LMFDB - Embedded newform 320.4.d.b.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,4,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.d (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:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.4.d.b.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35890i q^{3} -5.00000i q^{5} -16.7945 q^{7} +25.1534 q^{9} +O(q^{10})$	$q+1.35890i q^{3} -5.00000i q^{5} -16.7945 q^{7} +25.1534 q^{9} +0.717798i q^{11} +3.58899i q^{13} +6.79449 q^{15} -54.4602 q^{17} -55.7424i q^{19} -22.8220i q^{21} -143.972 q^{23} -25.0000 q^{25} +70.8712i q^{27} -254.356i q^{29} -213.589 q^{31} -0.975415 q^{33} +83.9725i q^{35} -210.000i q^{37} -4.87707 q^{39} +170.816 q^{41} -7.56146i q^{43} -125.767i q^{45} -336.028 q^{47} -60.9449 q^{49} -74.0059i q^{51} -229.233i q^{53} +3.58899 q^{55} +75.7483 q^{57} +54.0000i q^{59} -485.890i q^{61} -422.439 q^{63} +17.9449 q^{65} -715.856i q^{67} -195.644i q^{69} -40.7670 q^{71} -714.043 q^{73} -33.9725i q^{75} -12.0551i q^{77} -717.424 q^{79} +582.835 q^{81} +1221.71i q^{83} +272.301i q^{85} +345.644 q^{87} -162.920 q^{89} -60.2753i q^{91} -290.246i q^{93} -278.712 q^{95} +576.411 q^{97} +18.0551i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 20 * q^7 - 4 * q^9	$ 4 q + 20 q^{7} - 4 q^{9} - 60 q^{15} + 96 q^{17} - 140 q^{23} - 100 q^{25} - 680 q^{31} - 248 q^{33} - 1240 q^{39} - 328 q^{41} - 1780 q^{47} + 628 q^{49} - 160 q^{55} + 1384 q^{57} - 2300 q^{63} - 800 q^{65} + 360 q^{71} - 1008 q^{73} - 80 q^{79} - 284 q^{81} + 2080 q^{87} - 24 q^{89} + 280 q^{95} + 2480 q^{97}+O(q^{100}) $ 4 * q + 20 * q^7 - 4 * q^9 - 60 * q^15 + 96 * q^17 - 140 * q^23 - 100 * q^25 - 680 * q^31 - 248 * q^33 - 1240 * q^39 - 328 * q^41 - 1780 * q^47 + 628 * q^49 - 160 * q^55 + 1384 * q^57 - 2300 * q^63 - 800 * q^65 + 360 * q^71 - 1008 * q^73 - 80 * q^79 - 284 * q^81 + 2080 * q^87 - 24 * q^89 + 280 * q^95 + 2480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.35890i	0.261520i	0.991414	+	0.130760i	$0.0417418\pi$
$3$	1.35890i	0.261520i	−0.991414	+	0.130760i	$0.958258\pi$
$4$	0	0
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	−16.7945	−0.906817	−0.453409	−	0.891303i	$-0.649792\pi$
$7$	−16.7945	−0.906817	−0.453409	+	0.891303i	$0.649792\pi$
$8$	0	0
$9$	25.1534	0.931607
$10$	0	0
$11$	0.717798i	0.0196749i	0.999952	+	0.00983746i	$0.00313141\pi$
$11$	0.717798i	0.0196749i	−0.999952	+	0.00983746i	$0.996869\pi$
$12$	0	0
$13$	3.58899i	0.0765697i	0.999267	+	0.0382849i	$0.0121894\pi$
$13$	3.58899i	0.0765697i	−0.999267	+	0.0382849i	$0.987811\pi$
$14$	0	0
$15$	6.79449	0.116955
$16$	0	0
$17$	−54.4602	−0.776973	−0.388486	−	0.921455i	$-0.627002\pi$
$17$	−54.4602	−0.776973	−0.388486	+	0.921455i	$0.627002\pi$
$18$	0	0
$19$	− 55.7424i	− 0.673062i	−0.941672	−	0.336531i	$-0.890746\pi$
$19$	− 55.7424i	− 0.673062i	0.941672	−	0.336531i	$-0.109254\pi$
$20$	0	0
$21$	− 22.8220i	− 0.237151i
$22$	0	0
$23$	−143.972	−1.30523	−0.652616	−	0.757689i	$-0.726328\pi$
$23$	−143.972	−1.30523	−0.652616	+	0.757689i	$0.726328\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	70.8712i	0.505154i
$28$	0	0
$29$	− 254.356i	− 1.62871i	−0.580364	−	0.814357i	$-0.697090\pi$
$29$	− 254.356i	− 1.62871i	0.580364	−	0.814357i	$-0.302910\pi$
$30$	0	0
$31$	−213.589	−1.23747	−0.618737	−	0.785598i	$-0.712355\pi$
$31$	−213.589	−1.23747	−0.618737	+	0.785598i	$0.712355\pi$
$32$	0	0
$33$	−0.975415	−0.00514539
$34$	0	0
$35$	83.9725i	0.405541i
$36$	0	0
$37$	− 210.000i	− 0.933075i	−0.884501	−	0.466538i	$-0.845501\pi$
$37$	− 210.000i	− 0.933075i	0.884501	−	0.466538i	$-0.154499\pi$
$38$	0	0
$39$	−4.87707	−0.0200245
$40$	0	0
$41$	170.816	0.650659	0.325329	−	0.945601i	$-0.394525\pi$
$41$	170.816	0.650659	0.325329	+	0.945601i	$0.394525\pi$
$42$	0	0
$43$	− 7.56146i	− 0.0268166i	−0.999910	−	0.0134083i	$-0.995732\pi$
$43$	− 7.56146i	− 0.0268166i	0.999910	−	0.0134083i	$-0.00426812\pi$
$44$	0	0
$45$	− 125.767i	− 0.416627i
$46$	0	0
$47$	−336.028	−1.04286	−0.521432	−	0.853293i	$-0.674602\pi$
$47$	−336.028	−1.04286	−0.521432	+	0.853293i	$0.674602\pi$
$48$	0	0
$49$	−60.9449	−0.177682
$50$	0	0
$51$	− 74.0059i	− 0.203194i
$52$	0	0
$53$	− 229.233i	− 0.594105i	−0.954861	−	0.297053i	$-0.903996\pi$
$53$	− 229.233i	− 0.594105i	0.954861	−	0.297053i	$-0.0960037\pi$
$54$	0	0
$55$	3.58899	0.00879890
$56$	0	0
$57$	75.7483	0.176019
$58$	0	0
$59$	54.0000i	0.119156i	0.998224	+	0.0595780i	$0.0189755\pi$
$59$	54.0000i	0.119156i	−0.998224	+	0.0595780i	$0.981025\pi$
$60$	0	0
$61$	− 485.890i	− 1.01987i	−0.860214	−	0.509933i	$-0.829670\pi$
$61$	− 485.890i	− 1.01987i	0.860214	−	0.509933i	$-0.170330\pi$
$62$	0	0
$63$	−422.439	−0.844798
$64$	0	0
$65$	17.9449	0.0342430
$66$	0	0
$67$	− 715.856i	− 1.30531i	−0.757655	−	0.652655i	$-0.773655\pi$
$67$	− 715.856i	− 1.30531i	0.757655	−	0.652655i	$-0.226345\pi$
$68$	0	0
$69$	− 195.644i	− 0.341345i
