LMFDB - Embedded newform 4012.2.b.a.237.19

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.19
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.346659i q^{3} -2.94947i q^{5} -4.18117i q^{7} +2.87983 q^{9} +O(q^{10})$	$q-0.346659i q^{3} -2.94947i q^{5} -4.18117i q^{7} +2.87983 q^{9} -2.89035i q^{11} -3.88392 q^{13} -1.02246 q^{15} +(-3.65868 - 1.90106i) q^{17} -2.51034 q^{19} -1.44944 q^{21} +6.02484i q^{23} -3.69935 q^{25} -2.03829i q^{27} +2.25228i q^{29} -4.79291i q^{31} -1.00197 q^{33} -12.3322 q^{35} -7.22892i q^{37} +1.34639i q^{39} -1.10508i q^{41} -0.255876 q^{43} -8.49395i q^{45} -1.67852 q^{47} -10.4822 q^{49} +(-0.659021 + 1.26831i) q^{51} +3.70794 q^{53} -8.52499 q^{55} +0.870233i q^{57} -1.00000 q^{59} -4.76907i q^{61} -12.0410i q^{63} +11.4555i q^{65} +4.32764 q^{67} +2.08856 q^{69} +13.9504i q^{71} +8.35528i q^{73} +1.28241i q^{75} -12.0850 q^{77} +7.38519i q^{79} +7.93289 q^{81} +10.0409 q^{83} +(-5.60712 + 10.7912i) q^{85} +0.780772 q^{87} +4.00911 q^{89} +16.2393i q^{91} -1.66151 q^{93} +7.40417i q^{95} -6.67527i q^{97} -8.32371i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * 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
$3$		−	0.346659i		−	0.200144i	−0.994980	−	0.100072i	$-0.968093\pi$
$3$		−	0.346659i		−	0.200144i	0.994980	−	0.100072i	$-0.0319073\pi$
$4$		0			0
$5$		−	2.94947i		−	1.31904i	−0.751687	−	0.659521i	$-0.770759\pi$
$5$		−	2.94947i		−	1.31904i	0.751687	−	0.659521i	$-0.229241\pi$
$6$		0			0
$7$		−	4.18117i		−	1.58033i	−0.612893	−	0.790166i	$-0.709994\pi$
$7$		−	4.18117i		−	1.58033i	0.612893	−	0.790166i	$-0.290006\pi$
$8$		0			0
$9$	2.87983			0.959943
$10$		0			0
$11$		−	2.89035i		−	0.871473i	−0.900074	−	0.435737i	$-0.856488\pi$
$11$		−	2.89035i		−	0.871473i	0.900074	−	0.435737i	$-0.143512\pi$
$12$		0			0
$13$	−3.88392			−1.07720			−0.538602	−	0.842560i	$-0.681047\pi$
$13$	−3.88392			−1.07720			−0.538602	+	0.842560i	$0.681047\pi$
$14$		0			0
$15$	−1.02246			−0.263998
$16$		0			0
$17$	−3.65868	−	1.90106i	−0.887361	−	0.461076i
$17$	−3.65868	−	1.90106i	−0.887361	−	0.461076i
$18$		0			0
$19$	−2.51034			−0.575912			−0.287956	−	0.957644i	$-0.592976\pi$
$19$	−2.51034			−0.575912			−0.287956	+	0.957644i	$0.592976\pi$
$20$		0			0
$21$	−1.44944			−0.316293
$22$		0			0
$23$			6.02484i			1.25627i	0.778106	+	0.628133i	$0.216181\pi$
$23$			6.02484i			1.25627i	−0.778106	+	0.628133i	$0.783819\pi$
$24$		0			0
$25$	−3.69935			−0.739869
$26$		0			0
$27$		−	2.03829i		−	0.392270i
$28$		0			0
$29$			2.25228i			0.418237i	0.977890	+	0.209119i	$0.0670595\pi$
$29$			2.25228i			0.418237i	−0.977890	+	0.209119i	$0.932940\pi$
$30$		0			0
$31$		−	4.79291i		−	0.860833i	−0.902630	−	0.430416i	$-0.858367\pi$
$31$		−	4.79291i		−	0.860833i	0.902630	−	0.430416i	$-0.141633\pi$
$32$		0			0
$33$	−1.00197			−0.174420
$34$		0			0
$35$	−12.3322			−2.08452
$36$		0			0
$37$		−	7.22892i		−	1.18843i	−0.804307	−	0.594214i	$-0.797463\pi$
$37$		−	7.22892i		−	1.18843i	0.804307	−	0.594214i	$-0.202537\pi$
$38$		0			0
$39$			1.34639i			0.215596i
$40$		0			0
$41$		−	1.10508i		−	0.172585i	−0.996270	−	0.0862926i	$-0.972498\pi$
$41$		−	1.10508i		−	0.172585i	0.996270	−	0.0862926i	$-0.0275020\pi$
$42$		0			0
$43$	−0.255876			−0.0390208			−0.0195104	−	0.999810i	$-0.506211\pi$
$43$	−0.255876			−0.0390208			−0.0195104	+	0.999810i	$0.506211\pi$
$44$		0			0
$45$		−	8.49395i		−	1.26620i
$46$		0			0
$47$	−1.67852			−0.244837			−0.122419	−	0.992479i	$-0.539065\pi$
$47$	−1.67852			−0.244837			−0.122419	+	0.992479i	$0.539065\pi$
$48$		0			0
$49$	−10.4822			−1.49745
$50$		0			0
$51$	−0.659021	+	1.26831i	−0.0922814	+	0.177600i
$52$		0			0
$53$	3.70794			0.509324			0.254662	−	0.967030i	$-0.418036\pi$
$53$	3.70794			0.509324			0.254662	+	0.967030i	$0.418036\pi$
$54$		0			0
$55$	−8.52499			−1.14951
$56$		0			0
$57$			0.870233i			0.115265i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$		−	4.76907i		−	0.610617i	−0.952253	−	0.305309i	$-0.901240\pi$
$61$		−	4.76907i		−	0.610617i	0.952253	−	0.305309i	$-0.0987596\pi$
$62$		0			0
$63$		−	12.0410i		−	1.51703i
$64$		0			0
$65$			11.4555i			1.42088i
$66$		0			0
$67$	4.32764			0.528705			0.264353	−	0.964426i	$-0.414842\pi$
$67$	4.32764			0.528705			0.264353	+	0.964426i	$0.414842\pi$
$68$		0			0
$69$	2.08856			0.251434
$70$		0			0
$71$			13.9504i			1.65561i	0.561017	+	0.827804i	$0.310410\pi$
$71$			13.9504i			1.65561i	−0.561017	+	0.827804i	$0.689590\pi$
$72$		0			0
$73$			8.35528i			0.977912i	0.872308	+	0.488956i	$0.162622\pi$
$73$			8.35528i			0.977912i	−0.872308	+	0.488956i	$0.837378\pi$
$74$		0			0
$75$			1.28241i			0.148080i
$76$		0			0
$77$	−12.0850			−1.37722
$78$		0			0
$79$			7.38519i			0.830899i	0.909616	+	0.415449i	$0.136376\pi$
$79$			7.38519i			0.830899i	−0.909616	+	0.415449i	$0.863624\pi$
$80$		0			0
$81$	7.93289			0.881432
$82$		0			0
$83$	10.0409			1.10213			0.551066	−	0.834462i	$-0.314221\pi$
$83$	10.0409			1.10213			0.551066	+	0.834462i	$0.314221\pi$
$84$		0			0
$85$	−5.60712	+	10.7912i	−0.608178	+	1.17047i
$86$		0			0
$87$	0.780772			0.0837075
$88$		0			0
$89$	4.00911			0.424965			0.212482	−	0.977165i	$-0.431845\pi$
