LMFDB - Embedded newform 2475.4.a.s.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:	$3$
Coefficient field:	3.3.23612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32906$ 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)

$f(q)$	$=$	$q-2.32906 q^{2} -2.57547 q^{4} -22.4672 q^{7} +24.6309 q^{8} +O(q^{10})$	$q-2.32906 q^{2} -2.57547 q^{4} -22.4672 q^{7} +24.6309 q^{8} -11.0000 q^{11} +9.86030 q^{13} +52.3275 q^{14} -36.7633 q^{16} -128.137 q^{17} +7.04001 q^{19} +25.6197 q^{22} +0.654969 q^{23} -22.9653 q^{26} +57.8635 q^{28} +229.279 q^{29} +155.789 q^{31} -111.423 q^{32} +298.438 q^{34} +110.279 q^{37} -16.3966 q^{38} -154.749 q^{41} +401.014 q^{43} +28.3301 q^{44} -1.52546 q^{46} -277.532 q^{47} +161.774 q^{49} -25.3949 q^{52} -651.566 q^{53} -553.388 q^{56} -534.005 q^{58} +423.869 q^{59} +681.851 q^{61} -362.842 q^{62} +553.618 q^{64} -374.028 q^{67} +330.011 q^{68} -96.6950 q^{71} +19.9460 q^{73} -256.848 q^{74} -18.1313 q^{76} +247.139 q^{77} +24.4286 q^{79} +360.419 q^{82} -1127.35 q^{83} -933.987 q^{86} -270.940 q^{88} +639.624 q^{89} -221.533 q^{91} -1.68685 q^{92} +646.389 q^{94} +730.865 q^{97} -376.783 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} + 22 q^{4} + 4 q^{7} - 48 q^{8}+O(q^{10}) $ 3 * q - 4 * q^2 + 22 * q^4 + 4 * q^7 - 48 * q^8	$ 3 q - 4 q^{2} + 22 q^{4} + 4 q^{7} - 48 q^{8} - 33 q^{11} + 56 q^{14} + 50 q^{16} - 218 q^{17} + 146 q^{19} + 44 q^{22} - 200 q^{23} + 508 q^{26} + 340 q^{28} - 68 q^{29} - 68 q^{31} - 688 q^{32} - 176 q^{34} + 390 q^{37} - 316 q^{38} + 196 q^{41} + 524 q^{43} - 242 q^{44} + 1160 q^{46} - 60 q^{47} - 157 q^{49} - 1020 q^{52} - 158 q^{53} - 1368 q^{56} - 1092 q^{58} + 1044 q^{59} + 642 q^{61} + 88 q^{62} + 1166 q^{64} + 236 q^{67} + 144 q^{68} + 544 q^{71} - 900 q^{73} - 1536 q^{74} + 1996 q^{76} - 44 q^{77} - 1586 q^{79} - 380 q^{82} - 1582 q^{83} - 3568 q^{86} + 528 q^{88} + 2122 q^{89} - 8 q^{91} - 4128 q^{92} - 2152 q^{94} - 618 q^{97} + 572 q^{98}+O(q^{100}) $ 3 * q - 4 * q^2 + 22 * q^4 + 4 * q^7 - 48 * q^8 - 33 * q^11 + 56 * q^14 + 50 * q^16 - 218 * q^17 + 146 * q^19 + 44 * q^22 - 200 * q^23 + 508 * q^26 + 340 * q^28 - 68 * q^29 - 68 * q^31 - 688 * q^32 - 176 * q^34 + 390 * q^37 - 316 * q^38 + 196 * q^41 + 524 * q^43 - 242 * q^44 + 1160 * q^46 - 60 * q^47 - 157 * q^49 - 1020 * q^52 - 158 * q^53 - 1368 * q^56 - 1092 * q^58 + 1044 * q^59 + 642 * q^61 + 88 * q^62 + 1166 * q^64 + 236 * q^67 + 144 * q^68 + 544 * q^71 - 900 * q^73 - 1536 * q^74 + 1996 * q^76 - 44 * q^77 - 1586 * q^79 - 380 * q^82 - 1582 * q^83 - 3568 * q^86 + 528 * q^88 + 2122 * q^89 - 8 * q^91 - 4128 * q^92 - 2152 * q^94 - 618 * q^97 + 572 * 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 $

$2$	−2.32906	−0.823448	−0.411724	−	0.911309i	$-0.635073\pi$
$2$	−2.32906	−0.823448	−0.411724	+	0.911309i	$0.635073\pi$
$3$	0	0
$3$	0	0
$4$	−2.57547	−0.321933
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−22.4672	−1.21311	−0.606557	−	0.795040i	$-0.707450\pi$
$7$	−22.4672	−1.21311	−0.606557	+	0.795040i	$0.707450\pi$
$8$	24.6309	1.08854
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	9.86030	0.210366	0.105183	−	0.994453i	$-0.466457\pi$
$13$	9.86030	0.210366	0.105183	+	0.994453i	$0.466457\pi$
$14$	52.3275	0.998936
$15$	0	0
$16$	−36.7633	−0.574426
$17$	−128.137	−1.82810	−0.914049	−	0.405603i	$-0.867062\pi$
$17$	−128.137	−1.82810	−0.914049	+	0.405603i	$0.867062\pi$
$18$	0	0
$19$	7.04001	0.0850047	0.0425024	−	0.999096i	$-0.486467\pi$
$19$	7.04001	0.0850047	0.0425024	+	0.999096i	$0.486467\pi$
$20$	0	0
$21$	0	0
$22$	25.6197	0.248279
$23$	0.654969	0.00593785	0.00296892	−	0.999996i	$-0.499055\pi$
$23$	0.654969	0.00593785	0.00296892	+	0.999996i	$0.499055\pi$
$24$	0	0
$25$	0	0
$26$	−22.9653	−0.173225
$27$	0	0
$28$	57.8635	0.390542
$29$	229.279	1.46814	0.734069	−	0.679075i	$-0.237619\pi$
$29$	229.279	1.46814	0.734069	+	0.679075i	$0.237619\pi$
$30$	0	0
$31$	155.789	0.902596	0.451298	−	0.892373i	$-0.350961\pi$
$31$	155.789	0.902596	0.451298	+	0.892373i	$0.350961\pi$
$32$	−111.423	−0.615534
$33$	0	0
$34$	298.438	1.50534
$35$	0	0
$36$	0	0
$37$	110.279	0.489995	0.244998	−	0.969524i	$-0.421213\pi$
$37$	110.279	0.489995	0.244998	+	0.969524i	$0.421213\pi$
$38$	−16.3966	−0.0699970
$39$	0	0
$40$	0	0
$41$	−154.749	−0.589456	−0.294728	−	0.955581i	$-0.595229\pi$
$41$	−154.749	−0.589456	−0.294728	+	0.955581i	$0.595229\pi$
$42$	0	0
$43$	401.014	1.42219	0.711094	−	0.703097i	$-0.248200\pi$
$43$	401.014	1.42219	0.711094	+	0.703097i	$0.248200\pi$
$44$	28.3301	0.0970665
$45$	0	0
$46$	−1.52546	−0.00488951
$47$	−277.532	−0.861323	−0.430661	−	0.902514i	$-0.641720\pi$
$47$	−277.532	−0.861323	−0.430661	+	0.902514i	$0.641720\pi$
$48$	0	0
$49$	161.774	0.471645
$50$	0	0
$51$	0	0
$52$	−25.3949	−0.0677237
$53$	−651.566	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$53$	−651.566	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$54$	0	0
$55$	0	0
