LMFDB - Embedded newform 7448.2.a.bp.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.a (trivial)

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:	yes
Analytic conductor:	$59.4725794254$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 $ x^7 - x^6 - 14x^5 + 13x^4 + 50x^3 - 53x^2 - 25*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.11867$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11867 q^{3} +2.20110 q^{5} +6.72607 q^{9} +O(q^{10})$	$q+3.11867 q^{3} +2.20110 q^{5} +6.72607 q^{9} +4.34957 q^{11} -1.54473 q^{13} +6.86449 q^{15} +0.501805 q^{17} +1.00000 q^{19} -4.34060 q^{23} -0.155164 q^{25} +11.6204 q^{27} -0.841822 q^{29} -1.98025 q^{31} +13.5649 q^{33} -10.2903 q^{37} -4.81748 q^{39} +5.41225 q^{41} +7.20752 q^{43} +14.8047 q^{45} +6.28281 q^{47} +1.56496 q^{51} +10.0386 q^{53} +9.57384 q^{55} +3.11867 q^{57} -8.10666 q^{59} -2.28281 q^{61} -3.40009 q^{65} +1.01951 q^{67} -13.5369 q^{69} +15.7182 q^{71} +1.86509 q^{73} -0.483906 q^{75} -3.43201 q^{79} +16.0618 q^{81} +15.4222 q^{83} +1.10452 q^{85} -2.62536 q^{87} +16.1018 q^{89} -6.17573 q^{93} +2.20110 q^{95} -1.96899 q^{97} +29.2555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{3} + q^{5} + 8 q^{9}+O(q^{10}) $ 7 * q + q^3 + q^5 + 8 * q^9	$ 7 q + q^{3} + q^{5} + 8 q^{9} + 3 q^{11} + 6 q^{13} - 4 q^{15} + 10 q^{17} + 7 q^{19} - 2 q^{23} + 8 q^{25} - 2 q^{27} + 3 q^{29} - 6 q^{31} + 21 q^{33} - 7 q^{37} - 2 q^{39} + 9 q^{41} + q^{43} + 24 q^{45} + 15 q^{47} + 6 q^{51} + 5 q^{53} - 6 q^{55} + q^{57} - 9 q^{59} + 13 q^{61} - 6 q^{65} - 2 q^{67} - 20 q^{69} - q^{71} + 42 q^{73} + 40 q^{75} - 3 q^{79} - q^{81} - 12 q^{83} - 16 q^{85} - 2 q^{87} + 41 q^{89} - 2 q^{93} + q^{95} + 5 q^{97} + 9 q^{99}+O(q^{100}) $ 7 * q + q^3 + q^5 + 8 * q^9 + 3 * q^11 + 6 * q^13 - 4 * q^15 + 10 * q^17 + 7 * q^19 - 2 * q^23 + 8 * q^25 - 2 * q^27 + 3 * q^29 - 6 * q^31 + 21 * q^33 - 7 * q^37 - 2 * q^39 + 9 * q^41 + q^43 + 24 * q^45 + 15 * q^47 + 6 * q^51 + 5 * q^53 - 6 * q^55 + q^57 - 9 * q^59 + 13 * q^61 - 6 * q^65 - 2 * q^67 - 20 * q^69 - q^71 + 42 * q^73 + 40 * q^75 - 3 * q^79 - q^81 - 12 * q^83 - 16 * q^85 - 2 * q^87 + 41 * q^89 - 2 * q^93 + q^95 + 5 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.11867	1.80056	0.900281	−	0.435309i	$-0.143361\pi$
$3$	3.11867	1.80056	0.900281	+	0.435309i	$0.143361\pi$
$4$	0	0
$5$	2.20110	0.984361	0.492181	−	0.870493i	$-0.336200\pi$
$5$	2.20110	0.984361	0.492181	+	0.870493i	$0.336200\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.72607	2.24202
$10$	0	0
$11$	4.34957	1.31145	0.655723	−	0.755002i	$-0.272364\pi$
$11$	4.34957	1.31145	0.655723	+	0.755002i	$0.272364\pi$
$12$	0	0
$13$	−1.54473	−0.428430	−0.214215	−	0.976787i	$-0.568719\pi$
$13$	−1.54473	−0.428430	−0.214215	+	0.976787i	$0.568719\pi$
$14$	0	0
$15$	6.86449	1.77240
$16$	0	0
$17$	0.501805	0.121706	0.0608528	−	0.998147i	$-0.480618\pi$
$17$	0.501805	0.121706	0.0608528	+	0.998147i	$0.480618\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.34060	−0.905077	−0.452538	−	0.891745i	$-0.649481\pi$
$23$	−4.34060	−0.905077	−0.452538	+	0.891745i	$0.649481\pi$
$24$	0	0
$25$	−0.155164	−0.0310329
$26$	0	0
$27$	11.6204	2.23634
$28$	0	0
$29$	−0.841822	−0.156322	−0.0781612	−	0.996941i	$-0.524905\pi$
$29$	−0.841822	−0.156322	−0.0781612	+	0.996941i	$0.524905\pi$
$30$	0	0
$31$	−1.98025	−0.355663	−0.177831	−	0.984061i	$-0.556908\pi$
$31$	−1.98025	−0.355663	−0.177831	+	0.984061i	$0.556908\pi$
$32$	0	0
$33$	13.5649	2.36134
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.2903	−1.69172	−0.845858	−	0.533408i	$-0.820911\pi$
$37$	−10.2903	−1.69172	−0.845858	+	0.533408i	$0.820911\pi$
$38$	0	0
$39$	−4.81748	−0.771415
$40$	0	0
$41$	5.41225	0.845252	0.422626	−	0.906304i	$-0.361108\pi$
$41$	5.41225	0.845252	0.422626	+	0.906304i	$0.361108\pi$
$42$	0	0
$43$	7.20752	1.09914	0.549568	−	0.835449i	$-0.314792\pi$
$43$	7.20752	1.09914	0.549568	+	0.835449i	$0.314792\pi$
$44$	0	0
$45$	14.8047	2.20696
$46$	0	0
$47$	6.28281	0.916442	0.458221	−	0.888838i	$-0.348487\pi$
$47$	6.28281	0.916442	0.458221	+	0.888838i	$0.348487\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.56496	0.219138
$52$	0	0
$53$	10.0386	1.37890	0.689452	−	0.724332i	$-0.257852\pi$
$53$	10.0386	1.37890	0.689452	+	0.724332i	$0.257852\pi$
$54$	0	0
$55$	9.57384	1.29094
$56$	0	0
$57$	3.11867	0.413077
$58$	0	0
$59$	−8.10666	−1.05540	−0.527698	−	0.849432i	$-0.676945\pi$
$59$	−8.10666	−1.05540	−0.527698	+	0.849432i	$0.676945\pi$
$60$	0	0
$61$	−2.28281	−0.292283	−0.146142	−	0.989264i	$-0.546686\pi$
$61$	−2.28281	−0.292283	−0.146142	+	0.989264i	$0.546686\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.40009	−0.421730
