LMFDB - Embedded newform 1225.4.a.bs.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 82x^{10} + 2261x^{8} - 25924x^{6} + 124444x^{4} - 217392x^{2} + 51984 $ x^12 - 82x^10 + 2261x^8 - 25924x^6 + 124444x^4 - 217392*x^2 + 51984
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-3.43837$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.85259 q^{2} -1.14521 q^{3} +15.5476 q^{4} -5.55723 q^{6} +36.6254 q^{8} -25.6885 q^{9} +O(q^{10})$	\(q+4.85259 q^{2} -1.14521 q^{3} +15.5476 q^{4} -5.55723 q^{6} +36.6254 q^{8} -25.6885 q^{9} -39.7837 q^{11} -17.8053 q^{12} +61.1016 q^{13} +53.3473 q^{16} -79.0682 q^{17} -124.656 q^{18} -2.34584 q^{19} -193.054 q^{22} -186.511 q^{23} -41.9438 q^{24} +296.501 q^{26} +60.3394 q^{27} +62.5696 q^{29} +209.138 q^{31} -34.1312 q^{32} +45.5607 q^{33} -383.685 q^{34} -399.395 q^{36} -207.454 q^{37} -11.3834 q^{38} -69.9741 q^{39} -346.718 q^{41} -271.573 q^{43} -618.542 q^{44} -905.061 q^{46} -539.632 q^{47} -61.0938 q^{48} +90.5497 q^{51} +949.983 q^{52} +594.438 q^{53} +292.802 q^{54} +2.68648 q^{57} +303.624 q^{58} +224.672 q^{59} -15.5088 q^{61} +1014.86 q^{62} -592.403 q^{64} +221.087 q^{66} -503.432 q^{67} -1229.32 q^{68} +213.594 q^{69} -660.472 q^{71} -940.852 q^{72} +694.155 q^{73} -1006.69 q^{74} -36.4722 q^{76} -339.556 q^{78} +169.255 q^{79} +624.488 q^{81} -1682.48 q^{82} +335.518 q^{83} -1317.83 q^{86} -71.6553 q^{87} -1457.10 q^{88} -273.618 q^{89} -2899.80 q^{92} -239.507 q^{93} -2618.61 q^{94} +39.0874 q^{96} -251.125 q^{97} +1021.98 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 44 q^{4} - 140 q^{9}+O(q^{10}) $ 12 * q + 44 * q^4 - 140 * q^9	$ 12 q + 44 q^{4} - 140 q^{9} - 24 q^{11} + 84 q^{16} - 296 q^{29} - 1100 q^{36} + 184 q^{39} - 2760 q^{44} - 2440 q^{46} - 1928 q^{51} - 2332 q^{64} - 1840 q^{71} - 8656 q^{74} - 5032 q^{79} - 1284 q^{81} - 1680 q^{86} + 3504 q^{99}+O(q^{100}) $ 12 * q + 44 * q^4 - 140 * q^9 - 24 * q^11 + 84 * q^16 - 296 * q^29 - 1100 * q^36 + 184 * q^39 - 2760 * q^44 - 2440 * q^46 - 1928 * q^51 - 2332 * q^64 - 1840 * q^71 - 8656 * q^74 - 5032 * q^79 - 1284 * q^81 - 1680 * q^86 + 3504 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.85259	1.71565	0.857824	−	0.513943i	$-0.171816\pi$
$2$	4.85259	1.71565	0.857824	+	0.513943i	$0.171816\pi$
$3$	−1.14521	−0.220396	−0.110198	−	0.993910i	$-0.535148\pi$
$3$	−1.14521	−0.220396	−0.110198	+	0.993910i	$0.535148\pi$
$4$	15.5476	1.94345
$5$	0	0
$5$	0	0
$6$	−5.55723	−0.378122
$7$	0	0
$7$	0	0
$8$	36.6254	1.61863
$9$	−25.6885	−0.951426
$10$	0	0
$11$	−39.7837	−1.09048	−0.545238	−	0.838281i	$-0.683561\pi$
$11$	−39.7837	−1.09048	−0.545238	+	0.838281i	$0.683561\pi$
$12$	−17.8053	−0.428328
$13$	61.1016	1.30358	0.651789	−	0.758400i	$-0.274019\pi$
$13$	61.1016	1.30358	0.651789	+	0.758400i	$0.274019\pi$
$14$	0	0
$15$	0	0
$16$	53.3473	0.833551
$17$	−79.0682	−1.12805	−0.564025	−	0.825758i	$-0.690748\pi$
$17$	−79.0682	−1.12805	−0.564025	+	0.825758i	$0.690748\pi$
$18$	−124.656	−1.63231
$19$	−2.34584	−0.0283249	−0.0141624	−	0.999900i	$-0.504508\pi$
$19$	−2.34584	−0.0283249	−0.0141624	+	0.999900i	$0.504508\pi$
$20$	0	0
$21$	0	0
$22$	−193.054	−1.87087
$23$	−186.511	−1.69088	−0.845440	−	0.534071i	$-0.820662\pi$
$23$	−186.511	−1.69088	−0.845440	+	0.534071i	$0.820662\pi$
$24$	−41.9438	−0.356739
$25$	0	0
$26$	296.501	2.23648
$27$	60.3394	0.430086
$28$	0	0
$29$	62.5696	0.400651	0.200325	−	0.979729i	$-0.435800\pi$
$29$	62.5696	0.400651	0.200325	+	0.979729i	$0.435800\pi$
$30$	0	0
$31$	209.138	1.21169	0.605843	−	0.795584i	$-0.292836\pi$
$31$	209.138	1.21169	0.605843	+	0.795584i	$0.292836\pi$
$32$	−34.1312	−0.188550
$33$	45.5607	0.240336
$34$	−383.685	−1.93534
$35$	0	0
$36$	−399.395	−1.84905
$37$	−207.454	−0.921762	−0.460881	−	0.887462i	$-0.652467\pi$
$37$	−207.454	−0.921762	−0.460881	+	0.887462i	$0.652467\pi$
$38$	−11.3834	−0.0485955
$39$	−69.9741	−0.287303
$40$	0	0
$41$	−346.718	−1.32069	−0.660344	−	0.750964i	$-0.729589\pi$
$41$	−346.718	−1.32069	−0.660344	+	0.750964i	$0.729589\pi$
$42$	0	0
$43$	−271.573	−0.963129	−0.481564	−	0.876411i	$-0.659931\pi$
$43$	−271.573	−0.963129	−0.481564	+	0.876411i	$0.659931\pi$
$44$	−618.542	−2.11929
$45$	0	0
$46$	−905.061	−2.90096
$47$	−539.632	−1.67475	−0.837377	−	0.546625i	$-0.815912\pi$
$47$	−539.632	−1.67475	−0.837377	+	0.546625i	$0.815912\pi$
$48$	−61.0938	−0.183711
$49$	0	0
$50$	0	0
$51$	90.5497	0.248618
$52$	949.983	2.53344
$53$	594.438	1.54061	0.770305	−	0.637676i	$-0.220104\pi$
$53$	594.438	1.54061	0.770305	+	0.637676i	$0.220104\pi$
$54$	292.802	0.737877
$55$	0	0
$56$	0	0
$57$	2.68648	0.00624268
$58$	303.624	0.687376
$59$	224.672	0.495759	0.247879	−	0.968791i	$-0.420266\pi$
$59$	224.672	0.495759	0.247879	+	0.968791i	$0.420266\pi$
$60$	0	0
$61$	−15.5088	−0.0325525	−0.0162762	−	0.999868i	$-0.505181\pi$
$61$	−15.5088	−0.0325525	−0.0162762	+	0.999868i	$0.505181\pi$
$62$	1014.86	2.07883
$63$	0	0
$64$	−592.403	−1.15704
$65$	0	0
$66$	221.087	0.412333
$67$	−503.432	−0.917971	−0.458985	−	0.888444i	$-0.651787\pi$
$67$	−503.432	−0.917971	−0.458985	+	0.888444i	$0.651787\pi$
$68$	−1229.32	−2.19231
$69$	213.594	0.372663
$70$	0	0