$89$	4.00911			0.424965			0.212482	+	0.977165i	$0.431845\pi$
$90$		0			0
$91$			16.2393i			1.70234i
$92$		0			0
$93$	−1.66151			−0.172290
$94$		0			0
$95$			7.40417i			0.759652i
$96$		0			0
$97$		−	6.67527i		−	0.677771i	−0.940828	−	0.338885i	$-0.889950\pi$
$97$		−	6.67527i		−	0.677771i	0.940828	−	0.338885i	$-0.110050\pi$
$98$		0			0
$99$		−	8.32371i		−	0.836564i
$100$		0			0
$101$	2.59148			0.257862			0.128931	−	0.991654i	$-0.458845\pi$
$101$	2.59148			0.257862			0.128931	+	0.991654i	$0.458845\pi$
$102$		0			0
$103$	7.31202			0.720475			0.360238	−	0.932861i	$-0.382696\pi$
$103$	7.31202			0.720475			0.360238	+	0.932861i	$0.382696\pi$
$104$		0			0
$105$			4.27507i			0.417204i
$106$		0			0
$107$		−	5.65407i		−	0.546599i	−0.961929	−	0.273300i	$-0.911885\pi$
$107$		−	5.65407i		−	0.546599i	0.961929	−	0.273300i	$-0.0881151\pi$
$108$		0			0
$109$		−	2.15368i		−	0.206285i	−0.994667	−	0.103143i	$-0.967110\pi$
$109$		−	2.15368i		−	0.206285i	0.994667	−	0.103143i	$-0.0328898\pi$
$110$		0			0
$111$	−2.50597			−0.237856
$112$		0			0
$113$		−	4.16386i		−	0.391703i	−0.980634	−	0.195851i	$-0.937253\pi$
$113$		−	4.16386i		−	0.391703i	0.980634	−	0.195851i	$-0.0627471\pi$
$114$		0			0
$115$	17.7701			1.65707
$116$		0			0
$117$	−11.1850			−1.03405
$118$		0			0
$119$	−7.94867	+	15.2976i	−0.728653	+	1.40232i
$120$		0			0
$121$	2.64588			0.240535
$122$		0			0
$123$	−0.383087			−0.0345418
$124$		0			0
$125$		−	3.83623i		−	0.343123i
$126$		0			0
$127$	2.72290			0.241618			0.120809	−	0.992676i	$-0.461451\pi$
$127$	2.72290			0.241618			0.120809	+	0.992676i	$0.461451\pi$
$128$		0			0
$129$			0.0887018i			0.00780976i
$130$		0			0
$131$		−	6.04289i		−	0.527970i	−0.964527	−	0.263985i	$-0.914963\pi$
$131$		−	6.04289i		−	0.527970i	0.964527	−	0.263985i	$-0.0850369\pi$
$132$		0			0
$133$			10.4962i			0.910133i
$134$		0			0
$135$	−6.01188			−0.517420
$136$		0			0
$137$	10.9665			0.936934			0.468467	−	0.883481i	$-0.344806\pi$
$137$	10.9665			0.936934			0.468467	+	0.883481i	$0.344806\pi$
$138$		0			0
$139$			14.1081i			1.19663i	0.801259	+	0.598317i	$0.204164\pi$
$139$			14.1081i			1.19663i	−0.801259	+	0.598317i	$0.795836\pi$
$140$		0			0
$141$			0.581874i			0.0490026i
$142$		0			0
$143$			11.2259i			0.938755i
$144$		0			0
$145$	6.64301			0.551672
$146$		0			0
$147$			3.63373i			0.299705i
$148$		0			0
$149$	13.4486			1.10175			0.550874	−	0.834588i	$-0.314294\pi$
$149$	13.4486			1.10175			0.550874	+	0.834588i	$0.314294\pi$
$150$		0			0
$151$	−18.7831			−1.52855			−0.764275	−	0.644891i	$-0.776903\pi$
$151$	−18.7831			−1.52855			−0.764275	+	0.644891i	$0.776903\pi$
$152$		0			0
$153$	−10.5364	−	5.47474i	−0.851815	−	0.442606i
$154$		0			0
$155$	−14.1365			−1.13547
$156$		0			0
$157$	9.60673			0.766700			0.383350	−	0.923603i	$-0.374770\pi$
$157$	9.60673			0.766700			0.383350	+	0.923603i	$0.374770\pi$
$158$		0			0
$159$		−	1.28539i		−	0.101938i
$160$		0			0
$161$	25.1909			1.98532
$162$		0			0
$163$			13.8944i			1.08829i	0.838991	+	0.544145i	$0.183146\pi$
$163$			13.8944i			1.08829i	−0.838991	+	0.544145i	$0.816854\pi$
$164$		0			0
$165$			2.95526i			0.230067i
$166$		0			0
$167$		−	24.0266i		−	1.85924i	−0.368524	−	0.929618i	$-0.620137\pi$
$167$		−	24.0266i		−	1.85924i	0.368524	−	0.929618i	$-0.379863\pi$
$168$		0			0
$169$	2.08481			0.160370
$170$		0			0
$171$	−7.22936			−0.552843
$172$		0			0
$173$			3.36199i			0.255607i	0.991799	+	0.127804i	$0.0407927\pi$
$173$			3.36199i			0.255607i	−0.991799	+	0.127804i	$0.959207\pi$
$174$		0			0
$175$			15.4676i			1.16924i
$176$		0			0
$177$			0.346659i			0.0260565i
$178$		0			0
$179$	−11.6683			−0.872133			−0.436066	−	0.899915i	$-0.643629\pi$
$179$	−11.6683			−0.872133			−0.436066	+	0.899915i	$0.643629\pi$
$180$		0			0
$181$		−	11.7961i		−	0.876795i	−0.898781	−	0.438398i	$-0.855546\pi$
$181$		−	11.7961i		−	0.876795i	0.898781	−	0.438398i	$-0.144454\pi$
$182$		0			0
$183$	−1.65324			−0.122211
$184$		0			0
$185$	−21.3215			−1.56758
$186$		0			0
$187$	−5.49474	+	10.5749i	−0.401815	+	0.773311i
$188$		0			0
$189$	−8.52245			−0.619917
$190$		0			0
$191$	−21.9459			−1.58795			−0.793973	−	0.607953i	$-0.791991\pi$
$191$	−21.9459			−1.58795			−0.793973	+	0.607953i	$0.791991\pi$
$192$		0			0
$193$			15.0012i			1.07981i	0.841725	+	0.539906i	$0.181540\pi$
$193$			15.0012i			1.07981i	−0.841725	+	0.539906i	$0.818460\pi$
$194$		0			0
$195$	3.97114			0.284380
$196$		0			0
$197$		−	10.3526i		−	0.737592i	−0.929510	−	0.368796i	$-0.879770\pi$
$197$		−	10.3526i		−	0.737592i	0.929510	−	0.368796i	$-0.120230\pi$
$198$		0			0
$199$		−	11.1868i		−	0.793009i	−0.918033	−	0.396504i	$-0.870223\pi$
$199$		−	11.1868i		−	0.793009i	0.918033	−	0.396504i	$-0.129777\pi$
$200$		0			0
$201$		−	1.50021i		−	0.105817i
$202$		0			0
$203$	9.41715			0.660954
$204$		0			0
$205$	−3.25941			−0.227647
$206$		0			0
$207$			17.3505i			1.20594i
$208$		0			0
$209$			7.25577i			0.501892i
$210$		0			0
$211$		−	4.71267i		−	0.324434i	−0.986755	−	0.162217i	$-0.948136\pi$
$211$		−	4.71267i		−	0.324434i	0.986755	−	0.162217i	$-0.0518644\pi$
$212$		0			0
$213$	4.83603			0.331359
$214$		0			0