$66$	0	0
$67$	1.01951	0.124553	0.0622765	−	0.998059i	$-0.480164\pi$
$67$	1.01951	0.124553	0.0622765	+	0.998059i	$0.480164\pi$
$68$	0	0
$69$	−13.5369	−1.62965
$70$	0	0
$71$	15.7182	1.86541	0.932704	−	0.360643i	$-0.117443\pi$
$71$	15.7182	1.86541	0.932704	+	0.360643i	$0.117443\pi$
$72$	0	0
$73$	1.86509	0.218292	0.109146	−	0.994026i	$-0.465188\pi$
$73$	1.86509	0.218292	0.109146	+	0.994026i	$0.465188\pi$
$74$	0	0
$75$	−0.483906	−0.0558766
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.43201	−0.386131	−0.193065	−	0.981186i	$-0.561843\pi$
$79$	−3.43201	−0.386131	−0.193065	+	0.981186i	$0.561843\pi$
$80$	0	0
$81$	16.0618	1.78465
$82$	0	0
$83$	15.4222	1.69281	0.846403	−	0.532544i	$-0.178764\pi$
$83$	15.4222	1.69281	0.846403	+	0.532544i	$0.178764\pi$
$84$	0	0
$85$	1.10452	0.119802
$86$	0	0
$87$	−2.62536	−0.281468
$88$	0	0
$89$	16.1018	1.70679	0.853396	−	0.521263i	$-0.174539\pi$
$89$	16.1018	1.70679	0.853396	+	0.521263i	$0.174539\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.17573	−0.640393
$94$	0	0
$95$	2.20110	0.225828
$96$	0	0
$97$	−1.96899	−0.199921	−0.0999603	−	0.994991i	$-0.531872\pi$
$97$	−1.96899	−0.199921	−0.0999603	+	0.994991i	$0.531872\pi$
$98$	0	0
$99$	29.2555	2.94029
$100$	0	0
$101$	−6.55769	−0.652515	−0.326257	−	0.945281i	$-0.605788\pi$
$101$	−6.55769	−0.652515	−0.326257	+	0.945281i	$0.605788\pi$
$102$	0	0
$103$	−7.59106	−0.747970	−0.373985	−	0.927435i	$-0.622009\pi$
$103$	−7.59106	−0.747970	−0.373985	+	0.927435i	$0.622009\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.9802	1.35152	0.675761	−	0.737121i	$-0.263815\pi$
$107$	13.9802	1.35152	0.675761	+	0.737121i	$0.263815\pi$
$108$	0	0
$109$	5.39598	0.516841	0.258421	−	0.966032i	$-0.416798\pi$
$109$	5.39598	0.516841	0.258421	+	0.966032i	$0.416798\pi$
$110$	0	0
$111$	−32.0920	−3.04604
$112$	0	0
$113$	−6.25894	−0.588792	−0.294396	−	0.955684i	$-0.595118\pi$
$113$	−6.25894	−0.588792	−0.294396	+	0.955684i	$0.595118\pi$
$114$	0	0
$115$	−9.55408	−0.890923
$116$	0	0
$117$	−10.3899	−0.960550
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.91879	0.719890
$122$	0	0
$123$	16.8790	1.52193
$124$	0	0
$125$	−11.3470	−1.01491
$126$	0	0
$127$	−2.86813	−0.254505	−0.127252	−	0.991870i	$-0.540616\pi$
$127$	−2.86813	−0.254505	−0.127252	+	0.991870i	$0.540616\pi$
$128$	0	0
$129$	22.4778	1.97906
$130$	0	0
$131$	−15.4456	−1.34949	−0.674744	−	0.738052i	$-0.735746\pi$
$131$	−15.4456	−1.34949	−0.674744	+	0.738052i	$0.735746\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	25.5776	2.20137
$136$	0	0
$137$	−1.74936	−0.149458	−0.0747291	−	0.997204i	$-0.523809\pi$
$137$	−1.74936	−0.149458	−0.0747291	+	0.997204i	$0.523809\pi$
$138$	0	0
$139$	−19.7725	−1.67708	−0.838541	−	0.544838i	$-0.816591\pi$
$139$	−19.7725	−1.67708	−0.838541	+	0.544838i	$0.816591\pi$
$140$	0	0
$141$	19.5940	1.65011
$142$	0	0
$143$	−6.71890	−0.561863
$144$	0	0
$145$	−1.85293	−0.153878
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.360002	0.0294925	0.0147463	−	0.999891i	$-0.495306\pi$
$149$	0.360002	0.0294925	0.0147463	+	0.999891i	$0.495306\pi$
$150$	0	0
$151$	−11.2118	−0.912400	−0.456200	−	0.889877i	$-0.650790\pi$
$151$	−11.2118	−0.912400	−0.456200	+	0.889877i	$0.650790\pi$
$152$	0	0
$153$	3.37518	0.272867
$154$	0	0
$155$	−4.35872	−0.350101
$156$	0	0
$157$	11.9976	0.957510	0.478755	−	0.877949i	$-0.341088\pi$
$157$	11.9976	0.957510	0.478755	+	0.877949i	$0.341088\pi$
$158$	0	0
$159$	31.3069	2.48280
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.9969	−1.56627	−0.783137	−	0.621849i	$-0.786382\pi$
$163$	−19.9969	−1.56627	−0.783137	+	0.621849i	$0.786382\pi$
$164$	0	0
$165$	29.8576	2.32441
$166$	0	0
$167$	−17.0364	−1.31831	−0.659157	−	0.752005i	$-0.729087\pi$
$167$	−17.0364	−1.31831	−0.659157	+	0.752005i	$0.729087\pi$
$168$	0	0
$169$	−10.6138	−0.816448
$170$	0	0
$171$	6.72607	0.514356
$172$	0	0
$173$	−14.2051	−1.08000	−0.539998	−	0.841666i	$-0.681575\pi$
$173$	−14.2051	−1.08000	−0.539998	+	0.841666i	$0.681575\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.2819	−1.90031
$178$	0	0
$179$	−1.00060	−0.0747887	−0.0373943	−	0.999301i	$-0.511906\pi$
$179$	−1.00060	−0.0747887	−0.0373943	+	0.999301i	$0.511906\pi$
$180$	0	0
$181$	3.62945	0.269775	0.134887	−	0.990861i	$-0.456933\pi$
$181$	3.62945	0.269775	0.134887	+	0.990861i	$0.456933\pi$
$182$	0	0
$183$	−7.11931	−0.526275
$184$	0	0
$185$	−22.6500	−1.66526
$186$	0	0
$187$	2.18264	0.159610
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.3220	−1.47045	−0.735225	−	0.677823i	$-0.762924\pi$
$191$	−20.3220	−1.47045	−0.735225	+	0.677823i	$0.762924\pi$
$192$	0	0
$193$	12.8275	0.923345	0.461672	−	0.887051i	$-0.347250\pi$
