LMFDB - Embedded newform 2475.4.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
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} - 2x^{4} - 38x^{3} + 61x^{2} + 304x - 148 $ x^5 - 2x^4 - 38x^3 + 61x^2 + 304x - 148
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.75920$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75920 q^{2} -0.386826 q^{4} -13.2555 q^{7} +23.1409 q^{8} +O(q^{10})$	$q-2.75920 q^{2} -0.386826 q^{4} -13.2555 q^{7} +23.1409 q^{8} +11.0000 q^{11} +30.6714 q^{13} +36.5746 q^{14} -60.7558 q^{16} +35.1080 q^{17} -134.694 q^{19} -30.3512 q^{22} -6.51724 q^{23} -84.6285 q^{26} +5.12758 q^{28} +30.3054 q^{29} +331.803 q^{31} -17.4901 q^{32} -96.8700 q^{34} -172.426 q^{37} +371.649 q^{38} -58.1492 q^{41} -210.100 q^{43} -4.25508 q^{44} +17.9823 q^{46} -151.570 q^{47} -167.291 q^{49} -11.8645 q^{52} +318.384 q^{53} -306.745 q^{56} -83.6186 q^{58} -414.677 q^{59} +498.088 q^{61} -915.511 q^{62} +534.305 q^{64} -639.464 q^{67} -13.5807 q^{68} +267.420 q^{71} -540.297 q^{73} +475.756 q^{74} +52.1033 q^{76} -145.811 q^{77} +691.157 q^{79} +160.445 q^{82} +663.244 q^{83} +579.709 q^{86} +254.550 q^{88} +119.542 q^{89} -406.565 q^{91} +2.52103 q^{92} +418.211 q^{94} +284.049 q^{97} +461.589 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 40 q^{4} - 40 q^{7} + 21 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 40 * q^4 - 40 * q^7 + 21 * q^8	$ 5 q + 2 q^{2} + 40 q^{4} - 40 q^{7} + 21 q^{8} + 55 q^{11} - 211 q^{13} + 133 q^{14} + 208 q^{16} + 72 q^{17} + 23 q^{19} + 22 q^{22} + 57 q^{23} - 322 q^{26} - 1050 q^{28} + 183 q^{29} - q^{31} - 430 q^{32} - 650 q^{34} - 856 q^{37} + 348 q^{38} - 94 q^{41} - 497 q^{43} + 440 q^{44} + 1006 q^{46} + 26 q^{47} + 227 q^{49} - 2086 q^{52} + 264 q^{53} + 835 q^{56} - 843 q^{58} - 1174 q^{59} + 640 q^{61} + 1454 q^{62} - 949 q^{64} - 98 q^{67} + 1767 q^{68} + 673 q^{71} - 1462 q^{73} + 561 q^{74} + 1248 q^{76} - 440 q^{77} + 104 q^{79} + 221 q^{82} - 747 q^{83} - 2497 q^{86} + 231 q^{88} - 2821 q^{89} + 480 q^{91} - 1503 q^{92} - 1787 q^{94} - 615 q^{97} - 5586 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 40 * q^4 - 40 * q^7 + 21 * q^8 + 55 * q^11 - 211 * q^13 + 133 * q^14 + 208 * q^16 + 72 * q^17 + 23 * q^19 + 22 * q^22 + 57 * q^23 - 322 * q^26 - 1050 * q^28 + 183 * q^29 - q^31 - 430 * q^32 - 650 * q^34 - 856 * q^37 + 348 * q^38 - 94 * q^41 - 497 * q^43 + 440 * q^44 + 1006 * q^46 + 26 * q^47 + 227 * q^49 - 2086 * q^52 + 264 * q^53 + 835 * q^56 - 843 * q^58 - 1174 * q^59 + 640 * q^61 + 1454 * q^62 - 949 * q^64 - 98 * q^67 + 1767 * q^68 + 673 * q^71 - 1462 * q^73 + 561 * q^74 + 1248 * q^76 - 440 * q^77 + 104 * q^79 + 221 * q^82 - 747 * q^83 - 2497 * q^86 + 231 * q^88 - 2821 * q^89 + 480 * q^91 - 1503 * q^92 - 1787 * q^94 - 615 * q^97 - 5586 * 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 $

$2$	−2.75920	−0.975524	−0.487762	−	0.872977i	$-0.662187\pi$
$2$	−2.75920	−0.975524	−0.487762	+	0.872977i	$0.662187\pi$
$3$	0	0
$3$	0	0
$4$	−0.386826	−0.0483532
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−13.2555	−0.715731	−0.357865	−	0.933773i	$-0.616495\pi$
$7$	−13.2555	−0.715731	−0.357865	+	0.933773i	$0.616495\pi$
$8$	23.1409	1.02269
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	30.6714	0.654363	0.327181	−	0.944962i	$-0.393901\pi$
$13$	30.6714	0.654363	0.327181	+	0.944962i	$0.393901\pi$
$14$	36.5746	0.698213
$15$	0	0
$16$	−60.7558	−0.949309
$17$	35.1080	0.500879	0.250440	−	0.968132i	$-0.419425\pi$
$17$	35.1080	0.500879	0.250440	+	0.968132i	$0.419425\pi$
$18$	0	0
$19$	−134.694	−1.62637	−0.813185	−	0.582006i	$-0.802268\pi$
$19$	−134.694	−1.62637	−0.813185	+	0.582006i	$0.802268\pi$
$20$	0	0
$21$	0	0
$22$	−30.3512	−0.294132
$23$	−6.51724	−0.0590843	−0.0295421	−	0.999564i	$-0.509405\pi$
$23$	−6.51724	−0.0590843	−0.0295421	+	0.999564i	$0.509405\pi$
$24$	0	0
$25$	0	0
$26$	−84.6285	−0.638347
$27$	0	0
$28$	5.12758	0.0346079
$29$	30.3054	0.194054	0.0970271	−	0.995282i	$-0.469067\pi$
$29$	30.3054	0.194054	0.0970271	+	0.995282i	$0.469067\pi$
$30$	0	0
$31$	331.803	1.92237	0.961187	−	0.275897i	$-0.0889748\pi$
$31$	331.803	1.92237	0.961187	+	0.275897i	$0.0889748\pi$
$32$	−17.4901	−0.0966202
$33$	0	0
$34$	−96.8700	−0.488620
$35$	0	0
$36$	0	0
$37$	−172.426	−0.766124	−0.383062	−	0.923723i	$-0.625130\pi$
$37$	−172.426	−0.766124	−0.383062	+	0.923723i	$0.625130\pi$
$38$	371.649	1.58656
$39$	0	0
$40$	0	0
$41$	−58.1492	−0.221497	−0.110749	−	0.993848i	$-0.535325\pi$
$41$	−58.1492	−0.221497	−0.110749	+	0.993848i	$0.535325\pi$
$42$	0	0
$43$	−210.100	−0.745116	−0.372558	−	0.928009i	$-0.621519\pi$
$43$	−210.100	−0.745116	−0.372558	+	0.928009i	$0.621519\pi$
$44$	−4.25508	−0.0145790
$45$	0	0
$46$	17.9823	0.0576381
$47$	−151.570	−0.470398	−0.235199	−	0.971947i	$-0.575574\pi$
$47$	−151.570	−0.470398	−0.235199	+	0.971947i	$0.575574\pi$
$48$	0	0
$49$	−167.291	−0.487729
$50$	0	0
$51$	0	0
$52$	−11.8645	−0.0316406
