LMFDB - Embedded newform 1968.2.j.f.1393.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1968,2,Mod(1393,1968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1968, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1968.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1968 = 2^{4} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1968.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.7145591178$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 984)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.8
Root			$2.54814i$ of defining polynomial
Character	$\chi$	$=$	1968.1393
Dual form			1968.2.j.f.1393.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.54814 q^{5} +5.04115i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +3.54814 q^{5} +5.04115i q^{7} -1.00000 q^{9} -1.25627i q^{11} -4.25627i q^{13} +3.54814i q^{15} -0.236747i q^{17} +0.840013i q^{19} -5.04115 q^{21} +7.74928 q^{23} +7.58929 q^{25} -1.00000i q^{27} +10.3525i q^{29} +4.69415 q^{31} +1.25627 q^{33} +17.8867i q^{35} +5.58929 q^{37} +4.25627 q^{39} +(6.33303 + 0.944874i) q^{41} -7.82392 q^{43} -3.54814 q^{45} -2.07464i q^{47} -18.4132 q^{49} +0.236747 q^{51} +1.29187i q^{53} -4.45740i q^{55} -0.840013 q^{57} -7.40767 q^{59} -10.7709 q^{61} -5.04115i q^{63} -15.1018i q^{65} -5.18717i q^{67} +7.74928i q^{69} +6.74928i q^{71} +8.41625 q^{73} +7.58929i q^{75} +6.33303 q^{77} +11.1786i q^{79} +1.00000 q^{81} -10.0607 q^{83} -0.840013i q^{85} -10.3525 q^{87} -3.21511i q^{89} +21.4565 q^{91} +4.69415i q^{93} +2.98048i q^{95} -3.94487i q^{97} +1.25627i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q + 10 * q^5 - 8 * q^9	$ 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) $ 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1968\mathbb{Z}\right)^\times$.

$n$	$1231$	$1313$	$1441$	$1477$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	3.54814			1.58678			0.793388	−	0.608716i	$-0.208315\pi$
$5$	3.54814			1.58678			0.793388	+	0.608716i	$0.208315\pi$
$6$		0			0
$7$			5.04115i			1.90538i	0.303949	+	0.952688i	$0.401695\pi$
$7$			5.04115i			1.90538i	−0.303949	+	0.952688i	$0.598305\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	1.25627i		−	0.378778i	−0.981902	−	0.189389i	$-0.939349\pi$
$11$		−	1.25627i		−	0.378778i	0.981902	−	0.189389i	$-0.0606508\pi$
$12$		0			0
$13$		−	4.25627i		−	1.18048i	−0.807229	−	0.590238i	$-0.799034\pi$
$13$		−	4.25627i		−	1.18048i	0.807229	−	0.590238i	$-0.200966\pi$
$14$		0			0
$15$			3.54814i			0.916126i
$16$		0			0
$17$		−	0.236747i		−	0.0574197i	−0.999588	−	0.0287098i	$-0.990860\pi$
$17$		−	0.236747i		−	0.0574197i	0.999588	−	0.0287098i	$-0.00913988\pi$
$18$		0			0
$19$			0.840013i			0.192712i	0.995347	+	0.0963561i	$0.0307187\pi$
$19$			0.840013i			0.192712i	−0.995347	+	0.0963561i	$0.969281\pi$
$20$		0			0
$21$	−5.04115			−1.10007
$22$		0			0
$23$	7.74928			1.61584			0.807918	−	0.589295i	$-0.200594\pi$
$23$	7.74928			1.61584			0.807918	+	0.589295i	$0.200594\pi$
$24$		0			0
$25$	7.58929			1.51786
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			10.3525i			1.92242i	0.275819	+	0.961210i	$0.411051\pi$
$29$			10.3525i			1.92242i	−0.275819	+	0.961210i	$0.588949\pi$
$30$		0			0
$31$	4.69415			0.843095			0.421547	−	0.906806i	$-0.361487\pi$
$31$	4.69415			0.843095			0.421547	+	0.906806i	$0.361487\pi$
$32$		0			0
$33$	1.25627			0.218688
$34$		0			0
$35$			17.8867i			3.02341i
$36$		0			0
$37$	5.58929			0.918874			0.459437	−	0.888210i	$-0.348051\pi$
$37$	5.58929			0.918874			0.459437	+	0.888210i	$0.348051\pi$
$38$		0			0
$39$	4.25627			0.681548
$40$		0			0
$41$	6.33303	+	0.944874i	0.989052	+	0.147564i
$41$	6.33303	+	0.944874i	0.989052	+	0.147564i
$42$		0			0
$43$	−7.82392			−1.19314			−0.596569	−	0.802562i	$-0.703470\pi$
$43$	−7.82392			−1.19314			−0.596569	+	0.802562i	$0.703470\pi$
$44$		0			0
$45$	−3.54814			−0.528925
$46$		0			0
$47$		−	2.07464i		−	0.302618i	−0.988487	−	0.151309i	$-0.951651\pi$
$47$		−	2.07464i		−	0.302618i	0.988487	−	0.151309i	$-0.0483488\pi$
$48$		0			0
$49$	−18.4132			−2.63046
$50$		0			0
$51$	0.236747			0.0331513
$52$		0			0
$53$			1.29187i			0.177452i	0.996056	+	0.0887262i	$0.0282796\pi$
$53$			1.29187i			0.177452i	−0.996056	+	0.0887262i	$0.971720\pi$
$54$		0			0
$55$		−	4.45740i		−	0.601036i
$56$		0			0
$57$	−0.840013			−0.111262
$58$		0			0
$59$	−7.40767			−0.964396			−0.482198	−	0.876062i	$-0.660162\pi$
$59$	−7.40767			−0.964396			−0.482198	+	0.876062i	$0.660162\pi$
$60$		0			0
$61$	−10.7709			−1.37907			−0.689537	−	0.724250i	$-0.742186\pi$
$61$	−10.7709			−1.37907			−0.689537	+	0.724250i	$0.742186\pi$
$62$		0			0
$63$		−	5.04115i		−	0.635125i
$64$		0			0
$65$		−	15.1018i		−	1.87315i
$66$		0			0
$67$		−	5.18717i		−	0.633713i	−0.948473	−	0.316857i	$-0.897373\pi$
$67$		−	5.18717i		−	0.633713i	0.948473	−	0.316857i	$-0.102627\pi$
$68$		0			0
$69$			7.74928i			0.932904i
$70$		0			0
$71$			6.74928i			0.800992i	0.916299	+	0.400496i	$0.131162\pi$
$71$			6.74928i			0.800992i	−0.916299	+	0.400496i	$0.868838\pi$
$72$		0			0
$73$	8.41625			0.985048			0.492524	−	0.870299i	$-0.336074\pi$
$73$	8.41625			0.985048			0.492524	+	0.870299i	$0.336074\pi$
$74$		0			0
$75$			7.58929i			0.876336i
