LMFDB - Embedded newform 675.4.b.m.649.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.12559936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{3} + 49x^{2} - 42x + 18 $ x^6 - 6x^3 + 49x^2 - 42*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$0.455589 + 0.455589i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.m.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58488i q^{2} +1.31841 q^{4} -22.8935i q^{7} +24.0869i q^{8} +O(q^{10})$	$q+2.58488i q^{2} +1.31841 q^{4} -22.8935i q^{7} +24.0869i q^{8} -11.0828 q^{11} +11.6368i q^{13} +59.1769 q^{14} -51.7145 q^{16} -10.0643i q^{17} -117.865 q^{19} -28.6477i q^{22} +172.441i q^{23} -30.0798 q^{26} -30.1831i q^{28} +178.321 q^{29} +140.528 q^{31} +59.0200i q^{32} +26.0149 q^{34} +250.074i q^{37} -304.666i q^{38} -361.569 q^{41} +360.707i q^{43} -14.6117 q^{44} -445.738 q^{46} +600.121i q^{47} -181.114 q^{49} +15.3421i q^{52} +201.312i q^{53} +551.435 q^{56} +460.937i q^{58} +415.772 q^{59} -54.6270 q^{61} +363.248i q^{62} -566.275 q^{64} -531.079i q^{67} -13.2689i q^{68} +933.534 q^{71} +560.199i q^{73} -646.409 q^{74} -155.394 q^{76} +253.725i q^{77} -810.781 q^{79} -934.611i q^{82} +538.210i q^{83} -932.384 q^{86} -266.951i q^{88} +686.173 q^{89} +266.408 q^{91} +227.348i q^{92} -1551.24 q^{94} +714.655i q^{97} -468.156i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 34 q^{4}+O(q^{10}) $ 6 * q - 34 * q^4	$ 6 q - 34 q^{4} - 10 q^{11} + 120 q^{14} + 322 q^{16} + 100 q^{19} - 370 q^{26} + 230 q^{29} - 230 q^{31} - 826 q^{34} - 1160 q^{41} - 2830 q^{44} - 570 q^{46} - 1154 q^{49} + 4380 q^{56} + 760 q^{59} - 304 q^{61} - 5874 q^{64} - 80 q^{71} - 5440 q^{74} - 6552 q^{76} - 2026 q^{79} + 3110 q^{86} + 2040 q^{89} - 1264 q^{91} - 7666 q^{94}+O(q^{100}) $ 6 * q - 34 * q^4 - 10 * q^11 + 120 * q^14 + 322 * q^16 + 100 * q^19 - 370 * q^26 + 230 * q^29 - 230 * q^31 - 826 * q^34 - 1160 * q^41 - 2830 * q^44 - 570 * q^46 - 1154 * q^49 + 4380 * q^56 + 760 * q^59 - 304 * q^61 - 5874 * q^64 - 80 * q^71 - 5440 * q^74 - 6552 * q^76 - 2026 * q^79 + 3110 * q^86 + 2040 * q^89 - 1264 * q^91 - 7666 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.58488i	0.913892i	0.889494	+	0.456946i	$0.151057\pi$
$2$	2.58488i	0.913892i	−0.889494	+	0.456946i	$0.848943\pi$
$3$	0	0
$3$	0	0
$4$	1.31841	0.164802
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 22.8935i	− 1.23613i	−0.786125	−	0.618067i	$-0.787916\pi$
$7$	− 22.8935i	− 1.23613i	0.786125	−	0.618067i	$-0.212084\pi$
$8$	24.0869i	1.06450i
$9$	0	0
$10$	0	0
$11$	−11.0828	−0.303781	−0.151891	−	0.988397i	$-0.548536\pi$
$11$	−11.0828	−0.303781	−0.151891	+	0.988397i	$0.548536\pi$
$12$	0	0
$13$	11.6368i	0.248267i	0.992266	+	0.124134i	$0.0396152\pi$
$13$	11.6368i	0.248267i	−0.992266	+	0.124134i	$0.960385\pi$
$14$	59.1769	1.12969
$15$	0	0
$16$	−51.7145	−0.808039
$17$	− 10.0643i	− 0.143585i	−0.997420	−	0.0717926i	$-0.977128\pi$
$17$	− 10.0643i	− 0.143585i	0.997420	−	0.0717926i	$-0.0228720\pi$
$18$	0	0
$19$	−117.865	−1.42316	−0.711579	−	0.702606i	$-0.752020\pi$
$19$	−117.865	−1.42316	−0.711579	+	0.702606i	$0.752020\pi$
$20$	0	0
$21$	0	0
$22$	− 28.6477i	− 0.277623i
$23$	172.441i	1.56332i	0.623704	+	0.781661i	$0.285627\pi$
$23$	172.441i	1.56332i	−0.623704	+	0.781661i	$0.714373\pi$
$24$	0	0
$25$	0	0
$26$	−30.0798	−0.226889
$27$	0	0
$28$	− 30.1831i	− 0.203717i
$29$	178.321	1.14184	0.570919	−	0.821006i	$-0.306587\pi$
$29$	178.321	1.14184	0.570919	+	0.821006i	$0.306587\pi$
$30$	0	0
$31$	140.528	0.814182	0.407091	−	0.913388i	$-0.366543\pi$
$31$	140.528	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$32$	59.0200i	0.326043i
$33$	0	0
$34$	26.0149	0.131221
$35$	0	0
$36$	0	0
$37$	250.074i	1.11113i	0.831473	+	0.555566i	$0.187498\pi$
$37$	250.074i	1.11113i	−0.831473	+	0.555566i	$0.812502\pi$
$38$	− 304.666i	− 1.30061i
$39$	0	0
$40$	0	0
$41$	−361.569	−1.37726	−0.688629	−	0.725114i	$-0.741787\pi$
$41$	−361.569	−1.37726	−0.688629	+	0.725114i	$0.741787\pi$
$42$	0	0
$43$	360.707i	1.27924i	0.768691	+	0.639620i	$0.220908\pi$
$43$	360.707i	1.27924i	−0.768691	+	0.639620i	$0.779092\pi$
$44$	−14.6117	−0.0500636
$45$	0	0
$46$	−445.738	−1.42871
$47$	600.121i	1.86248i	0.364405	+	0.931241i	$0.381273\pi$
$47$	600.121i	1.86248i	−0.364405	+	0.931241i	$0.618727\pi$
$48$	0	0
$49$	−181.114	−0.528028
$50$	0	0
$51$	0	0
$52$	15.3421i	0.0409149i
$53$	201.312i	0.521742i	0.965374	+	0.260871i	$0.0840097\pi$
$53$	201.312i	0.521742i	−0.965374	+	0.260871i	$0.915990\pi$
$54$	0	0
$55$	0	0
$56$	551.435	1.31587
$57$	0	0
$58$	460.937i	1.04352i
$59$	415.772	0.917438	0.458719	−	0.888581i	$-0.348308\pi$
$59$	415.772	0.917438	0.458719	+	0.888581i	$0.348308\pi$
$60$	0	0
$61$	−54.6270	−0.114660	−0.0573301	−	0.998355i	$-0.518259\pi$
$61$	−54.6270	−0.114660	−0.0573301	+	0.998355i	$0.518259\pi$
$62$	363.248i	0.744074i
$63$	0	0
$64$	−566.275	−1.10601
$65$	0	0
$66$	0	0
$67$	− 531.079i	− 0.968382i	−0.874962	−	0.484191i	$-0.839114\pi$
$67$	− 531.079i	− 0.968382i	0.874962	−	0.484191i	$-0.160886\pi$
