# Properties

 Label 7524.2.l.b.2089.3 Level 7524 Weight 2 Character 7524.2089 Analytic conductor 60.079 Analytic rank 0 Dimension 8 CM discriminant -627 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7524.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$60.0794424808$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.488455618816.6 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 2089.3 Root $$-1.67945 - 1.15831i$$ of $$x^{8} - 4 x^{7} + 14 x^{5} + 105 x^{4} - 238 x^{3} - 426 x^{2} + 548 x + 3140$$ Character $$\chi$$ $$=$$ 7524.2089 Dual form 7524.2.l.b.2089.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ $$q-3.31662i q^{11} +2.79549 q^{13} +4.88004i q^{17} -4.35890 q^{19} -5.00000 q^{25} +7.00000 q^{49} +8.72842i q^{53} +2.72842i q^{59} +14.7284i q^{71} -17.1043 q^{79} +1.75321i q^{83} -3.27158i q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 40q^{25} + 56q^{49} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7524\mathbb{Z}\right)^\times$$.

 $$n$$ $$2377$$ $$3763$$ $$4105$$ $$6689$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 3.31662i − 1.00000i
$$12$$ 0 0
$$13$$ 2.79549 0.775329 0.387664 0.921801i $$-0.373282\pi$$
0.387664 + 0.921801i $$0.373282\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.88004i 1.18358i 0.806091 + 0.591791i $$0.201579\pi$$
−0.806091 + 0.591791i $$0.798421\pi$$
$$18$$ 0 0
$$19$$ −4.35890 −1.00000
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 8.72842i 1.19894i 0.800397 + 0.599470i $$0.204622\pi$$
−0.800397 + 0.599470i $$0.795378\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 2.72842i 0.355210i 0.984102 + 0.177605i $$0.0568349\pi$$
−0.984102 + 0.177605i $$0.943165\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 14.7284i 1.74794i 0.485979 + 0.873971i $$0.338463\pi$$
−0.485979 + 0.873971i $$0.661537\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −17.1043 −1.92438 −0.962190 0.272380i $$-0.912189\pi$$
−0.962190 + 0.272380i $$0.912189\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.75321i 0.192440i 0.995360 + 0.0962201i $$0.0306753\pi$$
−0.995360 + 0.0962201i $$0.969325\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 3.27158i − 0.346787i −0.984853 0.173394i $$-0.944527\pi$$
0.984853 0.173394i $$-0.0554733\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.2665i 1.32007i 0.751237 + 0.660033i $$0.229458\pi$$
−0.751237 + 0.660033i $$0.770542\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 17.4356 1.67003 0.835014 0.550229i $$-0.185460\pi$$
0.835014 + 0.550229i $$0.185460\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 20.7284i 1.94997i 0.222281 + 0.974983i $$0.428650\pi$$
−0.222281 + 0.974983i $$0.571350\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.92231 0.525520 0.262760 0.964861i $$-0.415367\pi$$
0.262760 + 0.964861i $$0.415367\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 21.2734i − 1.85866i −0.369248 0.929331i $$-0.620385\pi$$
0.369248 0.929331i $$-0.379615\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 9.27158i − 0.775329i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 18.1465i − 1.48662i −0.668946 0.743311i $$-0.733254\pi$$
0.668946 0.743311i $$-0.266746\pi$$
$$150$$ 0 0
$$151$$ −8.71780 −0.709444 −0.354722 0.934972i $$-0.615425\pi$$
−0.354722 + 0.934972i $$0.615425\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.1852 1.77058 0.885288 0.465044i $$-0.153961\pi$$
0.885288 + 0.465044i $$0.153961\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −24.1852 −1.89433 −0.947167 0.320740i $$-0.896069\pi$$
−0.947167 + 0.320740i $$0.896069\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −5.18525 −0.398865
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 26.7284i 1.99778i 0.0471502 + 0.998888i $$0.484986\pi$$
