LMFDB - Embedded newform 1575.4.a.bn.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.04851$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.04851 q^{2} +8.39045 q^{4} +7.00000 q^{7} +1.58074 q^{8} +O(q^{10})$	$q+4.04851 q^{2} +8.39045 q^{4} +7.00000 q^{7} +1.58074 q^{8} -6.78210 q^{11} +48.9221 q^{13} +28.3396 q^{14} -60.7239 q^{16} -92.4381 q^{17} -125.574 q^{19} -27.4574 q^{22} +32.2681 q^{23} +198.062 q^{26} +58.7332 q^{28} -282.778 q^{29} +205.434 q^{31} -258.488 q^{32} -374.237 q^{34} +190.627 q^{37} -508.388 q^{38} -123.269 q^{41} +35.0202 q^{43} -56.9049 q^{44} +130.638 q^{46} -419.030 q^{47} +49.0000 q^{49} +410.478 q^{52} +0.365379 q^{53} +11.0652 q^{56} -1144.83 q^{58} -328.317 q^{59} -515.707 q^{61} +831.704 q^{62} -560.699 q^{64} -828.957 q^{67} -775.597 q^{68} +496.231 q^{71} +701.132 q^{73} +771.757 q^{74} -1053.62 q^{76} -47.4747 q^{77} +199.388 q^{79} -499.056 q^{82} +194.923 q^{83} +141.780 q^{86} -10.7208 q^{88} -137.406 q^{89} +342.454 q^{91} +270.744 q^{92} -1696.45 q^{94} +220.440 q^{97} +198.377 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8}+O(q^{10}) $ 5 * q - 4 * q^2 + 18 * q^4 + 35 * q^7 - 42 * q^8	$ 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8} - 42 q^{11} + 34 q^{13} - 28 q^{14} + 74 q^{16} - 238 q^{17} - 36 q^{19} + 358 q^{22} - 152 q^{23} + 310 q^{26} + 126 q^{28} + 44 q^{29} + 60 q^{31} - 710 q^{32} - 482 q^{34} + 312 q^{37} - 280 q^{38} + 426 q^{41} + 304 q^{43} - 712 q^{44} + 88 q^{46} - 370 q^{47} + 245 q^{49} - 1156 q^{52} - 976 q^{53} - 294 q^{56} - 2722 q^{58} + 432 q^{59} - 442 q^{61} + 956 q^{62} + 1362 q^{64} - 804 q^{67} + 420 q^{68} - 440 q^{71} + 564 q^{73} + 1512 q^{74} - 1336 q^{76} - 294 q^{77} + 1790 q^{79} - 3480 q^{82} - 1656 q^{83} - 1216 q^{86} - 1092 q^{88} - 746 q^{89} + 238 q^{91} - 572 q^{92} - 826 q^{94} + 518 q^{97} - 196 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 18 * q^4 + 35 * q^7 - 42 * q^8 - 42 * q^11 + 34 * q^13 - 28 * q^14 + 74 * q^16 - 238 * q^17 - 36 * q^19 + 358 * q^22 - 152 * q^23 + 310 * q^26 + 126 * q^28 + 44 * q^29 + 60 * q^31 - 710 * q^32 - 482 * q^34 + 312 * q^37 - 280 * q^38 + 426 * q^41 + 304 * q^43 - 712 * q^44 + 88 * q^46 - 370 * q^47 + 245 * q^49 - 1156 * q^52 - 976 * q^53 - 294 * q^56 - 2722 * q^58 + 432 * q^59 - 442 * q^61 + 956 * q^62 + 1362 * q^64 - 804 * q^67 + 420 * q^68 - 440 * q^71 + 564 * q^73 + 1512 * q^74 - 1336 * q^76 - 294 * q^77 + 1790 * q^79 - 3480 * q^82 - 1656 * q^83 - 1216 * q^86 - 1092 * q^88 - 746 * q^89 + 238 * q^91 - 572 * q^92 - 826 * q^94 + 518 * q^97 - 196 * q^98

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 $

$2$	4.04851	1.43137	0.715683	−	0.698426i	$-0.246116\pi$
$2$	4.04851	1.43137	0.715683	+	0.698426i	$0.246116\pi$
$3$	0	0
$3$	0	0
$4$	8.39045	1.04881
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	1.58074	0.0698597
$9$	0	0
$10$	0	0
$11$	−6.78210	−0.185898	−0.0929491	−	0.995671i	$-0.529629\pi$
$11$	−6.78210	−0.185898	−0.0929491	+	0.995671i	$0.529629\pi$
$12$	0	0
$13$	48.9221	1.04373	0.521867	−	0.853027i	$-0.325236\pi$
$13$	48.9221	1.04373	0.521867	+	0.853027i	$0.325236\pi$
$14$	28.3396	0.541005
$15$	0	0
$16$	−60.7239	−0.948812
$17$	−92.4381	−1.31880	−0.659398	−	0.751794i	$-0.729189\pi$
$17$	−92.4381	−1.31880	−0.659398	+	0.751794i	$0.729189\pi$
$18$	0	0
$19$	−125.574	−1.51625	−0.758123	−	0.652112i	$-0.773883\pi$
$19$	−125.574	−1.51625	−0.758123	+	0.652112i	$0.773883\pi$
$20$	0	0
$21$	0	0
$22$	−27.4574	−0.266088
$23$	32.2681	0.292538	0.146269	−	0.989245i	$-0.453274\pi$
$23$	32.2681	0.292538	0.146269	+	0.989245i	$0.453274\pi$
$24$	0	0
$25$	0	0
$26$	198.062	1.49396
$27$	0	0
$28$	58.7332	0.396412
$29$	−282.778	−1.81071	−0.905354	−	0.424659i	$-0.860394\pi$
$29$	−282.778	−1.81071	−0.905354	+	0.424659i	$0.860394\pi$
$30$	0	0
$31$	205.434	1.19023	0.595115	−	0.803641i	$-0.297107\pi$
$31$	205.434	1.19023	0.595115	+	0.803641i	$0.297107\pi$
$32$	−258.488	−1.42796
$33$	0	0
$34$	−374.237	−1.88768
$35$	0	0
$36$	0	0
$37$	190.627	0.846998	0.423499	−	0.905897i	$-0.360802\pi$
$37$	190.627	0.846998	0.423499	+	0.905897i	$0.360802\pi$
$38$	−508.388	−2.17030
$39$	0	0
$40$	0	0
$41$	−123.269	−0.469546	−0.234773	−	0.972050i	$-0.575435\pi$
$41$	−123.269	−0.469546	−0.234773	+	0.972050i	$0.575435\pi$
$42$	0	0
$43$	35.0202	0.124198	0.0620991	−	0.998070i	$-0.480221\pi$
$43$	35.0202	0.124198	0.0620991	+	0.998070i	$0.480221\pi$
$44$	−56.9049	−0.194971
$45$	0	0
$46$	130.638	0.418728
$47$	−419.030	−1.30046	−0.650231	−	0.759736i	$-0.725328\pi$
$47$	−419.030	−1.30046	−0.650231	+	0.759736i	$0.725328\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	410.478	1.09467
$53$	0.365379	0.000946956 0	0.000473478	−	1.00000i	$-0.499849\pi$
$53$	0.365379	0.000946956 0	0.000473478		1.00000i	$0.499849\pi$
$54$	0	0
$55$	0	0
$56$	11.0652	0.0264045
$57$	0	0
$58$	−1144.83	−2.59178
$59$	−328.317	−0.724461	−0.362231	−	0.932088i	$-0.617985\pi$
$59$	−328.317	−0.724461	−0.362231	+	0.932088i	$0.617985\pi$
$60$	0	0
$61$	−515.707	−1.08245	−0.541226	−	0.840877i	$-0.682040\pi$