$68$	− 13.2689i	− 0.0236631i
$69$	0	0
$70$	0	0
$71$	933.534	1.56042	0.780212	−	0.625515i	$-0.215111\pi$
$71$	933.534	1.56042	0.780212	+	0.625515i	$0.215111\pi$
$72$	0	0
$73$	560.199i	0.898169i	0.893489	+	0.449085i	$0.148250\pi$
$73$	560.199i	0.898169i	−0.893489	+	0.449085i	$0.851750\pi$
$74$	−646.409	−1.01545
$75$	0	0
$76$	−155.394	−0.234539
$77$	253.725i	0.375514i
$78$	0	0
$79$	−810.781	−1.15468	−0.577342	−	0.816503i	$-0.695910\pi$
$79$	−810.781	−1.15468	−0.577342	+	0.816503i	$0.695910\pi$
$80$	0	0
$81$	0	0
$82$	− 934.611i	− 1.25867i
$83$	538.210i	0.711762i	0.934531	+	0.355881i	$0.115819\pi$
$83$	538.210i	0.711762i	−0.934531	+	0.355881i	$0.884181\pi$
$84$	0	0
$85$	0	0
$86$	−932.384	−1.16909
$87$	0	0
$88$	− 266.951i	− 0.323376i
$89$	686.173	0.817238	0.408619	−	0.912705i	$-0.366010\pi$
$89$	686.173	0.817238	0.408619	+	0.912705i	$0.366010\pi$
$90$	0	0
$91$	266.408	0.306892
$92$	227.348i	0.257638i
$93$	0	0
$94$	−1551.24	−1.70211
$95$	0	0
$96$	0	0
$97$	714.655i	0.748064i	0.927416	+	0.374032i	$0.122025\pi$
$97$	714.655i	0.748064i	−0.927416	+	0.374032i	$0.877975\pi$
$98$	− 468.156i	− 0.482560i
$99$	0	0
$100$	0	0
$101$	−973.907	−0.959479	−0.479739	−	0.877411i	$-0.659269\pi$
$101$	−973.907	−0.959479	−0.479739	+	0.877411i	$0.659269\pi$
$102$	0	0
$103$	759.229i	0.726301i	0.931730	+	0.363151i	$0.118299\pi$
$103$	759.229i	0.726301i	−0.931730	+	0.363151i	$0.881701\pi$
$104$	−280.296	−0.264281
$105$	0	0
$106$	−520.366	−0.476815
$107$	− 1832.06i	− 1.65525i	−0.561282	−	0.827625i	$-0.689692\pi$
$107$	− 1832.06i	− 1.65525i	0.561282	−	0.827625i	$-0.310308\pi$
$108$	0	0
$109$	−1370.91	−1.20467	−0.602335	−	0.798244i	$-0.705763\pi$
$109$	−1370.91	−1.20467	−0.602335	+	0.798244i	$0.705763\pi$
$110$	0	0
$111$	0	0
$112$	1183.93i	0.998844i
$113$	583.567i	0.485817i	0.970049	+	0.242909i	$0.0781015\pi$
$113$	583.567i	0.485817i	−0.970049	+	0.242909i	$0.921898\pi$
$114$	0	0
$115$	0	0
$116$	235.100	0.188177
$117$	0	0
$118$	1074.72i	0.838439i
$119$	−230.407	−0.177491
$120$	0	0
$121$	−1208.17	−0.907717
$122$	− 141.204i	− 0.104787i
$123$	0	0
$124$	185.274	0.134178
$125$	0	0
$126$	0	0
$127$	− 2432.76i	− 1.69978i	−0.526958	−	0.849891i	$-0.676667\pi$
$127$	− 2432.76i	− 1.69978i	0.526958	−	0.849891i	$-0.323333\pi$
$128$	− 991.592i	− 0.684728i
$129$	0	0
$130$	0	0
$131$	−501.073	−0.334191	−0.167095	−	0.985941i	$-0.553439\pi$
$131$	−501.073	−0.334191	−0.167095	+	0.985941i	$0.553439\pi$
$132$	0	0
$133$	2698.34i	1.75922i
$134$	1372.77	0.884996
$135$	0	0
$136$	242.418	0.152847
$137$	− 2523.07i	− 1.57343i	−0.617316	−	0.786716i	$-0.711780\pi$
$137$	− 2523.07i	− 1.57343i	0.617316	−	0.786716i	$-0.288220\pi$
$138$	0	0
$139$	638.842	0.389826	0.194913	−	0.980821i	$-0.437558\pi$
$139$	638.842	0.389826	0.194913	+	0.980821i	$0.437558\pi$
$140$	0	0
$141$	0	0
$142$	2413.07i	1.42606i
$143$	− 128.969i	− 0.0754189i
$144$	0	0
$145$	0	0
$146$	−1448.05	−0.820829
$147$	0	0
$148$	329.700i	0.183116i
$149$	−2674.13	−1.47029	−0.735146	−	0.677909i	$-0.762886\pi$
$149$	−2674.13	−1.47029	−0.735146	+	0.677909i	$0.762886\pi$
$150$	0	0
$151$	−1036.68	−0.558700	−0.279350	−	0.960189i	$-0.590119\pi$
$151$	−1036.68	−0.558700	−0.279350	+	0.960189i	$0.590119\pi$
$152$	− 2839.00i	− 1.51496i
$153$	0	0
$154$	−655.847	−0.343179
$155$	0	0
$156$	0	0
$157$	− 381.858i	− 0.194112i	−0.995279	−	0.0970560i	$-0.969057\pi$
$157$	− 381.858i	− 0.194112i	0.995279	−	0.0970560i	$-0.0309426\pi$
$158$	− 2095.77i	− 1.05526i
$159$	0	0
$160$	0	0
$161$	3947.78	1.93248
$162$	0	0
$163$	− 2421.17i	− 1.16344i	−0.813390	−	0.581719i	$-0.802380\pi$
$163$	− 2421.17i	− 1.16344i	0.813390	−	0.581719i	$-0.197620\pi$
$164$	−476.697	−0.226974
$165$	0	0
$166$	−1391.21	−0.650473
$167$	3535.76i	1.63836i	0.573540	+	0.819178i	$0.305570\pi$
$167$	3535.76i	1.63836i	−0.573540	+	0.819178i	$0.694430\pi$
$168$	0	0
$169$	2061.58	0.938363
$170$	0	0
$171$	0	0
$172$	475.561i	0.210821i
$173$	3143.88i	1.38165i	0.723023	+	0.690824i	$0.242752\pi$
$173$	3143.88i	1.38165i	−0.723023	+	0.690824i	$0.757248\pi$
$174$	0	0
$175$	0	0
$176$	573.142	0.245467
$177$	0	0
$178$	1773.67i	0.746867i
$179$	−3299.06	−1.37756	−0.688780	−	0.724970i	$-0.741853\pi$
$179$	−3299.06	−1.37756	−0.688780	+	0.724970i	$0.741853\pi$
$180$	0	0
$181$	1875.10	0.770026	0.385013	−	0.922911i	$-0.374197\pi$
$181$	1875.10	0.770026	0.385013	+	0.922911i	$0.374197\pi$
$182$	688.632i	0.280466i
$183$	0	0
$184$	−4153.57	−1.66416
$185$	0	0
$186$	0	0
$187$	111.541i	0.0436185i
$188$	791.207i	0.306940i
$189$	0	0
$190$	0	0
$191$	1361.62	0.515829	0.257915	−	0.966168i	$-0.416965\pi$
$191$	1361.62	0.515829	0.257915	+	0.966168i	$0.416965\pi$
$192$	0	0
$193$	− 2234.20i	− 0.833271i	−0.909074	−	0.416636i	$-0.863209\pi$
$193$	− 2234.20i	− 0.833271i	0.909074	−	0.416636i	$-0.136791\pi$
$194$	−1847.29	−0.683650
$195$	0	0
$196$	−238.782	−0.0870199
$197$	346.625i	0.125360i	0.998034	+	0.0626801i	$0.0199648\pi$
$197$	346.625i	0.125360i	−0.998034	+	0.0626801i	$0.980035\pi$
$198$	0	0