$76$		0			0
$77$	6.33303			0.721715
$78$		0			0
$79$			11.1786i			1.25769i	0.777531	+	0.628844i	$0.216472\pi$
$79$			11.1786i			1.25769i	−0.777531	+	0.628844i	$0.783528\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−10.0607			−1.10430			−0.552151	−	0.833744i	$-0.686193\pi$
$83$	−10.0607			−1.10430			−0.552151	+	0.833744i	$0.686193\pi$
$84$		0			0
$85$		−	0.840013i		−	0.0911122i
$86$		0			0
$87$	−10.3525			−1.10991
$88$		0			0
$89$		−	3.21511i		−	0.340801i	−0.985375	−	0.170401i	$-0.945494\pi$
$89$		−	3.21511i		−	0.340801i	0.985375	−	0.170401i	$-0.0545062\pi$
$90$		0			0
$91$	21.4565			2.24925
$92$		0			0
$93$			4.69415i			0.486761i
$94$		0			0
$95$			2.98048i			0.305791i
$96$		0			0
$97$		−	3.94487i		−	0.400541i	−0.979741	−	0.200271i	$-0.935818\pi$
$97$		−	3.94487i		−	0.400541i	0.979741	−	0.200271i	$-0.0641821\pi$
$98$		0			0
$99$			1.25627i			0.126259i
$100$		0			0
$101$		−	14.6304i		−	1.45578i	−0.685692	−	0.727892i	$-0.740500\pi$
$101$		−	14.6304i		−	1.45578i	0.685692	−	0.727892i	$-0.259500\pi$
$102$		0			0
$103$	−16.5809			−1.63376			−0.816880	−	0.576807i	$-0.804298\pi$
$103$	−16.5809			−1.63376			−0.816880	+	0.576807i	$0.804298\pi$
$104$		0			0
$105$	−17.8867			−1.74556
$106$		0			0
$107$	−2.46491			−0.238292			−0.119146	−	0.992877i	$-0.538016\pi$
$107$	−2.46491			−0.238292			−0.119146	+	0.992877i	$0.538016\pi$
$108$		0			0
$109$			5.15140i			0.493415i	0.969090	+	0.246708i	$0.0793487\pi$
$109$			5.15140i			0.493415i	−0.969090	+	0.246708i	$0.920651\pi$
$110$		0			0
$111$			5.58929i			0.530512i
$112$		0			0
$113$	−6.37206			−0.599433			−0.299717	−	0.954028i	$-0.596892\pi$
$113$	−6.37206			−0.599433			−0.299717	+	0.954028i	$0.596892\pi$
$114$		0			0
$115$	27.4955			2.56397
$116$		0			0
$117$			4.25627i			0.393492i
$118$		0			0
$119$	1.19348			0.109406
$120$		0			0
$121$	9.42180			0.856527
$122$		0			0
$123$	−0.944874	+	6.33303i	−0.0851964	+	0.571030i
$124$		0			0
$125$	9.18717			0.821725
$126$		0			0
$127$	8.47904			0.752393			0.376197	−	0.926540i	$-0.377232\pi$
$127$	8.47904			0.752393			0.376197	+	0.926540i	$0.377232\pi$
$128$		0			0
$129$		−	7.82392i		−	0.688858i
$130$		0			0
$131$	−17.9028			−1.56417			−0.782087	−	0.623169i	$-0.785845\pi$
$131$	−17.9028			−1.56417			−0.782087	+	0.623169i	$0.785845\pi$
$132$		0			0
$133$	−4.23463			−0.367189
$134$		0			0
$135$		−	3.54814i		−	0.305375i
$136$		0			0
$137$			10.5061i			0.897594i	0.893634	+	0.448797i	$0.148147\pi$
$137$			10.5061i			0.897594i	−0.893634	+	0.448797i	$0.851853\pi$
$138$		0			0
$139$	5.29187			0.448851			0.224425	−	0.974491i	$-0.427949\pi$
$139$	5.29187			0.448851			0.224425	+	0.974491i	$0.427949\pi$
$140$		0			0
$141$	2.07464			0.174717
$142$		0			0
$143$	−5.34700			−0.447139
$144$		0			0
$145$			36.7323i			3.05045i
$146$		0			0
$147$		−	18.4132i		−	1.51870i
$148$		0			0
$149$		−	0.986026i		−	0.0807784i	−0.999184	−	0.0403892i	$-0.987140\pi$
$149$		−	0.986026i		−	0.0807784i	0.999184	−	0.0403892i	$-0.0128598\pi$
$150$		0			0
$151$		−	14.7763i		−	1.20248i	−0.799069	−	0.601239i	$-0.794674\pi$
$151$		−	14.7763i		−	1.20248i	0.799069	−	0.601239i	$-0.205326\pi$
$152$		0			0
$153$			0.236747i			0.0191399i
$154$		0			0
$155$	16.6555			1.33780
$156$		0			0
$157$		−	17.6088i		−	1.40534i	−0.711518	−	0.702668i	$-0.751992\pi$
$157$		−	17.6088i		−	1.40534i	0.711518	−	0.702668i	$-0.248008\pi$
$158$		0			0
$159$	−1.29187			−0.102452
$160$		0			0
$161$			39.0653i			3.07878i
$162$		0			0
$163$	14.5612			1.14052			0.570260	−	0.821464i	$-0.306842\pi$
$163$	14.5612			1.14052			0.570260	+	0.821464i	$0.306842\pi$
$164$		0			0
$165$	4.45740			0.347009
$166$		0			0
$167$			0.909266i			0.0703611i	0.999381	+	0.0351805i	$0.0112006\pi$
$167$			0.909266i			0.0703611i	−0.999381	+	0.0351805i	$0.988799\pi$
$168$		0			0
$169$	−5.11580			−0.393523
$170$		0			0
$171$		−	0.840013i		−	0.0642374i
$172$		0			0
$173$	12.3721			0.940630			0.470315	−	0.882498i	$-0.344140\pi$
$173$	12.3721			0.940630			0.470315	+	0.882498i	$0.344140\pi$
$174$		0			0
$175$			38.2588i			2.89209i
$176$		0			0
$177$		−	7.40767i		−	0.556795i
$178$		0			0
$179$		−	18.8509i		−	1.40899i	−0.709711	−	0.704493i	$-0.751174\pi$
$179$		−	18.8509i		−	1.40899i	0.709711	−	0.704493i	$-0.248826\pi$
$180$		0			0
$181$		−	9.92020i		−	0.737363i	−0.929556	−	0.368681i	$-0.879809\pi$
$181$		−	9.92020i		−	0.737363i	0.929556	−	0.368681i	$-0.120191\pi$
$182$		0			0
$183$		−	10.7709i		−	0.796209i
$184$		0			0
$185$	19.8316			1.45805
$186$		0			0
$187$	−0.297418			−0.0217493
$188$		0			0
$189$	5.04115			0.366690
$190$		0			0
$191$		−	3.60881i		−	0.261124i	−0.991440	−	0.130562i	$-0.958322\pi$
$191$		−	3.60881i		−	0.261124i	0.991440	−	0.130562i	$-0.0416782\pi$
$192$		0			0
$193$			23.3872i			1.68345i	0.539907	+	0.841725i	$0.318459\pi$
$193$			23.3872i			1.68345i	−0.539907	+	0.841725i	$0.681541\pi$
$194$		0			0
$195$	15.1018			1.08146
$196$		0			0
$197$	5.63044			0.401152			0.200576	−	0.979678i	$-0.435719\pi$
$197$	5.63044			0.401152			0.200576	+	0.979678i	$0.435719\pi$
$198$		0			0
$199$		−	16.5397i		−	1.17247i	−0.810141	−	0.586234i	$-0.800610\pi$
$199$		−	16.5397i		−	1.17247i	0.810141	−	0.586234i	$-0.199390\pi$
$200$		0			0