$215$			0.754699i			0.0514700i
$216$		0			0
$217$	−20.0400			−1.36040
$218$		0			0
$219$	2.89643			0.195723
$220$		0			0
$221$	14.2100	+	7.38358i	0.955869	+	0.496673i
$222$		0			0
$223$	−13.1723			−0.882081			−0.441040	−	0.897487i	$-0.645390\pi$
$223$	−13.1723			−0.882081			−0.441040	+	0.897487i	$0.645390\pi$
$224$		0			0
$225$	−10.6535			−0.710232
$226$		0			0
$227$		−	4.43341i		−	0.294256i	−0.989117	−	0.147128i	$-0.952997\pi$
$227$		−	4.43341i		−	0.294256i	0.989117	−	0.147128i	$-0.0470029\pi$
$228$		0			0
$229$	5.35060			0.353578			0.176789	−	0.984249i	$-0.443429\pi$
$229$	5.35060			0.353578			0.176789	+	0.984249i	$0.443429\pi$
$230$		0			0
$231$			4.18938i			0.275641i
$232$		0			0
$233$			9.41883i			0.617048i	0.951217	+	0.308524i	$0.0998350\pi$
$233$			9.41883i			0.617048i	−0.951217	+	0.308524i	$0.900165\pi$
$234$		0			0
$235$			4.95074i			0.322951i
$236$		0			0
$237$	2.56014			0.166299
$238$		0			0
$239$	−5.01209			−0.324205			−0.162102	−	0.986774i	$-0.551827\pi$
$239$	−5.01209			−0.324205			−0.162102	+	0.986774i	$0.551827\pi$
$240$		0			0
$241$			15.1134i			0.973540i	0.873530	+	0.486770i	$0.161825\pi$
$241$			15.1134i			0.973540i	−0.873530	+	0.486770i	$0.838175\pi$
$242$		0			0
$243$		−	8.86489i		−	0.568683i
$244$		0			0
$245$			30.9168i			1.97520i
$246$		0			0
$247$	9.74997			0.620375
$248$		0			0
$249$		−	3.48077i		−	0.220585i
$250$		0			0
$251$	−23.4894			−1.48264			−0.741318	−	0.671154i	$-0.765799\pi$
$251$	−23.4894			−1.48264			−0.741318	+	0.671154i	$0.765799\pi$
$252$		0			0
$253$	17.4139			1.09480
$254$		0			0
$255$	3.74085	+	1.94376i	0.234261	+	0.121723i
$256$		0			0
$257$	−25.9200			−1.61685			−0.808423	−	0.588602i	$-0.799679\pi$
$257$	−25.9200			−1.61685			−0.808423	+	0.588602i	$0.799679\pi$
$258$		0			0
$259$	−30.2253			−1.87811
$260$		0			0
$261$			6.48617i			0.401484i
$262$		0			0
$263$	6.00599			0.370345			0.185173	−	0.982706i	$-0.440716\pi$
$263$	6.00599			0.370345			0.185173	+	0.982706i	$0.440716\pi$
$264$		0			0
$265$		−	10.9364i		−	0.671820i
$266$		0			0
$267$		−	1.38979i		−	0.0850540i
$268$		0			0
$269$		−	19.3456i		−	1.17952i	−0.807579	−	0.589760i	$-0.799223\pi$
$269$		−	19.3456i		−	1.17952i	0.807579	−	0.589760i	$-0.200777\pi$
$270$		0			0
$271$	−9.80891			−0.595849			−0.297924	−	0.954589i	$-0.596294\pi$
$271$	−9.80891			−0.595849			−0.297924	+	0.954589i	$0.596294\pi$
$272$		0			0
$273$	5.62950			0.340713
$274$		0			0
$275$			10.6924i			0.644776i
$276$		0			0
$277$		−	1.96044i		−	0.117791i	−0.998264	−	0.0588956i	$-0.981242\pi$
$277$		−	1.96044i		−	0.117791i	0.998264	−	0.0588956i	$-0.0187579\pi$
$278$		0			0
$279$		−	13.8028i		−	0.826350i
$280$		0			0
$281$	−23.2485			−1.38689			−0.693444	−	0.720511i	$-0.743907\pi$
$281$	−23.2485			−1.38689			−0.693444	+	0.720511i	$0.743907\pi$
$282$		0			0
$283$			17.6632i			1.04997i	0.851112	+	0.524984i	$0.175929\pi$
$283$			17.6632i			1.04997i	−0.851112	+	0.524984i	$0.824071\pi$
$284$		0			0
$285$	2.56672			0.152039
$286$		0			0
$287$	−4.62054			−0.272742
$288$		0			0
$289$	9.77191	+	13.9108i	0.574818	+	0.818281i
$290$		0			0
$291$	−2.31404			−0.135651
$292$		0			0
$293$	28.9633			1.69205			0.846027	−	0.533140i	$-0.178988\pi$
$293$	28.9633			1.69205			0.846027	+	0.533140i	$0.178988\pi$
$294$		0			0
$295$			2.94947i			0.171725i
$296$		0			0
$297$	−5.89138			−0.341853
$298$		0			0
$299$		−	23.4000i		−	1.35326i
$300$		0			0
$301$			1.06986i			0.0616658i
$302$		0			0
$303$		−	0.898359i		−	0.0516094i
$304$		0			0
$305$	−14.0662			−0.805429
$306$		0			0
$307$	−7.06072			−0.402977			−0.201488	−	0.979491i	$-0.564578\pi$
$307$	−7.06072			−0.402977			−0.201488	+	0.979491i	$0.564578\pi$
$308$		0			0
$309$		−	2.53478i		−	0.144199i
$310$		0			0
$311$			4.46316i			0.253083i	0.991961	+	0.126541i	$0.0403876\pi$
$311$			4.46316i			0.253083i	−0.991961	+	0.126541i	$0.959612\pi$
$312$		0			0
$313$		−	8.61716i		−	0.487071i	−0.969892	−	0.243535i	$-0.921693\pi$
$313$		−	8.61716i		−	0.487071i	0.969892	−	0.243535i	$-0.0783072\pi$
$314$		0			0
$315$	−35.5146			−2.00102
$316$		0			0
$317$		−	16.3043i		−	0.915738i	−0.889020	−	0.457869i	$-0.848613\pi$
$317$		−	16.3043i		−	0.915738i	0.889020	−	0.457869i	$-0.151387\pi$
$318$		0			0
$319$	6.50987			0.364483
$320$		0			0
$321$	−1.96003			−0.109398
$322$		0			0
$323$	9.18455	+	4.77233i	0.511042	+	0.265539i
$324$		0			0
$325$	14.3680			0.796991
$326$		0			0
$327$	−0.746592			−0.0412866
$328$		0			0
$329$			7.01817i			0.386924i
$330$		0			0
$331$	−5.65588			−0.310875			−0.155438	−	0.987846i	$-0.549679\pi$
$331$	−5.65588			−0.310875			−0.155438	+	0.987846i	$0.549679\pi$
$332$		0			0
$333$		−	20.8181i		−	1.14082i
$334$		0			0
$335$		−	12.7642i		−	0.697384i
$336$		0			0
$337$		−	25.6162i		−	1.39540i	−0.716388	−	0.697702i	$-0.754206\pi$
$337$		−	25.6162i		−	1.39540i	0.716388	−	0.697702i	$-0.245794\pi$
$338$		0			0
$339$	−1.44344			−0.0783968
$340$		0			0
$341$	−13.8532			−0.750193
$342$		0			0
$343$			14.5595i			0.786137i
$344$		0			0
$345$		−	6.16015i		−	0.331651i
$346$		0			0
$347$		−	15.9691i		−	0.857264i	−0.903479	−	0.428632i	$-0.858996\pi$