−0.0471502 + 0.998888i $$0.515014\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.1852 1.18358
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 25.8221 1.85871 0.929356 0.369184i $$-0.120363\pi$$
0.929356 + 0.369184i $$0.120363\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 27.9066i 1.98826i 0.108176 + 0.994132i $$0.465499\pi$$
−0.108176 + 0.994132i $$0.534501\pi$$
$$198$$ 0 0
$$199$$ 28.1852 1.99800 0.999000 0.0447181i $$-0.0142390\pi$$
0.999000 + 0.0447181i $$0.0142390\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 14.4568i 1.00000i
$$210$$ 0 0
$$211$$ 22.6952 1.56240 0.781202 0.624278i $$-0.214607\pi$$
0.781202 + 0.624278i $$0.214607\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 13.6421i 0.917666i
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −30.1852 −1.99470 −0.997349 0.0727709i $$-0.976816\pi$$
−0.997349 + 0.0727709i $$0.976816\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 26.5330i − 1.73823i −0.494606 0.869117i $$-0.664688\pi$$
0.494606 0.869117i $$-0.335312\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 6.63325i − 0.429069i −0.976716 0.214535i $$-0.931177\pi$$
0.976716 0.214535i $$-0.0688235\pi$$
$$240$$ 0 0
$$241$$ −20.2311 −1.30320 −0.651599 0.758563i $$-0.725902\pi$$
−0.651599 + 0.758563i $$0.725902\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −12.1852 −0.775329
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 28.9137i 1.80358i 0.432169 + 0.901792i $$0.357748\pi$$
−0.432169 + 0.901792i $$0.642252\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 8.00686i 0.493724i 0.969051 + 0.246862i $$0.0793994\pi$$
−0.969051 + 0.246862i $$0.920601\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 32.7284i 1.99549i 0.0671428 + 0.997743i $$0.478612\pi$$
−0.0671428 + 0.997743i $$0.521388\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 16.5831i 1.00000i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −6.81475 −0.400868
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −0.331335 −0.0189103 −0.00945514 0.999955i $$-0.503010\pi$$
−0.00945514 + 0.999955i $$0.503010\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 34.1852 1.93226 0.966132 0.258047i $$-0.0830791\pi$$
0.966132 + 0.258047i $$0.0830791\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 28.9137i 1.62395i 0.583690 + 0.811977i $$0.301608\pi$$
−0.583690 + 0.811977i $$0.698392\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 21.2716i − 1.18358i
$$324$$ 0 0
$$325$$ −13.9774 −0.775329
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −34.8712 −1.89955 −0.949777 0.312926i $$-0.898691\pi$$
−0.949777 + 0.312926i $$0.898691\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.7798i 1.33025i 0.746733 + 0.665124i $$0.231621\pi$$
−0.746733 + 0.665124i $$0.768379\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 31.0334i 1.63788i 0.573878 + 0.818941i $$0.305438\pi$$
−0.573878 + 0.818941i $$0.694562\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −36.1852 −1.88885 −0.944427 0.328720i $$-0.893383\pi$$
−0.944427 + 0.328720i $$0.893383\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 17.4356 0.902781 0.451390 0.892327i $$-0.350928\pi$$
0.451390 + 0.892327i $$0.350928\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 38.7284i 1.97893i 0.144774 + 0.989465i $$0.453755\pi$$
−0.144774 + 0.989465i $$0.546245\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.18525 −0.310429 −0.155214 0.987881i $$-0.549607\pi$$
−0.155214 + 0.987881i $$0.549607\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 28.9137i 1.44388i 0.691956 + 0.721940i $$0.256749\pi$$
−0.691956 + 0.721940i $$0.743251\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −37.0040 −1.82973 −0.914865 0.403759i $$-0.867703\pi$$
−0.914865 + 0.403759i $$0.867703\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 24.4002i − 1.18358i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −40.1308 −1.91534 −0.957670 0.287868i $$-0.907054\pi$$
