			0
$105$			17.9287i			1.74966i
$106$		0			0
$107$		−	8.68193i		−	0.839314i	−0.907683	−	0.419657i	$-0.862150\pi$
$107$		−	8.68193i		−	0.839314i	0.907683	−	0.419657i	$-0.137850\pi$
$108$		0			0
$109$			0.888172i			0.0850714i	0.999095	+	0.0425357i	$0.0135436\pi$
$109$			0.888172i			0.0850714i	−0.999095	+	0.0425357i	$0.986456\pi$
$110$		0			0
$111$	−1.90407			−0.180726
$112$		0			0
$113$			2.57544i			0.242277i	0.992636	+	0.121138i	$0.0386545\pi$
$113$			2.57544i			0.242277i	−0.992636	+	0.121138i	$0.961345\pi$
$114$		0			0
$115$	14.8707			1.38670
$116$		0			0
$117$	2.20884			0.204208
$118$		0			0
$119$	5.61217	+	12.5311i	0.514467	+	1.14873i
$120$		0			0
$121$	−11.6553			−1.05957
$122$		0			0
$123$	11.4881			1.03585
$124$		0			0
$125$		−	13.5291i		−	1.21008i
$126$		0			0
$127$	−15.5725			−1.38184			−0.690920	−	0.722931i	$-0.742794\pi$
$127$	−15.5725			−1.38184			−0.690920	+	0.722931i	$0.742794\pi$
$128$		0			0
$129$			14.3100i			1.25992i
$130$		0			0
$131$			1.44083i			0.125886i	0.998017	+	0.0629431i	$0.0200487\pi$
$131$			1.44083i			0.125886i	−0.998017	+	0.0629431i	$0.979951\pi$
$132$		0			0
$133$			3.27334i			0.283835i
$134$		0			0
$135$	−20.8793			−1.79700
$136$		0			0
$137$	14.2841			1.22037			0.610187	−	0.792257i	$-0.291094\pi$
$137$	14.2841			1.22037			0.610187	+	0.792257i	$0.291094\pi$
$138$		0			0
$139$			15.9480i			1.35269i	0.736586	+	0.676344i	$0.236437\pi$
$139$			15.9480i			1.35269i	−0.736586	+	0.676344i	$0.763563\pi$
$140$		0			0
$141$			19.3378i			1.62854i
$142$		0			0
$143$		−	11.9720i		−	1.00115i
$144$		0			0
$145$	1.38838			0.115299
$146$		0			0
$147$		−	5.95732i		−	0.491352i
$148$		0			0
$149$	−2.90208			−0.237748			−0.118874	−	0.992909i	$-0.537928\pi$
$149$	−2.90208			−0.237748			−0.118874	+	0.992909i	$0.537928\pi$
$150$		0			0
$151$	5.59425			0.455254			0.227627	−	0.973748i	$-0.426903\pi$
$151$	5.59425			0.455254			0.227627	+	0.973748i	$0.426903\pi$
$152$		0			0
$153$	−3.30457	+	1.47998i	−0.267158	+	0.119649i
$154$		0			0
$155$	11.7186			0.941261
$156$		0			0
$157$	22.6290			1.80599			0.902997	−	0.429646i	$-0.141362\pi$
$157$	22.6290			1.80599			0.902997	+	0.429646i	$0.141362\pi$
$158$		0			0
$159$			2.76635i			0.219386i
$160$		0			0
$161$	−13.3987			−1.05596
$162$		0			0
$163$		−	18.2245i		−	1.42745i	−0.700425	−	0.713726i	$-0.747006\pi$
$163$		−	18.2245i		−	1.42745i	0.700425	−	0.713726i	$-0.252994\pi$
$164$		0			0
$165$			25.6255i			1.99494i
$166$		0			0
$167$		−	2.07461i		−	0.160538i	−0.996773	−	0.0802690i	$-0.974422\pi$
$167$		−	2.07461i		−	0.160538i	0.996773	−	0.0802690i	$-0.0255779\pi$
$168$		0			0
$169$	−6.67355			−0.513350
$170$		0			0
$171$	−0.863208			−0.0660112
$172$		0			0
$173$		−	10.9810i		−	0.834872i	−0.908706	−	0.417436i	$-0.862929\pi$
$173$		−	10.9810i		−	0.834872i	0.908706	−	0.417436i	$-0.137071\pi$
$174$		0			0
$175$			28.8405i			2.18013i
$176$		0			0
$177$			1.45665i			0.109488i
$178$		0			0
$179$	9.18593			0.686588			0.343294	−	0.939228i	$-0.388457\pi$
$179$	9.18593			0.686588			0.343294	+	0.939228i	$0.388457\pi$
$180$		0			0
$181$			15.4332i			1.14714i	0.819156	+	0.573570i	$0.194442\pi$
$181$			15.4332i			1.14714i	−0.819156	+	0.573570i	$0.805558\pi$
$182$		0			0
$183$	1.21917			0.0901234
$184$		0			0
$185$	−4.83127			−0.355202
$186$		0			0
$187$	8.02149	+	17.9108i	0.586590	+	1.30977i
$188$		0			0
$189$	18.8124			1.36840
$190$		0			0
$191$	4.90307			0.354774			0.177387	−	0.984141i	$-0.443236\pi$
$191$	4.90307			0.354774			0.177387	+	0.984141i	$0.443236\pi$
$192$		0			0
$193$			9.14153i			0.658022i	0.944326	+	0.329011i	$0.106715\pi$
$193$			9.14153i			0.658022i	−0.944326	+	0.329011i	$0.893285\pi$
$194$		0			0
$195$	−13.5415			−0.969726
$196$		0			0
$197$		−	8.68513i		−	0.618790i	−0.950934	−	0.309395i	$-0.899873\pi$
$197$		−	8.68513i		−	0.618790i	0.950934	−	0.309395i	$-0.100127\pi$
$198$		0			0
$199$			18.1632i			1.28756i	0.765213	+	0.643778i	$0.222634\pi$
$199$			18.1632i			1.28756i	−0.765213	+	0.643778i	$0.777366\pi$
$200$		0			0
$201$			1.90268i			0.134205i
$202$		0			0
$203$	−1.25094			−0.0877987
$204$		0			0
$205$	29.1492			2.03587
$206$		0			0
$207$		−	3.53334i		−	0.245584i
$208$		0			0
$209$			4.67859i			0.323625i
$210$		0			0
$211$			4.81118i			0.331215i	0.986192	+	0.165608i	$0.0529586\pi$
$211$			4.81118i			0.331215i	−0.986192	+	0.165608i	$0.947041\pi$
$212$		0			0
$213$	15.7945			1.08222