$199$	−4198.77	−1.49569	−0.747846	−	0.663872i	$-0.768912\pi$
$199$	−4198.77	−1.49569	−0.747846	+	0.663872i	$0.768912\pi$
$200$	0	0
$201$	0	0
$202$	− 2517.43i	− 0.876860i
$203$	− 4082.39i	− 1.41146i
$204$	0	0
$205$	0	0
$206$	−1962.51	−0.663761
$207$	0	0
$208$	− 601.792i	− 0.200610i
$209$	1306.27	0.432329
$210$	0	0
$211$	4728.46	1.54275	0.771375	−	0.636380i	$-0.219569\pi$
$211$	4728.46	1.54275	0.771375	+	0.636380i	$0.219569\pi$
$212$	265.412i	0.0859839i
$213$	0	0
$214$	4735.64	1.51272
$215$	0	0
$216$	0	0
$217$	− 3217.19i	− 1.00644i
$218$	− 3543.62i	− 1.10094i
$219$	0	0
$220$	0	0
$221$	117.116	0.0356475
$222$	0	0
$223$	− 2430.16i	− 0.729756i	−0.931055	−	0.364878i	$-0.881111\pi$
$223$	− 2430.16i	− 0.729756i	0.931055	−	0.364878i	$-0.118889\pi$
$224$	1351.18	0.403032
$225$	0	0
$226$	−1508.45	−0.443985
$227$	584.687i	0.170956i	0.996340	+	0.0854780i	$0.0272417\pi$
$227$	584.687i	0.170956i	−0.996340	+	0.0854780i	$0.972758\pi$
$228$	0	0
$229$	4731.77	1.36543	0.682717	−	0.730683i	$-0.260798\pi$
$229$	4731.77	1.36543	0.682717	+	0.730683i	$0.260798\pi$
$230$	0	0
$231$	0	0
$232$	4295.20i	1.21549i
$233$	1228.45i	0.345401i	0.984974	+	0.172701i	$0.0552494\pi$
$233$	1228.45i	0.345401i	−0.984974	+	0.172701i	$0.944751\pi$
$234$	0	0
$235$	0	0
$236$	548.159	0.151195
$237$	0	0
$238$	− 595.574i	− 0.162207i
$239$	−120.545	−0.0326253	−0.0163126	−	0.999867i	$-0.505193\pi$
$239$	−120.545	−0.0326253	−0.0163126	+	0.999867i	$0.505193\pi$
$240$	0	0
$241$	1732.56	0.463086	0.231543	−	0.972825i	$-0.425623\pi$
$241$	1732.56	0.463086	0.231543	+	0.972825i	$0.425623\pi$
$242$	− 3122.97i	− 0.829555i
$243$	0	0
$244$	−72.0209	−0.0188962
$245$	0	0
$246$	0	0
$247$	− 1371.57i	− 0.353324i
$248$	3384.90i	0.866699i
$249$	0	0
$250$	0	0
$251$	3287.82	0.826793	0.413397	−	0.910551i	$-0.364342\pi$
$251$	3287.82	0.826793	0.413397	+	0.910551i	$0.364342\pi$
$252$	0	0
$253$	− 1911.13i	− 0.474908i
$254$	6288.38	1.55342
$255$	0	0
$256$	−1967.06	−0.480239
$257$	1489.82i	0.361605i	0.983519	+	0.180803i	$0.0578696\pi$
$257$	1489.82i	0.361605i	−0.983519	+	0.180803i	$0.942130\pi$
$258$	0	0
$259$	5725.07	1.37351
$260$	0	0
$261$	0	0
$262$	− 1295.21i	− 0.305414i
$263$	710.144i	0.166499i	0.996529	+	0.0832497i	$0.0265299\pi$
$263$	710.144i	0.166499i	−0.996529	+	0.0832497i	$0.973470\pi$
$264$	0	0
$265$	0	0
$266$	−6974.87	−1.60773
$267$	0	0
$268$	− 700.181i	− 0.159591i
$269$	3667.61	0.831294	0.415647	−	0.909526i	$-0.363555\pi$
$269$	3667.61	0.831294	0.415647	+	0.909526i	$0.363555\pi$
$270$	0	0
$271$	−1990.45	−0.446166	−0.223083	−	0.974799i	$-0.571612\pi$
$271$	−1990.45	−0.446166	−0.223083	+	0.974799i	$0.571612\pi$
$272$	520.469i	0.116022i
$273$	0	0
$274$	6521.81	1.43795
$275$	0	0
$276$	0	0
$277$	8314.68i	1.80354i	0.432215	+	0.901770i	$0.357732\pi$
$277$	8314.68i	1.80354i	−0.432215	+	0.901770i	$0.642268\pi$
$278$	1651.33i	0.356259i
$279$	0	0
$280$	0	0
$281$	−5765.13	−1.22391	−0.611955	−	0.790892i	$-0.709617\pi$
$281$	−5765.13	−1.22391	−0.611955	+	0.790892i	$0.709617\pi$
$282$	0	0
$283$	− 457.561i	− 0.0961102i	−0.998845	−	0.0480551i	$-0.984698\pi$
$283$	− 457.561i	− 0.0961102i	0.998845	−	0.0480551i	$-0.0153023\pi$
$284$	1230.78	0.257160
$285$	0	0
$286$	333.368	0.0689247
$287$	8277.59i	1.70248i
$288$	0	0
$289$	4811.71	0.979383
$290$	0	0
$291$	0	0
$292$	738.574i	0.148020i
$293$	5302.82i	1.05732i	0.848834	+	0.528659i	$0.177305\pi$
$293$	5302.82i	1.05732i	−0.848834	+	0.528659i	$0.822695\pi$
$294$	0	0
$295$	0	0
$296$	−6023.51	−1.18280
$297$	0	0
$298$	− 6912.30i	− 1.34369i
$299$	−2006.66	−0.388122
$300$	0	0
$301$	8257.86	1.58131
$302$	− 2679.68i	− 0.510591i
$303$	0	0
$304$	6095.31	1.14997
$305$	0	0
$306$	0	0
$307$	− 6583.54i	− 1.22392i	−0.790890	−	0.611959i	$-0.790382\pi$
$307$	− 6583.54i	− 1.22392i	0.790890	−	0.611959i	$-0.209618\pi$
$308$	334.514i	0.0618853i
$309$	0	0
$310$	0	0
$311$	8128.05	1.48199	0.740996	−	0.671510i	$-0.234354\pi$
$311$	8128.05	1.48199	0.740996	+	0.671510i	$0.234354\pi$
$312$	0	0
$313$	5207.63i	0.940424i	0.882554	+	0.470212i	$0.155823\pi$
$313$	5207.63i	0.940424i	−0.882554	+	0.470212i	$0.844177\pi$
$314$	987.056	0.177397
$315$	0	0
$316$	−1068.94	−0.190294
$317$	3262.34i	0.578016i	0.957327	+	0.289008i	$0.0933254\pi$
$317$	3262.34i	0.578016i	−0.957327	+	0.289008i	$0.906675\pi$
$318$	0	0
$319$	−1976.29	−0.346869
$320$	0	0
$321$	0	0
$322$	10204.5i	1.76607i
$323$	1186.22i	0.204345i
$324$	0	0
$325$	0	0
$326$	6258.41	1.06326
$327$	0	0
$328$	− 8709.09i	− 1.46610i
$329$	13738.9	2.30228
$330$	0	0
$331$	10360.5	1.72043	0.860216	−	0.509930i	$-0.170329\pi$
$331$	10360.5	1.72043	0.860216	+	0.509930i	$0.170329\pi$
$332$	709.583i	0.117299i
$333$	0	0
$334$	−9139.51	−1.49728
$335$	0	0
$336$	0	0
$337$	− 3735.26i	− 0.603777i	−0.953343	−	0.301888i	$-0.902383\pi$
$337$	− 3735.26i	− 0.603777i	0.953343	−	0.301888i	$-0.0976170\pi$
$338$	5328.94i	0.857563i
$339$	0	0
$340$	0	0
$341$	−1557.45	−0.247333
$342$	0	0
$343$	− 3706.15i	− 0.583421i
$344$	−8688.34	−1.36176
$345$	0	0
$346$	−8126.55	−1.26268