$53$	318.384	0.825159	0.412579	−	0.910922i	$-0.364628\pi$
$53$	318.384	0.825159	0.412579	+	0.910922i	$0.364628\pi$
$54$	0	0
$55$	0	0
$56$	−306.745	−0.731973
$57$	0	0
$58$	−83.6186	−0.189304
$59$	−414.677	−0.915023	−0.457512	−	0.889204i	$-0.651259\pi$
$59$	−414.677	−0.915023	−0.457512	+	0.889204i	$0.651259\pi$
$60$	0	0
$61$	498.088	1.04547	0.522735	−	0.852495i	$-0.324912\pi$
$61$	498.088	1.04547	0.522735	+	0.852495i	$0.324912\pi$
$62$	−915.511	−1.87532
$63$	0	0
$64$	534.305	1.04356
$65$	0	0
$66$	0	0
$67$	−639.464	−1.16601	−0.583007	−	0.812467i	$-0.698124\pi$
$67$	−639.464	−1.16601	−0.583007	+	0.812467i	$0.698124\pi$
$68$	−13.5807	−0.0242191
$69$	0	0
$70$	0	0
$71$	267.420	0.446998	0.223499	−	0.974704i	$-0.428252\pi$
$71$	267.420	0.446998	0.223499	+	0.974704i	$0.428252\pi$
$72$	0	0
$73$	−540.297	−0.866260	−0.433130	−	0.901331i	$-0.642591\pi$
$73$	−540.297	−0.866260	−0.433130	+	0.901331i	$0.642591\pi$
$74$	475.756	0.747373
$75$	0	0
$76$	52.1033	0.0786402
$77$	−145.811	−0.215801
$78$	0	0
$79$	691.157	0.984319	0.492160	−	0.870505i	$-0.336208\pi$
$79$	691.157	0.984319	0.492160	+	0.870505i	$0.336208\pi$
$80$	0	0
$81$	0	0
$82$	160.445	0.216076
$83$	663.244	0.877115	0.438557	−	0.898703i	$-0.355490\pi$
$83$	663.244	0.877115	0.438557	+	0.898703i	$0.355490\pi$
$84$	0	0
$85$	0	0
$86$	579.709	0.726879
$87$	0	0
$88$	254.550	0.308354
$89$	119.542	0.142376	0.0711880	−	0.997463i	$-0.477321\pi$
$89$	119.542	0.142376	0.0711880	+	0.997463i	$0.477321\pi$
$90$	0	0
$91$	−406.565	−0.468348
$92$	2.52103	0.00285691
$93$	0	0
$94$	418.211	0.458884
$95$	0	0
$96$	0	0
$97$	284.049	0.297328	0.148664	−	0.988888i	$-0.452503\pi$
$97$	284.049	0.297328	0.148664	+	0.988888i	$0.452503\pi$
$98$	461.589	0.475792
$99$	0	0
$100$	0	0
$101$	929.277	0.915510	0.457755	−	0.889078i	$-0.348654\pi$
$101$	929.277	0.915510	0.457755	+	0.889078i	$0.348654\pi$
$102$	0	0
$103$	1176.11	1.12511	0.562553	−	0.826761i	$-0.309819\pi$
$103$	1176.11	1.12511	0.562553	+	0.826761i	$0.309819\pi$
$104$	709.764	0.669213
$105$	0	0
$106$	−878.484	−0.804962
$107$	1104.32	0.997744	0.498872	−	0.866676i	$-0.333748\pi$
$107$	1104.32	0.997744	0.498872	+	0.866676i	$0.333748\pi$
$108$	0	0
$109$	−154.368	−0.135649	−0.0678246	−	0.997697i	$-0.521606\pi$
$109$	−154.368	−0.135649	−0.0678246	+	0.997697i	$0.521606\pi$
$110$	0	0
$111$	0	0
$112$	805.349	0.679450
$113$	−2066.25	−1.72014	−0.860071	−	0.510174i	$-0.829581\pi$
$113$	−2066.25	−1.72014	−0.860071	+	0.510174i	$0.829581\pi$
$114$	0	0
$115$	0	0
$116$	−11.7229	−0.00938314
$117$	0	0
$118$	1144.18	0.892627
$119$	−465.375	−0.358495
$120$	0	0
$121$	121.000	0.0909091
$122$	−1374.32	−1.01988
$123$	0	0
$124$	−128.350	−0.0929530
$125$	0	0
$126$	0	0
$127$	−41.1067	−0.0287215	−0.0143607	−	0.999897i	$-0.504571\pi$
$127$	−41.1067	−0.0287215	−0.0143607	+	0.999897i	$0.504571\pi$
$128$	−1334.33	−0.921401
$129$	0	0
$130$	0	0
$131$	886.633	0.591339	0.295670	−	0.955290i	$-0.404457\pi$
$131$	886.633	0.591339	0.295670	+	0.955290i	$0.404457\pi$
$132$	0	0
$133$	1785.44	1.16404
$134$	1764.41	1.13747
$135$	0	0
$136$	812.432	0.512246
$137$	1826.43	1.13900	0.569498	−	0.821993i	$-0.307138\pi$
$137$	1826.43	1.13900	0.569498	+	0.821993i	$0.307138\pi$
$138$	0	0
$139$	2741.65	1.67297	0.836487	−	0.547986i	$-0.184605\pi$
$139$	2741.65	1.67297	0.836487	+	0.547986i	$0.184605\pi$
$140$	0	0
$141$	0	0
$142$	−737.864	−0.436058
$143$	337.385	0.197298
$144$	0	0
$145$	0	0
$146$	1490.79	0.845058
$147$	0	0
$148$	66.6987	0.0370446
$149$	−2789.06	−1.53348	−0.766740	−	0.641958i	$-0.778122\pi$
$149$	−2789.06	−1.53348	−0.766740	+	0.641958i	$0.778122\pi$
$150$	0	0
$151$	−1743.59	−0.939679	−0.469840	−	0.882752i	$-0.655688\pi$
$151$	−1743.59	−0.939679	−0.469840	+	0.882752i	$0.655688\pi$
$152$	−3116.95	−1.66328
$153$	0	0
$154$	402.321	0.210519
$155$	0	0
$156$	0	0
$157$	−2509.69	−1.27576	−0.637882	−	0.770134i	$-0.720189\pi$
$157$	−2509.69	−1.27576	−0.637882	+	0.770134i	$0.720189\pi$
$158$	−1907.04	−0.960227
$159$	0	0
$160$	0	0
$161$	86.3894	0.0422884
$162$	0	0
$163$	−3254.12	−1.56370	−0.781848	−	0.623469i	$-0.785723\pi$
$163$	−3254.12	−1.56370	−0.781848	+	0.623469i	$0.785723\pi$
$164$	22.4936	0.0107101
$165$	0	0
$166$	−1830.02	−0.855646
$167$	−3198.42	−1.48204	−0.741021	−	0.671482i	$-0.765658\pi$
$167$	−3198.42	−1.48204	−0.741021	+	0.671482i	$0.765658\pi$
$168$	0	0
$169$	−1256.26	−0.571809
$170$	0	0
$171$	0	0
$172$	81.2722	0.0360288
$173$	4003.85	1.75958	0.879790	−	0.475362i	$-0.157683\pi$
$173$	4003.85	1.75958	0.879790	+	0.475362i	$0.157683\pi$
$174$	0	0
$175$	0	0
$176$	−668.313	−0.286227
$177$	0	0
$178$	−329.841	−0.138891
$179$	3671.58	1.53311	0.766555	−	0.642179i	$-0.221969\pi$
$179$	3671.58	1.53311	0.766555	+	0.642179i	$0.221969\pi$
$180$	0	0
$181$	1463.79	0.601119	0.300560	−	0.953763i	$-0.402827\pi$
$181$	1463.79	0.601119	0.300560	+	0.953763i	$0.402827\pi$
$182$	1121.79	0.456884
$183$	0	0
$184$	−150.815	−0.0604251
$185$	0	0
$186$	0	0
$187$	386.188	0.151021
$188$	58.6310	0.0227453
$189$	0	0