$193$	12.8275	0.923345	0.461672	+	0.887051i	$0.347250\pi$
$194$	0	0
$195$	−10.6038	−0.759351
$196$	0	0
$197$	−16.8124	−1.19783	−0.598916	−	0.800812i	$-0.704402\pi$
$197$	−16.8124	−1.19783	−0.598916	+	0.800812i	$0.704402\pi$
$198$	0	0
$199$	10.8072	0.766101	0.383050	−	0.923727i	$-0.374874\pi$
$199$	10.8072	0.766101	0.383050	+	0.923727i	$0.374874\pi$
$200$	0	0
$201$	3.17951	0.224265
$202$	0	0
$203$	0	0
$204$	0	0
$205$	11.9129	0.832034
$206$	0	0
$207$	−29.1952	−2.02920
$208$	0	0
$209$	4.34957	0.300866
$210$	0	0
$211$	2.73499	0.188285	0.0941424	−	0.995559i	$-0.469989\pi$
$211$	2.73499	0.188285	0.0941424	+	0.995559i	$0.469989\pi$
$212$	0	0
$213$	49.0198	3.35878
$214$	0	0
$215$	15.8645	1.08195
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.81659	0.393049
$220$	0	0
$221$	−0.775151	−0.0521423
$222$	0	0
$223$	−24.9765	−1.67255	−0.836274	−	0.548312i	$-0.815271\pi$
$223$	−24.9765	−1.67255	−0.836274	+	0.548312i	$0.815271\pi$
$224$	0	0
$225$	−1.04365	−0.0695764
$226$	0	0
$227$	22.3570	1.48388	0.741942	−	0.670464i	$-0.233905\pi$
$227$	22.3570	1.48388	0.741942	+	0.670464i	$0.233905\pi$
$228$	0	0
$229$	0.989096	0.0653613	0.0326807	−	0.999466i	$-0.489596\pi$
$229$	0.989096	0.0653613	0.0326807	+	0.999466i	$0.489596\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.9343	−0.912865	−0.456433	−	0.889758i	$-0.650873\pi$
$233$	−13.9343	−0.912865	−0.456433	+	0.889758i	$0.650873\pi$
$234$	0	0
$235$	13.8291	0.902110
$236$	0	0
$237$	−10.7033	−0.695253
$238$	0	0
$239$	27.0277	1.74827	0.874137	−	0.485680i	$-0.161428\pi$
$239$	27.0277	1.74827	0.874137	+	0.485680i	$0.161428\pi$
$240$	0	0
$241$	3.82837	0.246607	0.123304	−	0.992369i	$-0.460651\pi$
$241$	3.82837	0.246607	0.123304	+	0.992369i	$0.460651\pi$
$242$	0	0
$243$	15.2304	0.977028
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.54473	−0.0982886
$248$	0	0
$249$	48.0966	3.04800
$250$	0	0
$251$	−14.8968	−0.940277	−0.470138	−	0.882593i	$-0.655796\pi$
$251$	−14.8968	−0.940277	−0.470138	+	0.882593i	$0.655796\pi$
$252$	0	0
$253$	−18.8797	−1.18696
$254$	0	0
$255$	3.44464	0.215711
$256$	0	0
$257$	30.0665	1.87550	0.937748	−	0.347316i	$-0.112907\pi$
$257$	30.0665	1.87550	0.937748	+	0.347316i	$0.112907\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.66216	−0.350479
$262$	0	0
$263$	−1.43780	−0.0886587	−0.0443293	−	0.999017i	$-0.514115\pi$
$263$	−1.43780	−0.0886587	−0.0443293	+	0.999017i	$0.514115\pi$
$264$	0	0
$265$	22.0959	1.35734
$266$	0	0
$267$	50.2163	3.07319
$268$	0	0
$269$	−17.6249	−1.07461	−0.537304	−	0.843389i	$-0.680557\pi$
$269$	−17.6249	−1.07461	−0.537304	+	0.843389i	$0.680557\pi$
$270$	0	0
$271$	13.9275	0.846038	0.423019	−	0.906121i	$-0.360970\pi$
$271$	13.9275	0.846038	0.423019	+	0.906121i	$0.360970\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.674899	−0.0406979
$276$	0	0
$277$	−30.6415	−1.84107	−0.920534	−	0.390662i	$-0.872246\pi$
$277$	−30.6415	−1.84107	−0.920534	+	0.390662i	$0.872246\pi$
$278$	0	0
$279$	−13.3193	−0.797405
$280$	0	0
$281$	−25.0208	−1.49261	−0.746307	−	0.665602i	$-0.768175\pi$
$281$	−25.0208	−1.49261	−0.746307	+	0.665602i	$0.768175\pi$
$282$	0	0
$283$	10.1751	0.604848	0.302424	−	0.953173i	$-0.402204\pi$
$283$	10.1751	0.604848	0.302424	+	0.953173i	$0.402204\pi$
$284$	0	0
$285$	6.86449	0.406617
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.7482	−0.985188
$290$	0	0
$291$	−6.14062	−0.359969
$292$	0	0
$293$	−0.222616	−0.0130054	−0.00650269	−	0.999979i	$-0.502070\pi$
$293$	−0.222616	−0.0130054	−0.00650269	+	0.999979i	$0.502070\pi$
$294$	0	0
$295$	−17.8436	−1.03889
$296$	0	0
$297$	50.5437	2.93284
$298$	0	0
$299$	6.70503	0.387762
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.4512	−1.17489
$304$	0	0
$305$	−5.02468	−0.287713
$306$	0	0
$307$	−18.3870	−1.04940	−0.524702	−	0.851286i	$-0.675823\pi$
$307$	−18.3870	−1.04940	−0.524702	+	0.851286i	$0.675823\pi$
$308$	0	0
$309$	−23.6740	−1.34677
$310$	0	0
$311$	3.88655	0.220386	0.110193	−	0.993910i	$-0.464853\pi$
$311$	3.88655	0.220386	0.110193	+	0.993910i	$0.464853\pi$
$312$	0	0
$313$	−19.9972	−1.13031	−0.565153	−	0.824986i	$-0.691183\pi$
$313$	−19.9972	−1.13031	−0.565153	+	0.824986i	$0.691183\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.7799	0.886289	0.443145	−	0.896450i	$-0.353863\pi$
$317$	15.7799	0.886289	0.443145	+	0.896450i	$0.353863\pi$
$318$	0	0
$319$	−3.66157	−0.205008
$320$	0	0
$321$	43.5997	2.43350
$322$	0	0
$323$	0.501805	0.0279212
$324$	0	0
$325$	0.239686	0.0132954
$326$	0	0
$327$	16.8283	0.930605
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.9796	−0.713426	−0.356713	−	0.934214i	$-0.616103\pi$
$331$	−12.9796	−0.713426	−0.356713	+	0.934214i	$0.616103\pi$
$332$	0	0