$214$		0			0
$215$			36.3093i			2.47627i
$216$		0			0
$217$	−10.5585			−0.716761
$218$		0			0
$219$	−0.00788893			−0.000533085
$220$		0			0
$221$	−9.46475	+	4.23887i	−0.636668	+	0.285137i
$222$		0			0
$223$	−28.1279			−1.88358			−0.941792	−	0.336197i	$-0.890859\pi$
$223$	−28.1279			−1.88358			−0.941792	+	0.336197i	$0.890859\pi$
$224$		0			0
$225$	−7.60547			−0.507032
$226$		0			0
$227$		−	12.9136i		−	0.857103i	−0.903517	−	0.428551i	$-0.859024\pi$
$227$		−	12.9136i		−	0.857103i	0.903517	−	0.428551i	$-0.140976\pi$
$228$		0			0
$229$	18.8142			1.24327			0.621637	−	0.783305i	$-0.286468\pi$
$229$	18.8142			1.24327			0.621637	+	0.783305i	$0.286468\pi$
$230$		0			0
$231$		−	23.0887i		−	1.51913i
$232$		0			0
$233$		−	7.70533i		−	0.504793i	−0.967624	−	0.252396i	$-0.918781\pi$
$233$		−	7.70533i		−	0.504793i	0.967624	−	0.252396i	$-0.0812187\pi$
$234$		0			0
$235$			49.0666i			3.20075i
$236$		0			0
$237$	−15.5642			−1.01100
$238$		0			0
$239$	−21.3317			−1.37983			−0.689916	−	0.723889i	$-0.742353\pi$
$239$	−21.3317			−1.37983			−0.689916	+	0.723889i	$0.742353\pi$
$240$		0			0
$241$			10.7602i			0.693126i	0.938027	+	0.346563i	$0.112651\pi$
$241$			10.7602i			0.693126i	−0.938027	+	0.346563i	$0.887349\pi$
$242$		0			0
$243$			8.79859i			0.564430i
$244$		0			0
$245$		−	15.1158i		−	0.965711i
$246$		0			0
$247$	−2.47235			−0.157312
$248$		0			0
$249$			10.8106i			0.685095i
$250$		0			0
$251$	2.59142			0.163569			0.0817846	−	0.996650i	$-0.473938\pi$
$251$	2.59142			0.163569			0.0817846	+	0.996650i	$0.473938\pi$
$252$		0			0
$253$	−19.1507			−1.20400
$254$		0			0
$255$	20.2589	−	9.07312i	1.26866	−	0.568181i
$256$		0			0
$257$	15.9311			0.993754			0.496877	−	0.867821i	$-0.334480\pi$
$257$	15.9311			0.993754			0.496877	+	0.867821i	$0.334480\pi$
$258$		0			0
$259$	4.35301			0.270483
$260$		0			0
$261$		−	0.329883i		−	0.0204193i
$262$		0			0
$263$	−18.3184			−1.12956			−0.564781	−	0.825241i	$-0.691039\pi$
$263$	−18.3184			−1.12956			−0.564781	+	0.825241i	$0.691039\pi$
$264$		0			0
$265$			7.01916i			0.431184i
$266$		0			0
$267$		−	6.67636i		−	0.408587i
$268$		0			0
$269$		−	6.62647i		−	0.404023i	−0.979383	−	0.202011i	$-0.935252\pi$
$269$		−	6.62647i		−	0.404023i	0.979383	−	0.202011i	$-0.0647478\pi$
$270$		0			0
$271$	29.1979			1.77364			0.886822	−	0.462111i	$-0.152908\pi$
$271$	29.1979			1.77364			0.886822	+	0.462111i	$0.152908\pi$
$272$		0			0
$273$	12.2010			0.738437
$274$		0			0
$275$			41.2217i			2.48576i
$276$		0			0
$277$			20.9842i			1.26082i	0.776264	+	0.630408i	$0.217112\pi$
$277$			20.9842i			1.26082i	−0.776264	+	0.630408i	$0.782888\pi$
$278$		0			0
$279$		−	2.78438i		−	0.166696i
$280$		0			0
$281$	−18.3190			−1.09282			−0.546409	−	0.837519i	$-0.684005\pi$
$281$	−18.3190			−1.09282			−0.546409	+	0.837519i	$0.684005\pi$
$282$		0			0
$283$			9.67937i			0.575379i	0.957724	+	0.287689i	$0.0928871\pi$
$283$			9.67937i			0.575379i	−0.957724	+	0.287689i	$0.907113\pi$
$284$		0			0
$285$	5.29196			0.313469
$286$		0			0
$287$	−26.2636			−1.55029
$288$		0			0
$289$	11.3197	−	12.6832i	0.665865	−	0.746072i
$290$		0			0
$291$	15.2324			0.892941
$292$		0			0
$293$	−3.37269			−0.197035			−0.0985173	−	0.995135i	$-0.531410\pi$
$293$	−3.37269			−0.197035			−0.0985173	+	0.995135i	$0.531410\pi$
$294$		0			0
$295$			3.69601i			0.215190i
$296$		0			0
$297$	26.8885			1.56023
$298$		0			0
$299$		−	10.1200i		−	0.585254i
$300$		0			0
$301$		−	32.7149i		−	1.88566i
$302$		0			0
$303$			10.3245i			0.593125i
$304$		0			0
$305$	3.09344			0.177130
$306$		0			0
$307$	−13.9557			−0.796492			−0.398246	−	0.917279i	$-0.630381\pi$
$307$	−13.9557			−0.796492			−0.398246	+	0.917279i	$0.630381\pi$
$308$		0			0
$309$			13.6962i			0.779150i
$310$		0			0
$311$		−	1.75361i		−	0.0994380i	−0.998763	−	0.0497190i	$-0.984167\pi$
$311$		−	1.75361i		−	0.0994380i	0.998763	−	0.0497190i	$-0.0158326\pi$
$312$		0			0
$313$			9.60421i			0.542862i	0.962458	+	0.271431i	$0.0874969\pi$
$313$			9.60421i			0.542862i	−0.962458	+	0.271431i	$0.912503\pi$
$314$		0			0
$315$	10.8088			0.609009
$316$		0			0
$317$		−	6.48874i		−	0.364444i	−0.983257	−	0.182222i	$-0.941671\pi$
$317$		−	6.48874i		−	0.364444i	0.983257	−	0.182222i	$-0.0583290\pi$
$318$		0			0
$319$	−1.78797			−0.100107
$320$		0			0
$321$	12.6465			0.705859
$322$		0			0
$323$	3.69879	−	1.65653i	0.205806	−	0.0921720i
$324$		0			0