$61$	−515.707	−1.08245	−0.541226	+	0.840877i	$0.682040\pi$
$62$	831.704	1.70365
$63$	0	0
$64$	−560.699	−1.09511
$65$	0	0
$66$	0	0
$67$	−828.957	−1.51154	−0.755770	−	0.654837i	$-0.772737\pi$
$67$	−828.957	−1.51154	−0.755770	+	0.654837i	$0.772737\pi$
$68$	−775.597	−1.38316
$69$	0	0
$70$	0	0
$71$	496.231	0.829462	0.414731	−	0.909944i	$-0.363876\pi$
$71$	496.231	0.829462	0.414731	+	0.909944i	$0.363876\pi$
$72$	0	0
$73$	701.132	1.12413	0.562064	−	0.827094i	$-0.310008\pi$
$73$	701.132	1.12413	0.562064	+	0.827094i	$0.310008\pi$
$74$	771.757	1.21236
$75$	0	0
$76$	−1053.62	−1.59025
$77$	−47.4747	−0.0702629
$78$	0	0
$79$	199.388	0.283961	0.141981	−	0.989869i	$-0.454653\pi$
$79$	199.388	0.283961	0.141981	+	0.989869i	$0.454653\pi$
$80$	0	0
$81$	0	0
$82$	−499.056	−0.672092
$83$	194.923	0.257778	0.128889	−	0.991659i	$-0.458859\pi$
$83$	194.923	0.257778	0.128889	+	0.991659i	$0.458859\pi$
$84$	0	0
$85$	0	0
$86$	141.780	0.177773
$87$	0	0
$88$	−10.7208	−0.0129868
$89$	−137.406	−0.163651	−0.0818257	−	0.996647i	$-0.526075\pi$
$89$	−137.406	−0.163651	−0.0818257	+	0.996647i	$0.526075\pi$
$90$	0	0
$91$	342.454	0.394494
$92$	270.744	0.306815
$93$	0	0
$94$	−1696.45	−1.86144
$95$	0	0
$96$	0	0
$97$	220.440	0.230745	0.115372	−	0.993322i	$-0.463194\pi$
$97$	220.440	0.230745	0.115372	+	0.993322i	$0.463194\pi$
$98$	198.377	0.204481
$99$	0	0
$100$	0	0
$101$	591.358	0.582597	0.291298	−	0.956632i	$-0.405913\pi$
$101$	591.358	0.582597	0.291298	+	0.956632i	$0.405913\pi$
$102$	0	0
$103$	−476.494	−0.455829	−0.227914	−	0.973681i	$-0.573191\pi$
$103$	−476.494	−0.455829	−0.227914	+	0.973681i	$0.573191\pi$
$104$	77.3332	0.0729149
$105$	0	0
$106$	1.47924	0.00135544
$107$	225.584	0.203813	0.101907	−	0.994794i	$-0.467506\pi$
$107$	225.584	0.203813	0.101907	+	0.994794i	$0.467506\pi$
$108$	0	0
$109$	−1627.65	−1.43028	−0.715142	−	0.698979i	$-0.753638\pi$
$109$	−1627.65	−1.43028	−0.715142	+	0.698979i	$0.753638\pi$
$110$	0	0
$111$	0	0
$112$	−425.068	−0.358617
$113$	−357.040	−0.297235	−0.148617	−	0.988895i	$-0.547482\pi$
$113$	−357.040	−0.297235	−0.148617	+	0.988895i	$0.547482\pi$
$114$	0	0
$115$	0	0
$116$	−2372.63	−1.89908
$117$	0	0
$118$	−1329.19	−1.03697
$119$	−647.067	−0.498458
$120$	0	0
$121$	−1285.00	−0.965442
$122$	−2087.85	−1.54938
$123$	0	0
$124$	1723.69	1.24832
$125$	0	0
$126$	0	0
$127$	−1728.25	−1.20754	−0.603771	−	0.797158i	$-0.706336\pi$
$127$	−1728.25	−1.20754	−0.603771	+	0.797158i	$0.706336\pi$
$128$	−202.094	−0.139553
$129$	0	0
$130$	0	0
$131$	−1461.19	−0.974543	−0.487272	−	0.873250i	$-0.662008\pi$
$131$	−1461.19	−0.974543	−0.487272	+	0.873250i	$0.662008\pi$
$132$	0	0
$133$	−879.018	−0.573087
$134$	−3356.04	−2.16357
$135$	0	0
$136$	−146.121	−0.0921306
$137$	−1892.96	−1.18049	−0.590243	−	0.807226i	$-0.700968\pi$
$137$	−1892.96	−1.18049	−0.590243	+	0.807226i	$0.700968\pi$
$138$	0	0
$139$	−1715.03	−1.04652	−0.523262	−	0.852172i	$-0.675285\pi$
$139$	−1715.03	−1.04652	−0.523262	+	0.852172i	$0.675285\pi$
$140$	0	0
$141$	0	0
$142$	2009.00	1.18726
$143$	−331.794	−0.194028
$144$	0	0
$145$	0	0
$146$	2838.54	1.60904
$147$	0	0
$148$	1599.45	0.888337
$149$	2798.32	1.53857	0.769286	−	0.638904i	$-0.220612\pi$
$149$	2798.32	1.53857	0.769286	+	0.638904i	$0.220612\pi$
$150$	0	0
$151$	−2867.49	−1.54538	−0.772692	−	0.634781i	$-0.781090\pi$
$151$	−2867.49	−1.54538	−0.772692	+	0.634781i	$0.781090\pi$
$152$	−198.500	−0.105924
$153$	0	0
$154$	−192.202	−0.100572
$155$	0	0
$156$	0	0
$157$	−783.696	−0.398381	−0.199190	−	0.979961i	$-0.563831\pi$
$157$	−783.696	−0.398381	−0.199190	+	0.979961i	$0.563831\pi$
$158$	807.226	0.406452
$159$	0	0
$160$	0	0
$161$	225.877	0.110569
$162$	0	0
$163$	2416.93	1.16140	0.580702	−	0.814116i	$-0.302778\pi$
$163$	2416.93	1.16140	0.580702	+	0.814116i	$0.302778\pi$
$164$	−1034.28	−0.492463
$165$	0	0
$166$	789.149	0.368975
$167$	704.424	0.326407	0.163204	−	0.986592i	$-0.447817\pi$
$167$	704.424	0.326407	0.163204	+	0.986592i	$0.447817\pi$
$168$	0	0
$169$	196.369	0.0893803
$170$	0	0
$171$	0	0
$172$	293.835	0.130260
$173$	1398.49	0.614598	0.307299	−	0.951613i	$-0.400575\pi$
$173$	1398.49	0.614598	0.307299	+	0.951613i	$0.400575\pi$
$174$	0	0
$175$	0	0
$176$	411.836	0.176382
$177$	0	0
$178$	−556.289	−0.234245
$179$	−368.688	−0.153950	−0.0769749	−	0.997033i	$-0.524526\pi$
$179$	−368.688	−0.153950	−0.0769749	+	0.997033i	$0.524526\pi$
$180$	0	0
$181$	−315.621	−0.129613	−0.0648064	−	0.997898i	$-0.520643\pi$
$181$	−315.621	−0.129613	−0.0648064	+	0.997898i	$0.520643\pi$
$182$	1386.43	0.564665
$183$	0	0
$184$	51.0076	0.0204366
$185$	0	0
$186$	0	0
$187$	626.924	0.245162
$188$	−3515.85	−1.36393
$189$	0	0
$190$	0	0
$191$	151.629	0.0574424	0.0287212	−	0.999587i	$-0.490857\pi$
$191$	151.629	0.0574424	0.0287212	+	0.999587i	$0.490857\pi$
$192$	0	0
$193$	−690.689	−0.257600	−0.128800	−	0.991671i	$-0.541113\pi$
$193$	−690.689	−0.257600	−0.128800	+	0.991671i	$0.541113\pi$
$194$	892.453	0.330280
$195$	0	0
$196$	411.132	0.149829
$197$	834.136	0.301674	0.150837	−	0.988559i	$-0.451803\pi$