$71$	−660.472	−1.10399	−0.551997	−	0.833846i	$-0.686134\pi$
$71$	−660.472	−1.10399	−0.551997	+	0.833846i	$0.686134\pi$
$72$	−940.852	−1.54001
$73$	694.155	1.11294	0.556471	−	0.830867i	$-0.312155\pi$
$73$	694.155	1.11294	0.556471	+	0.830867i	$0.312155\pi$
$74$	−1006.69	−1.58142
$75$	0	0
$76$	−36.4722	−0.0550480
$77$	0	0
$78$	−339.556	−0.492912
$79$	169.255	0.241047	0.120524	−	0.992710i	$-0.461543\pi$
$79$	169.255	0.241047	0.120524	+	0.992710i	$0.461543\pi$
$80$	0	0
$81$	624.488	0.856637
$82$	−1682.48	−2.26584
$83$	335.518	0.443709	0.221854	−	0.975080i	$-0.428789\pi$
$83$	335.518	0.443709	0.221854	+	0.975080i	$0.428789\pi$
$84$	0	0
$85$	0	0
$86$	−1317.83	−1.65239
$87$	−71.6553	−0.0883018
$88$	−1457.10	−1.76508
$89$	−273.618	−0.325882	−0.162941	−	0.986636i	$-0.552098\pi$
$89$	−273.618	−0.325882	−0.162941	+	0.986636i	$0.552098\pi$
$90$	0	0
$91$	0	0
$92$	−2899.80	−3.28614
$93$	−239.507	−0.267051
$94$	−2618.61	−2.87329
$95$	0	0
$96$	39.0874	0.0415557
$97$	−251.125	−0.262865	−0.131433	−	0.991325i	$-0.541958\pi$
$97$	−251.125	−0.262865	−0.131433	+	0.991325i	$0.541958\pi$
$98$	0	0
$99$	1021.98	1.03751
$100$	0	0
$101$	−1402.99	−1.38220	−0.691101	−	0.722758i	$-0.742874\pi$
$101$	−1402.99	−1.38220	−0.691101	+	0.722758i	$0.742874\pi$
$102$	439.400	0.426540
$103$	402.593	0.385133	0.192566	−	0.981284i	$-0.438319\pi$
$103$	402.593	0.385133	0.192566	+	0.981284i	$0.438319\pi$
$104$	2237.87	2.11001
$105$	0	0
$106$	2884.56	2.64314
$107$	451.516	0.407941	0.203971	−	0.978977i	$-0.434615\pi$
$107$	451.516	0.407941	0.203971	+	0.978977i	$0.434615\pi$
$108$	938.133	0.835851
$109$	375.061	0.329581	0.164790	−	0.986329i	$-0.447305\pi$
$109$	375.061	0.329581	0.164790	+	0.986329i	$0.447305\pi$
$110$	0	0
$111$	237.578	0.203153
$112$	0	0
$113$	892.024	0.742607	0.371303	−	0.928512i	$-0.378911\pi$
$113$	892.024	0.742607	0.371303	+	0.928512i	$0.378911\pi$
$114$	13.0364	0.0107102
$115$	0	0
$116$	972.807	0.778645
$117$	−1569.61	−1.24026
$118$	1090.24	0.850548
$119$	0	0
$120$	0	0
$121$	251.744	0.189139
$122$	−75.2579	−0.0558486
$123$	397.064	0.291074
$124$	3251.59	2.35485
$125$	0	0
$126$	0	0
$127$	−1203.39	−0.840813	−0.420407	−	0.907336i	$-0.638113\pi$
$127$	−1203.39	−0.840813	−0.420407	+	0.907336i	$0.638113\pi$
$128$	−2601.64	−1.79652
$129$	311.009	0.212270
$130$	0	0
$131$	−1704.75	−1.13698	−0.568490	−	0.822690i	$-0.692472\pi$
$131$	−1704.75	−1.13698	−0.568490	+	0.822690i	$0.692472\pi$
$132$	708.360	0.467082
$133$	0	0
$134$	−2442.95	−1.57492
$135$	0	0
$136$	−2895.91	−1.82590
$137$	537.841	0.335408	0.167704	−	0.985837i	$-0.446365\pi$
$137$	537.841	0.335408	0.167704	+	0.985837i	$0.446365\pi$
$138$	1036.49	0.639359
$139$	1587.63	0.968786	0.484393	−	0.874851i	$-0.339041\pi$
$139$	1587.63	0.968786	0.484393	+	0.874851i	$0.339041\pi$
$140$	0	0
$141$	617.992	0.369109
$142$	−3205.00	−1.89407
$143$	−2430.85	−1.42152
$144$	−1370.41	−0.793062
$145$	0	0
$146$	3368.45	1.90942
$147$	0	0
$148$	−3225.41	−1.79140
$149$	−1752.59	−0.963610	−0.481805	−	0.876278i	$-0.660019\pi$
$149$	−1752.59	−0.963610	−0.481805	+	0.876278i	$0.660019\pi$
$150$	0	0
$151$	2072.99	1.11720	0.558602	−	0.829436i	$-0.311338\pi$
$151$	2072.99	1.11720	0.558602	+	0.829436i	$0.311338\pi$
$152$	−85.9173	−0.0458475
$153$	2031.14	1.07326
$154$	0	0
$155$	0	0
$156$	−1087.93	−0.558360
$157$	1091.08	0.554636	0.277318	−	0.960778i	$-0.410555\pi$
$157$	1091.08	0.554636	0.277318	+	0.960778i	$0.410555\pi$
$158$	821.327	0.413552
$159$	−680.756	−0.339544
$160$	0	0
$161$	0	0
$162$	3030.38	1.46969
$163$	1118.05	0.537254	0.268627	−	0.963244i	$-0.413430\pi$
$163$	1118.05	0.537254	0.268627	+	0.963244i	$0.413430\pi$
$164$	−5390.63	−2.56669
$165$	0	0
$166$	1628.13	0.761249
$167$	3355.23	1.55470	0.777352	−	0.629066i	$-0.216562\pi$
$167$	3355.23	1.55470	0.777352	+	0.629066i	$0.216562\pi$
$168$	0	0
$169$	1536.40	0.699318
$170$	0	0
$171$	60.2610	0.0269490
$172$	−4222.32	−1.87179
$173$	−3269.03	−1.43665	−0.718323	−	0.695709i	$-0.755090\pi$
$173$	−3269.03	−1.43665	−0.718323	+	0.695709i	$0.755090\pi$
$174$	−347.714	−0.151495
$175$	0	0
$176$	−2122.35	−0.908967
$177$	−257.296	−0.109263
$178$	−1327.76	−0.559099
$179$	−2128.04	−0.888587	−0.444293	−	0.895881i	$-0.646545\pi$
$179$	−2128.04	−0.888587	−0.444293	+	0.895881i	$0.646545\pi$
$180$	0	0
$181$	922.343	0.378769	0.189385	−	0.981903i	$-0.439351\pi$
$181$	922.343	0.378769	0.189385	+	0.981903i	$0.439351\pi$
$182$	0	0
$183$	17.7609	0.00717443
$184$	−6831.05	−2.73691
$185$	0	0
$186$	−1162.23	−0.458165
$187$	3145.63	1.23011
$188$	−8389.99	−3.25480
$189$	0	0
$190$	0	0
$191$	−1108.17	−0.419814	−0.209907	−	0.977721i	$-0.567316\pi$
$191$	−1108.17	−0.419814	−0.209907	+	0.977721i	$0.567316\pi$
$192$	678.426	0.255006
$193$	279.047	0.104074	0.0520368	−	0.998645i	$-0.483429\pi$
$193$	279.047	0.104074	0.0520368	+	0.998645i	$0.483429\pi$
$194$	−1218.61	−0.450984
$195$	0	0
$196$	0	0
$197$	1232.54	0.445759	0.222879	−	0.974846i	$-0.428454\pi$
$197$	1232.54	0.445759	0.222879	+	0.974846i	$0.428454\pi$
$198$	4959.26	1.78000
$199$	1500.64	0.534562	0.267281	−	0.963619i	$-0.413875\pi$
$199$	1500.64	0.534562	0.267281	+	0.963619i	$0.413875\pi$
$200$	0	0