$190$	0	0
$191$	3332.68	1.26254	0.631268	−	0.775565i	$-0.282535\pi$
$191$	3332.68	1.26254	0.631268	+	0.775565i	$0.282535\pi$
$192$	0	0
$193$	−4565.00	−1.70257	−0.851285	−	0.524703i	$-0.824176\pi$
$193$	−4565.00	−1.70257	−0.851285	+	0.524703i	$0.824176\pi$
$194$	−783.747	−0.290050
$195$	0	0
$196$	64.7125	0.0235833
$197$	−1094.45	−0.395821	−0.197910	−	0.980220i	$-0.563415\pi$
$197$	−1094.45	−0.395821	−0.197910	+	0.980220i	$0.563415\pi$
$198$	0	0
$199$	4326.27	1.54111	0.770555	−	0.637374i	$-0.219979\pi$
$199$	4326.27	1.54111	0.770555	+	0.637374i	$0.219979\pi$
$200$	0	0
$201$	0	0
$202$	−2564.06	−0.893102
$203$	−401.714	−0.138891
$204$	0	0
$205$	0	0
$206$	−3245.13	−1.09757
$207$	0	0
$208$	−1863.46	−0.621192
$209$	−1481.64	−0.490369
$210$	0	0
$211$	−1106.01	−0.360856	−0.180428	−	0.983588i	$-0.557748\pi$
$211$	−1106.01	−0.360856	−0.180428	+	0.983588i	$0.557748\pi$
$212$	−123.159	−0.0398991
$213$	0	0
$214$	−3047.04	−0.973323
$215$	0	0
$216$	0	0
$217$	−4398.22	−1.37590
$218$	425.932	0.132329
$219$	0	0
$220$	0	0
$221$	1076.81	0.327757
$222$	0	0
$223$	−668.500	−0.200745	−0.100372	−	0.994950i	$-0.532003\pi$
$223$	−668.500	−0.200745	−0.100372	+	0.994950i	$0.532003\pi$
$224$	231.841	0.0691541
$225$	0	0
$226$	5701.18	1.67804
$227$	−4381.01	−1.28096	−0.640480	−	0.767975i	$-0.721265\pi$
$227$	−4381.01	−1.28096	−0.640480	+	0.767975i	$0.721265\pi$
$228$	0	0
$229$	4173.47	1.20433	0.602164	−	0.798373i	$-0.294305\pi$
$229$	4173.47	1.20433	0.602164	+	0.798373i	$0.294305\pi$
$230$	0	0
$231$	0	0
$232$	701.295	0.198458
$233$	5684.75	1.59837	0.799185	−	0.601086i	$-0.205265\pi$
$233$	5684.75	1.59837	0.799185	+	0.601086i	$0.205265\pi$
$234$	0	0
$235$	0	0
$236$	160.408	0.0442443
$237$	0	0
$238$	1284.06	0.349720
$239$	4804.76	1.30039	0.650196	−	0.759766i	$-0.274687\pi$
$239$	4804.76	1.30039	0.650196	+	0.759766i	$0.274687\pi$
$240$	0	0
$241$	−429.892	−0.114904	−0.0574518	−	0.998348i	$-0.518298\pi$
$241$	−429.892	−0.114904	−0.0574518	+	0.998348i	$0.518298\pi$
$242$	−333.863	−0.0886840
$243$	0	0
$244$	−192.673	−0.0505518
$245$	0	0
$246$	0	0
$247$	−4131.27	−1.06424
$248$	7678.23	1.96600
$249$	0	0
$250$	0	0
$251$	423.188	0.106420	0.0532099	−	0.998583i	$-0.483055\pi$
$251$	423.188	0.106420	0.0532099	+	0.998583i	$0.483055\pi$
$252$	0	0
$253$	−71.6896	−0.0178146
$254$	113.422	0.0280185
$255$	0	0
$256$	−592.753	−0.144715
$257$	1376.87	0.334191	0.167095	−	0.985941i	$-0.446561\pi$
$257$	1376.87	0.334191	0.167095	+	0.985941i	$0.446561\pi$
$258$	0	0
$259$	2285.59	0.548339
$260$	0	0
$261$	0	0
$262$	−2446.39	−0.576866
$263$	3996.71	0.937063	0.468532	−	0.883447i	$-0.344783\pi$
$263$	3996.71	0.937063	0.468532	+	0.883447i	$0.344783\pi$
$264$	0	0
$265$	0	0
$266$	−4926.40	−1.13555
$267$	0	0
$268$	247.361	0.0563805
$269$	4844.04	1.09794	0.548970	−	0.835842i	$-0.315020\pi$
$269$	4844.04	1.09794	0.548970	+	0.835842i	$0.315020\pi$
$270$	0	0
$271$	−5141.48	−1.15248	−0.576242	−	0.817279i	$-0.695481\pi$
$271$	−5141.48	−1.15248	−0.576242	+	0.817279i	$0.695481\pi$
$272$	−2133.02	−0.475489
$273$	0	0
$274$	−5039.48	−1.11112
$275$	0	0
$276$	0	0
$277$	−5871.40	−1.27357	−0.636784	−	0.771042i	$-0.719736\pi$
$277$	−5871.40	−1.27357	−0.636784	+	0.771042i	$0.719736\pi$
$278$	−7564.75	−1.63203
$279$	0	0
$280$	0	0
$281$	−2091.86	−0.444093	−0.222046	−	0.975036i	$-0.571274\pi$
$281$	−2091.86	−0.444093	−0.222046	+	0.975036i	$0.571274\pi$
$282$	0	0
$283$	2899.13	0.608959	0.304479	−	0.952519i	$-0.401518\pi$
$283$	2899.13	0.608959	0.304479	+	0.952519i	$0.401518\pi$
$284$	−103.445	−0.0216138
$285$	0	0
$286$	−930.913	−0.192469
$287$	770.798	0.158532
$288$	0	0
$289$	−3680.43	−0.749120
$290$	0	0
$291$	0	0
$292$	209.001	0.0418865
$293$	−2470.76	−0.492640	−0.246320	−	0.969189i	$-0.579221\pi$
$293$	−2470.76	−0.492640	−0.246320	+	0.969189i	$0.579221\pi$
$294$	0	0
$295$	0	0
$296$	−3990.09	−0.783510
$297$	0	0
$298$	7695.56	1.49595
$299$	−199.893	−0.0386625
$300$	0	0
$301$	2784.99	0.533303
$302$	4810.92	0.916680
$303$	0	0
$304$	8183.46	1.54393
$305$	0	0
$306$	0	0
$307$	−8790.58	−1.63422	−0.817109	−	0.576484i	$-0.804424\pi$
$307$	−8790.58	−1.63422	−0.817109	+	0.576484i	$0.804424\pi$
$308$	56.4033	0.0104347
$309$	0	0
$310$	0	0
$311$	2085.43	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$311$	2085.43	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$312$	0	0
$313$	−7071.44	−1.27700	−0.638501	−	0.769621i	$-0.720445\pi$
$313$	−7071.44	−1.27700	−0.638501	+	0.769621i	$0.720445\pi$
$314$	6924.73	1.24454
$315$	0	0
$316$	−267.357	−0.0475950
$317$	−5877.25	−1.04132	−0.520661	−	0.853763i	$-0.674315\pi$
$317$	−5877.25	−1.04132	−0.520661	+	0.853763i	$0.674315\pi$
$318$	0	0
$319$	333.359	0.0585095
$320$	0	0
$321$	0	0
$322$	−238.365	−0.0412534
$323$	−4728.86	−0.814615
$324$	0	0
$325$	0	0
$326$	8978.77	1.52542
$327$	0	0
$328$	−1345.63	−0.226524
$329$	2009.13	0.336678
$330$	0	0
$331$	−4046.15	−0.671892	−0.335946	−	0.941881i	$-0.609056\pi$
$331$	−4046.15	−0.671892	−0.335946	+	0.941881i	$0.609056\pi$