$56$	−553.388	−1.32053
$57$	0	0
$58$	−534.005	−1.20894
$59$	423.869	0.935307	0.467653	−	0.883912i	$-0.345100\pi$
$59$	423.869	0.935307	0.467653	+	0.883912i	$0.345100\pi$
$60$	0	0
$61$	681.851	1.43118	0.715590	−	0.698520i	$-0.246158\pi$
$61$	681.851	1.43118	0.715590	+	0.698520i	$0.246158\pi$
$62$	−362.842	−0.743241
$63$	0	0
$64$	553.618	1.08129
$65$	0	0
$66$	0	0
$67$	−374.028	−0.682012	−0.341006	−	0.940061i	$-0.610768\pi$
$67$	−374.028	−0.682012	−0.341006	+	0.940061i	$0.610768\pi$
$68$	330.011	0.588526
$69$	0	0
$70$	0	0
$71$	−96.6950	−0.161628	−0.0808140	−	0.996729i	$-0.525752\pi$
$71$	−96.6950	−0.161628	−0.0808140	+	0.996729i	$0.525752\pi$
$72$	0	0
$73$	19.9460	0.0319795	0.0159897	−	0.999872i	$-0.494910\pi$
$73$	19.9460	0.0319795	0.0159897	+	0.999872i	$0.494910\pi$
$74$	−256.848	−0.403486
$75$	0	0
$76$	−18.1313	−0.0273658
$77$	247.139	0.365768
$78$	0	0
$79$	24.4286	0.0347903	0.0173951	−	0.999849i	$-0.494463\pi$
$79$	24.4286	0.0347903	0.0173951	+	0.999849i	$0.494463\pi$
$80$	0	0
$81$	0	0
$82$	360.419	0.485386
$83$	−1127.35	−1.49088	−0.745439	−	0.666574i	$-0.767760\pi$
$83$	−1127.35	−1.49088	−0.745439	+	0.666574i	$0.767760\pi$
$84$	0	0
$85$	0	0
$86$	−933.987	−1.17110
$87$	0	0
$88$	−270.940	−0.328208
$89$	639.624	0.761798	0.380899	−	0.924617i	$-0.375615\pi$
$89$	639.624	0.761798	0.380899	+	0.924617i	$0.375615\pi$
$90$	0	0
$91$	−221.533	−0.255198
$92$	−1.68685	−0.00191159
$93$	0	0
$94$	646.389	0.709255
$95$	0	0
$96$	0	0
$97$	730.865	0.765032	0.382516	−	0.923949i	$-0.375058\pi$
$97$	730.865	0.765032	0.382516	+	0.923949i	$0.375058\pi$
$98$	−376.783	−0.388375
$99$	0	0
$100$	0	0
$101$	810.342	0.798337	0.399168	−	0.916878i	$-0.369299\pi$
$101$	810.342	0.798337	0.399168	+	0.916878i	$0.369299\pi$
$102$	0	0
$103$	1461.89	1.39849	0.699245	−	0.714882i	$-0.253520\pi$
$103$	1461.89	1.39849	0.699245	+	0.714882i	$0.253520\pi$
$104$	242.868	0.228992
$105$	0	0
$106$	1517.54	1.39053
$107$	1690.40	1.52726	0.763630	−	0.645654i	$-0.223415\pi$
$107$	1690.40	1.52726	0.763630	+	0.645654i	$0.223415\pi$
$108$	0	0
$109$	−1409.41	−1.23851	−0.619254	−	0.785190i	$-0.712565\pi$
$109$	−1409.41	−1.23851	−0.619254	+	0.785190i	$0.712565\pi$
$110$	0	0
$111$	0	0
$112$	825.967	0.696844
$113$	2185.67	1.81956	0.909780	−	0.415090i	$-0.136250\pi$
$113$	2185.67	1.81956	0.909780	+	0.415090i	$0.136250\pi$
$114$	0	0
$115$	0	0
$116$	−590.499	−0.472642
$117$	0	0
$118$	−987.219	−0.770177
$119$	2878.87	2.21769
$120$	0	0
$121$	121.000	0.0909091
$122$	−1588.07	−1.17850
$123$	0	0
$124$	−401.228	−0.290576
$125$	0	0
$126$	0	0
$127$	1918.85	1.34071	0.670357	−	0.742038i	$-0.266141\pi$
$127$	1918.85	1.34071	0.670357	+	0.742038i	$0.266141\pi$
$128$	−398.024	−0.274849
$129$	0	0
$130$	0	0
$131$	−1339.41	−0.893320	−0.446660	−	0.894704i	$-0.647387\pi$
$131$	−1339.41	−0.893320	−0.446660	+	0.894704i	$0.647387\pi$
$132$	0	0
$133$	−158.169	−0.103120
$134$	871.135	0.561602
$135$	0	0
$136$	−3156.12	−1.98996
$137$	−1100.56	−0.686330	−0.343165	−	0.939275i	$-0.611499\pi$
$137$	−1100.56	−0.686330	−0.343165	+	0.939275i	$0.611499\pi$
$138$	0	0
$139$	−1284.51	−0.783819	−0.391910	−	0.920004i	$-0.628185\pi$
$139$	−1284.51	−0.783819	−0.391910	+	0.920004i	$0.628185\pi$
$140$	0	0
$141$	0	0
$142$	225.209	0.133092
$143$	−108.463	−0.0634277
$144$	0	0
$145$	0	0
$146$	−46.4554	−0.0263334
$147$	0	0
$148$	−284.021	−0.157746
$149$	−1277.21	−0.702236	−0.351118	−	0.936331i	$-0.614198\pi$
$149$	−1277.21	−0.702236	−0.351118	+	0.936331i	$0.614198\pi$
$150$	0	0
$151$	886.317	0.477665	0.238833	−	0.971061i	$-0.423235\pi$
$151$	886.317	0.477665	0.238833	+	0.971061i	$0.423235\pi$
$152$	173.402	0.0925313
$153$	0	0
$154$	−575.602	−0.301191
$155$	0	0
$156$	0	0
$157$	1681.12	0.854575	0.427288	−	0.904116i	$-0.359469\pi$
$157$	1681.12	0.854575	0.427288	+	0.904116i	$0.359469\pi$
$158$	−56.8958	−0.0286480
$159$	0	0
$160$	0	0
$161$	−14.7153	−0.00720329
$162$	0	0
$163$	622.100	0.298937	0.149468	−	0.988767i	$-0.452244\pi$
$163$	622.100	0.298937	0.149468	+	0.988767i	$0.452244\pi$
$164$	398.550	0.189765
$165$	0	0
$166$	2625.67	1.22766
$167$	−2611.82	−1.21023	−0.605115	−	0.796138i	$-0.706873\pi$
$167$	−2611.82	−1.21023	−0.605115	+	0.796138i	$0.706873\pi$
$168$	0	0
$169$	−2099.77	−0.955746
$170$	0	0
$171$	0	0
$172$	−1032.80	−0.457849
$173$	2342.97	1.02967	0.514835	−	0.857290i	$-0.327853\pi$
$173$	2342.97	1.02967	0.514835	+	0.857290i	$0.327853\pi$
$174$	0	0
$175$	0	0
$176$	404.396	0.173196
$177$	0	0
$178$	−1489.72	−0.627301
$179$	−1314.75	−0.548991	−0.274495	−	0.961588i	$-0.588511\pi$
$179$	−1314.75	−0.548991	−0.274495	+	0.961588i	$0.588511\pi$
$180$	0	0
$181$	8.69006	0.00356866	0.00178433	−	0.999998i	$-0.499432\pi$
$181$	8.69006	0.00356866	0.00178433	+	0.999998i	$0.499432\pi$
$182$	515.965	0.210142
$183$	0	0
$184$	16.1325	0.00646361
$185$	0	0
$186$	0	0
$187$	1409.50	0.551192
$188$	714.774	0.277288
$189$	0	0
$190$	0	0
$191$	−644.102	−0.244008	−0.122004	−	0.992530i	$-0.538932\pi$