$201$	576.536	0.202317
$202$	−6808.12	−2.37137
$203$	0	0
$204$	1407.83	0.483176
$205$	0	0
$206$	1953.62	0.660753
$207$	4791.19	1.60875
$208$	3259.60	1.08660
$209$	93.3261	0.0308876
$210$	0	0
$211$	−3971.90	−1.29591	−0.647955	−	0.761679i	$-0.724376\pi$
$211$	−3971.90	−1.29591	−0.647955	+	0.761679i	$0.724376\pi$
$212$	9242.08	2.99410
$213$	756.379	0.243316
$214$	2191.02	0.699884
$215$	0	0
$216$	2209.96	0.696150
$217$	0	0
$218$	1820.01	0.565445
$219$	−794.954	−0.245288
$220$	0	0
$221$	−4831.19	−1.47050
$222$	1152.87	0.348538
$223$	2861.98	0.859429	0.429715	−	0.902965i	$-0.358614\pi$
$223$	2861.98	0.859429	0.429715	+	0.902965i	$0.358614\pi$
$224$	0	0
$225$	0	0
$226$	4328.62	1.27405
$227$	−2070.96	−0.605525	−0.302763	−	0.953066i	$-0.597909\pi$
$227$	−2070.96	−0.605525	−0.302763	+	0.953066i	$0.597909\pi$
$228$	41.7683	0.0121323
$229$	5758.22	1.66163	0.830816	−	0.556547i	$-0.187874\pi$
$229$	5758.22	1.66163	0.830816	+	0.556547i	$0.187874\pi$
$230$	0	0
$231$	0	0
$232$	2291.64	0.648506
$233$	1649.99	0.463925	0.231962	−	0.972725i	$-0.425485\pi$
$233$	1649.99	0.463925	0.231962	+	0.972725i	$0.425485\pi$
$234$	−7616.66	−2.12785
$235$	0	0
$236$	3493.11	0.963482
$237$	−193.833	−0.0531258
$238$	0	0
$239$	−4008.31	−1.08484	−0.542419	−	0.840108i	$-0.682491\pi$
$239$	−4008.31	−1.08484	−0.542419	+	0.840108i	$0.682491\pi$
$240$	0	0
$241$	−3151.21	−0.842271	−0.421136	−	0.906998i	$-0.638368\pi$
$241$	−3151.21	−0.842271	−0.421136	+	0.906998i	$0.638368\pi$
$242$	1221.61	0.324496
$243$	−2344.33	−0.618885
$244$	−241.125	−0.0632641
$245$	0	0
$246$	1926.79	0.499381
$247$	−143.334	−0.0369237
$248$	7659.76	1.96127
$249$	−384.238	−0.0977916
$250$	0	0
$251$	−3531.54	−0.888084	−0.444042	−	0.896006i	$-0.646456\pi$
$251$	−3531.54	−0.888084	−0.444042	+	0.896006i	$0.646456\pi$
$252$	0	0
$253$	7420.10	1.84386
$254$	−5839.54	−1.44254
$255$	0	0
$256$	−7885.45	−1.92516
$257$	5685.53	1.37997	0.689987	−	0.723822i	$-0.257616\pi$
$257$	5685.53	1.37997	0.689987	+	0.723822i	$0.257616\pi$
$258$	1509.20	0.364180
$259$	0	0
$260$	0	0
$261$	−1607.32	−0.381189
$262$	−8272.43	−1.95066
$263$	4441.96	1.04146	0.520728	−	0.853723i	$-0.325661\pi$
$263$	4441.96	1.04146	0.520728	+	0.853723i	$0.325661\pi$
$264$	1668.68	0.389016
$265$	0	0
$266$	0	0
$267$	313.351	0.0718230
$268$	−7827.17	−1.78403
$269$	1679.13	0.380589	0.190294	−	0.981727i	$-0.439056\pi$
$269$	1679.13	0.380589	0.190294	+	0.981727i	$0.439056\pi$
$270$	0	0
$271$	1141.77	0.255931	0.127966	−	0.991779i	$-0.459155\pi$
$271$	1141.77	0.255931	0.127966	+	0.991779i	$0.459155\pi$
$272$	−4218.07	−0.940287
$273$	0	0
$274$	2609.92	0.575442
$275$	0	0
$276$	3320.88	0.724252
$277$	2714.61	0.588828	0.294414	−	0.955678i	$-0.404876\pi$
$277$	2714.61	0.588828	0.294414	+	0.955678i	$0.404876\pi$
$278$	7704.12	1.66210
$279$	−5372.44	−1.15283
$280$	0	0
$281$	−5343.68	−1.13444	−0.567219	−	0.823567i	$-0.691981\pi$
$281$	−5343.68	−1.13444	−0.567219	+	0.823567i	$0.691981\pi$
$282$	2998.86	0.633261
$283$	−6798.47	−1.42801	−0.714006	−	0.700140i	$-0.753121\pi$
$283$	−6798.47	−1.42801	−0.714006	+	0.700140i	$0.753121\pi$
$284$	−10268.8	−2.14556
$285$	0	0
$286$	−11795.9	−2.43883
$287$	0	0
$288$	876.780	0.179391
$289$	1338.78	0.272497
$290$	0	0
$291$	287.591	0.0579344
$292$	10792.5	2.16295
$293$	5119.59	1.02078	0.510392	−	0.859942i	$-0.329500\pi$
$293$	5119.59	1.02078	0.510392	+	0.859942i	$0.329500\pi$
$294$	0	0
$295$	0	0
$296$	−7598.09	−1.49199
$297$	−2400.53	−0.468999
$298$	−8504.61	−1.65322
$299$	−11396.1	−2.20420
$300$	0	0
$301$	0	0
$302$	10059.4	1.91673
$303$	1606.71	0.304632
$304$	−125.144	−0.0236102
$305$	0	0
$306$	9856.30	1.84133
$307$	−4188.66	−0.778695	−0.389347	−	0.921091i	$-0.627299\pi$
$307$	−4188.66	−0.778695	−0.389347	+	0.921091i	$0.627299\pi$
$308$	0	0
$309$	−461.054	−0.0848817
$310$	0	0
$311$	9759.18	1.77940	0.889699	−	0.456548i	$-0.150914\pi$
$311$	9759.18	1.77940	0.889699	+	0.456548i	$0.150914\pi$
$312$	−2562.83	−0.465038
$313$	5960.70	1.07642	0.538209	−	0.842811i	$-0.319101\pi$
$313$	5960.70	1.07642	0.538209	+	0.842811i	$0.319101\pi$
$314$	5294.57	0.951560
$315$	0	0
$316$	2631.52	0.468463
$317$	−2722.35	−0.482342	−0.241171	−	0.970483i	$-0.577531\pi$
$317$	−2722.35	−0.482342	−0.241171	+	0.970483i	$0.577531\pi$
$318$	−3303.43	−0.582538
$319$	−2489.25	−0.436900
$320$	0	0
$321$	−517.081	−0.0899085
$322$	0	0
$323$	185.481	0.0319519
$324$	9709.30	1.66483
$325$	0	0
$326$	5425.44	0.921740
$327$	−429.523	−0.0726382
$328$	−12698.7	−2.13771
$329$	0	0
$330$	0	0
$331$	378.753	0.0628947	0.0314474	−	0.999505i	$-0.489988\pi$
$331$	378.753	0.0628947	0.0314474	+	0.999505i	$0.489988\pi$
$332$	5216.50	0.862327
$333$	5329.18	0.876988
$334$	16281.6	2.66733
$335$	0	0
$336$	0	0
$337$	8031.54	1.29824	0.649118	−	0.760687i	$-0.275138\pi$
$337$	8031.54	1.29824	0.649118	+	0.760687i	$0.275138\pi$
$338$	7455.52	1.19978
$339$	−1021.55	−0.163667
$340$	0	0
$341$	−8320.28	−1.32132
$342$	292.422	0.0462350
$343$	0	0
$344$	−9946.49	−1.55895
$345$	0	0
$346$	−15863.3	−2.46478
$347$	−650.137	−0.100580	−0.0502899	−	0.998735i	$-0.516015\pi$
$347$	−650.137	−0.100580	−0.0502899	+	0.998735i	$0.516015\pi$
$348$	−1114.07	−0.171610