$197$	834.136	0.301674	0.150837	+	0.988559i	$0.451803\pi$
$198$	0	0
$199$	387.269	0.137954	0.0689769	−	0.997618i	$-0.478027\pi$
$199$	387.269	0.137954	0.0689769	+	0.997618i	$0.478027\pi$
$200$	0	0
$201$	0	0
$202$	2394.12	0.833909
$203$	−1979.44	−0.684383
$204$	0	0
$205$	0	0
$206$	−1929.09	−0.652457
$207$	0	0
$208$	−2970.74	−0.990307
$209$	851.656	0.281867
$210$	0	0
$211$	3070.54	1.00182	0.500912	−	0.865498i	$-0.332998\pi$
$211$	3070.54	1.00182	0.500912	+	0.865498i	$0.332998\pi$
$212$	3.06570	0.000993174 0
$213$	0	0
$214$	913.278	0.291731
$215$	0	0
$216$	0	0
$217$	1438.04	0.449865
$218$	−6589.58	−2.04726
$219$	0	0
$220$	0	0
$221$	−4522.26	−1.37647
$222$	0	0
$223$	1004.46	0.301631	0.150816	−	0.988562i	$-0.451810\pi$
$223$	1004.46	0.301631	0.150816	+	0.988562i	$0.451810\pi$
$224$	−1809.41	−0.539716
$225$	0	0
$226$	−1445.48	−0.425451
$227$	5374.22	1.57136	0.785681	−	0.618631i	$-0.212313\pi$
$227$	5374.22	1.57136	0.785681	+	0.618631i	$0.212313\pi$
$228$	0	0
$229$	3650.97	1.05355	0.526775	−	0.850005i	$-0.323401\pi$
$229$	3650.97	1.05355	0.526775	+	0.850005i	$0.323401\pi$
$230$	0	0
$231$	0	0
$232$	−446.999	−0.126495
$233$	−4582.88	−1.28856	−0.644280	−	0.764789i	$-0.722843\pi$
$233$	−4582.88	−1.28856	−0.644280	+	0.764789i	$0.722843\pi$
$234$	0	0
$235$	0	0
$236$	−2754.73	−0.759820
$237$	0	0
$238$	−2619.66	−0.713476
$239$	−696.769	−0.188578	−0.0942892	−	0.995545i	$-0.530058\pi$
$239$	−696.769	−0.188578	−0.0942892	+	0.995545i	$0.530058\pi$
$240$	0	0
$241$	5082.82	1.35856	0.679281	−	0.733878i	$-0.262292\pi$
$241$	5082.82	1.35856	0.679281	+	0.733878i	$0.262292\pi$
$242$	−5202.35	−1.38190
$243$	0	0
$244$	−4327.02	−1.13528
$245$	0	0
$246$	0	0
$247$	−6143.34	−1.58256
$248$	324.739	0.0831490
$249$	0	0
$250$	0	0
$251$	1207.32	0.303606	0.151803	−	0.988411i	$-0.451492\pi$
$251$	1207.32	0.303606	0.151803	+	0.988411i	$0.451492\pi$
$252$	0	0
$253$	−218.846	−0.0543822
$254$	−6996.86	−1.72843
$255$	0	0
$256$	3667.41	0.895363
$257$	−510.936	−0.124013	−0.0620064	−	0.998076i	$-0.519750\pi$
$257$	−510.936	−0.124013	−0.0620064	+	0.998076i	$0.519750\pi$
$258$	0	0
$259$	1334.39	0.320135
$260$	0	0
$261$	0	0
$262$	−5915.67	−1.39493
$263$	−2269.04	−0.531997	−0.265998	−	0.963974i	$-0.585702\pi$
$263$	−2269.04	−0.531997	−0.265998	+	0.963974i	$0.585702\pi$
$264$	0	0
$265$	0	0
$266$	−3558.72	−0.820297
$267$	0	0
$268$	−6955.32	−1.58531
$269$	7836.10	1.77612	0.888058	−	0.459731i	$-0.152054\pi$
$269$	7836.10	1.77612	0.888058	+	0.459731i	$0.152054\pi$
$270$	0	0
$271$	1466.28	0.328673	0.164336	−	0.986404i	$-0.447452\pi$
$271$	1466.28	0.328673	0.164336	+	0.986404i	$0.447452\pi$
$272$	5613.21	1.25129
$273$	0	0
$274$	−7663.67	−1.68971
$275$	0	0
$276$	0	0
$277$	4154.17	0.901083	0.450542	−	0.892755i	$-0.351231\pi$
$277$	4154.17	0.901083	0.450542	+	0.892755i	$0.351231\pi$
$278$	−6943.32	−1.49796
$279$	0	0
$280$	0	0
$281$	3490.40	0.740996	0.370498	−	0.928833i	$-0.379187\pi$
$281$	3490.40	0.740996	0.370498	+	0.928833i	$0.379187\pi$
$282$	0	0
$283$	−3125.29	−0.656464	−0.328232	−	0.944597i	$-0.606453\pi$
$283$	−3125.29	−0.656464	−0.328232	+	0.944597i	$0.606453\pi$
$284$	4163.60	0.869945
$285$	0	0
$286$	−1343.27	−0.277725
$287$	−862.883	−0.177472
$288$	0	0
$289$	3631.80	0.739223
$290$	0	0
$291$	0	0
$292$	5882.81	1.17899
$293$	1447.69	0.288652	0.144326	−	0.989530i	$-0.453899\pi$
$293$	1447.69	0.288652	0.144326	+	0.989530i	$0.453899\pi$
$294$	0	0
$295$	0	0
$296$	301.333	0.0591710
$297$	0	0
$298$	11329.0	2.20226
$299$	1578.62	0.305332
$300$	0	0
$301$	245.141	0.0469425
$302$	−11609.1	−2.21201
$303$	0	0
$304$	7625.35	1.43863
$305$	0	0
$306$	0	0
$307$	1591.43	0.295856	0.147928	−	0.988998i	$-0.452740\pi$
$307$	1591.43	0.295856	0.147928	+	0.988998i	$0.452740\pi$
$308$	−398.334	−0.0736922
$309$	0	0
$310$	0	0
$311$	8584.92	1.56529	0.782647	−	0.622466i	$-0.213869\pi$
$311$	8584.92	1.56529	0.782647	+	0.622466i	$0.213869\pi$
$312$	0	0
$313$	−7210.05	−1.30203	−0.651016	−	0.759064i	$-0.725657\pi$
$313$	−7210.05	−1.30203	−0.651016	+	0.759064i	$0.725657\pi$
$314$	−3172.80	−0.570228
$315$	0	0
$316$	1672.96	0.297820
$317$	−7787.24	−1.37973	−0.689865	−	0.723938i	$-0.742330\pi$
$317$	−7787.24	−1.37973	−0.689865	+	0.723938i	$0.742330\pi$
$318$	0	0
$319$	1917.83	0.336607
$320$	0	0
$321$	0	0
$322$	914.465	0.158264
$323$	11607.8	1.99962
$324$	0	0
$325$	0	0
$326$	9784.98	1.66239
$327$	0	0
$328$	−194.857	−0.0328023
$329$	−2933.21	−0.491529
$330$	0	0
$331$	−1729.78	−0.287243	−0.143621	−	0.989633i	$-0.545875\pi$
$331$	−1729.78	−0.287243	−0.143621	+	0.989633i	$0.545875\pi$
$332$	1635.49	0.270359
$333$	0	0
$334$	2851.87	0.467208
$335$	0	0
$336$	0	0
$337$	−7815.06	−1.26325	−0.631623	−	0.775276i	$-0.717611\pi$
$337$	−7815.06	−1.26325	−0.631623	+	0.775276i	$0.717611\pi$
$338$	795.000	0.127936
$339$	0	0
$340$	0	0
$341$	−1393.28	−0.221262
$342$	0	0
$343$	343.000	0.0539949
$344$	55.3579	0.00867645
$345$	0	0
$346$	5661.82	0.879714
$347$	2359.77	0.365070	0.182535	−	0.983199i	$-0.441570\pi$
$347$	2359.77	0.365070	0.182535	+	0.983199i	$0.441570\pi$
$348$	0	0
