LMFDB - Embedded newform 7448.2.a.bo.1.6

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:	$1$
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			1.6
Root			$-2.52291$ 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)

$f(q)$	$=$	$q+2.52291 q^{3} -0.138247 q^{5} +3.36506 q^{9} +O(q^{10})$	$q+2.52291 q^{3} -0.138247 q^{5} +3.36506 q^{9} -5.32196 q^{11} +2.03588 q^{13} -0.348783 q^{15} +2.22115 q^{17} -1.00000 q^{19} -4.84059 q^{23} -4.98089 q^{25} +0.921017 q^{27} -6.11878 q^{29} +0.190938 q^{31} -13.4268 q^{33} +1.53514 q^{37} +5.13633 q^{39} +2.46986 q^{41} -2.10852 q^{43} -0.465208 q^{45} +4.70457 q^{47} +5.60376 q^{51} -5.96134 q^{53} +0.735742 q^{55} -2.52291 q^{57} +10.5826 q^{59} -8.70457 q^{61} -0.281453 q^{65} -7.93608 q^{67} -12.2124 q^{69} -10.6692 q^{71} -12.7437 q^{73} -12.5663 q^{75} +2.66080 q^{79} -7.77154 q^{81} +9.33589 q^{83} -0.307067 q^{85} -15.4371 q^{87} -6.54811 q^{89} +0.481719 q^{93} +0.138247 q^{95} -13.5395 q^{97} -17.9087 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.52291	1.45660	0.728301	−	0.685258i	$-0.240310\pi$
$3$	2.52291	1.45660	0.728301	+	0.685258i	$0.240310\pi$
$4$	0	0
$5$	−0.138247	−0.0618258	−0.0309129	−	0.999522i	$-0.509841\pi$
$5$	−0.138247	−0.0618258	−0.0309129	+	0.999522i	$0.509841\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.36506	1.12169
$10$	0	0
$11$	−5.32196	−1.60463	−0.802315	−	0.596901i	$-0.796399\pi$
$11$	−5.32196	−1.60463	−0.802315	+	0.596901i	$0.796399\pi$
$12$	0	0
$13$	2.03588	0.564651	0.282325	−	0.959319i	$-0.408894\pi$
$13$	2.03588	0.564651	0.282325	+	0.959319i	$0.408894\pi$
$14$	0	0
$15$	−0.348783	−0.0900555
$16$	0	0
$17$	2.22115	0.538709	0.269354	−	0.963041i	$-0.413190\pi$
$17$	2.22115	0.538709	0.269354	+	0.963041i	$0.413190\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.84059	−1.00933	−0.504666	−	0.863314i	$-0.668384\pi$
$23$	−4.84059	−1.00933	−0.504666	+	0.863314i	$0.668384\pi$
$24$	0	0
$25$	−4.98089	−0.996178
$26$	0	0
$27$	0.921017	0.177250
$28$	0	0
$29$	−6.11878	−1.13623	−0.568115	−	0.822949i	$-0.692327\pi$
$29$	−6.11878	−1.13623	−0.568115	+	0.822949i	$0.692327\pi$
$30$	0	0
$31$	0.190938	0.0342935	0.0171467	−	0.999853i	$-0.494542\pi$
$31$	0.190938	0.0342935	0.0171467	+	0.999853i	$0.494542\pi$
$32$	0	0
$33$	−13.4268	−2.33731
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.53514	0.252376	0.126188	−	0.992006i	$-0.459726\pi$
$37$	1.53514	0.252376	0.126188	+	0.992006i	$0.459726\pi$
$38$	0	0
$39$	5.13633	0.822471
$40$	0	0
$41$	2.46986	0.385728	0.192864	−	0.981225i	$-0.438222\pi$
$41$	2.46986	0.385728	0.192864	+	0.981225i	$0.438222\pi$
$42$	0	0
$43$	−2.10852	−0.321546	−0.160773	−	0.986991i	$-0.551399\pi$
$43$	−2.10852	−0.321546	−0.160773	+	0.986991i	$0.551399\pi$
$44$	0	0
$45$	−0.465208	−0.0693492
$46$	0	0
$47$	4.70457	0.686231	0.343116	−	0.939293i	$-0.388518\pi$
$47$	4.70457	0.686231	0.343116	+	0.939293i	$0.388518\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.60376	0.784684
$52$	0	0
$53$	−5.96134	−0.818854	−0.409427	−	0.912343i	$-0.634271\pi$
$53$	−5.96134	−0.818854	−0.409427	+	0.912343i	$0.634271\pi$
$54$	0	0
$55$	0.735742	0.0992075
$56$	0	0
$57$	−2.52291	−0.334167
$58$	0	0
$59$	10.5826	1.37774	0.688868	−	0.724886i	$-0.258108\pi$
$59$	10.5826	1.37774	0.688868	+	0.724886i	$0.258108\pi$
$60$	0	0
$61$	−8.70457	−1.11451	−0.557253	−	0.830343i	$-0.688145\pi$
$61$	−8.70457	−1.11451	−0.557253	+	0.830343i	$0.688145\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.281453	−0.0349100
$66$	0	0
$67$	−7.93608	−0.969547	−0.484773	−	0.874640i	$-0.661098\pi$
$67$	−7.93608	−0.969547	−0.484773	+	0.874640i	$0.661098\pi$
$68$	0	0
$69$	−12.2124	−1.47020
$70$	0	0
$71$	−10.6692	−1.26620	−0.633099	−	0.774071i	$-0.718217\pi$
$71$	−10.6692	−1.26620	−0.633099	+	0.774071i	$0.718217\pi$
$72$	0	0
$73$	−12.7437	−1.49154	−0.745769	−	0.666205i	$-0.767918\pi$
$73$	−12.7437	−1.49154	−0.745769	+	0.666205i	$0.767918\pi$
$74$	0	0
$75$	−12.5663	−1.45103
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.66080	0.299364	0.149682	−	0.988734i	$-0.452175\pi$
$79$	2.66080	0.299364	0.149682	+	0.988734i	$0.452175\pi$
$80$	0	0
$81$	−7.77154	−0.863505
$82$	0	0
$83$	9.33589	1.02475	0.512374	−	0.858763i	$-0.328766\pi$
$83$	9.33589	1.02475	0.512374	+	0.858763i	$0.328766\pi$
$84$	0	0
$85$	−0.307067	−0.0333061
$86$	0	0
$87$	−15.4371	−1.65503
$88$	0	0
$89$	−6.54811	−0.694099	−0.347049	−	0.937847i	$-0.612816\pi$