$325$	−21.7832			−1.20831
$326$		0			0
$327$	−1.29375			−0.0715446
$328$		0			0
$329$		−	44.2094i		−	2.43734i
$330$		0			0
$331$	10.6209			0.583776			0.291888	−	0.956452i	$-0.405716\pi$
$331$	10.6209			0.583776			0.291888	+	0.956452i	$0.405716\pi$
$332$		0			0
$333$			1.14793i			0.0629059i
$334$		0			0
$335$			4.82775i			0.263768i
$336$		0			0
$337$			18.8193i			1.02515i	0.858641	+	0.512577i	$0.171309\pi$
$337$			18.8193i			1.02515i	−0.858641	+	0.512577i	$0.828691\pi$
$338$		0			0
$339$	−3.75150			−0.203754
$340$		0			0
$341$	−15.0914			−0.817242
$342$		0			0
$343$		−	9.69149i		−	0.523291i
$344$		0			0
$345$			21.6614i			1.16621i
$346$		0			0
$347$		−	19.1271i		−	1.02680i	−0.858150	−	0.513400i	$-0.828386\pi$
$347$		−	19.1271i		−	1.02680i	0.858150	−	0.513400i	$-0.171614\pi$
$348$		0			0
$349$	−5.15560			−0.275973			−0.137987	−	0.990434i	$-0.544063\pi$
$349$	−5.15560			−0.275973			−0.137987	+	0.990434i	$0.544063\pi$
$350$		0			0
$351$			14.2090i			0.758418i
$352$		0			0
$353$	19.9898			1.06395			0.531975	−	0.846760i	$-0.321450\pi$
$353$	19.9898			1.06395			0.531975	+	0.846760i	$0.321450\pi$
$354$		0			0
$355$	40.0760			2.12701
$356$		0			0
$357$	−18.2534	+	8.17495i	−0.966074	+	0.432664i
$358$		0			0
$359$	21.6478			1.14253			0.571263	−	0.820767i	$-0.306454\pi$
$359$	21.6478			1.14253			0.571263	+	0.820767i	$0.306454\pi$
$360$		0			0
$361$	−18.0338			−0.949148
$362$		0			0
$363$		−	16.9777i		−	0.891096i
$364$		0			0
$365$	−0.0200169			−0.00104773
$366$		0			0
$367$			25.4012i			1.32593i	0.748649	+	0.662967i	$0.230703\pi$
$367$			25.4012i			1.32593i	−0.748649	+	0.662967i	$0.769297\pi$
$368$		0			0
$369$		−	6.92594i		−	0.360550i
$370$		0			0
$371$		−	6.32432i		−	0.328342i
$372$		0			0
$373$	−2.83555			−0.146819			−0.0734096	−	0.997302i	$-0.523388\pi$
$373$	−2.83555			−0.146819			−0.0734096	+	0.997302i	$0.523388\pi$
$374$		0			0
$375$	19.7071			1.01767
$376$		0			0
$377$		−	0.944833i		−	0.0486614i
$378$		0			0
$379$		−	10.8572i		−	0.557696i	−0.960335	−	0.278848i	$-0.910047\pi$
$379$		−	10.8572i		−	0.557696i	0.960335	−	0.278848i	$-0.0899526\pi$
$380$		0			0
$381$		−	22.6837i		−	1.16212i
$382$		0			0
$383$	−16.3714			−0.836539			−0.418269	−	0.908323i	$-0.637363\pi$
$383$	−16.3714			−0.836539			−0.418269	+	0.908323i	$0.637363\pi$
$384$		0			0
$385$		−	58.5840i		−	2.98572i
$386$		0			0
$387$	8.62721			0.438546
$388$		0			0
$389$	4.97521			0.252253			0.126127	−	0.992014i	$-0.459745\pi$
$389$	4.97521			0.252253			0.126127	+	0.992014i	$0.459745\pi$
$390$		0			0
$391$	6.78063	+	15.1401i	0.342911	+	0.765669i
$392$		0			0
$393$	−2.09878			−0.105870
$394$		0			0
$395$	−39.4917			−1.98704
$396$		0			0
$397$		−	19.2233i		−	0.964791i	−0.875954	−	0.482395i	$-0.839767\pi$
$397$		−	19.2233i		−	0.964791i	0.875954	−	0.482395i	$-0.160233\pi$
$398$		0			0
$399$	−4.76810			−0.238703
$400$		0			0
$401$		−	14.9706i		−	0.747595i	−0.927510	−	0.373798i	$-0.878055\pi$
$401$		−	14.9706i		−	0.747595i	0.927510	−	0.373798i	$-0.121945\pi$
$402$		0			0
$403$		−	7.97486i		−	0.397256i
$404$		0			0
$405$		−	20.6764i		−	1.02742i
$406$		0			0
$407$	6.22176			0.308401
$408$		0			0
$409$	−18.3553			−0.907610			−0.453805	−	0.891101i	$-0.649934\pi$
$409$	−18.3553			−0.907610			−0.453805	+	0.891101i	$0.649934\pi$
$410$		0			0
$411$			20.8069i			1.02633i
$412$		0			0
$413$		−	3.33013i		−	0.163865i
$414$		0			0
$415$			27.4302i			1.34650i
$416$		0			0
$417$	−23.2305			−1.13760
$418$		0			0
$419$		−	5.34290i		−	0.261018i	−0.991447	−	0.130509i	$-0.958339\pi$
$419$		−	5.34290i		−	0.261018i	0.991447	−	0.130509i	$-0.0416611\pi$
$420$		0			0
$421$	−25.8543			−1.26006			−0.630031	−	0.776570i	$-0.716958\pi$
$421$	−25.8543			−1.26006			−0.630031	+	0.776570i	$0.716958\pi$
$422$		0			0
$423$	11.6584			0.566850
$424$		0			0
$425$	32.5890	−	14.5952i	1.58080	−	0.707973i
$426$		0			0
$427$	−2.78721			−0.134883
$428$		0			0
$429$	17.4389			0.841958
$430$		0			0
$431$		−	36.7833i		−	1.77179i	−0.463886	−	0.885895i	$-0.653545\pi$
$431$		−	36.7833i		−	1.77179i	0.463886	−	0.885895i	$-0.346455\pi$
$432$		0			0
$433$	12.3232			0.592214			0.296107	−	0.955155i	$-0.404312\pi$
$433$	12.3232			0.592214			0.296107	+	0.955155i	$0.404312\pi$
$434$		0			0
$435$			2.02238i			0.0969655i
$436$		0			0
$437$			3.95485i			0.189186i
$438$		0			0
$439$		−	16.6823i		−	0.796204i	−0.917341	−	0.398102i	$-0.869669\pi$