$349$	3212.73	0.492761	0.246380	−	0.969173i	$-0.420759\pi$
$349$	3212.73	0.492761	0.246380	+	0.969173i	$0.420759\pi$
$350$	0	0
$351$	0	0
$352$	1753.09	0.265454
$353$	−5582.04	−0.841649	−0.420824	−	0.907142i	$-0.638259\pi$
$353$	−5582.04	−0.841649	−0.420824	+	0.907142i	$0.638259\pi$
$354$	0	0
$355$	0	0
$356$	−1152.90	−0.171639
$357$	0	0
$358$	−1492.64	−0.220358
$359$	10630.6	1.56285	0.781425	−	0.623999i	$-0.214493\pi$
$359$	10630.6	1.56285	0.781425	+	0.623999i	$0.214493\pi$
$360$	0	0
$361$	8909.84	1.29900
$362$	−1277.80	−0.185523
$363$	0	0
$364$	2873.35	0.413748
$365$	0	0
$366$	0	0
$367$	−4514.69	−0.642139	−0.321070	−	0.947056i	$-0.604042\pi$
$367$	−4514.69	−0.642139	−0.321070	+	0.947056i	$0.604042\pi$
$368$	−1959.45	−0.277563
$369$	0	0
$370$	0	0
$371$	2.55765	0.000357916 0
$372$	0	0
$373$	−11445.8	−1.58885	−0.794426	−	0.607361i	$-0.792228\pi$
$373$	−11445.8	−1.58885	−0.794426	+	0.607361i	$0.792228\pi$
$374$	2538.11	0.350916
$375$	0	0
$376$	−662.378	−0.0908499
$377$	−13834.1	−1.88990
$378$	0	0
$379$	8146.48	1.10411	0.552054	−	0.833809i	$-0.313844\pi$
$379$	8146.48	1.10411	0.552054	+	0.833809i	$0.313844\pi$
$380$	0	0
$381$	0	0
$382$	613.872	0.0822210
$383$	−5261.56	−0.701967	−0.350983	−	0.936382i	$-0.614153\pi$
$383$	−5261.56	−0.701967	−0.350983	+	0.936382i	$0.614153\pi$
$384$	0	0
$385$	0	0
$386$	−2796.26	−0.368720
$387$	0	0
$388$	1849.59	0.242007
$389$	−13207.1	−1.72141	−0.860706	−	0.509103i	$-0.829977\pi$
$389$	−13207.1	−1.72141	−0.860706	+	0.509103i	$0.829977\pi$
$390$	0	0
$391$	−2982.80	−0.385798
$392$	77.4564	0.00997995
$393$	0	0
$394$	3377.01	0.431805
$395$	0	0
$396$	0	0
$397$	5663.15	0.715933	0.357967	−	0.933734i	$-0.383470\pi$
$397$	5663.15	0.715933	0.357967	+	0.933734i	$0.383470\pi$
$398$	1567.86	0.197462
$399$	0	0
$400$	0	0
$401$	11989.0	1.49302	0.746512	−	0.665372i	$-0.231727\pi$
$401$	11989.0	1.49302	0.746512	+	0.665372i	$0.231727\pi$
$402$	0	0
$403$	10050.3	1.24228
$404$	4961.76	0.611031
$405$	0	0
$406$	−8013.80	−0.979602
$407$	−1292.85	−0.157455
$408$	0	0
$409$	5249.67	0.634669	0.317334	−	0.948314i	$-0.397212\pi$
$409$	5249.67	0.634669	0.317334	+	0.948314i	$0.397212\pi$
$410$	0	0
$411$	0	0
$412$	−3998.00	−0.478076
$413$	−2298.22	−0.273821
$414$	0	0
$415$	0	0
$416$	−12645.7	−1.49041
$417$	0	0
$418$	3447.94	0.403455
$419$	14948.9	1.74297	0.871484	−	0.490424i	$-0.163158\pi$
$419$	14948.9	1.74297	0.871484	+	0.490424i	$0.163158\pi$
$420$	0	0
$421$	5840.31	0.676103	0.338051	−	0.941128i	$-0.390232\pi$
$421$	5840.31	0.676103	0.338051	+	0.941128i	$0.390232\pi$
$422$	12431.1	1.43398
$423$	0	0
$424$	0.577571	6.61540e−5 0
$425$	0	0
$426$	0	0
$427$	−3609.95	−0.409128
$428$	1892.75	0.213760
$429$	0	0
$430$	0	0
$431$	−7439.21	−0.831402	−0.415701	−	0.909501i	$-0.636464\pi$
$431$	−7439.21	−0.831402	−0.415701	+	0.909501i	$0.636464\pi$
$432$	0	0
$433$	877.657	0.0974077	0.0487038	−	0.998813i	$-0.484491\pi$
$433$	877.657	0.0974077	0.0487038	+	0.998813i	$0.484491\pi$
$434$	5821.93	0.643920
$435$	0	0
$436$	−13656.8	−1.50009
$437$	−4052.04	−0.443559
$438$	0	0
$439$	−10855.8	−1.18023	−0.590114	−	0.807320i	$-0.700917\pi$
$439$	−10855.8	−1.18023	−0.590114	+	0.807320i	$0.700917\pi$
$440$	0	0
$441$	0	0
$442$	−18308.4	−1.97023
$443$	−10797.0	−1.15798	−0.578988	−	0.815336i	$-0.696552\pi$
$443$	−10797.0	−1.15798	−0.578988	+	0.815336i	$0.696552\pi$
$444$	0	0
$445$	0	0
$446$	4066.58	0.431745
$447$	0	0
$448$	−3924.89	−0.413914
$449$	−8621.70	−0.906198	−0.453099	−	0.891460i	$-0.649682\pi$
$449$	−8621.70	−0.906198	−0.453099	+	0.891460i	$0.649682\pi$
$450$	0	0
$451$	836.023	0.0872878
$452$	−2995.73	−0.311742
$453$	0	0
$454$	21757.6	2.24919
$455$	0	0
$456$	0	0
$457$	3785.90	0.387520	0.193760	−	0.981049i	$-0.437932\pi$
$457$	3785.90	0.387520	0.193760	+	0.981049i	$0.437932\pi$
$458$	14781.0	1.50801
$459$	0	0
$460$	0	0
$461$	−5760.18	−0.581948	−0.290974	−	0.956731i	$-0.593979\pi$
$461$	−5760.18	−0.581948	−0.290974	+	0.956731i	$0.593979\pi$
$462$	0	0
$463$	−10760.9	−1.08014	−0.540068	−	0.841621i	$-0.681602\pi$
$463$	−10760.9	−1.08014	−0.540068	+	0.841621i	$0.681602\pi$
$464$	17171.4	1.71802
$465$	0	0
$466$	−18553.9	−1.84440
$467$	2153.58	0.213395	0.106698	−	0.994292i	$-0.465972\pi$
$467$	2153.58	0.213395	0.106698	+	0.994292i	$0.465972\pi$
$468$	0	0
$469$	−5802.70	−0.571309
$470$	0	0
$471$	0	0
$472$	−518.985	−0.0506106
$473$	−237.510	−0.0230882
$474$	0	0
$475$	0	0
$476$	−5429.18	−0.522786
$477$	0	0
$478$	−2820.88	−0.269924
$479$	6890.26	0.657253	0.328626	−	0.944460i	$-0.393414\pi$
$479$	6890.26	0.657253	0.328626	+	0.944460i	$0.393414\pi$
$480$	0	0
$481$	9325.88	0.884041
$482$	20577.9	1.94460
$483$	0	0
$484$	−10781.8	−1.01256
$485$	0	0
$486$	0	0
$487$	19006.4	1.76850	0.884251	−	0.467011i	$-0.154669\pi$
$487$	19006.4	1.76850	0.884251	+	0.467011i	$0.154669\pi$
$488$	−815.201	−0.0756197
$489$	0	0
$490$	0	0
$491$	−4530.05	−0.416371	−0.208186	−	0.978089i	$-0.566756\pi$
$491$	−4530.05	−0.416371	−0.208186	+	0.978089i	$0.566756\pi$
$492$	0	0
$493$	26139.4	2.38795
$494$	−24871.4	−2.26522
$495$	0	0
$496$	−12474.8	−1.12930