$89$	−6.54811	−0.694099	−0.347049	+	0.937847i	$0.612816\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.481719	0.0499519
$94$	0	0
$95$	0.138247	0.0141838
$96$	0	0
$97$	−13.5395	−1.37473	−0.687364	−	0.726314i	$-0.741232\pi$
$97$	−13.5395	−1.37473	−0.687364	+	0.726314i	$0.741232\pi$
$98$	0	0
$99$	−17.9087	−1.79989
$100$	0	0
$101$	−7.77311	−0.773453	−0.386727	−	0.922194i	$-0.626394\pi$
$101$	−7.77311	−0.773453	−0.386727	+	0.922194i	$0.626394\pi$
$102$	0	0
$103$	10.9318	1.07714	0.538572	−	0.842580i	$-0.318964\pi$
$103$	10.9318	1.07714	0.538572	+	0.842580i	$0.318964\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.24419	−0.120280	−0.0601401	−	0.998190i	$-0.519155\pi$
$107$	−1.24419	−0.120280	−0.0601401	+	0.998190i	$0.519155\pi$
$108$	0	0
$109$	−6.19178	−0.593065	−0.296532	−	0.955023i	$-0.595830\pi$
$109$	−6.19178	−0.593065	−0.296532	+	0.955023i	$0.595830\pi$
$110$	0	0
$111$	3.87302	0.367611
$112$	0	0
$113$	14.9869	1.40984	0.704922	−	0.709284i	$-0.250982\pi$
$113$	14.9869	1.40984	0.704922	+	0.709284i	$0.250982\pi$
$114$	0	0
$115$	0.669195	0.0624028
$116$	0	0
$117$	6.85085	0.633362
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.3232	1.57484
$122$	0	0
$123$	6.23124	0.561852
$124$	0	0
$125$	1.37982	0.123415
$126$	0	0
$127$	0.848376	0.0752812	0.0376406	−	0.999291i	$-0.488016\pi$
$127$	0.848376	0.0752812	0.0376406	+	0.999291i	$0.488016\pi$
$128$	0	0
$129$	−5.31960	−0.468365
$130$	0	0
$131$	2.85722	0.249636	0.124818	−	0.992180i	$-0.460165\pi$
$131$	2.85722	0.249636	0.124818	+	0.992180i	$0.460165\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.127328	−0.0109586
$136$	0	0
$137$	−12.9327	−1.10492	−0.552458	−	0.833541i	$-0.686310\pi$
$137$	−12.9327	−1.10492	−0.552458	+	0.833541i	$0.686310\pi$
$138$	0	0
$139$	10.0028	0.848429	0.424214	−	0.905562i	$-0.360550\pi$
$139$	10.0028	0.848429	0.424214	+	0.905562i	$0.360550\pi$
$140$	0	0
$141$	11.8692	0.999566
$142$	0	0
$143$	−10.8349	−0.906056
$144$	0	0
$145$	0.845901	0.0702483
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.52343	0.452497	0.226249	−	0.974070i	$-0.427354\pi$
$149$	5.52343	0.452497	0.226249	+	0.974070i	$0.427354\pi$
$150$	0	0
$151$	−4.77619	−0.388680	−0.194340	−	0.980934i	$-0.562257\pi$
$151$	−4.77619	−0.388680	−0.194340	+	0.980934i	$0.562257\pi$
$152$	0	0
$153$	7.47431	0.604263
$154$	0	0
$155$	−0.0263965	−0.00212022
$156$	0	0
$157$	−14.6003	−1.16523	−0.582616	−	0.812747i	$-0.697971\pi$
$157$	−14.6003	−1.16523	−0.582616	+	0.812747i	$0.697971\pi$
$158$	0	0
$159$	−15.0399	−1.19274
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.60929	−0.126049	−0.0630246	−	0.998012i	$-0.520075\pi$
$163$	−1.60929	−0.126049	−0.0630246	+	0.998012i	$0.520075\pi$
$164$	0	0
$165$	1.85621	0.144506
$166$	0	0
$167$	14.3841	1.11308	0.556539	−	0.830822i	$-0.312129\pi$
$167$	14.3841	1.11308	0.556539	+	0.830822i	$0.312129\pi$
$168$	0	0
$169$	−8.85520	−0.681169
$170$	0	0
$171$	−3.36506	−0.257333
$172$	0	0
$173$	20.1042	1.52849	0.764246	−	0.644925i	$-0.223112\pi$
$173$	20.1042	1.52849	0.764246	+	0.644925i	$0.223112\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.6989	2.00681
$178$	0	0
$179$	−2.44337	−0.182626	−0.0913129	−	0.995822i	$-0.529106\pi$
$179$	−2.44337	−0.182626	−0.0913129	+	0.995822i	$0.529106\pi$
$180$	0	0
$181$	14.9066	1.10800	0.554000	−	0.832517i	$-0.313101\pi$
$181$	14.9066	1.10800	0.554000	+	0.832517i	$0.313101\pi$
$182$	0	0
$183$	−21.9608	−1.62339
$184$	0	0
$185$	−0.212228	−0.0156033
$186$	0	0
$187$	−11.8209	−0.864428
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.6038	−0.767265	−0.383632	−	0.923486i	$-0.625327\pi$
$191$	−10.6038	−0.767265	−0.383632	+	0.923486i	$0.625327\pi$
$192$	0	0
$193$	−9.82540	−0.707248	−0.353624	−	0.935388i	$-0.615051\pi$
$193$	−9.82540	−0.707248	−0.353624	+	0.935388i	$0.615051\pi$
$194$	0	0
$195$	−0.710080	−0.0508499
$196$	0	0
$197$	−12.6952	−0.904498	−0.452249	−	0.891892i	$-0.649378\pi$
$197$	−12.6952	−0.904498	−0.452249	+	0.891892i	$0.649378\pi$
$198$	0	0
$199$	6.13512	0.434907	0.217453	−	0.976071i	$-0.430225\pi$
$199$	6.13512	0.434907	0.217453	+	0.976071i	$0.430225\pi$
$200$	0	0
$201$	−20.0220	−1.41224
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.341450	−0.0238479
$206$	0	0
$207$	−16.2889	−1.13216
$208$	0	0
$209$	5.32196	0.368127
$210$	0	0
$211$	−23.4098	−1.61160	−0.805800	−	0.592188i	$-0.798264\pi$
$211$	−23.4098	−1.61160	−0.805800	+	0.592188i	$0.798264\pi$