$191$	−644.102	−0.244008	−0.122004	+	0.992530i	$0.538932\pi$
$192$	0	0
$193$	−3970.76	−1.48094	−0.740470	−	0.672089i	$-0.765397\pi$
$193$	−3970.76	−1.48094	−0.740470	+	0.672089i	$0.765397\pi$
$194$	−1702.23	−0.629964
$195$	0	0
$196$	−416.644	−0.151838
$197$	3756.34	1.35852	0.679260	−	0.733898i	$-0.262301\pi$
$197$	3756.34	1.35852	0.679260	+	0.733898i	$0.262301\pi$
$198$	0	0
$199$	4825.48	1.71894	0.859470	−	0.511186i	$-0.170794\pi$
$199$	4825.48	1.71894	0.859470	+	0.511186i	$0.170794\pi$
$200$	0	0
$201$	0	0
$202$	−1887.34	−0.657389
$203$	−5151.25	−1.78102
$204$	0	0
$205$	0	0
$206$	−3404.84	−1.15158
$207$	0	0
$208$	−362.497	−0.120840
$209$	−77.4401	−0.0256299
$210$	0	0
$211$	−4394.02	−1.43363	−0.716817	−	0.697261i	$-0.754402\pi$
$211$	−4394.02	−1.43363	−0.716817	+	0.697261i	$0.754402\pi$
$212$	1678.08	0.543638
$213$	0	0
$214$	−3937.04	−1.25762
$215$	0	0
$216$	0	0
$217$	−3500.13	−1.09495
$218$	3282.62	1.01985
$219$	0	0
$220$	0	0
$221$	−1263.46	−0.384569
$222$	0	0
$223$	−2189.67	−0.657538	−0.328769	−	0.944410i	$-0.606634\pi$
$223$	−2189.67	−0.657538	−0.328769	+	0.944410i	$0.606634\pi$
$224$	2503.37	0.746712
$225$	0	0
$226$	−5090.56	−1.49831
$227$	−1139.27	−0.333110	−0.166555	−	0.986032i	$-0.553264\pi$
$227$	−1139.27	−0.333110	−0.166555	+	0.986032i	$0.553264\pi$
$228$	0	0
$229$	3416.10	0.985773	0.492886	−	0.870094i	$-0.335942\pi$
$229$	3416.10	0.985773	0.492886	+	0.870094i	$0.335942\pi$
$230$	0	0
$231$	0	0
$232$	5647.35	1.59813
$233$	6147.08	1.72836	0.864181	−	0.503181i	$-0.167837\pi$
$233$	6147.08	1.72836	0.864181	+	0.503181i	$0.167837\pi$
$234$	0	0
$235$	0	0
$236$	−1091.66	−0.301106
$237$	0	0
$238$	−6705.06	−1.82615
$239$	2080.03	0.562954	0.281477	−	0.959568i	$-0.409176\pi$
$239$	2080.03	0.562954	0.281477	+	0.959568i	$0.409176\pi$
$240$	0	0
$241$	1846.28	0.493484	0.246742	−	0.969081i	$-0.420640\pi$
$241$	1846.28	0.493484	0.246742	+	0.969081i	$0.420640\pi$
$242$	−281.817	−0.0748589
$243$	0	0
$244$	−1756.08	−0.460745
$245$	0	0
$246$	0	0
$247$	69.4166	0.0178821
$248$	3837.22	0.982515
$249$	0	0
$250$	0	0
$251$	2555.32	0.642592	0.321296	−	0.946979i	$-0.395882\pi$
$251$	2555.32	0.642592	0.321296	+	0.946979i	$0.395882\pi$
$252$	0	0
$253$	−7.20466	−0.00179033
$254$	−4469.13	−1.10401
$255$	0	0
$256$	−3501.92	−0.854962
$257$	−1819.39	−0.441598	−0.220799	−	0.975319i	$-0.570866\pi$
$257$	−1819.39	−0.441598	−0.220799	+	0.975319i	$0.570866\pi$
$258$	0	0
$259$	−2477.67	−0.594420
$260$	0	0
$261$	0	0
$262$	3119.57	0.735603
$263$	−6023.03	−1.41215	−0.706076	−	0.708136i	$-0.749536\pi$
$263$	−6023.03	−1.41215	−0.706076	+	0.708136i	$0.749536\pi$
$264$	0	0
$265$	0	0
$266$	368.386	0.0849143
$267$	0	0
$268$	963.297	0.219562
$269$	2978.38	0.675075	0.337537	−	0.941312i	$-0.390406\pi$
$269$	2978.38	0.675075	0.337537	+	0.941312i	$0.390406\pi$
$270$	0	0
$271$	−524.969	−0.117674	−0.0588369	−	0.998268i	$-0.518739\pi$
$271$	−524.969	−0.117674	−0.0588369	+	0.998268i	$0.518739\pi$
$272$	4710.72	1.05011
$273$	0	0
$274$	2563.28	0.565157
$275$	0	0
$276$	0	0
$277$	1693.07	0.367245	0.183623	−	0.982997i	$-0.441218\pi$
$277$	1693.07	0.367245	0.183623	+	0.982997i	$0.441218\pi$
$278$	2991.71	0.645435
$279$	0	0
$280$	0	0
$281$	−7346.60	−1.55965	−0.779824	−	0.625998i	$-0.784692\pi$
$281$	−7346.60	−1.55965	−0.779824	+	0.625998i	$0.784692\pi$
$282$	0	0
$283$	1501.69	0.315429	0.157714	−	0.987485i	$-0.449587\pi$
$283$	1501.69	0.315429	0.157714	+	0.987485i	$0.449587\pi$
$284$	249.035	0.0520334
$285$	0	0
$286$	252.618	0.0522294
$287$	3476.77	0.715077
$288$	0	0
$289$	11506.0	2.34194
$290$	0	0
$291$	0	0
$292$	−51.3702	−0.0102952
$293$	−4481.03	−0.893462	−0.446731	−	0.894668i	$-0.647412\pi$
$293$	−4481.03	−0.893462	−0.446731	+	0.894668i	$0.647412\pi$
$294$	0	0
$295$	0	0
$296$	2716.28	0.533381
$297$	0	0
$298$	2974.71	0.578255
$299$	6.45819	0.00124912
$300$	0	0
$301$	−9009.66	−1.72528
$302$	−2064.29	−0.393333
$303$	0	0
$304$	−258.814	−0.0488289
$305$	0	0
$306$	0	0
$307$	−3052.17	−0.567416	−0.283708	−	0.958911i	$-0.591565\pi$
$307$	−3052.17	−0.567416	−0.283708	+	0.958911i	$0.591565\pi$
$308$	−636.498	−0.117753
$309$	0	0
$310$	0	0
$311$	−10255.1	−1.86983	−0.934913	−	0.354878i	$-0.884523\pi$
$311$	−10255.1	−1.86983	−0.934913	+	0.354878i	$0.884523\pi$
$312$	0	0
$313$	6190.18	1.11786	0.558929	−	0.829215i	$-0.311212\pi$
$313$	6190.18	1.11786	0.558929	+	0.829215i	$0.311212\pi$
$314$	−3915.44	−0.703698
$315$	0	0
$316$	−62.9150	−0.0112001
$317$	−6735.38	−1.19337	−0.596683	−	0.802477i	$-0.703515\pi$
$317$	−6735.38	−1.19337	−0.596683	+	0.802477i	$0.703515\pi$
$318$	0	0
$319$	−2522.07	−0.442660
$320$	0	0
$321$	0	0
$322$	34.2729	0.00593153
$323$	−902.083	−0.155397
$324$	0	0
$325$	0	0
$326$	−1448.91	−0.246159
$327$	0	0
$328$	−3811.60	−0.641648
$329$	6235.36	1.04488
$330$	0	0
$331$	−4780.83	−0.793891	−0.396946	−	0.917842i	$-0.629930\pi$
$331$	−4780.83	−0.793891	−0.396946	+	0.917842i	$0.629930\pi$
$332$	2903.45	0.479963
$333$	0	0