−0.957670 + 0.287868i $$0.907054\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 27.2716i − 1.28703i −0.765435 0.643513i $$-0.777476\pi$$
0.765435 0.643513i $$-0.222524\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 1.37361i − 0.0639753i −0.999488 0.0319877i $$-0.989816\pi$$
0.999488 0.0319877i $$-0.0101837\pi$$
$$462$$ 0 0
$$463$$ 4.18525 0.194505 0.0972525 0.995260i $$-0.468995\pi$$
0.0972525 + 0.995260i $$0.468995\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 21.7945 1.00000
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 15.0197i − 0.686268i −0.939286 0.343134i $$-0.888511\pi$$
0.939286 0.343134i $$-0.111489\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 44.2999i − 1.99923i −0.0277858 0.999614i $$-0.508846\pi$$
0.0277858 0.999614i $$-0.491154\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 40.1852 1.79894 0.899469 0.436984i $$-0.143953\pi$$
0.899469 + 0.436984i $$0.143953\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 33.1662i 1.47881i 0.673261 + 0.739405i $$0.264893\pi$$
−0.673261 + 0.739405i $$0.735107\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 28.9137i 1.28158i 0.767718 + 0.640788i $$0.221392\pi$$
−0.767718 + 0.640788i $$0.778608\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 44.7284i 1.95959i 0.200011 + 0.979794i $$0.435902\pi$$
−0.200011 + 0.979794i $$0.564098\pi$$
$$522$$ 0 0
$$523$$ 45.7218 1.99928 0.999638 0.0269223i $$-0.00857067\pi$$
0.999638 + 0.0269223i $$0.00857067\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 23.2164i − 1.00000i
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −8.71780 −0.372746 −0.186373 0.982479i $$-0.559673\pi$$
−0.186373 + 0.982479i $$0.559673\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13.2665i 0.562120i 0.959690 + 0.281060i $$0.0906859\pi$$
−0.959690 + 0.281060i $$0.909314\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −42.1852 −1.75619 −0.878097 0.478482i $$-0.841187\pi$$
−0.878097 + 0.478482i $$0.841187\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 28.9489 1.19894
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 41.1731i − 1.69078i −0.534152 0.845388i $$-0.679369\pi$$
0.534152 0.845388i $$-0.320631\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 33.2716i − 1.35944i −0.733472 0.679720i $$-0.762101\pi$$
0.733472 0.679720i $$-0.237899\pi$$
$$600$$ 0 0
$$601$$ −34.8712 −1.42243 −0.711213 0.702977i $$-0.751854\pi$$
−0.711213 + 0.702977i $$0.751854\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 43.5890 1.76922 0.884611 0.466329i $$-0.154424\pi$$
0.884611 + 0.466329i $$0.154424\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −0.185248 −0.00744576 −0.00372288 0.999993i $$-0.501185\pi$$
−0.00372288 + 0.999993i $$0.501185\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 19.5684 0.775329
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 28.9137i 1.14202i 0.820943 + 0.571011i $$0.193448\pi$$
−0.820943 + 0.571011i $$0.806552\pi$$
$$642$$ 0 0
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 9.04913 0.355210
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 42.5950 1.64192 0.820958 0.570989i $$-0.193440\pi$$
0.820958 + 0.570989i $$0.193440\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 50.7284i 1.94107i 0.240962 + 0.970534i $$0.422537\pi$$
−0.240962 + 0.970534i $$0.577463\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 24.4002i 0.929573i
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 47.4268i − 1.79128i −0.444776 0.895642i $$-0.646717\pi$$
0.444776 0.895642i $$-0.353283\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.81475 −0.0681544 −0.0340772 0.999419i $$-0.510849\pi$$