$497$	3473.62	0.313507
$498$	0	0
$499$	3620.18	0.324773	0.162386	−	0.986727i	$-0.448081\pi$
$499$	3620.18	0.324773	0.162386	+	0.986727i	$0.448081\pi$
$500$	0	0
$501$	0	0
$502$	4887.84	0.434571
$503$	11761.8	1.04261	0.521303	−	0.853372i	$-0.325446\pi$
$503$	11761.8	1.04261	0.521303	+	0.853372i	$0.325446\pi$
$504$	0	0
$505$	0	0
$506$	−885.999	−0.0778408
$507$	0	0
$508$	−14500.8	−1.26648
$509$	5254.47	0.457564	0.228782	−	0.973478i	$-0.426526\pi$
$509$	5254.47	0.457564	0.228782	+	0.973478i	$0.426526\pi$
$510$	0	0
$511$	4907.92	0.424880
$512$	16464.3	1.42114
$513$	0	0
$514$	−2068.53	−0.177508
$515$	0	0
$516$	0	0
$517$	2841.90	0.241754
$518$	5402.30	0.458230
$519$	0	0
$520$	0	0
$521$	15511.8	1.30439	0.652193	−	0.758053i	$-0.273849\pi$
$521$	15511.8	1.30439	0.652193	+	0.758053i	$0.273849\pi$
$522$	0	0
$523$	−3814.73	−0.318942	−0.159471	−	0.987203i	$-0.550979\pi$
$523$	−3814.73	−0.318942	−0.159471	+	0.987203i	$0.550979\pi$
$524$	−12260.1	−1.02211
$525$	0	0
$526$	−9186.24	−0.761481
$527$	−18990.0	−1.56967
$528$	0	0
$529$	−11125.8	−0.914422
$530$	0	0
$531$	0	0
$532$	−7375.36	−0.601057
$533$	−6030.58	−0.490081
$534$	0	0
$535$	0	0
$536$	−1310.37	−0.105596
$537$	0	0
$538$	31724.5	2.54227
$539$	−332.323	−0.0265569
$540$	0	0
$541$	−4573.77	−0.363478	−0.181739	−	0.983347i	$-0.558173\pi$
$541$	−4573.77	−0.363478	−0.181739	+	0.983347i	$0.558173\pi$
$542$	5936.26	0.470451
$543$	0	0
$544$	23894.1	1.88318
$545$	0	0
$546$	0	0
$547$	−13327.6	−1.04177	−0.520885	−	0.853627i	$-0.674398\pi$
$547$	−13327.6	−1.04177	−0.520885	+	0.853627i	$0.674398\pi$
$548$	−15882.8	−1.23810
$549$	0	0
$550$	0	0
$551$	35509.5	2.74548
$552$	0	0
$553$	1395.72	0.107327
$554$	16818.2	1.28978
$555$	0	0
$556$	−14389.9	−1.09760
$557$	12096.1	0.920155	0.460077	−	0.887879i	$-0.347822\pi$
$557$	12096.1	0.920155	0.460077	+	0.887879i	$0.347822\pi$
$558$	0	0
$559$	1713.26	0.129630
$560$	0	0
$561$	0	0
$562$	14130.9	1.06064
$563$	22943.6	1.71751	0.858755	−	0.512387i	$-0.171239\pi$
$563$	22943.6	1.71751	0.858755	+	0.512387i	$0.171239\pi$
$564$	0	0
$565$	0	0
$566$	−12652.8	−0.939639
$567$	0	0
$568$	784.414	0.0579459
$569$	−485.307	−0.0357560	−0.0178780	−	0.999840i	$-0.505691\pi$
$569$	−485.307	−0.0357560	−0.0178780	+	0.999840i	$0.505691\pi$
$570$	0	0
$571$	12271.8	0.899401	0.449701	−	0.893179i	$-0.351531\pi$
$571$	12271.8	0.899401	0.449701	+	0.893179i	$0.351531\pi$
$572$	−2783.90	−0.203498
$573$	0	0
$574$	−3493.39	−0.254027
$575$	0	0
$576$	0	0
$577$	14122.8	1.01896	0.509478	−	0.860484i	$-0.329838\pi$
$577$	14122.8	1.01896	0.509478	+	0.860484i	$0.329838\pi$
$578$	14703.4	1.05810
$579$	0	0
$580$	0	0
$581$	1364.46	0.0974310
$582$	0	0
$583$	−2.47804	−0.000176037 0
$584$	1108.31	0.0785312
$585$	0	0
$586$	5861.00	0.413167
$587$	11005.3	0.773826	0.386913	−	0.922116i	$-0.373541\pi$
$587$	11005.3	0.773826	0.386913	+	0.922116i	$0.373541\pi$
$588$	0	0
$589$	−25797.2	−1.80468
$590$	0	0
$591$	0	0
$592$	−11575.6	−0.803642
$593$	−11233.1	−0.777889	−0.388944	−	0.921261i	$-0.627160\pi$
$593$	−11233.1	−0.777889	−0.388944	+	0.921261i	$0.627160\pi$
$594$	0	0
$595$	0	0
$596$	23479.2	1.61366
$597$	0	0
$598$	6391.08	0.437041
$599$	−14855.4	−1.01331	−0.506656	−	0.862149i	$-0.669118\pi$
$599$	−14855.4	−1.01331	−0.506656	+	0.862149i	$0.669118\pi$
$600$	0	0
$601$	25358.8	1.72115	0.860573	−	0.509327i	$-0.170106\pi$
$601$	25358.8	1.72115	0.860573	+	0.509327i	$0.170106\pi$
$602$	992.457	0.0671919
$603$	0	0
$604$	−24059.5	−1.62081
$605$	0	0
$606$	0	0
$607$	−14393.9	−0.962487	−0.481243	−	0.876587i	$-0.659815\pi$
$607$	−14393.9	−0.962487	−0.481243	+	0.876587i	$0.659815\pi$
$608$	32459.3	2.16513
$609$	0	0
$610$	0	0
$611$	−20499.8	−1.35734
$612$	0	0
$613$	−4769.94	−0.314284	−0.157142	−	0.987576i	$-0.550228\pi$
$613$	−4769.94	−0.314284	−0.157142	+	0.987576i	$0.550228\pi$
$614$	6442.92	0.423477
$615$	0	0
$616$	−75.0453	−0.00490854
$617$	−7769.86	−0.506974	−0.253487	−	0.967339i	$-0.581577\pi$
$617$	−7769.86	−0.506974	−0.253487	+	0.967339i	$0.581577\pi$
$618$	0	0
$619$	−5680.75	−0.368867	−0.184433	−	0.982845i	$-0.559045\pi$
$619$	−5680.75	−0.368867	−0.184433	+	0.982845i	$0.559045\pi$
$620$	0	0
$621$	0	0
$622$	34756.2	2.24051
$623$	−961.840	−0.0618544
$624$	0	0
$625$	0	0
$626$	−29190.0	−1.86368
$627$	0	0
$628$	−6575.56	−0.417824
$629$	−17621.2	−1.11702
$630$	0	0
$631$	−10395.8	−0.655864	−0.327932	−	0.944701i	$-0.606352\pi$
$631$	−10395.8	−0.655864	−0.327932	+	0.944701i	$0.606352\pi$
$632$	315.182	0.0198374
$633$	0	0
$634$	−31526.7	−1.97490
$635$	0	0
$636$	0	0
$637$	2397.18	0.149105
$638$	7764.34	0.481808
$639$	0	0
$640$	0	0
$641$	8194.28	0.504921	0.252461	−	0.967607i	$-0.418760\pi$
$641$	8194.28	0.504921	0.252461	+	0.967607i	$0.418760\pi$
$642$	0	0
$643$	32118.2	1.96985	0.984927	−	0.172970i	$-0.0553363\pi$
$643$	32118.2	1.96985	0.984927	+	0.172970i	$0.0553363\pi$
$644$	1895.21	0.115965
$645$	0	0
$646$	46994.4	2.86218
$647$	−22299.0	−1.35496	−0.677482	−	0.735539i	$-0.736929\pi$
$647$	−22299.0	−1.35496	−0.677482	+	0.735539i	$0.736929\pi$
$648$	0	0
$649$	2226.68	0.134676
$650$	0	0