$334$	6083.09	0.996562
$335$	0	0
$336$	0	0
$337$	−11890.3	−1.92197	−0.960984	−	0.276604i	$-0.910791\pi$
$337$	−11890.3	−1.92197	−0.960984	+	0.276604i	$0.910791\pi$
$338$	4890.51	0.787007
$339$	0	0
$340$	0	0
$341$	−1713.68	−0.272143
$342$	0	0
$343$	4071.63	0.640954
$344$	9877.35	1.54811
$345$	0	0
$346$	−5456.93	−0.847879
$347$	8462.47	1.30919	0.654595	−	0.755979i	$-0.272839\pi$
$347$	8462.47	1.30919	0.654595	+	0.755979i	$0.272839\pi$
$348$	0	0
$349$	−3291.90	−0.504903	−0.252452	−	0.967610i	$-0.581237\pi$
$349$	−3291.90	−0.504903	−0.252452	+	0.967610i	$0.581237\pi$
$350$	0	0
$351$	0	0
$352$	1225.66	0.185590
$353$	−8193.52	−1.23540	−0.617701	−	0.786413i	$-0.711936\pi$
$353$	−8193.52	−1.23540	−0.617701	+	0.786413i	$0.711936\pi$
$354$	0	0
$355$	0	0
$356$	−1647.33	−0.245248
$357$	0	0
$358$	3062.15	0.452066
$359$	−12817.6	−1.88437	−0.942185	−	0.335093i	$-0.891232\pi$
$359$	−12817.6	−1.88437	−0.942185	+	0.335093i	$0.891232\pi$
$360$	0	0
$361$	−6809.44	−0.992774
$362$	−20.2397	−0.00293861
$363$	0	0
$364$	570.551	0.0821566
$365$	0	0
$366$	0	0
$367$	2801.22	0.398427	0.199213	−	0.979956i	$-0.436161\pi$
$367$	2801.22	0.398427	0.199213	+	0.979956i	$0.436161\pi$
$368$	−24.0788	−0.00341085
$369$	0	0
$370$	0	0
$371$	14638.8	2.04855
$372$	0	0
$373$	−6838.03	−0.949222	−0.474611	−	0.880196i	$-0.657411\pi$
$373$	−6838.03	−0.949222	−0.474611	+	0.880196i	$0.657411\pi$
$374$	−3282.82	−0.453878
$375$	0	0
$376$	−6835.86	−0.937587
$377$	2260.76	0.308846
$378$	0	0
$379$	−7465.79	−1.01185	−0.505926	−	0.862577i	$-0.668849\pi$
$379$	−7465.79	−1.01185	−0.505926	+	0.862577i	$0.668849\pi$
$380$	0	0
$381$	0	0
$382$	1500.16	0.200928
$383$	−8646.55	−1.15357	−0.576786	−	0.816895i	$-0.695693\pi$
$383$	−8646.55	−1.15357	−0.576786	+	0.816895i	$0.695693\pi$
$384$	0	0
$385$	0	0
$386$	9248.15	1.21948
$387$	0	0
$388$	−1882.32	−0.246289
$389$	−4382.78	−0.571248	−0.285624	−	0.958342i	$-0.592201\pi$
$389$	−4382.78	−0.571248	−0.285624	+	0.958342i	$0.592201\pi$
$390$	0	0
$391$	−83.9255	−0.0108550
$392$	3984.65	0.513406
$393$	0	0
$394$	−8748.76	−1.11867
$395$	0	0
$396$	0	0
$397$	4432.58	0.560365	0.280182	−	0.959947i	$-0.409605\pi$
$397$	4432.58	0.560365	0.280182	+	0.959947i	$0.409605\pi$
$398$	−11238.8	−1.41546
$399$	0	0
$400$	0	0
$401$	5034.93	0.627013	0.313507	−	0.949586i	$-0.398496\pi$
$401$	5034.93	0.627013	0.313507	+	0.949586i	$0.398496\pi$
$402$	0	0
$403$	1536.12	0.189875
$404$	−2087.01	−0.257011
$405$	0	0
$406$	11997.6	1.46658
$407$	−1213.07	−0.147739
$408$	0	0
$409$	6474.64	0.782764	0.391382	−	0.920228i	$-0.371997\pi$
$409$	6474.64	0.782764	0.391382	+	0.920228i	$0.371997\pi$
$410$	0	0
$411$	0	0
$412$	−3765.05	−0.450220
$413$	−9523.15	−1.13463
$414$	0	0
$415$	0	0
$416$	−1098.67	−0.129487
$417$	0	0
$418$	180.363	0.0211049
$419$	8257.80	0.962816	0.481408	−	0.876497i	$-0.340126\pi$
$419$	8257.80	0.962816	0.481408	+	0.876497i	$0.340126\pi$
$420$	0	0
$421$	−3429.36	−0.397000	−0.198500	−	0.980101i	$-0.563607\pi$
$421$	−3429.36	−0.397000	−0.198500	+	0.980101i	$0.563607\pi$
$422$	10233.9	1.18052
$423$	0	0
$424$	−16048.7	−1.83819
$425$	0	0
$426$	0	0
$427$	−15319.3	−1.73618
$428$	−4353.56	−0.491676
$429$	0	0
$430$	0	0
$431$	11260.4	1.25846	0.629230	−	0.777219i	$-0.283370\pi$
$431$	11260.4	1.25846	0.629230	+	0.777219i	$0.283370\pi$
$432$	0	0
$433$	−12598.5	−1.39826	−0.699128	−	0.714996i	$-0.746428\pi$
$433$	−12598.5	−1.39826	−0.699128	+	0.714996i	$0.746428\pi$
$434$	8152.03	0.901636
$435$	0	0
$436$	3629.90	0.398717
$437$	4.61099	0.000504745 0
$438$	0	0
$439$	4176.90	0.454106	0.227053	−	0.973882i	$-0.427091\pi$
$439$	4176.90	0.454106	0.227053	+	0.973882i	$0.427091\pi$
$440$	0	0
$441$	0	0
$442$	2942.69	0.316673
$443$	2354.16	0.252482	0.126241	−	0.992000i	$-0.459709\pi$
$443$	2354.16	0.252482	0.126241	+	0.992000i	$0.459709\pi$
$444$	0	0
$445$	0	0
$446$	5099.88	0.541449
$447$	0	0
$448$	−12438.2	−1.31172
$449$	−9286.18	−0.976040	−0.488020	−	0.872832i	$-0.662281\pi$
$449$	−9286.18	−0.976040	−0.488020	+	0.872832i	$0.662281\pi$
$450$	0	0
$451$	1702.24	0.177728
$452$	−5629.11	−0.585777
$453$	0	0
$454$	2653.43	0.274299
$455$	0	0
$456$	0	0
$457$	−14378.4	−1.47176	−0.735878	−	0.677115i	$-0.763230\pi$
$457$	−14378.4	−1.47176	−0.735878	+	0.677115i	$0.763230\pi$
$458$	−7956.30	−0.811733
$459$	0	0
$460$	0	0
$461$	5383.19	0.543861	0.271931	−	0.962317i	$-0.412338\pi$
$461$	5383.19	0.543861	0.271931	+	0.962317i	$0.412338\pi$
$462$	0	0
$463$	−18360.5	−1.84294	−0.921471	−	0.388446i	$-0.873012\pi$
$463$	−18360.5	−1.84294	−0.921471	+	0.388446i	$0.873012\pi$
$464$	−8429.03	−0.843336
$465$	0	0
$466$	−14316.9	−1.42322
$467$	9063.65	0.898106	0.449053	−	0.893505i	$-0.351761\pi$
$467$	9063.65	0.898106	0.449053	+	0.893505i	$0.351761\pi$
$468$	0	0
$469$	8403.36	0.827358
$470$	0	0
$471$	0	0
$472$	10440.3	1.01812
$473$	−4411.15	−0.428806
$474$	0	0
$475$	0	0
$476$	−7414.42	−0.713949
$477$	0	0
$478$	−4844.53	−0.463564
$479$	−12608.2	−1.20268	−0.601341	−	0.798992i	$-0.705367\pi$