$439$		−	16.6823i		−	0.796204i	0.917341	−	0.398102i	$-0.130331\pi$
$440$		0			0
$441$	−3.59155			−0.171026
$442$		0			0
$443$	−13.5919			−0.645773			−0.322886	−	0.946438i	$-0.604653\pi$
$443$	−13.5919			−0.645773			−0.322886	+	0.946438i	$0.604653\pi$
$444$		0			0
$445$		−	16.9402i		−	0.803043i
$446$		0			0
$447$		−	4.22730i		−	0.199945i
$448$		0			0
$449$			34.3123i			1.61930i	0.586916	+	0.809648i	$0.300342\pi$
$449$			34.3123i			1.61930i	−0.586916	+	0.809648i	$0.699658\pi$
$450$		0			0
$451$	−37.5387			−1.76763
$452$		0			0
$453$			8.14884i			0.382866i
$454$		0			0
$455$	30.9581			1.45134
$456$		0			0
$457$	−23.5737			−1.10273			−0.551367	−	0.834263i	$-0.685893\pi$
$457$	−23.5737			−1.10273			−0.551367	+	0.834263i	$0.685893\pi$
$458$		0			0
$459$	−9.52034	−	21.2575i	−0.444371	−	0.992214i
$460$		0			0
$461$	−2.50142			−0.116503			−0.0582514	−	0.998302i	$-0.518552\pi$
$461$	−2.50142			−0.116503			−0.0582514	+	0.998302i	$0.518552\pi$
$462$		0			0
$463$	−14.2151			−0.660632			−0.330316	−	0.943870i	$-0.607155\pi$
$463$	−14.2151			−0.660632			−0.330316	+	0.943870i	$0.607155\pi$
$464$		0			0
$465$			17.0698i			0.791595i
$466$		0			0
$467$	−21.2130			−0.981623			−0.490811	−	0.871266i	$-0.663300\pi$
$467$	−21.2130			−0.981623			−0.490811	+	0.871266i	$0.663300\pi$
$468$		0			0
$469$		−	4.34984i		−	0.200857i
$470$		0			0
$471$			32.9625i			1.51883i
$472$		0			0
$473$		−	46.7596i		−	2.15001i
$474$		0			0
$475$	8.51278			0.390593
$476$		0			0
$477$	1.66778			0.0763622
$478$		0			0
$479$		−	8.51651i		−	0.389129i	−0.980890	−	0.194565i	$-0.937671\pi$
$479$		−	8.51651i		−	0.389129i	0.980890	−	0.194565i	$-0.0623294\pi$
$480$		0			0
$481$			3.28782i			0.149912i
$482$		0			0
$483$		−	19.5171i		−	0.888058i
$484$		0			0
$485$	38.6499			1.75500
$486$		0			0
$487$		−	8.90145i		−	0.403363i	−0.979451	−	0.201682i	$-0.935359\pi$
$487$		−	8.90145i		−	0.403363i	0.979451	−	0.201682i	$-0.0646406\pi$
$488$		0			0
$489$	26.5466			1.20048
$490$		0			0
$491$	19.9226			0.899093			0.449547	−	0.893257i	$-0.351586\pi$
$491$	19.9226			0.899093			0.449547	+	0.893257i	$0.351586\pi$
$492$		0			0
$493$	0.633061	+	1.41353i	0.0285116	+	0.0636621i
$494$		0			0
$495$	15.4491			0.694385
$496$		0			0
$497$	−36.1088			−1.61970
$498$		0			0
$499$			16.1979i			0.725119i	0.931961	+	0.362559i	$0.118097\pi$
$499$			16.1979i			0.725119i	−0.931961	+	0.362559i	$0.881903\pi$
$500$		0			0
$501$	3.02197			0.135012
$502$		0			0
$503$		−	39.8469i		−	1.77668i	−0.459183	−	0.888342i	$-0.651858\pi$
$503$		−	39.8469i		−	1.77668i	0.459183	−	0.888342i	$-0.348142\pi$
$504$		0			0
$505$			26.1967i			1.16574i
$506$		0			0
$507$		−	9.72100i		−	0.431725i
$508$		0			0
$509$	8.91248			0.395039			0.197519	−	0.980299i	$-0.436711\pi$
$509$	8.91248			0.395039			0.197519	+	0.980299i	$0.436711\pi$
$510$		0			0
$511$	0.0180354			0.000797838
$512$		0			0
$513$		−	5.55281i		−	0.245162i
$514$		0			0
$515$			34.7519i			1.53135i
$516$		0			0
$517$		−	63.1885i		−	2.77903i
$518$		0			0
$519$	15.9955			0.702123
$520$		0			0
$521$			1.16643i			0.0511022i	0.999674	+	0.0255511i	$0.00813406\pi$
$521$			1.16643i			0.0511022i	−0.999674	+	0.0255511i	$0.991866\pi$
$522$		0			0
$523$	−28.3461			−1.23949			−0.619744	−	0.784804i	$-0.712764\pi$
$523$	−28.3461			−1.23949			−0.619744	+	0.784804i	$0.712764\pi$
$524$		0			0
$525$	−42.0103			−1.83348
$526$		0			0
$527$	5.34334	+	11.9309i	0.232760	+	0.519717i
$528$		0			0
$529$	6.81174			0.296162
$530$		0			0
$531$	0.878183			0.0381099
$532$		0			0
$533$		−	19.8369i		−	0.859231i
$534$		0			0
$535$	32.0885			1.38731
$536$		0			0
$537$			13.3806i			0.577417i
$538$		0			0
$539$			19.4663i			0.838471i
$540$		0			0
$541$		−	16.8243i		−	0.723332i	−0.932308	−	0.361666i	$-0.882208\pi$
$541$		−	16.8243i		−	0.723332i	0.932308	−	0.361666i	$-0.117792\pi$
$542$		0			0
$543$	−22.4807			−0.964739
$544$		0			0
$545$	−3.28269			−0.140615
$546$		0			0
$547$			27.6741i			1.18326i	0.806210	+	0.591630i	$0.201515\pi$
$547$			27.6741i			1.18326i	−0.806210	+	0.591630i	$0.798485\pi$
$548$		0			0
$549$		−	0.735012i		−	0.0313695i
$550$		0			0
$551$			0.369237i			0.0157300i
$552$		0			0
$553$	35.5823			1.51311
$554$		0			0
$555$		−	7.03745i		−	0.298723i
$556$		0			0
$557$	−10.2906			−0.436028			−0.218014	−	0.975946i	$-0.569958\pi$
$557$	−10.2906			−0.436028			−0.218014	+	0.975946i	$0.569958\pi$
$558$		0			0