−0.0340772 + 0.999419i $$0.510849\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 20.0000 0.741759 0.370879 0.928681i $$-0.379056\pi$$
0.370879 + 0.928681i $$0.379056\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 46.1852 1.67863 0.839316 0.543644i $$-0.182956\pi$$
0.839316 + 0.543644i $$0.182956\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 53.0660i 1.92364i 0.273681 + 0.961820i $$0.411759\pi$$
−0.273681 + 0.961820i $$0.588241\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7.62725i 0.275404i
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 39.2716i − 1.41250i −0.707962 0.706250i $$-0.750385\pi$$
0.707962 0.706250i $$-0.249615\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 48.8486 1.74794
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −33.8772 −1.20759 −0.603796 0.797139i $$-0.706346\pi$$
−0.603796 + 0.797139i $$0.706346\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 28.9137i 1.02417i 0.858933 + 0.512087i $$0.171128\pi$$
−0.858933 + 0.512087i $$0.828872\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 21.6530i 0.761278i 0.924724 + 0.380639i $$0.124296\pi$$
−0.924724 + 0.380639i $$0.875704\pi$$
$$810$$ 0 0
$$811$$ −23.3579 −0.820207 −0.410104 0.912039i $$-0.634507\pi$$
−0.410104 + 0.912039i $$0.634507\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 11.1337i 0.388568i 0.980945 + 0.194284i $$0.0622384\pi$$
−0.980945 + 0.194284i $$0.937762\pi$$
$$822$$ 0 0
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 34.1603i 1.18358i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 57.8273i − 1.99642i −0.0597970 0.998211i $$-0.519045\pi$$
0.0597970 0.998211i $$-0.480955\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ −48.1852 −1.64406 −0.822030 0.569445i $$-0.807158\pi$$
−0.822030 + 0.569445i $$0.807158\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 57.8273i − 1.96847i −0.176879 0.984233i $$-0.556600\pi$$
0.176879 0.984233i $$-0.443400\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 56.7284i 1.92438i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −43.2577 −1.46071 −0.730354 0.683069i $$-0.760645\pi$$
−0.730354 + 0.683069i $$0.760645\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −42.5950 −1.41904
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 57.8273i − 1.91590i −0.286927 0.957952i $$-0.592634\pi$$
0.286927 0.957952i $$-0.407366\pi$$
$$912$$ 0 0
$$913$$ 5.81475 0.192440
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 41.1731i 1.35523i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ −30.5123 −1.00000
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 31.0000 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 45.2716i − 1.45283i −0.687254 0.726417i $$-0.741184\pi$$
0.687254 0.726417i $$-0.258816\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 28.9137i 0.925030i 0.886611 + 0.462515i $$0.153053\pi$$
−0.886611 + 0.462515i $$0.846947\pi$$
$$978$$ 0 0
$$979$$ −10.8506 −0.346787
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 57.8273i − 1.84441i −0.386707 0.922203i $$-0.626387\pi$$
0.386707 0.922203i $$-0.373613\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7524.2.l.b.2089.3 yes 8
3.2 odd 2 inner 7524.2.l.b.2089.7 yes 8
11.10 odd 2 inner 7524.2.l.b.2089.6 yes 8
19.18 odd 2 inner 7524.2.l.b.2089.2 8
33.32 even 2 inner 7524.2.l.b.2089.2 8
57.56 even 2 inner 7524.2.l.b.2089.6 yes 8
209.208 even 2 inner 7524.2.l.b.2089.7 yes 8
627.626 odd 2 CM 7524.2.l.b.2089.3 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
7524.2.l.b.2089.2 8 19.18 odd 2 inner
7524.2.l.b.2089.2 8 33.32 even 2 inner
7524.2.l.b.2089.3 yes 8 1.1 even 1 trivial
7524.2.l.b.2089.3 yes 8 627.626 odd 2 CM
7524.2.l.b.2089.6 yes 8 11.10 odd 2 inner
7524.2.l.b.2089.6 yes 8 57.56 even 2 inner
7524.2.l.b.2089.7 yes 8 3.2 odd 2 inner
7524.2.l.b.2089.7 yes 8 209.208 even 2 inner