$651$	0	0
$652$	20279.1	1.21809
$653$	920.410	0.0551584	0.0275792	−	0.999620i	$-0.491220\pi$
$653$	920.410	0.0551584	0.0275792	+	0.999620i	$0.491220\pi$
$654$	0	0
$655$	0	0
$656$	7485.38	0.445511
$657$	0	0
$658$	−11875.1	−0.703557
$659$	−17824.3	−1.05362	−0.526812	−	0.849982i	$-0.676613\pi$
$659$	−17824.3	−1.05362	−0.526812	+	0.849982i	$0.676613\pi$
$660$	0	0
$661$	11343.8	0.667510	0.333755	−	0.942660i	$-0.391684\pi$
$661$	11343.8	0.667510	0.333755	+	0.942660i	$0.391684\pi$
$662$	−7003.03	−0.411149
$663$	0	0
$664$	308.123	0.0180083
$665$	0	0
$666$	0	0
$667$	−9124.71	−0.529700
$668$	5910.44	0.342338
$669$	0	0
$670$	0	0
$671$	3497.58	0.201226
$672$	0	0
$673$	13422.9	0.768816	0.384408	−	0.923163i	$-0.374406\pi$
$673$	13422.9	0.768816	0.384408	+	0.923163i	$0.374406\pi$
$674$	−31639.4	−1.80817
$675$	0	0
$676$	1647.62	0.0937426
$677$	−1066.77	−0.0605602	−0.0302801	−	0.999541i	$-0.509640\pi$
$677$	−1066.77	−0.0605602	−0.0302801	+	0.999541i	$0.509640\pi$
$678$	0	0
$679$	1543.08	0.0872134
$680$	0	0
$681$	0	0
$682$	−5640.70	−0.316706
$683$	−19090.6	−1.06952	−0.534759	−	0.845005i	$-0.679598\pi$
$683$	−19090.6	−1.06952	−0.534759	+	0.845005i	$0.679598\pi$
$684$	0	0
$685$	0	0
$686$	1388.64	0.0772865
$687$	0	0
$688$	−2126.56	−0.117841
$689$	17.8751	0.000988370 0
$690$	0	0
$691$	−16878.4	−0.929208	−0.464604	−	0.885518i	$-0.653803\pi$
$691$	−16878.4	−0.929208	−0.464604	+	0.885518i	$0.653803\pi$
$692$	11734.0	0.644594
$693$	0	0
$694$	9553.57	0.522549
$695$	0	0
$696$	0	0
$697$	11394.8	0.619236
$698$	13006.8	0.705321
$699$	0	0
$700$	0	0
$701$	−30272.6	−1.63107	−0.815535	−	0.578707i	$-0.803557\pi$
$701$	−30272.6	−1.63107	−0.815535	+	0.578707i	$0.803557\pi$
$702$	0	0
$703$	−23937.8	−1.28426
$704$	3802.71	0.203580
$705$	0	0
$706$	−22599.0	−1.20471
$707$	4139.50	0.220201
$708$	0	0
$709$	−6593.32	−0.349248	−0.174624	−	0.984635i	$-0.555871\pi$
$709$	−6593.32	−0.349248	−0.174624	+	0.984635i	$0.555871\pi$
$710$	0	0
$711$	0	0
$712$	−217.203	−0.0114326
$713$	6628.99	0.348187
$714$	0	0
$715$	0	0
$716$	−3093.46	−0.161464
$717$	0	0
$718$	43038.2	2.23701
$719$	−3293.72	−0.170842	−0.0854208	−	0.996345i	$-0.527223\pi$
$719$	−3293.72	−0.170842	−0.0854208	+	0.996345i	$0.527223\pi$
$720$	0	0
$721$	−3335.46	−0.172287
$722$	36071.6	1.85934
$723$	0	0
$724$	−2648.20	−0.135939
$725$	0	0
$726$	0	0
$727$	−27757.8	−1.41606	−0.708032	−	0.706181i	$-0.750417\pi$
$727$	−27757.8	−1.41606	−0.708032	+	0.706181i	$0.750417\pi$
$728$	541.333	0.0275592
$729$	0	0
$730$	0	0
$731$	−3237.20	−0.163792
$732$	0	0
$733$	−38324.6	−1.93117	−0.965587	−	0.260080i	$-0.916251\pi$
$733$	−38324.6	−1.93117	−0.965587	+	0.260080i	$0.916251\pi$
$734$	−18277.8	−0.919136
$735$	0	0
$736$	−8340.91	−0.417731
$737$	5622.07	0.280993
$738$	0	0
$739$	21957.3	1.09298	0.546490	−	0.837466i	$-0.315964\pi$
$739$	21957.3	1.09298	0.546490	+	0.837466i	$0.315964\pi$
$740$	0	0
$741$	0	0
$742$	10.3547	0.000512308 0
$743$	−14695.0	−0.725580	−0.362790	−	0.931871i	$-0.618176\pi$
$743$	−14695.0	−0.725580	−0.362790	+	0.931871i	$0.618176\pi$
$744$	0	0
$745$	0	0
$746$	−46338.5	−2.27423
$747$	0	0
$748$	5260.18	0.257127
$749$	1579.09	0.0770341
$750$	0	0
$751$	−21439.1	−1.04171	−0.520855	−	0.853645i	$-0.674387\pi$
$751$	−21439.1	−1.04171	−0.520855	+	0.853645i	$0.674387\pi$
$752$	25445.1	1.23389
$753$	0	0
$754$	−56007.4	−2.70513
$755$	0	0
$756$	0	0
$757$	23896.8	1.14735	0.573675	−	0.819083i	$-0.305517\pi$
$757$	23896.8	1.14735	0.573675	+	0.819083i	$0.305517\pi$
$758$	32981.1	1.58038
$759$	0	0
$760$	0	0
$761$	24436.9	1.16404	0.582022	−	0.813173i	$-0.302262\pi$
$761$	24436.9	1.16404	0.582022	+	0.813173i	$0.302262\pi$
$762$	0	0
$763$	−11393.6	−0.540597
$764$	1272.24	0.0602459
$765$	0	0
$766$	−21301.5	−1.00477
$767$	−16061.9	−0.756145
$768$	0	0
$769$	−30689.3	−1.43912	−0.719560	−	0.694430i	$-0.755657\pi$
$769$	−30689.3	−1.43912	−0.719560	+	0.694430i	$0.755657\pi$
$770$	0	0
$771$	0	0
$772$	−5795.19	−0.270173
$773$	−5110.74	−0.237801	−0.118901	−	0.992906i	$-0.537937\pi$
$773$	−5110.74	−0.237801	−0.118901	+	0.992906i	$0.537937\pi$
$774$	0	0
$775$	0	0
$776$	348.459	0.0161198
$777$	0	0
$778$	−53469.3	−2.46397
$779$	15479.4	0.711947
$780$	0	0
$781$	−3365.49	−0.154195
$782$	−12075.9	−0.552217
$783$	0	0
$784$	−2975.47	−0.135545
$785$	0	0
$786$	0	0
$787$	−6991.30	−0.316662	−0.158331	−	0.987386i	$-0.550611\pi$
$787$	−6991.30	−0.316662	−0.158331	+	0.987386i	$0.550611\pi$
$788$	6998.78	0.316397
$789$	0	0
$790$	0	0
$791$	−2499.28	−0.112344
$792$	0	0
$793$	−25229.5	−1.12979
$794$	22927.3	1.02476
$795$	0	0
$796$	3249.36	0.144687
$797$	4020.31	0.178678	0.0893392	−	0.996001i	$-0.471524\pi$
$797$	4020.31	0.178678	0.0893392	+	0.996001i	$0.471524\pi$
$798$	0	0
$799$	38734.3	1.71505
$800$	0	0
$801$	0	0
$802$	48537.7	2.13706
$803$	−4755.15	−0.208973
$804$	0	0
$805$	0	0
$806$	40688.7	1.77816
$807$	0	0
$808$	934.785	0.0407000
$809$	41608.1	1.80824	0.904119	−	0.427281i	$-0.140528\pi$
$809$	41608.1	1.80824	0.904119	+	0.427281i	$0.140528\pi$
$810$	0	0
$811$	42271.3	1.83027	0.915133	−	0.403152i	$-0.132086\pi$