$201$	5.18717			0.365874
$202$		0			0
$203$	−52.1887			−3.66293
$204$		0			0
$205$	22.4705	+	3.35254i	1.56940	+	0.234152i
$206$		0			0
$207$	−7.74928			−0.538612
$208$		0			0
$209$	1.05528			0.0729952
$210$		0			0
$211$		−	5.14813i		−	0.354412i	−0.984174	−	0.177206i	$-0.943294\pi$
$211$		−	5.14813i		−	0.354412i	0.984174	−	0.177206i	$-0.0567059\pi$
$212$		0			0
$213$	−6.74928			−0.462453
$214$		0			0
$215$	−27.7604			−1.89324
$216$		0			0
$217$			23.6639i			1.60641i
$218$		0			0
$219$			8.41625i			0.568718i
$220$		0			0
$221$	−1.00766			−0.0677825
$222$		0			0
$223$	−15.1872			−1.01701			−0.508504	−	0.861060i	$-0.669801\pi$
$223$	−15.1872			−1.01701			−0.508504	+	0.861060i	$0.669801\pi$
$224$		0			0
$225$	−7.58929			−0.505953
$226$		0			0
$227$			13.2283i			0.877994i	0.898488	+	0.438997i	$0.144666\pi$
$227$			13.2283i			0.877994i	−0.898488	+	0.438997i	$0.855334\pi$
$228$		0			0
$229$			18.8425i			1.24515i	0.782561	+	0.622574i	$0.213913\pi$
$229$			18.8425i			1.24515i	−0.782561	+	0.622574i	$0.786087\pi$
$230$		0			0
$231$			6.33303i			0.416682i
$232$		0			0
$233$			28.5039i			1.86736i	0.358114	+	0.933678i	$0.383420\pi$
$233$			28.5039i			1.86736i	−0.358114	+	0.933678i	$0.616580\pi$
$234$		0			0
$235$		−	7.36113i		−	0.480187i
$236$		0			0
$237$	−11.1786			−0.726127
$238$		0			0
$239$		−	6.52111i		−	0.421816i	−0.977506	−	0.210908i	$-0.932358\pi$
$239$		−	6.52111i		−	0.421816i	0.977506	−	0.210908i	$-0.0676420\pi$
$240$		0			0
$241$	18.4565			1.18889			0.594443	−	0.804138i	$-0.297372\pi$
$241$	18.4565			1.18889			0.594443	+	0.804138i	$0.297372\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−65.3326			−4.17395
$246$		0			0
$247$	3.57532			0.227492
$248$		0			0
$249$		−	10.0607i		−	0.637569i
$250$		0			0
$251$	16.7051			1.05442			0.527208	−	0.849736i	$-0.323239\pi$
$251$	16.7051			1.05442			0.527208	+	0.849736i	$0.323239\pi$
$252$		0			0
$253$		−	9.73515i		−	0.612044i
$254$		0			0
$255$	0.840013			0.0526036
$256$		0			0
$257$		−	25.6555i		−	1.60035i	−0.599769	−	0.800173i	$-0.704741\pi$
$257$		−	25.6555i		−	1.60035i	0.599769	−	0.800173i	$-0.295259\pi$
$258$		0			0
$259$			28.1765i			1.75080i
$260$		0			0
$261$		−	10.3525i		−	0.640806i
$262$		0			0
$263$		−	22.8175i		−	1.40698i	−0.710703	−	0.703492i	$-0.751623\pi$
$263$		−	22.8175i		−	1.40698i	0.710703	−	0.703492i	$-0.248377\pi$
$264$		0			0
$265$			4.58375i			0.281577i
$266$		0			0
$267$	3.21511			0.196762
$268$		0			0
$269$	−22.8783			−1.39491			−0.697457	−	0.716627i	$-0.745685\pi$
$269$	−22.8783			−1.39491			−0.697457	+	0.716627i	$0.745685\pi$
$270$		0			0
$271$	24.6856			1.49954			0.749771	−	0.661698i	$-0.230164\pi$
$271$	24.6856			1.49954			0.749771	+	0.661698i	$0.230164\pi$
$272$		0			0
$273$			21.4565i			1.29861i
$274$		0			0
$275$		−	9.53416i		−	0.574932i
$276$		0			0
$277$	8.09073			0.486125			0.243063	−	0.970011i	$-0.421848\pi$
$277$	8.09073			0.486125			0.243063	+	0.970011i	$0.421848\pi$
$278$		0			0
$279$	−4.69415			−0.281032
$280$		0			0
$281$		−	14.0691i		−	0.839292i	−0.907688	−	0.419646i	$-0.862154\pi$
$281$		−	14.0691i		−	0.839292i	0.907688	−	0.419646i	$-0.137846\pi$
$282$		0			0
$283$	4.17304			0.248061			0.124031	−	0.992278i	$-0.460418\pi$
$283$	4.17304			0.248061			0.124031	+	0.992278i	$0.460418\pi$
$284$		0			0
$285$	−2.98048			−0.176549
$286$		0			0
$287$	−4.76325	+	31.9257i	−0.281166	+	1.88452i
$288$		0			0
$289$	16.9440			0.996703
$290$		0			0
$291$	3.94487			0.231253
$292$		0			0
$293$			8.77973i			0.512917i	0.966555	+	0.256459i	$0.0825558\pi$
$293$			8.77973i			0.512917i	−0.966555	+	0.256459i	$0.917444\pi$
$294$		0			0
$295$	−26.2834			−1.53028
$296$		0			0
$297$	−1.25627			−0.0728959
$298$		0			0
$299$		−	32.9830i		−	1.90746i
$300$		0			0
$301$		−	39.4416i		−	2.27338i
$302$		0			0
$303$	14.6304			0.840497
$304$		0			0
$305$	−38.2167			−2.18828
$306$		0			0
$307$	20.3656			1.16233			0.581163	−	0.813787i	$-0.302598\pi$
$307$	20.3656			1.16233			0.581163	+	0.813787i	$0.302598\pi$
$308$		0			0
$309$		−	16.5809i		−	0.943252i
$310$		0			0
$311$		−	7.95238i		−	0.450938i	−0.974250	−	0.225469i	$-0.927609\pi$
$311$		−	7.95238i		−	0.450938i	0.974250	−	0.225469i	$-0.0723915\pi$
$312$		0			0
$313$		−	12.9699i		−	0.733104i	−0.930398	−	0.366552i	$-0.880538\pi$
$313$		−	12.9699i		−	0.733104i	0.930398	−	0.366552i	$-0.119462\pi$
$314$		0			0
$315$		−	17.8867i		−	1.00780i
$316$		0			0
$317$		−	15.2391i		−	0.855913i	−0.903799	−	0.427957i	$-0.859234\pi$
$317$		−	15.2391i		−	0.855913i	0.903799	−	0.427957i	$-0.140766\pi$
$318$		0			0
$319$	13.0055			0.728171
$320$		0			0
$321$		−	2.46491i		−	0.137578i
$322$		0			0
$323$	0.198871			0.0110655
$324$		0			0
$325$		−	32.3020i		−	1.79179i
$326$		0			0
$327$	−5.15140			−0.284873
$328$		0			0
$329$	10.4586			0.576601
$330$		0			0
$331$		−	27.9560i		−	1.53660i	−0.640091	−	0.768299i	$-0.721103\pi$
$331$		−	27.9560i		−	1.53660i	0.640091	−	0.768299i	$-0.278897\pi$
$332$		0			0
$333$	−5.58929			−0.306291
$334$		0			0
$335$		−	18.4048i		−	1.00556i
$336$		0			0
$337$	−5.27790			−0.287506			−0.143753	−	0.989614i	$-0.545917\pi$
$337$	−5.27790			−0.287506			−0.143753	+	0.989614i	$0.545917\pi$
$338$		0			0
$339$		−	6.37206i		−	0.346083i