$479$	−12608.2	−1.20268	−0.601341	+	0.798992i	$0.705367\pi$
$480$	0	0
$481$	1087.39	0.103078
$482$	−4300.11	−0.406358
$483$	0	0
$484$	−311.631	−0.0292667
$485$	0	0
$486$	0	0
$487$	13214.2	1.22955	0.614775	−	0.788703i	$-0.289247\pi$
$487$	13214.2	1.22955	0.614775	+	0.788703i	$0.289247\pi$
$488$	16794.6	1.55790
$489$	0	0
$490$	0	0
$491$	−6553.27	−0.602332	−0.301166	−	0.953572i	$-0.597376\pi$
$491$	−6553.27	−0.602332	−0.301166	+	0.953572i	$0.597376\pi$
$492$	0	0
$493$	−29379.0	−2.68390
$494$	−161.676	−0.0147250
$495$	0	0
$496$	−5727.30	−0.518474
$497$	2172.46	0.196073
$498$	0	0
$499$	−2596.63	−0.232948	−0.116474	−	0.993194i	$-0.537159\pi$
$499$	−2596.63	−0.232948	−0.116474	+	0.993194i	$0.537159\pi$
$500$	0	0
$501$	0	0
$502$	−5951.51	−0.529141
$503$	−659.714	−0.0584795	−0.0292398	−	0.999572i	$-0.509309\pi$
$503$	−659.714	−0.0584795	−0.0292398	+	0.999572i	$0.509309\pi$
$504$	0	0
$505$	0	0
$506$	16.7801	0.00147424
$507$	0	0
$508$	−4941.94	−0.431621
$509$	−4825.41	−0.420201	−0.210101	−	0.977680i	$-0.567379\pi$
$509$	−4825.41	−0.420201	−0.210101	+	0.977680i	$0.567379\pi$
$510$	0	0
$511$	−448.130	−0.0387947
$512$	11340.4	0.978866
$513$	0	0
$514$	4237.48	0.363633
$515$	0	0
$516$	0	0
$517$	3052.85	0.259699
$518$	5770.64	0.489474
$519$	0	0
$520$	0	0
$521$	2329.24	0.195866	0.0979328	−	0.995193i	$-0.468777\pi$
$521$	2329.24	0.195866	0.0979328	+	0.995193i	$0.468777\pi$
$522$	0	0
$523$	15104.6	1.26287	0.631434	−	0.775429i	$-0.282467\pi$
$523$	15104.6	1.26287	0.631434	+	0.775429i	$0.282467\pi$
$524$	3449.61	0.287589
$525$	0	0
$526$	14028.0	1.16283
$527$	−19962.2	−1.65003
$528$	0	0
$529$	−12166.6	−0.999965
$530$	0	0
$531$	0	0
$532$	407.359	0.0331979
$533$	−1525.87	−0.124001
$534$	0	0
$535$	0	0
$536$	−9212.66	−0.742400
$537$	0	0
$538$	−6936.84	−0.555889
$539$	−1779.52	−0.142206
$540$	0	0
$541$	10712.1	0.851293	0.425647	−	0.904889i	$-0.360047\pi$
$541$	10712.1	0.851293	0.425647	+	0.904889i	$0.360047\pi$
$542$	1222.69	0.0968983
$543$	0	0
$544$	14277.4	1.12526
$545$	0	0
$546$	0	0
$547$	17251.6	1.34849	0.674245	−	0.738508i	$-0.264469\pi$
$547$	17251.6	1.34849	0.674245	+	0.738508i	$0.264469\pi$
$548$	2834.46	0.220953
$549$	0	0
$550$	0	0
$551$	1614.12	0.124799
$552$	0	0
$553$	−548.842	−0.0422046
$554$	−3943.27	−0.302407
$555$	0	0
$556$	3308.22	0.252337
$557$	−8179.34	−0.622208	−0.311104	−	0.950376i	$-0.600699\pi$
$557$	−8179.34	−0.622208	−0.311104	+	0.950376i	$0.600699\pi$
$558$	0	0
$559$	3954.12	0.299180
$560$	0	0
$561$	0	0
$562$	17110.7	1.28429
$563$	4939.38	0.369752	0.184876	−	0.982762i	$-0.440812\pi$
$563$	4939.38	0.369752	0.184876	+	0.982762i	$0.440812\pi$
$564$	0	0
$565$	0	0
$566$	−3497.54	−0.259739
$567$	0	0
$568$	−2381.69	−0.175939
$569$	7658.76	0.564274	0.282137	−	0.959374i	$-0.408957\pi$
$569$	7658.76	0.564274	0.282137	+	0.959374i	$0.408957\pi$
$570$	0	0
$571$	−1744.16	−0.127830	−0.0639149	−	0.997955i	$-0.520359\pi$
$571$	−1744.16	−0.127830	−0.0639149	+	0.997955i	$0.520359\pi$
$572$	279.344	0.0204195
$573$	0	0
$574$	−8097.61	−0.588829
$575$	0	0
$576$	0	0
$577$	−1264.61	−0.0912414	−0.0456207	−	0.998959i	$-0.514527\pi$
$577$	−1264.61	−0.0912414	−0.0456207	+	0.998959i	$0.514527\pi$
$578$	−26798.1	−1.92847
$579$	0	0
$580$	0	0
$581$	25328.4	1.80860
$582$	0	0
$583$	7167.22	0.509153
$584$	491.288	0.0348110
$585$	0	0
$586$	10436.6	0.735720
$587$	−17167.4	−1.20711	−0.603557	−	0.797320i	$-0.706250\pi$
$587$	−17167.4	−1.20711	−0.603557	+	0.797320i	$0.706250\pi$
$588$	0	0
$589$	1096.75	0.0767249
$590$	0	0
$591$	0	0
$592$	−4054.23	−0.281466
$593$	−21429.5	−1.48399	−0.741995	−	0.670406i	$-0.766120\pi$
$593$	−21429.5	−1.48399	−0.741995	+	0.670406i	$0.766120\pi$
$594$	0	0
$595$	0	0
$596$	3289.41	0.226073
$597$	0	0
$598$	−15.0415	−0.00102859
$599$	7994.61	0.545327	0.272664	−	0.962109i	$-0.412095\pi$
$599$	7994.61	0.545327	0.272664	+	0.962109i	$0.412095\pi$
$600$	0	0
$601$	−24313.4	−1.65019	−0.825094	−	0.564996i	$-0.808878\pi$
$601$	−24313.4	−1.65019	−0.825094	+	0.564996i	$0.808878\pi$
$602$	20984.1	1.42067
$603$	0	0
$604$	−2282.68	−0.153776
$605$	0	0
$606$	0	0
$607$	−24569.7	−1.64292	−0.821460	−	0.570266i	$-0.806840\pi$
$607$	−24569.7	−1.64292	−0.821460	+	0.570266i	$0.806840\pi$
$608$	−784.423	−0.0523232
$609$	0	0
$610$	0	0
$611$	−2736.55	−0.181193
$612$	0	0
$613$	12746.7	0.839859	0.419929	−	0.907557i	$-0.362055\pi$
$613$	12746.7	0.839859	0.419929	+	0.907557i	$0.362055\pi$
$614$	7108.70	0.467237
$615$	0	0
$616$	6087.26	0.398154
$617$	−15607.4	−1.01837	−0.509183	−	0.860658i	$-0.670052\pi$
$617$	−15607.4	−1.01837	−0.509183	+	0.860658i	$0.670052\pi$
$618$	0	0
$619$	−11909.7	−0.773329	−0.386665	−	0.922220i	$-0.626373\pi$
$619$	−11909.7	−0.773329	−0.386665	+	0.922220i	$0.626373\pi$
$620$	0	0
$621$	0	0
$622$	23884.9	1.53970
$623$	−14370.5	−0.924147
$624$	0	0
$625$	0	0
$626$	−14417.3	−0.920499
$627$	0	0
$628$	−4329.68	−0.275116
$629$	−14130.8	−0.895759
$630$	0	0