$332$	−256.560	−0.0424113
$333$	0	0
$334$	8825.07	1.44577
$335$	0	0
$336$	0	0
$337$	2827.44	0.457034	0.228517	−	0.973540i	$-0.426612\pi$
$337$	2827.44	0.457034	0.228517	+	0.973540i	$0.426612\pi$
$338$	3466.28	0.557814
$339$	0	0
$340$	0	0
$341$	3649.83	0.579618
$342$	0	0
$343$	6764.18	1.06481
$344$	−4861.91	−0.762026
$345$	0	0
$346$	−11047.4	−1.71651
$347$	−6698.19	−1.03625	−0.518123	−	0.855306i	$-0.673369\pi$
$347$	−6698.19	−1.03625	−0.518123	+	0.855306i	$0.673369\pi$
$348$	0	0
$349$	−10355.3	−1.58827	−0.794133	−	0.607744i	$-0.792075\pi$
$349$	−10355.3	−1.58827	−0.794133	+	0.607744i	$0.792075\pi$
$350$	0	0
$351$	0	0
$352$	−192.391	−0.0291321
$353$	−10077.5	−1.51946	−0.759732	−	0.650236i	$-0.774670\pi$
$353$	−10077.5	−1.51946	−0.759732	+	0.650236i	$0.774670\pi$
$354$	0	0
$355$	0	0
$356$	−46.2420	−0.00688434
$357$	0	0
$358$	−10130.6	−1.49558
$359$	−7877.23	−1.15806	−0.579031	−	0.815306i	$-0.696569\pi$
$359$	−7877.23	−1.15806	−0.579031	+	0.815306i	$0.696569\pi$
$360$	0	0
$361$	11283.6	1.64508
$362$	−4038.88	−0.586406
$363$	0	0
$364$	157.270	0.0226461
$365$	0	0
$366$	0	0
$367$	8558.21	1.21726	0.608630	−	0.793454i	$-0.291719\pi$
$367$	8558.21	1.21726	0.608630	+	0.793454i	$0.291719\pi$
$368$	395.960	0.0560892
$369$	0	0
$370$	0	0
$371$	−4220.35	−0.590591
$372$	0	0
$373$	−3842.48	−0.533395	−0.266697	−	0.963780i	$-0.585932\pi$
$373$	−3842.48	−0.533395	−0.266697	+	0.963780i	$0.585932\pi$
$374$	−1065.57	−0.147324
$375$	0	0
$376$	−3507.46	−0.481073
$377$	929.509	0.126982
$378$	0	0
$379$	2824.79	0.382849	0.191424	−	0.981507i	$-0.438689\pi$
$379$	2824.79	0.382849	0.191424	+	0.981507i	$0.438689\pi$
$380$	0	0
$381$	0	0
$382$	−9195.52	−1.23163
$383$	−518.110	−0.0691232	−0.0345616	−	0.999403i	$-0.511003\pi$
$383$	−518.110	−0.0691232	−0.0345616	+	0.999403i	$0.511003\pi$
$384$	0	0
$385$	0	0
$386$	12595.8	1.66090
$387$	0	0
$388$	−109.877	−0.0143767
$389$	−3371.34	−0.439419	−0.219709	−	0.975565i	$-0.570511\pi$
$389$	−3371.34	−0.439419	−0.219709	+	0.975565i	$0.570511\pi$
$390$	0	0
$391$	−228.807	−0.0295941
$392$	−3871.27	−0.498798
$393$	0	0
$394$	3019.82	0.386132
$395$	0	0
$396$	0	0
$397$	1593.40	0.201437	0.100718	−	0.994915i	$-0.467886\pi$
$397$	1593.40	0.201437	0.100718	+	0.994915i	$0.467886\pi$
$398$	−11937.0	−1.50339
$399$	0	0
$400$	0	0
$401$	−14258.7	−1.77567	−0.887834	−	0.460163i	$-0.847791\pi$
$401$	−14258.7	−1.77567	−0.887834	+	0.460163i	$0.847791\pi$
$402$	0	0
$403$	10176.9	1.25793
$404$	−359.468	−0.0442679
$405$	0	0
$406$	1108.41	0.135491
$407$	−1896.68	−0.230995
$408$	0	0
$409$	2543.37	0.307485	0.153743	−	0.988111i	$-0.450867\pi$
$409$	2543.37	0.307485	0.153743	+	0.988111i	$0.450867\pi$
$410$	0	0
$411$	0	0
$412$	−454.951	−0.0544025
$413$	5496.76	0.654910
$414$	0	0
$415$	0	0
$416$	−536.447	−0.0632247
$417$	0	0
$418$	4088.13	0.478366
$419$	−13302.3	−1.55098	−0.775492	−	0.631358i	$-0.782498\pi$
$419$	−13302.3	−1.55098	−0.775492	+	0.631358i	$0.782498\pi$
$420$	0	0
$421$	−10574.5	−1.22416	−0.612079	−	0.790797i	$-0.709667\pi$
$421$	−10574.5	−1.22416	−0.612079	+	0.790797i	$0.709667\pi$
$422$	3051.69	0.352024
$423$	0	0
$424$	7367.70	0.843884
$425$	0	0
$426$	0	0
$427$	−6602.41	−0.748275
$428$	−427.179	−0.0482441
$429$	0	0
$430$	0	0
$431$	−3472.31	−0.388064	−0.194032	−	0.980995i	$-0.562157\pi$
$431$	−3472.31	−0.388064	−0.194032	+	0.980995i	$0.562157\pi$
$432$	0	0
$433$	−6609.57	−0.733570	−0.366785	−	0.930306i	$-0.619541\pi$
$433$	−6609.57	−0.733570	−0.366785	+	0.930306i	$0.619541\pi$
$434$	12135.6	1.34223
$435$	0	0
$436$	59.7135	0.00655908
$437$	877.835	0.0960928
$438$	0	0
$439$	260.583	0.0283302	0.0141651	−	0.999900i	$-0.495491\pi$
$439$	260.583	0.0283302	0.0141651	+	0.999900i	$0.495491\pi$
$440$	0	0
$441$	0	0
$442$	−2971.14	−0.319735
$443$	−13460.4	−1.44362	−0.721808	−	0.692094i	$-0.756688\pi$
$443$	−13460.4	−1.44362	−0.721808	+	0.692094i	$0.756688\pi$
$444$	0	0
$445$	0	0
$446$	1844.52	0.195831
$447$	0	0
$448$	−7082.49	−0.746911
$449$	6838.00	0.718720	0.359360	−	0.933199i	$-0.382995\pi$
$449$	6838.00	0.718720	0.359360	+	0.933199i	$0.382995\pi$
$450$	0	0
$451$	−639.642	−0.0667839
$452$	799.277	0.0831744
$453$	0	0
$454$	12088.1	1.24961
$455$	0	0
$456$	0	0
$457$	−9562.19	−0.978776	−0.489388	−	0.872066i	$-0.662780\pi$
$457$	−9562.19	−0.978776	−0.489388	+	0.872066i	$0.662780\pi$
$458$	−11515.4	−1.17485
$459$	0	0
$460$	0	0
$461$	−16949.3	−1.71239	−0.856193	−	0.516657i	$-0.827176\pi$
$461$	−16949.3	−1.71239	−0.856193	+	0.516657i	$0.827176\pi$
$462$	0	0
$463$	12524.8	1.25718	0.628590	−	0.777737i	$-0.283632\pi$
$463$	12524.8	1.25718	0.628590	+	0.777737i	$0.283632\pi$
$464$	−1841.23	−0.184217
$465$	0	0
$466$	−15685.3	−1.55925
$467$	−6584.36	−0.652436	−0.326218	−	0.945295i	$-0.605774\pi$
$467$	−6584.36	−0.652436	−0.326218	+	0.945295i	$0.605774\pi$
$468$	0	0
$469$	8476.42	0.834552
$470$	0	0
$471$	0	0
$472$	−9596.01	−0.935788
$473$	−2311.10	−0.224661
$474$	0	0
$475$	0	0
$476$	180.019	0.0173344