$340$		0			0
$341$		−	5.89710i		−	0.319346i
$342$		0			0
$343$		−	57.5357i		−	3.10664i
$344$		0			0
$345$			27.4955i			1.48031i
$346$		0			0
$347$			8.26720i			0.443807i	0.975069	+	0.221903i	$0.0712269\pi$
$347$			8.26720i			0.443807i	−0.975069	+	0.221903i	$0.928773\pi$
$348$		0			0
$349$	−0.104861			−0.00561309			−0.00280654	−	0.999996i	$-0.500893\pi$
$349$	−0.104861			−0.00561309			−0.00280654	+	0.999996i	$0.500893\pi$
$350$		0			0
$351$	−4.25627			−0.227183
$352$		0			0
$353$	−20.8866			−1.11168			−0.555840	−	0.831290i	$-0.687603\pi$
$353$	−20.8866			−1.11168			−0.555840	+	0.831290i	$0.687603\pi$
$354$		0			0
$355$			23.9474i			1.27099i
$356$		0			0
$357$			1.19348i			0.0631656i
$358$		0			0
$359$	29.0814			1.53486			0.767428	−	0.641135i	$-0.221536\pi$
$359$	29.0814			1.53486			0.767428	+	0.641135i	$0.221536\pi$
$360$		0			0
$361$	18.2944			0.962862
$362$		0			0
$363$			9.42180i			0.494516i
$364$		0			0
$365$	29.8620			1.56305
$366$		0			0
$367$	−2.84213			−0.148358			−0.0741789	−	0.997245i	$-0.523634\pi$
$367$	−2.84213			−0.148358			−0.0741789	+	0.997245i	$0.523634\pi$
$368$		0			0
$369$	−6.33303	−	0.944874i	−0.329684	−	0.0491882i
$370$		0			0
$371$	−6.51253			−0.338114
$372$		0			0
$373$	4.88116			0.252737			0.126369	−	0.991983i	$-0.459668\pi$
$373$	4.88116			0.252737			0.126369	+	0.991983i	$0.459668\pi$
$374$		0			0
$375$			9.18717i			0.474423i
$376$		0			0
$377$	44.0632			2.26937
$378$		0			0
$379$	14.1286			0.725738			0.362869	−	0.931840i	$-0.381797\pi$
$379$	14.1286			0.725738			0.362869	+	0.931840i	$0.381797\pi$
$380$		0			0
$381$			8.47904i			0.434394i
$382$		0			0
$383$			25.8846i			1.32264i	0.750103	+	0.661320i	$0.230004\pi$
$383$			25.8846i			1.32264i	−0.750103	+	0.661320i	$0.769996\pi$
$384$		0			0
$385$	22.4705			1.14520
$386$		0			0
$387$	7.82392			0.397712
$388$		0			0
$389$	15.7797			0.800064			0.400032	−	0.916501i	$-0.368999\pi$
$389$	15.7797			0.800064			0.400032	+	0.916501i	$0.368999\pi$
$390$		0			0
$391$		−	1.83462i		−	0.0927808i
$392$		0			0
$393$		−	17.9028i		−	0.903077i
$394$		0			0
$395$			39.6632i			1.99567i
$396$		0			0
$397$		−	29.4459i		−	1.47785i	−0.673788	−	0.738925i	$-0.735334\pi$
$397$		−	29.4459i		−	1.47785i	0.673788	−	0.738925i	$-0.264666\pi$
$398$		0			0
$399$		−	4.23463i		−	0.211997i
$400$		0			0
$401$	−3.93302			−0.196405			−0.0982027	−	0.995166i	$-0.531309\pi$
$401$	−3.93302			−0.196405			−0.0982027	+	0.995166i	$0.531309\pi$
$402$		0			0
$403$		−	19.9796i		−	0.995253i
$404$		0			0
$405$	3.54814			0.176308
$406$		0			0
$407$		−	7.02163i		−	0.348049i
$408$		0			0
$409$	−1.29203			−0.0638866			−0.0319433	−	0.999490i	$-0.510170\pi$
$409$	−1.29203			−0.0638866			−0.0319433	+	0.999490i	$0.510170\pi$
$410$		0			0
$411$	−10.5061			−0.518226
$412$		0			0
$413$		−	37.3432i		−	1.83754i
$414$		0			0
$415$	−35.6967			−1.75228
$416$		0			0
$417$			5.29187i			0.259144i
$418$		0			0
$419$	3.34950			0.163634			0.0818170	−	0.996647i	$-0.473928\pi$
$419$	3.34950			0.163634			0.0818170	+	0.996647i	$0.473928\pi$
$420$		0			0
$421$		−	17.2193i		−	0.839220i	−0.907705	−	0.419610i	$-0.862167\pi$
$421$		−	17.2193i		−	0.839220i	0.907705	−	0.419610i	$-0.137833\pi$
$422$		0			0
$423$			2.07464i			0.100873i
$424$		0			0
$425$		−	1.79674i		−	0.0871549i
$426$		0			0
$427$		−	54.2978i		−	2.62766i
$428$		0			0
$429$		−	5.34700i		−	0.258156i
$430$		0			0
$431$	4.48839			0.216198			0.108099	−	0.994140i	$-0.465524\pi$
$431$	4.48839			0.216198			0.108099	+	0.994140i	$0.465524\pi$
$432$		0			0
$433$	−27.5918			−1.32598			−0.662989	−	0.748630i	$-0.730712\pi$
$433$	−27.5918			−1.32598			−0.662989	+	0.748630i	$0.730712\pi$
$434$		0			0
$435$	−36.7323			−1.76118
$436$		0			0
$437$			6.50949i			0.311391i
$438$		0			0
$439$		−	21.0079i		−	1.00265i	−0.865258	−	0.501326i	$-0.832846\pi$
$439$		−	21.0079i		−	1.00265i	0.865258	−	0.501326i	$-0.167154\pi$
$440$		0			0
$441$	18.4132			0.876820
$442$		0			0
$443$	3.39689			0.161391			0.0806955	−	0.996739i	$-0.474286\pi$
$443$	3.39689			0.161391			0.0806955	+	0.996739i	$0.474286\pi$
$444$		0			0
$445$		−	11.4077i		−	0.540775i
$446$		0			0
$447$	0.986026			0.0466374
$448$		0			0
$449$	−6.33529			−0.298981			−0.149491	−	0.988763i	$-0.547763\pi$
$449$	−6.33529			−0.298981			−0.149491	+	0.988763i	$0.547763\pi$
$450$		0			0
$451$	1.18701	−	7.95596i	0.0558942	−	0.374632i
$452$		0			0
$453$	14.7763			0.694251
$454$		0			0
$455$	76.1306			3.56906
$456$		0			0
$457$			22.3962i			1.04765i	0.851826	+	0.523825i	$0.175495\pi$
$457$			22.3962i			1.04765i	−0.851826	+	0.523825i	$0.824505\pi$
$458$		0			0
$459$	−0.236747			−0.0110504
$460$		0			0
$461$	−30.0934			−1.40159			−0.700795	−	0.713363i	$-0.747171\pi$
$461$	−30.0934			−1.40159			−0.700795	+	0.713363i	$0.747171\pi$
$462$		0			0
$463$		−	0.0830744i		−	0.00386080i	−0.999998	−	0.00193040i	$-0.999386\pi$
$463$		−	0.0830744i		−	0.00386080i	0.999998	−	0.00193040i	$-0.000614465\pi$
$464$		0			0
$465$			16.6555i			0.772381i
$466$		0			0
$467$	−10.1449			−0.469452			−0.234726	−	0.972062i	$-0.575419\pi$
$467$	−10.1449			−0.469452			−0.234726	+	0.972062i	$0.575419\pi$
$468$		0			0
$469$	26.1493			1.20746
$470$		0			0