$559$	24.7096			1.04510
$560$		0			0
$561$	−26.0897	+	11.6845i	−1.10151	+	0.493319i
$562$		0			0
$563$	40.7751			1.71846			0.859232	−	0.511585i	$-0.170942\pi$
$563$	40.7751			1.71846			0.859232	+	0.511585i	$0.170942\pi$
$564$		0			0
$565$	−9.51884			−0.400461
$566$		0			0
$567$			18.6296i			0.782368i
$568$		0			0
$569$	8.65394			0.362792			0.181396	−	0.983410i	$-0.441938\pi$
$569$	8.65394			0.362792			0.181396	+	0.983410i	$0.441938\pi$
$570$		0			0
$571$			44.4118i			1.85858i	0.369355	+	0.929288i	$0.379579\pi$
$571$			44.4118i			1.85858i	−0.369355	+	0.929288i	$0.620421\pi$
$572$		0			0
$573$			7.14204i			0.298363i
$574$		0			0
$575$			34.8451i			1.45314i
$576$		0			0
$577$	12.4836			0.519701			0.259850	−	0.965649i	$-0.416327\pi$
$577$	12.4836			0.519701			0.259850	+	0.965649i	$0.416327\pi$
$578$		0			0
$579$	−13.3160			−0.553393
$580$		0			0
$581$		−	24.7148i		−	1.02534i
$582$		0			0
$583$		−	9.03936i		−	0.374372i
$584$		0			0
$585$			8.16390i			0.337536i
$586$		0			0
$587$	−7.15235			−0.295209			−0.147605	−	0.989046i	$-0.547156\pi$
$587$	−7.15235			−0.295209			−0.147605	+	0.989046i	$0.547156\pi$
$588$		0			0
$589$			3.11654i			0.128415i
$590$		0			0
$591$	12.6512			0.520399
$592$		0			0
$593$	29.0737			1.19391			0.596957	−	0.802273i	$-0.296376\pi$
$593$	29.0737			1.19391			0.596957	+	0.802273i	$0.296376\pi$
$594$		0			0
$595$	−46.3152	+	20.7426i	−1.89874	+	0.850365i
$596$		0			0
$597$	−26.4573			−1.08283
$598$		0			0
$599$	32.9059			1.34450			0.672250	−	0.740324i	$-0.265328\pi$
$599$	32.9059			1.34450			0.672250	+	0.740324i	$0.265328\pi$
$600$		0			0
$601$		−	34.5604i		−	1.40975i	−0.709333	−	0.704873i	$-0.751004\pi$
$601$		−	34.5604i		−	1.40975i	0.709333	−	0.704873i	$-0.248996\pi$
$602$		0			0
$603$	1.14709			0.0467131
$604$		0			0
$605$		−	43.0781i		−	1.75137i
$606$		0			0
$607$			32.5505i			1.32118i	0.750745	+	0.660592i	$0.229695\pi$
$607$			32.5505i			1.32118i	−0.750745	+	0.660592i	$0.770305\pi$
$608$		0			0
$609$		−	1.82217i		−	0.0738382i
$610$		0			0
$611$	33.3913			1.35087
$612$		0			0
$613$	13.6816			0.552594			0.276297	−	0.961072i	$-0.410893\pi$
$613$	13.6816			0.552594			0.276297	+	0.961072i	$0.410893\pi$
$614$		0			0
$615$			42.4600i			1.71215i
$616$		0			0
$617$			48.7609i			1.96304i	0.191362	+	0.981520i	$0.438710\pi$
$617$			48.7609i			1.96304i	−0.191362	+	0.981520i	$0.561290\pi$
$618$		0			0
$619$		−	9.51877i		−	0.382592i	−0.981532	−	0.191296i	$-0.938731\pi$
$619$		−	9.51877i		−	0.382592i	0.981532	−	0.191296i	$-0.0612690\pi$
$620$		0			0
$621$	22.7291			0.912088
$622$		0			0
$623$			15.2633i			0.611509i
$624$		0			0
$625$	6.70129			0.268052
$626$		0			0
$627$	−6.81505			−0.272167
$628$		0			0
$629$	−2.20292	−	4.91879i	−0.0878362	−	0.196125i
$630$		0			0
$631$	21.5112			0.856348			0.428174	−	0.903696i	$-0.359157\pi$
$631$	21.5112			0.856348			0.428174	+	0.903696i	$0.359157\pi$
$632$		0			0
$633$	−7.00819			−0.278550
$634$		0			0
$635$		−	57.5562i		−	2.28405i
$636$		0			0
$637$	−10.2867			−0.407575
$638$		0			0
$639$		−	9.52219i		−	0.376692i
$640$		0			0
$641$			44.8593i			1.77184i	0.463843	+	0.885918i	$0.346470\pi$
$641$			44.8593i			1.77184i	−0.463843	+	0.885918i	$0.653530\pi$
$642$		0			0
$643$		−	39.4001i		−	1.55379i	−0.629632	−	0.776894i	$-0.716794\pi$
$643$		−	39.4001i		−	1.55379i	0.629632	−	0.776894i	$-0.283206\pi$
$644$		0			0
$645$	−52.8898			−2.08253
$646$		0			0
$647$	−38.2810			−1.50498			−0.752491	−	0.658602i	$-0.771148\pi$
$647$	−38.2810			−1.50498			−0.752491	+	0.658602i	$0.771148\pi$
$648$		0			0
$649$		−	4.75976i		−	0.186837i
$650$		0			0
$651$		−	15.3801i		−	0.602792i
$652$		0			0
$653$		−	47.7182i		−	1.86736i	−0.358111	−	0.933679i	$-0.616579\pi$
$653$		−	47.7182i		−	1.86736i	0.358111	−	0.933679i	$-0.383421\pi$
$654$		0			0
$655$	−5.32533			−0.208078
$656$		0			0
$657$			0.00475608i			0.000185553i
$658$		0			0
$659$	29.9368			1.16617			0.583087	−	0.812410i	$-0.301845\pi$
$659$	29.9368			1.16617			0.583087	+	0.812410i	$0.301845\pi$
$660$		0			0
$661$	7.71077			0.299914			0.149957	−	0.988693i	$-0.452086\pi$
$661$	7.71077			0.299914			0.149957	+	0.988693i	$0.452086\pi$
$662$		0			0
$663$	−6.17453	−	13.7868i	−0.239799	−	0.535435i
$664$		0			0
$665$	−12.0983			−0.469151
$666$		0			0
$667$	−1.51139			−0.0585211
$668$		0			0
$669$		−	40.9724i		−	1.58408i
$670$		0			0
$671$	−3.98377			−0.153792