$349$	1240.68	0.190292	0.0951462	−	0.995463i	$-0.469668\pi$
$349$	1240.68	0.190292	0.0951462	+	0.995463i	$0.469668\pi$
$350$	0	0
$351$	3686.83	0.560651
$352$	1357.87	0.205609
$353$	−7873.85	−1.18720	−0.593601	−	0.804759i	$-0.702294\pi$
$353$	−7873.85	−1.18720	−0.593601	+	0.804759i	$0.702294\pi$
$354$	−1248.55	−0.187457
$355$	0	0
$356$	−4254.11	−0.633336
$357$	0	0
$358$	−10326.5	−1.52450
$359$	−406.721	−0.0597936	−0.0298968	−	0.999553i	$-0.509518\pi$
$359$	−406.721	−0.0597936	−0.0298968	+	0.999553i	$0.509518\pi$
$360$	0	0
$361$	−6853.50	−0.999198
$362$	4475.75	0.649835
$363$	−288.299	−0.0416854
$364$	0	0
$365$	0	0
$366$	86.1861	0.0123088
$367$	−654.275	−0.0930597	−0.0465298	−	0.998917i	$-0.514816\pi$
$367$	−654.275	−0.0930597	−0.0465298	+	0.998917i	$0.514816\pi$
$368$	−9949.85	−1.40943
$369$	8906.65	1.25654
$370$	0	0
$371$	0	0
$372$	−3723.76	−0.519000
$373$	−5500.41	−0.763540	−0.381770	−	0.924257i	$-0.624685\pi$
$373$	−5500.41	−0.763540	−0.381770	+	0.924257i	$0.624685\pi$
$374$	15264.4	2.11044
$375$	0	0
$376$	−19764.3	−2.71081
$377$	3823.10	0.522280
$378$	0	0
$379$	−1771.48	−0.240091	−0.120046	−	0.992768i	$-0.538304\pi$
$379$	−1771.48	−0.240091	−0.120046	+	0.992768i	$0.538304\pi$
$380$	0	0
$381$	1378.13	0.185312
$382$	−5377.50	−0.720254
$383$	−8905.25	−1.18809	−0.594043	−	0.804433i	$-0.702469\pi$
$383$	−8905.25	−1.18809	−0.594043	+	0.804433i	$0.702469\pi$
$384$	2979.42	0.395945
$385$	0	0
$386$	1354.10	0.178554
$387$	6976.31	0.916346
$388$	−3904.40	−0.510865
$389$	9373.48	1.22173	0.610866	−	0.791734i	$-0.290821\pi$
$389$	9373.48	1.22173	0.610866	+	0.791734i	$0.290821\pi$
$390$	0	0
$391$	14747.1	1.90740
$392$	0	0
$393$	1952.29	0.250586
$394$	5980.98	0.764766
$395$	0	0
$396$	15889.4	2.01634
$397$	−10498.8	−1.32726	−0.663628	−	0.748063i	$-0.730984\pi$
$397$	−10498.8	−1.32726	−0.663628	+	0.748063i	$0.730984\pi$
$398$	7282.01	0.917121
$399$	0	0
$400$	0	0
$401$	−2798.54	−0.348509	−0.174255	−	0.984701i	$-0.555752\pi$
$401$	−2798.54	−0.348509	−0.174255	+	0.984701i	$0.555752\pi$
$402$	2797.69	0.347105
$403$	12778.7	1.57953
$404$	−21813.1	−2.68624
$405$	0	0
$406$	0	0
$407$	8253.28	1.00516
$408$	3316.42	0.402420
$409$	−7590.18	−0.917629	−0.458815	−	0.888532i	$-0.651726\pi$
$409$	−7590.18	−0.917629	−0.458815	+	0.888532i	$0.651726\pi$
$410$	0	0
$411$	−615.941	−0.0739225
$412$	6259.36	0.748487
$413$	0	0
$414$	23249.7	2.76004
$415$	0	0
$416$	−2085.47	−0.245790
$417$	−1818.17	−0.213516
$418$	452.873	0.0529922
$419$	11070.5	1.29076	0.645382	−	0.763860i	$-0.276698\pi$
$419$	11070.5	1.29076	0.645382	+	0.763860i	$0.276698\pi$
$420$	0	0
$421$	8891.82	1.02936	0.514680	−	0.857382i	$-0.327911\pi$
$421$	8891.82	1.02936	0.514680	+	0.857382i	$0.327911\pi$
$422$	−19274.0	−2.22333
$423$	13862.3	1.59340
$424$	21771.5	2.49368
$425$	0	0
$426$	3670.40	0.417444
$427$	0	0
$428$	7019.99	0.792814
$429$	2783.83	0.313297
$430$	0	0
$431$	15239.7	1.70318	0.851592	−	0.524205i	$-0.175638\pi$
$431$	15239.7	1.70318	0.851592	+	0.524205i	$0.175638\pi$
$432$	3218.94	0.358499
$433$	15959.0	1.77123	0.885615	−	0.464421i	$-0.153737\pi$
$433$	15959.0	1.77123	0.885615	+	0.464421i	$0.153737\pi$
$434$	0	0
$435$	0	0
$436$	5831.30	0.640524
$437$	437.525	0.0478939
$438$	−3857.58	−0.420828
$439$	−10834.5	−1.17791	−0.588953	−	0.808167i	$-0.700460\pi$
$439$	−10834.5	−1.17791	−0.588953	+	0.808167i	$0.700460\pi$
$440$	0	0
$441$	0	0
$442$	−23443.8	−2.52287
$443$	8253.23	0.885153	0.442577	−	0.896731i	$-0.354065\pi$
$443$	8253.23	0.885153	0.442577	+	0.896731i	$0.354065\pi$
$444$	3693.77	0.394817
$445$	0	0
$446$	13888.0	1.47448
$447$	2007.09	0.212376
$448$	0	0
$449$	−11022.5	−1.15853	−0.579267	−	0.815138i	$-0.696661\pi$
$449$	−11022.5	−1.15853	−0.579267	+	0.815138i	$0.696661\pi$
$450$	0	0
$451$	13793.7	1.44018
$452$	13868.8	1.44322
$453$	−2374.01	−0.246227
$454$	−10049.5	−1.03887
$455$	0	0
$456$	98.3934	0.0101046
$457$	9328.58	0.954864	0.477432	−	0.878669i	$-0.341568\pi$
$457$	9328.58	0.954864	0.477432	+	0.878669i	$0.341568\pi$
$458$	27942.3	2.85078
$459$	−4770.93	−0.485159
$460$	0	0
$461$	16806.4	1.69794	0.848972	−	0.528438i	$-0.177222\pi$
$461$	16806.4	1.69794	0.848972	+	0.528438i	$0.177222\pi$
$462$	0	0
$463$	8238.55	0.826950	0.413475	−	0.910515i	$-0.364315\pi$
$463$	8238.55	0.826950	0.413475	+	0.910515i	$0.364315\pi$
$464$	3337.91	0.333963
$465$	0	0
$466$	8006.73	0.795932
$467$	−19564.1	−1.93859	−0.969293	−	0.245909i	$-0.920914\pi$
$467$	−19564.1	−1.93859	−0.969293	+	0.245909i	$0.920914\pi$
$468$	−24403.6	−2.41038
$469$	0	0
$470$	0	0
$471$	−1249.52	−0.122239
$472$	8228.70	0.802450
$473$	10804.2	1.05027
$474$	−940.592	−0.0911452
$475$	0	0
$476$	0	0
$477$	−15270.2	−1.46578
$478$	−19450.7	−1.86120
$479$	−1253.34	−0.119554	−0.0597770	−	0.998212i	$-0.519039\pi$
$479$	−1253.34	−0.119554	−0.0597770	+	0.998212i	$0.519039\pi$
$480$	0	0
$481$	−12675.8	−1.20159
$482$	−15291.5	−1.44504
$483$	0	0
$484$	3914.01	0.367582
$485$	0	0
$486$	−11376.1	−1.06179
$487$	3563.34	0.331561	0.165781	−	0.986163i	$-0.446986\pi$
$487$	3563.34	0.331561	0.165781	+	0.986163i	$0.446986\pi$
$488$	−568.017	−0.0526904
$489$	−1280.40	−0.118409
$490$	0	0