$477$	0	0
$478$	−13257.3	−1.26856
$479$	19145.9	1.82630	0.913152	−	0.407620i	$-0.133641\pi$
$479$	19145.9	1.82630	0.913152	+	0.407620i	$0.133641\pi$
$480$	0	0
$481$	−5288.54	−0.501323
$482$	1186.16	0.112091
$483$	0	0
$484$	−46.8059	−0.00439575
$485$	0	0
$486$	0	0
$487$	11777.7	1.09589	0.547947	−	0.836513i	$-0.315409\pi$
$487$	11777.7	1.09589	0.547947	+	0.836513i	$0.315409\pi$
$488$	11526.2	1.06919
$489$	0	0
$490$	0	0
$491$	−2966.30	−0.272642	−0.136321	−	0.990665i	$-0.543528\pi$
$491$	−2966.30	−0.272642	−0.136321	+	0.990665i	$0.543528\pi$
$492$	0	0
$493$	1063.96	0.0971977
$494$	11399.0	1.03819
$495$	0	0
$496$	−20159.0	−1.82493
$497$	−3544.79	−0.319931
$498$	0	0
$499$	2696.53	0.241910	0.120955	−	0.992658i	$-0.461404\pi$
$499$	2696.53	0.241910	0.120955	+	0.992658i	$0.461404\pi$
$500$	0	0
$501$	0	0
$502$	−1167.66	−0.103815
$503$	−1834.68	−0.162633	−0.0813164	−	0.996688i	$-0.525912\pi$
$503$	−1834.68	−0.162633	−0.0813164	+	0.996688i	$0.525912\pi$
$504$	0	0
$505$	0	0
$506$	197.806	0.0173785
$507$	0	0
$508$	15.9011	0.00138878
$509$	−735.568	−0.0640540	−0.0320270	−	0.999487i	$-0.510196\pi$
$509$	−735.568	−0.0640540	−0.0320270	+	0.999487i	$0.510196\pi$
$510$	0	0
$511$	7161.92	0.620009
$512$	12310.2	1.06257
$513$	0	0
$514$	−3799.07	−0.326011
$515$	0	0
$516$	0	0
$517$	−1667.27	−0.141830
$518$	−6306.40	−0.534918
$519$	0	0
$520$	0	0
$521$	−6603.97	−0.555327	−0.277663	−	0.960678i	$-0.589560\pi$
$521$	−6603.97	−0.555327	−0.277663	+	0.960678i	$0.589560\pi$
$522$	0	0
$523$	−9477.96	−0.792433	−0.396216	−	0.918157i	$-0.629677\pi$
$523$	−9477.96	−0.792433	−0.396216	+	0.918157i	$0.629677\pi$
$524$	−342.972	−0.0285932
$525$	0	0
$526$	−11027.7	−0.914127
$527$	11649.0	0.962878
$528$	0	0
$529$	−12124.5	−0.996509
$530$	0	0
$531$	0	0
$532$	−690.656	−0.0562852
$533$	−1783.52	−0.144940
$534$	0	0
$535$	0	0
$536$	−14797.8	−1.19247
$537$	0	0
$538$	−13365.7	−1.07107
$539$	−1840.20	−0.147056
$540$	0	0
$541$	−7071.66	−0.561986	−0.280993	−	0.959710i	$-0.590664\pi$
$541$	−7071.66	−0.561986	−0.280993	+	0.959710i	$0.590664\pi$
$542$	14186.4	1.12427
$543$	0	0
$544$	−614.044	−0.0483951
$545$	0	0
$546$	0	0
$547$	277.112	0.0216608	0.0108304	−	0.999941i	$-0.496553\pi$
$547$	277.112	0.0216608	0.0108304	+	0.999941i	$0.496553\pi$
$548$	−706.511	−0.0550741
$549$	0	0
$550$	0	0
$551$	−4081.97	−0.315604
$552$	0	0
$553$	−9161.65	−0.704508
$554$	16200.4	1.24240
$555$	0	0
$556$	−1060.54	−0.0808937
$557$	25871.4	1.96805	0.984027	−	0.178017i	$-0.0569684\pi$
$557$	25871.4	1.96805	0.984027	+	0.178017i	$0.0569684\pi$
$558$	0	0
$559$	−6444.07	−0.487577
$560$	0	0
$561$	0	0
$562$	5771.86	0.433223
$563$	5398.71	0.404136	0.202068	−	0.979372i	$-0.435234\pi$
$563$	5398.71	0.404136	0.202068	+	0.979372i	$0.435234\pi$
$564$	0	0
$565$	0	0
$566$	−7999.27	−0.594054
$567$	0	0
$568$	6188.34	0.457142
$569$	−15946.6	−1.17490	−0.587450	−	0.809261i	$-0.699868\pi$
$569$	−15946.6	−1.17490	−0.587450	+	0.809261i	$0.699868\pi$
$570$	0	0
$571$	19962.6	1.46307	0.731533	−	0.681806i	$-0.238805\pi$
$571$	19962.6	1.46307	0.731533	+	0.681806i	$0.238805\pi$
$572$	−130.509	−0.00953999
$573$	0	0
$574$	−2126.79	−0.154652
$575$	0	0
$576$	0	0
$577$	−8151.97	−0.588165	−0.294082	−	0.955780i	$-0.595014\pi$
$577$	−8151.97	−0.588165	−0.294082	+	0.955780i	$0.595014\pi$
$578$	10155.0	0.730784
$579$	0	0
$580$	0	0
$581$	−8791.65	−0.627778
$582$	0	0
$583$	3502.22	0.248795
$584$	−12503.0	−0.885919
$585$	0	0
$586$	6817.32	0.480582
$587$	−16937.1	−1.19092	−0.595459	−	0.803386i	$-0.703030\pi$
$587$	−16937.1	−1.19092	−0.595459	+	0.803386i	$0.703030\pi$
$588$	0	0
$589$	−44692.0	−3.12649
$590$	0	0
$591$	0	0
$592$	10475.8	0.727288
$593$	18143.4	1.25643	0.628214	−	0.778040i	$-0.283786\pi$
$593$	18143.4	1.25643	0.628214	+	0.778040i	$0.283786\pi$
$594$	0	0
$595$	0	0
$596$	1078.88	0.0741487
$597$	0	0
$598$	551.544	0.0377162
$599$	−27810.1	−1.89698	−0.948488	−	0.316814i	$-0.897387\pi$
$599$	−27810.1	−1.89698	−0.948488	+	0.316814i	$0.897387\pi$
$600$	0	0
$601$	−6644.44	−0.450969	−0.225484	−	0.974247i	$-0.572396\pi$
$601$	−6644.44	−0.450969	−0.225484	+	0.974247i	$0.572396\pi$
$602$	−7684.34	−0.520250
$603$	0	0
$604$	674.467	0.0454365
$605$	0	0
$606$	0	0
$607$	5101.57	0.341131	0.170565	−	0.985346i	$-0.445441\pi$
$607$	5101.57	0.341131	0.170565	+	0.985346i	$0.445441\pi$
$608$	2355.82	0.157140
$609$	0	0
$610$	0	0
$611$	−4648.85	−0.307811
$612$	0	0
$613$	−19048.6	−1.25508	−0.627540	−	0.778585i	$-0.715938\pi$
$613$	−19048.6	−1.25508	−0.627540	+	0.778585i	$0.715938\pi$
$614$	24254.9	1.59422
$615$	0	0
$616$	−3374.19	−0.220698
$617$	−29249.6	−1.90850	−0.954251	−	0.299007i	$-0.903345\pi$
$617$	−29249.6	−1.90850	−0.954251	+	0.299007i	$0.903345\pi$
$618$	0	0
$619$	16904.8	1.09768	0.548838	−	0.835929i	$-0.315070\pi$
$619$	16904.8	1.09768	0.548838	+	0.835929i	$0.315070\pi$
$620$	0	0
$621$	0	0
$622$	−5754.11	−0.370931
$623$	−1584.60	−0.101903
$624$	0	0
$625$	0	0
$626$	19511.5	1.24575
$627$	0	0
$628$	970.813	0.0616873