$471$	17.6088			0.811371
$472$		0			0
$473$			9.82892i			0.451934i
$474$		0			0
$475$			6.37510i			0.292510i
$476$		0			0
$477$		−	1.29187i		−	0.0591508i
$478$		0			0
$479$		−	4.90161i		−	0.223960i	−0.993710	−	0.111980i	$-0.964281\pi$
$479$		−	4.90161i		−	0.223960i	0.993710	−	0.111980i	$-0.0357193\pi$
$480$		0			0
$481$		−	23.7895i		−	1.08471i
$482$		0			0
$483$	−39.0653			−1.77753
$484$		0			0
$485$		−	13.9970i		−	0.635569i
$486$		0			0
$487$	−11.1505			−0.505277			−0.252638	−	0.967561i	$-0.581298\pi$
$487$	−11.1505			−0.505277			−0.252638	+	0.967561i	$0.581298\pi$
$488$		0			0
$489$			14.5612i			0.658480i
$490$		0			0
$491$	−9.08573			−0.410033			−0.205017	−	0.978758i	$-0.565725\pi$
$491$	−9.08573			−0.410033			−0.205017	+	0.978758i	$0.565725\pi$
$492$		0			0
$493$	2.45094			0.110385
$494$		0			0
$495$			4.45740i			0.200345i
$496$		0			0
$497$	−34.0241			−1.52619
$498$		0			0
$499$			32.4048i			1.45064i	0.688413	+	0.725319i	$0.258308\pi$
$499$			32.4048i			1.45064i	−0.688413	+	0.725319i	$0.741692\pi$
$500$		0			0
$501$	−0.909266			−0.0406230
$502$		0			0
$503$		−	13.3192i		−	0.593874i	−0.954897	−	0.296937i	$-0.904035\pi$
$503$		−	13.3192i		−	0.593874i	0.954897	−	0.296937i	$-0.0959651\pi$
$504$		0			0
$505$		−	51.9108i		−	2.31000i
$506$		0			0
$507$		−	5.11580i		−	0.227200i
$508$		0			0
$509$		−	11.7949i		−	0.522800i	−0.965231	−	0.261400i	$-0.915816\pi$
$509$		−	11.7949i		−	0.522800i	0.965231	−	0.261400i	$-0.0841841\pi$
$510$		0			0
$511$			42.4276i			1.87689i
$512$		0			0
$513$	0.840013			0.0370875
$514$		0			0
$515$	−58.8312			−2.59241
$516$		0			0
$517$	−2.60630			−0.114625
$518$		0			0
$519$			12.3721i			0.543073i
$520$		0			0
$521$		−	12.2770i		−	0.537864i	−0.963159	−	0.268932i	$-0.913329\pi$
$521$		−	12.2770i		−	0.537864i	0.963159	−	0.268932i	$-0.0866707\pi$
$522$		0			0
$523$	29.7313			1.30006			0.650030	−	0.759908i	$-0.274756\pi$
$523$	29.7313			1.30006			0.650030	+	0.759908i	$0.274756\pi$
$524$		0			0
$525$	−38.2588			−1.66975
$526$		0			0
$527$		−	1.11133i		−	0.0484102i
$528$		0			0
$529$	37.0513			1.61093
$530$		0			0
$531$	7.40767			0.321465
$532$		0			0
$533$	4.02163	−	26.9550i	0.174196	−	1.16755i
$534$		0			0
$535$	−8.74585			−0.378116
$536$		0			0
$537$	18.8509			0.813478
$538$		0			0
$539$			23.1319i			0.996361i
$540$		0			0
$541$	−28.0153			−1.20447			−0.602236	−	0.798318i	$-0.705724\pi$
$541$	−28.0153			−1.20447			−0.602236	+	0.798318i	$0.705724\pi$
$542$		0			0
$543$	9.92020			0.425717
$544$		0			0
$545$			18.2779i			0.782939i
$546$		0			0
$547$			8.12307i			0.347317i	0.984806	+	0.173659i	$0.0555589\pi$
$547$			8.12307i			0.347317i	−0.984806	+	0.173659i	$0.944441\pi$
$548$		0			0
$549$	10.7709			0.459691
$550$		0			0
$551$	−8.69627			−0.370473
$552$		0			0
$553$	−56.3529			−2.39637
$554$		0			0
$555$			19.8316i			0.841804i
$556$		0			0
$557$			12.7269i			0.539255i	0.962965	+	0.269627i	$0.0869006\pi$
$557$			12.7269i			0.539255i	−0.962965	+	0.269627i	$0.913099\pi$
$558$		0			0
$559$			33.3007i			1.40847i
$560$		0			0
$561$		−	0.297418i		−	0.0125570i
$562$		0			0
$563$		−	27.3788i		−	1.15388i	−0.816787	−	0.576939i	$-0.804247\pi$
$563$		−	27.3788i		−	1.15388i	0.816787	−	0.576939i	$-0.195753\pi$
$564$		0			0
$565$	−22.6090			−0.951166
$566$		0			0
$567$			5.04115i			0.211708i
$568$		0			0
$569$	−5.47257			−0.229422			−0.114711	−	0.993399i	$-0.536594\pi$
$569$	−5.47257			−0.229422			−0.114711	+	0.993399i	$0.536594\pi$
$570$		0			0
$571$		−	14.7935i		−	0.619087i	−0.950885	−	0.309544i	$-0.899824\pi$
$571$		−	14.7935i		−	0.619087i	0.950885	−	0.309544i	$-0.100176\pi$
$572$		0			0
$573$	3.60881			0.150760
$574$		0			0
$575$	58.8115			2.45261
$576$		0			0
$577$			35.0515i			1.45921i	0.683868	+	0.729606i	$0.260296\pi$
$577$			35.0515i			1.45921i	−0.683868	+	0.729606i	$0.739704\pi$
$578$		0			0
$579$	−23.3872			−0.971940
$580$		0			0
$581$		−	50.7174i		−	2.10411i
$582$		0			0
$583$	1.62294			0.0672152
$584$		0			0
$585$			15.1018i			0.624384i
$586$		0			0
$587$			41.5062i			1.71314i	0.516027	+	0.856572i	$0.327410\pi$
$587$			41.5062i			1.71314i	−0.516027	+	0.856572i	$0.672590\pi$
$588$		0			0
$589$			3.94315i			0.162475i
$590$		0			0
$591$			5.63044i			0.231605i
$592$		0			0
$593$			44.5227i			1.82833i	0.405345	+	0.914164i	$0.367152\pi$
$593$			44.5227i			1.82833i	−0.405345	+	0.914164i	$0.632848\pi$
$594$		0			0
$595$	4.23463			0.173603
$596$		0			0
$597$	16.5397			0.676925
$598$		0			0
$599$	1.07029			0.0437310			0.0218655	−	0.999761i	$-0.493039\pi$
$599$	1.07029			0.0437310			0.0218655	+	0.999761i	$0.493039\pi$
$600$		0			0
$601$		−	27.6803i		−	1.12910i	−0.825397	−	0.564552i	$-0.809049\pi$
$601$		−	27.6803i		−	1.12910i	0.825397	−	0.564552i	$-0.190951\pi$
$602$		0			0
$603$			5.18717i			0.211238i
$604$		0			0
$605$	33.4298			1.35912
$606$		0			0
$607$	−20.9118			−0.848783			−0.424391	−	0.905479i	$-0.639512\pi$
$607$	−20.9118			−0.848783			−0.424391	+	0.905479i	$0.639512\pi$
$608$		0			0
$609$		−	52.1887i		−	2.11479i
$610$		0			0
$611$	−8.83024			−0.357233
$612$		0			0
$613$	−44.2006			−1.78525			−0.892623	−	0.450804i	$-0.851137\pi$
$613$	−44.2006			−1.78525			−0.892623	+	0.450804i	$0.851137\pi$
$614$		0			0