$212$	0	0
$213$	−26.9173	−1.84435
$214$	0	0
$215$	0.291496	0.0198798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−32.1512	−2.17258
$220$	0	0
$221$	4.52199	0.304182
$222$	0	0
$223$	−13.7479	−0.920628	−0.460314	−	0.887756i	$-0.652263\pi$
$223$	−13.7479	−0.920628	−0.460314	+	0.887756i	$0.652263\pi$
$224$	0	0
$225$	−16.7610	−1.11740
$226$	0	0
$227$	−7.67051	−0.509110	−0.254555	−	0.967058i	$-0.581929\pi$
$227$	−7.67051	−0.509110	−0.254555	+	0.967058i	$0.581929\pi$
$228$	0	0
$229$	7.44215	0.491791	0.245896	−	0.969296i	$-0.420918\pi$
$229$	7.44215	0.491791	0.245896	+	0.969296i	$0.420918\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.34426	0.219090	0.109545	−	0.993982i	$-0.465061\pi$
$233$	3.34426	0.219090	0.109545	+	0.993982i	$0.465061\pi$
$234$	0	0
$235$	−0.650390	−0.0424268
$236$	0	0
$237$	6.71296	0.436053
$238$	0	0
$239$	24.2819	1.57067	0.785334	−	0.619072i	$-0.212491\pi$
$239$	24.2819	1.57067	0.785334	+	0.619072i	$0.212491\pi$
$240$	0	0
$241$	−22.1689	−1.42803	−0.714014	−	0.700132i	$-0.753125\pi$
$241$	−22.1689	−1.42803	−0.714014	+	0.700132i	$0.753125\pi$
$242$	0	0
$243$	−22.3699	−1.43503
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.03588	−0.129540
$248$	0	0
$249$	23.5536	1.49265
$250$	0	0
$251$	18.9399	1.19548	0.597740	−	0.801690i	$-0.296066\pi$
$251$	18.9399	1.19548	0.597740	+	0.801690i	$0.296066\pi$
$252$	0	0
$253$	25.7614	1.61961
$254$	0	0
$255$	−0.774701	−0.0485137
$256$	0	0
$257$	1.74037	0.108561	0.0542807	−	0.998526i	$-0.482713\pi$
$257$	1.74037	0.108561	0.0542807	+	0.998526i	$0.482713\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−20.5901	−1.27449
$262$	0	0
$263$	20.2346	1.24772	0.623859	−	0.781537i	$-0.285564\pi$
$263$	20.2346	1.24772	0.623859	+	0.781537i	$0.285564\pi$
$264$	0	0
$265$	0.824136	0.0506263
$266$	0	0
$267$	−16.5203	−1.01103
$268$	0	0
$269$	11.1341	0.678857	0.339428	−	0.940632i	$-0.389766\pi$
$269$	11.1341	0.678857	0.339428	+	0.940632i	$0.389766\pi$
$270$	0	0
$271$	−25.9483	−1.57625	−0.788123	−	0.615517i	$-0.788947\pi$
$271$	−25.9483	−1.57625	−0.788123	+	0.615517i	$0.788947\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.5081	1.59850
$276$	0	0
$277$	−19.7091	−1.18421	−0.592104	−	0.805862i	$-0.701702\pi$
$277$	−19.7091	−1.18421	−0.592104	+	0.805862i	$0.701702\pi$
$278$	0	0
$279$	0.642518	0.0384665
$280$	0	0
$281$	19.0517	1.13653	0.568266	−	0.822845i	$-0.307614\pi$
$281$	19.0517	1.13653	0.568266	+	0.822845i	$0.307614\pi$
$282$	0	0
$283$	9.30114	0.552895	0.276448	−	0.961029i	$-0.410843\pi$
$283$	9.30114	0.552895	0.276448	+	0.961029i	$0.410843\pi$
$284$	0	0
$285$	0.348783	0.0206601
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.0665	−0.709793
$290$	0	0
$291$	−34.1589	−2.00243
$292$	0	0
$293$	−2.54463	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$293$	−2.54463	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$294$	0	0
$295$	−1.46301	−0.0851796
$296$	0	0
$297$	−4.90161	−0.284420
$298$	0	0
$299$	−9.85485	−0.569921
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−19.6108	−1.12661
$304$	0	0
$305$	1.20338	0.0689052
$306$	0	0
$307$	−25.8562	−1.47569	−0.737846	−	0.674969i	$-0.764157\pi$
$307$	−25.8562	−1.47569	−0.737846	+	0.674969i	$0.764157\pi$
$308$	0	0
$309$	27.5800	1.56897
$310$	0	0
$311$	−31.2960	−1.77463	−0.887316	−	0.461162i	$-0.847433\pi$
$311$	−31.2960	−1.77463	−0.887316	+	0.461162i	$0.847433\pi$
$312$	0	0
$313$	−22.6766	−1.28176	−0.640878	−	0.767643i	$-0.721430\pi$
$313$	−22.6766	−1.28176	−0.640878	+	0.767643i	$0.721430\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.7339	1.55769	0.778845	−	0.627216i	$-0.215806\pi$
$317$	27.7339	1.55769	0.778845	+	0.627216i	$0.215806\pi$
$318$	0	0
$319$	32.5639	1.82323
$320$	0	0
$321$	−3.13897	−0.175200
$322$	0	0
$323$	−2.22115	−0.123588
$324$	0	0
$325$	−10.1405	−0.562492
$326$	0	0
$327$	−15.6213	−0.863859
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3.68755	0.202686	0.101343	−	0.994852i	$-0.467686\pi$
$331$	3.68755	0.202686	0.101343	+	0.994852i	$0.467686\pi$
$332$	0	0
$333$	5.16585	0.283087
$334$	0	0
$335$	1.09714	0.0599430
$336$	0	0
$337$	8.31038	0.452695	0.226348	−	0.974047i	$-0.427321\pi$
$337$	8.31038	0.452695	0.226348	+	0.974047i	$0.427321\pi$
$338$	0	0
$339$	37.8105	2.05358
$340$	0	0
$341$	−1.01616	−0.0550283
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.68832	0.0908960
$346$	0	0
$347$	8.53352	0.458104	0.229052	−	0.973414i	$-0.426437\pi$