$70$	0	0
$71$	−40.7670	−0.0681429	−0.0340715	−	0.999419i	$-0.510847\pi$
$71$	−40.7670	−0.0681429	−0.0340715	+	0.999419i	$0.510847\pi$
$72$	0	0
$73$	−714.043	−1.14483	−0.572414	−	0.819965i	$-0.693993\pi$
$73$	−714.043	−1.14483	−0.572414	+	0.819965i	$0.693993\pi$
$74$	0	0
$75$	− 33.9725i	− 0.0523040i
$76$	0	0
$77$	− 12.0551i	− 0.0178416i
$78$	0	0
$79$	−717.424	−1.02173	−0.510864	−	0.859662i	$-0.670674\pi$
$79$	−717.424	−1.02173	−0.510864	+	0.859662i	$0.670674\pi$
$80$	0	0
$81$	582.835	0.799499
$82$	0	0
$83$	1221.71i	1.61566i	0.589414	+	0.807831i	$0.299359\pi$
$83$	1221.71i	1.61566i	−0.589414	+	0.807831i	$0.700641\pi$
$84$	0	0
$85$	272.301i	0.347473i
$86$	0	0
$87$	345.644	0.425942
$88$	0	0
$89$	−162.920	−0.194040	−0.0970198	−	0.995282i	$-0.530931\pi$
$89$	−162.920	−0.194040	−0.0970198	+	0.995282i	$0.530931\pi$
$90$	0	0
$91$	− 60.2753i	− 0.0694348i
$92$	0	0
$93$	− 290.246i	− 0.323625i
$94$	0	0
$95$	−278.712	−0.301003
$96$	0	0
$97$	576.411	0.603357	0.301679	−	0.953410i	$-0.402453\pi$
$97$	576.411	0.603357	0.301679	+	0.953410i	$0.402453\pi$
$98$	0	0
$99$	18.0551i	0.0183293i
$100$	0	0
$101$	880.492i	0.867448i	0.901046	+	0.433724i	$0.142801\pi$
$101$	880.492i	0.867448i	−0.901046	+	0.433724i	$0.857199\pi$
$102$	0	0
$103$	1564.74	1.49688	0.748439	−	0.663204i	$-0.230804\pi$
$103$	1564.74	1.49688	0.748439	+	0.663204i	$0.230804\pi$
$104$	0	0
$105$	−114.110	−0.106057
$106$	0	0
$107$	− 358.224i	− 0.323653i	−0.986819	−	0.161826i	$-0.948262\pi$
$107$	− 358.224i	− 0.323653i	0.986819	−	0.161826i	$-0.0517384\pi$
$108$	0	0
$109$	1757.67i	1.54453i	0.635298	+	0.772267i	$0.280877\pi$
$109$	1757.67i	1.54453i	−0.635298	+	0.772267i	$0.719123\pi$
$110$	0	0
$111$	285.369	0.244018
$112$	0	0
$113$	−1226.72	−1.02124	−0.510621	−	0.859806i	$-0.670585\pi$
$113$	−1226.72	−1.02124	−0.510621	+	0.859806i	$0.670585\pi$
$114$	0	0
$115$	719.862i	0.583717i
$116$	0	0
$117$	90.2753i	0.0713329i
$118$	0	0
$119$	914.631	0.704572
$120$	0	0
$121$	1330.48	0.999613
$122$	0	0
$123$	232.122i	0.170160i
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	261.672	0.182831	0.0914157	−	0.995813i	$-0.470861\pi$
$127$	261.672	0.182831	0.0914157	+	0.995813i	$0.470861\pi$
$128$	0	0
$129$	10.2753	0.00701307
$130$	0	0
$131$	− 2297.72i	− 1.53247i	−0.642562	−	0.766233i	$-0.722129\pi$
$131$	− 2297.72i	− 1.53247i	0.642562	−	0.766233i	$-0.277871\pi$
$132$	0	0
$133$	936.165i	0.610344i
$134$	0	0
$135$	354.356	0.225912
$136$	0	0
$137$	20.1710	0.0125790	0.00628952	−	0.999980i	$-0.497998\pi$
$137$	20.1710	0.0125790	0.00628952	+	0.999980i	$0.497998\pi$
$138$	0	0
$139$	− 972.970i	− 0.593714i	−0.954922	−	0.296857i	$-0.904062\pi$
$139$	− 972.970i	− 0.593714i	0.954922	−	0.296857i	$-0.0959384\pi$
$140$	0	0
$141$	− 456.627i	− 0.272730i
$142$	0	0
$143$	−2.57617	−0.00150650
$144$	0	0
$145$	−1271.78	−0.728383
$146$	0	0
$147$	− 82.8180i	− 0.0464675i
$148$	0	0
$149$	6.44042i	0.00354107i	0.999998	+	0.00177054i	$0.000563580\pi$
$149$	6.44042i	0.00354107i	−0.999998	+	0.00177054i	$0.999436\pi$
$150$	0	0
$151$	3086.41	1.66337	0.831684	−	0.555249i	$-0.187377\pi$
$151$	3086.41	1.66337	0.831684	+	0.555249i	$0.187377\pi$
$152$	0	0
$153$	−1369.86	−0.723833
$154$	0	0
$155$	1067.94i	0.553415i
$156$	0	0
$157$	2820.74i	1.43388i	0.697134	+	0.716941i	$0.254458\pi$
$157$	2820.74i	1.43388i	−0.697134	+	0.716941i	$0.745542\pi$
$158$	0	0
$159$	311.505	0.155371
$160$	0	0
$161$	2417.94	1.18361
$162$	0	0
$163$	3351.60i	1.61054i	0.592909	+	0.805270i	$0.297980\pi$
$163$	3351.60i	1.61054i	−0.592909	+	0.805270i	$0.702020\pi$
$164$	0	0
$165$	4.87707i	0.00230109i
$166$	0	0
$167$	2658.82	1.23201	0.616005	−	0.787742i	$-0.288750\pi$
$167$	2658.82	1.23201	0.616005	+	0.787742i	$0.288750\pi$
$168$	0	0
$169$	2184.12	0.994137
$170$	0	0
$171$	− 1402.11i	− 0.627029i
$172$	0	0
$173$	1582.27i	0.695363i	0.937613	+	0.347682i	$0.113031\pi$
$173$	1582.27i	0.695363i	−0.937613	+	0.347682i	$0.886969\pi$
$174$	0	0
$175$	419.862	0.181363
$176$	0	0
$177$	−73.3805	−0.0311617
$178$	0	0
$179$	− 158.614i	− 0.0662309i	−0.999452	−	0.0331155i	$-0.989457\pi$
$179$	− 158.614i	− 0.0662309i	0.999452	−	0.0331155i	$-0.0105429\pi$
$180$	0	0
$181$	2690.25i	1.10478i	0.833587	+	0.552388i	$0.186283\pi$
$181$	2690.25i	1.10478i	−0.833587	+	0.552388i	$0.813717\pi$
$182$	0	0
$183$	660.275	0.266716
$184$	0	0
$185$	−1050.00	−0.417284
$186$	0	0
$187$	− 39.0914i	− 0.0152869i
$188$	0	0
$189$	− 1190.25i	− 0.458083i
$190$	0	0
$191$	−4146.35	−1.57078	−0.785391	−	0.618999i	$-0.787538\pi$
$191$	−4146.35	−1.57078	−0.785391	+	0.618999i	$0.787538\pi$
$192$	0	0
$193$	−3834.67	−1.43018	−0.715092	−	0.699030i	$-0.753615\pi$
$193$	−3834.67	−1.43018	−0.715092	+	0.699030i	$0.753615\pi$
$194$	0	0
$195$	24.3854i	0.00895524i
$196$	0	0
$197$	− 2525.61i	− 0.913414i	−0.889617	−	0.456707i	$-0.849029\pi$
$197$	− 2525.61i	− 0.913414i	0.889617	−	0.456707i	$-0.150971\pi$
$198$	0	0