$615$	−3.35254	+	22.4705i	−0.135188	+	0.906096i
$616$		0			0
$617$	−18.9669			−0.763579			−0.381789	−	0.924249i	$-0.624692\pi$
$617$	−18.9669			−0.763579			−0.381789	+	0.924249i	$0.624692\pi$
$618$		0			0
$619$	−7.26932			−0.292178			−0.146089	−	0.989271i	$-0.546669\pi$
$619$	−7.26932			−0.292178			−0.146089	+	0.989271i	$0.546669\pi$
$620$		0			0
$621$		−	7.74928i		−	0.310968i
$622$		0			0
$623$	16.2079			0.649355
$624$		0			0
$625$	−5.34912			−0.213965
$626$		0			0
$627$			1.05528i			0.0421438i
$628$		0			0
$629$		−	1.32325i		−	0.0527614i
$630$		0			0
$631$	41.2328			1.64145			0.820726	−	0.571323i	$-0.193569\pi$
$631$	41.2328			1.64145			0.820726	+	0.571323i	$0.193569\pi$
$632$		0			0
$633$	5.14813			0.204620
$634$		0			0
$635$	30.0848			1.19388
$636$		0			0
$637$			78.3715i			3.10519i
$638$		0			0
$639$		−	6.74928i		−	0.266997i
$640$		0			0
$641$			25.4771i			1.00628i	0.864204	+	0.503142i	$0.167823\pi$
$641$			25.4771i			1.00628i	−0.864204	+	0.503142i	$0.832177\pi$
$642$		0			0
$643$			6.78489i			0.267570i	0.991010	+	0.133785i	$0.0427131\pi$
$643$			6.78489i			0.267570i	−0.991010	+	0.133785i	$0.957287\pi$
$644$		0			0
$645$		−	27.7604i		−	1.09306i
$646$		0			0
$647$	−48.7292			−1.91574			−0.957872	−	0.287196i	$-0.907277\pi$
$647$	−48.7292			−1.91574			−0.957872	+	0.287196i	$0.907277\pi$
$648$		0			0
$649$			9.30600i			0.365292i
$650$		0			0
$651$	−23.6639			−0.927463
$652$		0			0
$653$			1.14601i			0.0448470i	0.999749	+	0.0224235i	$0.00713821\pi$
$653$			1.14601i			0.0448470i	−0.999749	+	0.0224235i	$0.992862\pi$
$654$		0			0
$655$	−63.5216			−2.48200
$656$		0			0
$657$	−8.41625			−0.328349
$658$		0			0
$659$		−	25.0683i		−	0.976523i	−0.872697	−	0.488262i	$-0.837631\pi$
$659$		−	25.0683i		−	0.976523i	0.872697	−	0.488262i	$-0.162369\pi$
$660$		0			0
$661$	35.2425			1.37078			0.685388	−	0.728178i	$-0.259633\pi$
$661$	35.2425			1.37078			0.685388	+	0.728178i	$0.259633\pi$
$662$		0			0
$663$		−	1.00766i		−	0.0391343i
$664$		0			0
$665$	−15.0251			−0.582647
$666$		0			0
$667$			80.2247i			3.10631i
$668$		0			0
$669$		−	15.1872i		−	0.587170i
$670$		0			0
$671$			13.5311i			0.522363i
$672$		0			0
$673$		−	24.7441i		−	0.953816i	−0.878953	−	0.476908i	$-0.841758\pi$
$673$		−	24.7441i		−	0.953816i	0.878953	−	0.476908i	$-0.158242\pi$
$674$		0			0
$675$		−	7.58929i		−	0.292112i
$676$		0			0
$677$	−21.6025			−0.830251			−0.415126	−	0.909764i	$-0.636262\pi$
$677$	−21.6025			−0.830251			−0.415126	+	0.909764i	$0.636262\pi$
$678$		0			0
$679$	19.8867			0.763182
$680$		0			0
$681$	−13.2283			−0.506910
$682$		0			0
$683$		−	38.5638i		−	1.47560i	−0.675017	−	0.737802i	$-0.735864\pi$
$683$		−	38.5638i		−	1.47560i	0.675017	−	0.737802i	$-0.264136\pi$
$684$		0			0
$685$			37.2770i			1.42428i
$686$		0			0
$687$	−18.8425			−0.718887
$688$		0			0
$689$	5.49856			0.209478
$690$		0			0
$691$		−	6.72118i		−	0.255686i	−0.991794	−	0.127843i	$-0.959195\pi$
$691$		−	6.72118i		−	0.255686i	0.991794	−	0.127843i	$-0.0408053\pi$
$692$		0			0
$693$	−6.33303			−0.240572
$694$		0			0
$695$	18.7763			0.712226
$696$		0			0
$697$	0.223696	−	1.49933i	0.00847311	−	0.0567911i
$698$		0			0
$699$	−28.5039			−1.07812
$700$		0			0
$701$	−4.79347			−0.181047			−0.0905234	−	0.995894i	$-0.528854\pi$
$701$	−4.79347			−0.181047			−0.0905234	+	0.995894i	$0.528854\pi$
$702$		0			0
$703$			4.69508i			0.177078i
$704$		0			0
$705$	7.36113			0.277236
$706$		0			0
$707$	73.7543			2.77382
$708$		0			0
$709$		−	10.5583i		−	0.396525i	−0.980149	−	0.198263i	$-0.936470\pi$
$709$		−	10.5583i		−	0.396525i	0.980149	−	0.198263i	$-0.0635299\pi$
$710$		0			0
$711$		−	11.1786i		−	0.419230i
$712$		0			0
$713$	36.3763			1.36230
$714$		0			0
$715$	−18.9719			−0.709509
$716$		0			0
$717$	6.52111			0.243535
$718$		0			0
$719$		−	30.6922i		−	1.14463i	−0.820035	−	0.572313i	$-0.806046\pi$
$719$		−	30.6922i		−	1.14463i	0.820035	−	0.572313i	$-0.193954\pi$
$720$		0			0
$721$		−	83.5866i		−	3.11293i
$722$		0			0
$723$			18.4565i			0.686404i
$724$		0			0
$725$			78.5685i			2.91796i
$726$		0			0
$727$			4.27594i			0.158586i	0.996851	+	0.0792929i	$0.0252662\pi$
$727$			4.27594i			0.158586i	−0.996851	+	0.0792929i	$0.974734\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			1.85229i			0.0685095i
$732$		0			0
$733$	8.65643			0.319732			0.159866	−	0.987139i	$-0.448894\pi$
$733$	8.65643			0.319732			0.159866	+	0.987139i	$0.448894\pi$
$734$		0			0
$735$		−	65.3326i		−	2.40983i
$736$		0			0
$737$	−6.51646			−0.240037
$738$		0			0
$739$	−28.7820			−1.05876			−0.529382	−	0.848384i	$-0.677576\pi$
$739$	−28.7820			−1.05876			−0.529382	+	0.848384i	$0.677576\pi$
$740$		0			0
$741$			3.57532i			0.131343i
$742$		0			0
$743$	−20.3031			−0.744849			−0.372425	−	0.928062i	$-0.621473\pi$
$743$	−20.3031			−0.744849			−0.372425	+	0.928062i	$0.621473\pi$
$744$		0			0
$745$		−	3.49856i		−	0.128177i
$746$		0			0
$747$	10.0607			0.368101
$748$		0			0
$749$		−	12.4260i		−	0.454036i
$750$		0			0
$751$		−	39.6138i		−	1.44553i	−0.691095	−	0.722764i	$-0.742871\pi$
$751$		−	39.6138i		−	1.44553i	0.691095	−	0.722764i	$-0.257129\pi$
$752$		0			0
$753$			16.7051i			0.608767i
$754$		0			0
$755$		−	52.4284i		−	1.90806i
$756$		0			0