$672$		0			0
$673$		−	17.0441i		−	0.657001i	−0.944504	−	0.328501i	$-0.893457\pi$
$673$		−	17.0441i		−	0.657001i	0.944504	−	0.328501i	$-0.106543\pi$
$674$		0			0
$675$		−	48.9242i		−	1.88309i
$676$		0			0
$677$		−	34.9057i		−	1.34153i	−0.741668	−	0.670767i	$-0.765965\pi$
$677$		−	34.9057i		−	1.34153i	0.741668	−	0.670767i	$-0.234035\pi$
$678$		0			0
$679$	−34.8238			−1.33642
$680$		0			0
$681$	18.8105			0.720819
$682$		0			0
$683$		−	11.4915i		−	0.439709i	−0.975533	−	0.219855i	$-0.929442\pi$
$683$		−	11.4915i		−	0.439709i	0.975533	−	0.219855i	$-0.0705583\pi$
$684$		0			0
$685$			52.7942i			2.01716i
$686$		0			0
$687$			27.4056i			1.04559i
$688$		0			0
$689$	4.77675			0.181980
$690$		0			0
$691$			24.6427i			0.937451i	0.883344	+	0.468725i	$0.155287\pi$
$691$			24.6427i			0.937451i	−0.883344	+	0.468725i	$0.844713\pi$
$692$		0			0
$693$	−13.9197			−0.528767
$694$		0			0
$695$	−58.9437			−2.23586
$696$		0			0
$697$	13.2912	+	29.6772i	0.503440	+	1.12410i
$698$		0			0
$699$	11.2239			0.424528
$700$		0			0
$701$	−21.0000			−0.793157			−0.396579	−	0.918001i	$-0.629803\pi$
$701$	−21.0000			−0.793157			−0.396579	+	0.918001i	$0.629803\pi$
$702$		0			0
$703$		−	1.28487i		−	0.0484597i
$704$		0			0
$705$	−71.4726			−2.69182
$706$		0			0
$707$		−	23.6034i		−	0.887698i
$708$		0			0
$709$		−	11.8482i		−	0.444967i	−0.974936	−	0.222484i	$-0.928584\pi$
$709$		−	11.8482i		−	0.444967i	0.974936	−	0.222484i	$-0.0714164\pi$
$710$		0			0
$711$			9.38335i			0.351903i
$712$		0			0
$713$	−12.7568			−0.477747
$714$		0			0
$715$	44.2484			1.65480
$716$		0			0
$717$		−	31.0727i		−	1.16043i
$718$		0			0
$719$			37.9487i			1.41525i	0.706589	+	0.707625i	$0.250233\pi$
$719$			37.9487i			1.41525i	−0.706589	+	0.707625i	$0.749767\pi$
$720$		0			0
$721$		−	31.3117i		−	1.16611i
$722$		0			0
$723$	−15.6738			−0.582916
$724$		0			0
$725$			3.25324i			0.120822i
$726$		0			0
$727$	−0.354944			−0.0131641			−0.00658207	−	0.999978i	$-0.502095\pi$
$727$	−0.354944			−0.0131641			−0.00658207	+	0.999978i	$0.502095\pi$
$728$		0			0
$729$	−29.5992			−1.09626
$730$		0			0
$731$	−36.9670	+	16.5560i	−1.36728	+	0.612346i
$732$		0			0
$733$	−29.6022			−1.09338			−0.546692	−	0.837334i	$-0.684113\pi$
$733$	−29.6022			−1.09338			−0.546692	+	0.837334i	$0.684113\pi$
$734$		0			0
$735$	22.0183			0.812158
$736$		0			0
$737$		−	6.21724i		−	0.229015i
$738$		0			0
$739$	1.73448			0.0638040			0.0319020	−	0.999491i	$-0.489844\pi$
$739$	1.73448			0.0638040			0.0319020	+	0.999491i	$0.489844\pi$
$740$		0			0
$741$		−	3.60134i		−	0.132299i
$742$		0			0
$743$		−	47.4202i		−	1.73968i	−0.493335	−	0.869839i	$-0.664222\pi$
$743$		−	47.4202i		−	1.73968i	0.493335	−	0.869839i	$-0.335778\pi$
$744$		0			0
$745$		−	10.7261i		−	0.392974i
$746$		0			0
$747$	6.51751			0.238463
$748$		0			0
$749$	−28.9119			−1.05642
$750$		0			0
$751$		−	37.8765i		−	1.38213i	−0.722792	−	0.691066i	$-0.757141\pi$
$751$		−	37.8765i		−	1.38213i	0.722792	−	0.691066i	$-0.242859\pi$
$752$		0			0
$753$			3.77478i			0.137561i
$754$		0			0
$755$			20.6764i			0.752491i
$756$		0			0
$757$	−26.6011			−0.966834			−0.483417	−	0.875390i	$-0.660604\pi$
$757$	−26.6011			−0.966834			−0.483417	+	0.875390i	$0.660604\pi$
$758$		0			0
$759$		−	27.8958i		−	1.01255i
$760$		0			0
$761$	9.39591			0.340602			0.170301	−	0.985392i	$-0.445526\pi$
$761$	9.39591			0.340602			0.170301	+	0.985392i	$0.445526\pi$
$762$		0			0
$763$	2.95773			0.107077
$764$		0			0
$765$	−5.47001	−	12.2137i	−0.197769	−	0.441587i
$766$		0			0
$767$	2.51524			0.0908202
$768$		0			0
$769$	2.29722			0.0828400			0.0414200	−	0.999142i	$-0.486812\pi$
$769$	2.29722			0.0828400			0.0414200	+	0.999142i	$0.486812\pi$
$770$		0			0
$771$			23.2059i			0.835742i
$772$		0			0
$773$	−22.3295			−0.803135			−0.401568	−	0.915829i	$-0.631535\pi$
$773$	−22.3295			−0.803135			−0.401568	+	0.915829i	$0.631535\pi$
$774$		0			0
$775$			27.4590i			0.986355i
$776$		0			0
$777$			6.34079i			0.227475i
$778$		0			0
$779$			7.75218i			0.277751i
$780$		0			0
$781$	−51.6103			−1.84676
$782$		0			0
$783$	2.12206			0.0758362
$784$		0			0
$785$			83.6371i			2.98514i
$786$		0			0
$787$		−	28.5559i		−	1.01791i	−0.860794	−	0.508953i	$-0.830033\pi$
$787$		−	28.5559i		−	1.01791i	0.860794	−	0.508953i	$-0.169967\pi$
$788$		0			0
$789$		−	26.6834i		−	0.949955i
$790$		0			0
$791$	8.57654			0.304947
$792$		0			0
$793$		−	2.10518i		−	0.0747571i