$347$	− 197.010i	− 0.0304785i	−0.999884	−	0.0152393i	$-0.995149\pi$
$347$	− 197.010i	− 0.0304785i	0.999884	−	0.0152393i	$-0.00485100\pi$
$348$	0	0
$349$	7334.20	1.12490	0.562451	−	0.826831i	$-0.309859\pi$
$349$	7334.20	1.12490	0.562451	+	0.826831i	$0.309859\pi$
$350$	0	0
$351$	0	0
$352$	− 654.107i	− 0.0990456i
$353$	4616.28i	0.696034i	0.937488	+	0.348017i	$0.113145\pi$
$353$	4616.28i	0.696034i	−0.937488	+	0.348017i	$0.886855\pi$
$354$	0	0
$355$	0	0
$356$	904.659	0.134682
$357$	0	0
$358$	− 8527.66i	− 1.25894i
$359$	−1153.79	−0.169623	−0.0848115	−	0.996397i	$-0.527029\pi$
$359$	−1153.79	−0.169623	−0.0848115	+	0.996397i	$0.527029\pi$
$360$	0	0
$361$	7033.09	1.02538
$362$	4846.89i	0.703721i
$363$	0	0
$364$	351.236	0.0505763
$365$	0	0
$366$	0	0
$367$	− 3449.05i	− 0.490569i	−0.969451	−	0.245285i	$-0.921119\pi$
$367$	− 3449.05i	− 0.490569i	0.969451	−	0.245285i	$-0.0788814\pi$
$368$	− 8917.69i	− 1.26322i
$369$	0	0
$370$	0	0
$371$	4608.74	0.644943
$372$	0	0
$373$	− 362.880i	− 0.0503732i	−0.999683	−	0.0251866i	$-0.991982\pi$
$373$	− 362.880i	− 0.0503732i	0.999683	−	0.0251866i	$-0.00801799\pi$
$374$	−288.319	−0.0398626
$375$	0	0
$376$	−14455.1	−1.98262
$377$	2075.09i	0.283481i
$378$	0	0
$379$	7719.79	1.04628	0.523139	−	0.852248i	$-0.324761\pi$
$379$	7719.79	1.04628	0.523139	+	0.852248i	$0.324761\pi$
$380$	0	0
$381$	0	0
$382$	3519.62i	0.471412i
$383$	− 5673.66i	− 0.756946i	−0.925612	−	0.378473i	$-0.876449\pi$
$383$	− 5673.66i	− 0.756946i	0.925612	−	0.378473i	$-0.123551\pi$
$384$	0	0
$385$	0	0
$386$	5775.13	0.761520
$387$	0	0
$388$	942.210i	0.123282i
$389$	−8180.45	−1.06623	−0.533117	−	0.846041i	$-0.678980\pi$
$389$	−8180.45	−1.06623	−0.533117	+	0.846041i	$0.678980\pi$
$390$	0	0
$391$	1735.49	0.224470
$392$	− 4362.47i	− 0.562087i
$393$	0	0
$394$	−895.982	−0.114566
$395$	0	0
$396$	0	0
$397$	− 12940.5i	− 1.63594i	−0.575263	−	0.817968i	$-0.695100\pi$
$397$	− 12940.5i	− 1.63594i	0.575263	−	0.817968i	$-0.304900\pi$
$398$	− 10853.3i	− 1.36690i
$399$	0	0
$400$	0	0
$401$	−6581.29	−0.819586	−0.409793	−	0.912179i	$-0.634399\pi$
$401$	−6581.29	−0.819586	−0.409793	+	0.912179i	$0.634399\pi$
$402$	0	0
$403$	1635.30i	0.202135i
$404$	−1284.01	−0.158124
$405$	0	0
$406$	10552.5	1.28993
$407$	− 2771.52i	− 0.337541i
$408$	0	0
$409$	−4921.85	−0.595036	−0.297518	−	0.954716i	$-0.596159\pi$
$409$	−4921.85	−0.595036	−0.297518	+	0.954716i	$0.596159\pi$
$410$	0	0
$411$	0	0
$412$	1000.98i	0.119696i
$413$	− 9518.48i	− 1.13408i
$414$	0	0
$415$	0	0
$416$	−686.806	−0.0809457
$417$	0	0
$418$	3376.55i	0.395102i
$419$	−3914.29	−0.456385	−0.228193	−	0.973616i	$-0.573282\pi$
$419$	−3914.29	−0.456385	−0.228193	+	0.973616i	$0.573282\pi$
$420$	0	0
$421$	−5258.76	−0.608780	−0.304390	−	0.952547i	$-0.598453\pi$
$421$	−5258.76	−0.608780	−0.304390	+	0.952547i	$0.598453\pi$
$422$	12222.5i	1.40991i
$423$	0	0
$424$	−4848.99	−0.555395
$425$	0	0
$426$	0	0
$427$	1250.60i	0.141735i
$428$	− 2415.41i	− 0.272788i
$429$	0	0
$430$	0	0
$431$	−14350.1	−1.60376	−0.801881	−	0.597484i	$-0.796167\pi$
$431$	−14350.1	−1.60376	−0.801881	+	0.597484i	$0.796167\pi$
$432$	0	0
$433$	− 863.149i	− 0.0957974i	−0.998852	−	0.0478987i	$-0.984748\pi$
$433$	− 863.149i	− 0.0957974i	0.998852	−	0.0478987i	$-0.0152525\pi$
$434$	8316.04	0.919776
$435$	0	0
$436$	−1807.42	−0.198532
$437$	− 20324.7i	− 2.22486i
$438$	0	0
$439$	16142.1	1.75494	0.877470	−	0.479632i	$-0.159230\pi$
$439$	16142.1	1.75494	0.877470	+	0.479632i	$0.159230\pi$
$440$	0	0
$441$	0	0
$442$	302.731i	0.0325780i
$443$	5884.09i	0.631065i	0.948915	+	0.315532i	$0.102183\pi$
$443$	5884.09i	0.631065i	−0.948915	+	0.315532i	$0.897817\pi$
$444$	0	0
$445$	0	0
$446$	6281.67	0.666918
$447$	0	0
$448$	12964.0i	1.36717i
$449$	−16858.4	−1.77193	−0.885965	−	0.463753i	$-0.846503\pi$
$449$	−16858.4	−1.77193	−0.885965	+	0.463753i	$0.846503\pi$
$450$	0	0
$451$	4007.20	0.418385
$452$	769.382i	0.0800635i
$453$	0	0
$454$	−1511.34	−0.156235
$455$	0	0
$456$	0	0
$457$	7623.04i	0.780286i	0.920754	+	0.390143i	$0.127574\pi$
$457$	7623.04i	0.780286i	−0.920754	+	0.390143i	$0.872426\pi$
$458$	12231.1i	1.24786i
$459$	0	0
$460$	0	0
$461$	4079.39	0.412139	0.206070	−	0.978537i	$-0.433933\pi$
$461$	4079.39	0.412139	0.206070	+	0.978537i	$0.433933\pi$
$462$	0	0
$463$	5499.22i	0.551988i	0.961159	+	0.275994i	$0.0890069\pi$
$463$	5499.22i	0.551988i	−0.961159	+	0.275994i	$0.910993\pi$
$464$	−9221.75	−0.922649
$465$	0	0
$466$	−3175.40	−0.315660
$467$	− 6422.51i	− 0.636399i	−0.948024	−	0.318199i	$-0.896922\pi$
$467$	− 6422.51i	− 0.636399i	0.948024	−	0.318199i	$-0.103078\pi$
$468$	0	0
$469$	−12158.3	−1.19705
$470$	0	0
$471$	0	0
$472$	10014.7i	0.976615i
$473$	− 3997.65i	− 0.388609i
$474$	0	0
$475$	0	0
$476$	−303.772	−0.0292507
$477$	0	0
$478$	− 311.595i	− 0.0298160i
$479$	198.760	0.0189595	0.00947973	−	0.999955i	$-0.496982\pi$
$479$	198.760	0.0189595	0.00947973	+	0.999955i	$0.496982\pi$
$480$	0	0
$481$	−2910.06	−0.275858
$482$	4478.44i	0.423211i
$483$	0	0
$484$	−1592.87	−0.149593
$485$	0	0