$347$		−	15.9691i		−	0.857264i	0.903479	−	0.428632i	$-0.141004\pi$
$348$		0			0
$349$	−4.13978			−0.221597			−0.110799	−	0.993843i	$-0.535341\pi$
$349$	−4.13978			−0.221597			−0.110799	+	0.993843i	$0.535341\pi$
$350$		0			0
$351$			7.91657i			0.422555i
$352$		0			0
$353$	9.43290			0.502063			0.251031	−	0.967979i	$-0.419230\pi$
$353$	9.43290			0.502063			0.251031	+	0.967979i	$0.419230\pi$
$354$		0			0
$355$	41.1462			2.18382
$356$		0			0
$357$	5.30303	+	2.75548i	0.280666	+	0.145835i
$358$		0			0
$359$	−24.6933			−1.30326			−0.651630	−	0.758537i	$-0.725915\pi$
$359$	−24.6933			−1.30326			−0.651630	+	0.758537i	$0.725915\pi$
$360$		0			0
$361$	−12.6982			−0.668325
$362$		0			0
$363$		−	0.917219i		−	0.0481415i
$364$		0			0
$365$	24.6436			1.28991
$366$		0			0
$367$		−	28.2863i		−	1.47653i	−0.674511	−	0.738265i	$-0.735645\pi$
$367$		−	28.2863i		−	1.47653i	0.674511	−	0.738265i	$-0.264355\pi$
$368$		0			0
$369$		−	3.18245i		−	0.165672i
$370$		0			0
$371$		−	15.5035i		−	0.804902i
$372$		0			0
$373$	−19.7517			−1.02271			−0.511353	−	0.859371i	$-0.670856\pi$
$373$	−19.7517			−1.02271			−0.511353	+	0.859371i	$0.670856\pi$
$374$		0			0
$375$	−1.32986			−0.0686738
$376$		0			0
$377$		−	8.74766i		−	0.450527i
$378$		0			0
$379$		−	0.339107i		−	0.0174188i	−0.999962	−	0.00870938i	$-0.997228\pi$
$379$		−	0.339107i		−	0.0174188i	0.999962	−	0.00870938i	$-0.00277232\pi$
$380$		0			0
$381$		−	0.943918i		−	0.0483584i
$382$		0			0
$383$	32.4911			1.66022			0.830109	−	0.557601i	$-0.188278\pi$
$383$	32.4911			1.66022			0.830109	+	0.557601i	$0.188278\pi$
$384$		0			0
$385$			35.6444i			1.81661i
$386$		0			0
$387$	−0.736880			−0.0374577
$388$		0			0
$389$	12.0533			0.611129			0.305564	−	0.952171i	$-0.401155\pi$
$389$	12.0533			0.611129			0.305564	+	0.952171i	$0.401155\pi$
$390$		0			0
$391$	11.4536	−	22.0430i	0.579234	−	1.11476i
$392$		0			0
$393$	−2.09482			−0.105670
$394$		0			0
$395$	21.7824			1.09599
$396$		0			0
$397$		−	2.72746i		−	0.136887i	−0.997655	−	0.0684436i	$-0.978197\pi$
$397$		−	2.72746i		−	0.136887i	0.997655	−	0.0684436i	$-0.0218033\pi$
$398$		0			0
$399$	3.63859			0.182157
$400$		0			0
$401$			3.88343i			0.193929i	0.995288	+	0.0969645i	$0.0309133\pi$
$401$			3.88343i			0.193929i	−0.995288	+	0.0969645i	$0.969087\pi$
$402$		0			0
$403$			18.6153i			0.927293i
$404$		0			0
$405$		−	23.3978i		−	1.16265i
$406$		0			0
$407$	−20.8941			−1.03568
$408$		0			0
$409$	−16.8134			−0.831367			−0.415683	−	0.909509i	$-0.636458\pi$
$409$	−16.8134			−0.831367			−0.415683	+	0.909509i	$0.636458\pi$
$410$		0			0
$411$		−	3.80165i		−	0.187521i
$412$		0			0
$413$			4.18117i			0.205742i
$414$		0			0
$415$		−	29.6153i		−	1.45376i
$416$		0			0
$417$	4.89070			0.239499
$418$		0			0
$419$		−	11.8451i		−	0.578670i	−0.957228	−	0.289335i	$-0.906566\pi$
$419$		−	11.8451i		−	0.578670i	0.957228	−	0.289335i	$-0.0934341\pi$
$420$		0			0
$421$	−2.09529			−0.102118			−0.0510592	−	0.998696i	$-0.516260\pi$
$421$	−2.09529			−0.102118			−0.0510592	+	0.998696i	$0.516260\pi$
$422$		0			0
$423$	−4.83385			−0.235030
$424$		0			0
$425$	13.5347	+	7.03270i	0.656531	+	0.341136i
$426$		0			0
$427$	−19.9403			−0.964978
$428$		0			0
$429$	3.89155			0.187886
$430$		0			0
$431$			26.5166i			1.27726i	0.769514	+	0.638630i	$0.220499\pi$
$431$			26.5166i			1.27726i	−0.769514	+	0.638630i	$0.779501\pi$
$432$		0			0
$433$	29.2437			1.40536			0.702682	−	0.711504i	$-0.251986\pi$
$433$	29.2437			1.40536			0.702682	+	0.711504i	$0.251986\pi$
$434$		0			0
$435$		−	2.30286i		−	0.110414i
$436$		0			0
$437$		−	15.1244i		−	0.723499i
$438$		0			0
$439$			8.84428i			0.422114i	0.977474	+	0.211057i	$0.0676906\pi$
$439$			8.84428i			0.422114i	−0.977474	+	0.211057i	$0.932309\pi$
$440$		0			0
$441$	−30.1868			−1.43747
$442$		0			0
$443$	−17.1379			−0.814248			−0.407124	−	0.913373i	$-0.633468\pi$
$443$	−17.1379			−0.814248			−0.407124	+	0.913373i	$0.633468\pi$
$444$		0			0
$445$		−	11.8247i		−	0.560546i
$446$		0			0
$447$		−	4.66206i		−	0.220508i
$448$		0			0
$449$			30.9227i			1.45933i	0.683804	+	0.729665i	$0.260324\pi$
$449$			30.9227i			1.45933i	−0.683804	+	0.729665i	$0.739676\pi$
$450$		0			0
$451$	−3.19408			−0.150403
$452$		0			0
$453$			6.51134i			0.305930i
$454$		0			0
$455$	47.8973			2.24546
$456$		0			0
$457$	−21.5020			−1.00582			−0.502911	−	0.864338i	$-0.667738\pi$
$457$	−21.5020			−1.00582			−0.502911	+	0.864338i	$0.667738\pi$
$458$		0			0
$459$	−3.87493	+	7.45747i	−0.180866	+	0.348085i
$460$		0			0
$461$	5.64103			0.262729			0.131365	−	0.991334i	$-0.458064\pi$
$461$	5.64103			0.262729			0.131365	+	0.991334i	$0.458064\pi$
$462$		0			0
$463$	−11.9853			−0.557004			−0.278502	−	0.960436i	$-0.589838\pi$
$463$	−11.9853			−0.557004			−0.278502	+	0.960436i	$0.589838\pi$
$464$		0			0
$465$			4.90056i			0.227258i
$466$		0			0
$467$	3.12332			0.144530			0.0722651	−	0.997385i	$-0.476977\pi$
$467$	3.12332			0.144530			0.0722651	+	0.997385i	$0.476977\pi$
$468$		0			0
$469$		−	18.0946i		−	0.835530i
$470$		0			0
$471$		−	3.33026i		−	0.153450i
$472$		0			0
$473$			0.739572i			0.0340056i
$474$		0			0
$475$	9.28663			0.426100
$476$		0			0