$811$	42271.3	1.83027	0.915133	+	0.403152i	$0.132086\pi$
$812$	−16608.4	−0.717785
$813$	0	0
$814$	−5234.13	−0.225376
$815$	0	0
$816$	0	0
$817$	−4397.62	−0.188315
$818$	21253.4	0.908443
$819$	0	0
$820$	0	0
$821$	−27184.7	−1.15561	−0.577804	−	0.816176i	$-0.696090\pi$
$821$	−27184.7	−1.15561	−0.577804	+	0.816176i	$0.696090\pi$
$822$	0	0
$823$	12967.9	0.549250	0.274625	−	0.961551i	$-0.411446\pi$
$823$	12967.9	0.549250	0.274625	+	0.961551i	$0.411446\pi$
$824$	−753.215	−0.0318440
$825$	0	0
$826$	−9304.36	−0.391937
$827$	−33111.2	−1.39225	−0.696124	−	0.717921i	$-0.745094\pi$
$827$	−33111.2	−1.39225	−0.696124	+	0.717921i	$0.745094\pi$
$828$	0	0
$829$	3715.75	0.155673	0.0778367	−	0.996966i	$-0.475199\pi$
$829$	3715.75	0.155673	0.0778367	+	0.996966i	$0.475199\pi$
$830$	0	0
$831$	0	0
$832$	−27430.5	−1.14301
$833$	−4529.47	−0.188399
$834$	0	0
$835$	0	0
$836$	7145.77	0.295624
$837$	0	0
$838$	60521.0	2.49482
$839$	1460.56	0.0601005	0.0300502	−	0.999548i	$-0.490433\pi$
$839$	1460.56	0.0601005	0.0300502	+	0.999548i	$0.490433\pi$
$840$	0	0
$841$	55574.2	2.27866
$842$	23644.6	0.967750
$843$	0	0
$844$	25763.2	1.05072
$845$	0	0
$846$	0	0
$847$	−8995.02	−0.364903
$848$	−22.1873	−0.000898483 0
$849$	0	0
$850$	0	0
$851$	6151.19	0.247779
$852$	0	0
$853$	36027.4	1.44613	0.723067	−	0.690777i	$-0.242732\pi$
$853$	36027.4	1.44613	0.723067	+	0.690777i	$0.242732\pi$
$854$	−14614.9	−0.585612
$855$	0	0
$856$	356.590	0.0142383
$857$	23500.7	0.936719	0.468359	−	0.883538i	$-0.344845\pi$
$857$	23500.7	0.936719	0.468359	+	0.883538i	$0.344845\pi$
$858$	0	0
$859$	5551.11	0.220491	0.110245	−	0.993904i	$-0.464836\pi$
$859$	5551.11	0.220491	0.110245	+	0.993904i	$0.464836\pi$
$860$	0	0
$861$	0	0
$862$	−30117.7	−1.19004
$863$	27721.4	1.09345	0.546725	−	0.837312i	$-0.315874\pi$
$863$	27721.4	1.09345	0.546725	+	0.837312i	$0.315874\pi$
$864$	0	0
$865$	0	0
$866$	3553.21	0.139426
$867$	0	0
$868$	12065.8	0.471821
$869$	−1352.27	−0.0527878
$870$	0	0
$871$	−40554.3	−1.57765
$872$	−2572.90	−0.0999192
$873$	0	0
$874$	−16404.7	−0.634895
$875$	0	0
$876$	0	0
$877$	−47255.6	−1.81951	−0.909754	−	0.415149i	$-0.863730\pi$
$877$	−47255.6	−1.81951	−0.909754	+	0.415149i	$0.863730\pi$
$878$	−43950.0	−1.68934
$879$	0	0
$880$	0	0
$881$	−32267.0	−1.23394	−0.616972	−	0.786985i	$-0.711641\pi$
$881$	−32267.0	−1.23394	−0.616972	+	0.786985i	$0.711641\pi$
$882$	0	0
$883$	−5062.08	−0.192925	−0.0964623	−	0.995337i	$-0.530753\pi$
$883$	−5062.08	−0.192925	−0.0964623	+	0.995337i	$0.530753\pi$
$884$	−37943.8	−1.44365
$885$	0	0
$886$	−43712.0	−1.65749
$887$	7626.08	0.288679	0.144340	−	0.989528i	$-0.453894\pi$
$887$	7626.08	0.288679	0.144340	+	0.989528i	$0.453894\pi$
$888$	0	0
$889$	−12097.8	−0.456408
$890$	0	0
$891$	0	0
$892$	8427.89	0.316353
$893$	52619.2	1.97182
$894$	0	0
$895$	0	0
$896$	−1414.66	−0.0527460
$897$	0	0
$898$	−34905.0	−1.29710
$899$	−58092.3	−2.15516
$900$	0	0
$901$	−33.7750	−0.00124884
$902$	3384.65	0.124941
$903$	0	0
$904$	−564.389	−0.0207647
$905$	0	0
$906$	0	0
$907$	8546.31	0.312873	0.156436	−	0.987688i	$-0.449999\pi$
$907$	8546.31	0.312873	0.156436	+	0.987688i	$0.449999\pi$
$908$	45092.1	1.64806
$909$	0	0
$910$	0	0
$911$	−8778.92	−0.319274	−0.159637	−	0.987176i	$-0.551032\pi$
$911$	−8778.92	−0.319274	−0.159637	+	0.987176i	$0.551032\pi$
$912$	0	0
$913$	−1321.99	−0.0479205
$914$	15327.3	0.554683
$915$	0	0
$916$	30633.3	1.10497
$917$	−10228.4	−0.368343
$918$	0	0
$919$	21626.0	0.776253	0.388126	−	0.921606i	$-0.373122\pi$
$919$	21626.0	0.776253	0.388126	+	0.921606i	$0.373122\pi$
$920$	0	0
$921$	0	0
$922$	−23320.1	−0.832981
$923$	24276.7	0.865738
$924$	0	0
$925$	0	0
$926$	−43565.8	−1.54607
$927$	0	0
$928$	73094.5	2.58561
$929$	−12262.6	−0.433073	−0.216536	−	0.976275i	$-0.569476\pi$
$929$	−12262.6	−0.433073	−0.216536	+	0.976275i	$0.569476\pi$
$930$	0	0
$931$	−6153.13	−0.216606
$932$	−38452.5	−1.35145
$933$	0	0
$934$	8718.78	0.305447
$935$	0	0
$936$	0	0
$937$	13362.6	0.465889	0.232945	−	0.972490i	$-0.425164\pi$
$937$	13362.6	0.465889	0.232945	+	0.972490i	$0.425164\pi$
$938$	−23492.3	−0.817751
$939$	0	0
$940$	0	0
$941$	46330.1	1.60501	0.802506	−	0.596643i	$-0.203499\pi$
$941$	46330.1	1.60501	0.802506	+	0.596643i	$0.203499\pi$
$942$	0	0
$943$	−3977.66	−0.137360
$944$	19936.7	0.687377
$945$	0	0
$946$	−961.563	−0.0330477
$947$	−17387.6	−0.596643	−0.298321	−	0.954465i	$-0.596427\pi$
$947$	−17387.6	−0.596643	−0.298321	+	0.954465i	$0.596427\pi$
$948$	0	0
$949$	34300.8	1.17329
$950$	0	0
$951$	0	0
$952$	−1022.85	−0.0348221
$953$	−42384.6	−1.44068	−0.720342	−	0.693619i	$-0.756015\pi$
$953$	−42384.6	−1.44068	−0.720342	+	0.693619i	$0.756015\pi$
$954$	0	0
$955$	0	0
$956$	−5846.20	−0.197782
$957$	0	0
$958$	27895.3	0.940769
$959$	−13250.7	−0.446182
$960$	0	0
$961$	12412.3	0.416647
$962$	37755.9	1.26539
$963$	0	0
$964$	42647.2	1.42487
$965$	0	0
$966$	0	0
$967$	15771.6	0.524489	0.262245	−	0.965001i	$-0.415537\pi$
$967$	15771.6	0.524489	0.262245	+	0.965001i	$0.415537\pi$
$968$	−2031.26	−0.0674454
$969$	0	0
$970$	0	0