$486$	0	0
$487$	− 1308.73i	− 0.121774i	−0.998145	−	0.0608871i	$-0.980607\pi$
$487$	− 1308.73i	− 0.121774i	0.998145	−	0.0608871i	$-0.0193930\pi$
$488$	− 1315.80i	− 0.122056i
$489$	0	0
$490$	0	0
$491$	12474.8	1.14660	0.573300	−	0.819346i	$-0.305663\pi$
$491$	12474.8	1.14660	0.573300	+	0.819346i	$0.305663\pi$
$492$	0	0
$493$	− 1794.67i	− 0.163951i
$494$	3545.34	0.322900
$495$	0	0
$496$	−7267.35	−0.657890
$497$	− 21371.9i	− 1.92889i
$498$	0	0
$499$	−3146.82	−0.282307	−0.141153	−	0.989988i	$-0.545081\pi$
$499$	−3146.82	−0.282307	−0.141153	+	0.989988i	$0.545081\pi$
$500$	0	0
$501$	0	0
$502$	8498.60i	0.755600i
$503$	5419.35i	0.480391i	0.970725	+	0.240196i	$0.0772116\pi$
$503$	5419.35i	0.480391i	−0.970725	+	0.240196i	$0.922788\pi$
$504$	0	0
$505$	0	0
$506$	4940.03	0.434014
$507$	0	0
$508$	− 3207.38i	− 0.280127i
$509$	17194.6	1.49732	0.748660	−	0.662954i	$-0.230698\pi$
$509$	17194.6	1.49732	0.748660	+	0.662954i	$0.230698\pi$
$510$	0	0
$511$	12824.9	1.11026
$512$	− 13017.3i	− 1.12361i
$513$	0	0
$514$	−3851.01	−0.330468
$515$	0	0
$516$	0	0
$517$	− 6651.02i	− 0.565787i
$518$	14798.6i	1.25524i
$519$	0	0
$520$	0	0
$521$	−19829.5	−1.66746	−0.833730	−	0.552172i	$-0.813799\pi$
$521$	−19829.5	−1.66746	−0.833730	+	0.552172i	$0.813799\pi$
$522$	0	0
$523$	− 5392.82i	− 0.450882i	−0.974257	−	0.225441i	$-0.927618\pi$
$523$	− 5392.82i	− 0.450882i	0.974257	−	0.225441i	$-0.0723824\pi$
$524$	−660.621	−0.0550752
$525$	0	0
$526$	−1835.63	−0.152162
$527$	− 1414.32i	− 0.116904i
$528$	0	0
$529$	−17568.8	−1.44397
$530$	0	0
$531$	0	0
$532$	3557.53i	0.289922i
$533$	− 4207.52i	− 0.341928i
$534$	0	0
$535$	0	0
$536$	12792.1	1.03085
$537$	0	0
$538$	9480.32i	0.759713i
$539$	2007.25	0.160405
$540$	0	0
$541$	−11475.8	−0.911984	−0.455992	−	0.889984i	$-0.650715\pi$
$541$	−11475.8	−0.911984	−0.455992	+	0.889984i	$0.650715\pi$
$542$	− 5145.06i	− 0.407747i
$543$	0	0
$544$	593.994	0.0468149
$545$	0	0
$546$	0	0
$547$	− 7929.09i	− 0.619787i	−0.950771	−	0.309893i	$-0.899707\pi$
$547$	− 7929.09i	− 0.619787i	0.950771	−	0.309893i	$-0.100293\pi$
$548$	− 3326.44i	− 0.259304i
$549$	0	0
$550$	0	0
$551$	−21017.7	−1.62502
$552$	0	0
$553$	18561.6i	1.42734i
$554$	−21492.4	−1.64824
$555$	0	0
$556$	842.257	0.0642440
$557$	9519.36i	0.724144i	0.932150	+	0.362072i	$0.117931\pi$
$557$	9519.36i	0.724144i	−0.932150	+	0.362072i	$0.882069\pi$
$558$	0	0
$559$	−4197.49	−0.317594
$560$	0	0
$561$	0	0
$562$	− 14902.1i	− 1.11852i
$563$	− 11922.3i	− 0.892476i	−0.894914	−	0.446238i	$-0.852763\pi$
$563$	− 11922.3i	− 0.892476i	0.894914	−	0.446238i	$-0.147237\pi$
$564$	0	0
$565$	0	0
$566$	1182.74	0.0878343
$567$	0	0
$568$	22486.0i	1.66108i
$569$	17591.4	1.29608	0.648042	−	0.761605i	$-0.275588\pi$
$569$	17591.4	1.29608	0.648042	+	0.761605i	$0.275588\pi$
$570$	0	0
$571$	8250.73	0.604698	0.302349	−	0.953197i	$-0.402229\pi$
$571$	8250.73	0.604698	0.302349	+	0.953197i	$0.402229\pi$
$572$	− 170.034i	− 0.0124292i
$573$	0	0
$574$	−21396.5	−1.55588
$575$	0	0
$576$	0	0
$577$	− 15922.4i	− 1.14880i	−0.818574	−	0.574401i	$-0.805235\pi$
$577$	− 15922.4i	− 1.14880i	0.818574	−	0.574401i	$-0.194765\pi$
$578$	12437.7i	0.895050i
$579$	0	0
$580$	0	0
$581$	12321.5	0.879833
$582$	0	0
$583$	− 2231.10i	− 0.158495i
$584$	−13493.5	−0.956103
$585$	0	0
$586$	−13707.1	−0.966274
$587$	9589.65i	0.674288i	0.941453	+	0.337144i	$0.109461\pi$
$587$	9589.65i	0.674288i	−0.941453	+	0.337144i	$0.890539\pi$
$588$	0	0
$589$	−16563.3	−1.15871
$590$	0	0
$591$	0	0
$592$	− 12932.4i	− 0.897837i
$593$	5311.77i	0.367838i	0.982941	+	0.183919i	$0.0588785\pi$
$593$	5311.77i	0.367838i	−0.982941	+	0.183919i	$0.941122\pi$
$594$	0	0
$595$	0	0
$596$	−3525.61	−0.242307
$597$	0	0
$598$	− 5186.98i	− 0.354701i
$599$	−6346.04	−0.432875	−0.216437	−	0.976296i	$-0.569444\pi$
$599$	−6346.04	−0.432875	−0.216437	+	0.976296i	$0.569444\pi$
$600$	0	0
$601$	−9813.33	−0.666047	−0.333023	−	0.942919i	$-0.608069\pi$
$601$	−9813.33	−0.666047	−0.333023	+	0.942919i	$0.608069\pi$
$602$	21345.6i	1.44515i
$603$	0	0
$604$	−1366.77	−0.0920746
$605$	0	0
$606$	0	0
$607$	− 4170.95i	− 0.278902i	−0.990229	−	0.139451i	$-0.955466\pi$
$607$	− 4170.95i	− 0.278902i	0.990229	−	0.139451i	$-0.0445338\pi$
$608$	− 6956.38i	− 0.464011i
$609$	0	0
$610$	0	0
$611$	−6983.50	−0.462393
$612$	0	0
$613$	3157.32i	0.208031i	0.994576	+	0.104016i	$0.0331692\pi$
$613$	3157.32i	0.208031i	−0.994576	+	0.104016i	$0.966831\pi$
$614$	17017.6	1.11853
$615$	0	0
$616$	−6111.45	−0.399736
$617$	2962.95i	0.193329i	0.995317	+	0.0966643i	$0.0308173\pi$
$617$	2962.95i	0.193329i	−0.995317	+	0.0966643i	$0.969183\pi$
$618$	0	0
$619$	−4695.72	−0.304906	−0.152453	−	0.988311i	$-0.548717\pi$
$619$	−4695.72	−0.304906	−0.152453	+	0.988311i	$0.548717\pi$
$620$	0	0
$621$	0	0
$622$	21010.0i	1.35438i
$623$	− 15708.9i	− 1.01022i
$624$	0	0
$625$	0	0
$626$	−13461.1	−0.859446
$627$	0	0
$628$	− 503.447i	− 0.0319900i
$629$	2516.81	0.159542
$630$	0	0
$631$	−17397.6	−1.09760	−0.548800	−	0.835954i	$-0.684915\pi$