$477$	10.6782			0.488922
$478$		0			0
$479$		−	9.69817i		−	0.443121i	−0.975147	−	0.221560i	$-0.928885\pi$
$479$		−	9.69817i		−	0.443121i	0.975147	−	0.221560i	$-0.0711150\pi$
$480$		0			0
$481$			28.0765i			1.28018i
$482$		0			0
$483$		−	8.73263i		−	0.397349i
$484$		0			0
$485$	−19.6885			−0.894008
$486$		0			0
$487$		−	11.2051i		−	0.507754i	−0.967237	−	0.253877i	$-0.918294\pi$
$487$		−	11.2051i		−	0.507754i	0.967237	−	0.253877i	$-0.0817058\pi$
$488$		0			0
$489$	4.81660			0.217814
$490$		0			0
$491$	−10.2964			−0.464670			−0.232335	−	0.972636i	$-0.574637\pi$
$491$	−10.2964			−0.464670			−0.232335	+	0.972636i	$0.574637\pi$
$492$		0			0
$493$	4.28172	−	8.24037i	0.192839	−	0.371127i
$494$		0			0
$495$	−24.5505			−1.10346
$496$		0			0
$497$	58.3290			2.61641
$498$		0			0
$499$			39.1342i			1.75189i	0.482412	+	0.875944i	$0.339761\pi$
$499$			39.1342i			1.75189i	−0.482412	+	0.875944i	$0.660239\pi$
$500$		0			0
$501$	−8.32904			−0.372114
$502$		0			0
$503$		−	41.4526i		−	1.84828i	−0.382053	−	0.924140i	$-0.624783\pi$
$503$		−	41.4526i		−	1.84828i	0.382053	−	0.924140i	$-0.375217\pi$
$504$		0			0
$505$		−	7.64347i		−	0.340130i
$506$		0			0
$507$		−	0.722718i		−	0.0320970i
$508$		0			0
$509$	30.3369			1.34466			0.672329	−	0.740252i	$-0.265294\pi$
$509$	30.3369			1.34466			0.672329	+	0.740252i	$0.265294\pi$
$510$		0			0
$511$	34.9348			1.54543
$512$		0			0
$513$			5.11682i			0.225913i
$514$		0			0
$515$		−	21.5666i		−	0.950336i
$516$		0			0
$517$			4.85151i			0.213369i
$518$		0			0
$519$	1.16546			0.0511581
$520$		0			0
$521$			24.2476i			1.06231i	0.847276	+	0.531153i	$0.178241\pi$
$521$			24.2476i			1.06231i	−0.847276	+	0.531153i	$0.821759\pi$
$522$		0			0
$523$	−44.8200			−1.95984			−0.979920	−	0.199392i	$-0.936103\pi$
$523$	−44.8200			−1.95984			−0.979920	+	0.199392i	$0.936103\pi$
$524$		0			0
$525$	5.36198			0.234016
$526$		0			0
$527$	−9.11164	+	17.5357i	−0.396909	+	0.763869i
$528$		0			0
$529$	−13.2987			−0.578204
$530$		0			0
$531$	−2.87983			−0.124974
$532$		0			0
$533$			4.29206i			0.185910i
$534$		0			0
$535$	−16.6765			−0.720987
$536$		0			0
$537$			4.04493i			0.174552i
$538$		0			0
$539$			30.2971i			1.30499i
$540$		0			0
$541$		−	5.54946i		−	0.238590i	−0.992859	−	0.119295i	$-0.961937\pi$
$541$		−	5.54946i		−	0.238590i	0.992859	−	0.119295i	$-0.0380634\pi$
$542$		0			0
$543$	−4.08921			−0.175485
$544$		0			0
$545$	−6.35220			−0.272098
$546$		0			0
$547$			22.2415i			0.950979i	0.879721	+	0.475489i	$0.157729\pi$
$547$			22.2415i			0.950979i	−0.879721	+	0.475489i	$0.842271\pi$
$548$		0			0
$549$		−	13.7341i		−	0.586157i
$550$		0			0
$551$		−	5.65399i		−	0.240868i
$552$		0			0
$553$	30.8787			1.31310
$554$		0			0
$555$			7.39127i			0.313742i
$556$		0			0
$557$	42.9254			1.81881			0.909405	−	0.415912i	$-0.136538\pi$
$557$	42.9254			1.81881			0.909405	+	0.415912i	$0.136538\pi$
$558$		0			0
$559$	0.993803			0.0420334
$560$		0			0
$561$	3.66587	+	1.90480i	0.154773	+	0.0804207i
$562$		0			0
$563$	16.2781			0.686039			0.343020	−	0.939328i	$-0.388550\pi$
$563$	16.2781			0.686039			0.343020	+	0.939328i	$0.388550\pi$
$564$		0			0
$565$	−12.2812			−0.516672
$566$		0			0
$567$		−	33.1687i		−	1.39296i
$568$		0			0
$569$	13.6412			0.571868			0.285934	−	0.958249i	$-0.407696\pi$
$569$	13.6412			0.571868			0.285934	+	0.958249i	$0.407696\pi$
$570$		0			0
$571$			0.837358i			0.0350423i	0.999846	+	0.0175212i	$0.00557744\pi$
$571$			0.837358i			0.0350423i	−0.999846	+	0.0175212i	$0.994423\pi$
$572$		0			0
$573$			7.60773i			0.317817i
$574$		0			0
$575$		−	22.2880i		−	0.929473i
$576$		0			0
$577$	16.8630			0.702014			0.351007	−	0.936373i	$-0.385839\pi$
$577$	16.8630			0.702014			0.351007	+	0.936373i	$0.385839\pi$
$578$		0			0
$579$	5.20031			0.216118
$580$		0			0
$581$		−	41.9827i		−	1.74173i
$582$		0			0
$583$		−	10.7172i		−	0.443862i
$584$		0			0
$585$			32.9898i			1.36396i
$586$		0			0
$587$	22.2431			0.918071			0.459036	−	0.888418i	$-0.348195\pi$
$587$	22.2431			0.918071			0.459036	+	0.888418i	$0.348195\pi$
$588$		0			0
$589$			12.0319i			0.495764i
$590$		0			0
$591$	−3.58882			−0.147624
$592$		0			0
$593$	−3.19377			−0.131152			−0.0655762	−	0.997848i	$-0.520889\pi$
$593$	−3.19377			−0.131152			−0.0655762	+	0.997848i	$0.520889\pi$
$594$		0			0
$595$	45.1196	+	23.4443i	1.84972	+	0.961123i
$596$		0			0
$597$	−3.87799			−0.158716
$598$		0			0
$599$	27.0469			1.10511			0.552553	−	0.833478i	$-0.313654\pi$
$599$	27.0469			1.10511			0.552553	+	0.833478i	$0.313654\pi$
$600$		0			0
$601$			12.2487i			0.499637i	0.968293	+	0.249818i	$0.0803709\pi$
$601$			12.2487i			0.499637i	−0.968293	+	0.249818i	$0.919629\pi$
$602$		0			0
$603$	12.4628			0.507526
$604$		0			0
$605$		−	7.80394i		−	0.317275i
$606$		0			0
$607$		−	7.04526i		−	0.285958i	−0.989726	−	0.142979i	$-0.954332\pi$
$607$		−	7.04526i		−	0.285958i	0.989726	−	0.142979i	$-0.0456682\pi$
$608$		0			0
$609$		−	3.26454i		−	0.132286i
$610$		0			0
$611$	6.51923			0.263740
$612$		0			0
$613$	−39.8578			−1.60984			−0.804920	−	0.593383i	$-0.797792\pi$
$613$	−39.8578			−1.60984			−0.804920	+	0.593383i	$0.797792\pi$
$614$		0			0