$199$	4216.38	1.50197	0.750983	−	0.660321i	$-0.229580\pi$
$199$	4216.38	1.50197	0.750983	+	0.660321i	$0.229580\pi$
$200$	0	0
$201$	972.777	0.341365
$202$	0	0
$203$	4271.78i	1.47695i
$204$	0	0
$205$	− 854.081i	− 0.290983i
$206$	0	0
$207$	−3621.40	−1.21596
$208$	0	0
$209$	40.0118	0.0132424
$210$	0	0
$211$	− 3880.17i	− 1.26598i	−0.774160	−	0.632990i	$-0.781827\pi$
$211$	− 3880.17i	− 1.26598i	0.774160	−	0.632990i	$-0.218173\pi$
$212$	0	0
$213$	− 55.3982i	− 0.0178208i
$214$	0	0
$215$	−37.8073	−0.0119927
$216$	0	0
$217$	3587.12	1.12216
$218$	0	0
$219$	− 970.313i	− 0.299396i
$220$	0	0
$221$	− 195.457i	− 0.0594926i
$222$	0	0
$223$	−3702.10	−1.11171	−0.555855	−	0.831279i	$-0.687609\pi$
$223$	−3702.10	−1.11171	−0.555855	+	0.831279i	$0.687609\pi$
$224$	0	0
$225$	−628.835	−0.186321
$226$	0	0
$227$	− 3779.12i	− 1.10497i	−0.833521	−	0.552487i	$-0.813679\pi$
$227$	− 3779.12i	− 1.10497i	0.833521	−	0.552487i	$-0.186321\pi$
$228$	0	0
$229$	− 1190.25i	− 0.343466i	−0.985144	−	0.171733i	$-0.945063\pi$
$229$	− 1190.25i	− 0.343466i	0.985144	−	0.171733i	$-0.0549366\pi$
$230$	0	0
$231$	16.3816	0.00466593
$232$	0	0
$233$	−6429.67	−1.80782	−0.903909	−	0.427726i	$-0.859315\pi$
$233$	−6429.67	−1.80782	−0.903909	+	0.427726i	$0.859315\pi$
$234$	0	0
$235$	1680.14i	0.466383i
$236$	0	0
$237$	− 974.906i	− 0.267202i
$238$	0	0
$239$	−4327.18	−1.17114	−0.585569	−	0.810623i	$-0.699129\pi$
$239$	−4327.18	−1.17114	−0.585569	+	0.810623i	$0.699129\pi$
$240$	0	0
$241$	3257.40	0.870655	0.435327	−	0.900272i	$-0.356633\pi$
$241$	3257.40	0.870655	0.435327	+	0.900272i	$0.356633\pi$
$242$	0	0
$243$	2705.54i	0.714240i
$244$	0	0
$245$	304.725i	0.0794618i
$246$	0	0
$247$	200.059	0.0515362
$248$	0	0
$249$	−1660.18	−0.422528
$250$	0	0
$251$	− 6051.00i	− 1.52166i	−0.648953	−	0.760829i	$-0.724793\pi$
$251$	− 6051.00i	− 1.52166i	0.648953	−	0.760829i	$-0.275207\pi$
$252$	0	0
$253$	− 103.343i	− 0.0256803i
$254$	0	0
$255$	−370.029	−0.0908711
$256$	0	0
$257$	6369.39	1.54596	0.772980	−	0.634431i	$-0.218765\pi$
$257$	6369.39	1.54596	0.772980	+	0.634431i	$0.218765\pi$
$258$	0	0
$259$	3526.84i	0.846129i
$260$	0	0
$261$	− 6397.92i	− 1.51732i
$262$	0	0
$263$	−871.976	−0.204442	−0.102221	−	0.994762i	$-0.532595\pi$
$263$	−871.976	−0.204442	−0.102221	+	0.994762i	$0.532595\pi$
$264$	0	0
$265$	−1146.17	−0.265692
$266$	0	0
$267$	− 221.392i	− 0.0507453i
$268$	0	0
$269$	− 3394.60i	− 0.769415i	−0.923039	−	0.384707i	$-0.874302\pi$
$269$	− 3394.60i	− 0.769415i	0.923039	−	0.384707i	$-0.125698\pi$
$270$	0	0
$271$	836.598	0.187527	0.0937633	−	0.995595i	$-0.470110\pi$
$271$	836.598	0.187527	0.0937633	+	0.995595i	$0.470110\pi$
$272$	0	0
$273$	81.9080	0.0181586
$274$	0	0
$275$	− 17.9449i	− 0.00393499i
$276$	0	0
$277$	2115.64i	0.458905i	0.973320	+	0.229453i	$0.0736936\pi$
$277$	2115.64i	0.458905i	−0.973320	+	0.229453i	$0.926306\pi$
$278$	0	0
$279$	−5372.49	−1.15284
$280$	0	0
$281$	3996.50	0.848437	0.424219	−	0.905560i	$-0.360549\pi$
$281$	3996.50	0.848437	0.424219	+	0.905560i	$0.360549\pi$
$282$	0	0
$283$	6248.34i	1.31246i	0.754563	+	0.656228i	$0.227849\pi$
$283$	6248.34i	1.31246i	−0.754563	+	0.656228i	$0.772151\pi$
$284$	0	0
$285$	− 378.741i	− 0.0787182i
$286$	0	0
$287$	−2868.77	−0.590029
$288$	0	0
$289$	−1947.09	−0.396314
$290$	0	0
$291$	783.284i	0.157790i
$292$	0	0
$293$	− 7788.41i	− 1.55291i	−0.630171	−	0.776457i	$-0.717015\pi$
$293$	− 7788.41i	− 1.55291i	0.630171	−	0.776457i	$-0.282985\pi$
$294$	0	0
$295$	270.000	0.0532882
$296$	0	0
$297$	−50.8712	−0.00993888
$298$	0	0
$299$	− 516.716i	− 0.0999413i
$300$	0	0
$301$	126.991i	0.0243177i
$302$	0	0
$303$	−1196.50	−0.226855
$304$	0	0
$305$	−2429.45	−0.456098
$306$	0	0
$307$	− 3882.83i	− 0.721839i	−0.932597	−	0.360920i	$-0.882463\pi$
$307$	− 3882.83i	− 0.721839i	0.932597	−	0.360920i	$-0.117537\pi$
$308$	0	0
$309$	2126.32i	0.391464i
$310$	0	0
$311$	−4420.22	−0.805940	−0.402970	−	0.915213i	$-0.632022\pi$
$311$	−4420.22	−0.805940	−0.402970	+	0.915213i	$0.632022\pi$
$312$	0	0
$313$	−6303.25	−1.13828	−0.569139	−	0.822241i	$-0.692723\pi$
$313$	−6303.25	−1.13828	−0.569139	+	0.822241i	$0.692723\pi$
$314$	0	0
$315$	2112.19i	0.377805i
$316$	0	0
$317$	− 9788.56i	− 1.73432i	−0.498027	−	0.867162i	$-0.665942\pi$
$317$	− 9788.56i	− 1.73432i	0.498027	−	0.867162i	$-0.334058\pi$
$318$	0	0
$319$	182.576	0.0320448
$320$	0	0
$321$	486.790	0.0846417
$322$	0	0
$323$	3035.74i	0.522951i
$324$	0	0
$325$	− 89.7247i	− 0.0153139i
$326$	0	0
$327$	−2388.50	−0.403927
$328$	0	0
$329$	5643.41	0.945688
$330$	0	0
$331$	− 8560.60i	− 1.42155i	−0.703419	−	0.710775i	$-0.748344\pi$
$331$	− 8560.60i	− 1.42155i	0.703419	−	0.710775i	$-0.251656\pi$
$332$	0	0
$333$	− 5282.21i	− 0.869260i
$334$	0	0
$335$	−3579.28	−0.583753
$336$	0	0
$337$	−4996.43	−0.807635	−0.403818	−	0.914839i	$-0.632317\pi$
$337$	−4996.43	−0.807635	−0.403818	+	0.914839i	$0.632317\pi$
$338$	0	0
$339$	− 1666.99i	− 0.267076i