$631$	−17397.6	−1.09760	−0.548800	+	0.835954i	$0.684915\pi$
$632$	− 19529.2i	− 1.22916i
$633$	0	0
$634$	−8432.74	−0.528244
$635$	0	0
$636$	0	0
$637$	− 2107.59i	− 0.131092i
$638$	− 5108.47i	− 0.317000i
$639$	0	0
$640$	0	0
$641$	6903.46	0.425383	0.212691	−	0.977119i	$-0.431777\pi$
$641$	6903.46	0.425383	0.212691	+	0.977119i	$0.431777\pi$
$642$	0	0
$643$	12132.1i	0.744079i	0.928217	+	0.372039i	$0.121341\pi$
$643$	12132.1i	0.744079i	−0.928217	+	0.372039i	$0.878659\pi$
$644$	5204.80	0.318475
$645$	0	0
$646$	−3066.24	−0.186749
$647$	− 16784.3i	− 1.01988i	−0.860211	−	0.509939i	$-0.829668\pi$
$647$	− 16784.3i	− 1.01988i	0.860211	−	0.509939i	$-0.170332\pi$
$648$	0	0
$649$	−4607.92	−0.278700
$650$	0	0
$651$	0	0
$652$	− 3192.10i	− 0.191736i
$653$	− 15711.5i	− 0.941561i	−0.882250	−	0.470781i	$-0.843972\pi$
$653$	− 15711.5i	− 0.941561i	0.882250	−	0.470781i	$-0.156028\pi$
$654$	0	0
$655$	0	0
$656$	18698.4	1.11288
$657$	0	0
$658$	35513.3i	2.10403i
$659$	−20227.0	−1.19565	−0.597825	−	0.801626i	$-0.703968\pi$
$659$	−20227.0	−1.19565	−0.597825	+	0.801626i	$0.703968\pi$
$660$	0	0
$661$	14986.8	0.881875	0.440938	−	0.897538i	$-0.354646\pi$
$661$	14986.8	0.881875	0.440938	+	0.897538i	$0.354646\pi$
$662$	26780.5i	1.57229i
$663$	0	0
$664$	−12963.8	−0.757672
$665$	0	0
$666$	0	0
$667$	30749.7i	1.78506i
$668$	4661.59i	0.270004i
$669$	0	0
$670$	0	0
$671$	605.420	0.0348316
$672$	0	0
$673$	− 6833.74i	− 0.391414i	−0.980662	−	0.195707i	$-0.937300\pi$
$673$	− 6833.74i	− 0.391414i	0.980662	−	0.195707i	$-0.0627001\pi$
$674$	9655.19	0.551787
$675$	0	0
$676$	2718.02	0.154644
$677$	30439.3i	1.72803i	0.503467	+	0.864015i	$0.332058\pi$
$677$	30439.3i	1.72803i	−0.503467	+	0.864015i	$0.667942\pi$
$678$	0	0
$679$	16361.0	0.924707
$680$	0	0
$681$	0	0
$682$	− 4025.81i	− 0.226036i
$683$	− 9675.88i	− 0.542075i	−0.962569	−	0.271038i	$-0.912633\pi$
$683$	− 9675.88i	− 0.542075i	0.962569	−	0.271038i	$-0.0873667\pi$
$684$	0	0
$685$	0	0
$686$	9579.94	0.533184
$687$	0	0
$688$	− 18653.8i	− 1.03368i
$689$	−2342.63	−0.129531
$690$	0	0
$691$	13410.6	0.738295	0.369147	−	0.929371i	$-0.379650\pi$
$691$	13410.6	0.738295	0.369147	+	0.929371i	$0.379650\pi$
$692$	4144.94i	0.227698i
$693$	0	0
$694$	509.247	0.0278541
$695$	0	0
$696$	0	0
$697$	3638.93i	0.197754i
$698$	18958.0i	1.02804i
$699$	0	0
$700$	0	0
$701$	12796.5	0.689466	0.344733	−	0.938701i	$-0.387969\pi$
$701$	12796.5	0.689466	0.344733	+	0.938701i	$0.387969\pi$
$702$	0	0
$703$	− 29474.9i	− 1.58132i
$704$	6275.92	0.335984
$705$	0	0
$706$	−11932.5	−0.636100
$707$	22296.2i	1.18604i
$708$	0	0
$709$	−4849.33	−0.256869	−0.128435	−	0.991718i	$-0.540995\pi$
$709$	−4849.33	−0.256869	−0.128435	+	0.991718i	$0.540995\pi$
$710$	0	0
$711$	0	0
$712$	16527.8i	0.869952i
$713$	24232.8i	1.27283i
$714$	0	0
$715$	0	0
$716$	−4349.52	−0.227024
$717$	0	0
$718$	− 2982.40i	− 0.155017i
$719$	33988.7	1.76296	0.881479	−	0.472224i	$-0.156549\pi$
$719$	33988.7	1.76296	0.881479	+	0.472224i	$0.156549\pi$
$720$	0	0
$721$	17381.4	0.897806
$722$	18179.7i	0.937088i
$723$	0	0
$724$	2472.15	0.126902
$725$	0	0
$726$	0	0
$727$	− 16312.3i	− 0.832170i	−0.909326	−	0.416085i	$-0.863402\pi$
$727$	− 16312.3i	− 0.832170i	0.909326	−	0.416085i	$-0.136598\pi$
$728$	6416.96i	0.326687i
$729$	0	0
$730$	0	0
$731$	3630.26	0.183680
$732$	0	0
$733$	37831.8i	1.90634i	0.302430	+	0.953172i	$0.402202\pi$
$733$	37831.8i	1.90634i	−0.302430	+	0.953172i	$0.597798\pi$
$734$	8915.36	0.448327
$735$	0	0
$736$	−10177.5	−0.509710
$737$	5885.84i	0.294176i
$738$	0	0
$739$	16860.9	0.839295	0.419647	−	0.907687i	$-0.362154\pi$
$739$	16860.9	0.839295	0.419647	+	0.907687i	$0.362154\pi$
$740$	0	0
$741$	0	0
$742$	11913.0i	0.589408i
$743$	406.077i	0.0200505i	0.999950	+	0.0100253i	$0.00319119\pi$
$743$	406.077i	0.0200505i	−0.999950	+	0.0100253i	$0.996809\pi$
$744$	0	0
$745$	0	0
$746$	937.999	0.0460357
$747$	0	0
$748$	147.056i	0.00718839i
$749$	−41942.3	−2.04611
$750$	0	0
$751$	−15106.3	−0.734004	−0.367002	−	0.930220i	$-0.619616\pi$
$751$	−15106.3	−0.734004	−0.367002	+	0.930220i	$0.619616\pi$
$752$	− 31034.9i	− 1.50496i
$753$	0	0
$754$	−5363.84	−0.259071
$755$	0	0
$756$	0	0
$757$	− 14265.4i	− 0.684919i	−0.939533	−	0.342460i	$-0.888740\pi$
$757$	− 14265.4i	− 0.684919i	0.939533	−	0.342460i	$-0.111260\pi$
$758$	19954.7i	0.956184i
$759$	0	0
$760$	0	0
$761$	−23001.3	−1.09566	−0.547828	−	0.836591i	$-0.684545\pi$
$761$	−23001.3	−1.09566	−0.547828	+	0.836591i	$0.684545\pi$
$762$	0	0
$763$	31384.9i	1.48913i
$764$	1795.18	0.0850095
$765$	0	0
$766$	14665.7	0.691767
$767$	4838.26i	0.227770i
$768$	0	0
$769$	−12294.6	−0.576535	−0.288267	−	0.957550i	$-0.593079\pi$
$769$	−12294.6	−0.576535	−0.288267	+	0.957550i	$0.593079\pi$
$770$	0	0
$771$	0	0
$772$	− 2945.60i	− 0.137324i
$773$	5938.46i	0.276315i	0.990410	+	0.138158i	$0.0441181\pi$
$773$	5938.46i	0.276315i	−0.990410	+	0.138158i	$0.955882\pi$
$774$	0	0
$775$	0	0
$776$	−17213.9	−0.796316
$777$	0	0
$778$	− 21145.5i	− 0.974423i
$779$	42616.2	1.96006
$780$	0	0
$781$	−10346.2	−0.474027