$615$			1.12990i			0.0455621i
$616$		0			0
$617$			28.3456i			1.14115i	0.821245	+	0.570576i	$0.193280\pi$
$617$			28.3456i			1.14115i	−0.821245	+	0.570576i	$0.806720\pi$
$618$		0			0
$619$			20.0983i			0.807820i	0.914799	+	0.403910i	$0.132349\pi$
$619$			20.0983i			0.807820i	−0.914799	+	0.403910i	$0.867651\pi$
$620$		0			0
$621$	12.2804			0.492795
$622$		0			0
$623$		−	16.7628i		−	0.671586i
$624$		0			0
$625$	−29.8116			−1.19246
$626$		0			0
$627$	2.51528			0.100450
$628$		0			0
$629$	−13.7426	+	26.4483i	−0.547955	+	1.05456i
$630$		0			0
$631$	−11.1714			−0.444728			−0.222364	−	0.974964i	$-0.571377\pi$
$631$	−11.1714			−0.444728			−0.222364	+	0.974964i	$0.571377\pi$
$632$		0			0
$633$	−1.63369			−0.0649333
$634$		0			0
$635$		−	8.03110i		−	0.318705i
$636$		0			0
$637$	40.7118			1.61306
$638$		0			0
$639$			40.1748i			1.58929i
$640$		0			0
$641$		−	24.6014i		−	0.971699i	−0.874043	−	0.485849i	$-0.838510\pi$
$641$		−	24.6014i		−	0.971699i	0.874043	−	0.485849i	$-0.161490\pi$
$642$		0			0
$643$			4.75279i			0.187432i	0.995599	+	0.0937160i	$0.0298746\pi$
$643$			4.75279i			0.187432i	−0.995599	+	0.0937160i	$0.970125\pi$
$644$		0			0
$645$	0.261623			0.0103014
$646$		0			0
$647$	−45.5624			−1.79124			−0.895621	−	0.444819i	$-0.853268\pi$
$647$	−45.5624			−1.79124			−0.895621	+	0.444819i	$0.853268\pi$
$648$		0			0
$649$			2.89035i			0.113456i
$650$		0			0
$651$			6.94703i			0.272276i
$652$		0			0
$653$		−	17.8204i		−	0.697366i	−0.937241	−	0.348683i	$-0.886629\pi$
$653$		−	17.8204i		−	0.697366i	0.937241	−	0.348683i	$-0.113371\pi$
$654$		0			0
$655$	−17.8233			−0.696414
$656$		0			0
$657$			24.0618i			0.938739i
$658$		0			0
$659$	−34.5301			−1.34510			−0.672550	−	0.740052i	$-0.734801\pi$
$659$	−34.5301			−1.34510			−0.672550	+	0.740052i	$0.734801\pi$
$660$		0			0
$661$	−19.2934			−0.750428			−0.375214	−	0.926938i	$-0.622431\pi$
$661$	−19.2934			−0.750428			−0.375214	+	0.926938i	$0.622431\pi$
$662$		0			0
$663$	2.55958	−	4.92603i	0.0994059	−	0.191311i
$664$		0			0
$665$	30.9581			1.20050
$666$		0			0
$667$	−13.5696			−0.525417
$668$		0			0
$669$			4.56629i			0.176543i
$670$		0			0
$671$	−13.7843			−0.532136
$672$		0			0
$673$		−	45.8597i		−	1.76776i	−0.467711	−	0.883882i	$-0.654921\pi$
$673$		−	45.8597i		−	1.76776i	0.467711	−	0.883882i	$-0.345079\pi$
$674$		0			0
$675$			7.54036i			0.290229i
$676$		0			0
$677$		−	31.4025i		−	1.20690i	−0.797402	−	0.603448i	$-0.793793\pi$
$677$		−	31.4025i		−	1.20690i	0.797402	−	0.603448i	$-0.206207\pi$
$678$		0			0
$679$	−27.9104			−1.07110
$680$		0			0
$681$	−1.53688			−0.0588934
$682$		0			0
$683$		−	28.7357i		−	1.09954i	−0.835316	−	0.549770i	$-0.814715\pi$
$683$		−	28.7357i		−	1.09954i	0.835316	−	0.549770i	$-0.185285\pi$
$684$		0			0
$685$		−	32.3454i		−	1.23585i
$686$		0			0
$687$		−	1.85483i		−	0.0707663i
$688$		0			0
$689$	−14.4013			−0.548647
$690$		0			0
$691$			20.1474i			0.766444i	0.923656	+	0.383222i	$0.125186\pi$
$691$			20.1474i			0.766444i	−0.923656	+	0.383222i	$0.874814\pi$
$692$		0			0
$693$	−34.8028			−1.32205
$694$		0			0
$695$	41.6114			1.57841
$696$		0			0
$697$	−2.10084	+	4.04315i	−0.0795749	+	0.153145i
$698$		0			0
$699$	3.26512			0.123498
$700$		0			0
$701$	6.21763			0.234837			0.117418	−	0.993083i	$-0.462538\pi$
$701$	6.21763			0.234837			0.117418	+	0.993083i	$0.462538\pi$
$702$		0			0
$703$			18.1471i			0.684430i
$704$		0			0
$705$	1.71622			0.0646365
$706$		0			0
$707$		−	10.8354i		−	0.407507i
$708$		0			0
$709$		−	40.0594i		−	1.50446i	−0.658900	−	0.752231i	$-0.728978\pi$
$709$		−	40.0594i		−	1.50446i	0.658900	−	0.752231i	$-0.271022\pi$
$710$		0			0
$711$			21.2681i			0.797615i
$712$		0			0
$713$	28.8765			1.08143
$714$		0			0
$715$	33.1103			1.23826
$716$		0			0
$717$			1.73748i			0.0648875i
$718$		0			0
$719$			36.3046i			1.35393i	0.736013	+	0.676967i	$0.236706\pi$
$719$			36.3046i			1.35393i	−0.736013	+	0.676967i	$0.763294\pi$
$720$		0			0
$721$		−	30.5728i		−	1.13859i
$722$		0			0
$723$	5.23920			0.194848
$724$		0			0
$725$		−	8.33196i		−	0.309441i
$726$		0			0
$727$	−8.49373			−0.315015			−0.157507	−	0.987518i	$-0.550346\pi$
$727$	−8.49373			−0.315015			−0.157507	+	0.987518i	$0.550346\pi$
$728$		0			0
$729$	20.7256			0.767614
$730$		0			0
$731$	0.936170	+	0.486437i	0.0346255	+	0.0179915i
$732$		0			0
$733$	−25.4341			−0.939430			−0.469715	−	0.882818i	$-0.655643\pi$
$733$	−25.4341			−0.939430			−0.469715	+	0.882818i	$0.655643\pi$
$734$		0			0
$735$	10.7176			0.395323
$736$		0			0
$737$		−	12.5084i		−	0.460752i
$738$		0			0
$739$	−1.41780			−0.0521547			−0.0260774	−	0.999660i	$-0.508302\pi$
$739$	−1.41780			−0.0521547			−0.0260774	+	0.999660i	$0.508302\pi$
$740$		0			0
$741$		−	3.37991i		−	0.124164i
$742$		0			0
$743$			12.1454i			0.445570i	0.974868	+	0.222785i	$0.0715147\pi$
$743$			12.1454i			0.445570i	−0.974868	+	0.222785i	$0.928485\pi$
$744$		0			0
$745$		−	39.6661i		−	1.45325i
$746$		0			0
$747$	28.9161			1.05798
$748$		0			0
$749$	−23.6406			−0.863809
$750$		0			0
$751$			25.2379i			0.920944i	0.887674	+	0.460472i	$0.152320\pi$
$751$			25.2379i			0.920944i	−0.887674	+	0.460472i	$0.847680\pi$