$333$	−69.2134	−3.79287
$334$	0	0
$335$	2.24404	0.122605
$336$	0	0
$337$	0.777311	0.0423428	0.0211714	−	0.999776i	$-0.493260\pi$
$337$	0.777311	0.0423428	0.0211714	+	0.999776i	$0.493260\pi$
$338$	0	0
$339$	−19.5195	−1.06016
$340$	0	0
$341$	−8.61323	−0.466433
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−29.7960	−1.60416
$346$	0	0
$347$	9.44512	0.507041	0.253520	−	0.967330i	$-0.418412\pi$
$347$	9.44512	0.507041	0.253520	+	0.967330i	$0.418412\pi$
$348$	0	0
$349$	−13.7146	−0.734124	−0.367062	−	0.930197i	$-0.619636\pi$
$349$	−13.7146	−0.734124	−0.367062	+	0.930197i	$0.619636\pi$
$350$	0	0
$351$	−17.9503	−0.958116
$352$	0	0
$353$	21.1481	1.12560	0.562800	−	0.826593i	$-0.309724\pi$
$353$	21.1481	1.12560	0.562800	+	0.826593i	$0.309724\pi$
$354$	0	0
$355$	34.5973	1.83624
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.0313	1.74333	0.871663	−	0.490105i	$-0.163042\pi$
$359$	33.0313	1.74333	0.871663	+	0.490105i	$0.163042\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	24.6961	1.29621
$364$	0	0
$365$	4.10525	0.214878
$366$	0	0
$367$	−5.13804	−0.268203	−0.134102	−	0.990968i	$-0.542815\pi$
$367$	−5.13804	−0.268203	−0.134102	+	0.990968i	$0.542815\pi$
$368$	0	0
$369$	36.4032	1.89508
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.7740	0.609637	0.304818	−	0.952411i	$-0.401404\pi$
$373$	11.7740	0.609637	0.304818	+	0.952411i	$0.401404\pi$
$374$	0	0
$375$	−35.3876	−1.82741
$376$	0	0
$377$	1.30038	0.0669732
$378$	0	0
$379$	−9.74286	−0.500457	−0.250229	−	0.968187i	$-0.580506\pi$
$379$	−9.74286	−0.500457	−0.250229	+	0.968187i	$0.580506\pi$
$380$	0	0
$381$	−8.94472	−0.458252
$382$	0	0
$383$	7.06400	0.360954	0.180477	−	0.983579i	$-0.442236\pi$
$383$	7.06400	0.360954	0.180477	+	0.983579i	$0.442236\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	48.4783	2.46429
$388$	0	0
$389$	38.8735	1.97096	0.985482	−	0.169780i	$-0.0543058\pi$
$389$	38.8735	1.97096	0.985482	+	0.169780i	$0.0543058\pi$
$390$	0	0
$391$	−2.17813	−0.110153
$392$	0	0
$393$	−48.1697	−2.42984
$394$	0	0
$395$	−7.55419	−0.380092
$396$	0	0
$397$	0.128974	0.00647300	0.00323650	−	0.999995i	$-0.498970\pi$
$397$	0.128974	0.00647300	0.00323650	+	0.999995i	$0.498970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.8905	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$401$	23.8905	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$402$	0	0
$403$	3.05894	0.152377
$404$	0	0
$405$	35.3537	1.75674
$406$	0	0
$407$	−44.7585	−2.21859
$408$	0	0
$409$	−16.2603	−0.804019	−0.402010	−	0.915635i	$-0.631688\pi$
$409$	−16.2603	−0.804019	−0.402010	+	0.915635i	$0.631688\pi$
$410$	0	0
$411$	−5.45568	−0.269109
$412$	0	0
$413$	0	0
$414$	0	0
$415$	33.9458	1.66633
$416$	0	0
$417$	−61.6638	−3.01969
$418$	0	0
$419$	33.4603	1.63464	0.817321	−	0.576183i	$-0.195458\pi$
$419$	33.4603	1.63464	0.817321	+	0.576183i	$0.195458\pi$
$420$	0	0
$421$	3.51777	0.171446	0.0857228	−	0.996319i	$-0.472680\pi$
$421$	3.51777	0.171446	0.0857228	+	0.996319i	$0.472680\pi$
$422$	0	0
$423$	42.2586	2.05468
$424$	0	0
$425$	−0.0778622	−0.00377687
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−20.9540	−1.01167
$430$	0	0
$431$	37.3251	1.79789	0.898944	−	0.438063i	$-0.144335\pi$
$431$	37.3251	1.79789	0.898944	+	0.438063i	$0.144335\pi$
$432$	0	0
$433$	39.8020	1.91276	0.956381	−	0.292122i	$-0.0943614\pi$
$433$	39.8020	1.91276	0.956381	+	0.292122i	$0.0943614\pi$
$434$	0	0
$435$	−5.77868	−0.277067
$436$	0	0
$437$	−4.34060	−0.207639
$438$	0	0
$439$	−3.08123	−0.147059	−0.0735295	−	0.997293i	$-0.523426\pi$
$439$	−3.08123	−0.147059	−0.0735295	+	0.997293i	$0.523426\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.09741	−0.147162	−0.0735811	−	0.997289i	$-0.523443\pi$
$443$	−3.09741	−0.147162	−0.0735811	+	0.997289i	$0.523443\pi$
$444$	0	0
$445$	35.4418	1.68010
$446$	0	0
$447$	1.12273	0.0531031
$448$	0	0
$449$	−22.0834	−1.04218	−0.521090	−	0.853502i	$-0.674474\pi$
$449$	−22.0834	−1.04218	−0.521090	+	0.853502i	$0.674474\pi$
$450$	0	0
$451$	23.5410	1.10850
$452$	0	0
$453$	−34.9657	−1.64283
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.7201	−1.48380	−0.741901	−	0.670509i	$-0.766076\pi$
$457$	−31.7201	−1.48380	−0.741901	+	0.670509i	$0.766076\pi$
$458$	0	0
$459$	5.83116	0.272175
$460$	0	0
$461$	34.0976	1.58808	0.794042	−	0.607862i	$-0.207973\pi$
$461$	34.0976	1.58808	0.794042	+	0.607862i	$0.207973\pi$
$462$	0	0
$463$	−10.2606	−0.476849	−0.238425	−	0.971161i	$-0.576631\pi$
$463$	−10.2606	−0.476849	−0.238425	+	0.971161i	$0.576631\pi$
$464$	0	0
$465$	−13.5934	−0.630378
$466$	0	0
$467$	−26.8661	−1.24322	−0.621608	−	0.783329i	$-0.713520\pi$