$340$	0	0
$341$	− 153.314i	− 0.0243472i
$342$	0	0
$343$	6784.05	1.06794
$344$	0	0
$345$	−978.220	−0.152654
$346$	0	0
$347$	2924.57i	0.452447i	0.974075	+	0.226223i	$0.0726379\pi$
$347$	2924.57i	0.452447i	−0.974075	+	0.226223i	$0.927362\pi$
$348$	0	0
$349$	5116.38i	0.784738i	0.919808	+	0.392369i	$0.128344\pi$
$349$	5116.38i	0.784738i	−0.919808	+	0.392369i	$0.871656\pi$
$350$	0	0
$351$	−254.356	−0.0386795
$352$	0	0
$353$	−4474.60	−0.674671	−0.337336	−	0.941384i	$-0.609526\pi$
$353$	−4474.60	−0.674671	−0.337336	+	0.941384i	$0.609526\pi$
$354$	0	0
$355$	203.835i	0.0304745i
$356$	0	0
$357$	1242.89i	0.184260i
$358$	0	0
$359$	−831.347	−0.122220	−0.0611098	−	0.998131i	$-0.519464\pi$
$359$	−831.347	−0.122220	−0.0611098	+	0.998131i	$0.519464\pi$
$360$	0	0
$361$	3751.79	0.546987
$362$	0	0
$363$	1807.99i	0.261419i
$364$	0	0
$365$	3570.22i	0.511983i
$366$	0	0
$367$	6842.16	0.973183	0.486591	−	0.873630i	$-0.338240\pi$
$367$	6842.16	0.973183	0.486591	+	0.873630i	$0.338240\pi$
$368$	0	0
$369$	4296.61	0.606158
$370$	0	0
$371$	3849.85i	0.538745i
$372$	0	0
$373$	11494.1i	1.59555i	0.602957	+	0.797774i	$0.293989\pi$
$373$	11494.1i	1.59555i	−0.602957	+	0.797774i	$0.706011\pi$
$374$	0	0
$375$	−169.862	−0.0233911
$376$	0	0
$377$	912.881	0.124710
$378$	0	0
$379$	− 9900.29i	− 1.34180i	−0.741546	−	0.670902i	$-0.765907\pi$
$379$	− 9900.29i	− 1.34180i	0.741546	−	0.670902i	$-0.234093\pi$
$380$	0	0
$381$	355.585i	0.0478141i
$382$	0	0
$383$	−68.7907	−0.00917765	−0.00458883	−	0.999989i	$-0.501461\pi$
$383$	−68.7907	−0.00917765	−0.00458883	+	0.999989i	$0.501461\pi$
$384$	0	0
$385$	−60.2753	−0.00797899
$386$	0	0
$387$	− 190.196i	− 0.0249825i
$388$	0	0
$389$	− 2390.06i	− 0.311519i	−0.987795	−	0.155759i	$-0.950218\pi$
$389$	− 2390.06i	− 0.311519i	0.987795	−	0.155759i	$-0.0497824\pi$
$390$	0	0
$391$	7840.77	1.01413
$392$	0	0
$393$	3122.38	0.400771
$394$	0	0
$395$	3587.12i	0.456931i
$396$	0	0
$397$	10021.0i	1.26685i	0.773803	+	0.633426i	$0.218352\pi$
$397$	10021.0i	1.26685i	−0.773803	+	0.633426i	$0.781648\pi$
$398$	0	0
$399$	−1272.15	−0.159617
$400$	0	0
$401$	3698.40	0.460571	0.230286	−	0.973123i	$-0.426034\pi$
$401$	3698.40	0.460571	0.230286	+	0.973123i	$0.426034\pi$
$402$	0	0
$403$	− 766.569i	− 0.0947531i
$404$	0	0
$405$	− 2914.17i	− 0.357547i
$406$	0	0
$407$	150.738	0.0183582
$408$	0	0
$409$	527.296	0.0637484	0.0318742	−	0.999492i	$-0.489852\pi$
$409$	527.296	0.0637484	0.0318742	+	0.999492i	$0.489852\pi$
$410$	0	0
$411$	27.4104i	0.00328967i
$412$	0	0
$413$	− 906.903i	− 0.108053i
$414$	0	0
$415$	6108.54	0.722546
$416$	0	0
$417$	1322.17	0.155268
$418$	0	0
$419$	6191.63i	0.721912i	0.932583	+	0.360956i	$0.117550\pi$
$419$	6191.63i	0.721912i	−0.932583	+	0.360956i	$0.882450\pi$
$420$	0	0
$421$	1336.75i	0.154748i	0.997002	+	0.0773741i	$0.0246536\pi$
$421$	1336.75i	0.154748i	−0.997002	+	0.0773741i	$0.975346\pi$
$422$	0	0
$423$	−8452.23	−0.971540
$424$	0	0
$425$	1361.50	0.155395
$426$	0	0
$427$	8160.28i	0.924832i
$428$	0	0
$429$	− 3.50075i	0 0.000393981i
$430$	0	0
$431$	2062.36	0.230488	0.115244	−	0.993337i	$-0.463235\pi$
$431$	2062.36	0.230488	0.115244	+	0.993337i	$0.463235\pi$
$432$	0	0
$433$	1924.42	0.213584	0.106792	−	0.994281i	$-0.465942\pi$
$433$	1924.42	0.213584	0.106792	+	0.994281i	$0.465942\pi$
$434$	0	0
$435$	− 1728.22i	− 0.190487i
$436$	0	0
$437$	8025.37i	0.878502i
$438$	0	0
$439$	8462.03	0.919978	0.459989	−	0.887925i	$-0.347853\pi$
$439$	8462.03	0.919978	0.459989	+	0.887925i	$0.347853\pi$
$440$	0	0
$441$	−1532.97	−0.165530
$442$	0	0
$443$	− 5302.36i	− 0.568674i	−0.958724	−	0.284337i	$-0.908227\pi$
$443$	− 5302.36i	− 0.568674i	0.958724	−	0.284337i	$-0.0917735\pi$
$444$	0	0
$445$	814.602i	0.0867771i
$446$	0	0
$447$	−8.75188	−0.000926062 0
$448$	0	0
$449$	−11866.6	−1.24726	−0.623631	−	0.781719i	$-0.714343\pi$
$449$	−11866.6	−1.24726	−0.623631	+	0.781719i	$0.714343\pi$
$450$	0	0
$451$	122.611i	0.0128017i
$452$	0	0
$453$	4194.12i	0.435004i
$454$	0	0
$455$	−301.376	−0.0310522
$456$	0	0
$457$	−4714.41	−0.482562	−0.241281	−	0.970455i	$-0.577568\pi$
$457$	−4714.41	−0.482562	−0.241281	+	0.970455i	$0.577568\pi$
$458$	0	0
$459$	− 3859.66i	− 0.392491i
$460$	0	0
$461$	13233.1i	1.33694i	0.743740	+	0.668469i	$0.233050\pi$
$461$	13233.1i	1.33694i	−0.743740	+	0.668469i	$0.766950\pi$
$462$	0	0
$463$	18060.0	1.81278	0.906390	−	0.422441i	$-0.138827\pi$
$463$	18060.0	1.81278	0.906390	+	0.422441i	$0.138827\pi$
$464$	0	0
$465$	−1451.23	−0.144729
$466$	0	0
$467$	− 16999.4i	− 1.68445i	−0.539127	−	0.842224i	$-0.681246\pi$
$467$	− 16999.4i	− 1.68445i	0.539127	−	0.842224i	$-0.318754\pi$
$468$	0	0
$469$	12022.4i	1.18368i
$470$	0	0
$471$	−3833.10	−0.374989
$472$	0	0
$473$	5.42760	0.000527614 0
$474$	0	0
$475$	1393.56i	0.134612i
$476$	0	0
$477$	− 5765.99i	− 0.553473i
$478$	0	0
$479$	18926.9	1.80541	0.902705	−	0.430260i	$-0.141578\pi$