$491$	−10148.4	−0.932774	−0.466387	−	0.884581i	$-0.654445\pi$
$491$	−10148.4	−0.932774	−0.466387	+	0.884581i	$0.654445\pi$
$492$	6173.40	0.565688
$493$	−4947.26	−0.451954
$494$	−695.543	−0.0633481
$495$	0	0
$496$	11156.9	1.01000
$497$	0	0
$498$	−1864.55	−0.167776
$499$	8210.56	0.736583	0.368292	−	0.929710i	$-0.379943\pi$
$499$	8210.56	0.736583	0.368292	+	0.929710i	$0.379943\pi$
$500$	0	0
$501$	−3842.45	−0.342650
$502$	−17137.1	−1.52364
$503$	8479.88	0.751688	0.375844	−	0.926683i	$-0.377353\pi$
$503$	8479.88	0.751688	0.375844	+	0.926683i	$0.377353\pi$
$504$	0	0
$505$	0	0
$506$	36006.7	3.16342
$507$	−1759.50	−0.154127
$508$	−18709.8	−1.63408
$509$	−7023.01	−0.611570	−0.305785	−	0.952101i	$-0.598919\pi$
$509$	−7023.01	−0.611570	−0.305785	+	0.952101i	$0.598919\pi$
$510$	0	0
$511$	0	0
$512$	−17451.7	−1.50638
$513$	−141.546	−0.0121821
$514$	27589.5	2.36755
$515$	0	0
$516$	4835.44	0.412536
$517$	21468.6	1.82628
$518$	0	0
$519$	3743.73	0.316631
$520$	0	0
$521$	−6558.50	−0.551503	−0.275751	−	0.961229i	$-0.588927\pi$
$521$	−6558.50	−0.551503	−0.275751	+	0.961229i	$0.588927\pi$
$522$	−7799.65	−0.653987
$523$	106.128	0.00887313	0.00443656	−	0.999990i	$-0.498588\pi$
$523$	106.128	0.00887313	0.00443656	+	0.999990i	$0.498588\pi$
$524$	−26504.7	−2.20966
$525$	0	0
$526$	21555.0	1.78677
$527$	−16536.2	−1.36684
$528$	2430.54	0.200333
$529$	22619.4	1.85907
$530$	0	0
$531$	−5771.48	−0.471677
$532$	0	0
$533$	−21185.0	−1.72162
$534$	1520.56	0.123223
$535$	0	0
$536$	−18438.4	−1.48586
$537$	2437.05	0.195841
$538$	8148.12	0.652956
$539$	0	0
$540$	0	0
$541$	−1370.91	−0.108946	−0.0544732	−	0.998515i	$-0.517348\pi$
$541$	−1370.91	−0.108946	−0.0544732	+	0.998515i	$0.517348\pi$
$542$	5540.52	0.439088
$543$	−1056.28	−0.0834791
$544$	2698.69	0.212694
$545$	0	0
$546$	0	0
$547$	−5324.93	−0.416229	−0.208115	−	0.978104i	$-0.566733\pi$
$547$	−5324.93	−0.416229	−0.208115	+	0.978104i	$0.566733\pi$
$548$	8362.14	0.651849
$549$	398.398	0.0309712
$550$	0	0
$551$	−146.778	−0.0113484
$552$	7822.98	0.603203
$553$	0	0
$554$	13172.9	1.01022
$555$	0	0
$556$	24683.9	1.88279
$557$	−15886.0	−1.20846	−0.604229	−	0.796811i	$-0.706519\pi$
$557$	−15886.0	−1.20846	−0.604229	+	0.796811i	$0.706519\pi$
$558$	−26070.2	−1.97785
$559$	−16593.6	−1.25551
$560$	0	0
$561$	−3602.40	−0.271112
$562$	−25930.7	−1.94630
$563$	−17941.9	−1.34309	−0.671546	−	0.740963i	$-0.734370\pi$
$563$	−17941.9	−1.34309	−0.671546	+	0.740963i	$0.734370\pi$
$564$	9608.30	0.717345
$565$	0	0
$566$	−32990.2	−2.44997
$567$	0	0
$568$	−24190.1	−1.78696
$569$	−18807.0	−1.38565	−0.692823	−	0.721108i	$-0.743633\pi$
$569$	−18807.0	−1.38565	−0.692823	+	0.721108i	$0.743633\pi$
$570$	0	0
$571$	−5931.50	−0.434721	−0.217360	−	0.976091i	$-0.569745\pi$
$571$	−5931.50	−0.434721	−0.217360	+	0.976091i	$0.569745\pi$
$572$	−37793.9	−2.76266
$573$	1269.09	0.0925253
$574$	0	0
$575$	0	0
$576$	15217.9	1.10083
$577$	−11489.1	−0.828935	−0.414467	−	0.910064i	$-0.636032\pi$
$577$	−11489.1	−0.828935	−0.414467	+	0.910064i	$0.636032\pi$
$578$	6496.54	0.467510
$579$	−319.567	−0.0229374
$580$	0	0
$581$	0	0
$582$	1395.56	0.0993950
$583$	−23648.9	−1.68000
$584$	25423.7	1.80144
$585$	0	0
$586$	24843.2	1.75131
$587$	4853.49	0.341269	0.170634	−	0.985334i	$-0.445418\pi$
$587$	4853.49	0.341269	0.170634	+	0.985334i	$0.445418\pi$
$588$	0	0
$589$	−490.604	−0.0343208
$590$	0	0
$591$	−1411.51	−0.0982434
$592$	−11067.1	−0.768336
$593$	−2688.13	−0.186152	−0.0930762	−	0.995659i	$-0.529670\pi$
$593$	−2688.13	−0.186152	−0.0930762	+	0.995659i	$0.529670\pi$
$594$	−11648.8	−0.804637
$595$	0	0
$596$	−27248.6	−1.87273
$597$	−1718.55	−0.117815
$598$	−55300.7	−3.78163
$599$	−22958.1	−1.56602	−0.783008	−	0.622012i	$-0.786316\pi$
$599$	−22958.1	−1.56602	−0.783008	+	0.622012i	$0.786316\pi$
$600$	0	0
$601$	−7566.12	−0.513525	−0.256762	−	0.966475i	$-0.582656\pi$
$601$	−7566.12	−0.513525	−0.256762	+	0.966475i	$0.582656\pi$
$602$	0	0
$603$	12932.4	0.873381
$604$	32230.1	2.17123
$605$	0	0
$606$	7796.73	0.522641
$607$	23195.2	1.55101	0.775505	−	0.631341i	$-0.217495\pi$
$607$	23195.2	1.55101	0.775505	+	0.631341i	$0.217495\pi$
$608$	80.0663	0.00534065
$609$	0	0
$610$	0	0
$611$	−32972.4	−2.18317
$612$	31579.4	2.08582
$613$	−9426.74	−0.621113	−0.310557	−	0.950555i	$-0.600515\pi$
$613$	−9426.74	−0.621113	−0.310557	+	0.950555i	$0.600515\pi$
$614$	−20325.8	−1.33597
$615$	0	0
$616$	0	0
$617$	811.353	0.0529398	0.0264699	−	0.999650i	$-0.491573\pi$
$617$	811.353	0.0529398	0.0264699	+	0.999650i	$0.491573\pi$
$618$	−2237.30	−0.145627
$619$	−9282.87	−0.602763	−0.301381	−	0.953504i	$-0.597448\pi$
$619$	−9282.87	−0.602763	−0.301381	+	0.953504i	$0.597448\pi$
$620$	0	0
$621$	−11254.0	−0.727224
$622$	47357.3	3.05282
$623$	0	0
$624$	−3732.93	−0.239482
$625$	0	0
$626$	28924.8	1.84676
$627$	−106.878	−0.00680749
$628$	16963.7	1.07791
$629$	16403.0	1.03979
$630$	0	0
$631$	−26078.4	−1.64527	−0.822634	−	0.568572i	$-0.807496\pi$
$631$	−26078.4	−1.64527	−0.822634	+	0.568572i	$0.807496\pi$
$632$	6199.05	0.390166
$633$	4548.66	0.285613
$634$	−13210.4	−0.827530
$635$	0	0
$636$	−10584.1	−0.659887
$637$	0	0
$638$	−12079.3	−0.749567
$639$	16966.5	1.05037
$640$	0	0