$629$	−6053.52	−0.383736
$630$	0	0
$631$	16678.7	1.05225	0.526124	−	0.850408i	$-0.323645\pi$
$631$	16678.7	1.05225	0.526124	+	0.850408i	$0.323645\pi$
$632$	15994.0	1.00666
$633$	0	0
$634$	16216.5	1.01584
$635$	0	0
$636$	0	0
$637$	−5131.06	−0.319152
$638$	−919.805	−0.0570774
$639$	0	0
$640$	0	0
$641$	−24000.9	−1.47891	−0.739453	−	0.673209i	$-0.764916\pi$
$641$	−24000.9	−1.47891	−0.739453	+	0.673209i	$0.764916\pi$
$642$	0	0
$643$	−14805.6	−0.908052	−0.454026	−	0.890988i	$-0.650013\pi$
$643$	−14805.6	−0.908052	−0.454026	+	0.890988i	$0.650013\pi$
$644$	−33.4176	−0.00204478
$645$	0	0
$646$	13047.9	0.794676
$647$	11072.9	0.672828	0.336414	−	0.941714i	$-0.390786\pi$
$647$	11072.9	0.672828	0.336414	+	0.941714i	$0.390786\pi$
$648$	0	0
$649$	−4561.45	−0.275890
$650$	0	0
$651$	0	0
$652$	1258.78	0.0756098
$653$	−3186.71	−0.190973	−0.0954866	−	0.995431i	$-0.530441\pi$
$653$	−3186.71	−0.190973	−0.0954866	+	0.995431i	$0.530441\pi$
$654$	0	0
$655$	0	0
$656$	3532.90	0.210269
$657$	0	0
$658$	−5543.60	−0.328438
$659$	−28165.0	−1.66488	−0.832438	−	0.554118i	$-0.813056\pi$
$659$	−28165.0	−1.66488	−0.832438	+	0.554118i	$0.813056\pi$
$660$	0	0
$661$	−1989.21	−0.117052	−0.0585260	−	0.998286i	$-0.518640\pi$
$661$	−1989.21	−0.117052	−0.0585260	+	0.998286i	$0.518640\pi$
$662$	11164.1	0.655447
$663$	0	0
$664$	15348.1	0.897020
$665$	0	0
$666$	0	0
$667$	−197.507	−0.0114655
$668$	1237.23	0.0716615
$669$	0	0
$670$	0	0
$671$	5478.97	0.315221
$672$	0	0
$673$	−28620.0	−1.63926	−0.819629	−	0.572894i	$-0.805821\pi$
$673$	−28620.0	−1.63926	−0.819629	+	0.572894i	$0.805821\pi$
$674$	−7801.47	−0.445848
$675$	0	0
$676$	485.956	0.0276488
$677$	2211.83	0.125565	0.0627826	−	0.998027i	$-0.480003\pi$
$677$	2211.83	0.125565	0.0627826	+	0.998027i	$0.480003\pi$
$678$	0	0
$679$	−3765.21	−0.212807
$680$	0	0
$681$	0	0
$682$	−10070.6	−0.565431
$683$	−31139.9	−1.74456	−0.872281	−	0.489004i	$-0.837360\pi$
$683$	−31139.9	−1.74456	−0.872281	+	0.489004i	$0.837360\pi$
$684$	0	0
$685$	0	0
$686$	−18663.7	−1.03875
$687$	0	0
$688$	12764.8	0.707346
$689$	9765.28	0.539953
$690$	0	0
$691$	−14916.8	−0.821220	−0.410610	−	0.911811i	$-0.634684\pi$
$691$	−14916.8	−0.821220	−0.410610	+	0.911811i	$0.634684\pi$
$692$	−1548.79	−0.0850814
$693$	0	0
$694$	18481.6	1.01088
$695$	0	0
$696$	0	0
$697$	−2041.51	−0.110943
$698$	28572.2	1.54939
$699$	0	0
$700$	0	0
$701$	−5082.12	−0.273822	−0.136911	−	0.990583i	$-0.543717\pi$
$701$	−5082.12	−0.273822	−0.136911	+	0.990583i	$0.543717\pi$
$702$	0	0
$703$	23224.8	1.24600
$704$	5877.35	0.314646
$705$	0	0
$706$	27805.8	1.48227
$707$	−12318.1	−0.655259
$708$	0	0
$709$	−519.556	−0.0275209	−0.0137605	−	0.999905i	$-0.504380\pi$
$709$	−519.556	−0.0275209	−0.0137605	+	0.999905i	$0.504380\pi$
$710$	0	0
$711$	0	0
$712$	2766.32	0.145607
$713$	−2162.44	−0.113582
$714$	0	0
$715$	0	0
$716$	−1420.26	−0.0741308
$717$	0	0
$718$	21734.8	1.12972
$719$	−11584.1	−0.600856	−0.300428	−	0.953805i	$-0.597129\pi$
$719$	−11584.1	−0.600856	−0.300428	+	0.953805i	$0.597129\pi$
$720$	0	0
$721$	−15590.0	−0.805273
$722$	−31133.6	−1.60481
$723$	0	0
$724$	−566.231	−0.0290661
$725$	0	0
$726$	0	0
$727$	−6482.92	−0.330727	−0.165363	−	0.986233i	$-0.552880\pi$
$727$	−6482.92	−0.330727	−0.165363	+	0.986233i	$0.552880\pi$
$728$	−9408.30	−0.478976
$729$	0	0
$730$	0	0
$731$	−7376.21	−0.373213
$732$	0	0
$733$	2520.65	0.127015	0.0635077	−	0.997981i	$-0.479771\pi$
$733$	2520.65	0.127015	0.0635077	+	0.997981i	$0.479771\pi$
$734$	−23613.8	−1.18747
$735$	0	0
$736$	113.987	0.00570873
$737$	−7034.10	−0.351566
$738$	0	0
$739$	−1466.11	−0.0729791	−0.0364895	−	0.999334i	$-0.511618\pi$
$739$	−1466.11	−0.0729791	−0.0364895	+	0.999334i	$0.511618\pi$
$740$	0	0
$741$	0	0
$742$	11644.8	0.576136
$743$	−11215.3	−0.553769	−0.276885	−	0.960903i	$-0.589302\pi$
$743$	−11215.3	−0.553769	−0.276885	+	0.960903i	$0.589302\pi$
$744$	0	0
$745$	0	0
$746$	10602.2	0.520339
$747$	0	0
$748$	−149.388	−0.00730234
$749$	−14638.3	−0.714116
$750$	0	0
$751$	−29927.3	−1.45414	−0.727072	−	0.686561i	$-0.759119\pi$
$751$	−29927.3	−1.45414	−0.727072	+	0.686561i	$0.759119\pi$
$752$	9208.73	0.446553
$753$	0	0
$754$	−2564.70	−0.123874
$755$	0	0
$756$	0	0
$757$	−18872.9	−0.906138	−0.453069	−	0.891476i	$-0.649671\pi$
$757$	−18872.9	−0.906138	−0.453069	+	0.891476i	$0.649671\pi$
$758$	−7794.15	−0.373478
$759$	0	0
$760$	0	0
$761$	32135.0	1.53074	0.765370	−	0.643591i	$-0.222556\pi$
$761$	32135.0	1.53074	0.765370	+	0.643591i	$0.222556\pi$
$762$	0	0
$763$	2046.23	0.0970883
$764$	−1289.17	−0.0610476
$765$	0	0
$766$	1429.57	0.0674314
$767$	−12718.7	−0.598757
$768$	0	0
$769$	−11339.9	−0.531763	−0.265881	−	0.964006i	$-0.585663\pi$
$769$	−11339.9	−0.531763	−0.265881	+	0.964006i	$0.585663\pi$
$770$	0	0
$771$	0	0
$772$	1765.86	0.0823248
$773$	25434.8	1.18347	0.591737	−	0.806131i	$-0.298442\pi$
$773$	25434.8	1.18347	0.591737	+	0.806131i	$0.298442\pi$
$774$	0	0
$775$	0	0
$776$	6573.15	0.304075
$777$	0	0
$778$	9302.21	0.428664