$467$	−26.8661	−1.24322	−0.621608	+	0.783329i	$0.713520\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	37.4164	1.72406
$472$	0	0
$473$	31.3496	1.44146
$474$	0	0
$475$	−0.155164	−0.00711943
$476$	0	0
$477$	67.5201	3.09153
$478$	0	0
$479$	−40.9824	−1.87253	−0.936266	−	0.351292i	$-0.885743\pi$
$479$	−40.9824	−1.87253	−0.936266	+	0.351292i	$0.885743\pi$
$480$	0	0
$481$	15.8957	0.724782
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.33394	−0.196794
$486$	0	0
$487$	−25.4823	−1.15471	−0.577357	−	0.816492i	$-0.695916\pi$
$487$	−25.4823	−1.15471	−0.577357	+	0.816492i	$0.695916\pi$
$488$	0	0
$489$	−62.3635	−2.82017
$490$	0	0
$491$	−14.2626	−0.643662	−0.321831	−	0.946797i	$-0.604298\pi$
$491$	−14.2626	−0.643662	−0.321831	+	0.946797i	$0.604298\pi$
$492$	0	0
$493$	−0.422431	−0.0190253
$494$	0	0
$495$	64.3943	2.89431
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.5353	−1.41171	−0.705857	−	0.708354i	$-0.749438\pi$
$499$	−31.5353	−1.41171	−0.705857	+	0.708354i	$0.749438\pi$
$500$	0	0
$501$	−53.1308	−2.37371
$502$	0	0
$503$	4.55492	0.203094	0.101547	−	0.994831i	$-0.467621\pi$
$503$	4.55492	0.203094	0.101547	+	0.994831i	$0.467621\pi$
$504$	0	0
$505$	−14.4341	−0.642310
$506$	0	0
$507$	−33.1010	−1.47007
$508$	0	0
$509$	30.3362	1.34463	0.672315	−	0.740265i	$-0.265300\pi$
$509$	30.3362	1.34463	0.672315	+	0.740265i	$0.265300\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	11.6204	0.513052
$514$	0	0
$515$	−16.7087	−0.736272
$516$	0	0
$517$	27.3275	1.20186
$518$	0	0
$519$	−44.3011	−1.94460
$520$	0	0
$521$	13.7954	0.604389	0.302194	−	0.953246i	$-0.402281\pi$
$521$	13.7954	0.604389	0.302194	+	0.953246i	$0.402281\pi$
$522$	0	0
$523$	11.4907	0.502453	0.251226	−	0.967928i	$-0.419166\pi$
$523$	11.4907	0.502453	0.251226	+	0.967928i	$0.419166\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.993698	−0.0432862
$528$	0	0
$529$	−4.15922	−0.180836
$530$	0	0
$531$	−54.5260	−2.36623
$532$	0	0
$533$	−8.36045	−0.362131
$534$	0	0
$535$	30.7719	1.33039
$536$	0	0
$537$	−3.12055	−0.134662
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.98566	0.300337	0.150168	−	0.988660i	$-0.452018\pi$
$541$	6.98566	0.300337	0.150168	+	0.988660i	$0.452018\pi$
$542$	0	0
$543$	11.3190	0.485746
$544$	0	0
$545$	11.8771	0.508759
$546$	0	0
$547$	6.85429	0.293068	0.146534	−	0.989206i	$-0.453188\pi$
$547$	6.85429	0.293068	0.146534	+	0.989206i	$0.453188\pi$
$548$	0	0
$549$	−15.3543	−0.655307
$550$	0	0
$551$	−0.841822	−0.0358628
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−70.6377	−2.99840
$556$	0	0
$557$	−27.5860	−1.16885	−0.584427	−	0.811446i	$-0.698681\pi$
$557$	−27.5860	−1.16885	−0.584427	+	0.811446i	$0.698681\pi$
$558$	0	0
$559$	−11.1336	−0.470903
$560$	0	0
$561$	6.80692	0.287388
$562$	0	0
$563$	25.1433	1.05967	0.529833	−	0.848102i	$-0.322255\pi$
$563$	25.1433	1.05967	0.529833	+	0.848102i	$0.322255\pi$
$564$	0	0
$565$	−13.7766	−0.579584
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.67717	−0.196077	−0.0980386	−	0.995183i	$-0.531257\pi$
$569$	−4.67717	−0.196077	−0.0980386	+	0.995183i	$0.531257\pi$
$570$	0	0
$571$	−42.3339	−1.77162	−0.885810	−	0.464048i	$-0.846397\pi$
$571$	−42.3339	−1.77162	−0.885810	+	0.464048i	$0.846397\pi$
$572$	0	0
$573$	−63.3776	−2.64764
$574$	0	0
$575$	0.673506	0.0280871
$576$	0	0
$577$	−9.41497	−0.391950	−0.195975	−	0.980609i	$-0.562787\pi$
$577$	−9.41497	−0.391950	−0.195975	+	0.980609i	$0.562787\pi$
$578$	0	0
$579$	40.0047	1.66254
$580$	0	0
$581$	0	0
$582$	0	0
$583$	43.6635	1.80836
$584$	0	0
$585$	−22.8693	−0.945528
$586$	0	0
$587$	−38.8912	−1.60521	−0.802605	−	0.596511i	$-0.796553\pi$
$587$	−38.8912	−1.60521	−0.802605	+	0.596511i	$0.796553\pi$
$588$	0	0
$589$	−1.98025	−0.0815947
$590$	0	0
$591$	−52.4322	−2.15677
$592$	0	0
$593$	24.4540	1.00421	0.502103	−	0.864808i	$-0.332560\pi$
$593$	24.4540	1.00421	0.502103	+	0.864808i	$0.332560\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.7040	1.37941
$598$	0	0
$599$	−34.1789	−1.39651	−0.698256	−	0.715848i	$-0.746040\pi$
$599$	−34.1789	−1.39651	−0.698256	+	0.715848i	$0.746040\pi$
$600$	0	0
$601$	−41.6474	−1.69883	−0.849416	−	0.527723i	$-0.823046\pi$
$601$	−41.6474	−1.69883	−0.849416	+	0.527723i	$0.823046\pi$
$602$	0	0
$603$	6.85730	0.279251
$604$	0	0
$605$	17.4300	0.708632
$606$	0	0
$607$	−10.4909	−0.425813	−0.212907	−	0.977073i	$-0.568293\pi$
$607$	−10.4909	−0.425813	−0.212907	+	0.977073i	$0.568293\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.70522	−0.392631
$612$	0	0
$613$	27.6237	1.11571	0.557856	−	0.829938i	$-0.311624\pi$
$613$	27.6237	1.11571	0.557856	+	0.829938i	$0.311624\pi$
$614$	0	0
$615$	37.1524	1.49813
$616$	0	0
$617$	−39.1542	−1.57629	−0.788145	−	0.615490i	$-0.788958\pi$