$347$	8.53352	0.458104	0.229052	+	0.973414i	$0.426437\pi$
$348$	0	0
$349$	18.9510	1.01442	0.507211	−	0.861822i	$-0.330676\pi$
$349$	18.9510	1.01442	0.507211	+	0.861822i	$0.330676\pi$
$350$	0	0
$351$	1.87508	0.100084
$352$	0	0
$353$	5.05195	0.268888	0.134444	−	0.990921i	$-0.457075\pi$
$353$	5.05195	0.268888	0.134444	+	0.990921i	$0.457075\pi$
$354$	0	0
$355$	1.47498	0.0782837
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.33143	0.123048	0.0615240	−	0.998106i	$-0.480404\pi$
$359$	2.33143	0.123048	0.0615240	+	0.998106i	$0.480404\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	43.7049	2.29391
$364$	0	0
$365$	1.76177	0.0922155
$366$	0	0
$367$	29.1910	1.52376	0.761880	−	0.647719i	$-0.224277\pi$
$367$	29.1910	1.52376	0.761880	+	0.647719i	$0.224277\pi$
$368$	0	0
$369$	8.31125	0.432666
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.8843	0.926011	0.463006	−	0.886355i	$-0.346771\pi$
$373$	17.8843	0.926011	0.463006	+	0.886355i	$0.346771\pi$
$374$	0	0
$375$	3.48117	0.179767
$376$	0	0
$377$	−12.4571	−0.641573
$378$	0	0
$379$	12.0838	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$379$	12.0838	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$380$	0	0
$381$	2.14037	0.109655
$382$	0	0
$383$	−26.0134	−1.32922	−0.664612	−	0.747189i	$-0.731403\pi$
$383$	−26.0134	−1.32922	−0.664612	+	0.747189i	$0.731403\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.09530	−0.360674
$388$	0	0
$389$	−18.2294	−0.924268	−0.462134	−	0.886810i	$-0.652916\pi$
$389$	−18.2294	−0.924268	−0.462134	+	0.886810i	$0.652916\pi$
$390$	0	0
$391$	−10.7517	−0.543736
$392$	0	0
$393$	7.20850	0.363621
$394$	0	0
$395$	−0.367847	−0.0185084
$396$	0	0
$397$	2.88488	0.144788	0.0723939	−	0.997376i	$-0.476936\pi$
$397$	2.88488	0.144788	0.0723939	+	0.997376i	$0.476936\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.4962	−0.823781	−0.411891	−	0.911233i	$-0.635131\pi$
$401$	−16.4962	−0.823781	−0.411891	+	0.911233i	$0.635131\pi$
$402$	0	0
$403$	0.388726	0.0193638
$404$	0	0
$405$	1.07439	0.0533869
$406$	0	0
$407$	−8.16997	−0.404970
$408$	0	0
$409$	−14.1085	−0.697619	−0.348810	−	0.937194i	$-0.613414\pi$
$409$	−14.1085	−0.697619	−0.348810	+	0.937194i	$0.613414\pi$
$410$	0	0
$411$	−32.6280	−1.60942
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.29066	−0.0633558
$416$	0	0
$417$	25.2362	1.23582
$418$	0	0
$419$	18.6899	0.913061	0.456530	−	0.889708i	$-0.349092\pi$
$419$	18.6899	0.913061	0.456530	+	0.889708i	$0.349092\pi$
$420$	0	0
$421$	26.4149	1.28738	0.643692	−	0.765285i	$-0.277402\pi$
$421$	26.4149	1.28738	0.643692	+	0.765285i	$0.277402\pi$
$422$	0	0
$423$	15.8312	0.769737
$424$	0	0
$425$	−11.0633	−0.536649
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−27.3353	−1.31976
$430$	0	0
$431$	38.0935	1.83490	0.917449	−	0.397854i	$-0.130245\pi$
$431$	38.0935	1.83490	0.917449	+	0.397854i	$0.130245\pi$
$432$	0	0
$433$	−20.0271	−0.962440	−0.481220	−	0.876600i	$-0.659806\pi$
$433$	−20.0271	−0.962440	−0.481220	+	0.876600i	$0.659806\pi$
$434$	0	0
$435$	2.13413	0.102324
$436$	0	0
$437$	4.84059	0.231557
$438$	0	0
$439$	−12.8747	−0.614476	−0.307238	−	0.951633i	$-0.599405\pi$
$439$	−12.8747	−0.614476	−0.307238	+	0.951633i	$0.599405\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.33036	−0.0632072	−0.0316036	−	0.999500i	$-0.510061\pi$
$443$	−1.33036	−0.0632072	−0.0316036	+	0.999500i	$0.510061\pi$
$444$	0	0
$445$	0.905255	0.0429132
$446$	0	0
$447$	13.9351	0.659108
$448$	0	0
$449$	−16.8986	−0.797494	−0.398747	−	0.917061i	$-0.630555\pi$
$449$	−16.8986	−0.797494	−0.398747	+	0.917061i	$0.630555\pi$
$450$	0	0
$451$	−13.1445	−0.618951
$452$	0	0
$453$	−12.0499	−0.566152
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.8043	−1.62807	−0.814037	−	0.580812i	$-0.802735\pi$
$457$	−34.8043	−1.62807	−0.814037	+	0.580812i	$0.802735\pi$
$458$	0	0
$459$	2.04572	0.0954860
$460$	0	0
$461$	4.61954	0.215153	0.107577	−	0.994197i	$-0.465691\pi$
$461$	4.61954	0.215153	0.107577	+	0.994197i	$0.465691\pi$
$462$	0	0
$463$	17.7987	0.827177	0.413588	−	0.910464i	$-0.364275\pi$
$463$	17.7987	0.827177	0.413588	+	0.910464i	$0.364275\pi$
$464$	0	0
$465$	−0.0665960	−0.00308831
$466$	0	0
$467$	−21.5522	−0.997318	−0.498659	−	0.866798i	$-0.666174\pi$
$467$	−21.5522	−0.997318	−0.498659	+	0.866798i	$0.666174\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−36.8353	−1.69728
$472$	0	0
$473$	11.2214	0.515963
$474$	0	0
$475$	4.98089	0.228539