$794$		0			0
$795$	−10.2244			−0.362623
$796$		0			0
$797$	−41.5950			−1.47337			−0.736685	−	0.676236i	$-0.763610\pi$
$797$	−41.5950			−1.47337			−0.736685	+	0.676236i	$0.763610\pi$
$798$		0			0
$799$	−49.9554	+	22.3730i	−1.76730	+	0.791498i
$800$		0			0
$801$	−4.02505			−0.142218
$802$		0			0
$803$	0.0257780			0.000909686
$804$		0			0
$805$		−	49.5215i		−	1.74540i
$806$		0			0
$807$	9.65242			0.339781
$808$		0			0
$809$			3.50157i			0.123108i	0.998104	+	0.0615542i	$0.0196057\pi$
$809$			3.50157i			0.123108i	−0.998104	+	0.0615542i	$0.980394\pi$
$810$		0			0
$811$		−	43.1330i		−	1.51460i	−0.653064	−	0.757302i	$-0.726517\pi$
$811$		−	43.1330i		−	1.51460i	0.653064	−	0.757302i	$-0.273483\pi$
$812$		0			0
$813$			42.5310i			1.49163i
$814$		0			0
$815$	67.3578			2.35944
$816$		0			0
$817$	−9.65640			−0.337835
$818$		0			0
$819$		−	7.35574i		−	0.257030i
$820$		0			0
$821$			51.4467i			1.79550i	0.440503	+	0.897751i	$0.354800\pi$
$821$			51.4467i			1.79550i	−0.440503	+	0.897751i	$0.645200\pi$
$822$		0			0
$823$			28.3998i			0.989956i	0.868905	+	0.494978i	$0.164824\pi$
$823$			28.3998i			0.989956i	−0.868905	+	0.494978i	$0.835176\pi$
$824$		0			0
$825$	−60.0454			−2.09051
$826$		0			0
$827$		−	18.8102i		−	0.654095i	−0.945008	−	0.327048i	$-0.893946\pi$
$827$		−	18.8102i		−	0.654095i	0.945008	−	0.327048i	$-0.106054\pi$
$828$		0			0
$829$	−24.8576			−0.863341			−0.431670	−	0.902031i	$-0.642076\pi$
$829$	−24.8576			−0.863341			−0.431670	+	0.902031i	$0.642076\pi$
$830$		0			0
$831$	−30.5665			−1.06034
$832$		0			0
$833$	15.3896	−	6.89235i	0.533217	−	0.238806i
$834$		0			0
$835$	7.66777			0.265354
$836$		0			0
$837$	17.9112			0.619102
$838$		0			0
$839$		−	42.6796i		−	1.47346i	−0.676185	−	0.736732i	$-0.736368\pi$
$839$		−	42.6796i		−	1.47346i	0.676185	−	0.736732i	$-0.263632\pi$
$840$		0			0
$841$	28.8589			0.995134
$842$		0			0
$843$		−	26.6842i		−	0.919054i
$844$		0			0
$845$		−	24.6655i		−	0.848519i
$846$		0			0
$847$			38.8137i			1.33365i
$848$		0			0
$849$	−14.0994			−0.483890
$850$		0			0
$851$	5.25931			0.180287
$852$		0			0
$853$			24.9533i			0.854386i	0.904160	+	0.427193i	$0.140497\pi$
$853$			24.9533i			0.854386i	−0.904160	+	0.427193i	$0.859503\pi$
$854$		0			0
$855$		−	3.19042i		−	0.109110i
$856$		0			0
$857$		−	8.12879i		−	0.277674i	−0.990315	−	0.138837i	$-0.955664\pi$
$857$		−	8.12879i		−	0.277674i	0.990315	−	0.138837i	$-0.0443364\pi$
$858$		0			0
$859$	10.3754			0.354003			0.177001	−	0.984211i	$-0.443360\pi$
$859$	10.3754			0.354003			0.177001	+	0.984211i	$0.443360\pi$
$860$		0			0
$861$		−	38.2568i		−	1.30379i
$862$		0			0
$863$	−42.0982			−1.43304			−0.716519	−	0.697567i	$-0.754266\pi$
$863$	−42.0982			−1.43304			−0.716519	+	0.697567i	$0.754266\pi$
$864$		0			0
$865$	40.5860			1.37996
$866$		0			0
$867$	18.4750	+	16.4888i	0.627443	+	0.559989i
$868$		0			0
$869$	50.8578			1.72523
$870$		0			0
$871$	3.28543			0.111323
$872$		0			0
$873$		−	9.18334i		−	0.310809i
$874$		0			0
$875$	−45.0536			−1.52309
$876$		0			0
$877$		−	18.6769i		−	0.630674i	−0.948980	−	0.315337i	$-0.897882\pi$
$877$		−	18.6769i		−	0.630674i	0.948980	−	0.315337i	$-0.102118\pi$
$878$		0			0
$879$		−	4.91281i		−	0.165705i
$880$		0			0
$881$			6.62826i			0.223312i	0.993747	+	0.111656i	$0.0356154\pi$
$881$			6.62826i			0.223312i	−0.993747	+	0.111656i	$0.964385\pi$
$882$		0			0
$883$	45.5066			1.53142			0.765709	−	0.643187i	$-0.222388\pi$
$883$	45.5066			1.53142			0.765709	+	0.643187i	$0.222388\pi$
$884$		0			0
$885$	−5.38377			−0.180974
$886$		0			0
$887$			18.4439i			0.619284i	0.950853	+	0.309642i	$0.100209\pi$
$887$			18.4439i			0.619284i	−0.950853	+	0.309642i	$0.899791\pi$
$888$		0			0
$889$			51.8586i			1.73928i
$890$		0			0
$891$			26.6273i			0.892047i
$892$		0			0
$893$	−13.0492			−0.436674
$894$		0			0
$895$			33.9512i			1.13487i
$896$		0			0
$897$	14.7412			0.492196
$898$		0			0
$899$	−1.19102			−0.0397227
$900$		0			0
$901$	−7.14631	+	3.20054i	−0.238078	+	0.106625i
$902$		0			0
$903$	47.6541			1.58583
$904$		0			0
$905$	−57.0412			−1.89611
$906$		0			0
$907$			32.1170i			1.06643i	0.845980	+	0.533214i	$0.179016\pi$
$907$			32.1170i			1.06643i	−0.845980	+	0.533214i	$0.820984\pi$
$908$		0			0
$909$	6.22442			0.206451
$910$		0			0
$911$			38.7751i			1.28468i	0.766421	+	0.642339i	$0.222036\pi$
$911$			38.7751i			1.28468i	−0.766421	+	0.642339i	$0.777964\pi$