$617$	−39.1542	−1.57629	−0.788145	+	0.615490i	$0.788958\pi$
$618$	0	0
$619$	12.2527	0.492479	0.246240	−	0.969209i	$-0.420805\pi$
$619$	12.2527	0.492479	0.246240	+	0.969209i	$0.420805\pi$
$620$	0	0
$621$	−50.4393	−2.02406
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−24.2001	−0.968004
$626$	0	0
$627$	13.5649	0.541728
$628$	0	0
$629$	−5.16373	−0.205891
$630$	0	0
$631$	−1.41216	−0.0562171	−0.0281086	−	0.999605i	$-0.508948\pi$
$631$	−1.41216	−0.0562171	−0.0281086	+	0.999605i	$0.508948\pi$
$632$	0	0
$633$	8.52953	0.339018
$634$	0	0
$635$	−6.31303	−0.250525
$636$	0	0
$637$	0	0
$638$	0	0
$639$	105.722	4.18229
$640$	0	0
$641$	−11.3286	−0.447455	−0.223727	−	0.974652i	$-0.571823\pi$
$641$	−11.3286	−0.447455	−0.223727	+	0.974652i	$0.571823\pi$
$642$	0	0
$643$	28.3850	1.11939	0.559697	−	0.828698i	$-0.310918\pi$
$643$	28.3850	1.11939	0.559697	+	0.828698i	$0.310918\pi$
$644$	0	0
$645$	49.4760	1.94811
$646$	0	0
$647$	−6.50503	−0.255739	−0.127869	−	0.991791i	$-0.540814\pi$
$647$	−6.50503	−0.255739	−0.127869	+	0.991791i	$0.540814\pi$
$648$	0	0
$649$	−35.2605	−1.38410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.7378	1.08546	0.542732	−	0.839906i	$-0.317390\pi$
$653$	27.7378	1.08546	0.542732	+	0.839906i	$0.317390\pi$
$654$	0	0
$655$	−33.9973	−1.32838
$656$	0	0
$657$	12.5447	0.489416
$658$	0	0
$659$	−8.46434	−0.329724	−0.164862	−	0.986317i	$-0.552718\pi$
$659$	−8.46434	−0.329724	−0.164862	+	0.986317i	$0.552718\pi$
$660$	0	0
$661$	12.3620	0.480825	0.240413	−	0.970671i	$-0.422717\pi$
$661$	12.3620	0.480825	0.240413	+	0.970671i	$0.422717\pi$
$662$	0	0
$663$	−2.41744	−0.0938855
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.65401	0.141484
$668$	0	0
$669$	−77.8932	−3.01153
$670$	0	0
$671$	−9.92924	−0.383314
$672$	0	0
$673$	37.2778	1.43696	0.718478	−	0.695550i	$-0.244839\pi$
$673$	37.2778	1.43696	0.718478	+	0.695550i	$0.244839\pi$
$674$	0	0
$675$	−1.80307	−0.0694001
$676$	0	0
$677$	7.98829	0.307015	0.153507	−	0.988147i	$-0.450943\pi$
$677$	7.98829	0.307015	0.153507	+	0.988147i	$0.450943\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	69.7239	2.67183
$682$	0	0
$683$	14.0709	0.538406	0.269203	−	0.963083i	$-0.413240\pi$
$683$	14.0709	0.538406	0.269203	+	0.963083i	$0.413240\pi$
$684$	0	0
$685$	−3.85052	−0.147121
$686$	0	0
$687$	3.08466	0.117687
$688$	0	0
$689$	−15.5068	−0.590763
$690$	0	0
$691$	−40.1189	−1.52620	−0.763098	−	0.646283i	$-0.776322\pi$
$691$	−40.1189	−1.52620	−0.763098	+	0.646283i	$0.776322\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−43.5212	−1.65085
$696$	0	0
$697$	2.71590	0.102872
$698$	0	0
$699$	−43.4564	−1.64367
$700$	0	0
$701$	−31.3588	−1.18440	−0.592202	−	0.805789i	$-0.701741\pi$
$701$	−31.3588	−1.18440	−0.592202	+	0.805789i	$0.701741\pi$
$702$	0	0
$703$	−10.2903	−0.388106
$704$	0	0
$705$	43.1283	1.62430
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.7701	−0.704927	−0.352463	−	0.935826i	$-0.614656\pi$
$709$	−18.7701	−0.704927	−0.352463	+	0.935826i	$0.614656\pi$
$710$	0	0
$711$	−23.0839	−0.865715
$712$	0	0
$713$	8.59546	0.321902
$714$	0	0
$715$	−14.7890	−0.553076
$716$	0	0
$717$	84.2902	3.14788
$718$	0	0
$719$	−7.68396	−0.286563	−0.143282	−	0.989682i	$-0.545765\pi$
$719$	−7.68396	−0.286563	−0.143282	+	0.989682i	$0.545765\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.9394	0.444032
$724$	0	0
$725$	0.130621	0.00485113
$726$	0	0
$727$	17.4165	0.645940	0.322970	−	0.946409i	$-0.395319\pi$
$727$	17.4165	0.645940	0.322970	+	0.946409i	$0.395319\pi$
$728$	0	0
$729$	−0.687103	−0.0254482
$730$	0	0
$731$	3.61677	0.133771
$732$	0	0
$733$	−23.9404	−0.884259	−0.442129	−	0.896951i	$-0.645777\pi$
$733$	−23.9404	−0.884259	−0.442129	+	0.896951i	$0.645777\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.43443	0.163345
$738$	0	0
$739$	2.36626	0.0870443	0.0435222	−	0.999052i	$-0.486142\pi$
$739$	2.36626	0.0870443	0.0435222	+	0.999052i	$0.486142\pi$
$740$	0	0
$741$	−4.81748	−0.176975
$742$	0	0
$743$	−33.4020	−1.22540	−0.612701	−	0.790315i	$-0.709917\pi$
$743$	−33.4020	−1.22540	−0.612701	+	0.790315i	$0.709917\pi$
$744$	0	0
$745$	0.792400	0.0290313
$746$	0	0
$747$	103.731	3.79531
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.6008	−0.934187	−0.467093	−	0.884208i	$-0.654699\pi$
$751$	−25.6008	−0.934187	−0.467093	+	0.884208i	$0.654699\pi$
$752$	0	0
$753$	−46.4581	−1.69303
$754$	0	0
$755$	−24.6782	−0.898131
$756$	0	0
$757$	−14.5879	−0.530206	−0.265103	−	0.964220i	$-0.585406\pi$
$757$	−14.5879	−0.530206	−0.265103	+	0.964220i	$0.585406\pi$
$758$	0	0
$759$	−58.8796	−2.13719
$760$	0	0
$761$	37.2613	1.35072	0.675360	−	0.737488i	$-0.263988\pi$