$476$	0	0
$477$	−20.0603	−0.918498
$478$	0	0
$479$	11.9030	0.543863	0.271932	−	0.962317i	$-0.412338\pi$
$479$	11.9030	0.543863	0.271932	+	0.962317i	$0.412338\pi$
$480$	0	0
$481$	3.12536	0.142504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.87179	0.0849936
$486$	0	0
$487$	26.4746	1.19968	0.599839	−	0.800121i	$-0.295231\pi$
$487$	26.4746	1.19968	0.599839	+	0.800121i	$0.295231\pi$
$488$	0	0
$489$	−4.06009	−0.183603
$490$	0	0
$491$	−3.39719	−0.153313	−0.0766566	−	0.997058i	$-0.524425\pi$
$491$	−3.39719	−0.153313	−0.0766566	+	0.997058i	$0.524425\pi$
$492$	0	0
$493$	−13.5907	−0.612096
$494$	0	0
$495$	2.47582	0.111280
$496$	0	0
$497$	0	0
$498$	0	0
$499$	23.3972	1.04740	0.523702	−	0.851901i	$-0.324550\pi$
$499$	23.3972	1.04740	0.523702	+	0.851901i	$0.324550\pi$
$500$	0	0
$501$	36.2898	1.62131
$502$	0	0
$503$	7.51301	0.334988	0.167494	−	0.985873i	$-0.446432\pi$
$503$	7.51301	0.334988	0.167494	+	0.985873i	$0.446432\pi$
$504$	0	0
$505$	1.07461	0.0478193
$506$	0	0
$507$	−22.3409	−0.992192
$508$	0	0
$509$	13.9696	0.619193	0.309596	−	0.950868i	$-0.399806\pi$
$509$	13.9696	0.619193	0.309596	+	0.950868i	$0.399806\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.921017	−0.0406639
$514$	0	0
$515$	−1.51129	−0.0665952
$516$	0	0
$517$	−25.0375	−1.10115
$518$	0	0
$519$	50.7209	2.22640
$520$	0	0
$521$	−27.8722	−1.22111	−0.610553	−	0.791976i	$-0.709053\pi$
$521$	−27.8722	−1.22111	−0.610553	+	0.791976i	$0.709053\pi$
$522$	0	0
$523$	1.46574	0.0640922	0.0320461	−	0.999486i	$-0.489798\pi$
$523$	1.46574	0.0640922	0.0320461	+	0.999486i	$0.489798\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.424102	0.0184742
$528$	0	0
$529$	0.431307	0.0187525
$530$	0	0
$531$	35.6111	1.54539
$532$	0	0
$533$	5.02834	0.217802
$534$	0	0
$535$	0.172005	0.00743641
$536$	0	0
$537$	−6.16439	−0.266013
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.8828	0.940815	0.470407	−	0.882449i	$-0.344107\pi$
$541$	21.8828	0.940815	0.470407	+	0.882449i	$0.344107\pi$
$542$	0	0
$543$	37.6080	1.61391
$544$	0	0
$545$	0.855992	0.0366667
$546$	0	0
$547$	−19.8197	−0.847431	−0.423715	−	0.905795i	$-0.639274\pi$
$547$	−19.8197	−0.847431	−0.423715	+	0.905795i	$0.639274\pi$
$548$	0	0
$549$	−29.2914	−1.25013
$550$	0	0
$551$	6.11878	0.260669
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.535432	−0.0227278
$556$	0	0
$557$	15.0063	0.635838	0.317919	−	0.948118i	$-0.397016\pi$
$557$	15.0063	0.635838	0.317919	+	0.948118i	$0.397016\pi$
$558$	0	0
$559$	−4.29269	−0.181561
$560$	0	0
$561$	−29.8230	−1.25913
$562$	0	0
$563$	3.44546	0.145209	0.0726044	−	0.997361i	$-0.476869\pi$
$563$	3.44546	0.145209	0.0726044	+	0.997361i	$0.476869\pi$
$564$	0	0
$565$	−2.07188	−0.0871647
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.5055	−1.53039	−0.765194	−	0.643799i	$-0.777357\pi$
$569$	−36.5055	−1.53039	−0.765194	+	0.643799i	$0.777357\pi$
$570$	0	0
$571$	1.60085	0.0669933	0.0334967	−	0.999439i	$-0.489336\pi$
$571$	1.60085	0.0669933	0.0334967	+	0.999439i	$0.489336\pi$
$572$	0	0
$573$	−26.7524	−1.11760
$574$	0	0
$575$	24.1104	1.00547
$576$	0	0
$577$	−4.45128	−0.185309	−0.0926547	−	0.995698i	$-0.529535\pi$
$577$	−4.45128	−0.185309	−0.0926547	+	0.995698i	$0.529535\pi$
$578$	0	0
$579$	−24.7886	−1.03018
$580$	0	0
$581$	0	0
$582$	0	0
$583$	31.7260	1.31396
$584$	0	0
$585$	−0.947107	−0.0391581
$586$	0	0
$587$	−0.0528446	−0.00218113	−0.00109057	−	0.999999i	$-0.500347\pi$
$587$	−0.0528446	−0.00218113	−0.00109057	+	0.999999i	$0.500347\pi$
$588$	0	0
$589$	−0.190938	−0.00786746
$590$	0	0
$591$	−32.0289	−1.31749
$592$	0	0
$593$	37.3026	1.53183	0.765917	−	0.642939i	$-0.222285\pi$
$593$	37.3026	1.53183	0.765917	+	0.642939i	$0.222285\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.4783	0.633486
$598$	0	0
$599$	−26.5097	−1.08316	−0.541578	−	0.840650i	$-0.682173\pi$
$599$	−26.5097	−1.08316	−0.541578	+	0.840650i	$0.682173\pi$
$600$	0	0
$601$	−34.5164	−1.40795	−0.703977	−	0.710222i	$-0.748594\pi$
$601$	−34.5164	−1.40795	−0.703977	+	0.710222i	$0.748594\pi$
$602$	0	0
$603$	−26.7054	−1.08753
$604$	0	0
$605$	−2.39488	−0.0973656
$606$	0	0
$607$	36.0678	1.46395	0.731974	−	0.681333i	$-0.238599\pi$
$607$	36.0678	1.46395	0.731974	+	0.681333i	$0.238599\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.57792	0.387481
$612$	0	0
$613$	−11.3296	−0.457598	−0.228799	−	0.973474i	$-0.573480\pi$
$613$	−11.3296	−0.457598	−0.228799	+	0.973474i	$0.573480\pi$
$614$	0	0