$641$	−30971.4	−1.90842	−0.954210	−	0.299136i	$-0.903302\pi$
$641$	−30971.4	−1.90842	−0.954210	+	0.299136i	$0.903302\pi$
$642$	−2509.18	−0.154251
$643$	7482.92	0.458939	0.229469	−	0.973316i	$-0.426301\pi$
$643$	7482.92	0.458939	0.229469	+	0.973316i	$0.426301\pi$
$644$	0	0
$645$	0	0
$646$	900.064	0.0548182
$647$	27020.3	1.64185	0.820925	−	0.571037i	$-0.193459\pi$
$647$	27020.3	1.64185	0.820925	+	0.571037i	$0.193459\pi$
$648$	22872.1	1.38658
$649$	−8938.27	−0.540613
$650$	0	0
$651$	0	0
$652$	17383.0	1.04413
$653$	31321.5	1.87704	0.938518	−	0.345231i	$-0.112199\pi$
$653$	31321.5	1.87704	0.938518	+	0.345231i	$0.112199\pi$
$654$	−2084.30	−0.124622
$655$	0	0
$656$	−18496.4	−1.10086
$657$	−17831.8	−1.05888
$658$	0	0
$659$	−14051.9	−0.830627	−0.415314	−	0.909678i	$-0.636328\pi$
$659$	−14051.9	−0.830627	−0.415314	+	0.909678i	$0.636328\pi$
$660$	0	0
$661$	1196.10	0.0703828	0.0351914	−	0.999381i	$-0.488796\pi$
$661$	1196.10	0.0703828	0.0351914	+	0.999381i	$0.488796\pi$
$662$	1837.93	0.107905
$663$	5532.73	0.324093
$664$	12288.5	0.718201
$665$	0	0
$666$	25860.3	1.50460
$667$	−11669.9	−0.677452
$668$	52165.8	3.02149
$669$	−3277.57	−0.189415
$670$	0	0
$671$	616.998	0.0354977
$672$	0	0
$673$	4801.54	0.275016	0.137508	−	0.990501i	$-0.456091\pi$
$673$	4801.54	0.275016	0.137508	+	0.990501i	$0.456091\pi$
$674$	38973.7	2.22732
$675$	0	0
$676$	23887.4	1.35909
$677$	1446.40	0.0821117	0.0410558	−	0.999157i	$-0.486928\pi$
$677$	1446.40	0.0821117	0.0410558	+	0.999157i	$0.486928\pi$
$678$	−4957.18	−0.280796
$679$	0	0
$680$	0	0
$681$	2371.68	0.133455
$682$	−40374.9	−2.26691
$683$	−13408.7	−0.751202	−0.375601	−	0.926781i	$-0.622564\pi$
$683$	−13408.7	−0.751202	−0.375601	+	0.926781i	$0.622564\pi$
$684$	936.915	0.0523740
$685$	0	0
$686$	0	0
$687$	−6594.37	−0.366217
$688$	−14487.7	−0.802817
$689$	36321.1	2.00831
$690$	0	0
$691$	2613.48	0.143881	0.0719403	−	0.997409i	$-0.477081\pi$
$691$	2613.48	0.143881	0.0719403	+	0.997409i	$0.477081\pi$
$692$	−50825.6	−2.79205
$693$	0	0
$694$	−3154.85	−0.172560
$695$	0	0
$696$	−2624.41	−0.142928
$697$	27414.3	1.48980
$698$	6020.51	0.326475
$699$	−1889.59	−0.102247
$700$	0	0
$701$	−4957.94	−0.267131	−0.133566	−	0.991040i	$-0.542643\pi$
$701$	−4957.94	−0.267131	−0.133566	+	0.991040i	$0.542643\pi$
$702$	17890.7	0.961880
$703$	486.653	0.0261088
$704$	23568.0	1.26172
$705$	0	0
$706$	−38208.5	−2.03682
$707$	0	0
$708$	−4000.34	−0.212347
$709$	−31741.5	−1.68135	−0.840676	−	0.541539i	$-0.817842\pi$
$709$	−31741.5	−1.68135	−0.840676	+	0.541539i	$0.817842\pi$
$710$	0	0
$711$	−4347.92	−0.229338
$712$	−10021.4	−0.527483
$713$	−39006.5	−2.04882
$714$	0	0
$715$	0	0
$716$	−33085.9	−1.72692
$717$	4590.36	0.239094
$718$	−1973.65	−0.102585
$719$	24879.1	1.29045	0.645224	−	0.763993i	$-0.276764\pi$
$719$	24879.1	1.29045	0.645224	+	0.763993i	$0.276764\pi$
$720$	0	0
$721$	0	0
$722$	−33257.2	−1.71427
$723$	3608.80	0.185633
$724$	14340.2	0.736119
$725$	0	0
$726$	−1399.00	−0.0715175
$727$	−5429.27	−0.276974	−0.138487	−	0.990364i	$-0.544224\pi$
$727$	−5429.27	−0.276974	−0.138487	+	0.990364i	$0.544224\pi$
$728$	0	0
$729$	−14176.4	−0.720237
$730$	0	0
$731$	21472.8	1.08646
$732$	276.139	0.0139431
$733$	−12390.7	−0.624365	−0.312183	−	0.950022i	$-0.601060\pi$
$733$	−12390.7	−0.624365	−0.312183	+	0.950022i	$0.601060\pi$
$734$	−3174.93	−0.159658
$735$	0	0
$736$	6365.85	0.318816
$737$	20028.4	1.00103
$738$	43220.3	2.15577
$739$	−8174.44	−0.406903	−0.203452	−	0.979085i	$-0.565216\pi$
$739$	−8174.44	−0.406903	−0.203452	+	0.979085i	$0.565216\pi$
$740$	0	0
$741$	164.148	0.00813782
$742$	0	0
$743$	17213.0	0.849911	0.424956	−	0.905214i	$-0.360290\pi$
$743$	17213.0	0.849911	0.424956	+	0.905214i	$0.360290\pi$
$744$	−8772.04	−0.432256
$745$	0	0
$746$	−26691.2	−1.30997
$747$	−8618.94	−0.422156
$748$	48907.0	2.39066
$749$	0	0
$750$	0	0
$751$	2974.99	0.144553	0.0722763	−	0.997385i	$-0.476974\pi$
$751$	2974.99	0.144553	0.0722763	+	0.997385i	$0.476974\pi$
$752$	−28787.9	−1.39599
$753$	4044.36	0.195730
$754$	18551.9	0.896049
$755$	0	0
$756$	0	0
$757$	1108.45	0.0532198	0.0266099	−	0.999646i	$-0.491529\pi$
$757$	1108.45	0.0532198	0.0266099	+	0.999646i	$0.491529\pi$
$758$	−8596.24	−0.411912
$759$	−8497.57	−0.406380
$760$	0	0
$761$	−26924.3	−1.28253	−0.641264	−	0.767320i	$-0.721590\pi$
$761$	−26924.3	−1.28253	−0.641264	+	0.767320i	$0.721590\pi$
$762$	6687.50	0.317930
$763$	0	0
$764$	−17229.4	−0.815888
$765$	0	0
$766$	−43213.5	−2.03834
$767$	13727.8	0.646260
$768$	9030.50	0.424297
$769$	15160.3	0.710914	0.355457	−	0.934693i	$-0.384325\pi$
$769$	15160.3	0.710914	0.355457	+	0.934693i	$0.384325\pi$
$770$	0	0
$771$	−6511.12	−0.304141
$772$	4338.51	0.202262
$773$	9245.48	0.430190	0.215095	−	0.976593i	$-0.430994\pi$
$773$	9245.48	0.430190	0.215095	+	0.976593i	$0.430994\pi$
$774$	33853.2	1.57213
$775$	0	0
$776$	−9197.57	−0.425481
$777$	0	0
$778$	45485.6	2.09606
$779$	813.343	0.0374083
$780$	0	0
$781$	26276.0	1.20388
$782$	71561.5	3.27242
$783$	3775.41	0.172314
$784$	0	0
$785$	0	0
$786$	9473.67	0.429917
$787$	−30170.5	−1.36653	−0.683266	−	0.730169i	$-0.739441\pi$
$787$	−30170.5	−1.36653	−0.683266	+	0.730169i	$0.739441\pi$