$631$	−19304.2	−1.21789	−0.608946	−	0.793212i	$-0.708407\pi$
$631$	−19304.2	−1.21789	−0.608946	+	0.793212i	$0.708407\pi$
$632$	601.699	0.0378707
$633$	0	0
$634$	15687.1	0.982674
$635$	0	0
$636$	0	0
$637$	1595.14	0.0992180
$638$	5874.05	0.364508
$639$	0	0
$640$	0	0
$641$	−26678.6	−1.64390	−0.821950	−	0.569560i	$-0.807114\pi$
$641$	−26678.6	−1.64390	−0.821950	+	0.569560i	$0.807114\pi$
$642$	0	0
$643$	26456.2	1.62260	0.811299	−	0.584631i	$-0.198761\pi$
$643$	26456.2	1.62260	0.811299	+	0.584631i	$0.198761\pi$
$644$	37.8988	0.00231898
$645$	0	0
$646$	2101.01	0.127961
$647$	−23523.7	−1.42939	−0.714694	−	0.699438i	$-0.753434\pi$
$647$	−23523.7	−1.42939	−0.714694	+	0.699438i	$0.753434\pi$
$648$	0	0
$649$	−4662.56	−0.282006
$650$	0	0
$651$	0	0
$652$	−1602.20	−0.0962376
$653$	18071.1	1.08296	0.541482	−	0.840712i	$-0.317863\pi$
$653$	18071.1	1.08296	0.541482	+	0.840712i	$0.317863\pi$
$654$	0	0
$655$	0	0
$656$	5689.07	0.338599
$657$	0	0
$658$	−14522.5	−0.860407
$659$	−17023.1	−1.00626	−0.503130	−	0.864210i	$-0.667818\pi$
$659$	−17023.1	−1.00626	−0.503130	+	0.864210i	$0.667818\pi$
$660$	0	0
$661$	2137.74	0.125792	0.0628959	−	0.998020i	$-0.479966\pi$
$661$	2137.74	0.125792	0.0628959	+	0.998020i	$0.479966\pi$
$662$	11134.8	0.653728
$663$	0	0
$664$	−27767.7	−1.62288
$665$	0	0
$666$	0	0
$667$	150.171	0.00871758
$668$	6726.65	0.389614
$669$	0	0
$670$	0	0
$671$	−7500.36	−0.431517
$672$	0	0
$673$	−31790.1	−1.82083	−0.910414	−	0.413698i	$-0.864237\pi$
$673$	−31790.1	−1.82083	−0.910414	+	0.413698i	$0.864237\pi$
$674$	27693.1	1.58264
$675$	0	0
$676$	5407.90	0.307686
$677$	−10225.1	−0.580476	−0.290238	−	0.956955i	$-0.593734\pi$
$677$	−10225.1	−0.580476	−0.290238	+	0.956955i	$0.593734\pi$
$678$	0	0
$679$	−16420.5	−0.928071
$680$	0	0
$681$	0	0
$682$	3991.26	0.224096
$683$	21274.0	1.19184	0.595919	−	0.803044i	$-0.296788\pi$
$683$	21274.0	1.19184	0.595919	+	0.803044i	$0.296788\pi$
$684$	0	0
$685$	0	0
$686$	−9483.08	−0.527793
$687$	0	0
$688$	−14742.6	−0.816941
$689$	−6424.63	−0.355238
$690$	0	0
$691$	−22568.1	−1.24245	−0.621224	−	0.783633i	$-0.713364\pi$
$691$	−22568.1	−1.24245	−0.621224	+	0.783633i	$0.713364\pi$
$692$	−6034.24	−0.331485
$693$	0	0
$694$	−19709.6	−1.07805
$695$	0	0
$696$	0	0
$697$	19829.0	1.07758
$698$	7667.03	0.415761
$699$	0	0
$700$	0	0
$701$	7735.03	0.416759	0.208380	−	0.978048i	$-0.433181\pi$
$701$	7735.03	0.416759	0.208380	+	0.978048i	$0.433181\pi$
$702$	0	0
$703$	776.368	0.0416519
$704$	−6089.80	−0.326020
$705$	0	0
$706$	19083.2	1.01729
$707$	−18206.1	−0.968473
$708$	0	0
$709$	35115.1	1.86005	0.930025	−	0.367497i	$-0.119785\pi$
$709$	35115.1	1.86005	0.930025	+	0.367497i	$0.119785\pi$
$710$	0	0
$711$	0	0
$712$	15754.5	0.829250
$713$	102.037	0.00535948
$714$	0	0
$715$	0	0
$716$	3386.11	0.176738
$717$	0	0
$718$	29853.1	1.55168
$719$	−15334.3	−0.795370	−0.397685	−	0.917522i	$-0.630186\pi$
$719$	−15334.3	−0.795370	−0.397685	+	0.917522i	$0.630186\pi$
$720$	0	0
$721$	−32844.6	−1.69653
$722$	15859.6	0.817498
$723$	0	0
$724$	−22.3810	−0.00114887
$725$	0	0
$726$	0	0
$727$	12360.4	0.630567	0.315284	−	0.948997i	$-0.397900\pi$
$727$	12360.4	0.630567	0.315284	+	0.948997i	$0.397900\pi$
$728$	−5456.57	−0.277794
$729$	0	0
$730$	0	0
$731$	−51384.6	−2.59990
$732$	0	0
$733$	15097.6	0.760769	0.380384	−	0.924828i	$-0.375792\pi$
$733$	15097.6	0.760769	0.380384	+	0.924828i	$0.375792\pi$
$734$	−6524.23	−0.328084
$735$	0	0
$736$	−72.9790	−0.00365495
$737$	4114.31	0.205634
$738$	0	0
$739$	−3667.49	−0.182559	−0.0912793	−	0.995825i	$-0.529096\pi$
$739$	−3667.49	−0.182559	−0.0912793	+	0.995825i	$0.529096\pi$
$740$	0	0
$741$	0	0
$742$	−34094.8	−1.68687
$743$	10172.1	0.502257	0.251128	−	0.967954i	$-0.419198\pi$
$743$	10172.1	0.502257	0.251128	+	0.967954i	$0.419198\pi$
$744$	0	0
$745$	0	0
$746$	15926.2	0.781635
$747$	0	0
$748$	−3630.12	−0.177447
$749$	−37978.5	−1.85274
$750$	0	0
$751$	31430.3	1.52718	0.763588	−	0.645704i	$-0.223436\pi$
$751$	31430.3	1.52718	0.763588	+	0.645704i	$0.223436\pi$
$752$	10203.0	0.494766
$753$	0	0
$754$	−5265.45	−0.254319
$755$	0	0
$756$	0	0
$757$	28362.0	1.36174	0.680868	−	0.732406i	$-0.261603\pi$
$757$	28362.0	1.36174	0.680868	+	0.732406i	$0.261603\pi$
$758$	17388.3	0.833207
$759$	0	0
$760$	0	0
$761$	−30722.6	−1.46346	−0.731731	−	0.681594i	$-0.761287\pi$
$761$	−30722.6	−1.46346	−0.731731	+	0.681594i	$0.761287\pi$
$762$	0	0
$763$	31665.6	1.50245
$764$	1658.86	0.0785544
$765$	0	0
$766$	20138.4	0.949906
$767$	4179.48	0.196757
$768$	0	0
$769$	−18443.1	−0.864855	−0.432427	−	0.901669i	$-0.642343\pi$
$769$	−18443.1	−0.864855	−0.432427	+	0.901669i	$0.642343\pi$
$770$	0	0
$771$	0	0
$772$	10226.6	0.476764
$773$	−545.742	−0.0253932	−0.0126966	−	0.999919i	$-0.504042\pi$
$773$	−545.742	−0.0253932	−0.0126966	+	0.999919i	$0.504042\pi$
$774$	0	0
$775$	0	0
$776$	18001.9	0.832770
$777$	0	0
$778$	10207.8	0.470393
$779$	−1089.43	−0.0501065
$780$	0	0
$781$	1063.65	0.0487327
$782$	195.468	0.00893851
$783$	0	0