$912$		0			0
$913$		−	35.3250i		−	1.16909i
$914$		0			0
$915$			4.50605i			0.148965i
$916$		0			0
$917$	4.79816			0.158449
$918$		0			0
$919$	−22.7858			−0.751634			−0.375817	−	0.926694i	$-0.622638\pi$
$919$	−22.7858			−0.751634			−0.375817	+	0.926694i	$0.622638\pi$
$920$		0			0
$921$		−	20.3285i		−	0.669846i
$922$		0			0
$923$		−	27.2729i		−	0.897699i
$924$		0			0
$925$		−	11.3206i		−	0.372219i
$926$		0			0
$927$	8.25718			0.271201
$928$		0			0
$929$		−	3.72672i		−	0.122270i	−0.998130	−	0.0611348i	$-0.980528\pi$
$929$		−	3.72672i		−	0.122270i	0.998130	−	0.0611348i	$-0.0194719\pi$
$930$		0			0
$931$	4.02001			0.131751
$932$		0			0
$933$	2.55439			0.0836269
$934$		0			0
$935$	−66.1984	+	29.6475i	−2.16492	+	0.969577i
$936$		0			0
$937$	56.7552			1.85411			0.927056	−	0.374923i	$-0.122331\pi$
$937$	56.7552			1.85411			0.927056	+	0.374923i	$0.122331\pi$
$938$		0			0
$939$	−13.9899			−0.456544
$940$		0			0
$941$			57.3417i			1.86928i	0.355589	+	0.934642i	$0.384280\pi$
$941$			57.3417i			1.86928i	−0.355589	+	0.934642i	$0.615720\pi$
$942$		0			0
$943$	−31.7317			−1.03333
$944$		0			0
$945$			69.5306i			2.26183i
$946$		0			0
$947$			37.0170i			1.20289i	0.798914	+	0.601445i	$0.205408\pi$
$947$			37.0170i			1.20289i	−0.798914	+	0.601445i	$0.794592\pi$
$948$		0			0
$949$			0.0136221i			0.000442192i
$950$		0			0
$951$	9.45180			0.306496
$952$		0			0
$953$	12.1133			0.392387			0.196193	−	0.980565i	$-0.437142\pi$
$953$	12.1133			0.392387			0.196193	+	0.980565i	$0.437142\pi$
$954$		0			0
$955$			18.1218i			0.586407i
$956$		0			0
$957$		−	2.60444i		−	0.0841895i
$958$		0			0
$959$		−	47.5679i		−	1.53605i
$960$		0			0
$961$	20.9472			0.675717
$962$		0			0
$963$		−	7.62433i		−	0.245690i
$964$		0			0
$965$	−33.7872			−1.08765
$966$		0			0
$967$	−20.6470			−0.663961			−0.331981	−	0.943286i	$-0.607717\pi$
$967$	−20.6470			−0.663961			−0.331981	+	0.943286i	$0.607717\pi$
$968$		0			0
$969$	2.41298	+	5.38783i	0.0775162	+	0.173082i
$970$		0			0
$971$	43.1334			1.38422			0.692108	−	0.721794i	$-0.256682\pi$
$971$	43.1334			1.38422			0.692108	+	0.721794i	$0.256682\pi$
$972$		0			0
$973$	53.1087			1.70259
$974$		0			0
$975$		−	31.7304i		−	1.01618i
$976$		0			0
$977$	20.9393			0.669906			0.334953	−	0.942235i	$-0.391280\pi$
$977$	20.9393			0.669906			0.334953	+	0.942235i	$0.391280\pi$
$978$		0			0
$979$			21.8158i			0.697236i
$980$		0			0
$981$			0.779978i			0.0249028i
$982$		0			0
$983$		−	24.8458i		−	0.792458i	−0.918152	−	0.396229i	$-0.870319\pi$
$983$		−	24.8458i		−	0.792458i	0.918152	−	0.396229i	$-0.129681\pi$
$984$		0			0
$985$	32.1003			1.02280
$986$		0			0
$987$	64.3974			2.04979
$988$		0			0
$989$		−	39.5262i		−	1.25686i
$990$		0			0
$991$			16.3177i			0.518350i	0.965830	+	0.259175i	$0.0834506\pi$
$991$			16.3177i			0.518350i	−0.965830	+	0.259175i	$0.916549\pi$
$992$		0			0
$993$			15.4709i			0.490953i
$994$		0			0
$995$	−67.1313			−2.12821
$996$		0			0
$997$			35.6608i			1.12939i	0.825300	+	0.564694i	$0.191006\pi$
$997$			35.6608i			1.12939i	−0.825300	+	0.564694i	$0.808994\pi$
$998$		0			0
$999$	−7.38432			−0.233630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.32	yes	46
17.16	even	2	inner	4012.2.b.b.237.15	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.15	✓	46	17.16	even	2	inner
4012.2.b.b.237.32	yes	46	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.45665i q^{3} +3.69601i q^{5} -3.33013i q^{7} +0.878183 q^{9} +O(q^{10})\)	\(q+1.45665i q^{3} +3.69601i q^{5} -3.33013i q^{7} +0.878183 q^{9} -4.75976i q^{11} +2.51524 q^{13} -5.38377 q^{15} +(-3.76296 + 1.68527i) q^{17} -0.982947 q^{19} +4.85082 q^{21} -4.02346i q^{23} -8.66046 q^{25} +5.64914i q^{27} -0.375643i q^{29} -3.17061i q^{31} +6.93328 q^{33} +12.3082 q^{35} +1.30716i q^{37} +3.66382i q^{39} -7.88667i q^{41} +9.82393 q^{43} +3.24577i q^{45} +13.2756 q^{47} -4.08976 q^{49} +(-2.45484 - 5.48130i) q^{51} +1.89912 q^{53} +17.5921 q^{55} -1.43181i q^{57} +1.00000 q^{59} -0.836969i q^{61} -2.92446i q^{63} +9.29635i q^{65} +1.30621 q^{67} +5.86076 q^{69} -10.8431i q^{71} +0.00541582i q^{73} -12.6152i q^{75} -15.8506 q^{77} +10.6850i q^{79} -5.59424 q^{81} +7.42159 q^{83} +(-6.22878 - 13.9079i) q^{85} +0.547179 q^{87} -4.58338 q^{89} -8.37608i q^{91} +4.61846 q^{93} -3.63298i q^{95} -10.4572i q^{97} -4.17994i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more