$761$	37.2613	1.35072	0.675360	+	0.737488i	$0.263988\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	7.42910	0.268600
$766$	0	0
$767$	12.5226	0.452164
$768$	0	0
$769$	−32.0987	−1.15751	−0.578754	−	0.815502i	$-0.696461\pi$
$769$	−32.0987	−1.15751	−0.578754	+	0.815502i	$0.696461\pi$
$770$	0	0
$771$	93.7673	3.37695
$772$	0	0
$773$	0.277139	0.00996798	0.00498399	−	0.999988i	$-0.498414\pi$
$773$	0.277139	0.00996798	0.00498399	+	0.999988i	$0.498414\pi$
$774$	0	0
$775$	0.307264	0.0110372
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.41225	0.193914
$780$	0	0
$781$	68.3675	2.44638
$782$	0	0
$783$	−9.78229	−0.349590
$784$	0	0
$785$	26.4078	0.942536
$786$	0	0
$787$	43.9949	1.56825	0.784125	−	0.620603i	$-0.213112\pi$
$787$	43.9949	1.56825	0.784125	+	0.620603i	$0.213112\pi$
$788$	0	0
$789$	−4.48402	−0.159635
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.52631	0.125223
$794$	0	0
$795$	68.9096	2.44397
$796$	0	0
$797$	−10.0980	−0.357691	−0.178846	−	0.983877i	$-0.557236\pi$
$797$	−10.0980	−0.357691	−0.178846	+	0.983877i	$0.557236\pi$
$798$	0	0
$799$	3.15274	0.111536
$800$	0	0
$801$	108.302	3.82667
$802$	0	0
$803$	8.11234	0.286278
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−54.9661	−1.93490
$808$	0	0
$809$	−2.53900	−0.0892663	−0.0446332	−	0.999003i	$-0.514212\pi$
$809$	−2.53900	−0.0892663	−0.0446332	+	0.999003i	$0.514212\pi$
$810$	0	0
$811$	36.5896	1.28483	0.642417	−	0.766355i	$-0.277932\pi$
$811$	36.5896	1.28483	0.642417	+	0.766355i	$0.277932\pi$
$812$	0	0
$813$	43.4353	1.52334
$814$	0	0
$815$	−44.0150	−1.54178
$816$	0	0
$817$	7.20752	0.252159
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.1081	1.12058	0.560291	−	0.828296i	$-0.310689\pi$
$821$	32.1081	1.12058	0.560291	+	0.828296i	$0.310689\pi$
$822$	0	0
$823$	−2.43495	−0.0848769	−0.0424385	−	0.999099i	$-0.513513\pi$
$823$	−2.43495	−0.0848769	−0.0424385	+	0.999099i	$0.513513\pi$
$824$	0	0
$825$	−2.10478	−0.0732791
$826$	0	0
$827$	16.3568	0.568782	0.284391	−	0.958708i	$-0.408209\pi$
$827$	16.3568	0.568782	0.284391	+	0.958708i	$0.408209\pi$
$828$	0	0
$829$	19.1942	0.666641	0.333320	−	0.942814i	$-0.391831\pi$
$829$	19.1942	0.666641	0.333320	+	0.942814i	$0.391831\pi$
$830$	0	0
$831$	−95.5606	−3.31496
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−37.4987	−1.29770
$836$	0	0
$837$	−23.0112	−0.795384
$838$	0	0
$839$	−56.4871	−1.95015	−0.975075	−	0.221875i	$-0.928782\pi$
$839$	−56.4871	−1.95015	−0.975075	+	0.221875i	$0.928782\pi$
$840$	0	0
$841$	−28.2913	−0.975563
$842$	0	0
$843$	−78.0314	−2.68755
$844$	0	0
$845$	−23.3621	−0.803680
$846$	0	0
$847$	0	0
$848$	0	0
$849$	31.7328	1.08907
$850$	0	0
$851$	44.6661	1.53113
$852$	0	0
$853$	30.3163	1.03801	0.519004	−	0.854772i	$-0.326303\pi$
$853$	30.3163	1.03801	0.519004	+	0.854772i	$0.326303\pi$
$854$	0	0
$855$	14.8047	0.506312
$856$	0	0
$857$	48.5463	1.65831	0.829155	−	0.559018i	$-0.188822\pi$
$857$	48.5463	1.65831	0.829155	+	0.559018i	$0.188822\pi$
$858$	0	0
$859$	3.60363	0.122954	0.0614772	−	0.998108i	$-0.480419\pi$
$859$	3.60363	0.122954	0.0614772	+	0.998108i	$0.480419\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.8163	0.436272	0.218136	−	0.975918i	$-0.430002\pi$
$863$	12.8163	0.436272	0.218136	+	0.975918i	$0.430002\pi$
$864$	0	0
$865$	−31.2669	−1.06311
$866$	0	0
$867$	−52.2320	−1.77389
$868$	0	0
$869$	−14.9278	−0.506390
$870$	0	0
$871$	−1.57486	−0.0533622
$872$	0	0
$873$	−13.2436	−0.448227
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.0967	−0.644849	−0.322424	−	0.946595i	$-0.604498\pi$
$877$	−19.0967	−0.644849	−0.322424	+	0.946595i	$0.604498\pi$
$878$	0	0
$879$	−0.694265	−0.0234170
$880$	0	0
$881$	1.01077	0.0340538	0.0170269	−	0.999855i	$-0.494580\pi$
$881$	1.01077	0.0340538	0.0170269	+	0.999855i	$0.494580\pi$
$882$	0	0
$883$	−44.4066	−1.49440	−0.747201	−	0.664598i	$-0.768603\pi$
$883$	−44.4066	−1.49440	−0.747201	+	0.664598i	$0.768603\pi$
$884$	0	0
$885$	−55.6481	−1.87059
$886$	0	0
$887$	−36.2738	−1.21795	−0.608977	−	0.793188i	$-0.708420\pi$
$887$	−36.2738	−1.21795	−0.608977	+	0.793188i	$0.708420\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	69.8621	2.34047
$892$	0	0
$893$	6.28281	0.210246
$894$	0	0
$895$	−2.20243	−0.0736191
$896$	0	0
$897$	20.9107	0.698190
$898$	0	0
$899$	1.66702	0.0555981
$900$	0	0
$901$	5.03740	0.167820
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.98877	0.265556
$906$	0	0
$907$	42.4622	1.40993	0.704967	−	0.709240i	$-0.250962\pi$
$907$	42.4622	1.40993	0.704967	+	0.709240i	$0.250962\pi$
$908$	0	0
$909$	−44.1075	−1.46295
$910$	0	0
$911$	−42.2424	−1.39955	−0.699777	−	0.714361i	$-0.746717\pi$
$911$	−42.2424	−1.39955	−0.699777	+	0.714361i	$0.746717\pi$
$912$	0	0
$913$	67.0799	2.22002
$914$	0	0