$615$	−0.861448	−0.0347369
$616$	0	0
$617$	47.6870	1.91981	0.959904	−	0.280330i	$-0.0904438\pi$
$617$	47.6870	1.91981	0.959904	+	0.280330i	$0.0904438\pi$
$618$	0	0
$619$	17.7335	0.712771	0.356386	−	0.934339i	$-0.384009\pi$
$619$	17.7335	0.712771	0.356386	+	0.934339i	$0.384009\pi$
$620$	0	0
$621$	−4.45827	−0.178904
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.7137	0.988547
$626$	0	0
$627$	13.4268	0.536215
$628$	0	0
$629$	3.40979	0.135957
$630$	0	0
$631$	29.1183	1.15918	0.579590	−	0.814908i	$-0.303213\pi$
$631$	29.1183	1.15918	0.579590	+	0.814908i	$0.303213\pi$
$632$	0	0
$633$	−59.0609	−2.34746
$634$	0	0
$635$	−0.117285	−0.00465432
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−35.9024	−1.42028
$640$	0	0
$641$	−2.71096	−0.107076	−0.0535382	−	0.998566i	$-0.517050\pi$
$641$	−2.71096	−0.107076	−0.0535382	+	0.998566i	$0.517050\pi$
$642$	0	0
$643$	−14.5519	−0.573872	−0.286936	−	0.957950i	$-0.592637\pi$
$643$	−14.5519	−0.573872	−0.286936	+	0.957950i	$0.592637\pi$
$644$	0	0
$645$	0.735417	0.0289570
$646$	0	0
$647$	−21.0359	−0.827005	−0.413502	−	0.910503i	$-0.635695\pi$
$647$	−21.0359	−0.827005	−0.413502	+	0.910503i	$0.635695\pi$
$648$	0	0
$649$	−56.3201	−2.21076
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.8907	0.465321	0.232660	−	0.972558i	$-0.425257\pi$
$653$	11.8907	0.465321	0.232660	+	0.972558i	$0.425257\pi$
$654$	0	0
$655$	−0.395001	−0.0154340
$656$	0	0
$657$	−42.8834	−1.67304
$658$	0	0
$659$	27.9741	1.08971	0.544857	−	0.838529i	$-0.316584\pi$
$659$	27.9741	1.08971	0.544857	+	0.838529i	$0.316584\pi$
$660$	0	0
$661$	−16.8339	−0.654764	−0.327382	−	0.944892i	$-0.606166\pi$
$661$	−16.8339	−0.654764	−0.327382	+	0.944892i	$0.606166\pi$
$662$	0	0
$663$	11.4086	0.443072
$664$	0	0
$665$	0	0
$666$	0	0
$667$	29.6185	1.14683
$668$	0	0
$669$	−34.6847	−1.34099
$670$	0	0
$671$	46.3253	1.78837
$672$	0	0
$673$	22.0461	0.849813	0.424907	−	0.905237i	$-0.360307\pi$
$673$	22.0461	0.849813	0.424907	+	0.905237i	$0.360307\pi$
$674$	0	0
$675$	−4.58748	−0.176572
$676$	0	0
$677$	−37.4039	−1.43755	−0.718775	−	0.695243i	$-0.755297\pi$
$677$	−37.4039	−1.43755	−0.718775	+	0.695243i	$0.755297\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−19.3520	−0.741570
$682$	0	0
$683$	−31.5835	−1.20851	−0.604254	−	0.796792i	$-0.706529\pi$
$683$	−31.5835	−1.20851	−0.604254	+	0.796792i	$0.706529\pi$
$684$	0	0
$685$	1.78790	0.0683122
$686$	0	0
$687$	18.7759	0.716344
$688$	0	0
$689$	−12.1366	−0.462366
$690$	0	0
$691$	49.7525	1.89268	0.946338	−	0.323180i	$-0.104752\pi$
$691$	49.7525	1.89268	0.946338	+	0.323180i	$0.104752\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.38286	−0.0524548
$696$	0	0
$697$	5.48595	0.207795
$698$	0	0
$699$	8.43725	0.319126
$700$	0	0
$701$	4.47795	0.169130	0.0845649	−	0.996418i	$-0.473050\pi$
$701$	4.47795	0.169130	0.0845649	+	0.996418i	$0.473050\pi$
$702$	0	0
$703$	−1.53514	−0.0578990
$704$	0	0
$705$	−1.64087	−0.0617989
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.26024	0.122441	0.0612205	−	0.998124i	$-0.480501\pi$
$709$	3.26024	0.122441	0.0612205	+	0.998124i	$0.480501\pi$
$710$	0	0
$711$	8.95376	0.335792
$712$	0	0
$713$	−0.924252	−0.0346135
$714$	0	0
$715$	1.49788	0.0560176
$716$	0	0
$717$	61.2611	2.28784
$718$	0	0
$719$	−45.2952	−1.68923	−0.844613	−	0.535378i	$-0.820169\pi$
$719$	−45.2952	−1.68923	−0.844613	+	0.535378i	$0.820169\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−55.9302	−2.08007
$724$	0	0
$725$	30.4770	1.13189
$726$	0	0
$727$	8.99725	0.333690	0.166845	−	0.985983i	$-0.446642\pi$
$727$	8.99725	0.333690	0.166845	+	0.985983i	$0.446642\pi$
$728$	0	0
$729$	−33.1227	−1.22676
$730$	0	0
$731$	−4.68334	−0.173220
$732$	0	0
$733$	34.7582	1.28382	0.641912	−	0.766778i	$-0.278141\pi$
$733$	34.7582	1.28382	0.641912	+	0.766778i	$0.278141\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.2355	1.55576
$738$	0	0
$739$	−33.0301	−1.21503	−0.607516	−	0.794307i	$-0.707834\pi$
$739$	−33.0301	−1.21503	−0.607516	+	0.794307i	$0.707834\pi$
$740$	0	0
$741$	−5.13633	−0.188688
$742$	0	0
$743$	−17.3301	−0.635781	−0.317890	−	0.948127i	$-0.602974\pi$
$743$	−17.3301	−0.635781	−0.317890	+	0.948127i	$0.602974\pi$
$744$	0	0
$745$	−0.763596	−0.0279760
$746$	0	0
$747$	31.4159	1.14945
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.1470	−0.625703	−0.312851	−	0.949802i	$-0.601284\pi$
$751$	−17.1470	−0.625703	−0.312851	+	0.949802i	$0.601284\pi$
$752$	0	0