$782$	4486.04i	0.205141i
$783$	0	0
$784$	9366.19	0.426667
$785$	0	0
$786$	0	0
$787$	32425.0i	1.46865i	0.678798	+	0.734325i	$0.262501\pi$
$787$	32425.0i	1.46865i	−0.678798	+	0.734325i	$0.737499\pi$
$788$	456.994i	0.0206596i
$789$	0	0
$790$	0	0
$791$	13359.9	0.600535
$792$	0	0
$793$	− 635.685i	− 0.0284664i
$794$	33449.7	1.49507
$795$	0	0
$796$	−5535.71	−0.246492
$797$	20990.4i	0.932897i	0.884548	+	0.466449i	$0.154467\pi$
$797$	20990.4i	0.932897i	−0.884548	+	0.466449i	$0.845533\pi$
$798$	0	0
$799$	6039.79	0.267425
$800$	0	0
$801$	0	0
$802$	− 17011.8i	− 0.749013i
$803$	− 6208.58i	− 0.272847i
$804$	0	0
$805$	0	0
$806$	−4227.06	−0.184729
$807$	0	0
$808$	− 23458.4i	− 1.02137i
$809$	−4943.95	−0.214858	−0.107429	−	0.994213i	$-0.534262\pi$
$809$	−4943.95	−0.214858	−0.107429	+	0.994213i	$0.534262\pi$
$810$	0	0
$811$	−19844.5	−0.859230	−0.429615	−	0.903012i	$-0.641351\pi$
$811$	−19844.5	−0.859230	−0.429615	+	0.903012i	$0.641351\pi$
$812$	− 5382.27i	− 0.232612i
$813$	0	0
$814$	7164.03	0.308476
$815$	0	0
$816$	0	0
$817$	− 42514.7i	− 1.82056i
$818$	− 12722.4i	− 0.543798i
$819$	0	0
$820$	0	0
$821$	−12608.2	−0.535969	−0.267985	−	0.963423i	$-0.586358\pi$
$821$	−12608.2	−0.535969	−0.267985	+	0.963423i	$0.586358\pi$
$822$	0	0
$823$	− 24843.4i	− 1.05223i	−0.850413	−	0.526116i	$-0.823648\pi$
$823$	− 24843.4i	− 1.05223i	0.850413	−	0.526116i	$-0.176352\pi$
$824$	−18287.5	−0.773149
$825$	0	0
$826$	24604.1	1.03642
$827$	− 33361.1i	− 1.40276i	−0.712789	−	0.701379i	$-0.752568\pi$
$827$	− 33361.1i	− 1.40276i	0.712789	−	0.701379i	$-0.247432\pi$
$828$	0	0
$829$	5049.62	0.211557	0.105778	−	0.994390i	$-0.466267\pi$
$829$	5049.62	0.211557	0.105778	+	0.994390i	$0.466267\pi$
$830$	0	0
$831$	0	0
$832$	− 6589.65i	− 0.274585i
$833$	1822.78i	0.0758170i
$834$	0	0
$835$	0	0
$836$	1722.21	0.0712485
$837$	0	0
$838$	− 10117.9i	− 0.417087i
$839$	−17464.7	−0.718650	−0.359325	−	0.933213i	$-0.616993\pi$
$839$	−17464.7	−0.718650	−0.359325	+	0.933213i	$0.616993\pi$
$840$	0	0
$841$	7409.22	0.303793
$842$	− 13593.3i	− 0.556359i
$843$	0	0
$844$	6234.06	0.254248
$845$	0	0
$846$	0	0
$847$	27659.3i	1.12206i
$848$	− 10410.7i	− 0.421587i
$849$	0	0
$850$	0	0
$851$	−43122.9	−1.73706
$852$	0	0
$853$	− 2746.47i	− 0.110243i	−0.998480	−	0.0551216i	$-0.982445\pi$
$853$	− 2746.47i	− 0.110243i	0.998480	−	0.0551216i	$-0.0175546\pi$
$854$	−3232.66	−0.129531
$855$	0	0
$856$	44128.7	1.76202
$857$	− 15656.8i	− 0.624069i	−0.950071	−	0.312034i	$-0.898990\pi$
$857$	− 15656.8i	− 0.624069i	0.950071	−	0.312034i	$-0.101010\pi$
$858$	0	0
$859$	34507.3	1.37063	0.685317	−	0.728245i	$-0.259664\pi$
$859$	34507.3	1.37063	0.685317	+	0.728245i	$0.259664\pi$
$860$	0	0
$861$	0	0
$862$	− 37093.3i	− 1.46567i
$863$	26576.7i	1.04830i	0.851626	+	0.524150i	$0.175617\pi$
$863$	26576.7i	1.04830i	−0.851626	+	0.524150i	$0.824383\pi$
$864$	0	0
$865$	0	0
$866$	2231.13	0.0875485
$867$	0	0
$868$	− 4241.58i	− 0.165863i
$869$	8985.73	0.350771
$870$	0	0
$871$	6180.07	0.240418
$872$	− 33020.9i	− 1.28237i
$873$	0	0
$874$	52536.8	2.03328
$875$	0	0
$876$	0	0
$877$	− 13399.5i	− 0.515927i	−0.966155	−	0.257963i	$-0.916949\pi$
$877$	− 13399.5i	− 0.515927i	0.966155	−	0.257963i	$-0.0830514\pi$
$878$	41725.2i	1.60383i
$879$	0	0
$880$	0	0
$881$	44435.4	1.69928	0.849641	−	0.527362i	$-0.176819\pi$
$881$	44435.4	1.69928	0.849641	+	0.527362i	$0.176819\pi$
$882$	0	0
$883$	11162.1i	0.425408i	0.977117	+	0.212704i	$0.0682270\pi$
$883$	11162.1i	0.425408i	−0.977117	+	0.212704i	$0.931773\pi$
$884$	154.408	0.00587477
$885$	0	0
$886$	−15209.7	−0.576725
$887$	− 11591.2i	− 0.438776i	−0.975638	−	0.219388i	$-0.929594\pi$
$887$	− 11591.2i	− 0.438776i	0.975638	−	0.219388i	$-0.0704061\pi$
$888$	0	0
$889$	−55694.4	−2.10116
$890$	0	0
$891$	0	0
$892$	− 3203.96i	− 0.120265i
$893$	− 70733.1i	− 2.65061i
$894$	0	0
$895$	0	0
$896$	−22701.0	−0.846415
$897$	0	0
$898$	− 43576.8i	− 1.61935i
$899$	25059.1	0.929663
$900$	0	0
$901$	2026.06	0.0749144
$902$	10358.1i	0.382359i
$903$	0	0
$904$	−14056.3	−0.517154
$905$	0	0
$906$	0	0
$907$	8652.39i	0.316756i	0.987379	+	0.158378i	$0.0506265\pi$
$907$	8652.39i	0.316756i	−0.987379	+	0.158378i	$0.949374\pi$
$908$	770.859i	0.0281738i
$909$	0	0
$910$	0	0
$911$	36712.4	1.33517	0.667583	−	0.744536i	$-0.267329\pi$
$911$	36712.4	1.33517	0.667583	+	0.744536i	$0.267329\pi$
$912$	0	0
$913$	− 5964.88i	− 0.216220i
$914$	−19704.6	−0.713097
$915$	0	0
$916$	6238.43	0.225026
$917$	11471.3i	0.413104i
$918$	0	0
$919$	42003.3	1.50768	0.753841	−	0.657057i	$-0.228199\pi$
$919$	42003.3	1.50768	0.753841	+	0.657057i	$0.228199\pi$
$920$	0	0
$921$	0	0
$922$	10544.7i	0.376651i
$923$	10863.4i	0.387402i
$924$	0	0
$925$	0	0
$926$	−14214.8	−0.504457
$927$	0	0
$928$	10524.5i	0.372288i
$929$	−13112.6	−0.463090	−0.231545	−	0.972824i	$-0.574378\pi$
$929$	−13112.6	−0.463090	−0.231545	+	0.972824i	$0.574378\pi$
$930$	0	0
$931$	21346.9	0.751468
$932$	1619.61i	0.0569227i
$933$	0	0
$934$	16601.4	0.581600
$935$	0	0
$936$	0	0
$937$	49863.5i	1.73849i	0.494378	+	0.869247i	$0.335396\pi$