$479$	18926.9	1.80541	0.902705	+	0.430260i	$0.141578\pi$
$480$	0	0
$481$	753.688	0.0714453
$482$	0	0
$483$	3285.74i	0.309537i
$484$	0	0
$485$	− 2882.06i	− 0.269830i
$486$	0	0
$487$	3391.43	0.315565	0.157783	−	0.987474i	$-0.449566\pi$
$487$	3391.43	0.315565	0.157783	+	0.987474i	$0.449566\pi$
$488$	0	0
$489$	−4554.49	−0.421189
$490$	0	0
$491$	11092.0i	1.01950i	0.860323	+	0.509749i	$0.170262\pi$
$491$	11092.0i	1.01950i	−0.860323	+	0.509749i	$0.829738\pi$
$492$	0	0
$493$	13852.3i	1.26547i
$494$	0	0
$495$	90.2753	0.00819711
$496$	0	0
$497$	684.661	0.0617932
$498$	0	0
$499$	15719.1i	1.41019i	0.709112	+	0.705095i	$0.249096\pi$
$499$	15719.1i	1.41019i	−0.709112	+	0.705095i	$0.750904\pi$
$500$	0	0
$501$	3613.07i	0.322196i
$502$	0	0
$503$	−9128.52	−0.809186	−0.404593	−	0.914497i	$-0.632587\pi$
$503$	−9128.52	−0.809186	−0.404593	+	0.914497i	$0.632587\pi$
$504$	0	0
$505$	4402.46	0.387934
$506$	0	0
$507$	2968.00i	0.259987i
$508$	0	0
$509$	− 12016.8i	− 1.04643i	−0.852200	−	0.523215i	$-0.824732\pi$
$509$	− 12016.8i	− 1.04643i	0.852200	−	0.523215i	$-0.175268\pi$
$510$	0	0
$511$	11992.0	1.03815
$512$	0	0
$513$	3950.53	0.340000
$514$	0	0
$515$	− 7823.70i	− 0.669424i
$516$	0	0
$517$	− 241.200i	− 0.0205183i
$518$	0	0
$519$	−2150.15	−0.181852
$520$	0	0
$521$	−11920.4	−1.00238	−0.501191	−	0.865336i	$-0.667105\pi$
$521$	−11920.4	−1.00238	−0.501191	+	0.865336i	$0.667105\pi$
$522$	0	0
$523$	− 14741.9i	− 1.23254i	−0.787534	−	0.616271i	$-0.788643\pi$
$523$	− 14741.9i	− 1.23254i	0.787534	−	0.616271i	$-0.211357\pi$
$524$	0	0
$525$	570.551i	0.0474302i
$526$	0	0
$527$	11632.1	0.961484
$528$	0	0
$529$	8561.07	0.703631
$530$	0	0
$531$	1358.28i	0.111007i
$532$	0	0
$533$	613.057i	0.0498208i
$534$	0	0
$535$	−1791.12	−0.144742
$536$	0	0
$537$	215.540	0.0173207
$538$	0	0
$539$	− 43.7462i	− 0.00349588i
$540$	0	0
$541$	3280.49i	0.260701i	0.991468	+	0.130351i	$0.0416103\pi$
$541$	3280.49i	0.260701i	−0.991468	+	0.130351i	$0.958390\pi$
$542$	0	0
$543$	−3655.77	−0.288921
$544$	0	0
$545$	8788.35	0.690737
$546$	0	0
$547$	− 6271.38i	− 0.490210i	−0.969497	−	0.245105i	$-0.921178\pi$
$547$	− 6271.38i	− 0.490210i	0.969497	−	0.245105i	$-0.0788224\pi$
$548$	0	0
$549$	− 12221.8i	− 0.950114i
$550$	0	0
$551$	−14178.4	−1.09623
$552$	0	0
$553$	12048.8	0.926521
$554$	0	0
$555$	− 1426.84i	− 0.109128i
$556$	0	0
$557$	− 1042.45i	− 0.0792997i	−0.999214	−	0.0396499i	$-0.987376\pi$
$557$	− 1042.45i	− 0.0792997i	0.999214	−	0.0396499i	$-0.0126242\pi$
$558$	0	0
$559$	27.1380	0.00205334
$560$	0	0
$561$	53.1213	0.00399783
$562$	0	0
$563$	4076.37i	0.305148i	0.988292	+	0.152574i	$0.0487563\pi$
$563$	4076.37i	0.305148i	−0.988292	+	0.152574i	$0.951244\pi$
$564$	0	0
$565$	6133.62i	0.456714i
$566$	0	0
$567$	−9788.42	−0.725000
$568$	0	0
$569$	−2072.62	−0.152705	−0.0763523	−	0.997081i	$-0.524327\pi$
$569$	−2072.62	−0.152705	−0.0763523	+	0.997081i	$0.524327\pi$
$570$	0	0
$571$	19757.9i	1.44806i	0.689769	+	0.724029i	$0.257712\pi$
$571$	19757.9i	1.44806i	−0.689769	+	0.724029i	$0.742288\pi$
$572$	0	0
$573$	− 5634.47i	− 0.410791i
$574$	0	0
$575$	3599.31	0.261046
$576$	0	0
$577$	−808.717	−0.0583489	−0.0291745	−	0.999574i	$-0.509288\pi$
$577$	−808.717	−0.0583489	−0.0291745	+	0.999574i	$0.509288\pi$
$578$	0	0
$579$	− 5210.93i	− 0.374022i
$580$	0	0
$581$	− 20518.0i	− 1.46511i
$582$	0	0
$583$	164.543	0.0116890
$584$	0	0
$585$	451.376	0.0319010
$586$	0	0
$587$	16454.8i	1.15700i	0.815682	+	0.578501i	$0.196362\pi$
$587$	16454.8i	1.15700i	−0.815682	+	0.578501i	$0.803638\pi$
$588$	0	0
$589$	11906.0i	0.832897i
$590$	0	0
$591$	3432.06	0.238876
$592$	0	0
$593$	9485.67	0.656880	0.328440	−	0.944525i	$-0.393477\pi$
$593$	9485.67	0.656880	0.328440	+	0.944525i	$0.393477\pi$
$594$	0	0
$595$	− 4573.16i	− 0.315094i
$596$	0	0
$597$	5729.64i	0.392795i
$598$	0	0
$599$	−26035.2	−1.77591	−0.887955	−	0.459930i	$-0.847875\pi$
$599$	−26035.2	−1.77591	−0.887955	+	0.459930i	$0.847875\pi$
$600$	0	0
$601$	−10539.8	−0.715354	−0.357677	−	0.933845i	$-0.616431\pi$
$601$	−10539.8	−0.715354	−0.357677	+	0.933845i	$0.616431\pi$
$602$	0	0
$603$	− 18006.2i	− 1.21604i
$604$	0	0
$605$	− 6652.42i	− 0.447040i
$606$	0	0
$607$	104.956	0.00701817	0.00350908	−	0.999994i	$-0.498883\pi$
$607$	104.956	0.00701817	0.00350908	+	0.999994i	$0.498883\pi$
$608$	0	0
$609$	−5804.92	−0.386251
$610$	0	0
$611$	− 1206.00i	− 0.0798519i
$612$	0	0
$613$	− 24300.8i	− 1.60114i	−0.599236	−	0.800572i	$-0.704529\pi$
$613$	− 24300.8i	− 1.60114i	0.599236	−	0.800572i	$-0.295471\pi$
$614$	0	0
$615$	1160.61	0.0760980
$616$	0	0
$617$	−4788.13	−0.312420	−0.156210	−	0.987724i	$-0.549928\pi$
$617$	−4788.13	−0.312420	−0.156210	+	0.987724i	$0.549928\pi$
$618$	0	0
$619$	− 18888.8i	− 1.22650i	−0.789888	−	0.613251i	$-0.789861\pi$
$619$	− 18888.8i	− 1.22650i	0.789888	−	0.613251i	$-0.210139\pi$
$620$	0	0
$621$	− 10203.5i	− 0.659344i
$622$	0	0
$623$	2736.17	0.175958