$753$	47.7837	1.74134
$754$	0	0
$755$	0.660291	0.0240305
$756$	0	0
$757$	2.06947	0.0752162	0.0376081	−	0.999293i	$-0.488026\pi$
$757$	2.06947	0.0752162	0.0376081	+	0.999293i	$0.488026\pi$
$758$	0	0
$759$	64.9936	2.35912
$760$	0	0
$761$	9.34068	0.338599	0.169300	−	0.985565i	$-0.445849\pi$
$761$	9.34068	0.338599	0.169300	+	0.985565i	$0.445849\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.03330	−0.0373590
$766$	0	0
$767$	21.5449	0.777940
$768$	0	0
$769$	−9.27772	−0.334563	−0.167282	−	0.985909i	$-0.553499\pi$
$769$	−9.27772	−0.334563	−0.167282	+	0.985909i	$0.553499\pi$
$770$	0	0
$771$	4.39080	0.158131
$772$	0	0
$773$	0.719316	0.0258720	0.0129360	−	0.999916i	$-0.495882\pi$
$773$	0.719316	0.0258720	0.0129360	+	0.999916i	$0.495882\pi$
$774$	0	0
$775$	−0.951040	−0.0341624
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.46986	−0.0884921
$780$	0	0
$781$	56.7809	2.03178
$782$	0	0
$783$	−5.63550	−0.201397
$784$	0	0
$785$	2.01845	0.0720414
$786$	0	0
$787$	−23.4170	−0.834724	−0.417362	−	0.908740i	$-0.637045\pi$
$787$	−23.4170	−0.834724	−0.417362	+	0.908740i	$0.637045\pi$
$788$	0	0
$789$	51.0500	1.81743
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−17.7214	−0.629306
$794$	0	0
$795$	2.07922	0.0737423
$796$	0	0
$797$	−35.1123	−1.24374	−0.621871	−	0.783120i	$-0.713627\pi$
$797$	−35.1123	−1.24374	−0.621871	+	0.783120i	$0.713627\pi$
$798$	0	0
$799$	10.4496	0.369679
$800$	0	0
$801$	−22.0348	−0.778562
$802$	0	0
$803$	67.8215	2.39337
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.0902	0.988823
$808$	0	0
$809$	−28.0080	−0.984709	−0.492355	−	0.870395i	$-0.663864\pi$
$809$	−28.0080	−0.984709	−0.492355	+	0.870395i	$0.663864\pi$
$810$	0	0
$811$	10.6680	0.374603	0.187301	−	0.982303i	$-0.440026\pi$
$811$	10.6680	0.374603	0.187301	+	0.982303i	$0.440026\pi$
$812$	0	0
$813$	−65.4652	−2.29596
$814$	0	0
$815$	0.222479	0.00779309
$816$	0	0
$817$	2.10852	0.0737677
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−53.2576	−1.85870	−0.929352	−	0.369194i	$-0.879634\pi$
$821$	−53.2576	−1.85870	−0.929352	+	0.369194i	$0.879634\pi$
$822$	0	0
$823$	−3.52809	−0.122982	−0.0614908	−	0.998108i	$-0.519585\pi$
$823$	−3.52809	−0.122982	−0.0614908	+	0.998108i	$0.519585\pi$
$824$	0	0
$825$	66.8774	2.32837
$826$	0	0
$827$	31.0651	1.08024	0.540120	−	0.841588i	$-0.318379\pi$
$827$	31.0651	1.08024	0.540120	+	0.841588i	$0.318379\pi$
$828$	0	0
$829$	8.10706	0.281570	0.140785	−	0.990040i	$-0.455037\pi$
$829$	8.10706	0.281570	0.140785	+	0.990040i	$0.455037\pi$
$830$	0	0
$831$	−49.7243	−1.72492
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.98856	−0.0688168
$836$	0	0
$837$	0.175857	0.00607851
$838$	0	0
$839$	11.6918	0.403647	0.201823	−	0.979422i	$-0.435313\pi$
$839$	11.6918	0.403647	0.201823	+	0.979422i	$0.435313\pi$
$840$	0	0
$841$	8.43950	0.291017
$842$	0	0
$843$	48.0658	1.65547
$844$	0	0
$845$	1.22420	0.0421138
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23.4659	0.805348
$850$	0	0
$851$	−7.43100	−0.254731
$852$	0	0
$853$	29.7562	1.01883	0.509417	−	0.860520i	$-0.329861\pi$
$853$	29.7562	1.01883	0.509417	+	0.860520i	$0.329861\pi$
$854$	0	0
$855$	0.465208	0.0159098
$856$	0	0
$857$	13.6331	0.465697	0.232849	−	0.972513i	$-0.425195\pi$
$857$	13.6331	0.465697	0.232849	+	0.972513i	$0.425195\pi$
$858$	0	0
$859$	1.09828	0.0374729	0.0187365	−	0.999824i	$-0.494036\pi$
$859$	1.09828	0.0374729	0.0187365	+	0.999824i	$0.494036\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.3391	1.10084	0.550419	−	0.834889i	$-0.314468\pi$
$863$	32.3391	1.10084	0.550419	+	0.834889i	$0.314468\pi$
$864$	0	0
$865$	−2.77933	−0.0945001
$866$	0	0
$867$	−30.4426	−1.03389
$868$	0	0
$869$	−14.1607	−0.480368
$870$	0	0
$871$	−16.1569	−0.547455
$872$	0	0
$873$	−45.5612	−1.54201
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−51.3230	−1.73305	−0.866527	−	0.499130i	$-0.833653\pi$
$877$	−51.3230	−1.73305	−0.866527	+	0.499130i	$0.833653\pi$
$878$	0	0
$879$	−6.41986	−0.216537
$880$	0	0
$881$	37.5197	1.26407	0.632035	−	0.774940i	$-0.282220\pi$
$881$	37.5197	1.26407	0.632035	+	0.774940i	$0.282220\pi$
$882$	0	0
$883$	−14.6984	−0.494640	−0.247320	−	0.968934i	$-0.579550\pi$
$883$	−14.6984	−0.494640	−0.247320	+	0.968934i	$0.579550\pi$
$884$	0	0
$885$	−3.69103	−0.124073
$886$	0	0
$887$	56.6361	1.90165	0.950827	−	0.309722i	$-0.100236\pi$
$887$	56.6361	1.90165	0.950827	+	0.309722i	$0.100236\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	41.3598	1.38561