$784$	−5947.35	−0.270925
$785$	0	0
$786$	0	0
$787$	−17365.2	−0.786536	−0.393268	−	0.919424i	$-0.628655\pi$
$787$	−17365.2	−0.786536	−0.393268	+	0.919424i	$0.628655\pi$
$788$	−9674.33	−0.437352
$789$	0	0
$790$	0	0
$791$	−49105.8	−2.20733
$792$	0	0
$793$	6723.25	0.301071
$794$	−10323.8	−0.461431
$795$	0	0
$796$	−12427.9	−0.553384
$797$	7055.12	0.313557	0.156779	−	0.987634i	$-0.449889\pi$
$797$	7055.12	0.313557	0.156779	+	0.987634i	$0.449889\pi$
$798$	0	0
$799$	35562.0	1.57458
$800$	0	0
$801$	0	0
$802$	−11726.7	−0.516313
$803$	−219.406	−0.00964217
$804$	0	0
$805$	0	0
$806$	−3577.73	−0.156353
$807$	0	0
$808$	19959.5	0.869024
$809$	6937.17	0.301481	0.150740	−	0.988573i	$-0.451834\pi$
$809$	6937.17	0.301481	0.150740	+	0.988573i	$0.451834\pi$
$810$	0	0
$811$	5610.44	0.242921	0.121461	−	0.992596i	$-0.461242\pi$
$811$	5610.44	0.242921	0.121461	+	0.992596i	$0.461242\pi$
$812$	13266.9	0.573369
$813$	0	0
$814$	2825.32	0.121655
$815$	0	0
$816$	0	0
$817$	2823.14	0.120893
$818$	−15079.8	−0.644565
$819$	0	0
$820$	0	0
$821$	17001.8	0.722735	0.361368	−	0.932423i	$-0.382310\pi$
$821$	17001.8	0.722735	0.361368	+	0.932423i	$0.382310\pi$
$822$	0	0
$823$	−14567.3	−0.616991	−0.308496	−	0.951226i	$-0.599825\pi$
$823$	−14567.3	−0.616991	−0.308496	+	0.951226i	$0.599825\pi$
$824$	36007.7	1.52232
$825$	0	0
$826$	22180.0	0.934312
$827$	−7345.87	−0.308877	−0.154438	−	0.988002i	$-0.549357\pi$
$827$	−7345.87	−0.308877	−0.154438	+	0.988002i	$0.549357\pi$
$828$	0	0
$829$	−13903.2	−0.582482	−0.291241	−	0.956650i	$-0.594068\pi$
$829$	−13903.2	−0.582482	−0.291241	+	0.956650i	$0.594068\pi$
$830$	0	0
$831$	0	0
$832$	5458.84	0.227466
$833$	−20729.2	−0.862214
$834$	0	0
$835$	0	0
$836$	199.444	0.00825111
$837$	0	0
$838$	−19232.9	−0.792829
$839$	25111.9	1.03332	0.516662	−	0.856190i	$-0.327175\pi$
$839$	25111.9	1.03332	0.516662	+	0.856190i	$0.327175\pi$
$840$	0	0
$841$	28179.7	1.15543
$842$	7987.20	0.326909
$843$	0	0
$844$	11316.6	0.461534
$845$	0	0
$846$	0	0
$847$	−2718.53	−0.110283
$848$	23953.7	0.970014
$849$	0	0
$850$	0	0
$851$	72.2296	0.00290952
$852$	0	0
$853$	27545.3	1.10567	0.552833	−	0.833292i	$-0.313547\pi$
$853$	27545.3	1.10567	0.552833	+	0.833292i	$0.313547\pi$
$854$	35679.5	1.42966
$855$	0	0
$856$	41636.1	1.66249
$857$	1808.04	0.0720669	0.0360334	−	0.999351i	$-0.488528\pi$
$857$	1808.04	0.0720669	0.0360334	+	0.999351i	$0.488528\pi$
$858$	0	0
$859$	32160.8	1.27743	0.638716	−	0.769443i	$-0.279466\pi$
$859$	32160.8	1.27743	0.638716	+	0.769443i	$0.279466\pi$
$860$	0	0
$861$	0	0
$862$	−26226.3	−1.03628
$863$	33734.8	1.33064	0.665321	−	0.746557i	$-0.268294\pi$
$863$	33734.8	1.33064	0.665321	+	0.746557i	$0.268294\pi$
$864$	0	0
$865$	0	0
$866$	29342.7	1.15139
$867$	0	0
$868$	9014.47	0.352501
$869$	−268.715	−0.0104897
$870$	0	0
$871$	−3688.03	−0.143472
$872$	−34715.2	−1.34817
$873$	0	0
$874$	−10.7393	−0.000415631 0
$875$	0	0
$876$	0	0
$877$	−46573.5	−1.79325	−0.896623	−	0.442795i	$-0.853987\pi$
$877$	−46573.5	−1.79325	−0.896623	+	0.442795i	$0.853987\pi$
$878$	−9728.26	−0.373933
$879$	0	0
$880$	0	0
$881$	9949.72	0.380493	0.190247	−	0.981736i	$-0.439071\pi$
$881$	9949.72	0.380493	0.190247	+	0.981736i	$0.439071\pi$
$882$	0	0
$883$	49269.1	1.87773	0.938866	−	0.344282i	$-0.111878\pi$
$883$	49269.1	1.87773	0.938866	+	0.344282i	$0.111878\pi$
$884$	3254.01	0.123806
$885$	0	0
$886$	−5483.00	−0.207906
$887$	27347.5	1.03522	0.517609	−	0.855617i	$-0.326822\pi$
$887$	27347.5	1.03522	0.517609	+	0.855617i	$0.326822\pi$
$888$	0	0
$889$	−43111.2	−1.62644
$890$	0	0
$891$	0	0
$892$	5639.42	0.211683
$893$	−1953.83	−0.0732165
$894$	0	0
$895$	0	0
$896$	8942.48	0.333423
$897$	0	0
$898$	21628.1	0.803718
$899$	35719.0	1.32514
$900$	0	0
$901$	83489.4	3.08705
$902$	−3964.61	−0.146349
$903$	0	0
$904$	53835.0	1.98067
$905$	0	0
$906$	0	0
$907$	25516.6	0.934141	0.467071	−	0.884220i	$-0.345309\pi$
$907$	25516.6	0.934141	0.467071	+	0.884220i	$0.345309\pi$
$908$	2934.15	0.107239
$909$	0	0
$910$	0	0
$911$	−22379.0	−0.813885	−0.406942	−	0.913454i	$-0.633405\pi$
$911$	−22379.0	−0.813885	−0.406942	+	0.913454i	$0.633405\pi$
$912$	0	0
$913$	12400.9	0.449516
$914$	33488.1	1.21191
$915$	0	0
$916$	−8798.04	−0.317353
$917$	30092.8	1.08370
$918$	0	0
$919$	−6244.97	−0.224159	−0.112080	−	0.993699i	$-0.535751\pi$
$919$	−6244.97	−0.224159	−0.112080	+	0.993699i	$0.535751\pi$
$920$	0	0
$921$	0	0
$922$	−12537.8	−0.447841
$923$	−953.442	−0.0340010
$924$	0	0
$925$	0	0
$926$	42762.6	1.51757
$927$	0	0
$928$	−25547.0	−0.903688
$929$	−16122.2	−0.569378	−0.284689	−	0.958620i	$-0.591890\pi$
$929$	−16122.2	−0.569378	−0.284689	+	0.958620i	$0.591890\pi$
$930$	0	0
$931$	1138.89	0.0400921
$932$	−15831.6	−0.556417
$933$	0	0
$934$	−21109.8	−0.739544
$935$	0	0
$936$	0	0
$937$	−56379.6	−1.96568	−0.982839	−	0.184466i	$-0.940945\pi$
$937$	−56379.6	−1.96568	−0.982839	+	0.184466i	$0.940945\pi$
$938$	−19572.0	−0.681287
$939$	0	0
$940$	0	0
$941$	−25527.0	−0.884332	−0.442166	−	0.896933i	$-0.645790\pi$