$752$		0			0
$753$			8.14280i			0.296740i
$754$		0			0
$755$			55.4002i			2.01622i
$756$		0			0
$757$	41.1756			1.49655			0.748277	−	0.663387i	$-0.230882\pi$
$757$	41.1756			1.49655			0.748277	+	0.663387i	$0.230882\pi$
$758$		0			0
$759$		−	6.03668i		−	0.219118i
$760$		0			0
$761$	12.4046			0.449666			0.224833	−	0.974397i	$-0.427816\pi$
$761$	12.4046			0.449666			0.224833	+	0.974397i	$0.427816\pi$
$762$		0			0
$763$	−9.00489			−0.325999
$764$		0			0
$765$	−16.1476	+	31.0767i	−0.583816	+	1.12358i
$766$		0			0
$767$	3.88392			0.140240
$768$		0			0
$769$	8.96779			0.323387			0.161693	−	0.986841i	$-0.448304\pi$
$769$	8.96779			0.323387			0.161693	+	0.986841i	$0.448304\pi$
$770$		0			0
$771$			8.98540i			0.323601i
$772$		0			0
$773$	26.6787			0.959566			0.479783	−	0.877387i	$-0.340715\pi$
$773$	26.6787			0.959566			0.479783	+	0.877387i	$0.340715\pi$
$774$		0			0
$775$			17.7307i			0.636904i
$776$		0			0
$777$			10.4779i			0.375892i
$778$		0			0
$779$			2.77414i			0.0993940i
$780$		0			0
$781$	40.3215			1.44282
$782$		0			0
$783$	4.59080			0.164062
$784$		0			0
$785$		−	28.3347i		−	1.01131i
$786$		0			0
$787$			43.7841i			1.56074i	0.625320	+	0.780368i	$0.284968\pi$
$787$			43.7841i			1.56074i	−0.625320	+	0.780368i	$0.715032\pi$
$788$		0			0
$789$		−	2.08203i		−	0.0741222i
$790$		0			0
$791$	−17.4098			−0.619021
$792$		0			0
$793$			18.5227i			0.657760i
$794$		0			0
$795$	−3.79121			−0.134460
$796$		0			0
$797$	−8.69433			−0.307969			−0.153985	−	0.988073i	$-0.549211\pi$
$797$	−8.69433			−0.307969			−0.153985	+	0.988073i	$0.549211\pi$
$798$		0			0
$799$	6.14117	+	3.19098i	0.217259	+	0.112889i
$800$		0			0
$801$	11.5456			0.407942
$802$		0			0
$803$	24.1497			0.852224
$804$		0			0
$805$		−	74.2996i		−	2.61872i
$806$		0			0
$807$	−6.70631			−0.236073
$808$		0			0
$809$		−	19.0585i		−	0.670060i	−0.942208	−	0.335030i	$-0.891254\pi$
$809$		−	19.0585i		−	0.670060i	0.942208	−	0.335030i	$-0.108746\pi$
$810$		0			0
$811$		−	1.76441i		−	0.0619570i	−0.999520	−	0.0309785i	$-0.990138\pi$
$811$		−	1.76441i		−	0.0619570i	0.999520	−	0.0309785i	$-0.00986234\pi$
$812$		0			0
$813$			3.40035i			0.119255i
$814$		0			0
$815$	40.9809			1.43550
$816$		0			0
$817$	0.642338			0.0224725
$818$		0			0
$819$			46.7664i			1.63415i
$820$		0			0
$821$			15.6370i			0.545734i	0.962052	+	0.272867i	$0.0879719\pi$
$821$			15.6370i			0.545734i	−0.962052	+	0.272867i	$0.912028\pi$
$822$		0			0
$823$		−	25.9618i		−	0.904970i	−0.891772	−	0.452485i	$-0.850538\pi$
$823$		−	25.9618i		−	0.904970i	0.891772	−	0.452485i	$-0.149462\pi$
$824$		0			0
$825$	3.70662			0.129048
$826$		0			0
$827$		−	35.6121i		−	1.23835i	−0.785252	−	0.619177i	$-0.787466\pi$
$827$		−	35.6121i		−	1.23835i	0.785252	−	0.619177i	$-0.212534\pi$
$828$		0			0
$829$	33.3007			1.15658			0.578290	−	0.815831i	$-0.303720\pi$
$829$	33.3007			1.15658			0.578290	+	0.815831i	$0.303720\pi$
$830$		0			0
$831$	−0.679603			−0.0235752
$832$		0			0
$833$	38.3509	+	19.9273i	1.32878	+	0.690438i
$834$		0			0
$835$	−70.8657			−2.45241
$836$		0			0
$837$	−9.76937			−0.337679
$838$		0			0
$839$		−	26.0804i		−	0.900394i	−0.892929	−	0.450197i	$-0.851354\pi$
$839$		−	26.0804i		−	0.900394i	0.892929	−	0.450197i	$-0.148646\pi$
$840$		0			0
$841$	23.9272			0.825077
$842$		0			0
$843$			8.05929i			0.277577i
$844$		0			0
$845$		−	6.14908i		−	0.211535i
$846$		0			0
$847$		−	11.0629i		−	0.380125i
$848$		0			0
$849$	6.12311			0.210144
$850$		0			0
$851$	43.5531			1.49298
$852$		0			0
$853$		−	22.1037i		−	0.756816i	−0.925639	−	0.378408i	$-0.876472\pi$
$853$		−	22.1037i		−	0.756816i	0.925639	−	0.378408i	$-0.123528\pi$
$854$		0			0
$855$			21.3227i			0.729222i
$856$		0			0
$857$			38.2558i			1.30679i	0.757015	+	0.653397i	$0.226657\pi$
$857$			38.2558i			1.30679i	−0.757015	+	0.653397i	$0.773343\pi$
$858$		0			0
$859$	0.169166			0.00577188			0.00288594	−	0.999996i	$-0.499081\pi$
$859$	0.169166			0.00577188			0.00288594	+	0.999996i	$0.499081\pi$
$860$		0			0
$861$			1.60175i			0.0545876i
$862$		0			0
$863$	34.3834			1.17043			0.585213	−	0.810880i	$-0.301011\pi$
$863$	34.3834			1.17043			0.585213	+	0.810880i	$0.301011\pi$
$864$		0			0
$865$	9.91607			0.337156
$866$		0			0
$867$	4.82230	−	3.38752i	0.163774	−	0.115046i
$868$		0			0
$869$	21.3458			0.724106
$870$		0			0
$871$	−16.8082			−0.569524
$872$		0			0
$873$		−	19.2236i		−	0.650621i
$874$		0			0
$875$	−16.0399			−0.542248
$876$		0			0
$877$		−	54.0454i		−	1.82498i	−0.409094	−	0.912492i	$-0.634155\pi$
$877$		−	54.0454i		−	1.82498i	0.409094	−	0.912492i	$-0.365845\pi$
$878$		0			0
$879$		−	10.0404i		−	0.338654i
$880$		0			0
$881$		−	56.0183i		−	1.88730i	−0.330940	−	0.943652i	$-0.607366\pi$
$881$		−	56.0183i		−	1.88730i	0.330940	−	0.943652i	$-0.392634\pi$
$882$		0			0
$883$	12.5079			0.420924			0.210462	−	0.977602i	$-0.432503\pi$
$883$	12.5079			0.420924			0.210462	+	0.977602i	$0.432503\pi$
$884$		0			0
$885$	1.02246			0.0343696
$886$		0			0
$887$			5.04243i			0.169308i	0.996410	+	0.0846541i	$0.0269785\pi$
$887$			5.04243i			0.169308i	−0.996410	+	0.0846541i	$0.973021\pi$
$888$		0			0
$889$		−	11.3849i		−	0.381837i