$779$	7832.38	0.360236
$780$	0	0
$781$	2941.62	0.134775
$782$	631.325	0.0288697
$783$	0	0
$784$	10163.9	0.463006
$785$	0	0
$786$	0	0
$787$	−5909.97	−0.267684	−0.133842	−	0.991003i	$-0.542732\pi$
$787$	−5909.97	−0.267684	−0.133842	+	0.991003i	$0.542732\pi$
$788$	423.363	0.0191392
$789$	0	0
$790$	0	0
$791$	27389.2	1.23116
$792$	0	0
$793$	15277.1	0.684116
$794$	−4396.50	−0.196506
$795$	0	0
$796$	−1673.51	−0.0745176
$797$	−13869.1	−0.616396	−0.308198	−	0.951322i	$-0.599726\pi$
$797$	−13869.1	−0.616396	−0.308198	+	0.951322i	$0.599726\pi$
$798$	0	0
$799$	−5321.31	−0.235613
$800$	0	0
$801$	0	0
$802$	39342.5	1.73221
$803$	−5943.27	−0.261187
$804$	0	0
$805$	0	0
$806$	−28080.0	−1.22714
$807$	0	0
$808$	21504.3	0.936286
$809$	13809.7	0.600154	0.300077	−	0.953915i	$-0.402988\pi$
$809$	13809.7	0.600154	0.300077	+	0.953915i	$0.402988\pi$
$810$	0	0
$811$	19727.1	0.854145	0.427073	−	0.904217i	$-0.359545\pi$
$811$	19727.1	0.854145	0.427073	+	0.904217i	$0.359545\pi$
$812$	155.393	0.00671581
$813$	0	0
$814$	5233.32	0.225341
$815$	0	0
$816$	0	0
$817$	28299.3	1.21183
$818$	−7017.65	−0.299959
$819$	0	0
$820$	0	0
$821$	33427.7	1.42099	0.710496	−	0.703701i	$-0.248471\pi$
$821$	33427.7	1.42099	0.710496	+	0.703701i	$0.248471\pi$
$822$	0	0
$823$	37516.3	1.58899	0.794493	−	0.607273i	$-0.207737\pi$
$823$	37516.3	1.58899	0.794493	+	0.607273i	$0.207737\pi$
$824$	27216.3	1.15064
$825$	0	0
$826$	−15166.7	−0.638881
$827$	6754.82	0.284024	0.142012	−	0.989865i	$-0.454643\pi$
$827$	6754.82	0.284024	0.142012	+	0.989865i	$0.454643\pi$
$828$	0	0
$829$	24148.1	1.01170	0.505849	−	0.862622i	$-0.331179\pi$
$829$	24148.1	1.01170	0.505849	+	0.862622i	$0.331179\pi$
$830$	0	0
$831$	0	0
$832$	16387.9	0.682870
$833$	−5873.26	−0.244294
$834$	0	0
$835$	0	0
$836$	573.136	0.0237109
$837$	0	0
$838$	36703.8	1.51302
$839$	−6252.65	−0.257289	−0.128644	−	0.991691i	$-0.541063\pi$
$839$	−6252.65	−0.257289	−0.128644	+	0.991691i	$0.541063\pi$
$840$	0	0
$841$	−23470.6	−0.962343
$842$	29177.2	1.19420
$843$	0	0
$844$	427.832	0.0174485
$845$	0	0
$846$	0	0
$847$	−1603.92	−0.0650664
$848$	−19343.7	−0.783330
$849$	0	0
$850$	0	0
$851$	1123.74	0.0452659
$852$	0	0
$853$	7394.49	0.296814	0.148407	−	0.988926i	$-0.452585\pi$
$853$	7394.49	0.296814	0.148407	+	0.988926i	$0.452585\pi$
$854$	18217.4	0.729960
$855$	0	0
$856$	25555.0	1.02039
$857$	43009.6	1.71433	0.857164	−	0.515043i	$-0.172224\pi$
$857$	43009.6	1.71433	0.857164	+	0.515043i	$0.172224\pi$
$858$	0	0
$859$	−27843.6	−1.10595	−0.552976	−	0.833197i	$-0.686508\pi$
$859$	−27843.6	−1.10595	−0.552976	+	0.833197i	$0.686508\pi$
$860$	0	0
$861$	0	0
$862$	9580.80	0.378565
$863$	−17729.9	−0.699343	−0.349672	−	0.936872i	$-0.613707\pi$
$863$	−17729.9	−0.699343	−0.349672	+	0.936872i	$0.613707\pi$
$864$	0	0
$865$	0	0
$866$	18237.1	0.715615
$867$	0	0
$868$	1701.35	0.0665293
$869$	7602.73	0.296783
$870$	0	0
$871$	−19613.3	−0.762996
$872$	−3572.21	−0.138728
$873$	0	0
$874$	−2422.12	−0.0937408
$875$	0	0
$876$	0	0
$877$	−18067.8	−0.695674	−0.347837	−	0.937555i	$-0.613084\pi$
$877$	−18067.8	−0.695674	−0.347837	+	0.937555i	$0.613084\pi$
$878$	−719.000	−0.0276368
$879$	0	0
$880$	0	0
$881$	−23640.0	−0.904030	−0.452015	−	0.892010i	$-0.649295\pi$
$881$	−23640.0	−0.904030	−0.452015	+	0.892010i	$0.649295\pi$
$882$	0	0
$883$	−40722.8	−1.55202	−0.776008	−	0.630723i	$-0.782758\pi$
$883$	−40722.8	−1.55202	−0.776008	+	0.630723i	$0.782758\pi$
$884$	−416.539	−0.0158481
$885$	0	0
$886$	37139.8	1.40828
$887$	5237.95	0.198279	0.0991394	−	0.995074i	$-0.468391\pi$
$887$	5237.95	0.198279	0.0991394	+	0.995074i	$0.468391\pi$
$888$	0	0
$889$	544.891	0.0205569
$890$	0	0
$891$	0	0
$892$	258.593	0.00970666
$893$	20415.6	0.765041
$894$	0	0
$895$	0	0
$896$	17687.3	0.659475
$897$	0	0
$898$	−18867.4	−0.701128
$899$	10055.4	0.373045
$900$	0	0
$901$	11177.8	0.413305
$902$	1764.90	0.0651493
$903$	0	0
$904$	−47814.8	−1.75918
$905$	0	0
$906$	0	0
$907$	24778.5	0.907118	0.453559	−	0.891226i	$-0.350154\pi$
$907$	24778.5	0.907118	0.453559	+	0.891226i	$0.350154\pi$
$908$	1694.69	0.0619386
$909$	0	0
$910$	0	0
$911$	22488.4	0.817865	0.408933	−	0.912565i	$-0.365901\pi$
$911$	22488.4	0.817865	0.408933	+	0.912565i	$0.365901\pi$
$912$	0	0
$913$	7295.69	0.264460
$914$	26384.0	0.954819
$915$	0	0
$916$	−1614.41	−0.0582331
$917$	−11752.8	−0.423240
$918$	0	0
$919$	24214.0	0.869147	0.434574	−	0.900636i	$-0.356899\pi$
$919$	24214.0	0.869147	0.434574	+	0.900636i	$0.356899\pi$
$920$	0	0
$921$	0	0
$922$	46766.6	1.67047
$923$	8202.14	0.292499
$924$	0	0
$925$	0	0
$926$	−34558.3	−1.22641
$927$	0	0
$928$	−530.045	−0.0187496
$929$	−11588.7	−0.409273	−0.204636	−	0.978838i	$-0.565601\pi$
$929$	−11588.7	−0.409273	−0.204636	+	0.978838i	$0.565601\pi$
$930$	0	0
$931$	22533.2	0.793228
$932$	−2199.01	−0.0772863
$933$	0	0
$934$	18167.5	0.636467
$935$	0	0
$936$	0	0
$937$	36689.9	1.27920	0.639598	−	0.768709i	$-0.279101\pi$
$937$	36689.9	1.27920	0.639598	+	0.768709i	$0.279101\pi$
$938$	−23388.1	−0.814125