$892$	0	0
$893$	−4.70457	−0.157432
$894$	0	0
$895$	0.337787	0.0112910
$896$	0	0
$897$	−24.8629	−0.830147
$898$	0	0
$899$	−1.16831	−0.0389652
$900$	0	0
$901$	−13.2411	−0.441123
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.06079	−0.0685029
$906$	0	0
$907$	−7.60265	−0.252442	−0.126221	−	0.992002i	$-0.540285\pi$
$907$	−7.60265	−0.252442	−0.126221	+	0.992002i	$0.540285\pi$
$908$	0	0
$909$	−26.1570	−0.867573
$910$	0	0
$911$	21.4709	0.711363	0.355682	−	0.934607i	$-0.384249\pi$
$911$	21.4709	0.711363	0.355682	+	0.934607i	$0.384249\pi$
$912$	0	0
$913$	−49.6852	−1.64434
$914$	0	0
$915$	3.03601	0.100367
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−56.4076	−1.86071	−0.930357	−	0.366654i	$-0.880503\pi$
$919$	−56.4076	−1.86071	−0.930357	+	0.366654i	$0.880503\pi$
$920$	0	0
$921$	−65.2328	−2.14949
$922$	0	0
$923$	−21.7211	−0.714960
$924$	0	0
$925$	−7.64638	−0.251411
$926$	0	0
$927$	36.7862	1.20822
$928$	0	0
$929$	−37.1859	−1.22003	−0.610015	−	0.792390i	$-0.708837\pi$
$929$	−37.1859	−1.22003	−0.610015	+	0.792390i	$0.708837\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−78.9568	−2.58493
$934$	0	0
$935$	1.63420	0.0534439
$936$	0	0
$937$	45.7718	1.49530	0.747649	−	0.664094i	$-0.231182\pi$
$937$	45.7718	1.49530	0.747649	+	0.664094i	$0.231182\pi$
$938$	0	0
$939$	−57.2109	−1.86701
$940$	0	0
$941$	13.2502	0.431944	0.215972	−	0.976400i	$-0.430708\pi$
$941$	13.2502	0.431944	0.215972	+	0.976400i	$0.430708\pi$
$942$	0	0
$943$	−11.9556	−0.389328
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.2551	1.86054	0.930270	−	0.366875i	$-0.119572\pi$
$947$	57.2551	1.86054	0.930270	+	0.366875i	$0.119572\pi$
$948$	0	0
$949$	−25.9446	−0.842198
$950$	0	0
$951$	69.9700	2.26893
$952$	0	0
$953$	40.6347	1.31629	0.658143	−	0.752893i	$-0.271342\pi$
$953$	40.6347	1.31629	0.658143	+	0.752893i	$0.271342\pi$
$954$	0	0
$955$	1.46594	0.0474367
$956$	0	0
$957$	82.1557	2.65572
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9635	−0.998824
$962$	0	0
$963$	−4.18677	−0.134917
$964$	0	0
$965$	1.35833	0.0437261
$966$	0	0
$967$	12.9245	0.415624	0.207812	−	0.978169i	$-0.433366\pi$
$967$	12.9245	0.415624	0.207812	+	0.978169i	$0.433366\pi$
$968$	0	0
$969$	−5.60376	−0.180019
$970$	0	0
$971$	−54.1376	−1.73736	−0.868679	−	0.495375i	$-0.835031\pi$
$971$	−54.1376	−1.73736	−0.868679	+	0.495375i	$0.835031\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−25.5835	−0.819327
$976$	0	0
$977$	7.93688	0.253923	0.126962	−	0.991908i	$-0.459477\pi$
$977$	7.93688	0.253923	0.126962	+	0.991908i	$0.459477\pi$
$978$	0	0
$979$	34.8488	1.11377
$980$	0	0
$981$	−20.8357	−0.665233
$982$	0	0
$983$	32.4524	1.03507	0.517535	−	0.855662i	$-0.326850\pi$
$983$	32.4524	1.03507	0.517535	+	0.855662i	$0.326850\pi$
$984$	0	0
$985$	1.75507	0.0559213
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.2065	0.324547
$990$	0	0
$991$	0.804755	0.0255639	0.0127819	−	0.999918i	$-0.495931\pi$
$991$	0.804755	0.0255639	0.0127819	+	0.999918i	$0.495931\pi$
$992$	0	0
$993$	9.30336	0.295233
$994$	0	0
$995$	−0.848159	−0.0268885
$996$	0	0
$997$	10.2728	0.325344	0.162672	−	0.986680i	$-0.447989\pi$
$997$	10.2728	0.325344	0.162672	+	0.986680i	$0.447989\pi$
$998$	0	0
$999$	1.41389	0.0447336

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bo.1.6	✓	7
7.6	odd	2		7448.2.a.bp.1.2	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bo.1.6	✓	7	1.1	even	1	trivial
7448.2.a.bp.1.2	yes	7	7.6	odd	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52291 q^{3} -0.138247 q^{5} +3.36506 q^{9} +O(q^{10})\)	\(q+2.52291 q^{3} -0.138247 q^{5} +3.36506 q^{9} -5.32196 q^{11} +2.03588 q^{13} -0.348783 q^{15} +2.22115 q^{17} -1.00000 q^{19} -4.84059 q^{23} -4.98089 q^{25} +0.921017 q^{27} -6.11878 q^{29} +0.190938 q^{31} -13.4268 q^{33} +1.53514 q^{37} +5.13633 q^{39} +2.46986 q^{41} -2.10852 q^{43} -0.465208 q^{45} +4.70457 q^{47} +5.60376 q^{51} -5.96134 q^{53} +0.735742 q^{55} -2.52291 q^{57} +10.5826 q^{59} -8.70457 q^{61} -0.281453 q^{65} -7.93608 q^{67} -12.2124 q^{69} -10.6692 q^{71} -12.7437 q^{73} -12.5663 q^{75} +2.66080 q^{79} -7.77154 q^{81} +9.33589 q^{83} -0.307067 q^{85} -15.4371 q^{87} -6.54811 q^{89} +0.481719 q^{93} +0.138247 q^{95} -13.5395 q^{97} -17.9087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) \) 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

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