$788$	19163.0	0.866310
$789$	−5086.97	−0.229532
$790$	0	0
$791$	0	0
$792$	37430.6	1.67934
$793$	−947.613	−0.0424347
$794$	−50946.4	−2.27710
$795$	0	0
$796$	23331.4	1.03890
$797$	−14025.3	−0.623340	−0.311670	−	0.950190i	$-0.600888\pi$
$797$	−14025.3	−0.623340	−0.311670	+	0.950190i	$0.600888\pi$
$798$	0	0
$799$	42667.8	1.88921
$800$	0	0
$801$	7028.84	0.310052
$802$	−13580.1	−0.597920
$803$	−27616.1	−1.21364
$804$	8963.75	0.393193
$805$	0	0
$806$	62009.5	2.70992
$807$	−1922.96	−0.0838801
$808$	−51385.0	−2.23727
$809$	−8426.55	−0.366207	−0.183104	−	0.983094i	$-0.558614\pi$
$809$	−8426.55	−0.366207	−0.183104	+	0.983094i	$0.558614\pi$
$810$	0	0
$811$	16791.3	0.727031	0.363515	−	0.931588i	$-0.381576\pi$
$811$	16791.3	0.727031	0.363515	+	0.931588i	$0.381576\pi$
$812$	0	0
$813$	−1307.56	−0.0564062
$814$	40049.8	1.72450
$815$	0	0
$816$	4830.58	0.207235
$817$	637.067	0.0272805
$818$	−36832.0	−1.57433
$819$	0	0
$820$	0	0
$821$	10508.1	0.446694	0.223347	−	0.974739i	$-0.428302\pi$
$821$	10508.1	0.446694	0.223347	+	0.974739i	$0.428302\pi$
$822$	−2988.91	−0.126825
$823$	−33984.6	−1.43940	−0.719701	−	0.694284i	$-0.755721\pi$
$823$	−33984.6	−1.43940	−0.719701	+	0.694284i	$0.755721\pi$
$824$	14745.2	0.623388
$825$	0	0
$826$	0	0
$827$	31679.9	1.33207	0.666033	−	0.745922i	$-0.267991\pi$
$827$	31679.9	1.33207	0.666033	+	0.745922i	$0.267991\pi$
$828$	74491.5	3.12652
$829$	12685.8	0.531480	0.265740	−	0.964045i	$-0.414384\pi$
$829$	12685.8	0.531480	0.265740	+	0.964045i	$0.414384\pi$
$830$	0	0
$831$	−3108.80	−0.129775
$832$	−36196.7	−1.50829
$833$	0	0
$834$	−8822.84	−0.366319
$835$	0	0
$836$	1451.00	0.0600285
$837$	12619.3	0.521129
$838$	53720.6	2.21450
$839$	−31557.8	−1.29857	−0.649283	−	0.760547i	$-0.724931\pi$
$839$	−31557.8	−1.29857	−0.649283	+	0.760547i	$0.724931\pi$
$840$	0	0
$841$	−20474.1	−0.839479
$842$	43148.3	1.76602
$843$	6119.63	0.250025
$844$	−61753.6	−2.51854
$845$	0	0
$846$	67268.2	2.73372
$847$	0	0
$848$	31711.6	1.28418
$849$	7785.68	0.314728
$850$	0	0
$851$	38692.4	1.55859
$852$	11759.9	0.472872
$853$	5455.96	0.219002	0.109501	−	0.993987i	$-0.465075\pi$
$853$	5455.96	0.219002	0.109501	+	0.993987i	$0.465075\pi$
$854$	0	0
$855$	0	0
$856$	16537.0	0.660306
$857$	26328.5	1.04943	0.524717	−	0.851277i	$-0.324171\pi$
$857$	26328.5	1.04943	0.524717	+	0.851277i	$0.324171\pi$
$858$	13508.8	0.537508
$859$	−26489.0	−1.05214	−0.526072	−	0.850440i	$-0.676336\pi$
$859$	−26489.0	−1.05214	−0.526072	+	0.850440i	$0.676336\pi$
$860$	0	0
$861$	0	0
$862$	73952.2	2.92207
$863$	−8167.65	−0.322167	−0.161083	−	0.986941i	$-0.551499\pi$
$863$	−8167.65	−0.322167	−0.161083	+	0.986941i	$0.551499\pi$
$864$	−2059.46	−0.0810928
$865$	0	0
$866$	77442.6	3.03881
$867$	−1533.18	−0.0600573
$868$	0	0
$869$	−6733.61	−0.262856
$870$	0	0
$871$	−30760.5	−1.19665
$872$	13736.8	0.533469
$873$	6451.03	0.250097
$874$	2123.13	0.0821691
$875$	0	0
$876$	−12359.6	−0.476705
$877$	−5173.23	−0.199188	−0.0995939	−	0.995028i	$-0.531754\pi$
$877$	−5173.23	−0.199188	−0.0995939	+	0.995028i	$0.531754\pi$
$878$	−52575.2	−2.02087
$879$	−5863.00	−0.224976
$880$	0	0
$881$	−8713.88	−0.333233	−0.166616	−	0.986022i	$-0.553284\pi$
$881$	−8713.88	−0.333233	−0.166616	+	0.986022i	$0.553284\pi$
$882$	0	0
$883$	−32124.1	−1.22431	−0.612153	−	0.790740i	$-0.709696\pi$
$883$	−32124.1	−1.22431	−0.612153	+	0.790740i	$0.709696\pi$
$884$	−75113.5	−2.85785
$885$	0	0
$886$	40049.5	1.51861
$887$	15520.9	0.587531	0.293765	−	0.955878i	$-0.405092\pi$
$887$	15520.9	0.587531	0.293765	+	0.955878i	$0.405092\pi$
$888$	8701.41	0.328829
$889$	0	0
$890$	0	0
$891$	−24844.5	−0.934142
$892$	44497.0	1.67026
$893$	1265.89	0.0474372
$894$	9739.56	0.364362
$895$	0	0
$896$	0	0
$897$	13050.9	0.485795
$898$	−53487.4	−1.98764
$899$	13085.7	0.485463
$900$	0	0
$901$	−47001.1	−1.73788
$902$	66935.2	2.47084
$903$	0	0
$904$	32670.8	1.20201
$905$	0	0
$906$	−11520.1	−0.422439
$907$	−52670.5	−1.92822	−0.964109	−	0.265506i	$-0.914461\pi$
$907$	−52670.5	−1.92822	−0.964109	+	0.265506i	$0.914461\pi$
$908$	−32198.4	−1.17681
$909$	36040.6	1.31506
$910$	0	0
$911$	−19502.8	−0.709283	−0.354641	−	0.935002i	$-0.615397\pi$
$911$	−19502.8	−0.709283	−0.354641	+	0.935002i	$0.615397\pi$
$912$	143.316	0.00520359
$913$	−13348.1	−0.483854
$914$	45267.8	1.63821
$915$	0	0
$916$	89526.5	3.22930
$917$	0	0
$918$	−23151.3	−0.832362
$919$	−45280.4	−1.62531	−0.812656	−	0.582743i	$-0.801979\pi$
$919$	−45280.4	−1.62531	−0.812656	+	0.582743i	$0.801979\pi$
$920$	0	0
$921$	4796.89	0.171621
$922$	81554.5	2.91308
$923$	−40355.9	−1.43914
$924$	0	0
$925$	0	0
$926$	39978.3	1.41876
$927$	−10342.0	−0.366425
$928$	−2135.58	−0.0755428
$929$	3201.56	0.113067	0.0565337	−	0.998401i	$-0.481995\pi$
$929$	3201.56	0.113067	0.0565337	+	0.998401i	$0.481995\pi$
$930$	0	0
$931$	0	0
$932$	25653.4	0.901615
$933$	−11176.3	−0.392172
$934$	−94936.6	−3.32593
$935$	0	0
$936$	−57487.5	−2.00752
$937$	−23678.3	−0.825546	−0.412773	−	0.910834i	$-0.635440\pi$
$937$	−23678.3	−0.825546	−0.412773	+	0.910834i	$0.635440\pi$
$938$	0	0
$939$	−6826.26	−0.237238
$940$	0	0
$941$	−34726.7	−1.20304	−0.601518	−	0.798859i	$-0.705437\pi$