$890$		0			0
$891$		−	22.9288i		−	0.768144i
$892$		0			0
$893$	4.21366			0.141005
$894$		0			0
$895$			34.4154i			1.15038i
$896$		0			0
$897$	−8.11181			−0.270845
$898$		0			0
$899$	10.7950			0.360032
$900$		0			0
$901$	−13.5662	−	7.04903i	−0.451954	−	0.234837i
$902$		0			0
$903$	0.370877			0.0123420
$904$		0			0
$905$	−34.7921			−1.15653
$906$		0			0
$907$		−	35.5918i		−	1.18181i	−0.806742	−	0.590904i	$-0.798771\pi$
$907$		−	35.5918i		−	1.18181i	0.806742	−	0.590904i	$-0.201229\pi$
$908$		0			0
$909$	7.46301			0.247532
$910$		0			0
$911$			18.7816i			0.622261i	0.950367	+	0.311131i	$0.100708\pi$
$911$			18.7816i			0.622261i	−0.950367	+	0.311131i	$0.899292\pi$
$912$		0			0
$913$		−	29.0217i		−	0.960478i
$914$		0			0
$915$			4.87618i			0.161202i
$916$		0			0
$917$	−25.2663			−0.834368
$918$		0			0
$919$	−6.96486			−0.229750			−0.114875	−	0.993380i	$-0.536647\pi$
$919$	−6.96486			−0.229750			−0.114875	+	0.993380i	$0.536647\pi$
$920$		0			0
$921$			2.44766i			0.0806532i
$922$		0			0
$923$		−	54.1822i		−	1.78343i
$924$		0			0
$925$			26.7423i			0.879281i
$926$		0			0
$927$	21.0574			0.691615
$928$		0			0
$929$		−	33.9588i		−	1.11415i	−0.830462	−	0.557076i	$-0.811923\pi$
$929$		−	33.9588i		−	1.11415i	0.830462	−	0.557076i	$-0.188077\pi$
$930$		0			0
$931$	26.3138			0.862400
$932$		0			0
$933$	1.54719			0.0506529
$934$		0			0
$935$	31.1902	+	16.2065i	1.02003	+	0.530011i
$936$		0			0
$937$	24.1263			0.788173			0.394086	−	0.919073i	$-0.371061\pi$
$937$	24.1263			0.788173			0.394086	+	0.919073i	$0.371061\pi$
$938$		0			0
$939$	−2.98722			−0.0974841
$940$		0			0
$941$			27.0873i			0.883020i	0.897256	+	0.441510i	$0.145557\pi$
$941$			27.0873i			0.883020i	−0.897256	+	0.441510i	$0.854443\pi$
$942$		0			0
$943$	6.65796			0.216813
$944$		0			0
$945$			25.1367i			0.817696i
$946$		0			0
$947$		−	9.35004i		−	0.303835i	−0.988393	−	0.151918i	$-0.951455\pi$
$947$		−	9.35004i		−	0.303835i	0.988393	−	0.151918i	$-0.0485449\pi$
$948$		0			0
$949$		−	32.4512i		−	1.05341i
$950$		0			0
$951$	−5.65202			−0.183279
$952$		0			0
$953$	−14.6159			−0.473454			−0.236727	−	0.971576i	$-0.576075\pi$
$953$	−14.6159			−0.473454			−0.236727	+	0.971576i	$0.576075\pi$
$954$		0			0
$955$			64.7285i			2.09457i
$956$		0			0
$957$		−	2.25670i		−	0.0729489i
$958$		0			0
$959$		−	45.8529i		−	1.48067i
$960$		0			0
$961$	8.02798			0.258967
$962$		0			0
$963$		−	16.2827i		−	0.524704i
$964$		0			0
$965$	44.2456			1.42432
$966$		0			0
$967$	−11.9435			−0.384076			−0.192038	−	0.981387i	$-0.561510\pi$
$967$	−11.9435			−0.384076			−0.192038	+	0.981387i	$0.561510\pi$
$968$		0			0
$969$	1.65437	−	3.18391i	0.0531460	−	0.102282i
$970$		0			0
$971$	15.2934			0.490790			0.245395	−	0.969423i	$-0.421082\pi$
$971$	15.2934			0.490790			0.245395	+	0.969423i	$0.421082\pi$
$972$		0			0
$973$	58.9884			1.89108
$974$		0			0
$975$		−	4.98078i		−	0.159513i
$976$		0			0
$977$	−32.7536			−1.04788			−0.523940	−	0.851755i	$-0.675538\pi$
$977$	−32.7536			−1.04788			−0.523940	+	0.851755i	$0.675538\pi$
$978$		0			0
$979$		−	11.5877i		−	0.370346i
$980$		0			0
$981$		−	6.20222i		−	0.198022i
$982$		0			0
$983$			48.6061i			1.55030i	0.631780	+	0.775148i	$0.282325\pi$
$983$			48.6061i			1.55030i	−0.631780	+	0.775148i	$0.717675\pi$
$984$		0			0
$985$	−30.5346			−0.972914
$986$		0			0
$987$	2.43291			0.0774405
$988$		0			0
$989$		−	1.54161i		−	0.0490205i
$990$		0			0
$991$			16.1016i			0.511485i	0.966745	+	0.255743i	$0.0823200\pi$
$991$			16.1016i			0.511485i	−0.966745	+	0.255743i	$0.917680\pi$
$992$		0			0
$993$			1.96066i			0.0622197i
$994$		0			0
$995$	−32.9950			−1.04601
$996$		0			0
$997$			3.34282i			0.105868i	0.998598	+	0.0529341i	$0.0168573\pi$
$997$			3.34282i			0.105868i	−0.998598	+	0.0529341i	$0.983143\pi$
$998$		0			0
$999$	−14.7347			−0.466184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.19	✓	40
17.16	even	2	inner	4012.2.b.a.237.22	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.19	✓	40	1.1	even	1	trivial
4012.2.b.a.237.22	yes	40	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.346659i q^{3} -2.94947i q^{5} -4.18117i q^{7} +2.87983 q^{9} +O(q^{10})\)	\(q-0.346659i q^{3} -2.94947i q^{5} -4.18117i q^{7} +2.87983 q^{9} -2.89035i q^{11} -3.88392 q^{13} -1.02246 q^{15} +(-3.65868 - 1.90106i) q^{17} -2.51034 q^{19} -1.44944 q^{21} +6.02484i q^{23} -3.69935 q^{25} -2.03829i q^{27} +2.25228i q^{29} -4.79291i q^{31} -1.00197 q^{33} -12.3322 q^{35} -7.22892i q^{37} +1.34639i q^{39} -1.10508i q^{41} -0.255876 q^{43} -8.49395i q^{45} -1.67852 q^{47} -10.4822 q^{49} +(-0.659021 + 1.26831i) q^{51} +3.70794 q^{53} -8.52499 q^{55} +0.870233i q^{57} -1.00000 q^{59} -4.76907i q^{61} -12.0410i q^{63} +11.4555i q^{65} +4.32764 q^{67} +2.08856 q^{69} +13.9504i q^{71} +8.35528i q^{73} +1.28241i q^{75} -12.0850 q^{77} +7.38519i q^{79} +7.93289 q^{81} +10.0409 q^{83} +(-5.60712 + 10.7912i) q^{85} +0.780772 q^{87} +4.00911 q^{89} +16.2393i q^{91} -1.66151 q^{93} +7.40417i q^{95} -6.67527i q^{97} -8.32371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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