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	54.3719i	0.00346317i
$628$	0	0
$629$	11436.6i	0.724974i
$630$	0	0
$631$	−2995.12	−0.188960	−0.0944802	−	0.995527i	$-0.530119\pi$
$631$	−2995.12	−0.188960	−0.0944802	+	0.995527i	$0.530119\pi$
$632$	0	0
$633$	5272.76	0.331079
$634$	0	0
$635$	− 1308.36i	− 0.0817647i
$636$	0	0
$637$	− 218.731i	− 0.0136051i
$638$	0	0
$639$	−1025.43	−0.0634825
$640$	0	0
$641$	−25688.0	−1.58286	−0.791431	−	0.611259i	$-0.790663\pi$
$641$	−25688.0	−1.58286	−0.791431	+	0.611259i	$0.790663\pi$
$642$	0	0
$643$	− 14410.7i	− 0.883828i	−0.897057	−	0.441914i	$-0.854300\pi$
$643$	− 14410.7i	− 0.883828i	0.897057	−	0.441914i	$-0.145700\pi$
$644$	0	0
$645$	− 51.3763i	− 0.00313634i
$646$	0	0
$647$	−28091.3	−1.70693	−0.853464	−	0.521152i	$-0.825503\pi$
$647$	−28091.3	−1.70693	−0.853464	+	0.521152i	$0.825503\pi$
$648$	0	0
$649$	−38.7611	−0.00234438
$650$	0	0
$651$	4874.53i	0.293468i
$652$	0	0
$653$	− 726.342i	− 0.0435282i	−0.999763	−	0.0217641i	$-0.993072\pi$
$653$	− 726.342i	− 0.0435282i	0.999763	−	0.0217641i	$-0.00692828\pi$
$654$	0	0
$655$	−11488.6	−0.685340
$656$	0	0
$657$	−17960.6	−1.06653
$658$	0	0
$659$	23580.2i	1.39386i	0.717138	+	0.696931i	$0.245452\pi$
$659$	23580.2i	1.39386i	−0.717138	+	0.696931i	$0.754548\pi$
$660$	0	0
$661$	14478.6i	0.851970i	0.904730	+	0.425985i	$0.140072\pi$
$661$	14478.6i	0.851970i	−0.904730	+	0.425985i	$0.859928\pi$
$662$	0	0
$663$	265.606	0.0155585
$664$	0	0
$665$	4680.83	0.272954
$666$	0	0
$667$	36620.3i	2.12585i
$668$	0	0
$669$	− 5030.79i	− 0.290735i
$670$	0	0
$671$	348.771	0.0200658
$672$	0	0
$673$	26226.0	1.50214	0.751070	−	0.660223i	$-0.229538\pi$
$673$	26226.0	1.50214	0.751070	+	0.660223i	$0.229538\pi$
$674$	0	0
$675$	− 1771.78i	− 0.101031i
$676$	0	0
$677$	− 33030.5i	− 1.87513i	−0.347808	−	0.937566i	$-0.613074\pi$
$677$	− 33030.5i	− 1.87513i	0.347808	−	0.937566i	$-0.386926\pi$
$678$	0	0
$679$	−9680.53	−0.547135
$680$	0	0
$681$	5135.45	0.288973
$682$	0	0
$683$	− 33369.0i	− 1.86944i	−0.355380	−	0.934722i	$-0.615648\pi$
$683$	− 33369.0i	− 1.86944i	0.355380	−	0.934722i	$-0.384352\pi$
$684$	0	0
$685$	− 100.855i	− 0.00562552i
$686$	0	0
$687$	1617.42	0.0898232
$688$	0	0
$689$	822.715	0.0454905
$690$	0	0
$691$	21497.0i	1.18348i	0.806129	+	0.591739i	$0.201559\pi$
$691$	21497.0i	1.18348i	−0.806129	+	0.591739i	$0.798441\pi$
$692$	0	0
$693$	− 303.225i	− 0.0166213i
$694$	0	0
$695$	−4864.85	−0.265517
$696$	0	0
$697$	−9302.68	−0.505544
$698$	0	0
$699$	− 8737.27i	− 0.472781i
$700$	0	0
$701$	− 23462.6i	− 1.26415i	−0.774907	−	0.632075i	$-0.782203\pi$
$701$	− 23462.6i	− 1.26415i	0.774907	−	0.632075i	$-0.217797\pi$
$702$	0	0
$703$	−11705.9	−0.628018
$704$	0	0
$705$	−2283.14	−0.121969
$706$	0	0
$707$	− 14787.4i	− 0.786617i
$708$	0	0
$709$	− 19461.4i	− 1.03087i	−0.856929	−	0.515435i	$-0.827630\pi$
$709$	− 19461.4i	− 1.03087i	0.856929	−	0.515435i	$-0.172370\pi$
$710$	0	0
$711$	−18045.6	−0.951849
$712$	0	0
$713$	30750.9	1.61519
$714$	0	0
$715$	12.8808i	0 0.000673729i
$716$	0	0
$717$	− 5880.20i	− 0.306276i
$718$	0	0
$719$	−15310.5	−0.794138	−0.397069	−	0.917789i	$-0.629973\pi$
$719$	−15310.5	−0.794138	−0.397069	+	0.917789i	$0.629973\pi$
$720$	0	0
$721$	−26279.0	−1.35739
$722$	0	0
$723$	4426.48i	0.227694i
$724$	0	0
$725$	6358.90i	0.325743i
$726$	0	0
$727$	15300.3	0.780547	0.390274	−	0.920699i	$-0.372380\pi$
$727$	15300.3	0.780547	0.390274	+	0.920699i	$0.372380\pi$
$728$	0	0
$729$	12060.0	0.612711
$730$	0	0
$731$	411.799i	0.0208357i
$732$	0	0
$733$	− 16691.8i	− 0.841099i	−0.907270	−	0.420549i	$-0.861837\pi$
$733$	− 16691.8i	− 0.841099i	0.907270	−	0.420549i	$-0.138163\pi$
$734$	0	0
$735$	−414.090	−0.0207809
$736$	0	0
$737$	513.840	0.0256819
$738$	0	0
$739$	19822.5i	0.986714i	0.869827	+	0.493357i	$0.164230\pi$
$739$	19822.5i	0.986714i	−0.869827	+	0.493357i	$0.835770\pi$
$740$	0	0
$741$	271.860i	0.0134778i
$742$	0	0
$743$	−9668.24	−0.477380	−0.238690	−	0.971096i	$-0.576718\pi$
$743$	−9668.24	−0.477380	−0.238690	+	0.971096i	$0.576718\pi$
$744$	0	0
$745$	32.2021	0.00158362
$746$	0	0
$747$	30730.1i	1.50516i
$748$	0	0
$749$	6016.19i	0.293494i
$750$	0	0
$751$	−5583.95	−0.271320	−0.135660	−	0.990755i	$-0.543315\pi$
$751$	−5583.95	−0.271320	−0.135660	+	0.990755i	$0.543315\pi$
$752$	0	0
$753$	8222.70	0.397944
$754$	0	0
$755$	− 15432.1i	− 0.743881i
$756$	0	0
$757$	5209.68i	0.250131i	0.992148	+	0.125066i	$0.0399141\pi$
$757$	5209.68i	0.250131i	−0.992148	+	0.125066i	$0.960086\pi$
$758$	0	0
$759$	140.433	0.00671593
$760$	0	0
$761$	−7928.58	−0.377675	−0.188838	−	0.982008i	$-0.560472\pi$
$761$	−7928.58	−0.377675	−0.188838	+	0.982008i	$0.560472\pi$
$762$	0	0
$763$	− 29519.2i	− 1.40061i
$764$	0	0
$765$	6849.29i	0.323708i
$766$	0	0
$767$	−193.805	−0.00912374
$768$	0	0
$769$	10718.2	0.502611	0.251306	−	0.967908i	$-0.419140\pi$
$769$	10718.2	0.502611	0.251306	+	0.967908i	$0.419140\pi$
$770$	0	0
$771$	8655.36i	0.404300i