$915$	−15.6703	−0.518044
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−29.5501	−0.974769	−0.487385	−	0.873187i	$-0.662049\pi$
$919$	−29.5501	−0.974769	−0.487385	+	0.873187i	$0.662049\pi$
$920$	0	0
$921$	−57.3430	−1.88952
$922$	0	0
$923$	−24.2803	−0.799196
$924$	0	0
$925$	1.59669	0.0524988
$926$	0	0
$927$	−51.0580	−1.67697
$928$	0	0
$929$	2.71420	0.0890500	0.0445250	−	0.999008i	$-0.485823\pi$
$929$	2.71420	0.0890500	0.0445250	+	0.999008i	$0.485823\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.1208	0.396819
$934$	0	0
$935$	4.80420	0.157114
$936$	0	0
$937$	41.9699	1.37110	0.685548	−	0.728027i	$-0.259562\pi$
$937$	41.9699	1.37110	0.685548	+	0.728027i	$0.259562\pi$
$938$	0	0
$939$	−62.3644	−2.03519
$940$	0	0
$941$	16.6164	0.541680	0.270840	−	0.962624i	$-0.412699\pi$
$941$	16.6164	0.541680	0.270840	+	0.962624i	$0.412699\pi$
$942$	0	0
$943$	−23.4924	−0.765018
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.6824	−0.769575	−0.384788	−	0.923005i	$-0.625725\pi$
$947$	−23.6824	−0.769575	−0.384788	+	0.923005i	$0.625725\pi$
$948$	0	0
$949$	−2.88105	−0.0935229
$950$	0	0
$951$	49.2123	1.59582
$952$	0	0
$953$	52.2096	1.69124	0.845618	−	0.533789i	$-0.179232\pi$
$953$	52.2096	1.69124	0.845618	+	0.533789i	$0.179232\pi$
$954$	0	0
$955$	−44.7308	−1.44745
$956$	0	0
$957$	−11.4192	−0.369130
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0786	−0.873504
$962$	0	0
$963$	94.0321	3.03014
$964$	0	0
$965$	28.2346	0.908905
$966$	0	0
$967$	−47.9479	−1.54190	−0.770951	−	0.636894i	$-0.780219\pi$
$967$	−47.9479	−1.54190	−0.770951	+	0.636894i	$0.780219\pi$
$968$	0	0
$969$	1.56496	0.0502738
$970$	0	0
$971$	−10.9180	−0.350375	−0.175188	−	0.984535i	$-0.556053\pi$
$971$	−10.9180	−0.350375	−0.175188	+	0.984535i	$0.556053\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.747502	0.0239392
$976$	0	0
$977$	21.3984	0.684596	0.342298	−	0.939592i	$-0.388795\pi$
$977$	21.3984	0.684596	0.342298	+	0.939592i	$0.388795\pi$
$978$	0	0
$979$	70.0362	2.23837
$980$	0	0
$981$	36.2938	1.15877
$982$	0	0
$983$	8.57428	0.273477	0.136739	−	0.990607i	$-0.456338\pi$
$983$	8.57428	0.273477	0.136739	+	0.990607i	$0.456338\pi$
$984$	0	0
$985$	−37.0057	−1.17910
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−31.2849	−0.994803
$990$	0	0
$991$	−50.5165	−1.60471	−0.802355	−	0.596847i	$-0.796420\pi$
$991$	−50.5165	−1.60471	−0.802355	+	0.596847i	$0.796420\pi$
$992$	0	0
$993$	−40.4792	−1.28457
$994$	0	0
$995$	23.7877	0.754120
$996$	0	0
$997$	25.5524	0.809253	0.404627	−	0.914482i	$-0.367401\pi$
$997$	25.5524	0.809253	0.404627	+	0.914482i	$0.367401\pi$
$998$	0	0
$999$	−119.577	−3.78326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bp.1.7	yes	7
7.6	odd	2		7448.2.a.bo.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bo.1.1	✓	7	7.6	odd	2
7448.2.a.bp.1.7	yes	7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+3.11867 q^{3} +2.20110 q^{5} +6.72607 q^{9} +O(q^{10})\)	\(q+3.11867 q^{3} +2.20110 q^{5} +6.72607 q^{9} +4.34957 q^{11} -1.54473 q^{13} +6.86449 q^{15} +0.501805 q^{17} +1.00000 q^{19} -4.34060 q^{23} -0.155164 q^{25} +11.6204 q^{27} -0.841822 q^{29} -1.98025 q^{31} +13.5649 q^{33} -10.2903 q^{37} -4.81748 q^{39} +5.41225 q^{41} +7.20752 q^{43} +14.8047 q^{45} +6.28281 q^{47} +1.56496 q^{51} +10.0386 q^{53} +9.57384 q^{55} +3.11867 q^{57} -8.10666 q^{59} -2.28281 q^{61} -3.40009 q^{65} +1.01951 q^{67} -13.5369 q^{69} +15.7182 q^{71} +1.86509 q^{73} -0.483906 q^{75} -3.43201 q^{79} +16.0618 q^{81} +15.4222 q^{83} +1.10452 q^{85} -2.62536 q^{87} +16.1018 q^{89} -6.17573 q^{93} +2.20110 q^{95} -1.96899 q^{97} +29.2555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) 7 * q + q^3 + q^5 + 8 * q^9	\( 7 q + q^{3} + q^{5} + 8 q^{9} + 3 q^{11} + 6 q^{13} - 4 q^{15} + 10 q^{17} + 7 q^{19} - 2 q^{23} + 8 q^{25} - 2 q^{27} + 3 q^{29} - 6 q^{31} + 21 q^{33} - 7 q^{37} - 2 q^{39} + 9 q^{41} + q^{43} + 24 q^{45} + 15 q^{47} + 6 q^{51} + 5 q^{53} - 6 q^{55} + q^{57} - 9 q^{59} + 13 q^{61} - 6 q^{65} - 2 q^{67} - 20 q^{69} - q^{71} + 42 q^{73} + 40 q^{75} - 3 q^{79} - q^{81} - 12 q^{83} - 16 q^{85} - 2 q^{87} + 41 q^{89} - 2 q^{93} + q^{95} + 5 q^{97} + 9 q^{99}+O(q^{100}) \) 7 * q + q^3 + q^5 + 8 * q^9 + 3 * q^11 + 6 * q^13 - 4 * q^15 + 10 * q^17 + 7 * q^19 - 2 * q^23 + 8 * q^25 - 2 * q^27 + 3 * q^29 - 6 * q^31 + 21 * q^33 - 7 * q^37 - 2 * q^39 + 9 * q^41 + q^43 + 24 * q^45 + 15 * q^47 + 6 * q^51 + 5 * q^53 - 6 * q^55 + q^57 - 9 * q^59 + 13 * q^61 - 6 * q^65 - 2 * q^67 - 20 * q^69 - q^71 + 42 * q^73 + 40 * q^75 - 3 * q^79 - q^81 - 12 * q^83 - 16 * q^85 - 2 * q^87 + 41 * q^89 - 2 * q^93 + q^95 + 5 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more