$941$	−34726.7	−1.20304	−0.601518	+	0.798859i	$0.705437\pi$
$942$	−6063.39	−0.209720
$943$	64666.6	2.23312
$944$	11985.6	0.413240
$945$	0	0
$946$	52428.3	1.80189
$947$	29290.9	1.00510	0.502549	−	0.864549i	$-0.332396\pi$
$947$	29290.9	1.00510	0.502549	+	0.864549i	$0.332396\pi$
$948$	−3013.64	−0.103247
$949$	42414.0	1.45081
$950$	0	0
$951$	3117.66	0.106306
$952$	0	0
$953$	−23209.3	−0.788900	−0.394450	−	0.918917i	$-0.629065\pi$
$953$	−23209.3	−0.788900	−0.394450	+	0.918917i	$0.629065\pi$
$954$	−74100.0	−2.51476
$955$	0	0
$956$	−62319.7	−2.10833
$957$	2850.71	0.0962910
$958$	−6081.93	−0.205113
$959$	0	0
$960$	0	0
$961$	13947.6	0.468183
$962$	−61510.2	−2.06151
$963$	−11598.8	−0.388126
$964$	−48993.8	−1.63691
$965$	0	0
$966$	0	0
$967$	−25503.9	−0.848140	−0.424070	−	0.905629i	$-0.639399\pi$
$967$	−25503.9	−0.848140	−0.424070	+	0.905629i	$0.639399\pi$
$968$	9220.22	0.306146
$969$	−212.415	−0.00704205
$970$	0	0
$971$	−13831.6	−0.457135	−0.228568	−	0.973528i	$-0.573404\pi$
$971$	−13831.6	−0.457135	−0.228568	+	0.973528i	$0.573404\pi$
$972$	−36448.8	−1.20277
$973$	0	0
$974$	17291.4	0.568843
$975$	0	0
$976$	−827.353	−0.0271341
$977$	−38668.5	−1.26624	−0.633119	−	0.774054i	$-0.718226\pi$
$977$	−38668.5	−1.26624	−0.633119	+	0.774054i	$0.718226\pi$
$978$	−6213.27	−0.203148
$979$	10885.6	0.355367
$980$	0	0
$981$	−9634.74	−0.313572
$982$	−49246.1	−1.60031
$983$	28144.9	0.913207	0.456603	−	0.889670i	$-0.349066\pi$
$983$	28144.9	0.913207	0.456603	+	0.889670i	$0.349066\pi$
$984$	14542.7	0.471141
$985$	0	0
$986$	−24007.0	−0.775395
$987$	0	0
$988$	−2228.51	−0.0717594
$989$	50651.4	1.62854
$990$	0	0
$991$	−31197.9	−1.00003	−0.500017	−	0.866016i	$-0.666673\pi$
$991$	−31197.9	−1.00003	−0.500017	+	0.866016i	$0.666673\pi$
$992$	−7138.13	−0.228464
$993$	−433.752	−0.0138617
$994$	0	0
$995$	0	0
$996$	−5973.98	−0.190053
$997$	−41031.6	−1.30339	−0.651696	−	0.758480i	$-0.725942\pi$
$997$	−41031.6	−1.30339	−0.651696	+	0.758480i	$0.725942\pi$
$998$	39842.4	1.26372
$999$	−12517.6	−0.396437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bs.1.11		12
5.2	odd	4		245.4.b.g.99.12	yes	12
5.3	odd	4		245.4.b.g.99.1	✓	12
5.4	even	2	inner	1225.4.a.bs.1.2		12
7.6	odd	2	inner	1225.4.a.bs.1.12		12
35.2	odd	12		245.4.j.g.214.11		24
35.3	even	12		245.4.j.g.79.12		24
35.12	even	12		245.4.j.g.214.12		24
35.13	even	4		245.4.b.g.99.2	yes	12
35.17	even	12		245.4.j.g.79.1		24
35.18	odd	12		245.4.j.g.79.11		24
35.23	odd	12		245.4.j.g.214.2		24
35.27	even	4		245.4.b.g.99.11	yes	12
35.32	odd	12		245.4.j.g.79.2		24
35.33	even	12		245.4.j.g.214.1		24
35.34	odd	2	inner	1225.4.a.bs.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.b.g.99.1	✓	12	5.3	odd	4
245.4.b.g.99.2	yes	12	35.13	even	4
245.4.b.g.99.11	yes	12	35.27	even	4
245.4.b.g.99.12	yes	12	5.2	odd	4
245.4.j.g.79.1		24	35.17	even	12
245.4.j.g.79.2		24	35.32	odd	12
245.4.j.g.79.11		24	35.18	odd	12
245.4.j.g.79.12		24	35.3	even	12
245.4.j.g.214.1		24	35.33	even	12
245.4.j.g.214.2		24	35.23	odd	12
245.4.j.g.214.11		24	35.2	odd	12
245.4.j.g.214.12		24	35.12	even	12
1225.4.a.bs.1.1		12	35.34	odd	2	inner
1225.4.a.bs.1.2		12	5.4	even	2	inner
1225.4.a.bs.1.11		12	1.1	even	1	trivial
1225.4.a.bs.1.12		12	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+4.85259 q^{2} -1.14521 q^{3} +15.5476 q^{4} -5.55723 q^{6} +36.6254 q^{8} -25.6885 q^{9} +O(q^{10})\)	\(q+4.85259 q^{2} -1.14521 q^{3} +15.5476 q^{4} -5.55723 q^{6} +36.6254 q^{8} -25.6885 q^{9} -39.7837 q^{11} -17.8053 q^{12} +61.1016 q^{13} +53.3473 q^{16} -79.0682 q^{17} -124.656 q^{18} -2.34584 q^{19} -193.054 q^{22} -186.511 q^{23} -41.9438 q^{24} +296.501 q^{26} +60.3394 q^{27} +62.5696 q^{29} +209.138 q^{31} -34.1312 q^{32} +45.5607 q^{33} -383.685 q^{34} -399.395 q^{36} -207.454 q^{37} -11.3834 q^{38} -69.9741 q^{39} -346.718 q^{41} -271.573 q^{43} -618.542 q^{44} -905.061 q^{46} -539.632 q^{47} -61.0938 q^{48} +90.5497 q^{51} +949.983 q^{52} +594.438 q^{53} +292.802 q^{54} +2.68648 q^{57} +303.624 q^{58} +224.672 q^{59} -15.5088 q^{61} +1014.86 q^{62} -592.403 q^{64} +221.087 q^{66} -503.432 q^{67} -1229.32 q^{68} +213.594 q^{69} -660.472 q^{71} -940.852 q^{72} +694.155 q^{73} -1006.69 q^{74} -36.4722 q^{76} -339.556 q^{78} +169.255 q^{79} +624.488 q^{81} -1682.48 q^{82} +335.518 q^{83} -1317.83 q^{86} -71.6553 q^{87} -1457.10 q^{88} -273.618 q^{89} -2899.80 q^{92} -239.507 q^{93} -2618.61 q^{94} +39.0874 q^{96} -251.125 q^{97} +1021.98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 44 q^{4} - 140 q^{9}+O(q^{10}) \) 12 * q + 44 * q^4 - 140 * q^9	\( 12 q + 44 q^{4} - 140 q^{9} - 24 q^{11} + 84 q^{16} - 296 q^{29} - 1100 q^{36} + 184 q^{39} - 2760 q^{44} - 2440 q^{46} - 1928 q^{51} - 2332 q^{64} - 1840 q^{71} - 8656 q^{74} - 5032 q^{79} - 1284 q^{81} - 1680 q^{86} + 3504 q^{99}+O(q^{100}) \) 12 * q + 44 * q^4 - 140 * q^9 - 24 * q^11 + 84 * q^16 - 296 * q^29 - 1100 * q^36 + 184 * q^39 - 2760 * q^44 - 2440 * q^46 - 1928 * q^51 - 2332 * q^64 - 1840 * q^71 - 8656 * q^74 - 5032 * q^79 - 1284 * q^81 - 1680 * q^86 + 3504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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