$772$	0	0
$773$	− 708.859i	− 0.0329830i	−0.999864	−	0.0164915i	$-0.994750\pi$
$773$	− 708.859i	− 0.0329830i	0.999864	−	0.0164915i	$-0.00524965\pi$
$774$	0	0
$775$	5339.72	0.247495
$776$	0	0
$777$	−4792.62	−0.221280
$778$	0	0
$779$	− 9521.70i	− 0.437934i
$780$	0	0
$781$	− 29.2624i	− 0.00134071i
$782$	0	0
$783$	18026.5	0.822752
$784$	0	0
$785$	14103.7	0.641251
$786$	0	0
$787$	− 25542.4i	− 1.15691i	−0.815714	−	0.578456i	$-0.803656\pi$
$787$	− 25542.4i	− 1.15691i	0.815714	−	0.578456i	$-0.196344\pi$
$788$	0	0
$789$	− 1184.93i	− 0.0534658i
$790$	0	0
$791$	20602.2	0.926081
$792$	0	0
$793$	1743.85	0.0780909
$794$	0	0
$795$	− 1557.52i	− 0.0694838i
$796$	0	0
$797$	32127.5i	1.42787i	0.700210	+	0.713937i	$0.253090\pi$
$797$	32127.5i	1.42787i	−0.700210	+	0.713937i	$0.746910\pi$
$798$	0	0
$799$	18300.1	0.810277
$800$	0	0
$801$	−4098.00	−0.180769
$802$	0	0
$803$	− 512.539i	− 0.0225244i
$804$	0	0
$805$	− 12089.7i	− 0.529325i
$806$	0	0
$807$	4612.92	0.201217
$808$	0	0
$809$	−25417.8	−1.10463	−0.552313	−	0.833637i	$-0.686255\pi$
$809$	−25417.8	−1.10463	−0.552313	+	0.833637i	$0.686255\pi$
$810$	0	0
$811$	− 113.761i	− 0.00492563i	−0.999997	−	0.00246282i	$-0.999216\pi$
$811$	− 113.761i	− 0.00492563i	0.999997	−	0.00246282i	$-0.000783939\pi$
$812$	0	0
$813$	1136.85i	0.0490420i
$814$	0	0
$815$	16758.0	0.720255
$816$	0	0
$817$	−421.494	−0.0180492
$818$	0	0
$819$	− 1516.13i	− 0.0646859i
$820$	0	0
$821$	20116.2i	0.855128i	0.903985	+	0.427564i	$0.140628\pi$
$821$	20116.2i	0.855128i	−0.903985	+	0.427564i	$0.859372\pi$
$822$	0	0
$823$	−6359.51	−0.269354	−0.134677	−	0.990890i	$-0.543000\pi$
$823$	−6359.51	−0.269354	−0.134677	+	0.990890i	$0.543000\pi$
$824$	0	0
$825$	24.3854	0.00102908
$826$	0	0
$827$	10914.5i	0.458931i	0.973317	+	0.229466i	$0.0736978\pi$
$827$	10914.5i	0.458931i	−0.973317	+	0.229466i	$0.926302\pi$
$828$	0	0
$829$	− 5775.09i	− 0.241951i	−0.992655	−	0.120975i	$-0.961398\pi$
$829$	− 5775.09i	− 0.241951i	0.992655	−	0.120975i	$-0.0386022\pi$
$830$	0	0
$831$	−2874.95	−0.120013
$832$	0	0
$833$	3319.07	0.138054
$834$	0	0
$835$	− 13294.1i	− 0.550972i
$836$	0	0
$837$	− 15137.3i	− 0.625116i
$838$	0	0
$839$	19868.3	0.817556	0.408778	−	0.912634i	$-0.365955\pi$
$839$	19868.3	0.817556	0.408778	+	0.912634i	$0.365955\pi$
$840$	0	0
$841$	−40308.0	−1.65271
$842$	0	0
$843$	5430.83i	0.221884i
$844$	0	0
$845$	− 10920.6i	− 0.444592i
$846$	0	0
$847$	−22344.8	−0.906466
$848$	0	0
$849$	−8490.86	−0.343234
$850$	0	0
$851$	30234.2i	1.21788i
$852$	0	0
$853$	47859.3i	1.92107i	0.278161	+	0.960535i	$0.410275\pi$
$853$	47859.3i	1.92107i	−0.278161	+	0.960535i	$0.589725\pi$
$854$	0	0
$855$	−7010.55	−0.280416
$856$	0	0
$857$	25036.1	0.997919	0.498960	−	0.866625i	$-0.333716\pi$
$857$	25036.1	0.997919	0.498960	+	0.866625i	$0.333716\pi$
$858$	0	0
$859$	− 16742.8i	− 0.665025i	−0.943099	−	0.332512i	$-0.892104\pi$
$859$	− 16742.8i	− 0.665025i	0.943099	−	0.332512i	$-0.107896\pi$
$860$	0	0
$861$	− 3898.37i	− 0.154304i
$862$	0	0
$863$	8244.07	0.325181	0.162591	−	0.986694i	$-0.448015\pi$
$863$	8244.07	0.325181	0.162591	+	0.986694i	$0.448015\pi$
$864$	0	0
$865$	7911.36	0.310976
$866$	0	0
$867$	− 2645.90i	− 0.103644i
$868$	0	0
$869$	− 514.965i	− 0.0201024i
$870$	0	0
$871$	2569.20	0.0999473
$872$	0	0
$873$	14498.7	0.562092
$874$	0	0
$875$	− 2099.31i	− 0.0811082i
$876$	0	0
$877$	− 2593.82i	− 0.0998711i	−0.998752	−	0.0499355i	$-0.984098\pi$
$877$	− 2593.82i	− 0.0998711i	0.998752	−	0.0499355i	$-0.0159016\pi$
$878$	0	0
$879$	10583.7	0.406118
$880$	0	0
$881$	−1536.94	−0.0587749	−0.0293874	−	0.999568i	$-0.509356\pi$
$881$	−1536.94	−0.0587749	−0.0293874	+	0.999568i	$0.509356\pi$
$882$	0	0
$883$	− 17973.1i	− 0.684986i	−0.939520	−	0.342493i	$-0.888729\pi$
$883$	− 17973.1i	− 0.684986i	0.939520	−	0.342493i	$-0.111271\pi$
$884$	0	0
$885$	366.903i	0.0139359i
$886$	0	0
$887$	1407.55	0.0532818	0.0266409	−	0.999645i	$-0.491519\pi$
$887$	1407.55	0.0532818	0.0266409	+	0.999645i	$0.491519\pi$
$888$	0	0
$889$	−4394.64	−0.165795
$890$	0	0
$891$	418.358i	0.0157301i
$892$	0	0
$893$	18731.0i	0.701913i
$894$	0	0
$895$	−793.068	−0.0296194
$896$	0	0
$897$	702.164	0.0261367
$898$	0	0
$899$	54327.6i	2.01549i
$900$	0	0
$901$	12484.1i	0.461603i
$902$	0	0
$903$	−172.568	−0.00635958
$904$	0	0
$905$	13451.2	0.494071
$906$	0	0
$907$	31765.4i	1.16290i	0.813581	+	0.581451i	$0.197515\pi$
$907$	31765.4i	1.16290i	−0.813581	+	0.581451i	$0.802485\pi$
$908$	0	0
$909$	22147.4i	0.808120i
$910$	0	0
$911$	21973.2	0.799125	0.399563	−	0.916706i	$-0.369162\pi$
$911$	21973.2	0.799125	0.399563	+	0.916706i	$0.369162\pi$
$912$	0	0
$913$	−876.940	−0.0317880
$914$	0	0
$915$	− 3301.38i	− 0.119279i
$916$	0	0
$917$	38589.1i	1.38967i
$918$	0	0
$919$	29205.3	1.04831	0.524153	−	0.851624i	$-0.324382\pi$
$919$	29205.3	1.04831	0.524153	+	0.851624i	$0.324382\pi$
$920$	0	0
$921$	5276.37	0.188776
$922$	0	0