$939$	0	0
$940$	0	0
$941$	6743.11	0.233602	0.116801	−	0.993155i	$-0.462736\pi$
$941$	6743.11	0.233602	0.116801	+	0.993155i	$0.462736\pi$
$942$	0	0
$943$	378.972	0.0130870
$944$	25194.0	0.868639
$945$	0	0
$946$	6376.79	0.219162
$947$	10797.8	0.370520	0.185260	−	0.982690i	$-0.440687\pi$
$947$	10797.8	0.370520	0.185260	+	0.982690i	$0.440687\pi$
$948$	0	0
$949$	−16571.7	−0.566849
$950$	0	0
$951$	0	0
$952$	−10769.2	−0.366630
$953$	27513.8	0.935215	0.467608	−	0.883936i	$-0.345116\pi$
$953$	27513.8	0.935215	0.467608	+	0.883936i	$0.345116\pi$
$954$	0	0
$955$	0	0
$956$	−1858.60	−0.0628782
$957$	0	0
$958$	−52827.4	−1.78160
$959$	−24210.3	−0.815215
$960$	0	0
$961$	80302.3	2.69552
$962$	14592.1	0.489053
$963$	0	0
$964$	166.293	0.00555596
$965$	0	0
$966$	0	0
$967$	−32699.8	−1.08744	−0.543720	−	0.839267i	$-0.682985\pi$
$967$	−32699.8	−1.08744	−0.543720	+	0.839267i	$0.682985\pi$
$968$	2800.05	0.0929721
$969$	0	0
$970$	0	0
$971$	40436.6	1.33643	0.668214	−	0.743969i	$-0.267059\pi$
$971$	40436.6	1.33643	0.668214	+	0.743969i	$0.267059\pi$
$972$	0	0
$973$	−36342.0	−1.19740
$974$	−32497.1	−1.06907
$975$	0	0
$976$	−30261.7	−0.992473
$977$	−3686.09	−0.120705	−0.0603524	−	0.998177i	$-0.519222\pi$
$977$	−3686.09	−0.120705	−0.0603524	+	0.998177i	$0.519222\pi$
$978$	0	0
$979$	1314.97	0.0429280
$980$	0	0
$981$	0	0
$982$	8184.60	0.265969
$983$	−4059.64	−0.131722	−0.0658608	−	0.997829i	$-0.520979\pi$
$983$	−4059.64	−0.131722	−0.0658608	+	0.997829i	$0.520979\pi$
$984$	0	0
$985$	0	0
$986$	−2935.68	−0.0948187
$987$	0	0
$988$	1598.08	0.0514592
$989$	1369.27	0.0440246
$990$	0	0
$991$	34662.0	1.11107	0.555537	−	0.831492i	$-0.312513\pi$
$991$	34662.0	1.11107	0.555537	+	0.831492i	$0.312513\pi$
$992$	−5803.28	−0.185740
$993$	0	0
$994$	9780.77	0.312100
$995$	0	0
$996$	0	0
$997$	−54915.9	−1.74444	−0.872219	−	0.489115i	$-0.837320\pi$
$997$	−54915.9	−1.74444	−0.872219	+	0.489115i	$0.837320\pi$
$998$	−7440.26	−0.235989
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bl.1.2		5
3.2	odd	2		275.4.a.g.1.4	✓	5
5.4	even	2		2475.4.a.bh.1.4		5
15.2	even	4		275.4.b.f.199.7		10
15.8	even	4		275.4.b.f.199.4		10
15.14	odd	2		275.4.a.h.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.g.1.4	✓	5	3.2	odd	2
275.4.a.h.1.2	yes	5	15.14	odd	2
275.4.b.f.199.4		10	15.8	even	4
275.4.b.f.199.7		10	15.2	even	4
2475.4.a.bh.1.4		5	5.4	even	2
2475.4.a.bl.1.2		5	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75920 q^{2} -0.386826 q^{4} -13.2555 q^{7} +23.1409 q^{8} +O(q^{10})\)	\(q-2.75920 q^{2} -0.386826 q^{4} -13.2555 q^{7} +23.1409 q^{8} +11.0000 q^{11} +30.6714 q^{13} +36.5746 q^{14} -60.7558 q^{16} +35.1080 q^{17} -134.694 q^{19} -30.3512 q^{22} -6.51724 q^{23} -84.6285 q^{26} +5.12758 q^{28} +30.3054 q^{29} +331.803 q^{31} -17.4901 q^{32} -96.8700 q^{34} -172.426 q^{37} +371.649 q^{38} -58.1492 q^{41} -210.100 q^{43} -4.25508 q^{44} +17.9823 q^{46} -151.570 q^{47} -167.291 q^{49} -11.8645 q^{52} +318.384 q^{53} -306.745 q^{56} -83.6186 q^{58} -414.677 q^{59} +498.088 q^{61} -915.511 q^{62} +534.305 q^{64} -639.464 q^{67} -13.5807 q^{68} +267.420 q^{71} -540.297 q^{73} +475.756 q^{74} +52.1033 q^{76} -145.811 q^{77} +691.157 q^{79} +160.445 q^{82} +663.244 q^{83} +579.709 q^{86} +254.550 q^{88} +119.542 q^{89} -406.565 q^{91} +2.52103 q^{92} +418.211 q^{94} +284.049 q^{97} +461.589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 40 q^{4} - 40 q^{7} + 21 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 40 * q^4 - 40 * q^7 + 21 * q^8	\( 5 q + 2 q^{2} + 40 q^{4} - 40 q^{7} + 21 q^{8} + 55 q^{11} - 211 q^{13} + 133 q^{14} + 208 q^{16} + 72 q^{17} + 23 q^{19} + 22 q^{22} + 57 q^{23} - 322 q^{26} - 1050 q^{28} + 183 q^{29} - q^{31} - 430 q^{32} - 650 q^{34} - 856 q^{37} + 348 q^{38} - 94 q^{41} - 497 q^{43} + 440 q^{44} + 1006 q^{46} + 26 q^{47} + 227 q^{49} - 2086 q^{52} + 264 q^{53} + 835 q^{56} - 843 q^{58} - 1174 q^{59} + 640 q^{61} + 1454 q^{62} - 949 q^{64} - 98 q^{67} + 1767 q^{68} + 673 q^{71} - 1462 q^{73} + 561 q^{74} + 1248 q^{76} - 440 q^{77} + 104 q^{79} + 221 q^{82} - 747 q^{83} - 2497 q^{86} + 231 q^{88} - 2821 q^{89} + 480 q^{91} - 1503 q^{92} - 1787 q^{94} - 615 q^{97} - 5586 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 40 * q^4 - 40 * q^7 + 21 * q^8 + 55 * q^11 - 211 * q^13 + 133 * q^14 + 208 * q^16 + 72 * q^17 + 23 * q^19 + 22 * q^22 + 57 * q^23 - 322 * q^26 - 1050 * q^28 + 183 * q^29 - q^31 - 430 * q^32 - 650 * q^34 - 856 * q^37 + 348 * q^38 - 94 * q^41 - 497 * q^43 + 440 * q^44 + 1006 * q^46 + 26 * q^47 + 227 * q^49 - 2086 * q^52 + 264 * q^53 + 835 * q^56 - 843 * q^58 - 1174 * q^59 + 640 * q^61 + 1454 * q^62 - 949 * q^64 - 98 * q^67 + 1767 * q^68 + 673 * q^71 - 1462 * q^73 + 561 * q^74 + 1248 * q^76 - 440 * q^77 + 104 * q^79 + 221 * q^82 - 747 * q^83 - 2497 * q^86 + 231 * q^88 - 2821 * q^89 + 480 * q^91 - 1503 * q^92 - 1787 * q^94 - 615 * q^97 - 5586 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more