$757$		−	5.42626i		−	0.197221i	−0.995126	−	0.0986105i	$-0.968560\pi$
$757$		−	5.42626i		−	0.197221i	0.995126	−	0.0986105i	$-0.0314398\pi$
$758$		0			0
$759$	9.73515			0.353364
$760$		0			0
$761$	0.815340			0.0295560			0.0147780	−	0.999891i	$-0.495296\pi$
$761$	0.815340			0.0295560			0.0147780	+	0.999891i	$0.495296\pi$
$762$		0			0
$763$	−25.9690			−0.940141
$764$		0			0
$765$			0.840013i			0.0303707i
$766$		0			0
$767$			31.5290i			1.13845i
$768$		0			0
$769$	40.8729			1.47391			0.736957	−	0.675940i	$-0.236262\pi$
$769$	40.8729			1.47391			0.736957	+	0.675940i	$0.236262\pi$
$770$		0			0
$771$	25.6555			0.923960
$772$		0			0
$773$			18.9191i			0.680474i	0.940340	+	0.340237i	$0.110507\pi$
$773$			18.9191i			0.680474i	−0.940340	+	0.340237i	$0.889493\pi$
$774$		0			0
$775$	35.6253			1.27970
$776$		0			0
$777$	−28.1765			−1.01083
$778$		0			0
$779$	−0.793706	+	5.31982i	−0.0284375	+	0.190602i
$780$		0			0
$781$	8.47889			0.303398
$782$		0			0
$783$	10.3525			0.369970
$784$		0			0
$785$		−	62.4785i		−	2.22995i
$786$		0			0
$787$	−45.1238			−1.60849			−0.804246	−	0.594297i	$-0.797430\pi$
$787$	−45.1238			−1.60849			−0.804246	+	0.594297i	$0.797430\pi$
$788$		0			0
$789$	22.8175			0.812323
$790$		0			0
$791$		−	32.1225i		−	1.14215i
$792$		0			0
$793$			45.8439i			1.62796i
$794$		0			0
$795$	−4.58375			−0.162569
$796$		0			0
$797$	−10.6575			−0.377507			−0.188753	−	0.982025i	$-0.560445\pi$
$797$	−10.6575			−0.377507			−0.188753	+	0.982025i	$0.560445\pi$
$798$		0			0
$799$	−0.491167			−0.0173762
$800$		0			0
$801$			3.21511i			0.113600i
$802$		0			0
$803$		−	10.5730i		−	0.373115i
$804$		0			0
$805$			138.609i			4.88533i
$806$		0			0
$807$		−	22.8783i		−	0.805354i
$808$		0			0
$809$			20.5211i			0.721484i	0.932666	+	0.360742i	$0.117477\pi$
$809$			20.5211i			0.721484i	−0.932666	+	0.360742i	$0.882523\pi$
$810$		0			0
$811$	15.2973			0.537160			0.268580	−	0.963257i	$-0.413446\pi$
$811$	15.2973			0.537160			0.268580	+	0.963257i	$0.413446\pi$
$812$		0			0
$813$			24.6856i			0.865761i
$814$		0			0
$815$	51.6651			1.80975
$816$		0			0
$817$		−	6.57219i		−	0.229932i
$818$		0			0
$819$	−21.4565			−0.749750
$820$		0			0
$821$	−1.94356			−0.0678308			−0.0339154	−	0.999425i	$-0.510798\pi$
$821$	−1.94356			−0.0678308			−0.0339154	+	0.999425i	$0.510798\pi$
$822$		0			0
$823$			14.4566i			0.503927i	0.967737	+	0.251963i	$0.0810762\pi$
$823$			14.4566i			0.503927i	−0.967737	+	0.251963i	$0.918924\pi$
$824$		0			0
$825$	9.53416			0.331937
$826$		0			0
$827$		−	20.9443i		−	0.728306i	−0.931339	−	0.364153i	$-0.881359\pi$
$827$		−	20.9443i		−	0.728306i	0.931339	−	0.364153i	$-0.118641\pi$
$828$		0			0
$829$	36.6048			1.27134			0.635668	−	0.771963i	$-0.280725\pi$
$829$	36.6048			1.27134			0.635668	+	0.771963i	$0.280725\pi$
$830$		0			0
$831$			8.09073i			0.280664i
$832$		0			0
$833$			4.35928i			0.151040i
$834$		0			0
$835$			3.22620i			0.111647i
$836$		0			0
$837$		−	4.69415i		−	0.162254i
$838$		0			0
$839$		−	13.6496i		−	0.471236i	−0.971846	−	0.235618i	$-0.924289\pi$
$839$		−	13.6496i		−	0.471236i	0.971846	−	0.235618i	$-0.0757113\pi$
$840$		0			0
$841$	−78.1752			−2.69570
$842$		0			0
$843$	14.0691			0.484566
$844$		0			0
$845$	−18.1516			−0.624433
$846$		0			0
$847$			47.4967i			1.63201i
$848$		0			0
$849$			4.17304i			0.143218i
$850$		0			0
$851$	43.3130			1.48475
$852$		0			0
$853$	45.4568			1.55641			0.778205	−	0.628010i	$-0.216130\pi$
$853$	45.4568			1.55641			0.778205	+	0.628010i	$0.216130\pi$
$854$		0			0
$855$		−	2.98048i		−	0.101930i
$856$		0			0
$857$	−30.5455			−1.04341			−0.521707	−	0.853125i	$-0.674705\pi$
$857$	−30.5455			−1.04341			−0.521707	+	0.853125i	$0.674705\pi$
$858$		0			0
$859$	−13.7305			−0.468480			−0.234240	−	0.972179i	$-0.575260\pi$
$859$	−13.7305			−0.468480			−0.234240	+	0.972179i	$0.575260\pi$
$860$		0			0
$861$	−31.9257	−	4.76325i	−1.08803	−	0.162331i
$862$		0			0
$863$	27.1595			0.924519			0.462259	−	0.886745i	$-0.347039\pi$
$863$	27.1595			0.924519			0.462259	+	0.886745i	$0.347039\pi$
$864$		0			0
$865$	43.8978			1.49257
$866$		0			0
$867$			16.9440i			0.575447i
$868$		0			0
$869$	14.0433			0.476385
$870$		0			0
$871$	−22.0780			−0.748083
$872$		0			0
$873$			3.94487i			0.133514i
$874$		0			0
$875$			46.3139i			1.56570i
$876$		0			0
$877$	−45.5973			−1.53971			−0.769856	−	0.638217i	$-0.779672\pi$
$877$	−45.5973			−1.53971			−0.769856	+	0.638217i	$0.779672\pi$
$878$		0			0
$879$	−8.77973			−0.296133
$880$		0			0
$881$	11.2219			0.378074			0.189037	−	0.981970i	$-0.439463\pi$
$881$	11.2219			0.378074			0.189037	+	0.981970i	$0.439463\pi$
$882$		0			0
$883$		−	56.0339i		−	1.88569i	−0.333230	−	0.942846i	$-0.608138\pi$
$883$		−	56.0339i		−	1.88569i	0.333230	−	0.942846i	$-0.391862\pi$
$884$		0			0
$885$		−	26.2834i		−	0.883508i
$886$		0			0
$887$			36.1204i			1.21280i	0.795158	+	0.606402i	$0.207388\pi$
$887$			36.1204i			1.21280i	−0.795158	+	0.606402i	$0.792612\pi$
$888$		0			0
$889$			42.7441i			1.43359i
$890$		0			0
$891$		−	1.25627i		−	0.0420865i
$892$		0			0
$893$	1.74273			0.0583182
$894$		0			0
$895$		−	66.8858i		−	2.23575i
$896$		0			0
$897$	32.9830			1.10127
$898$		0			0
$899$			48.5964i			1.62078i
$900$		0			0