$971$	−36370.5	−1.20204	−0.601022	−	0.799232i	$-0.705240\pi$
$971$	−36370.5	−1.20204	−0.601022	+	0.799232i	$0.705240\pi$
$972$	0	0
$973$	−12005.2	−0.395549
$974$	76947.5	2.53137
$975$	0	0
$976$	31315.8	1.02704
$977$	−35122.8	−1.15013	−0.575065	−	0.818108i	$-0.695023\pi$
$977$	−35122.8	−1.15013	−0.575065	+	0.818108i	$0.695023\pi$
$978$	0	0
$979$	931.899	0.0304225
$980$	0	0
$981$	0	0
$982$	−18340.0	−0.595979
$983$	−12798.9	−0.415282	−0.207641	−	0.978205i	$-0.566579\pi$
$983$	−12798.9	−0.415282	−0.207641	+	0.978205i	$0.566579\pi$
$984$	0	0
$985$	0	0
$986$	105826.	3.41803
$987$	0	0
$988$	−51545.4	−1.65980
$989$	1130.03	0.0363327
$990$	0	0
$991$	−46594.9	−1.49358	−0.746789	−	0.665061i	$-0.768406\pi$
$991$	−46594.9	−1.49358	−0.746789	+	0.665061i	$0.768406\pi$
$992$	−53102.3	−1.69960
$993$	0	0
$994$	14063.0	0.448743
$995$	0	0
$996$	0	0
$997$	−47534.1	−1.50995	−0.754975	−	0.655753i	$-0.772351\pi$
$997$	−47534.1	−1.50995	−0.754975	+	0.655753i	$0.772351\pi$
$998$	14656.4	0.464869
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bn.1.5		5
3.2	odd	2		175.4.a.j.1.1		5
5.2	odd	4		315.4.d.c.64.9		10
5.3	odd	4		315.4.d.c.64.2		10
5.4	even	2		1575.4.a.bq.1.1		5
15.2	even	4		35.4.b.a.29.2	✓	10
15.8	even	4		35.4.b.a.29.9	yes	10
15.14	odd	2		175.4.a.i.1.5		5
21.20	even	2		1225.4.a.bh.1.1		5
60.23	odd	4		560.4.g.f.449.2		10
60.47	odd	4		560.4.g.f.449.9		10
105.2	even	12		245.4.j.e.214.2		20
105.17	odd	12		245.4.j.f.79.9		20
105.23	even	12		245.4.j.e.214.9		20
105.32	even	12		245.4.j.e.79.9		20
105.38	odd	12		245.4.j.f.79.2		20
105.47	odd	12		245.4.j.f.214.2		20
105.53	even	12		245.4.j.e.79.2		20
105.62	odd	4		245.4.b.d.99.2		10
105.68	odd	12		245.4.j.f.214.9		20
105.83	odd	4		245.4.b.d.99.9		10
105.104	even	2		1225.4.a.be.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.2	✓	10	15.2	even	4
35.4.b.a.29.9	yes	10	15.8	even	4
175.4.a.i.1.5		5	15.14	odd	2
175.4.a.j.1.1		5	3.2	odd	2
245.4.b.d.99.2		10	105.62	odd	4
245.4.b.d.99.9		10	105.83	odd	4
245.4.j.e.79.2		20	105.53	even	12
245.4.j.e.79.9		20	105.32	even	12
245.4.j.e.214.2		20	105.2	even	12
245.4.j.e.214.9		20	105.23	even	12
245.4.j.f.79.2		20	105.38	odd	12
245.4.j.f.79.9		20	105.17	odd	12
245.4.j.f.214.2		20	105.47	odd	12
245.4.j.f.214.9		20	105.68	odd	12
315.4.d.c.64.2		10	5.3	odd	4
315.4.d.c.64.9		10	5.2	odd	4
560.4.g.f.449.2		10	60.23	odd	4
560.4.g.f.449.9		10	60.47	odd	4
1225.4.a.be.1.5		5	105.104	even	2
1225.4.a.bh.1.1		5	21.20	even	2
1575.4.a.bn.1.5		5	1.1	even	1	trivial
1575.4.a.bq.1.1		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+4.04851 q^{2} +8.39045 q^{4} +7.00000 q^{7} +1.58074 q^{8} +O(q^{10})\)	\(q+4.04851 q^{2} +8.39045 q^{4} +7.00000 q^{7} +1.58074 q^{8} -6.78210 q^{11} +48.9221 q^{13} +28.3396 q^{14} -60.7239 q^{16} -92.4381 q^{17} -125.574 q^{19} -27.4574 q^{22} +32.2681 q^{23} +198.062 q^{26} +58.7332 q^{28} -282.778 q^{29} +205.434 q^{31} -258.488 q^{32} -374.237 q^{34} +190.627 q^{37} -508.388 q^{38} -123.269 q^{41} +35.0202 q^{43} -56.9049 q^{44} +130.638 q^{46} -419.030 q^{47} +49.0000 q^{49} +410.478 q^{52} +0.365379 q^{53} +11.0652 q^{56} -1144.83 q^{58} -328.317 q^{59} -515.707 q^{61} +831.704 q^{62} -560.699 q^{64} -828.957 q^{67} -775.597 q^{68} +496.231 q^{71} +701.132 q^{73} +771.757 q^{74} -1053.62 q^{76} -47.4747 q^{77} +199.388 q^{79} -499.056 q^{82} +194.923 q^{83} +141.780 q^{86} -10.7208 q^{88} -137.406 q^{89} +342.454 q^{91} +270.744 q^{92} -1696.45 q^{94} +220.440 q^{97} +198.377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8}+O(q^{10}) \) 5 * q - 4 * q^2 + 18 * q^4 + 35 * q^7 - 42 * q^8	\( 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8} - 42 q^{11} + 34 q^{13} - 28 q^{14} + 74 q^{16} - 238 q^{17} - 36 q^{19} + 358 q^{22} - 152 q^{23} + 310 q^{26} + 126 q^{28} + 44 q^{29} + 60 q^{31} - 710 q^{32} - 482 q^{34} + 312 q^{37} - 280 q^{38} + 426 q^{41} + 304 q^{43} - 712 q^{44} + 88 q^{46} - 370 q^{47} + 245 q^{49} - 1156 q^{52} - 976 q^{53} - 294 q^{56} - 2722 q^{58} + 432 q^{59} - 442 q^{61} + 956 q^{62} + 1362 q^{64} - 804 q^{67} + 420 q^{68} - 440 q^{71} + 564 q^{73} + 1512 q^{74} - 1336 q^{76} - 294 q^{77} + 1790 q^{79} - 3480 q^{82} - 1656 q^{83} - 1216 q^{86} - 1092 q^{88} - 746 q^{89} + 238 q^{91} - 572 q^{92} - 826 q^{94} + 518 q^{97} - 196 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 18 * q^4 + 35 * q^7 - 42 * q^8 - 42 * q^11 + 34 * q^13 - 28 * q^14 + 74 * q^16 - 238 * q^17 - 36 * q^19 + 358 * q^22 - 152 * q^23 + 310 * q^26 + 126 * q^28 + 44 * q^29 + 60 * q^31 - 710 * q^32 - 482 * q^34 + 312 * q^37 - 280 * q^38 + 426 * q^41 + 304 * q^43 - 712 * q^44 + 88 * q^46 - 370 * q^47 + 245 * q^49 - 1156 * q^52 - 976 * q^53 - 294 * q^56 - 2722 * q^58 + 432 * q^59 - 442 * q^61 + 956 * q^62 + 1362 * q^64 - 804 * q^67 + 420 * q^68 - 440 * q^71 + 564 * q^73 + 1512 * q^74 - 1336 * q^76 - 294 * q^77 + 1790 * q^79 - 3480 * q^82 - 1656 * q^83 - 1216 * q^86 - 1092 * q^88 - 746 * q^89 + 238 * q^91 - 572 * q^92 - 826 * q^94 + 518 * q^97 - 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more