$937$	49863.5i	1.73849i	−0.494378	+	0.869247i	$0.664604\pi$
$938$	− 31427.6i	− 1.09397i
$939$	0	0
$940$	0	0
$941$	9352.28	0.323991	0.161996	−	0.986791i	$-0.448207\pi$
$941$	9352.28	0.323991	0.161996	+	0.986791i	$0.448207\pi$
$942$	0	0
$943$	− 62349.3i	− 2.15310i
$944$	−21501.4	−0.741326
$945$	0	0
$946$	10333.4	0.355147
$947$	4011.39i	0.137648i	0.997629	+	0.0688239i	$0.0219247\pi$
$947$	4011.39i	0.137648i	−0.997629	+	0.0688239i	$0.978075\pi$
$948$	0	0
$949$	−6518.94	−0.222986
$950$	0	0
$951$	0	0
$952$	− 5549.80i	− 0.188939i
$953$	− 39128.6i	− 1.33001i	−0.746839	−	0.665005i	$-0.768429\pi$
$953$	− 39128.6i	− 1.33001i	0.746839	−	0.665005i	$-0.231571\pi$
$954$	0	0
$955$	0	0
$956$	−158.929	−0.00537670
$957$	0	0
$958$	513.770i	0.0173269i
$959$	−57761.9	−1.94497
$960$	0	0
$961$	−10042.8	−0.337108
$962$	− 7522.15i	− 0.252104i
$963$	0	0
$964$	2284.22	0.0763174
$965$	0	0
$966$	0	0
$967$	17635.0i	0.586456i	0.956043	+	0.293228i	$0.0947294\pi$
$967$	17635.0i	0.586456i	−0.956043	+	0.293228i	$0.905271\pi$
$968$	− 29101.2i	− 0.966267i
$969$	0	0
$970$	0	0
$971$	−26827.3	−0.886642	−0.443321	−	0.896363i	$-0.646200\pi$
$971$	−26827.3	−0.886642	−0.443321	+	0.896363i	$0.646200\pi$
$972$	0	0
$973$	− 14625.3i	− 0.481877i
$974$	3382.90	0.111288
$975$	0	0
$976$	2825.01	0.0926498
$977$	− 8343.63i	− 0.273221i	−0.990625	−	0.136610i	$-0.956379\pi$
$977$	− 8343.63i	− 0.273221i	0.990625	−	0.136610i	$-0.0436208\pi$
$978$	0	0
$979$	−7604.72	−0.248261
$980$	0	0
$981$	0	0
$982$	32245.8i	1.04787i
$983$	36890.1i	1.19696i	0.801138	+	0.598479i	$0.204228\pi$
$983$	36890.1i	1.19696i	−0.801138	+	0.598479i	$0.795772\pi$
$984$	0	0
$985$	0	0
$986$	4639.00	0.149833
$987$	0	0
$988$	− 1808.30i	− 0.0582283i
$989$	−62200.7	−1.99987
$990$	0	0
$991$	−24259.3	−0.777619	−0.388810	−	0.921318i	$-0.627114\pi$
$991$	−24259.3	−0.777619	−0.388810	+	0.921318i	$0.627114\pi$
$992$	8293.99i	0.265458i
$993$	0	0
$994$	55243.7	1.76280
$995$	0	0
$996$	0	0
$997$	31700.2i	1.00698i	0.864002	+	0.503488i	$0.167950\pi$
$997$	31700.2i	1.00698i	−0.864002	+	0.503488i	$0.832050\pi$
$998$	− 8134.15i	− 0.257998i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.m.649.5		6
3.2	odd	2		675.4.b.n.649.2		6
5.2	odd	4		675.4.a.s.1.1		3
5.3	odd	4		135.4.a.e.1.3	✓	3
5.4	even	2	inner	675.4.b.m.649.2		6
15.2	even	4		675.4.a.p.1.3		3
15.8	even	4		135.4.a.h.1.1	yes	3
15.14	odd	2		675.4.b.n.649.5		6
20.3	even	4		2160.4.a.bi.1.3		3
45.13	odd	12		405.4.e.v.136.1		6
45.23	even	12		405.4.e.q.136.3		6
45.38	even	12		405.4.e.q.271.3		6
45.43	odd	12		405.4.e.v.271.1		6
60.23	odd	4		2160.4.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.3	✓	3	5.3	odd	4
135.4.a.h.1.1	yes	3	15.8	even	4
405.4.e.q.136.3		6	45.23	even	12
405.4.e.q.271.3		6	45.38	even	12
405.4.e.v.136.1		6	45.13	odd	12
405.4.e.v.271.1		6	45.43	odd	12
675.4.a.p.1.3		3	15.2	even	4
675.4.a.s.1.1		3	5.2	odd	4
675.4.b.m.649.2		6	5.4	even	2	inner
675.4.b.m.649.5		6	1.1	even	1	trivial
675.4.b.n.649.2		6	3.2	odd	2
675.4.b.n.649.5		6	15.14	odd	2
2160.4.a.bi.1.3		3	20.3	even	4
2160.4.a.bq.1.3		3	60.23	odd	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.58488i q^{2} +1.31841 q^{4} -22.8935i q^{7} +24.0869i q^{8} +O(q^{10})\)	\(q+2.58488i q^{2} +1.31841 q^{4} -22.8935i q^{7} +24.0869i q^{8} -11.0828 q^{11} +11.6368i q^{13} +59.1769 q^{14} -51.7145 q^{16} -10.0643i q^{17} -117.865 q^{19} -28.6477i q^{22} +172.441i q^{23} -30.0798 q^{26} -30.1831i q^{28} +178.321 q^{29} +140.528 q^{31} +59.0200i q^{32} +26.0149 q^{34} +250.074i q^{37} -304.666i q^{38} -361.569 q^{41} +360.707i q^{43} -14.6117 q^{44} -445.738 q^{46} +600.121i q^{47} -181.114 q^{49} +15.3421i q^{52} +201.312i q^{53} +551.435 q^{56} +460.937i q^{58} +415.772 q^{59} -54.6270 q^{61} +363.248i q^{62} -566.275 q^{64} -531.079i q^{67} -13.2689i q^{68} +933.534 q^{71} +560.199i q^{73} -646.409 q^{74} -155.394 q^{76} +253.725i q^{77} -810.781 q^{79} -934.611i q^{82} +538.210i q^{83} -932.384 q^{86} -266.951i q^{88} +686.173 q^{89} +266.408 q^{91} +227.348i q^{92} -1551.24 q^{94} +714.655i q^{97} -468.156i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 34 q^{4}+O(q^{10}) \) 6 * q - 34 * q^4	\( 6 q - 34 q^{4} - 10 q^{11} + 120 q^{14} + 322 q^{16} + 100 q^{19} - 370 q^{26} + 230 q^{29} - 230 q^{31} - 826 q^{34} - 1160 q^{41} - 2830 q^{44} - 570 q^{46} - 1154 q^{49} + 4380 q^{56} + 760 q^{59} - 304 q^{61} - 5874 q^{64} - 80 q^{71} - 5440 q^{74} - 6552 q^{76} - 2026 q^{79} + 3110 q^{86} + 2040 q^{89} - 1264 q^{91} - 7666 q^{94}+O(q^{100}) \) 6 * q - 34 * q^4 - 10 * q^11 + 120 * q^14 + 322 * q^16 + 100 * q^19 - 370 * q^26 + 230 * q^29 - 230 * q^31 - 826 * q^34 - 1160 * q^41 - 2830 * q^44 - 570 * q^46 - 1154 * q^49 + 4380 * q^56 + 760 * q^59 - 304 * q^61 - 5874 * q^64 - 80 * q^71 - 5440 * q^74 - 6552 * q^76 - 2026 * q^79 + 3110 * q^86 + 2040 * q^89 - 1264 * q^91 - 7666 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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