$901$	0.305848			0.0101893
$902$		0			0
$903$	39.4416			1.31253
$904$		0			0
$905$		−	35.1983i		−	1.17003i
$906$		0			0
$907$	18.3452			0.609142			0.304571	−	0.952490i	$-0.401487\pi$
$907$	18.3452			0.609142			0.304571	+	0.952490i	$0.401487\pi$
$908$		0			0
$909$			14.6304i			0.485261i
$910$		0			0
$911$	−50.1842			−1.66268			−0.831339	−	0.555766i	$-0.812425\pi$
$911$	−50.1842			−1.66268			−0.831339	+	0.555766i	$0.812425\pi$
$912$		0			0
$913$			12.6389i			0.418286i
$914$		0			0
$915$		−	38.2167i		−	1.26340i
$916$		0			0
$917$		−	90.2507i		−	2.98034i
$918$		0			0
$919$			31.1736i			1.02832i	0.857694	+	0.514161i	$0.171897\pi$
$919$			31.1736i			1.02832i	−0.857694	+	0.514161i	$0.828103\pi$
$920$		0			0
$921$			20.3656i			0.671069i
$922$		0			0
$923$	28.7267			0.945552
$924$		0			0
$925$	42.4188			1.39472
$926$		0			0
$927$	16.5809			0.544587
$928$		0			0
$929$			33.2699i			1.09155i	0.837932	+	0.545774i	$0.183764\pi$
$929$			33.2699i			1.09155i	−0.837932	+	0.545774i	$0.816236\pi$
$930$		0			0
$931$		−	15.4673i		−	0.506921i
$932$		0			0
$933$	7.95238			0.260349
$934$		0			0
$935$	−1.05528			−0.0345113
$936$		0			0
$937$			1.97205i			0.0644241i	0.999481	+	0.0322121i	$0.0102552\pi$
$937$			1.97205i			0.0644241i	−0.999481	+	0.0322121i	$0.989745\pi$
$938$		0			0
$939$	12.9699			0.423258
$940$		0			0
$941$	8.57940			0.279680			0.139840	−	0.990174i	$-0.455341\pi$
$941$	8.57940			0.279680			0.139840	+	0.990174i	$0.455341\pi$
$942$		0			0
$943$	49.0764	+	7.32209i	1.59815	+	0.238440i
$944$		0			0
$945$	17.8867			0.581855
$946$		0			0
$947$	2.36729			0.0769265			0.0384633	−	0.999260i	$-0.487754\pi$
$947$	2.36729			0.0769265			0.0384633	+	0.999260i	$0.487754\pi$
$948$		0			0
$949$		−	35.8218i		−	1.16283i
$950$		0			0
$951$	15.2391			0.494162
$952$		0			0
$953$	51.8427			1.67935			0.839674	−	0.543090i	$-0.182746\pi$
$953$	51.8427			1.67935			0.839674	+	0.543090i	$0.182746\pi$
$954$		0			0
$955$		−	12.8046i		−	0.414346i
$956$		0			0
$957$			13.0055i			0.420410i
$958$		0			0
$959$	−52.9627			−1.71025
$960$		0			0
$961$	−8.96493			−0.289191
$962$		0			0
$963$	2.46491			0.0794307
$964$		0			0
$965$			82.9811i			2.67126i
$966$		0			0
$967$		−	41.0971i		−	1.32159i	−0.750565	−	0.660796i	$-0.770219\pi$
$967$		−	41.0971i		−	1.32159i	0.750565	−	0.660796i	$-0.229781\pi$
$968$		0			0
$969$			0.198871i			0.00638865i
$970$		0			0
$971$			49.1931i			1.57868i	0.613955	+	0.789341i	$0.289578\pi$
$971$			49.1931i			1.57868i	−0.613955	+	0.789341i	$0.710422\pi$
$972$		0			0
$973$			26.6771i			0.855230i
$974$		0			0
$975$	32.3020			1.03449
$976$		0			0
$977$		−	25.7031i		−	0.822316i	−0.911564	−	0.411158i	$-0.865124\pi$
$977$		−	25.7031i		−	0.822316i	0.911564	−	0.411158i	$-0.134876\pi$
$978$		0			0
$979$	−4.03904			−0.129088
$980$		0			0
$981$		−	5.15140i		−	0.164472i
$982$		0			0
$983$	17.0729			0.544542			0.272271	−	0.962221i	$-0.412225\pi$
$983$	17.0729			0.544542			0.272271	+	0.962221i	$0.412225\pi$
$984$		0			0
$985$	19.9776			0.636539
$986$		0			0
$987$			10.4586i			0.332901i
$988$		0			0
$989$	−60.6298			−1.92791
$990$		0			0
$991$			42.2641i			1.34256i	0.741203	+	0.671281i	$0.234256\pi$
$991$			42.2641i			1.34256i	−0.741203	+	0.671281i	$0.765744\pi$
$992$		0			0
$993$	27.9560			0.887156
$994$		0			0
$995$		−	58.6852i		−	1.86045i
$996$		0			0
$997$		−	52.6203i		−	1.66650i	−0.552896	−	0.833251i	$-0.686477\pi$
$997$		−	52.6203i		−	1.66650i	0.552896	−	0.833251i	$-0.313523\pi$
$998$		0			0
$999$		−	5.58929i		−	0.176837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.j.f.1393.8		8
4.3	odd	2		984.2.j.b.409.4	✓	8
12.11	even	2		2952.2.j.d.2377.1		8
41.40	even	2	inner	1968.2.j.f.1393.4		8
164.163	odd	2		984.2.j.b.409.8	yes	8
492.491	even	2		2952.2.j.d.2377.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.4	✓	8	4.3	odd	2
984.2.j.b.409.8	yes	8	164.163	odd	2
1968.2.j.f.1393.4		8	41.40	even	2	inner
1968.2.j.f.1393.8		8	1.1	even	1	trivial
2952.2.j.d.2377.1		8	12.11	even	2
2952.2.j.d.2377.2		8	492.491	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.54814 q^{5} +5.04115i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +3.54814 q^{5} +5.04115i q^{7} -1.00000 q^{9} -1.25627i q^{11} -4.25627i q^{13} +3.54814i q^{15} -0.236747i q^{17} +0.840013i q^{19} -5.04115 q^{21} +7.74928 q^{23} +7.58929 q^{25} -1.00000i q^{27} +10.3525i q^{29} +4.69415 q^{31} +1.25627 q^{33} +17.8867i q^{35} +5.58929 q^{37} +4.25627 q^{39} +(6.33303 + 0.944874i) q^{41} -7.82392 q^{43} -3.54814 q^{45} -2.07464i q^{47} -18.4132 q^{49} +0.236747 q^{51} +1.29187i q^{53} -4.45740i q^{55} -0.840013 q^{57} -7.40767 q^{59} -10.7709 q^{61} -5.04115i q^{63} -15.1018i q^{65} -5.18717i q^{67} +7.74928i q^{69} +6.74928i q^{71} +8.41625 q^{73} +7.58929i q^{75} +6.33303 q^{77} +11.1786i q^{79} +1.00000 q^{81} -10.0607 q^{83} -0.840013i q^{85} -10.3525 q^{87} -3.21511i q^{89} +21.4565 q^{91} +4.69415i q^{93} +2.98048i q^{95} -3.94487i q^{97} +1.25627i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q + 10 * q^5 - 8 * q^9	\( 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) \) 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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