$923$	− 146.312i	− 0.00521769i
$924$	0	0
$925$	5250.00i	0.186615i
$926$	0	0
$927$	39358.5	1.39450
$928$	0	0
$929$	1338.77	0.0472805	0.0236403	−	0.999721i	$-0.492474\pi$
$929$	1338.77	0.0472805	0.0236403	+	0.999721i	$0.492474\pi$
$930$	0	0
$931$	3397.22i	0.119591i
$932$	0	0
$933$	− 6006.63i	− 0.210770i
$934$	0	0
$935$	−195.457	−0.00683650
$936$	0	0
$937$	−17176.9	−0.598873	−0.299436	−	0.954116i	$-0.596799\pi$
$937$	−17176.9	−0.598873	−0.299436	+	0.954116i	$0.596799\pi$
$938$	0	0
$939$	− 8565.49i	− 0.297683i
$940$	0	0
$941$	15713.9i	0.544378i	0.962244	+	0.272189i	$0.0877475\pi$
$941$	15713.9i	0.544378i	−0.962244	+	0.272189i	$0.912252\pi$
$942$	0	0
$943$	−24592.8	−0.849260
$944$	0	0
$945$	−5951.23	−0.204861
$946$	0	0
$947$	− 44864.9i	− 1.53951i	−0.638342	−	0.769753i	$-0.720379\pi$
$947$	− 44864.9i	− 1.53951i	0.638342	−	0.769753i	$-0.279621\pi$
$948$	0	0
$949$	− 2562.69i	− 0.0876592i
$950$	0	0
$951$	13301.7	0.453561
$952$	0	0
$953$	−3234.61	−0.109947	−0.0549734	−	0.998488i	$-0.517507\pi$
$953$	−3234.61	−0.109947	−0.0549734	+	0.998488i	$0.517507\pi$
$954$	0	0
$955$	20731.8i	0.702475i
$956$	0	0
$957$	248.103i	0.00838037i
$958$	0	0
$959$	−338.762	−0.0114069
$960$	0	0
$961$	15829.3	0.531344
$962$	0	0
$963$	− 9010.55i	− 0.301517i
$964$	0	0
$965$	19173.3i	0.639598i
$966$	0	0
$967$	37522.1	1.24781	0.623904	−	0.781501i	$-0.285546\pi$
$967$	37522.1	1.24781	0.623904	+	0.781501i	$0.285546\pi$
$968$	0	0
$969$	−4125.26	−0.136762
$970$	0	0
$971$	− 6678.20i	− 0.220714i	−0.993892	−	0.110357i	$-0.964801\pi$
$971$	− 6678.20i	− 0.220714i	0.993892	−	0.110357i	$-0.0351995\pi$
$972$	0	0
$973$	16340.5i	0.538390i
$974$	0	0
$975$	121.927	0.00400491
$976$	0	0
$977$	20857.9	0.683014	0.341507	−	0.939879i	$-0.389063\pi$
$977$	20857.9	0.683014	0.341507	+	0.939879i	$0.389063\pi$
$978$	0	0
$979$	− 116.944i	− 0.00381771i
$980$	0	0
$981$	44211.4i	1.43890i
$982$	0	0
$983$	37184.8	1.20652	0.603261	−	0.797544i	$-0.293868\pi$
$983$	37184.8	1.20652	0.603261	+	0.797544i	$0.293868\pi$
$984$	0	0
$985$	−12628.1	−0.408491
$986$	0	0
$987$	7668.83i	0.247317i
$988$	0	0
$989$	1088.64i	0.0350018i
$990$	0	0
$991$	−52903.2	−1.69579	−0.847893	−	0.530167i	$-0.822129\pi$
$991$	−52903.2	−1.69579	−0.847893	+	0.530167i	$0.822129\pi$
$992$	0	0
$993$	11633.0	0.371764
$994$	0	0
$995$	− 21081.9i	− 0.671700i
$996$	0	0
$997$	− 8517.14i	− 0.270552i	−0.990808	−	0.135276i	$-0.956808\pi$
$997$	− 8517.14i	− 0.270552i	0.990808	−	0.135276i	$-0.0431921\pi$
$998$	0	0
$999$	14883.0	0.471347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.d.b.161.3	yes	4
4.3	odd	2		320.4.d.a.161.2	✓	4
8.3	odd	2		320.4.d.a.161.3	yes	4
8.5	even	2	inner	320.4.d.b.161.2	yes	4
16.3	odd	4		1280.4.a.a.1.2		2
16.5	even	4		1280.4.a.b.1.2		2
16.11	odd	4		1280.4.a.p.1.1		2
16.13	even	4		1280.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.4.d.a.161.2	✓	4	4.3	odd	2
320.4.d.a.161.3	yes	4	8.3	odd	2
320.4.d.b.161.2	yes	4	8.5	even	2	inner
320.4.d.b.161.3	yes	4	1.1	even	1	trivial
1280.4.a.a.1.2		2	16.3	odd	4
1280.4.a.b.1.2		2	16.5	even	4
1280.4.a.o.1.1		2	16.13	even	4
1280.4.a.p.1.1		2	16.11	odd	4

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.35890i q^{3} -5.00000i q^{5} -16.7945 q^{7} +25.1534 q^{9} +O(q^{10})\)	\(q+1.35890i q^{3} -5.00000i q^{5} -16.7945 q^{7} +25.1534 q^{9} +0.717798i q^{11} +3.58899i q^{13} +6.79449 q^{15} -54.4602 q^{17} -55.7424i q^{19} -22.8220i q^{21} -143.972 q^{23} -25.0000 q^{25} +70.8712i q^{27} -254.356i q^{29} -213.589 q^{31} -0.975415 q^{33} +83.9725i q^{35} -210.000i q^{37} -4.87707 q^{39} +170.816 q^{41} -7.56146i q^{43} -125.767i q^{45} -336.028 q^{47} -60.9449 q^{49} -74.0059i q^{51} -229.233i q^{53} +3.58899 q^{55} +75.7483 q^{57} +54.0000i q^{59} -485.890i q^{61} -422.439 q^{63} +17.9449 q^{65} -715.856i q^{67} -195.644i q^{69} -40.7670 q^{71} -714.043 q^{73} -33.9725i q^{75} -12.0551i q^{77} -717.424 q^{79} +582.835 q^{81} +1221.71i q^{83} +272.301i q^{85} +345.644 q^{87} -162.920 q^{89} -60.2753i q^{91} -290.246i q^{93} -278.712 q^{95} +576.411 q^{97} +18.0551i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 20 * q^7 - 4 * q^9	\( 4 q + 20 q^{7} - 4 q^{9} - 60 q^{15} + 96 q^{17} - 140 q^{23} - 100 q^{25} - 680 q^{31} - 248 q^{33} - 1240 q^{39} - 328 q^{41} - 1780 q^{47} + 628 q^{49} - 160 q^{55} + 1384 q^{57} - 2300 q^{63} - 800 q^{65} + 360 q^{71} - 1008 q^{73} - 80 q^{79} - 284 q^{81} + 2080 q^{87} - 24 q^{89} + 280 q^{95} + 2480 q^{97}+O(q^{100}) \) 4 * q + 20 * q^7 - 4 * q^9 - 60 * q^15 + 96 * q^17 - 140 * q^23 - 100 * q^25 - 680 * q^31 - 248 * q^33 - 1240 * q^39 - 328 * q^41 - 1780 * q^47 + 628 * q^49 - 160 * q^55 + 1384 * q^57 - 2300 * q^63 - 800 * q^65 + 360 * q^71 - 1008 * q^73 - 80 * q^79 - 284 * q^81 + 2080 * q^87 - 24 * q^89 + 280 * q^95 + 2480 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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