$941$	−25527.0	−0.884332	−0.442166	+	0.896933i	$0.645790\pi$
$942$	0	0
$943$	−101.356	−0.00350010
$944$	−15582.8	−0.537264
$945$	0	0
$946$	10273.9	0.353099
$947$	46411.3	1.59257	0.796285	−	0.604921i	$-0.206795\pi$
$947$	46411.3	1.59257	0.796285	+	0.604921i	$0.206795\pi$
$948$	0	0
$949$	196.673	0.00672738
$950$	0	0
$951$	0	0
$952$	70909.2	2.41405
$953$	−21266.1	−0.722850	−0.361425	−	0.932401i	$-0.617710\pi$
$953$	−21266.1	−0.722850	−0.361425	+	0.932401i	$0.617710\pi$
$954$	0	0
$955$	0	0
$956$	−5357.05	−0.181234
$957$	0	0
$958$	29365.4	0.990347
$959$	24726.5	0.832597
$960$	0	0
$961$	−5520.88	−0.185320
$962$	−2532.60	−0.0848796
$963$	0	0
$964$	−4755.04	−0.158869
$965$	0	0
$966$	0	0
$967$	−20035.9	−0.666300	−0.333150	−	0.942874i	$-0.608112\pi$
$967$	−20035.9	−0.666300	−0.333150	+	0.942874i	$0.608112\pi$
$968$	2980.34	0.0989585
$969$	0	0
$970$	0	0
$971$	−21354.1	−0.705751	−0.352876	−	0.935670i	$-0.614796\pi$
$971$	−21354.1	−0.705751	−0.352876	+	0.935670i	$0.614796\pi$
$972$	0	0
$973$	28859.4	0.950862
$974$	−30776.6	−1.01247
$975$	0	0
$976$	−25067.0	−0.822107
$977$	−34057.7	−1.11525	−0.557626	−	0.830092i	$-0.688288\pi$
$977$	−34057.7	−1.11525	−0.557626	+	0.830092i	$0.688288\pi$
$978$	0	0
$979$	−7035.86	−0.229691
$980$	0	0
$981$	0	0
$982$	15263.0	0.495989
$983$	31846.5	1.03331	0.516657	−	0.856193i	$-0.327176\pi$
$983$	31846.5	1.03331	0.516657	+	0.856193i	$0.327176\pi$
$984$	0	0
$985$	0	0
$986$	68425.5	2.21005
$987$	0	0
$988$	−178.780	−0.00575684
$989$	262.652	0.00844474
$990$	0	0
$991$	−20462.5	−0.655915	−0.327957	−	0.944693i	$-0.606360\pi$
$991$	−20462.5	−0.655915	−0.327957	+	0.944693i	$0.606360\pi$
$992$	−17358.5	−0.555578
$993$	0	0
$994$	−5059.81	−0.161456
$995$	0	0
$996$	0	0
$997$	−35227.4	−1.11902	−0.559510	−	0.828824i	$-0.689011\pi$
$997$	−35227.4	−1.11902	−0.559510	+	0.828824i	$0.689011\pi$
$998$	6047.71	0.191820
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.s.1.2		3
3.2	odd	2		825.4.a.s.1.2		3
5.4	even	2		495.4.a.l.1.2		3
15.2	even	4		825.4.c.l.199.4		6
15.8	even	4		825.4.c.l.199.3		6
15.14	odd	2		165.4.a.d.1.2	✓	3
165.164	even	2		1815.4.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.2	✓	3	15.14	odd	2
495.4.a.l.1.2		3	5.4	even	2
825.4.a.s.1.2		3	3.2	odd	2
825.4.c.l.199.3		6	15.8	even	4
825.4.c.l.199.4		6	15.2	even	4
1815.4.a.s.1.2		3	165.164	even	2
2475.4.a.s.1.2		3	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)

\(f(q)\)	\(=\)	\(q-2.32906 q^{2} -2.57547 q^{4} -22.4672 q^{7} +24.6309 q^{8} +O(q^{10})\)	\(q-2.32906 q^{2} -2.57547 q^{4} -22.4672 q^{7} +24.6309 q^{8} -11.0000 q^{11} +9.86030 q^{13} +52.3275 q^{14} -36.7633 q^{16} -128.137 q^{17} +7.04001 q^{19} +25.6197 q^{22} +0.654969 q^{23} -22.9653 q^{26} +57.8635 q^{28} +229.279 q^{29} +155.789 q^{31} -111.423 q^{32} +298.438 q^{34} +110.279 q^{37} -16.3966 q^{38} -154.749 q^{41} +401.014 q^{43} +28.3301 q^{44} -1.52546 q^{46} -277.532 q^{47} +161.774 q^{49} -25.3949 q^{52} -651.566 q^{53} -553.388 q^{56} -534.005 q^{58} +423.869 q^{59} +681.851 q^{61} -362.842 q^{62} +553.618 q^{64} -374.028 q^{67} +330.011 q^{68} -96.6950 q^{71} +19.9460 q^{73} -256.848 q^{74} -18.1313 q^{76} +247.139 q^{77} +24.4286 q^{79} +360.419 q^{82} -1127.35 q^{83} -933.987 q^{86} -270.940 q^{88} +639.624 q^{89} -221.533 q^{91} -1.68685 q^{92} +646.389 q^{94} +730.865 q^{97} -376.783 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} + 22 q^{4} + 4 q^{7} - 48 q^{8}+O(q^{10}) \) 3 * q - 4 * q^2 + 22 * q^4 + 4 * q^7 - 48 * q^8	\( 3 q - 4 q^{2} + 22 q^{4} + 4 q^{7} - 48 q^{8} - 33 q^{11} + 56 q^{14} + 50 q^{16} - 218 q^{17} + 146 q^{19} + 44 q^{22} - 200 q^{23} + 508 q^{26} + 340 q^{28} - 68 q^{29} - 68 q^{31} - 688 q^{32} - 176 q^{34} + 390 q^{37} - 316 q^{38} + 196 q^{41} + 524 q^{43} - 242 q^{44} + 1160 q^{46} - 60 q^{47} - 157 q^{49} - 1020 q^{52} - 158 q^{53} - 1368 q^{56} - 1092 q^{58} + 1044 q^{59} + 642 q^{61} + 88 q^{62} + 1166 q^{64} + 236 q^{67} + 144 q^{68} + 544 q^{71} - 900 q^{73} - 1536 q^{74} + 1996 q^{76} - 44 q^{77} - 1586 q^{79} - 380 q^{82} - 1582 q^{83} - 3568 q^{86} + 528 q^{88} + 2122 q^{89} - 8 q^{91} - 4128 q^{92} - 2152 q^{94} - 618 q^{97} + 572 q^{98}+O(q^{100}) \) 3 * q - 4 * q^2 + 22 * q^4 + 4 * q^7 - 48 * q^8 - 33 * q^11 + 56 * q^14 + 50 * q^16 - 218 * q^17 + 146 * q^19 + 44 * q^22 - 200 * q^23 + 508 * q^26 + 340 * q^28 - 68 * q^29 - 68 * q^31 - 688 * q^32 - 176 * q^34 + 390 * q^37 - 316 * q^38 + 196 * q^41 + 524 * q^43 - 242 * q^44 + 1160 * q^46 - 60 * q^47 - 157 * q^49 - 1020 * q^52 - 158 * q^53 - 1368 * q^56 - 1092 * q^58 + 1044 * q^59 + 642 * q^61 + 88 * q^62 + 1166 * q^64 + 236 * q^67 + 144 * q^68 + 544 * q^71 - 900 * q^73 - 1536 * q^74 + 1996 * q^76 - 44 * q^77 - 1586 * q^79 - 380 * q^82 - 1582 * q^83 - 3568 * q^86 + 528 * q^88 + 2122 * q^89 - 8 * q^91 - 4128 * q^92 - 2152 * q^94 - 618 * q^97 + 572 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more